Talk:Density matrix

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one instance of '2'. But why do we need a 2 x 2 density matrix to capture that? Surely the probabilities add to one, so a single number would suffice to capture the state probabilities? Even if you want to explicitly (and redundantly) write the two spin probabilities, you still only need two numbers. So why are four numbers needed, and why organized in two rows, two cols? My objective in this paragraph is not to press on that specific question, but rather to illustrate the type of shortcoming that besets this article, and the Example section provides the most concrete instances.

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or one oriented horizontally, or at any angle in between. We can ask whether the photon will pass through a left-handed circular polarizer, or a right-handed one. In each experiment, there will be some probability of success, and the probability of failure will be 1 minus that of course, but that probability of success can be different for each measurement. The remarkable thing is that the probabilities for all the different possible measurements hang together so well — that all the information we need to compute them fits into a single 2 × 2 matrix.

6480:(e) The example could be further clarified to show how it exemplifies "The density matrix is a representation of a linear operator called the density operator. The density matrix is obtained from the density operator by choice of basis in the underlying space." What is the "density operator" for this example, and how is the density matrix derived from it? I suspect this relates to your comment "matrix is written in the basis of energy eigenstates.", though neither energy nor eigenstates are mentioned in the article. 84: 1086:

of thermodynamics requiring the entropy to increase. You can find this under Feynman's set of lectures on statistical mechanics. Two things to note: 1) The entropy defined by density matrices for single particles stays constant under Schroedinger's time evolution and when measured (because it is always pure), in total agreement with Hamiltonian dynamics. 2) The same entropy defined by density matrices, but this time for ensembles, increases, in agreement with thermodynamics but against Hamiltonian dynamics.

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indivisible particle / system, then a mixed state makes no sense; rather, a single particle can only be in a pure state, and merely changed from one into another by a measurement. It is particularly confusing because of the terminology being overloaded to mean something completely different. In this case, we tend to denote the pure state of a single particle, but specifically a linear combination of the eigenbasis of an incompatible measuring operator, as a "mixed state".

182: 158: 7075:, one small piece at a time. I often read wikipedia with the intent of finding errors, or unclear statements, and then either research the topic - or if I already know, rephrase or add text to lower the bar for others to obtain the knowledge. So this makes me a bit sad... I guess, that my real only response could be: If you want an easy and probably adequate intro to this specific topic that this article is about, then see: 6477:(d) what do the specific example values mean? In other words, how do the values in the slots of the matrix correspond to the values mentioned in the main narrative? Obviously there are "0.5" entries in the "before" matrix, and there are "50%"s mentioned in the main narrative, but for two different quantities. One of which is probability, which you say is not what the matrix captures directly. 22: 6506:, among others), but I've never had the time. Apparently it's even worse than I thought, if it's this confusing! I'm not quite sure what your questions are asking for (which is probably an indication that the current article is so confusing it's hard to ask clear questions about it). The matrix in the figure is two-dimensional — i.e., a square rather than a cube, a 4899:. This lead, particularly the first sentence, failed this requirement to an almost comical degree. I have rearranged the text and streamlined it to improve readability without changing any of the content. However, someone should take this even further and replace some of the technical terminology in the last paragraph by a more accessible discussion. 7037:) before reading about how to handle quantum states that are noisy, statistical ensembles of a large number of states, suffered from decoherence, et.c. (i.e. density matrices). That said, I have only skimmed the article. Finding what seems to me to be an error in the intro, which was corrected (see above). · · · 1128:"There is a theorem stating that a system with a Hamiltonian, be it classical or quantum, has an entropy that is never-increasing". I am awfully confused about the details of this theorem. The universe is a system with a Hamiltonian -- do you agree? The universe has an entropy that increases -- do you agree? 3845:, which the link defines as all the elements are greater than zero. This is clearly not true of a density matrix. Also shouldn't there be a condition on the density matrix that keeps the coherences (off-diagonals) smaller than the populations (on-diagonal elements). Otherwise the matrix isn't physical?-- 7070:

Hmm. Here you essentially give me the choice of either answering, or refraining from making further updates. What can I say? This wikipedia-thing is a community project, and in that regard, it is like the Linux operating system during its infancy, when only unpaid nerds were contributing: We build it

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A density matrix tells us more than the probabilities for what might happen in a single experiment. It's a catalogue of probabilities for all the measurements we can imagine! (See the "Measurement" section.) We can ask whether the photon will get through a linear polarized filter oriented vertically,

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For example (!), when I ask why does the example show a 2D matrix, with 2x2 entries, your response "because photons spin-1" omits saying why "photons spin-1" has anything to do with 2D or 2x2. Now, I had already inferred that photons having a spin property which can take on one of two states explains

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of the idea of a PDF, with the term "density" hanging around for historical reasons. The indices simply label whatever basis the matrix is written in, for example the basis of energy eigenstates. The "set of density matrices" is exactly what it says on the tin: the set of all matrices which meet the

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would be on the line between R and L, four-fifths of the way towards R, if memory serves. Suppose you put light through a linear polarizer that allowed (1,0,0) on the sphere to pass 100%, and (-1,0,0) on the sphere is blocked 100%. I bet the intensity fraction passing through for light at (x,y,z) on

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I'm not sure whether I understood the difference between a superposition and a mixed state - is it true or not that a superposition means that the photon is (in some way) in both quantum states at once until measurement, but in a mixed state, the photon is physically in one of the pure states but we

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I'm sorry I still don't believe what you said. In fact, to be honest, I don't understand what you said about the relation between ensembles and factorizations. I can't believe that it's standard; if indeed it were standard it would be easy to come up with a source in the form page X of Y which says

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I disagree with the prescription, even though it is technically correct. The original explanation is clearly talking about pure ensemble states being converted into a mixed ensemble state by measurement. It might have been motivated by hindsight. If, instead, the measurement had been conducted on an

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Third, in the case of entangled states, it should explain exactly why a system cannot be in a pure state. It seems to me that entangled objects either share the same quantum state or have states that sum to zero. Either way, entangled objects may have pure or mixed states. I don't know if I'm right

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two-dimensional. Quantum mechanics uses linear operators, and the action of a linear operator upon a basis for a vector space is naturally written as a matrix. One can then ask, "Well, why does quantum mechanics use linear operators?" And, sooner or later, that series of "why?" questions will hit a

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Hi, surely there is some thing missing in the text and is bringing confusion to the reader. Confusion is arising because explanation started with the passing of vertical polarized through a circular polarizer as given below: "If we pass (|R\rangle +|L\rangle )/{\sqrt {2}} polarized light through a

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There is a theorem stating that a system with a Hamiltonian, be it classical or quantum, has an entropy that is never-increasing (luckily, it tends to be constant). The immediate consequence is that the entropy defined by density matrices would then be never-increasing, directly against the 2nd law

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so what's the problem with self adjoint operators as observables? in the finite dim case, the C* algebraic and SA operator formulations are the same. if one is to completely abandon the SA operator formulation, how does one come up with the C*-algebra of observables in the first place? As stated in

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Yes, I have become lazy during the last few months and stopped logging in. But here I am with my accountable account. I am not sure how to read your comment - I am not sure what to say in response. (WP:NOFORUM or whatever its called might be relevant.) However: 1) if its not a quantum system, them

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Cool, I think I get it now :D One clarification: when you say "Unlike linearly polarized light, passes through a polarizer at any angle", do you mean that it all passes through or half of it passes through? Also, maybe mixed states should be mentioned in the lead by name - I realise now that this

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Frankly, I don't believe this is true. (If the projections are assumed pairwise orthogonal, of coure it's just the spectral theorem). This claim would imply that all probability measures on the compact convex set of states, supported on the extreme points, are "unitarily equivalent" provided they

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It states in the context that "Therefore a pure state may be converted into a mixture by a measurement, but a proper mixture can never be converted into a pure state." So can anybody comment on how to produce a pure state? Reducing temperature to zero, or get Bosons under critical temperature, any

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Density matrices are a semi-advanced topic in quantum mechanics; at least in my day, an introductory course wasn't likely to cover them. So, most explanations that one will find are likely to presume at least a moderate background in mathematics, and questions like "why 2D" simply won't be on the

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I think most physicists think that mixed states are not a truly fundamental part of quantum physics or part of our universe, but rather a convenient tool to keep track of uncertainties that are there in various realistic situations. But I'm not sure that's universally agreed...starts getting into

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There is another theorem, the very important Fluctuation-Dissipation theorem, of thermodynamics, that governs the entropy increasing process. Actually, from the random fluctuations, there is no reason why entropy cannot spontaneously decrease. Indeed, it actually does do that from time to time in

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I agree that QM is complex and requires lots of reading. It may be even worse than that: even after reading all of the QM garden, one still may not really understand much of it. But, yes, this is not a forum. As to whether you corrected a problem in the article, I will have to disagree, for your

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2. “The indices simply label whatever basis the matrix is written in, for example the basis of energy eigenstates.” I may be missing something, but this appears to say that the indices of the matrix work like indices of a matrix, although I am not quite sure how to parse “label whatever basis”.

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CSTAR: ok, obviously they're not measures in the measure theoretic sense, but one can see the formal resemblence. and they are called quantum probability measures (was probably your contribution, if i have to guess) in their own right anyway. calling them just that makes the heuristic comment on

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Actually, it's more complicated then that. For uncountably infinite dimensional systems, you can't write the operator as a matrix, even if you have a 'basis' .... so the term 'operator' covers a much more general case than the term 'matrix' does ... this article should actually be called Density

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one should not try to explain it as a quantum system. 2) "...but my point is that the existing explanation isn't detailed enough" - this article can not be read in isolation, it requires prior knowledge of other topics. It is probably a good idea to start reading about normal "pure state" (e.g.

6660:'s recent improvements, I'm willing to upgrade my estimation of the article to "basically fine, as far as it goes". Bits here and there can doubtless be tuned up, the later sections might benefit from expansion, and some topics could still be added, but I think the page is now more in line with 6592:

you are right. i wish i had read the talk page a few hours ago. this entire page is a disaster, and it's wrong. it shouldn't be using 'little p' to represent the general density operator. every definition i've seen leaves it as a dyadic product which you can determine will sum to one after some

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Well, a diagonal density matrix isn't really a "classical state"; it's a quantum state that is diagonal in the given basis (and would generally have coherences with respect to a rotated basis). I'll go back and read some standard references like Schlosshauer; maybe that'll suggest some phrasing

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The article says that the density matrix was first developed by von Neumann, Landau and Bloch in 1927, but Bloch's articles only go as far back as 1945 on ISI web of knowledge, and I can't find anything by von Neumann. Can someone please provide the references for where the density matrix first

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where μ is a probability measure on the compact convex space of states. We can regard the measure μ as an ensemble representing the state τ. The non-uniqueness follows from the failure of unique representation by probability measures, even probability measures which are supported on the set of

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here is a more elementary issue, but could also be a problem. as stated in article, if your observables are compact operators, then the states are precisely what's defined in the article. but if you wanna define a quantum operation in the Schrodinger picture, then it's a map between trace-class

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I completely agree: the time evolution of a density matrix is, in general, more general that the von Neumann equation. And this time evolution simply “generates“ density matrices/operators (or mixed states) from pure states. Just adding “Lindblad equation“ to the further references list, is not

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The operator is the actual mathematical object, where the matrix is one of its many possible representations. Infact, in a different basis, the same operator is represented by a different matrix. This is true for finite and infinite dimensions. So I suggest to change the name of the article to

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In fact, if you take the unique positive square root, you get a positive operator. The columns of this operator are not going to be orthogonal (except in some trivial cases). How do you get an ensemble of orthonormal states. Do you mean the spectral decomposition of A? Could you please be more

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I again appreciate your comments, and candor! You ask what I'm looking for: I'm looking for the article to have explanations that are digestible without knowledge of concepts that are more advanced than the article at hand. Where an example is given, for it to be useful it can't just describe

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here. Entropy can't go down. But wait, my freezer lowers the entropy of water by making it into ice! How does that work? Well, it decreases the entropy of the water, but increases the entropy in other places (outside the freezer and inside the power plant making the electricity). In the light

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The whole "C*-algebraic formulation of density states" section seems pretty unnecessary to me - right now, it's a lot of jargon that doesn't even make sense to a physicist. Also, a statement like "It is now generally accepted that the description of quantum mechanics in which all self-adjoint

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which cryptically suggests "The interior points are called the mixed states."... but doesn't explain further. Do you know how a mixed state would be represented on a Bloch sphere? I'm guessing that your unpolarized light example would be right in the centre of the sphere... But how would you

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The page includes the usual method of obtaining the density matrix from spinors but does not show the less well known method of obtaining spinors from density matrices. I'll go ahead and add it in. If you haven't seen the method, see Julian Schwinger's "Quantum Kinematics and Dynamics" or

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my proposal is that, if we do propose a rewrite, we adopt a more mathematical notation. i don't mind either McWeeny (1955) and Lowdin (1955). i found davidson to be a great source that adopted McWeeny and Lowdin for bra-ket, but it's not a drop-in replacement for what we have right now.

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It may be better to alter the first sentence to say that the density matrix describes a state prior to measurement, which in practice is always a mixed state, since measurement can never be guaranteed to exactly align with the singularly pure aspect (orientation) of the state prepared.

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So are the values at the coordinates of the matrix samples of some large multi-dimensional probability density function, within which one can interpolate? Or are the values at each matrix coordinate actually independent probability density functions in their own right?

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Thanks for explaining it to me! This seems quite a good way a writing an article - a knowledgable person repeatedly rewriting it until the beginner finally understands :) There is one sentence that's still a bit unclear "By contrast, an example of a mixed state would be

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But this isn't true. To take a trivial example; any state ρ is representable as a trivial convex combination of 1 times itself ρ= 1 × ρ (cardinbality 1) and if it's a non-extremal state, at least one other non-trivial convex combination of two other states (cardinality

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True. How about, "we know the matrix is gradually becoming diagonal"? We do know that there is a transition period between the existence of a quantum state and its replacement by a classical state (the diagonal matrix). Collapse doesn't happen in a femtosecond, right?

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Why? Ensembles have the structure of a convex set. One might naturally associate to an ensemble to a probability measure supported on the convex set of density operators, but from there to the characterization given in the most recent edit, is in my view unjustified.

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circular polarizer which allows either only |R\rangle polarized light, or only |L\rangle polarized light, intensity would be reduced by half in both cases. This may make it seem like half of the photons are in state |R\rangle and the other half in state |L\rangle ."

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Mixed states redirects here, but I can't find any definition here of what a mixed state is, or an explanation of what a mixed state physically means. Could someone who understands it better than me either add one, or redirect mixed state to somewhere that explains

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of density matrices, to give us an idea of what that looks like in practice. Also, the article uses "convex" seven times, as though it's important, but never explains why it's important, so perhaps the significance to the example could be pointed out.

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in the formulation of my comments above, this is precisely the non uniqueness of square root factorization of positive semidefinite operators restated in this context. take for instance the finite dim case. given a mixed state described by the ensemble

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I will be incorporating the ideas from here soon. Giving thought as to the implications on Hamiltonian dynamics, entropy, how single particle dynamics relates to the ensemble and whether it can be mathematically smoothed over, is returning headaches.

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CSTAR, i don't know, man, it's probably in most linear algebra books. It's pretty standard. For instance, Choi used this to show how the Kraus operators of a CP map are related by a unitary matrix. Again, the claim is simply the following: Let

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This is also explained in the article it seems. The section "Example: light polarization" contains an explanation of this (see esp. the last sentence in that section). So, it seems the intro is in contradiction with the rest of the article?

5605:. The Von Neumann entropy is zero for this case (you can regard this density matrix to have an eigenvalue of 1, with the rest zero). On the other hand looking at the right hand side we can obtain a non zero result by expanding (for example): 6636:

doesn't mention them at all. As I recall, I didn't encounter them until my third year of undergrad. So, they're treated as a more advanced topic than solving the harmonic oscillator, for example, but not nearly as much as many other things.

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So far this article has ignored the idea of various orders of density matrices. Also, some of the particular expressions are in fact for reduced density operators where the true expressions for the reduced density *matrices* is not given.

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is absolutely standard, used in tens of thousands of physics papers, and if this article does not show density matrices using it then it is doing a disservice to our readers. Whatever problems the article has, that's not among them.

1227:"But this is not correct: Both |R\rangle and |L\rangle photons are partly absorbed by a vertical linear polarizer, but the (|R\rangle +|L\rangle )/{\sqrt {2}} light will pass through that polarizer with no absorption whatsoever." 636: 7011:

or wrong, but my point is that the existing explanation isn't detailed enough. If it were, then I would understand it. Isn't the point of a physics article to explain the physics, rather than use words in vague or confusing ways?

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between such functions and density operators. A probability mesure on states as mentioned in my remark above (in this talk page) is a probability measure in the classical sense, that is, a set function on the σ-algebra of Borel

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And what are these "extreme points" and entire "set of density matrices" that we already meet in the 3rd sentence? What is even a point in this context? Perhaps a particular coordinate of the matrix? But then what is extreme?

4618:! to put it in another way: a pure state is a normalized, positive, weakly continuous functional on the kinematical algebra which is represented by the scalar product of a unique vector in Hilbert Space so how can we represent 1262:

Quite briefly, I don't believe it. Is there a citation? The assumption is that ensembles are in 1-1 correspondence with operators of the form $ A U$ where $ A$ is non-negative trace class of trace $ 1$ and $ U$ unitary.

4408: 4290: 4159: 1853:, we simply append columns of zeros to get a square matrix . as a (somewhat common) abuse of notation from linear algebra, above i have put . all i am doing is stating a fact from linear algebra and relate to this context. 6599:

i am hoping 'semi-advanced' is your way of conveying subtly flexing your proficiency. as someone with a better math background who had less trouble with the Lowdin or McWeeny definitions, i find the bra-ket analogues less

5346: 3060: 364:"It is now generally accepted" again! This is not good writing style. Was there a vote that I missed on that acceptance? ** However at last these hidden arguments against the "adjoint operator style" should be made clear. 6734:

Löwdin, Per-Olov (1955). "Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configurational interaction".

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If the other system is a known state, ie together they make a pure entangled state, then per def they are a pure state. The intro makes it sound like this is not the case. Perhaps a better phrasing is something like..

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the sphere (or inside the sphere) is x/2+0.5. But that's just a guess. If you can find the book "Density Matrix Theory and Applications" it has a good simple discussion of the Bloch sphere interior as I recall. :-) --

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edited sentence is no clearer than the one it replaced. I think some parts of all the QM articles will have to wait for an expert to come along, someone who can explain to those of us who want better explanations.

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I'm assuming here A = M*M. What's the ensemble (understood as some convex combination of pure states) that gives the state A? Presumably, they're related to the columns or rows of M, but I'm sorry I don't see

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I guess we're back to square 1. My original question above remains as far as I can see, unanswered: What is the relation between "square root factorizations" and ensembles? Do you have a citation for this

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Dear Unsigned editor, The article now says "Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a

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No, the entries in a density matrix are not probabilities or values of a probability density function, as they can be complex-valued. Instead, probabilities are calculated from a density matrix using the

3947:

Thanks, that was quick! I'm still somewhat confused... can a single photon be in a mixed state? Or does a mixed state just mean that the beam of light contains a range of photons in different pure states?

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Any mixture of kets can be analysed by treating each ket separately. I suggest deletion. The point (about the merit of density matrices in statistical analysis) seems adequately made by the surrounding

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First, it seems to me that there is a third case of mixed states: when the system is large, hot, or otherwise only suitable for description by classical mechanics. Such a case can never be a pure state.

6263:, something like that could be a good addition, though I'd be wary of the "see the matrix gradually becoming diagonal" phrasing — "see" and "observe" are, after all, loaded words in quantum mechanics. 1367:. from this we see that there is a one-to-one correspondence between such factors and ensembles describing ρ. combine this with the unitary freedom of square roots and that gives what's being claimed. 514: 2869: 1199: 6136: 3416: 1438: 5603: 2665: 5376:

The very first sentence of this topic contradicts the section on 'Pure and mixed states'. One says the density matirx describes mixed states; the other says it describes pure states or mixed.

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CSTAR, i am done with this particular discussion. one can't get much more explicit than the explanations i've given. i am sorry you're not convinced. this is completely elementary and standard.

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1. Apparently the entries in a density matrix are not any of the things I suggested; instead “probabilities are calculated from a density matrix using the Born rule.” I note that neither this

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I don't know what you mean by "flexing"; when I wrote "semi-advanced", I meant simply what I said. One isn't likely to see density matrices in an introductory course. For example, Griffiths'

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Now I thought you just claimed that two representations as convex combinations of extreme poinys must be equivalent "unitarily"; What else could you possibly mean other than what I wrote? --

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Ok, i updated the text in the intro myself now. Why burdon others when i can do the labor myself. If you get upset about this or think i am wrong plz undo and also plz clarify. Thanks

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what i find most concerning about literature post 1970 (lecture notes and such) is the remarkably brief discussion of these important concepts with each lecturer varying in notation.

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is a randomly-varying real number, changing from photon to photon"? I don't like "an ensemble of mixed states", because an ensemble of mixed states is always another mixed state :-) --

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of density matrices is again a density matrix. Think of a ball, and pick any two points within or on the surface of the ball: the line between them always lies within the ball too.

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I agree with the first comment and I do not understand what "untenable means" here. The density matrix formalism is used constantly in magnetic resonance and it's perfectly viable.

4821: 4616: 2173: 752:, not Density Operator, for the title of his textbook on the subject. ...As long as the article explains the relation between the matrix and the operator, and I think it does. -- 6200:

is not of trace 1, so this is no counter-example. The problem with the original text comes from the word "equivalently", which attaches to the earlier sentence about rank one.

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I trust your judgement on replacements and deletions (and yes I prefer your non-normalized form of Nielsen and Chuang Theorem 2.6.) I feel I've done enough damage for one day.--

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I've made some additions to the article consisting of expansion & clarifications of exisiting material and also linking measurement with entropy. See what you think. --

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I would suggest to include at least a reference to the Löwdin paper that explains density matrices and natural orbitals. It has helped me a lot in my PhD work. Here it is:

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This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class.

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Thanks, sounds like good advice. Kudos on your clear adjustment to the lead. Please smooth out any other clunky attempts I make to explain more for non-expert readers.

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doesn't yet link anywhere. I believe readers need more explanation since it's so intimately related to measurement and so often used in QM. And to define, for example

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The counter-example given above is not valid, because the right-hand side is not actually an expansion into multiple different pure states. The qualification that the

2976: 5846: 2058: 390: 6519:, and photons are massless spin-1 particles ... again, there's a whole chain of possible "why?" questions, and I'm not sure which kind of answer you're looking for. 2179:

have the same center of mass. I would be surprised f that were true in anything but the abelian case. However, if you provide me with a reference, I'll believe it.--

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It's lacking an explanation here. The two references date back to 1963 and 1972, is this "now"? or the "now" more recent? would it be possible to give an example?

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As Nielsen and Chuang point out, it is actually more convenient to absorb the probability coefficient into the state by replacing thekets with "renormalized" kets:

6483:(f) The example figure starts with a mixed state and displays it in a single matrix. It would be helpful if it could also illustrate the mixed state captured as a 5030: 791:

sufficient, I think. Although I know, of course, that there are lots of discussions which is the “correct” (or “most general form”) of such an equation of motion.

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I have no preference between "density matrix" and "density operator" as the article title. I don't think there's anything wrong with "density matrix", for example

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Yikes No. That article is formulated in terms of the Schrodinger picture. The dual of a quantum operation (on trace class operators) is ultraweakly continuous.--

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The "Definition" section shows how to write a density matrix in Dirac notation, and the figure caption gives two examples written as square arrays of numbers.

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Do we really want C* algebra in the wikipedia article. This seems quite technical and probably not of interest to many people. Maybe it needs its own page.

6696:"The Density Matrix in Many-Electron Quantum Mechanics. I. Generalized Product Functions. Factorization and Physical Interpretation of the Density Matrices" 6613:'s work, since his approach follows Lowdin and McWeeny and uses bra-ket sparingly. but when you step out of that comfortable spot, things look quite scary 1974:

that was clear? Um, it was not clear to me. Now with that caveat, the result you are claiming is equivalent to the following: Representations of the form

1044:

Put a beam of unpolarized light (mixed state) through a polarizer. Then you have polarized light (pure state)! Very easy. We're basically talking about the

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was calling a completely additive function on the orthocomplemented lattice of projections on a Hilbert space a probability measure. There is an affine

1057:, not a subsystem of a larger system. The latter can certainly be made into a pure state. I am adding some text to the article to clarify this point... -- 560: 7127: 1956:

Yes, it's always compact in w* topology (even for C*-algebras) That's a result of fundamnetal importance. This follows from the Banach-ALouglu theorem.--

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I am also not sure how to understand your statement "a single particle can only be in a pure state". If I have a photon which is one-half of an EPR pair

821:

For one thing, superselection rules. For another, one needs to tie in observables to geometrical structure and symmetry; Geometrical structure as in a "

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The section has been replaced. It could be more explanatory, but as it stands is better than nothing. Those who find it useless may simply ignore it.

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are density states, then this is an ensemble (the "this" being the pair consisting of the mixture coefficients and the finite sequence of mixed states τ

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this entire page is in need of a huge rewrite. but as you've stated it's an advanced topic and resources for something like that are hard to come by.

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3. Example: light polarization. There seem to be a few bridges missing between the main narrative and the inset figure. Specifically, in the figure:

7142: 6515:"we don't know". Why is the matrix in the example 2 × 2 instead of 3 × 3 or 9 × 9? Well, because the example is talking about the polarization of a 4326: 4208: 4077: 4056:

Those are good suggestions, I tried again. Yes, unpolarized light is the center of the Bloch sphere. Light that had 80% probability of being |R: -->

252: 3357:

polarization? If so, I think this should be stated explicitly. If not, it might be better to change the example to one with non-orthogonal states.

5257: 2987: 2491:

This applies to couple other related objects as well. Besides Kraus operators, purifications of a given mixed state are related in a similar way.

1053:. (By a factor of 2.) The entropy does not decrease if you take the lost light into account too. Whoever wrote that sentence was referring to the 841:

operators. but the dual map, between observables, is now between the full space of bounded operators. there seems to be some inconsistency there.

444:

The terms "density matrix" and "density operator" are used interchangeably in everyday physics. Is there some technical difference? What is it? --

6877:"It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable. " 6245:

I will give this a week for feedback before I edit the article, since I am not an expert in the field, and this might include a mistake or two.

4621: 2717: 2531:

PS If somewhat has understood mct mht's point about the relation between ensembles and factorizations and I'm just being dense, I'll listen. --

709:. In my experience, people who say "matrix" in the context of "density matrix" are usually...but not always...thinking of finite dimensions. -- 106: 6651: 6304: 6272: 191: 7122: 7089: 6310:

Since a week has elapsed, I will edit the article as I indicated. But if an expert visits here, please feel free to improve what I've added.

5391: 5358: 4323:

of mixed states" or something similar? Illustrates the basic idea that mixed states come from lack of knowledge (statistical probabilities).

4185:." What does it randomly vary with? - between photons? I thought at first it meant randomly-varying with time but that doesn't seem to fit.-- 1246: 893: 779: 1001:

I switched Von Neumann Entropy to use ln instead of log_2. I found ln in lots of sources. It's possible that both definitions are in use. --

6673: 5815: 2880: 1107: 349: 6458:

article mention each other, which seems like a significant omission if this relationship is key to the significance of the Density Matrix.

2060:

are selfadjoint projections (that are however not assumed to be pairwise orthogonal) are uniquely determined up to unitaries. That is if

862:

so it seems one needs to use the Heisenberg picture map, between observables, in general. it's probably a good idea to modify the article

6570: 6528: 6438: 6618: 6447:@XOR'easter Thanks for your comments, though, for the most part, I’m not seeing how they fit with the existing article. Point by point: 6344: 6207: 6173: 4699: 4681: 3358: 2368:(the probabilities can be recovered easily). if you take the unique positive square root, you get an ensemble of orthonormal states ... 3569: 6899: 6052:{\displaystyle P({\vec {x}}):=\langle {\vec {v}}\mid {\vec {x}}\rangle {\vec {v}}+\langle {\vec {w}}\mid {\vec {x}}\rangle {\vec {w}}} 3901: 3651: 2476:

Just give me a citation (with the statement of the result about the relation between ensembles and factorizations) and I'll go away.--

214: 6981: 6960: 6939: 7137: 4459: 4411: 1376:

As I said, I don't see what this has to do with ensembles. Please define what you mean by an ensemble. Could you provide a citation?

1029: 238: 97: 58: 3789:

This is an ensemble representing A. Conversely revcersing the argument, one sees that any ensemble representing A has this form. --

6839: 6835: 6797: 6471:(b) what does each dimension mean, and why does its index have two levels? What do the rows capture? What do the columns capture? 6235: 5413: 434: 315:

operators represent observables is untenable," definitely needs a reference. I will delete the section unless someone objects. --

6816: 7101: 7042: 6664:; students who are one stage less experienced than those who typically work with density matrices can get something out of it. 4190: 4031: 3962: 3923: 1892:

in all my comments, ensemble elements are pure states (vectors). i thought that was clear. that's not so in the e.g. you gave.

6361: 2343:

OK that I believe and is easy to prove. What that has to do with representations of ensembles however, is not at all clear.--

465: 6421:

criteria for being valid density matrices (Hermitian, trace equal to 1). The set of all such matrices of a given size is a

2827: 1144: 205: 163: 6834:

I don't think such an article would be advisable. Projective measurements should be explained in context, in the article

6106: 3386: 1390: 6385: 3861: 1045: 33: 5563: 2606: 829:

for instance. There's more to it than this of course, but this is the 2 minute summary I could most quickly think of.--

4954:

I don't see why this is necessary as density matrices apply just as well to pure states: If I have the superposition,

3421:

Recall that an ensemble representing a density matrix is a sequence of positive rank one operators which add up to A.

6502:

Well, the existing article is kind of a train wreck that I've wanted to rewrite from scratch (like we had to do with

2207: 2066: 1980: 1710: 1573: 6812: 3075:

OK, that seems clear. I'd rather not work with renormalised states, though, and re-express your final equation as:

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I added more text, hope it helps: A bit about quantum superpositions, and a discussion of how you get mixed states.

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Anyway, please provide a citation with whatever it is you are claiming. That will save everybody a lot of time.--

1242: 6372:

The topic is "Density Matrix". So the very least this article needs to do is to answer basic questions such as:

5362: 3511:{\displaystyle \langle e_{i}\mid \phi \rangle \langle \phi \mid e_{j}\rangle =\phi _{i}{\overline {\phi _{j}}}} 3189:{\displaystyle \mid \psi _{i}\rangle {\sqrt {p}}_{i}=\sum _{j}u_{ij}\mid \psi '_{j}\rangle {\sqrt {p_{j}^{'}}}} 897: 783: 353: 1111: 826: 7014:

Note: I login and sign my edits, because I want to be accountable for my opinions, even when they are wrong.

6211: 6169: 4685: 3362: 2792: 2676: 7061: 7019: 6903: 6622: 6340: 6315: 6284: 6250: 4904: 5688:{\displaystyle \rho ={\frac {1}{2}}|\psi \rangle \langle \psi |+{\frac {1}{2}}|\psi \rangle \langle \psi |} 4463: 4415: 3275: 1343:

This might be true for a very special kind of ensemble, in which the components are linearly independent.--

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Yes, the technical difference is whether or not one has chosen a basis. In Dirac notation an operator e.g.

325:

Done. If someone feels like rewriting the section to be more explanatory and less jargon-filled, go ahead.

6062: 5884: 4526:! so how does this artificial distinction of a mixed state and a superposition make sense? If I calculate 3897: 1871:

PS BTW, in the example I suggested, no matter how much appending you do, that trick isn't going to work.--

680: 430: 326: 316: 6921:....physical system which is entangled with another, as its state can not be described by a pure state. 5351:

Thanks (sorry, couldnt be bothered to rewrite the page during exam period, might try during the summer)

6861: 6824: 6707: 6669: 6647: 6566: 6524: 6434: 6357: 6336: 6300: 6268: 6165: 5811: 3850: 1224:

but, suddenly started asserting that photons are abosrbed by vertical linear polarizer as given below:

39: 6828: 6551: 6497: 6238:

happens, and it is even possible to see the matrix gradually becoming diagonal as the system naturally

5748: 83: 4926: 4773: 4568: 2711:

a unit ket vector. The corresponding density matrix is the convex combination of rank one projections

2125: 811:

the article, the GNS lets you recover the Hilbert space, is this not the state space you start with?

6998: 6994: 6973: 6952: 6895: 6744: 6332: 6203: 6161: 5763: 5383: 5379: 5354: 4940: 4936: 4873: 4868:

as expected. I think you may have done the calculation wrong...could you please give more details? --

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in this definition are orthogonal (i.e. states in some orthonormal basis), as in the example with

213:

on Knowledge. If you would like to participate, please visit the project page, where you can join

105:

on Knowledge. If you would like to participate, please visit the project page, where you can join

7057: 7015: 6547: 6493: 6426: 6401: 6311: 6280: 6260: 6246: 5881:), however, I believe is wrong, which is the reason I removed it. My argument is as follows: Let 5698: 4922: 4900: 4896: 3983:

philosophy of quantum mechanics.... I'm not confident enough to write anything about that. :-) --

3250: 3201: 1831:

that's not true. let me clarify. we don't assume is full rank. same goes for . when an ensemble

1201:, would you describe it as a single particle? Would you describe it as being in a pure state? -- 970: 89: 6695: 1769:

What does this have to do with non-uniquess of factorizations (except in very special cases?) --

705:

There is such a thing as a matrix with infinitely or even uncountably-infinitely many rows. See

73: 52: 5040: 7086: 6801: 6792:

For explaining 'projection', which is essential to the motivation, this article links only to

6661: 6503: 5782: 3893: 3235: 1787:}, where the probabilities are absorbed into the states, for notational convenience. so ρ = Σ 1449: 913:

mixed states as probability distribution on states (common in physics literature) more clear.

863: 738: 676: 648:

OK, that makes sense. I just made an edit that hopefully corrects and clarifies this issue. --

6857: 6847: 6820: 6752: 6665: 6643: 6638: 6587: 6562: 6520: 6430: 6384:

Taking a guess, I suspect "density" refers to "probability density", and is thus related to

6353: 6296: 6264: 5807: 4427: 4295: 4164: 3860:

The link was wrong. The way it's intended to be used here, "nonnegative" is synonymous with

3846: 3231: 1501:

yes, that's the, i believe pretty common in physical context, definition of an ensemble. if

6997:

of possible preparations, and second when one wants to describe a physical system which is

6536:

It needs to connect the elements of the example to the more abstract version already given.

2951: 772:

I think it's necesary to add to the article the dynamical equation of the density matrix:

6720: 6231: 5831: 5759: 4869: 4445: 4060: 4023: 3984: 3934: 3865: 2403:

if you take the unique positive square root, you get an ensemble of orthonormal states ...

2036: 1481:) and the failure of uniqueness is due to the multiplicity of such convex representations. 1202: 1058: 1002: 753: 710: 649: 445: 375: 6819:? Seems like a linkable section of one of these QM articles would work well. Thoughts? -- 4957: 1968:

In all my comments, ensemble elements are pure states (vectors). i thought that was clear

6842:, it's just not well-explained. By the way, thanks for finding the mistake in the lead. 6748: 6775: 6451: 3227: 244: 197: 6183: 6141: 1824:

This would imply that all convex representations of a state have the same cardinality.

7116: 7030: 6610: 5209: 4892: 1704:

then the non-uniqueness follows from the multiplicity of representations of the form

631:{\displaystyle O_{mn}=\langle \phi _{m}|\psi \rangle \langle \psi |\phi _{n}\rangle } 337: 285: 7007:

Second, this sentence is quite long, so it could be confusing for that reason alone.

2945:. Two ensembles ψ, ψ' define the same density state iff there is a unitary matrix 4016: 3842: 3366: 3069: 2512: 2492: 2456: 2369: 2334: 2194: 1933: 1921: 1893: 1854: 1816: 1664: 1534: 1368: 1296: 1272: 979:

For von Neumann Entropy: I think Log should be to the base 2, not natural log? See

936: 914: 867: 842: 812: 734: 4919: 4026:) then tell you what would happen if put that light through a linear polarizer?-- 1663:

yes, agreed. didn't you object to calling those things probability measures. :-)

6843: 6657: 3805: 3239: 932: 749: 297:

Any objections to moving this page back to "Density matrix" where it belongs? --

181: 157: 102: 5772:

The formula is correct. See, for example, Theorem 11.8 of Nielsen and Chuang's

4403:{\displaystyle (|\psi _{1}\rangle +e^{i\theta }|\psi _{2}\rangle )/{\sqrt {2}}} 4285:{\displaystyle (|\psi _{1}\rangle +e^{i\theta }|\psi _{2}\rangle )/{\sqrt {2}}} 4154:{\displaystyle (|\psi _{1}\rangle +e^{i\theta }|\psi _{2}\rangle )/{\sqrt {2}}} 6541:

Just connecting the Example narrative to the inset figure would go a long way.

6422: 5406:

A formula relating Shannon and Von Neumann entropies, and why it is incorrect.

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I think what you are trying to say is that if we have a convex representation

1354:

let ρ be a mixed state. a square root factorization of ρ is of the from ρ = Σ

1344: 1334: 1317: 946: 928: 877: 851: 830: 289: 187: 79: 7096:, pages 288-295. This is now clearly over the boundary of WP:NOFORUM. · · · 5341:{\displaystyle \rho ={\begin{bmatrix}1/2&1/2\\1/2&1/2\end{bmatrix}}.} 4935:

Unfortunately it's not useful to most readers because it's behind a paywall.

3055:{\displaystyle \mid \psi _{i}\rangle =\sum _{j}u_{ij}\mid \psi '_{j}\rangle } 2874:

Thus with this renormalized formulation, the corresponding density matrix is

6756: 6507: 6455: 6413: 4667:{\displaystyle \left|\psi _{1}\right\rangle +\left|\psi _{2}\right\rangle } 3998:

seems to be what parts of the second and third paragraph are talking about.

2548:

First of all definition of an ensemble: this is an indexed family of pairs

6329:

An article on a matrix, but there is no matrix to be found!!! Poor again.

4696:

When you say "the suggested "pure" state", are you referring to the state

3521:

Conversely any operator whose matrix has that form is rank 1. Now suppose

2776:{\displaystyle \sum _{i}p_{i}\mid \psi _{i}\rangle \langle \psi _{i}\mid } 2261:

where the A_i are extreme points, then the A_i must be rank 1 projections.

2202:

If a states is an extreme point, it is a rank 1 projection. therefore, if

5348:

which is fully valid and something we do in quantum optics all the time.

1096:

regions. The theorem then says that the average entropy always increases.

1049:

example, you can't turn a beam of unpolarized light into polarized light

822: 307: 298: 6242:

due to connection with the environment without a measurement happening.

2329:, where all matrices are square (If we drop the square assumption, then 1761:

by probability measures μ supported on the compact convex set of states

7105: 7065: 7046: 7023: 6985: 6964: 6943: 6907: 6865: 6851: 6626: 6405: 6319: 6288: 6254: 6215: 5767: 5752: 5395: 5366: 4944: 4930: 4908: 4877: 4689: 4467: 4453: 4419: 4194: 4068: 4035: 3992: 3966: 3942: 3927: 3905: 3873: 3854: 3831: 3808: 3793: 3262: 3253: 3204: 2535: 2526: 2515: 2505: 2495: 2480: 2459: 2372: 2347: 2337: 2277: 2268: 2197: 2183: 1960: 1945: 1936: 1924: 1914: 1896: 1875: 1866: 1857: 1819: 1773: 1685: 1667: 1642: 1537: 1488: 1371: 1347: 1337: 1320: 1299: 1275: 1250: 1210: 1115: 1066: 1037: 1010: 993: 973: 958: 949: 939: 917: 901: 880: 870: 854: 845: 833: 815: 800: 761: 742: 718: 684: 657: 453: 438: 357: 340: 329: 319: 292: 210: 6381:-- What are the values associated with the coordinates in the matrix? 5560:

This formula is clearly incorrect. Consider the case of a pure state:

2932:{\displaystyle \sum _{i}\mid \psi _{i}\rangle \langle \psi _{i}\mid .} 6516: 284:

Would be helpful to tie this in to density operator as discusssed in

5410:

The formula which I have deleted from the article is the following:

3933:

I added some text to try to explain it more simply, does it help? --

3380:

By definition, a positive rank one operator on H is one of the form

4763:{\displaystyle (|\psi _{1}\rangle +|\psi _{2}\rangle )/{\sqrt {2}}} 4319:

I'm not sure about this, but shouldn't that read "an example of an

3424:

Fix an orthonormal basis for H. The matrix of such an operator is

7034: 6918:

The last sentence in the introduction feels quite non-pedagogic:

3635:{\displaystyle A_{ij}=\sum _{k}\psi _{ik}{\overline {\psi _{jk}}}} 3377:

OK it may be standard, but here is the the explicit relationship:

1941:

In the finite dimensional case, all TVS topologies are the same.--

7001:

with another, as its state can not be described by a pure state."

3712:{\displaystyle \Phi _{ij}^{k}=\psi _{ik}{\overline {\psi _{jk}}}} 3222:

I have another query; this line in the introduction seems false:

825:" kind of association between open sets and observables (in the 733:"Density operator", and have "Density matrix" redirected to it. 1932:

appended after CTAR's reply below: i meant norm compact above.

348:

I'd rather not edit it as I am not sure what the author means.

5806:

must have support on orthogonal subspaces is quite important.

4020:

represent light that had, say 80% probability of being |R: -->

247:

in the banner shell. Please resolve this conflict if possible.

243:

This article has been given a rating which conflicts with the

15: 5828:

There are all kinds of criteria under which a density matrix

5496:{\displaystyle S(\rho )=H(p_{i})+\sum _{i}p_{i}S(\rho _{i})} 6928:...physical system which is entangled with another system, 6227:

I'd like to add something like the following to the lead:

5925:

be two mutually orthogonal, normed vectors. The operator

5743:, and the Von Neumann ones on the r.h.s are still zero. -- 3645:

Consider the sequence of rank one operators with matrices

2501:

What's the "this" that applies to other related objects?--

2356:

it is related exactly in the sense i describe before. if

6788:

Link or explanation for 'projector' in quantum mechanics

6230:

The density matrix for an experimental system becomes a

372:

Is it just me or does this article use “density matrix,

6840:

Measurement in quantum mechanics#Projective measurement

1673: 509:{\displaystyle {\hat {O}}=|\psi \rangle \langle \psi |} 5272: 4950:

A density matrix represents a mixed state (first line)

2364:

gives an ensemble describing the state, same goes for

1807:} describes the same state iff there exists a unitary 6609:

i consider myself fortunate that i was starting with

6186: 6144: 6109: 6065: 5931: 5887: 5854: 5834: 5785: 5701: 5611: 5566: 5511: 5416: 5260: 5214: 5048: 4963: 4829: 4776: 4702: 4624: 4571: 4532: 4487: 4430: 4329: 4298: 4211: 4167: 4080: 3731: 3654: 3572: 3530: 3433: 3389: 3337: 3311: 3278: 3087: 2990: 2954: 2883: 2864:{\displaystyle \mid {\sqrt {p_{i}}}\psi _{i}\rangle } 2830: 2795: 2720: 2679: 2609: 2557: 2210: 2128: 2069: 2039: 1983: 1920:

BTW, in general the set of states ain't compact, no?

1713: 1576: 1452: 1393: 1194:{\displaystyle (|00\rangle +|11\rangle )/{\sqrt {2}}} 1147: 563: 522: 468: 398: 378: 6131:{\displaystyle \mid \phi \rangle \langle \phi \mid } 6103:. It cannot be written as a pure state (in the form 6059:

is an orthogonal projection on the plane spanned by

3411:{\displaystyle \mid \phi \rangle \langle \phi \mid } 3234:

vectors is not sufficient to describe the effect of

1433:{\displaystyle \rho =\sum _{i}\lambda _{i}\tau _{i}} 209:, a collaborative effort to improve the coverage of 101:, a collaborative effort to improve the coverage of 6474:(c) in what sense do these indices "label a basis"? 4205:"By contrast, an example of a mixed state would be 3272:Question: Is it implicitly assumed that the states 3213:

lost me) with the above crystal-clear construction?

6192: 6150: 6130: 6095: 6051: 5917: 5873: 5840: 5798: 5735: 5687: 5598:{\displaystyle \rho =|\psi \rangle \langle \psi |} 5597: 5550: 5495: 5340: 5242: 5186: 5021: 4860: 4815: 4762: 4666: 4610: 4557: 4518: 4436: 4402: 4304: 4284: 4173: 4153: 3778: 3711: 3634: 3552: 3510: 3410: 3349: 3323: 3297: 3188: 3054: 2970: 2931: 2863: 2816: 2775: 2700: 2660:{\displaystyle p_{i}\geq 0,\quad \sum _{i}p_{i}=1} 2659: 2589: 2249: 2167: 2108: 2052: 2022: 1750: 1613: 1465: 1432: 1193: 675:Operator, NOT the specific case of density matrix 630: 549: 508: 413: 384: 7133:B-Class articles with conflicting quality ratings 5848:represents a pure state. The one currently used ( 6930:without also describing the other systems state. 2250:{\displaystyle \tau =\sum _{i}\lambda _{i}A_{i}} 2109:{\displaystyle \tau =\sum _{i}\lambda _{i}F_{i}} 2023:{\displaystyle \tau =\sum _{i}\lambda _{i}E_{i}} 1751:{\displaystyle \rho =\int _{S}\tau d\mu (\tau )} 1614:{\displaystyle \rho =\int _{S}\tau d\mu (\tau )} 4920:http://prola.aps.org/abstract/PR/v97/i6/p1474_1 3779:{\displaystyle A_{ij}=\sum _{k}\Phi _{ij}^{k}} 3209:Are you going to replace the MM* stuff (which 1672:No. If I recall correctly, what I objected to 6787: 6772:Reduced Density Matrices in Quantum Chemistry 5695:Such that the Shannon entopy on the r.h.s is 5551:{\displaystyle \rho =\sum _{i}p_{i}\rho _{i}} 3065:This is Theorem 2.6 of Nielsen and Chuang. -- 2455:Σ , then it's trivial to verify what i said. 1862:Could you provide a citation for this fact?-- 516:can be represented by a matrix given a basis 8: 7077:Yanofsky, Noson S.; Mannucci, Mirco (2013). 6838:. Well, it is already there, in the section 6468:(a) why does the matrix have two dimensions? 6119: 6116: 6034: 6004: 5986: 5956: 5674: 5671: 5639: 5636: 5584: 5581: 4742: 4721: 4382: 4348: 4264: 4230: 4133: 4099: 3475: 3456: 3453: 3434: 3399: 3396: 3344: 3318: 3292: 3160: 3101: 3049: 3004: 2910: 2907: 2858: 2809: 2757: 2754: 2693: 2590:{\displaystyle p_{i},\mid \psi _{i}\rangle } 2584: 1173: 1159: 625: 604: 601: 580: 544: 541: 523: 495: 492: 5824:Removed incorrect criterion for pure states 5775:Quantum Computation and Quantum Information 4565:for the suggested "pure" state I arrive at 7148:B-Class physics articles of Mid-importance 6971: 6950: 6893: 6817:Projective measurement (quantum mechanics) 6462:Which prompts me to look to the example... 6330: 6201: 6159: 5377: 2333:need not be square, but it is isometric). 850:Ultraweak continuity takes care of this.-- 152: 47: 7079:Quantum computing for computer scientists 6811:Q: Is it advisable to make a new article 6534:something that happens to be an example. 6185: 6143: 6108: 6082: 6081: 6067: 6066: 6064: 6038: 6037: 6023: 6022: 6008: 6007: 5990: 5989: 5975: 5974: 5960: 5959: 5939: 5938: 5930: 5904: 5903: 5889: 5888: 5886: 5859: 5853: 5833: 5790: 5784: 5712: 5700: 5680: 5663: 5653: 5645: 5628: 5618: 5610: 5590: 5573: 5565: 5542: 5532: 5522: 5510: 5484: 5468: 5458: 5442: 5415: 5319: 5306: 5291: 5278: 5267: 5259: 5229: 5215: 5213: 5176: 5161: 5153: 5138: 5130: 5115: 5107: 5092: 5081: 5070: 5055: 5047: 5005: 4991: 4976: 4964: 4962: 4843: 4828: 4805: 4790: 4775: 4753: 4748: 4736: 4727: 4715: 4706: 4701: 4654: 4633: 4623: 4600: 4585: 4570: 4546: 4531: 4501: 4486: 4429: 4393: 4388: 4376: 4367: 4358: 4342: 4333: 4328: 4297: 4275: 4270: 4258: 4249: 4240: 4224: 4215: 4210: 4166: 4144: 4139: 4127: 4118: 4109: 4093: 4084: 4079: 3770: 3762: 3752: 3736: 3730: 3695: 3689: 3680: 3667: 3659: 3653: 3618: 3612: 3603: 3593: 3577: 3571: 3544: 3529: 3497: 3491: 3485: 3469: 3441: 3432: 3388: 3336: 3310: 3286: 3277: 3174: 3169: 3163: 3151: 3135: 3125: 3112: 3105: 3095: 3086: 3040: 3024: 3014: 2998: 2989: 2959: 2953: 2917: 2901: 2888: 2882: 2852: 2840: 2834: 2829: 2803: 2794: 2764: 2748: 2735: 2725: 2719: 2687: 2678: 2645: 2635: 2614: 2608: 2578: 2562: 2556: 2241: 2231: 2221: 2209: 2159: 2149: 2133: 2127: 2100: 2090: 2080: 2068: 2044: 2038: 2014: 2004: 1994: 1982: 1767:non-uniqueness of convex representations. 1724: 1712: 1587: 1575: 1457: 1451: 1424: 1414: 1404: 1392: 1184: 1179: 1165: 1151: 1146: 619: 610: 593: 587: 568: 562: 535: 526: 521: 501: 484: 470: 469: 467: 400: 399: 397: 377: 6416:. Density matrices can be thought of as 2817:{\displaystyle \mid \psi _{i}\rangle \,} 2701:{\displaystyle \mid \psi _{i}\rangle \,} 2299:be a positive semidefinite matrix. Then 1508:is a discrete probability distribution, 6686: 6378:-- What are the indices of the matrix? 3841:The article claims a density matrix is 2812: 2696: 2395:gives an ensemble describing the state, 2193:at all equivalent to what was claimed! 1258:Recent edit on equivalence of ensembles 154: 49: 19: 6716: 6705: 4057:, and 20% probability of being |L: --> 4021:, and 20% probability of being |L: --> 3298:{\displaystyle \mid \psi _{i}\rangle } 1554:Well, then more generally we can write 707:Matrix (mathematics)#Infinite matrices 550:{\displaystyle \{|\phi _{k}\rangle \}} 6978:2001:9B1:26FD:8D00:81D:350A:F27A:DE70 6957:2001:9B1:26FD:8D00:81D:350A:F27A:DE70 6936:2001:9B1:26FD:8D00:81D:350A:F27A:DE70 6593:swapping and playing with the duals. 6096:{\displaystyle {\vec {v}},{\vec {w}}} 5918:{\displaystyle {\vec {v}},{\vec {w}}} 4481:Btw.: a state is pure if and only if 4458:Yes, that sounds like a good idea :) 4022:. Would the Bloch sphere (or perhaps 2119:then there is a unitary U such that 7: 4895:says that an article should have an 4410:by itself is a pure state, I think? 935:can easily be merged with this one. 203:This article is within the scope of 95:This article is within the scope of 3957:just don't know which one it is? -- 3242:, on a quantum mechanical ensemble. 38:It is of interest to the following 3759: 3656: 3541: 3537: 2360:is a state, the column vectors of 806:question on C* algebra formulation 368:Density matrix vs Density operator 245:project-independent quality rating 14: 7128:Mid-priority mathematics articles 6634:Introduction to Quantum Mechanics 6510:, etc. — because, well, matrices 4816:{\displaystyle tr(\rho ^{2})=1/2} 4611:{\displaystyle tr(\rho ^{2})=1/2} 2408:specific, saying how you do this? 2168:{\displaystyle E_{i}=UF_{i}U^{*}} 115:Knowledge:WikiProject Mathematics 6836:Measurement in quantum mechanics 6798:Measurement in quantum mechanics 5187:{\displaystyle \rho =|\psi : --> 3553:{\displaystyle A=\Psi \Psi ^{*}} 190: 180: 156: 118:Template:WikiProject Mathematics 82: 72: 51: 20: 7143:Mid-importance physics articles 6662:the "write one level down" idea 5874:{\displaystyle \rho ^{2}=\rho } 5031:={\frac {1}{\sqrt {2}}}(|1: --> 5023:={\frac {1}{\sqrt {2}}}(|1: --> 4958:={\frac {1}{\sqrt {2}}}(|1: --> 4861:{\displaystyle tr(\rho ^{2})=1} 4770:? If so, how did you calculate 4519:{\displaystyle tr(\rho ^{2})=1} 1515:a family of pure states, then { 257:This article has been rated as 135:This article has been rated as 6813:Projection (quantum mechanics) 6138:) but it is idempotent. Thus, 6125: 6110: 6087: 6072: 6043: 6028: 6013: 5995: 5980: 5965: 5950: 5944: 5935: 5909: 5894: 5724: 5718: 5681: 5664: 5646: 5629: 5591: 5574: 5490: 5477: 5448: 5435: 5426: 5420: 5230: 5216: 5181: 5177: 5162: 5154: 5139: 5131: 5116: 5108: 5093: 5089: 5071: 5056: 5016: 5006: 4992: 4988: 4965: 4849: 4836: 4796: 4783: 4745: 4728: 4707: 4703: 4591: 4578: 4552: 4539: 4507: 4494: 4385: 4368: 4334: 4330: 4267: 4250: 4216: 4212: 4136: 4119: 4085: 4081: 3906:00:04, 26 September 2010 (UTC) 3405: 3390: 3350:{\displaystyle \mid L\rangle } 3338: 3324:{\displaystyle \mid R\rangle } 3312: 3279: 3088: 2991: 2831: 2796: 2770: 2680: 1745: 1739: 1608: 1602: 1176: 1166: 1152: 1148: 1051:without reducing the intensity 981:Von_Neumann_entropy#Definition 786:) 04:28, August 23, 2007 (UTC) 750:Karl Blum chose Density Matrix 685:23:58, 25 September 2010 (UTC) 658:20:13, 12 September 2008 (UTC) 611: 594: 527: 502: 485: 475: 454:20:10, 10 September 2008 (UTC) 439:12:22, 10 September 2008 (UTC) 414:{\displaystyle {\hat {\rho }}} 405: 1: 7106:20:45, 10 December 2022 (UTC) 7066:20:00, 10 December 2022 (UTC) 7047:19:48, 10 December 2022 (UTC) 7024:14:44, 10 December 2022 (UTC) 6986:13:35, 10 December 2022 (UTC) 6965:13:08, 10 December 2022 (UTC) 6944:12:44, 10 December 2022 (UTC) 6770:Davidson, Ernest Roy (1976). 6571:16:01, 28 February 2021 (UTC) 6552:11:43, 28 February 2021 (UTC) 6529:20:59, 25 February 2021 (UTC) 6498:18:00, 25 February 2021 (UTC) 6439:15:57, 24 February 2021 (UTC) 6406:03:33, 24 February 2021 (UTC) 6255:14:38, 29 November 2019 (UTC) 6216:12:20, 9 September 2019 (UTC) 5816:01:29, 3 September 2019 (UTC) 4945:10:53, 17 February 2021 (UTC) 4558:{\displaystyle tr(\rho ^{2})} 4195:14:29, 28 December 2010 (UTC) 4069:22:50, 27 December 2010 (UTC) 4036:20:50, 27 December 2010 (UTC) 3993:16:29, 27 December 2010 (UTC) 3967:11:13, 27 December 2010 (UTC) 3943:06:22, 27 December 2010 (UTC) 3928:21:54, 26 December 2010 (UTC) 3832:09:47, 10 November 2007 (UTC) 3799:Spinors from Density Matrices 3226:The description of states in 2629: 1251:11:18, 14 February 2014 (UTC) 1230:Kindly, address this issue. 392:” for the “density operator, 358:21:49, 27 February 2024 (UTC) 223:Knowledge:WikiProject Physics 217:and see a list of open tasks. 109:and see a list of open tasks. 7123:C-Class mathematics articles 6908:01:23, 9 December 2022 (UTC) 6386:Probability density function 6362:16:32, 2 November 2020 (UTC) 6320:12:02, 7 December 2019 (UTC) 6305:15:46, 3 December 2019 (UTC) 6289:19:07, 2 December 2019 (UTC) 6273:17:13, 2 December 2019 (UTC) 5736:{\displaystyle log_{2}(2)=1} 5396:08:50, 10 January 2017 (UTC) 4314:randomly-varying real number 4183:randomly-varying real number 3874:01:59, 19 January 2008 (UTC) 3862:positive-semidefinite matrix 3855:23:20, 18 January 2008 (UTC) 3704: 3627: 3503: 3367:21:50, 23 October 2019 (UTC) 1316:I still don't believe it. -- 1046:second law of thermodynamics 902:10:12, 2 November 2014 (UTC) 801:10:30, 8 December 2023 (UTC) 762:12:08, 17 October 2012 (UTC) 743:01:07, 17 October 2012 (UTC) 719:19:05, 13 January 2011 (UTC) 330:16:22, 30 January 2006 (UTC) 320:17:20, 27 January 2006 (UTC) 226:Template:WikiProject Physics 6223:I will add: diagonal matrix 5254:basis gives me the matrix: 3809:01:16, 6 October 2006 (UTC) 1765:. This can be described as 1291:it's also not claimed that 1211:21:24, 30 August 2012 (UTC) 1116:03:07, 30 August 2012 (UTC) 310:13:08, August 7, 2005 (UTC) 7164: 7083:Cambridge University Press 6796:(linear algebra). So does 6368:What density? What matrix? 5367:17:12, 26 March 2013 (UTC) 5022:{\displaystyle |\psi : --> 3864:. I corrected the link. -- 3814:Orders of density matrices 1011:20:23, 28 April 2012 (UTC) 994:17:05, 28 April 2012 (UTC) 263:project's importance scale 6674:16:19, 6 April 2021 (UTC) 6652:14:35, 3 April 2021 (UTC) 6627:23:56, 6 March 2021 (UTC) 5799:{\displaystyle \rho _{i}} 5768:00:20, 25 July 2018 (UTC) 5753:15:19, 24 July 2018 (UTC) 4909:16:31, 5 April 2012 (UTC) 4454:20:29, 9 April 2011 (UTC) 4420:20:22, 8 April 2011 (UTC) 2543: 2522:Z. But, suit yourself. -- 2506:13:43, 30 June 2006 (UTC) 2496:05:31, 30 June 2006 (UTC) 2481:13:43, 30 June 2006 (UTC) 2373:05:19, 30 June 2006 (UTC) 2348:05:06, 30 June 2006 (UTC) 2338:04:53, 30 June 2006 (UTC) 2278:04:16, 30 June 2006 (UTC) 2269:04:15, 30 June 2006 (UTC) 2198:04:05, 30 June 2006 (UTC) 2184:03:49, 30 June 2006 (UTC) 1961:03:17, 30 June 2006 (UTC) 1946:03:24, 30 June 2006 (UTC) 1937:03:23, 30 June 2006 (UTC) 1925:03:14, 30 June 2006 (UTC) 1915:02:20, 30 June 2006 (UTC) 1897:03:30, 30 June 2006 (UTC) 1876:03:19, 30 June 2006 (UTC) 1867:03:17, 30 June 2006 (UTC) 1858:03:13, 30 June 2006 (UTC) 1820:01:39, 30 June 2006 (UTC) 1774:00:45, 30 June 2006 (UTC) 1686:21:48, 29 June 2006 (UTC) 1668:21:38, 29 June 2006 (UTC) 1643:21:21, 29 June 2006 (UTC) 1538:21:11, 29 June 2006 (UTC) 1489:14:31, 29 June 2006 (UTC) 1466:{\displaystyle \tau _{i}} 1372:05:11, 29 June 2006 (UTC) 1348:05:04, 29 June 2006 (UTC) 1338:04:56, 29 June 2006 (UTC) 1321:21:07, 29 June 2006 (UTC) 1300:21:00, 29 June 2006 (UTC) 1280:I still don't believe it, 1276:05:15, 29 June 2006 (UTC) 1067:05:05, 17 July 2011 (UTC) 1038:16:54, 16 July 2011 (UTC) 1017:Mixed State to pure state 974:20:51, 23 June 2006 (UTC) 881:02:36, 18 June 2006 (UTC) 871:02:26, 18 June 2006 (UTC) 855:02:25, 18 June 2006 (UTC) 846:00:50, 18 June 2006 (UTC) 768:on the dynamical equation 256: 242: 175: 134: 67: 46: 7138:B-Class physics articles 6866:07:49, 21 May 2021 (UTC) 6852:08:59, 20 May 2021 (UTC) 6829:23:01, 19 May 2021 (UTC) 4931:11:42, 5 June 2012 (UTC) 3823:WikiProject class rating 3794:21:21, 1 July 2006 (UTC) 3263:06:10, 1 July 2006 (UTC) 3254:06:03, 1 July 2006 (UTC) 3205:05:18, 1 July 2006 (UTC) 3070:04:43, 1 July 2006 (UTC) 2536:04:07, 1 July 2006 (UTC) 2527:04:03, 1 July 2006 (UTC) 2516:02:05, 1 July 2006 (UTC) 2460:02:05, 1 July 2006 (UTC) 2427:has spectral resolution 959:15:25, 8 June 2006 (UTC) 950:17:03, 22 May 2006 (UTC) 940:05:20, 22 May 2006 (UTC) 918:16:57, 22 May 2006 (UTC) 834:02:03, 22 May 2006 (UTC) 827:Streater-Wightman axioms 816:01:41, 22 May 2006 (UTC) 301:16:35, 2005 Jun 15 (UTC) 293:23:03, 19 May 2004 (UTC) 141:project's priority scale 6757:10.1103/PhysRev.97.1474 4897:accessible introduction 4878:06:17, 3 May 2011 (UTC) 4690:14:00, 2 May 2011 (UTC) 4468:14:04, 7 May 2011 (UTC) 4437:{\displaystyle \theta } 4305:{\displaystyle \theta } 4174:{\displaystyle \theta } 3804:www.DensityMatrix.com . 1533:is called an ensemble. 341:16:58, 5 May 2006 (UTC) 98:WikiProject Mathematics 6806:projective measurement 6715:Cite journal requires 6325:Where is the matrix??? 6194: 6158:is a counter-example. 6152: 6132: 6097: 6053: 5919: 5875: 5842: 5800: 5737: 5689: 5599: 5552: 5497: 5342: 5246: 5243:{\displaystyle |1: --> 5198:<\psi |=1/2(|1: --> 5193: 5188:<\psi |=1/2(|1: --> 5041:<\psi |=1/2(|1: --> 5026: 4862: 4817: 4764: 4668: 4612: 4559: 4520: 4438: 4404: 4306: 4286: 4175: 4155: 3780: 3713: 3636: 3554: 3512: 3412: 3373:Beating the dead horse 3351: 3325: 3299: 3190: 3056: 2972: 2971:{\displaystyle u_{ij}} 2933: 2865: 2818: 2777: 2702: 2661: 2591: 2391:the column vectors of 2251: 2169: 2110: 2054: 2024: 1752: 1615: 1467: 1434: 1267:it's not claimed that 1195: 908:Comment on recent edit 632: 551: 510: 415: 386: 28:This article is rated 7098:Omnissiahs hierophant 7039:Omnissiahs hierophant 6914:more pedagogic intro? 6295:that's really solid. 6195: 6153: 6133: 6098: 6054: 5920: 5876: 5843: 5841:{\displaystyle \rho } 5801: 5758:Thanks! I endorse. -- 5738: 5690: 5600: 5553: 5498: 5343: 5247: 5194: 5027: 4863: 4818: 4765: 4674:by a unique vector? 4669: 4613: 4560: 4521: 4439: 4405: 4307: 4287: 4187:Physics is all gnomes 4176: 4156: 4028:Physics is all gnomes 3959:Physics is all gnomes 3920:Physics is all gnomes 3781: 3714: 3637: 3555: 3513: 3413: 3352: 3326: 3300: 3191: 3057: 2973: 2934: 2866: 2819: 2778: 2703: 2662: 2592: 2252: 2170: 2111: 2055: 2053:{\displaystyle E_{i}} 2025: 1753: 1616: 1468: 1435: 1196: 778:comment was added by 633: 552: 511: 416: 387: 385:{\displaystyle \rho } 6995:statistical ensemble 6375:-- Density of what? 6184: 6142: 6107: 6063: 5929: 5885: 5852: 5832: 5783: 5699: 5609: 5564: 5509: 5414: 5258: 5212: 5046: 4961: 4827: 4774: 4700: 4622: 4569: 4530: 4485: 4428: 4327: 4296: 4209: 4165: 4078: 3880:Help with references 3729: 3652: 3570: 3528: 3431: 3387: 3335: 3309: 3276: 3085: 2988: 2952: 2881: 2828: 2793: 2718: 2677: 2607: 2555: 2208: 2126: 2067: 2037: 1981: 1800:. another ensemble { 1711: 1574: 1450: 1391: 1145: 561: 520: 466: 396: 376: 121:mathematics articles 6794:projection operator 6749:1955PhRv...97.1474L 3775: 3672: 3183: 3159: 3048: 866:a bit accordingly. 206:WikiProject Physics 6427:affine combination 6190: 6148: 6128: 6093: 6049: 5915: 5871: 5838: 5796: 5733: 5685: 5595: 5548: 5527: 5493: 5463: 5338: 5329: 5240: 5184: 5019: 4858: 4813: 4760: 4664: 4608: 4555: 4516: 4434: 4400: 4302: 4282: 4171: 4151: 3776: 3758: 3757: 3709: 3655: 3632: 3598: 3550: 3508: 3408: 3347: 3321: 3295: 3236:quantum operations 3186: 3165: 3147: 3130: 3052: 3036: 3019: 2968: 2929: 2893: 2861: 2814: 2813: 2773: 2730: 2698: 2697: 2657: 2640: 2630: 2587: 2247: 2226: 2165: 2106: 2085: 2050: 2020: 1999: 1748: 1638:extreme points. -- 1611: 1463: 1430: 1409: 1191: 628: 547: 506: 411: 382: 90:Mathematics portal 34:content assessment 7091:978-0-521-87996-5 6988: 6976:comment added by 6967: 6955:comment added by 6910: 6898:comment added by 6802:Quantum mechanics 6504:quantum mechanics 6388:. Is that right? 6348: 6335:comment added by 6218: 6206:comment added by 6193:{\displaystyle P} 6177: 6164:comment added by 6151:{\displaystyle P} 6090: 6075: 6046: 6031: 6016: 5998: 5983: 5968: 5947: 5912: 5897: 5661: 5626: 5518: 5454: 5398: 5382:comment added by 5357:comment added by 4986: 4985: 4758: 4680:comment added by 4424:How about "where 4398: 4280: 4149: 3909: 3892:comment added by 3748: 3707: 3630: 3589: 3506: 3184: 3121: 3110: 3010: 2884: 2846: 2721: 2631: 2325:for some unitary 2217: 2076: 1990: 1400: 1254: 1237:comment added by 1189: 1028:comment added by 892:comment added by 864:quantum operation 787: 478: 425:comment added by 408: 277: 276: 273: 272: 269: 268: 151: 150: 147: 146: 7155: 7095: 7035:pure qubit state 6889: 6888: 6884: 6780: 6779: 6767: 6761: 6760: 6743:(6): 1474–1489. 6731: 6725: 6724: 6718: 6713: 6711: 6703: 6691: 6591: 6561:authors' minds. 6454:article nor the 6199: 6197: 6196: 6191: 6157: 6155: 6154: 6149: 6137: 6135: 6134: 6129: 6102: 6100: 6099: 6094: 6092: 6091: 6083: 6077: 6076: 6068: 6058: 6056: 6055: 6050: 6048: 6047: 6039: 6033: 6032: 6024: 6018: 6017: 6009: 6000: 5999: 5991: 5985: 5984: 5976: 5970: 5969: 5961: 5949: 5948: 5940: 5924: 5922: 5921: 5916: 5914: 5913: 5905: 5899: 5898: 5890: 5880: 5878: 5877: 5872: 5864: 5863: 5847: 5845: 5844: 5839: 5805: 5803: 5802: 5797: 5795: 5794: 5742: 5740: 5739: 5734: 5717: 5716: 5694: 5692: 5691: 5686: 5684: 5667: 5662: 5654: 5649: 5632: 5627: 5619: 5604: 5602: 5601: 5596: 5594: 5577: 5557: 5555: 5554: 5549: 5547: 5546: 5537: 5536: 5526: 5502: 5500: 5499: 5494: 5489: 5488: 5473: 5472: 5462: 5447: 5446: 5369: 5347: 5345: 5344: 5339: 5334: 5333: 5323: 5310: 5295: 5282: 5253: 5249: 5248: 5241: 5233: 5219: 5203: 5202:<2|)}" /: --> 5196: 5195: 5185: 5180: 5165: 5157: 5142: 5134: 5119: 5111: 5096: 5085: 5074: 5059: 5034: 5029: 5028: 5020: 5009: 4995: 4987: 4981: 4977: 4968: 4887:Accessible intro 4867: 4865: 4864: 4859: 4848: 4847: 4822: 4820: 4819: 4814: 4809: 4795: 4794: 4769: 4767: 4766: 4761: 4759: 4754: 4752: 4741: 4740: 4731: 4720: 4719: 4710: 4692: 4673: 4671: 4670: 4665: 4663: 4659: 4658: 4642: 4638: 4637: 4617: 4615: 4614: 4609: 4604: 4590: 4589: 4564: 4562: 4561: 4556: 4551: 4550: 4525: 4523: 4522: 4517: 4506: 4505: 4443: 4441: 4440: 4435: 4409: 4407: 4406: 4401: 4399: 4394: 4392: 4381: 4380: 4371: 4366: 4365: 4347: 4346: 4337: 4311: 4309: 4308: 4303: 4291: 4289: 4288: 4283: 4281: 4276: 4274: 4263: 4262: 4253: 4248: 4247: 4229: 4228: 4219: 4180: 4178: 4177: 4172: 4160: 4158: 4157: 4152: 4150: 4145: 4143: 4132: 4131: 4122: 4117: 4116: 4098: 4097: 4088: 3908: 3886: 3785: 3783: 3782: 3777: 3774: 3769: 3756: 3744: 3743: 3718: 3716: 3715: 3710: 3708: 3703: 3702: 3690: 3688: 3687: 3671: 3666: 3641: 3639: 3638: 3633: 3631: 3626: 3625: 3613: 3611: 3610: 3597: 3585: 3584: 3559: 3557: 3556: 3551: 3549: 3548: 3517: 3515: 3514: 3509: 3507: 3502: 3501: 3492: 3490: 3489: 3474: 3473: 3446: 3445: 3417: 3415: 3414: 3409: 3356: 3354: 3353: 3348: 3330: 3328: 3327: 3322: 3304: 3302: 3301: 3296: 3291: 3290: 3251:Michael C. Price 3202:Michael C. Price 3195: 3193: 3192: 3187: 3185: 3182: 3181: 3173: 3164: 3155: 3143: 3142: 3129: 3117: 3116: 3111: 3106: 3100: 3099: 3061: 3059: 3058: 3053: 3044: 3032: 3031: 3018: 3003: 3002: 2977: 2975: 2974: 2969: 2967: 2966: 2938: 2936: 2935: 2930: 2922: 2921: 2906: 2905: 2892: 2870: 2868: 2867: 2862: 2857: 2856: 2847: 2845: 2844: 2835: 2823: 2821: 2820: 2815: 2808: 2807: 2782: 2780: 2779: 2774: 2769: 2768: 2753: 2752: 2740: 2739: 2729: 2707: 2705: 2704: 2699: 2692: 2691: 2666: 2664: 2663: 2658: 2650: 2649: 2639: 2619: 2618: 2596: 2594: 2593: 2588: 2583: 2582: 2567: 2566: 2544:Here's an answer 2256: 2254: 2253: 2248: 2246: 2245: 2236: 2235: 2225: 2174: 2172: 2171: 2166: 2164: 2163: 2154: 2153: 2138: 2137: 2115: 2113: 2112: 2107: 2105: 2104: 2095: 2094: 2084: 2059: 2057: 2056: 2051: 2049: 2048: 2029: 2027: 2026: 2021: 2019: 2018: 2009: 2008: 1998: 1757: 1755: 1754: 1749: 1729: 1728: 1620: 1618: 1617: 1612: 1592: 1591: 1472: 1470: 1469: 1464: 1462: 1461: 1439: 1437: 1436: 1431: 1429: 1428: 1419: 1418: 1408: 1271:is nonnegative. 1253: 1231: 1200: 1198: 1197: 1192: 1190: 1185: 1183: 1169: 1155: 1040: 997: 971:Michael C. Price 904: 773: 637: 635: 634: 629: 624: 623: 614: 597: 592: 591: 576: 575: 557:by the elements 556: 554: 553: 548: 540: 539: 530: 515: 513: 512: 507: 505: 488: 480: 479: 471: 420: 418: 417: 412: 410: 409: 401: 391: 389: 388: 383: 280:Initial comments 231: 230: 229:physics articles 227: 224: 221: 200: 195: 194: 184: 177: 176: 171: 168: 160: 153: 123: 122: 119: 116: 113: 92: 87: 86: 76: 69: 68: 63: 55: 48: 31: 25: 24: 16: 7163: 7162: 7158: 7157: 7156: 7154: 7153: 7152: 7113: 7112: 7092: 7076: 6916: 6890: 6886: 6882: 6880: 6879: 6790: 6785: 6784: 6783: 6769: 6768: 6764: 6737:Physical Review 6733: 6732: 6728: 6714: 6704: 6693: 6692: 6688: 6585: 6418:generalizations 6370: 6327: 6232:diagonal matrix 6225: 6182: 6181: 6140: 6139: 6105: 6104: 6061: 6060: 5927: 5926: 5883: 5882: 5855: 5850: 5849: 5830: 5829: 5826: 5786: 5781: 5780: 5708: 5697: 5696: 5607: 5606: 5562: 5561: 5538: 5528: 5507: 5506: 5480: 5464: 5438: 5412: 5411: 5408: 5384:Stephiefaulkner 5359:142.150.226.247 5352: 5328: 5327: 5314: 5300: 5299: 5286: 5268: 5256: 5255: 5208: 5207: 5039: 5038: 4956: 4955: 4952: 4916: 4893:Manual of Style 4889: 4839: 4825: 4824: 4823:? I calculated 4786: 4772: 4771: 4732: 4711: 4698: 4697: 4675: 4650: 4646: 4629: 4625: 4620: 4619: 4581: 4567: 4566: 4542: 4528: 4527: 4497: 4483: 4482: 4426: 4425: 4372: 4354: 4338: 4325: 4324: 4294: 4293: 4254: 4236: 4220: 4207: 4206: 4163: 4162: 4123: 4105: 4089: 4076: 4075: 4024:Poincare sphere 3915: 3887: 3882: 3839: 3825: 3816: 3801: 3732: 3727: 3726: 3691: 3676: 3650: 3649: 3614: 3599: 3573: 3568: 3567: 3540: 3526: 3525: 3493: 3481: 3465: 3437: 3429: 3428: 3385: 3384: 3375: 3369:Anonymous User 3333: 3332: 3307: 3306: 3282: 3274: 3273: 3175: 3131: 3104: 3091: 3083: 3082: 3020: 2994: 2986: 2985: 2955: 2950: 2949: 2913: 2897: 2879: 2878: 2848: 2836: 2826: 2825: 2799: 2791: 2790: 2760: 2744: 2731: 2716: 2715: 2683: 2675: 2674: 2641: 2610: 2605: 2604: 2574: 2558: 2553: 2552: 2546: 2237: 2227: 2206: 2205: 2189:CSTAR, that is 2155: 2145: 2129: 2124: 2123: 2096: 2086: 2065: 2064: 2040: 2035: 2034: 2010: 2000: 1979: 1978: 1844: 1837: 1806: 1799: 1793: 1786: 1720: 1709: 1708: 1583: 1572: 1571: 1532: 1528: 1521: 1514: 1507: 1480: 1453: 1448: 1447: 1420: 1410: 1389: 1388: 1366: 1360: 1260: 1239:Sharma yamijala 1232: 1143: 1142: 1023: 1019: 987: 967: 925: 910: 894:109.153.172.124 887: 808: 780:201.232.162.150 774:—The preceding 770: 615: 583: 564: 559: 558: 531: 518: 517: 464: 463: 421:”? —Preceding 394: 393: 374: 373: 370: 327:130.126.230.244 317:130.126.230.244 282: 228: 225: 222: 219: 218: 196: 189: 169: 166: 120: 117: 114: 111: 110: 88: 81: 61: 32:on Knowledge's 29: 12: 11: 5: 7161: 7159: 7151: 7150: 7145: 7140: 7135: 7130: 7125: 7115: 7114: 7111: 7110: 7109: 7108: 7090: 7052: 7051: 7050: 7049: 7012: 7008: 7005: 7002: 6915: 6912: 6878: 6875: 6873: 6871: 6870: 6869: 6868: 6789: 6786: 6782: 6781: 6776:Academic Press 6762: 6726: 6694:McWeeny, Roy. 6685: 6684: 6680: 6679: 6678: 6677: 6676: 6639:Dirac notation 6604: 6603: 6602: 6601: 6597: 6582: 6581: 6580: 6579: 6578: 6577: 6576: 6575: 6574: 6573: 6558: 6481: 6478: 6475: 6472: 6469: 6466: 6463: 6459: 6452:Density matrix 6448: 6442: 6441: 6425:, because the 6369: 6366: 6365: 6364: 6326: 6323: 6308: 6307: 6276: 6275: 6224: 6221: 6220: 6219: 6189: 6147: 6127: 6124: 6121: 6118: 6115: 6112: 6089: 6086: 6080: 6074: 6071: 6045: 6042: 6036: 6030: 6027: 6021: 6015: 6012: 6006: 6003: 5997: 5994: 5988: 5982: 5979: 5973: 5967: 5964: 5958: 5955: 5952: 5946: 5943: 5937: 5934: 5911: 5908: 5902: 5896: 5893: 5870: 5867: 5862: 5858: 5837: 5825: 5822: 5821: 5820: 5819: 5818: 5793: 5789: 5732: 5729: 5726: 5723: 5720: 5715: 5711: 5707: 5704: 5683: 5679: 5676: 5673: 5670: 5666: 5660: 5657: 5652: 5648: 5644: 5641: 5638: 5635: 5631: 5625: 5622: 5617: 5614: 5593: 5589: 5586: 5583: 5580: 5576: 5572: 5569: 5545: 5541: 5535: 5531: 5525: 5521: 5517: 5514: 5492: 5487: 5483: 5479: 5476: 5471: 5467: 5461: 5457: 5453: 5450: 5445: 5441: 5437: 5434: 5431: 5428: 5425: 5422: 5419: 5407: 5404: 5400: 5399: 5375: 5337: 5332: 5326: 5322: 5318: 5315: 5313: 5309: 5305: 5302: 5301: 5298: 5294: 5290: 5287: 5285: 5281: 5277: 5274: 5273: 5271: 5266: 5263: 5239: 5236: 5232: 5228: 5225: 5222: 5218: 5201:<1|+|2: --> 5200:<2|+|2: --> 5199:<1|+|1: --> 5191:<1|+|2: --> 5190:<2|+|2: --> 5189:<1|+|1: --> 5183: 5179: 5175: 5172: 5168: 5164: 5160: 5156: 5152: 5149: 5145: 5141: 5137: 5133: 5129: 5126: 5122: 5118: 5114: 5110: 5106: 5103: 5099: 5095: 5091: 5088: 5084: 5080: 5077: 5073: 5069: 5066: 5062: 5058: 5054: 5051: 5045:<2|)}": --> 5044:<1|+|2: --> 5043:<2|+|2: --> 5042:<1|+|1: --> 5035:, I can write 5018: 5015: 5012: 5008: 5004: 5001: 4998: 4994: 4990: 4984: 4980: 4975: 4971: 4967: 4951: 4948: 4915: 4912: 4888: 4885: 4883: 4881: 4880: 4857: 4854: 4851: 4846: 4842: 4838: 4835: 4832: 4812: 4808: 4804: 4801: 4798: 4793: 4789: 4785: 4782: 4779: 4757: 4751: 4747: 4744: 4739: 4735: 4730: 4726: 4723: 4718: 4714: 4709: 4705: 4662: 4657: 4653: 4649: 4645: 4641: 4636: 4632: 4628: 4607: 4603: 4599: 4596: 4593: 4588: 4584: 4580: 4577: 4574: 4554: 4549: 4545: 4541: 4538: 4535: 4515: 4512: 4509: 4504: 4500: 4496: 4493: 4490: 4479: 4478: 4477: 4476: 4475: 4474: 4473: 4472: 4471: 4470: 4433: 4397: 4391: 4387: 4384: 4379: 4375: 4370: 4364: 4361: 4357: 4353: 4350: 4345: 4341: 4336: 4332: 4317: 4301: 4279: 4273: 4269: 4266: 4261: 4257: 4252: 4246: 4243: 4239: 4235: 4232: 4227: 4223: 4218: 4214: 4203: 4202: 4201: 4200: 4199: 4198: 4197: 4170: 4148: 4142: 4138: 4135: 4130: 4126: 4121: 4115: 4112: 4108: 4104: 4101: 4096: 4092: 4087: 4083: 4045: 4044: 4043: 4042: 4041: 4040: 4039: 4038: 4015:I was reading 4006: 4005: 4004: 4003: 4002: 4001: 4000: 3999: 3980: 3972: 3971: 3970: 3969: 3951: 3950: 3949: 3948: 3914: 3911: 3881: 3878: 3877: 3876: 3838: 3835: 3829:BetacommandBot 3824: 3821: 3815: 3812: 3800: 3797: 3787: 3786: 3773: 3768: 3765: 3761: 3755: 3751: 3747: 3742: 3739: 3735: 3722:Then clearly 3720: 3719: 3706: 3701: 3698: 3694: 3686: 3683: 3679: 3675: 3670: 3665: 3662: 3658: 3643: 3642: 3629: 3624: 3621: 3617: 3609: 3606: 3602: 3596: 3592: 3588: 3583: 3580: 3576: 3561: 3560: 3547: 3543: 3539: 3536: 3533: 3519: 3518: 3505: 3500: 3496: 3488: 3484: 3480: 3477: 3472: 3468: 3464: 3461: 3458: 3455: 3452: 3449: 3444: 3440: 3436: 3419: 3418: 3407: 3404: 3401: 3398: 3395: 3392: 3374: 3371: 3346: 3343: 3340: 3320: 3317: 3314: 3294: 3289: 3285: 3281: 3270: 3269: 3268: 3267: 3266: 3265: 3246: 3245: 3244: 3228:Dirac notation 3217: 3216: 3215: 3214: 3197: 3196: 3180: 3177: 3172: 3168: 3162: 3158: 3154: 3150: 3146: 3141: 3138: 3134: 3128: 3124: 3120: 3115: 3109: 3103: 3098: 3094: 3090: 3080: 3079: 3078: 3063: 3062: 3051: 3047: 3043: 3039: 3035: 3030: 3027: 3023: 3017: 3013: 3009: 3006: 3001: 2997: 2993: 2979: 2978: 2965: 2962: 2958: 2940: 2939: 2928: 2925: 2920: 2916: 2912: 2909: 2904: 2900: 2896: 2891: 2887: 2872: 2871: 2860: 2855: 2851: 2843: 2839: 2833: 2811: 2806: 2802: 2798: 2784: 2783: 2772: 2767: 2763: 2759: 2756: 2751: 2747: 2743: 2738: 2734: 2728: 2724: 2709: 2708: 2695: 2690: 2686: 2682: 2668: 2667: 2656: 2653: 2648: 2644: 2638: 2634: 2628: 2625: 2622: 2617: 2613: 2598: 2597: 2586: 2581: 2577: 2573: 2570: 2565: 2561: 2545: 2542: 2541: 2540: 2539: 2538: 2509: 2508: 2490: 2488: 2487: 2486: 2485: 2484: 2483: 2469: 2468: 2467: 2466: 2465: 2464: 2463: 2462: 2414: 2413: 2412: 2411: 2410: 2409: 2405: 2399: 2387: 2378: 2377: 2376: 2375: 2351: 2350: 2285: 2284: 2283: 2282: 2281: 2280: 2262: 2259: 2258: 2257: 2244: 2240: 2234: 2230: 2224: 2220: 2216: 2213: 2176: 2175: 2162: 2158: 2152: 2148: 2144: 2141: 2136: 2132: 2117: 2116: 2103: 2099: 2093: 2089: 2083: 2079: 2075: 2072: 2047: 2043: 2031: 2030: 2017: 2013: 2007: 2003: 1997: 1993: 1989: 1986: 1964: 1963: 1953: 1952: 1951: 1950: 1949: 1948: 1918: 1917: 1906: 1905: 1904: 1903: 1902: 1901: 1900: 1899: 1883: 1882: 1881: 1880: 1879: 1878: 1869: 1842: 1835: 1826: 1825: 1822: 1804: 1797: 1791: 1784: 1759: 1758: 1747: 1744: 1741: 1738: 1735: 1732: 1727: 1723: 1719: 1716: 1702:In conclusion, 1699: 1698: 1697: 1696: 1695: 1694: 1693: 1692: 1691: 1690: 1689: 1688: 1652: 1651: 1650: 1649: 1648: 1647: 1646: 1645: 1628: 1627: 1626: 1625: 1624: 1623: 1622: 1621: 1610: 1607: 1604: 1601: 1598: 1595: 1590: 1586: 1582: 1579: 1562: 1561: 1560: 1559: 1558: 1557: 1556: 1555: 1545: 1544: 1543: 1542: 1541: 1540: 1530: 1526: 1519: 1512: 1505: 1494: 1493: 1492: 1491: 1482: 1478: 1475: 1474: 1473: 1460: 1456: 1442: 1441: 1440: 1427: 1423: 1417: 1413: 1407: 1403: 1399: 1396: 1380: 1379: 1378: 1377: 1364: 1358: 1351: 1350: 1330: 1329: 1328: 1327: 1326: 1325: 1324: 1323: 1307: 1306: 1305: 1304: 1303: 1302: 1284: 1283: 1282: 1281: 1259: 1256: 1218: 1217: 1216: 1215: 1214: 1213: 1188: 1182: 1178: 1175: 1172: 1168: 1164: 1161: 1158: 1154: 1150: 1134: 1133: 1132: 1131: 1130: 1129: 1121: 1120: 1119: 1118: 1108:94.194.100.255 1100: 1099: 1098: 1097: 1090: 1089: 1088: 1087: 1080: 1079: 1078: 1077: 1070: 1069: 1018: 1015: 1014: 1013: 992:comment added 978: 966: 963: 962: 961: 952: 924: 921: 909: 906: 884: 883: 860: 859: 858: 857: 837: 836: 807: 804: 769: 766: 765: 764: 730: 729: 728: 727: 726: 725: 724: 723: 722: 721: 694: 693: 692: 691: 690: 689: 688: 687: 665: 664: 663: 662: 661: 660: 641: 640: 639: 638: 627: 622: 618: 613: 609: 606: 603: 600: 596: 590: 586: 582: 579: 574: 571: 567: 546: 543: 538: 534: 529: 525: 504: 500: 497: 494: 491: 487: 483: 477: 474: 457: 456: 407: 404: 381: 369: 366: 363: 361: 360: 350:144.174.214.46 346: 343: 333: 332: 312: 311: 302: 281: 278: 275: 274: 271: 270: 267: 266: 259:Mid-importance 255: 249: 248: 241: 235: 234: 232: 215:the discussion 202: 201: 198:Physics portal 185: 173: 172: 170:Mid‑importance 161: 149: 148: 145: 144: 133: 127: 126: 124: 107:the discussion 94: 93: 77: 65: 64: 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 7160: 7149: 7146: 7144: 7141: 7139: 7136: 7134: 7131: 7129: 7126: 7124: 7121: 7120: 7118: 7107: 7103: 7099: 7093: 7088: 7084: 7080: 7074: 7069: 7068: 7067: 7063: 7059: 7058:David Spector 7054: 7053: 7048: 7044: 7040: 7036: 7032: 7031:quantum state 7027: 7026: 7025: 7021: 7017: 7016:David Spector 7013: 7009: 7006: 7003: 7000: 6996: 6991: 6990: 6989: 6987: 6983: 6979: 6975: 6968: 6966: 6962: 6958: 6954: 6946: 6945: 6941: 6937: 6932: 6931: 6926: 6922: 6919: 6913: 6911: 6909: 6905: 6901: 6897: 6885: 6876: 6874: 6867: 6863: 6859: 6855: 6854: 6853: 6849: 6845: 6841: 6837: 6833: 6832: 6831: 6830: 6826: 6822: 6818: 6814: 6809: 6807: 6803: 6799: 6795: 6777: 6773: 6766: 6763: 6758: 6754: 6750: 6746: 6742: 6738: 6730: 6727: 6722: 6709: 6701: 6697: 6690: 6687: 6683: 6675: 6671: 6667: 6663: 6659: 6655: 6654: 6653: 6649: 6645: 6640: 6635: 6631: 6630: 6629: 6628: 6624: 6620: 6619:198.53.159.44 6614: 6612: 6611:Richard Bader 6607: 6598: 6595: 6594: 6589: 6584: 6583: 6572: 6568: 6564: 6559: 6555: 6554: 6553: 6549: 6545: 6542: 6537: 6532: 6531: 6530: 6526: 6522: 6518: 6513: 6509: 6505: 6501: 6500: 6499: 6495: 6491: 6486: 6482: 6479: 6476: 6473: 6470: 6467: 6464: 6460: 6457: 6453: 6449: 6446: 6445: 6444: 6443: 6440: 6436: 6432: 6428: 6424: 6419: 6415: 6410: 6409: 6408: 6407: 6403: 6399: 6393: 6389: 6387: 6382: 6379: 6376: 6373: 6367: 6363: 6359: 6355: 6351: 6350: 6349: 6346: 6342: 6338: 6337:Koitus~nlwiki 6334: 6324: 6322: 6321: 6317: 6313: 6312:David Spector 6306: 6302: 6298: 6293: 6292: 6291: 6290: 6286: 6282: 6281:David Spector 6274: 6270: 6266: 6262: 6261:David spector 6259: 6258: 6257: 6256: 6252: 6248: 6247:David Spector 6243: 6241: 6237: 6233: 6228: 6222: 6217: 6213: 6209: 6208:217.95.163.80 6205: 6187: 6180: 6179: 6178: 6175: 6171: 6167: 6166:217.95.160.47 6163: 6145: 6122: 6113: 6084: 6078: 6069: 6040: 6025: 6019: 6010: 6001: 5992: 5977: 5971: 5962: 5953: 5941: 5932: 5906: 5900: 5891: 5868: 5865: 5860: 5856: 5835: 5823: 5817: 5813: 5809: 5791: 5787: 5778: 5776: 5771: 5770: 5769: 5765: 5761: 5757: 5756: 5755: 5754: 5750: 5746: 5730: 5727: 5721: 5713: 5709: 5705: 5702: 5677: 5668: 5658: 5655: 5650: 5642: 5633: 5623: 5620: 5615: 5612: 5587: 5578: 5570: 5567: 5558: 5543: 5539: 5533: 5529: 5523: 5519: 5515: 5512: 5503: 5485: 5481: 5474: 5469: 5465: 5459: 5455: 5451: 5443: 5439: 5432: 5429: 5423: 5417: 5405: 5403: 5397: 5393: 5389: 5385: 5381: 5374: 5373:CONTRADICTION 5370: 5368: 5364: 5360: 5356: 5349: 5335: 5330: 5324: 5320: 5316: 5311: 5307: 5303: 5296: 5292: 5288: 5283: 5279: 5275: 5269: 5264: 5261: 5237: 5234: 5226: 5223: 5220: 5206:which in the 5204: 5173: 5169: 5166: 5158: 5150: 5146: 5143: 5135: 5127: 5123: 5120: 5112: 5104: 5100: 5097: 5086: 5082: 5078: 5075: 5067: 5063: 5060: 5052: 5049: 5036: 5013: 5010: 5002: 4999: 4996: 4982: 4978: 4972: 4969: 4949: 4947: 4946: 4942: 4938: 4933: 4932: 4928: 4924: 4921: 4913: 4911: 4910: 4906: 4902: 4901:RockMagnetist 4898: 4894: 4886: 4884: 4879: 4875: 4871: 4855: 4852: 4844: 4840: 4833: 4830: 4810: 4806: 4802: 4799: 4791: 4787: 4780: 4777: 4755: 4749: 4737: 4733: 4724: 4716: 4712: 4695: 4694: 4693: 4691: 4687: 4683: 4682:130.75.25.219 4679: 4660: 4655: 4651: 4647: 4643: 4639: 4634: 4630: 4626: 4605: 4601: 4597: 4594: 4586: 4582: 4575: 4572: 4547: 4543: 4536: 4533: 4513: 4510: 4502: 4498: 4491: 4488: 4469: 4465: 4461: 4457: 4456: 4455: 4451: 4447: 4431: 4423: 4422: 4421: 4417: 4413: 4395: 4389: 4377: 4373: 4362: 4359: 4355: 4351: 4343: 4339: 4322: 4318: 4315: 4299: 4277: 4271: 4259: 4255: 4244: 4241: 4237: 4233: 4225: 4221: 4204: 4196: 4192: 4188: 4184: 4168: 4146: 4140: 4128: 4124: 4113: 4110: 4106: 4102: 4094: 4090: 4072: 4071: 4070: 4066: 4062: 4055: 4054: 4053: 4052: 4051: 4050: 4049: 4048: 4047: 4046: 4037: 4033: 4029: 4025: 4018: 4014: 4013: 4012: 4011: 4010: 4009: 4008: 4007: 3996: 3995: 3994: 3990: 3986: 3981: 3978: 3977: 3976: 3975: 3974: 3973: 3968: 3964: 3960: 3955: 3954: 3953: 3952: 3946: 3945: 3944: 3940: 3936: 3932: 3931: 3930: 3929: 3925: 3921: 3912: 3910: 3907: 3903: 3899: 3895: 3891: 3879: 3875: 3871: 3867: 3863: 3859: 3858: 3857: 3856: 3852: 3848: 3844: 3836: 3834: 3833: 3830: 3822: 3820: 3813: 3811: 3810: 3807: 3798: 3796: 3795: 3792: 3771: 3766: 3763: 3753: 3749: 3745: 3740: 3737: 3733: 3725: 3724: 3723: 3699: 3696: 3692: 3684: 3681: 3677: 3673: 3668: 3663: 3660: 3648: 3647: 3646: 3622: 3619: 3615: 3607: 3604: 3600: 3594: 3590: 3586: 3581: 3578: 3574: 3566: 3565: 3564: 3545: 3534: 3531: 3524: 3523: 3522: 3498: 3494: 3486: 3482: 3478: 3470: 3466: 3462: 3459: 3450: 3447: 3442: 3438: 3427: 3426: 3425: 3422: 3402: 3393: 3383: 3382: 3381: 3378: 3372: 3370: 3368: 3364: 3360: 3359:72.229.22.106 3341: 3315: 3287: 3283: 3264: 3261: 3257: 3256: 3255: 3252: 3247: 3243: 3241: 3237: 3233: 3229: 3224: 3223: 3221: 3220: 3219: 3218: 3212: 3208: 3207: 3206: 3203: 3199: 3198: 3178: 3176: 3170: 3166: 3156: 3152: 3148: 3144: 3139: 3136: 3132: 3126: 3122: 3118: 3113: 3107: 3096: 3092: 3081: 3077: 3076: 3074: 3073: 3072: 3071: 3068: 3045: 3041: 3037: 3033: 3028: 3025: 3021: 3015: 3011: 3007: 2999: 2995: 2984: 2983: 2982: 2963: 2960: 2956: 2948: 2947: 2946: 2944: 2926: 2923: 2918: 2914: 2902: 2898: 2894: 2889: 2885: 2877: 2876: 2875: 2853: 2849: 2841: 2837: 2804: 2800: 2789: 2788: 2787: 2765: 2761: 2749: 2745: 2741: 2736: 2732: 2726: 2722: 2714: 2713: 2712: 2688: 2684: 2673: 2672: 2671: 2654: 2651: 2646: 2642: 2636: 2632: 2626: 2623: 2620: 2615: 2611: 2603: 2602: 2601: 2579: 2575: 2571: 2568: 2563: 2559: 2551: 2550: 2549: 2537: 2534: 2530: 2529: 2528: 2525: 2520: 2519: 2518: 2517: 2514: 2507: 2504: 2500: 2499: 2498: 2497: 2494: 2482: 2479: 2475: 2474: 2473: 2472: 2471: 2470: 2461: 2458: 2454: 2450: 2446: 2442: 2438: 2434: 2430: 2426: 2422: 2421: 2420: 2419: 2418: 2417: 2416: 2415: 2406: 2404: 2400: 2396: 2392: 2388: 2384: 2383: 2382: 2381: 2380: 2379: 2374: 2371: 2367: 2363: 2359: 2355: 2354: 2353: 2352: 2349: 2346: 2342: 2341: 2340: 2339: 2336: 2332: 2328: 2324: 2321: 2317: 2313: 2310: 2306: 2302: 2298: 2295: 2291: 2279: 2276: 2272: 2271: 2270: 2267: 2263: 2260: 2242: 2238: 2232: 2228: 2222: 2218: 2214: 2211: 2204: 2203: 2201: 2200: 2199: 2196: 2192: 2188: 2187: 2186: 2185: 2182: 2160: 2156: 2150: 2146: 2142: 2139: 2134: 2130: 2122: 2121: 2120: 2101: 2097: 2091: 2087: 2081: 2077: 2073: 2070: 2063: 2062: 2061: 2045: 2041: 2015: 2011: 2005: 2001: 1995: 1991: 1987: 1984: 1977: 1976: 1975: 1973: 1969: 1962: 1959: 1955: 1954: 1947: 1944: 1940: 1939: 1938: 1935: 1931: 1930: 1929: 1928: 1927: 1926: 1923: 1916: 1913: 1908: 1907: 1898: 1895: 1891: 1890: 1889: 1888: 1887: 1886: 1885: 1884: 1877: 1874: 1870: 1868: 1865: 1861: 1860: 1859: 1856: 1852: 1848: 1845:is such that 1841: 1834: 1830: 1829: 1828: 1827: 1823: 1821: 1818: 1814: 1810: 1803: 1796: 1790: 1783: 1778: 1777: 1776: 1775: 1772: 1768: 1764: 1742: 1736: 1733: 1730: 1725: 1721: 1717: 1714: 1707: 1706: 1705: 1703: 1687: 1684: 1679: 1675: 1671: 1670: 1669: 1666: 1662: 1661: 1660: 1659: 1658: 1657: 1656: 1655: 1654: 1653: 1644: 1641: 1636: 1635: 1634: 1633: 1632: 1631: 1630: 1629: 1605: 1599: 1596: 1593: 1588: 1584: 1580: 1577: 1570: 1569: 1568: 1567: 1566: 1565: 1564: 1563: 1553: 1552: 1551: 1550: 1549: 1548: 1547: 1546: 1539: 1536: 1525: 1518: 1511: 1504: 1500: 1499: 1498: 1497: 1496: 1495: 1490: 1487: 1483: 1476: 1458: 1454: 1446: 1445: 1443: 1425: 1421: 1415: 1411: 1405: 1401: 1397: 1394: 1387: 1386: 1384: 1383: 1382: 1381: 1375: 1374: 1373: 1370: 1363: 1357: 1353: 1352: 1349: 1346: 1342: 1341: 1340: 1339: 1336: 1322: 1319: 1315: 1314: 1313: 1312: 1311: 1310: 1309: 1308: 1301: 1298: 1295:has trace 1. 1294: 1290: 1289: 1288: 1287: 1286: 1285: 1279: 1278: 1277: 1274: 1270: 1266: 1265: 1264: 1257: 1255: 1252: 1248: 1244: 1240: 1236: 1228: 1225: 1222: 1212: 1208: 1204: 1186: 1180: 1170: 1162: 1156: 1140: 1139: 1138: 1137: 1136: 1135: 1127: 1126: 1125: 1124: 1123: 1122: 1117: 1113: 1109: 1104: 1103: 1102: 1101: 1094: 1093: 1092: 1091: 1084: 1083: 1082: 1081: 1074: 1073: 1072: 1071: 1068: 1064: 1060: 1056: 1052: 1047: 1043: 1042: 1041: 1039: 1035: 1031: 1027: 1022:other ways? 1016: 1012: 1008: 1004: 1000: 999: 998: 995: 991: 986: 985:Mark Moriarty 982: 976: 975: 972: 964: 960: 957: 953: 951: 948: 944: 943: 942: 941: 938: 934: 930: 927:the articles 922: 920: 919: 916: 907: 905: 903: 899: 895: 891: 882: 879: 875: 874: 873: 872: 869: 865: 856: 853: 849: 848: 847: 844: 839: 838: 835: 832: 828: 824: 820: 819: 818: 817: 814: 805: 803: 802: 798: 794: 788: 785: 781: 777: 767: 763: 759: 755: 751: 747: 746: 745: 744: 740: 736: 720: 716: 712: 708: 704: 703: 702: 701: 700: 699: 698: 697: 696: 695: 686: 682: 678: 673: 672: 671: 670: 669: 668: 667: 666: 659: 655: 651: 647: 646: 645: 644: 643: 642: 620: 616: 607: 598: 588: 584: 577: 572: 569: 565: 536: 532: 498: 489: 481: 472: 461: 460: 459: 458: 455: 451: 447: 443: 442: 441: 440: 436: 432: 428: 424: 402: 379: 367: 365: 359: 355: 351: 347: 344: 342: 339: 335: 334: 331: 328: 324: 323: 322: 321: 318: 309: 305: 304: 303: 300: 295: 294: 291: 287: 286:quantum logic 279: 264: 260: 254: 251: 250: 246: 240: 237: 236: 233: 216: 212: 208: 207: 199: 193: 188: 186: 183: 179: 178: 174: 165: 162: 159: 155: 142: 138: 132: 129: 128: 125: 108: 104: 100: 99: 91: 85: 80: 78: 75: 71: 70: 66: 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 7078: 7072: 6972:— Preceding 6969: 6951:— Preceding 6947: 6933: 6929: 6927: 6923: 6920: 6917: 6900:89.3.212.183 6894:— Preceding 6891: 6881:"": --> 6872: 6810: 6805: 6791: 6771: 6765: 6740: 6736: 6729: 6708:cite journal 6699: 6689: 6681: 6633: 6615: 6608: 6605: 6540: 6535: 6511: 6484: 6417: 6394: 6390: 6383: 6380: 6377: 6374: 6371: 6331:— Preceding 6328: 6309: 6277: 6244: 6229: 6226: 6202:— Preceding 6160:— Preceding 5827: 5773: 5559: 5504: 5409: 5378:— Preceding 5372: 5371: 5353:— Preceding 5350: 5205: 5037: 4953: 4934: 4917: 4914:Löwdin paper 4890: 4882: 4480: 4320: 4313: 4182: 4017:Bloch sphere 3916: 3894:Dr. Universe 3885:developed. 3883: 3843:non-negative 3840: 3837:Non-negative 3826: 3817: 3802: 3788: 3721: 3644: 3562: 3520: 3423: 3420: 3379: 3376: 3271: 3225: 3210: 3064: 2980: 2942: 2941: 2873: 2785: 2710: 2669: 2599: 2547: 2510: 2489: 2452: 2448: 2444: 2440: 2436: 2432: 2428: 2424: 2402: 2394: 2390: 2365: 2361: 2357: 2330: 2326: 2322: 2319: 2315: 2311: 2308: 2304: 2300: 2296: 2293: 2289: 2286: 2190: 2177: 2118: 2032: 1971: 1967: 1965: 1919: 1850: 1846: 1839: 1832: 1812: 1808: 1801: 1794: 1788: 1781: 1766: 1762: 1760: 1701: 1700: 1677: 1523: 1516: 1509: 1502: 1361: 1355: 1331: 1292: 1268: 1261: 1233:— Preceding 1229: 1226: 1223: 1219: 1055:whole system 1054: 1050: 1024:— Preceding 1020: 977: 968: 926: 911: 888:— Preceding 885: 861: 809: 789: 771: 731: 677:Dr. Universe 371: 362: 313: 296: 283: 258: 204: 137:Mid-priority 136: 96: 62:Mid‑priority 40:WikiProjects 6858:MadeOfAtoms 6821:MadeOfAtoms 6423:convex body 6236:measurement 4676:—Preceding 4460:137.73.4.92 4412:2.25.206.58 3913:Mixed State 3888:—Preceding 3847:J S Lundeen 3240:measurement 2981:such that 2423:reply: say 1811:such that 1030:173.20.49.9 988:—Preceding 933:mixed state 112:Mathematics 103:mathematics 59:Mathematics 7117:Categories 6682:References 6666:XOR'easter 6644:XOR'easter 6600:intuitive. 6588:XOR'easter 6563:XOR'easter 6521:XOR'easter 6431:XOR'easter 6354:XOR'easter 6297:XOR'easter 6265:XOR'easter 5808:XOR'easter 5033:)}" /: --> 4937:Longitude2 3238:, such as 3211:completely 2303:satisfies 2033:where the 929:pure state 793:DieHenkels 427:Laser Lars 6999:entangled 6778:, London. 6717:|journal= 6508:tesseract 6456:Born rule 6414:Born rule 6240:decoheres 5252:}" /: --> 2386:relation? 1678:bijection 954:Support. 7073:together 6974:unsigned 6953:unsigned 6896:unsigned 6544:Gwideman 6490:Gwideman 6398:Gwideman 6345:contribs 6333:unsigned 6204:unsigned 6174:contribs 6162:unsigned 5392:contribs 5380:unsigned 5355:unsigned 5251:,|2: --> 5244:,|2: --> 5210:,|2: --> 5192:<2|)} 5032:+|2: --> 5024:+|2: --> 4960:)}": --> 4959:+|2: --> 4678:unsigned 4321:ensemble 4292:, where 4161:, where 3902:contribs 3890:unsigned 1247:contribs 1235:unsigned 1026:unsigned 956:Nick Mks 890:unsigned 823:presheaf 776:unsigned 435:contribs 423:unsigned 338:Archelon 6934:Thanks 6745:Bibcode 6702:(1273). 6234:when a 5745:Nomadbl 5211:}": --> 3249:text.-- 2943:Theorem 2513:Mct mht 2493:Mct mht 2457:Mct mht 2370:Mct mht 2335:Mct mht 2195:Mct mht 1972:thought 1934:Mct mht 1922:Mct mht 1910:two).-- 1894:Mct mht 1855:Mct mht 1817:Mct mht 1681:sets.-- 1665:Mct mht 1535:Mct mht 1369:Mct mht 1297:Mct mht 1273:Mct mht 990:undated 965:Entropy 937:Mct mht 915:Mct mht 868:Mct mht 843:Mct mht 813:Mct mht 735:Oakwood 306:Done -- 261:on the 220:Physics 211:Physics 167:B‑class 164:Physics 139:on the 30:C-class 6844:Tercer 6800:, and 6658:Tercer 6656:After 6517:photon 5505:Where 4923:Jocasa 3806:CarlAB 2600:with 2447:. Let 2443:Σ Σ 1444:where 945:Yep.-- 36:scale. 5760:Steve 5238:: --> 5224:: --> 5170:: --> 5147:: --> 5124:: --> 5101:: --> 5064:: --> 5014:: --> 5000:: --> 4973:: --> 4870:Steve 4446:Steve 4312:is a 4181:is a 4061:Steve 3985:Steve 3935:Steve 3918:it?-- 3866:Steve 3791:CSTAR 3563:Thus 3260:CSTAR 3067:CSTAR 2670:and 2533:CSTAR 2524:CSTAR 2503:CSTAR 2478:CSTAR 2398:this. 2345:CSTAR 2275:CSTAR 2266:CSTAR 2181:CSTAR 1958:CSTAR 1943:CSTAR 1912:CSTAR 1873:CSTAR 1864:CSTAR 1771:CSTAR 1683:CSTAR 1640:CSTAR 1486:CSTAR 1345:CSTAR 1335:CSTAR 1318:CSTAR 1203:Steve 1059:Steve 1003:Steve 947:CSTAR 923:merge 878:CSTAR 852:CSTAR 831:CSTAR 754:Steve 711:Steve 650:Steve 446:Steve 290:CSTAR 7102:talk 7087:ISBN 7062:talk 7043:talk 7020:talk 6982:talk 6961:talk 6940:talk 6904:talk 6883:edit 6862:talk 6848:talk 6825:talk 6721:help 6670:talk 6648:talk 6623:talk 6567:talk 6548:talk 6525:talk 6494:talk 6435:talk 6402:talk 6358:talk 6341:talk 6316:talk 6301:talk 6285:talk 6269:talk 6251:talk 6212:talk 6170:talk 5812:talk 5764:talk 5749:talk 5388:talk 5363:talk 5171:< 5148:< 5125:< 5102:< 5065:< 4941:talk 4927:talk 4905:talk 4891:The 4874:talk 4686:talk 4464:talk 4450:talk 4416:talk 4191:talk 4065:talk 4032:talk 3989:talk 3963:talk 3939:talk 3924:talk 3898:talk 3870:talk 3851:talk 3363:talk 3331:and 2401:Re: 2389:Re: 2314:iff 1970:You 1815:= . 1674:here 1243:talk 1207:talk 1112:talk 1063:talk 1034:talk 1007:talk 931:and 898:talk 797:talk 784:talk 758:talk 739:talk 715:talk 681:talk 654:talk 450:talk 431:talk 354:talk 6815:or 6753:doi 6700:253 6512:are 6485:set 3232:ket 3230:by 2824:by 2191:not 1966:Re: 1838:... 308:V79 299:V79 253:Mid 131:Mid 7119:: 7104:) 7085:. 7081:. 7064:) 7045:) 7033:, 7022:) 6984:) 6963:) 6942:) 6906:) 6864:) 6850:) 6827:) 6808:. 6774:. 6751:. 6741:97 6739:. 6712:: 6710:}} 6706:{{ 6698:. 6672:) 6650:) 6625:) 6569:) 6550:) 6527:) 6496:) 6437:) 6404:) 6360:) 6347:) 6343:• 6318:) 6303:) 6287:) 6271:) 6253:) 6214:) 6176:) 6172:• 6126:∣ 6123:ϕ 6120:⟨ 6117:⟩ 6114:ϕ 6111:∣ 6088:→ 6073:→ 6044:→ 6035:⟩ 6029:→ 6020:∣ 6014:→ 6005:⟨ 5996:→ 5987:⟩ 5981:→ 5972:∣ 5966:→ 5957:⟨ 5954::= 5945:→ 5910:→ 5895:→ 5869:ρ 5857:ρ 5836:ρ 5814:) 5788:ρ 5766:) 5751:) 5678:ψ 5675:⟨ 5672:⟩ 5669:ψ 5643:ψ 5640:⟨ 5637:⟩ 5634:ψ 5613:ρ 5588:ψ 5585:⟨ 5582:⟩ 5579:ψ 5568:ρ 5540:ρ 5520:∑ 5513:ρ 5482:ρ 5456:∑ 5424:ρ 5394:) 5390:• 5365:) 5262:ρ 5068:ψ 5061:ψ 5050:ρ 5025:)} 4970:ψ 4943:) 4929:) 4907:) 4876:) 4841:ρ 4788:ρ 4743:⟩ 4734:ψ 4722:⟩ 4713:ψ 4688:) 4652:ψ 4631:ψ 4583:ρ 4544:ρ 4499:ρ 4466:) 4452:) 4432:θ 4418:) 4383:⟩ 4374:ψ 4363:θ 4349:⟩ 4340:ψ 4316:." 4300:θ 4265:⟩ 4256:ψ 4245:θ 4231:⟩ 4222:ψ 4193:) 4169:θ 4134:⟩ 4125:ψ 4114:θ 4100:⟩ 4091:ψ 4067:) 4034:) 3991:) 3965:) 3941:) 3926:) 3904:) 3900:• 3872:) 3853:) 3760:Φ 3750:∑ 3705:¯ 3693:ψ 3678:ψ 3657:Φ 3628:¯ 3616:ψ 3601:ψ 3591:∑ 3546:∗ 3542:Ψ 3538:Ψ 3504:¯ 3495:ϕ 3483:ϕ 3476:⟩ 3463:∣ 3460:ϕ 3457:⟨ 3454:⟩ 3451:ϕ 3448:∣ 3435:⟨ 3406:∣ 3403:ϕ 3400:⟨ 3397:⟩ 3394:ϕ 3391:∣ 3365:) 3345:⟩ 3339:∣ 3319:⟩ 3313:∣ 3293:⟩ 3284:ψ 3280:∣ 3200:-- 3161:⟩ 3149:ψ 3145:∣ 3123:∑ 3102:⟩ 3093:ψ 3089:∣ 3050:⟩ 3038:ψ 3034:∣ 3012:∑ 3005:⟩ 2996:ψ 2992:∣ 2924:∣ 2915:ψ 2911:⟨ 2908:⟩ 2899:ψ 2895:∣ 2886:∑ 2859:⟩ 2850:ψ 2832:∣ 2810:⟩ 2801:ψ 2797:∣ 2771:∣ 2762:ψ 2758:⟨ 2755:⟩ 2746:ψ 2742:∣ 2723:∑ 2694:⟩ 2685:ψ 2681:∣ 2633:∑ 2621:≥ 2585:⟩ 2576:ψ 2572:∣ 2451:= 2439:= 2435:Σ 2431:= 2318:= 2307:= 2292:= 2229:λ 2219:∑ 2212:τ 2161:∗ 2088:λ 2078:∑ 2071:τ 2002:λ 1992:∑ 1985:τ 1849:≤ 1743:τ 1737:μ 1731:τ 1722:∫ 1715:ρ 1606:τ 1600:μ 1594:τ 1585:∫ 1578:ρ 1522:, 1484:-- 1455:τ 1422:τ 1412:λ 1402:∑ 1395:ρ 1333:-- 1249:) 1245:• 1209:) 1174:⟩ 1171:11 1160:⟩ 1157:00 1114:) 1065:) 1036:) 1009:) 900:) 799:) 760:) 741:) 717:) 683:) 656:) 626:⟩ 617:ϕ 608:ψ 605:⟨ 602:⟩ 599:ψ 585:ϕ 581:⟨ 542:⟩ 533:ϕ 499:ψ 496:⟨ 493:⟩ 490:ψ 476:^ 452:) 437:) 433:• 406:^ 403:ρ 380:ρ 356:) 288:. 7100:( 7094:. 7060:( 7041:( 7018:( 6980:( 6959:( 6938:( 6902:( 6887:] 6860:( 6846:( 6823:( 6759:. 6755:: 6747:: 6723:) 6719:( 6668:( 6646:( 6621:( 6590:: 6586:@ 6565:( 6546:( 6523:( 6492:( 6433:( 6400:( 6356:( 6339:( 6314:( 6299:( 6283:( 6267:( 6249:( 6210:( 6188:P 6168:( 6146:P 6085:w 6079:, 6070:v 6041:w 6026:x 6011:w 6002:+ 5993:v 5978:x 5963:v 5951:) 5942:x 5936:( 5933:P 5907:w 5901:, 5892:v 5866:= 5861:2 5810:( 5792:i 5777:. 5762:( 5747:( 5731:1 5728:= 5725:) 5722:2 5719:( 5714:2 5710:g 5706:o 5703:l 5682:| 5665:| 5659:2 5656:1 5651:+ 5647:| 5630:| 5624:2 5621:1 5616:= 5592:| 5575:| 5571:= 5544:i 5534:i 5530:p 5524:i 5516:= 5491:) 5486:i 5478:( 5475:S 5470:i 5466:p 5460:i 5452:+ 5449:) 5444:i 5440:p 5436:( 5433:H 5430:= 5427:) 5421:( 5418:S 5386:( 5361:( 5336:. 5331:] 5325:2 5321:/ 5317:1 5312:2 5308:/ 5304:1 5297:2 5293:/ 5289:1 5284:2 5280:/ 5276:1 5270:[ 5265:= 5245:} 5235:2 5231:| 5227:, 5221:1 5217:| 5182:) 5178:| 5174:2 5167:2 5163:| 5159:+ 5155:| 5151:1 5144:2 5140:| 5136:+ 5132:| 5128:2 5121:1 5117:| 5113:+ 5109:| 5105:1 5098:1 5094:| 5090:( 5087:2 5083:/ 5079:1 5076:= 5072:| 5057:| 5053:= 5017:) 5011:2 5007:| 5003:+ 4997:1 4993:| 4989:( 4983:2 4979:1 4974:= 4966:| 4939:( 4925:( 4903:( 4872:( 4856:1 4853:= 4850:) 4845:2 4837:( 4834:r 4831:t 4811:2 4807:/ 4803:1 4800:= 4797:) 4792:2 4784:( 4781:r 4778:t 4756:2 4750:/ 4746:) 4738:2 4729:| 4725:+ 4717:1 4708:| 4704:( 4684:( 4661:⟩ 4656:2 4648:| 4644:+ 4640:⟩ 4635:1 4627:| 4606:2 4602:/ 4598:1 4595:= 4592:) 4587:2 4579:( 4576:r 4573:t 4553:) 4548:2 4540:( 4537:r 4534:t 4514:1 4511:= 4508:) 4503:2 4495:( 4492:r 4489:t 4462:( 4448:( 4414:( 4396:2 4390:/ 4386:) 4378:2 4369:| 4360:i 4356:e 4352:+ 4344:1 4335:| 4331:( 4278:2 4272:/ 4268:) 4260:2 4251:| 4242:i 4238:e 4234:+ 4226:1 4217:| 4213:( 4189:( 4147:2 4141:/ 4137:) 4129:2 4120:| 4111:i 4107:e 4103:+ 4095:1 4086:| 4082:( 4063:( 4030:( 3987:( 3961:( 3937:( 3922:( 3896:( 3868:( 3849:( 3772:k 3767:j 3764:i 3754:k 3746:= 3741:j 3738:i 3734:A 3700:k 3697:j 3685:k 3682:i 3674:= 3669:k 3664:j 3661:i 3623:k 3620:j 3608:k 3605:i 3595:k 3587:= 3582:j 3579:i 3575:A 3535:= 3532:A 3499:j 3487:i 3479:= 3471:j 3467:e 3443:i 3439:e 3361:( 3342:L 3316:R 3288:i 3179:′ 3171:j 3167:p 3157:′ 3153:j 3140:j 3137:i 3133:u 3127:j 3119:= 3114:i 3108:p 3097:i 3046:′ 3042:j 3029:j 3026:i 3022:u 3016:j 3008:= 3000:i 2964:j 2961:i 2957:u 2927:. 2919:i 2903:i 2890:i 2854:i 2842:i 2838:p 2805:i 2766:i 2750:i 2737:i 2733:p 2727:i 2689:i 2655:1 2652:= 2647:i 2643:p 2637:i 2627:, 2624:0 2616:i 2612:p 2580:i 2569:, 2564:i 2560:p 2453:U 2449:M 2445:U 2441:U 2437:U 2433:U 2429:A 2425:A 2393:M 2366:N 2362:M 2358:A 2331:U 2327:U 2323:U 2320:M 2316:N 2312:N 2309:N 2305:A 2301:N 2297:M 2294:M 2290:A 2243:i 2239:A 2233:i 2223:i 2215:= 2157:U 2151:i 2147:F 2143:U 2140:= 2135:i 2131:E 2102:i 2098:F 2092:i 2082:i 2074:= 2046:i 2042:E 2016:i 2012:E 2006:i 1996:i 1988:= 1851:n 1847:m 1843:m 1840:v 1836:1 1833:v 1813:U 1809:U 1805:i 1802:w 1798:i 1795:v 1792:i 1789:v 1785:i 1782:v 1780:{ 1763:S 1746:) 1740:( 1734:d 1726:S 1718:= 1609:) 1603:( 1597:d 1589:S 1581:= 1531:i 1529:} 1527:i 1524:v 1520:i 1517:p 1513:i 1510:v 1506:i 1503:p 1479:i 1459:i 1426:i 1416:i 1406:i 1398:= 1365:i 1362:v 1359:i 1356:v 1293:A 1269:A 1241:( 1205:( 1187:2 1181:/ 1177:) 1167:| 1163:+ 1153:| 1149:( 1110:( 1061:( 1032:( 1005:( 996:. 983:- 896:( 795:( 782:( 756:( 737:( 713:( 679:( 652:( 621:n 612:| 595:| 589:m 578:= 573:n 570:m 566:O 545:} 537:k 528:| 524:{ 503:| 486:| 482:= 473:O 448:( 429:( 352:( 265:. 239:B 143:. 42::

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