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38:
1145:, began to be developed, making it possible to isolate single atoms and arrange them in arrays. Prior to these developments, precise control over single quantum systems was not possible, and experiments utilized coarser, simultaneous control over a large number of quantum systems. The development of viable single-state manipulation techniques led to increased interest in the field of quantum information and computation.
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polarizations. This necessitates that the two distant parties have exact directionality synchronization. However, from quantum mechanics theory the quantum state is completely random so that it is impossible for Alice to predict if she will get vertical polarization or horizontal polarization results.
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quantum cryptographic protocol. The key idea was the use of the fundamental principle of quantum mechanics that observation disturbs the observed, and the introduction of an eavesdropper in a secure communication line will immediately let the two parties trying to communicate know of the presence of
2783:
The entangled states are perfectly correlated which means that if Alice and Bob both measure their particles having either a vertical or horizontal polarization, they always get the same answer with 100% probability. The same is true if they both measure any other pair of complementary (orthogonal)
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Despite all the excitement and interest over studying isolated quantum systems and trying to find a way to circumvent the theory of relativity, research in quantum information theory became stagnant in the 1980s. However, around the same time another avenue started dabbling into quantum information
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doubles every two years. As transistors began to become smaller and smaller in order to pack more power per surface area, quantum effects started to show up in the electronics resulting in inadvertent interference. This led to the advent of quantum computing, which used quantum mechanics to design
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If a quantum system were perfectly isolated, it would maintain coherence perfectly, but it would be impossible to test the entire system. If it is not perfectly isolated, for example during a measurement, coherence is shared with the environment and appears to be lost with time; this process is
2732:
are also applications of quantum communication. They are two opposite ways to communicate using qubits. While teleportation transfers one qubit from Alice and Bob by communicating two classical bits under the assumption that Alice and Bob have a pre-shared Bell state, dense coding transfers two
2810:
Like the BB84, Alice transmits to Bob a string of photons encoded with randomly chosen bits but this time the bits Alice chooses the bases she must use. Bob still randomly chooses a basis by which to measure but if he chooses the wrong basis, he will not measure anything which is guaranteed by
120:
cannot be precisely measured simultaneously, as an eigenstate in one basis is not an eigenstate in the other basis. According to the eigenstate–eigenvalue link, an observable is well-defined (definite) when the state of the system is an eigenstate of the observable. Since any two non-commuting
1699:
Shannon entropy is the quantification of the information gained by measuring the value of a random variable. Another way of thinking about it is by looking at the uncertainty of a system prior to measurement. As a result, entropy, as pictured by
Shannon, can be seen either as a measure of the
213:
using matrix mechanics. The equivalence of these methods was proven later. Their formulations described the dynamics of microscopic systems but had several unsatisfactory aspects in describing measurement processes. Von
Neumann formulated quantum theory using operator algebra in a way that it
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was made by Artur Ekert in 1991. His scheme uses entangled pairs of photons. These two photons can be created by Alice, Bob, or by a third party including eavesdropper Eve. One of the photons is distributed to Alice and the other to Bob so that each one ends up with one photon from the pair.
199:, or electrons spiraling into the nucleus. At first these problems were brushed aside by adding ad hoc hypotheses to classical physics. Soon, it became apparent that a new theory must be created in order to make sense of these absurdities, and the theory of quantum mechanics was born.
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One of the best known applications of quantum cryptography is quantum key distribution which provide a theoretical solution to the security issue of a classical key. The advantage of quantum key distribution is that it is impossible to copy a quantum key because of the
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This definition of entropy can be used to quantify the physical resources required to store the output of an information source. The ways of interpreting
Shannon entropy discussed above are usually only meaningful when the number of samples of an experiment is large.
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1104:
104:. Its main focus is in extracting information from matter at the microscopic scale. Observation in science is one of the most important ways of acquiring information and measurement is required in order to quantify the observation, making this crucial to the
1524:
Classical bits can be encoded into and subsequently retrieved from configurations of qubits, through the use of quantum gates. By itself, a single qubit can convey no more than one bit of accessible classical information about its preparation. This is
1611:(QKD) allows unconditionally secure transmission of classical information, unlike classical encryption, which can always be broken in principle, if not in practice. Do note that certain subtle points regarding the safety of QKD are still hotly debated.
2904:. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum information, but also with faulty quantum gates, faulty quantum preparation, and faulty measurements.
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is the study of how microscopic physical systems change dynamically in nature. In the field of quantum information theory, the quantum systems studied are abstracted away from any real world counterpart. A qubit might for instance physically be a
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and Gordon proposed a formulation of optical communications using quantum mechanics. This was the first historical appearance of quantum information theory. They mainly studied error probabilities and channel capacities for communication. Later,
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One interpretation of
Shannon entropy was the uncertainty associated with a probability distribution. When we want to describe the information or the uncertainty of a quantum state, the probability distributions are simply replaced by
2325:
1152:. If it were possible to clone an unknown quantum state, it would be possible to use entangled quantum states to transmit information faster than the speed of light, disproving Einstein's theory. However, the
2041:
2764:, was developed by Charles Bennett and Gilles Brassard in 1984. It is usually explained as a method of securely communicating a private key from a third party to another for use in one-time pad encryption.
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Due to the volatility of quantum systems and the impossibility of copying states, the storing of quantum information is much more difficult than storing classical information. Nevertheless, with the use of
41:
Optical lattices use lasers to separate rubidium atoms (red) for use as information bits in neutral-atom quantum processors—prototype devices which designers are trying to develop into full-fledged quantum
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is one of the applications of quantum physics and quantum information. There are some famous theorems such as the no-cloning theorem that illustrate some important properties in quantum communication.
1643:. Regardless of the physical implementation, the limits and features of qubits implied by quantum information theory hold as all these systems are mathematically described by the same apparatus of
1219:
Soon enough, the first computers were made, and computer hardware grew at such a fast pace that the growth, through experience in production, was codified into an empirical relationship called
1278:. A theory of error-correction also developed, which allows quantum computers to make efficient computations regardless of noise and make reliable communication over noisy quantum channels.
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described measurement as well as dynamics. These studies emphasized the philosophical aspects of measurement rather than a quantitative approach to extracting information via measurements.
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problem as it was called, could be solved efficiently on a quantum computer but not on a classical computer hence showing that quantum computers are more powerful than Turing machines.
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Quantum information theory is largely an extension of classical information theory to quantum systems. Classical information is produced when measurements of quantum systems are made.
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Bennett and
Brassard developed a communication channel on which it is impossible to eavesdrop without being detected, a way of communicating secretly at long distances using the
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is the technical term for the statement that quantum information within the universe is conserved. The five theorems open up possibilities in quantum information processing.
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that can factor numbers in polynomial time, compared to the best classical algorithms that take sub-exponential time. As factorization is an important part of the safety of
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Entropy measures the uncertainty in the state of a physical system. Entropy can be studied from the point of view of both the classical and quantum information theories.
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At this point, quantum computers showed promise of being much faster than classical computers for certain specific problems. One such example problem was developed by
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called quantum decoherence. As a result of this process, quantum behavior is apparently lost, just as energy appears to be lost by friction in classical mechanics.
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1267:. Shannon developed two fundamental theorems of information theory: noiseless channel coding theorem and noisy channel coding theorem. He also showed that
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Quantum information theory also followed a similar trajectory, Ben
Schumacher in 1995 made an analogue to Shannon's noiseless coding theorem using the
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is a generalization of
Shannon entropy defined above. The RĂ©nyi entropy of order r, written as a function of a discrete probability distribution,
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quantum information can still be reliably stored in principle. The existence of quantum error correcting codes has also led to the possibility of
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observables are not simultaneously well-defined, a quantum state can never contain definitive information about both non-commuting observables.
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that tries to find encryption schemes that remain safe even when quantum computers are in play. Other examples of algorithms that demonstrate
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1212:, he showed that any real-world computation can be translated into an equivalent computation involving a Turing machine. This is known as the
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classical bits from Alice to Bob by using one qubit, again under the same assumption, that Alice and Bob have a pre-shared Bell state.
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2752:. If someone tries to read encoded data, the quantum state being transmitted will change. This could be used to detect eavesdropping.
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1447:, which prevents an arbitrary qubit from being delivered to multiple recipients, although it can be transported from place to place (
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1679:. Classical information, in principle, can be stored in a bit of binary strings. Any system having two states is a capable bit.
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cryptography is the problem of doing communication or computation involving two or more parties who may not trust one another.
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Von
Neumann entropy plays a role in quantum information similar to the role Shannon entropy plays in classical information.
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of the 1 and 0 states. However, when qubits are measured the result of the measurement is always either a 0 or a 1; the
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quantum mechanics theories. Bob can simply tell Alice after each bit she sends whether or not he measured it correctly.
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2649:{\displaystyle S(\rho )\equiv -\mathrm {tr} (\rho \ \log _{2}\ \rho )=-\sum _{i}\lambda _{i}\ \log _{2}\ \lambda _{i},}
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Unlike classical digital states (which are discrete), a qubit is continuous-valued, describable by a direction on the
1400:
1099:{\displaystyle \rho _{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }\rho _{\rm {S}}(t)e^{-iH_{0,\mathrm {S} }~t/\hbar }}
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can be used to perform computations faster than in any known classical algorithm. The most famous example of this is
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showed that such cloning is impossible. The theorem was one of the earliest results of quantum information theory.
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obtained an upper bound of communication speed in the transmission of a classical message via a quantum channel.
5135:
4749:
1659:, quantum information theory is concerned with both continuous-variable systems and finite-dimensional systems.
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Quantum information, and changes in quantum information, can be quantitatively measured by using an analogue of
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has become an active research area because of the possibility to disrupt modern computation, communication, and
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Quantum
Information, Computation and Cryptography: An Introductory Survey of Theory, Technology and Experiments
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uncertainty prior to making a measurement or as a measure of information gained after making said measurement.
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are described as rotations on the Bloch Sphere. While classical gates correspond to the familiar operations of
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Around the time computer science was making a revolution, so was information theory and communication, through
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The state of a qubit contains all of its information. This state is frequently expressed as a vector on the
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31:
801:{\displaystyle A_{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }A_{\rm {S}}e^{-iH_{0,\mathrm {S} }~t/\hbar }}
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Any attempt at eavesdropping by Eve destroys this quantum entanglement such that Alice and Bob can detect.
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Studies in
History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics
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946:{\displaystyle \rho _{\rm {S}}(t)=e^{-iH_{\rm {S}}~t/\hbar }\rho _{\rm {S}}(0)e^{iH_{\rm {S}}~t/\hbar }}
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the value precisely. Five famous theorems describe the limits on manipulation of quantum information.
521:{\displaystyle |\psi _{\rm {I}}(t)\rangle =e^{iH_{0,\mathrm {S} }~t/\hbar }|\psi _{\rm {S}}(t)\rangle }
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1651:. Another important difference with quantum mechanics is that while quantum mechanics often studies
1290:, in many striking and unfamiliar ways. While the fundamental unit of classical information is the
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possible unit of quantum information, and despite the qubit state being continuous-valued, it is
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397:{\displaystyle |\psi _{\rm {S}}(t)\rangle =e^{-iH_{\rm {S}}~t/\hbar }|\psi _{\rm {S}}(0)\rangle }
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In the 1980s, interest arose in whether it might be possible to use quantum effects to disprove
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Georgiev, Danko D. (2020). "Quantum information theoretic approach to the mind-brain problem".
1847:, can be seen as the average information associated with this set of events, in units of bits:
159:. Just like the basic unit of classical information is the bit, quantum information deals with
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Hayashi, Masahito; Ishizaka, Satoshi; Kawachi, Akinori; Kimura, Gen; Ogawa, Tomohiro (2015).
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660:{\displaystyle A_{\rm {H}}(t)=e^{iH_{\rm {S}}~t/\hbar }A_{\rm {S}}e^{-iH_{\rm {S}}~t/\hbar }}
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qubits, can convey two bits of accessible information about their joint state to a receiver.
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The study of all of the above topics and differences comprises quantum information theory.
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4118:(1985). "Quantum theory, the Church–Turing principle and the universal quantum computer".
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1548:. Quantum messages have a finite size, measured in qubits; quantum channels have a finite
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Information is something physical that is encoded in the state of a quantum system. While
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Quantum Mechanics: The Theoretical Minimum. What You Need to Know to Start Doing Physics
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3959:"Bounds for the quantity of information transmitted by a quantum communication channel"
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The history of quantum information theory began at the turn of the 20th century when
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1223:. This 'law' is a projective trend that states that the number of transistors in an
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techniques. Quantum information refers to both the technical definition in terms of
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3909:. Mathematics in Science and Engineering. Vol. 123. New York: Academic Press.
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Quantum information differs strongly from classical information, epitomized by the
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89:
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17:
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Perlman, H. S. (1964). "Equivalence of the Schroedinger and Heisenberg pictures".
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In the 1970s, techniques for manipulating single-atom quantum states, such as the
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Shannon entropy, written as a functional of a discrete probability distribution,
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Proceedings of the Royal Society of London A: Mathematical and Physical Sciences
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3290:. Theoretical and Mathematical Physics. Dordrecht: Springer. pp. 255–315.
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2918:. A quantum error correcting code protects quantum information against errors.
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Benatti, Fabio; Fannes, Mark; Floreanini, Roberto; Petritis, Dimitri (2010).
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can also be generalized to the quantum case, such as Holevo entropy and the
1138:
195:. The theories of classical physics were predicting absurdities such as the
152:
136:
manipulates and processes information – performs logical operations – using
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4045:(2014). "Quantum cryptography: public key distribution and coin tossing".
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2320:{\displaystyle H_{r}(A)={1 \over 1-r}\log _{2}\sum _{i=1}^{n}P^{r}(a_{i})}
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Classical information is based on the concepts of information laid out by
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by storing the information of one qubit onto a highly entangled state of
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Quantum information, like classical information, can be processed using
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Gilton, Marian J. R. (2016). "Whence the eigenstate–eigenvalue link?".
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among other fields. Its study is also relevant to disciplines such as
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3852:
Helstrom, Carl W. (1969). "Quantum detection and estimation theory".
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Any quantum computation algorithm can be represented as a network of
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3130:(10th anniversary ed.). Cambridge: Cambridge University Press.
128:
deals with examining properties of matter at the microscopic level,
4556:(1992). "Quantum cryptography using any two nonorthogonal states".
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3817:
Gordon, J. P. (1962). "Quantum effects in communications systems".
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2930:, although only a few are dedicated to this area. Among these are:
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1410:. Despite being continuously valued in this way, a qubit is the
1243:. This problem however held little to no practical applications.
163:. Quantum information can be measured using Von Neumann entropy.
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2036:{\displaystyle H(X)=H=-\sum _{i=1}^{n}P(x_{i})\log _{2}P(x_{i})}
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Weedbrook, Christian; Pirandola, Stefano; GarcĂa-PatrĂłn, RaĂşl;
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We arrive at the definition of Shannon entropy from RĂ©nyi when
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The Feynman Lectures on Physics. Volume III. Quantum Mechanics
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that the qubits were in immediately prior to the measurement.
2834:
2779:
This scheme relies on two properties of quantum entanglement:
1602:
1388:{\displaystyle S(\rho )=-\operatorname {Tr} (\rho \ln \rho ).}
1291:
1287:
4164:(1998). "Cramming more components onto integrated circuits".
3257:. Lecture Notes in Physics. Vol. 808. Berlin: Springer.
1461:, which demonstrates the conservation of quantum information.
132:
focuses on extracting information from those properties, and
4610:
A Survey of the Prominent Quantum Key Distribution Protocols
3405:
Quantum Information and Consciousness: A Gentle Introduction
3575:. Pasadena, California: California Institute of Technology.
3547:. Pasadena, California: California Institute of Technology.
3790:
Mathematical Foundations of Quantum Mechanics: New Edition
2825:
The most widely used model in quantum computation is the
3686:"Quantum computing: Opening new realms of possibilities"
4466:(2nd ed.). Cambridge: Cambridge University Press.
1441:, which prevents an arbitrary qubit from being deleted.
3544:
Quantum Computation (Physics 219/Computer Science 219)
1435:, which prevents an arbitrary qubit from being copied.
1251:, one of finding the prime factors of an integer. The
1208:'s revolutionary ideas of a programmable computer, or
3286:
Benatti, Fabio (2009). "Quantum Information Theory".
2691:
2664:
2530:
2508:
2456:
2426:
2396:
2367:
2335:
2216:
2162:
2070:
1855:
1801:
1709:
1639:, or it might be a large collection of atoms as in a
1339:
1319:
963:
822:
674:
545:
414:
293:
3589:
Lo, Hoi-Kwong; Popescu, Sandu; Spiller, Tim (1998).
2837:
in classical computation. Qubits can be in a 1 or 0
1294:, the most basic unit of quantum information is the
1247:
in 1994 came up with a very important and practical
6519:
6349:
6342:
6246:
6198:
6031:
5963:
5897:
5810:
5774:
5728:
5593:
5550:
5425:
5388:
5354:
5331:
5298:
5289:
5222:
5151:
5089:
5049:
4961:
4902:
4828:
4737:
4665:
3592:
Introduction to Quantum Computation and Information
3371:
Quantum Information Theory: Mathematical Foundation
4232:(1948). "A mathematical theory of communication".
4202:(1948). "A mathematical theory of communication".
3219:Philosophy of Quantum Information and Entanglement
2896:to protect quantum information from errors due to
2697:
2677:
2648:
2514:
2468:
2438:
2408:
2379:
2353:
2319:
2200:
2148:
2035:
1839:
1787:
1387:
1325:
1098:
945:
800:
659:
520:
396:
4001:"On capacity of a quantum communications channel"
1271:could be used to protect information being sent.
1190:Development from computer science and mathematics
27:Information held in the state of a quantum system
3373:. Graduate Texts in Physics. Berlin: Springer.
1395:Many of the same entropy measures in classical
151:from one location to another, manipulated with
72:It is an interdisciplinary field that involves
4414:
4412:
3126:Nielsen, Michael A.; Chuang, Isaac L. (2010).
3121:
3119:
2910:first discovered this method of formulating a
2149:{\displaystyle P(a_{1}),P(a_{2}),...,P(a_{n})}
1788:{\displaystyle P(x_{1}),P(x_{2}),...,P(x_{n})}
1133:Development from atomic physics and relativity
183:Development from fundamental quantum mechanics
6320:
5528:
4643:
4505:
4503:
4501:
3812:
3810:
3364:
3362:
3360:
3358:
3356:
3117:
3115:
3113:
3111:
3109:
3107:
3105:
3103:
3101:
3099:
1540:Quantum information can be moved about, in a
244:
8:
3455:Progress in Biophysics and Molecular Biology
2936:International Journal of Quantum Information
1581:, Shor's algorithm sparked the new field of
515:
441:
391:
320:
4259:Shannon, Claude E.; Weaver, Warren (1964).
3321:Introduction to Quantum Information Science
3128:Quantum Computation and Quantum Information
3064:Introduction to Quantum Information Science
2760:The first quantum key distribution scheme,
1302:, while the quantum mechanical analogue is
6591:
6346:
6327:
6313:
6305:
5535:
5521:
5513:
5295:
4899:
4650:
4636:
4628:
2799:The main difference between B92 and BB84:
1544:, analogous to the concept of a classical
1298:. Classical information is measured using
251:
237:
4471:
4425:. Cambridge: Cambridge University Press.
4376:
4058:
3881:
3707:. Princeton: Princeton University Press.
3660:
3584:
3582:
3466:
3222:. Cambridge: Cambridge University Press.
3072:10.1093/acprof:oso/9780199215706.001.0001
2690:
2669:
2663:
2637:
2621:
2608:
2598:
2570:
2549:
2529:
2507:
2455:
2425:
2395:
2366:
2334:
2308:
2295:
2285:
2274:
2261:
2239:
2221:
2215:
2192:
2167:
2161:
2137:
2103:
2081:
2069:
2024:
2005:
1992:
1976:
1965:
1943:
1909:
1887:
1854:
1831:
1806:
1800:
1776:
1742:
1720:
1708:
1338:
1318:
1086:
1073:
1066:
1055:
1035:
1034:
1020:
1007:
1000:
992:
969:
968:
962:
933:
920:
919:
911:
891:
890:
876:
863:
862:
851:
828:
827:
821:
788:
775:
768:
757:
746:
745:
731:
718:
711:
703:
680:
679:
673:
647:
634:
633:
622:
611:
610:
596:
583:
582:
574:
551:
550:
544:
499:
498:
489:
479:
466:
459:
451:
425:
424:
415:
413:
375:
374:
365:
355:
342:
341:
330:
304:
303:
294:
292:
155:, and analyzed with computer science and
4363:(2012). "Gaussian quantum information".
4265:. Urbana: University of Illinois Press.
4262:The Mathematical Theory of Communication
1485:. This state can be changed by applying
222:
5168:Continuous-variable quantum information
3907:Quantum Detection and Estimation Theory
3649:IEEE Transactions on Information Theory
3216:Bokulich, Alisa; Jaeger, Gregg (2010).
3051:
3000:Positive Operator Valued Measure (POVM)
1310:of quantum mechanical systems with the
1091:
1025:
938:
881:
793:
736:
652:
601:
484:
360:
4613:. Washington University in St. Louis.
3647:(1998). "Quantum information theory".
2833:". Qubit is somewhat analogous to the
2829:, which are based on the quantum bit "
1597:of problems efficiently solvable by a
57:. It is the basic entity of study in
2957:Quantum Information & Computation
7:
4005:Problems of Information Transmission
3963:Problems of Information Transmission
3177:Quantum Information: An Introduction
2995:Interpretations of quantum mechanics
2849:of these two outcomes depend on the
2469:{\displaystyle r\rightarrow \infty }
69:and the general computational term.
3066:. Oxford: Oxford University Press.
1259:Development from information theory
4246:10.1002/j.1538-7305.1948.tb00917.x
4216:10.1002/j.1538-7305.1948.tb01338.x
2926:Many journals publish research in
2796:B92 is a simpler version of BB84.
2553:
2550:
2463:
2348:
1533:a sender, by acting on one of two
1074:
1036:
1008:
970:
921:
892:
864:
829:
776:
747:
719:
681:
635:
612:
584:
552:
500:
467:
426:
376:
343:
305:
25:
4422:The Theory of Quantum Information
4234:The Bell System Technical Journal
4204:The Bell System Technical Journal
3402:Georgiev, Danko D. (2017-12-06).
2354:{\displaystyle 0<r<\infty }
6590:
6289:
6288:
5497:
5496:
5487:
5486:
4513:Quantum Information: An Overview
3787:Neumann, John von (2018-02-27).
3690:Discovery: Research at Princeton
3477:10.1016/j.pbiomolbio.2020.08.002
2806:BB84 needs 4 polarization states
1641:superconducting quantum computer
1552:, measured in qubits per second.
3705:Quantum Mechanics in a Nutshell
3595:. Singapore: World Scientific.
2201:{\displaystyle a_{1},...,a_{n}}
1840:{\displaystyle x_{1},...,x_{n}}
1695:Shannon's source coding theorem
1633:linear optical quantum computer
1465:These theorems are proven from
1150:Einstein's theory of relativity
61:, and can be manipulated using
6238:Relativistic quantum mechanics
4271:11858/00-001M-0000-002C-4314-2
3854:Journal of Statistical Physics
3793:. Princeton University Press.
2990:Einstein's thought experiments
2971:Quantum Science and Technology
2964:Quantum Information Processing
2585:
2557:
2540:
2534:
2460:
2439:{\displaystyle r\rightarrow 0}
2430:
2409:{\displaystyle r\rightarrow 1}
2400:
2314:
2301:
2233:
2227:
2143:
2130:
2109:
2096:
2087:
2074:
2030:
2017:
1998:
1985:
1952:
1949:
1936:
1915:
1902:
1893:
1880:
1874:
1865:
1859:
1782:
1769:
1748:
1735:
1726:
1713:
1477:Quantum information processing
1379:
1364:
1349:
1343:
1112:Development from communication
1048:
1042:
982:
976:
904:
898:
841:
835:
693:
687:
564:
558:
512:
506:
490:
438:
432:
416:
388:
382:
366:
317:
311:
295:
138:quantum information processing
63:quantum information processing
1:
6216:Quantum statistical mechanics
5993:Quantum differential calculus
5915:Delayed-choice quantum eraser
5698:Symmetry in quantum mechanics
5163:Adiabatic quantum computation
3915:10.1016/s0076-5392(08)x6017-5
3025:Quantum statistical mechanics
2985:Categorical quantum mechanics
2912:quantum error correcting code
1619:Relation to quantum mechanics
1501:, quantum gates are physical
1282:Qubits and information theory
1160:Development from cryptography
1143:scanning tunneling microscope
5214:Topological quantum computer
4047:Theoretical Computer Science
3569:(2013). "Quantum behavior".
2678:{\displaystyle \lambda _{i}}
1689:Entropy (information theory)
1671:Classical information theory
1637:trapped ion quantum computer
6018:Quantum stochastic calculus
6008:Quantum measurement problem
5930:Mach–Zehnder interferometer
5492:Quantum information science
4659:Quantum information science
4578:10.1103/PhysRevLett.68.3121
3528:10.1016/j.shpsb.2016.08.005
3296:10.1007/978-1-4020-9306-7_6
3020:Quantum information science
2928:quantum information science
1401:conditional quantum entropy
130:quantum information science
6635:
6619:Quantum information theory
6574:Thermoacoustic heat engine
4887:quantum gate teleportation
4464:Quantum Information Theory
3905:Helstrom, Carl W. (1976).
3831:10.1109/jrproc.1962.288169
3684:Garlinghouse, Tom (2020).
3369:Hayashi, Masahito (2017).
3175:Hayashi, Masahito (2006).
2882:
2867:
2818:
2740:
2491:
2480:Quantum information theory
2053:
1692:
1686:
1193:
1163:
59:quantum information theory
49:is the information of the
29:
6586:
6559:Immersive virtual reality
6284:
6078:Quantum complexity theory
6056:Quantum cellular automata
5761:Path integral formulation
5482:
5016:Quantum Fourier transform
4912:Post-quantum cryptography
4855:Entanglement distillation
4522:10.1007/978-0-387-36944-0
4395:10.1103/RevModPhys.84.621
4365:Reviews of Modern Physics
4318:. New York: Basic Books.
4178:10.1109/jproc.1998.658762
4069:10.1016/j.tcs.2014.05.025
3703:Mahan, Gerald D. (2009).
3563:Leighton, Robert Benjamin
3559:Feynman, Richard Phillips
3408:. Boca Raton: CRC Press.
3379:10.1007/978-3-662-49725-8
3337:10.1007/978-3-662-43502-1
3263:10.1007/978-3-642-11914-9
2803:B92 only needs two states
2156:, associated with events
1591:Grover's search algorithm
1583:post-quantum cryptography
228:
225:
209:using wave mechanics and
6542:Digital scent technology
6145:Quantum machine learning
6125:Quantum key distribution
6115:Quantum image processing
6105:Quantum error correction
5955:Wheeler's delayed choice
5502:Quantum mechanics topics
5197:Quantum machine learning
5173:One-way quantum computer
5026:Quantum phase estimation
4927:Quantum key distribution
4860:Monogamy of entanglement
4312:; Friedman, Art (2014).
3228:10.1017/CBO9780511676550
3136:10.1017/cbo9780511976667
2885:Quantum error correction
2879:Quantum error correction
2743:Quantum key distribution
2737:Quantum key distribution
1609:Quantum key distribution
1512:quantum error correction
1427:no-teleportation theorem
191:was revolutionized into
6061:Quantum finite automata
5109:Randomized benchmarking
4971:Amplitude amplification
4607:Haitjema, Mart (2007).
4558:Physical Review Letters
4462:Wilde, Mark M. (2017).
4166:Proceedings of the IEEE
2943:npj Quantum Information
2685:are the eigenvalues of
2380:{\displaystyle r\neq 1}
1795:associated with events
1663:Entropy and information
1495:unitary transformations
1241:Deutsch–Jozsa algorithm
197:ultraviolet catastrophe
178:History and development
166:Recently, the field of
32:npj Quantum Information
6564:Magnetic refrigeration
6165:Quantum neural network
5209:Quantum Turing machine
5202:quantum neural network
4949:Quantum secret sharing
4516:. New York: Springer.
4510:Jaeger, Gregg (2007).
4419:Watrous, John (2018).
4140:10.1098/rspa.1985.0070
4092:"Church–Turing Thesis"
3819:Proceedings of the IRE
2841:, or they can be in a
2699:
2679:
2650:
2516:
2470:
2440:
2420:(or max-entropy) when
2410:
2381:
2355:
2321:
2290:
2202:
2150:
2037:
1981:
1841:
1789:
1546:communications channel
1487:linear transformations
1389:
1327:
1269:error correcting codes
1175:. In a general sense,
1100:
947:
802:
661:
522:
398:
43:
6537:Cloak of invisibility
6336:Emerging technologies
6190:Quantum teleportation
5718:Wave–particle duality
5281:Entanglement-assisted
5242:quantum convolutional
4917:Quantum coin flipping
4882:Quantum teleportation
4843:entanglement-assisted
4673:DiVincenzo's criteria
4482:10.1017/9781316809976
4431:10.1017/9781316848142
4096:mathworld.wolfram.com
3645:Shor, Peter Williston
3567:Sands, Matthew Linzee
3414:10.1201/9780203732519
3185:10.1007/3-540-30266-2
2730:quantum teleportation
2722:Quantum communication
2717:Quantum communication
2700:
2698:{\displaystyle \rho }
2680:
2651:
2517:
2515:{\displaystyle \rho }
2471:
2441:
2411:
2382:
2356:
2322:
2270:
2203:
2151:
2038:
1961:
1842:
1790:
1469:, which according to
1453:quantum teleportation
1390:
1328:
1326:{\displaystyle \rho }
1101:
948:
803:
662:
523:
399:
114:uncertainty principle
40:
30:For the journal, see
6221:Quantum field theory
6150:Quantum metamaterial
6095:Quantum cryptography
5825:Consistent histories
5092:processor benchmarks
5021:Quantum optimization
4904:Quantum cryptography
4715:physical vs. logical
4285:"Alexandr S. Holevo"
3997:Holevo, Alexander S.
3955:Holevo, Alexander S.
3323:. Berlin: Springer.
3179:. Berlin: Springer.
3010:Quantum entanglement
2689:
2662:
2528:
2506:
2454:
2424:
2394:
2365:
2333:
2214:
2160:
2068:
1853:
1799:
1707:
1653:infinite-dimensional
1445:no-broadcast theorem
1337:
1317:
1308:statistical ensemble
1214:Church–Turing thesis
1166:Quantum cryptography
961:
820:
672:
543:
412:
291:
6569:Phased-array optics
6527:Acoustic levitation
6206:Quantum fluctuation
6175:Quantum programming
6135:Quantum logic gates
6120:Quantum information
6100:Quantum electronics
5575:Classical mechanics
4805:Quantum speed limit
4700:Quantum programming
4695:Quantum information
4570:1992PhRvL..68.3121B
4554:Bennett, Charles H.
4387:2012RvMP...84..621W
4357:Shapiro, Jeffrey H.
4162:Moore, Gordon Earle
4132:1985RSPSA.400...97D
4090:Weisstein, Eric W.
4039:Bennett, Charles H.
3866:1969JSP.....1..231H
3758:1964Natur.204..771P
3641:Bennett, Charles H.
3601:1998iqci.book.....S
3520:2016SHPMP..55...92G
3329:2015iqis.book.....H
3015:Quantum foundations
2870:Quantum decoherence
2864:Quantum decoherence
2858:quantum logic gates
2815:Quantum computation
2494:Von Neumann entropy
2488:Von Neumann entropy
1657:harmonic oscillator
1564:von Neumann entropy
1519:quantum computation
1439:no-deleting theorem
1304:Von Neumann entropy
1204:With the advent of
134:quantum computation
67:Von Neumann entropy
47:Quantum information
18:Quantum Information
6259:in popular culture
6041:Quantum algorithms
5889:Von Neumann–Wigner
5869:Objective collapse
5580:Old quantum theory
5454:Forest/Rigetti QCS
5190:quantum logic gate
4976:Bernstein–Vazirani
4963:Quantum algorithms
4838:Classical capacity
4722:Quantum processors
4705:Quantum simulation
4230:Shannon, Claude E.
4200:Shannon, Claude E.
3874:10.1007/bf01007479
3713:10.2307/j.ctt7s8nw
2750:no-cloning theorem
2695:
2675:
2646:
2603:
2512:
2466:
2436:
2406:
2377:
2351:
2317:
2198:
2146:
2033:
1837:
1785:
1655:systems such as a
1571:quantum algorithms
1433:no-cloning theorem
1397:information theory
1385:
1323:
1253:discrete logarithm
1225:integrated circuit
1186:the eavesdropper.
1154:no-cloning theorem
1096:
943:
798:
657:
518:
394:
219:Dynamical Pictures
205:was formulated by
82:information theory
44:
6606:
6605:
6582:
6581:
6389:complexity theory
6374:cellular automata
6302:
6301:
6276:Quantum mysticism
6254:Schrödinger's cat
6185:Quantum simulator
6155:Quantum metrology
6083:Quantum computing
6046:Quantum amplifier
6023:Quantum spacetime
5988:Quantum cosmology
5978:Quantum chemistry
5693:Scattering theory
5641:Zero-point energy
5636:Degenerate levels
5544:Quantum mechanics
5510:
5509:
5421:
5420:
5318:Linear optical QC
5099:Quantum supremacy
5053:complexity theory
5006:Quantum annealing
4957:
4956:
4894:Superdense coding
4683:Quantum computing
4564:(21): 3121–3124.
4531:978-0-387-36944-0
4353:Ralph, Timothy C.
4325:978-0-465-08061-8
4310:Susskind, Leonard
3800:978-0-691-17856-1
3752:(4960): 771–772.
3722:978-1-4008-3338-2
3671:10.1109/18.720553
3618:978-981-4496-35-3
3388:978-3-662-49725-8
3346:978-3-662-43502-1
3305:978-1-4020-9306-7
3288:Quantum Entropies
3272:978-3-642-11914-9
3194:978-3-540-30266-7
2894:quantum computing
2821:Quantum computing
2632:
2616:
2594:
2581:
2565:
2501:density operators
2255:
2208:, is defined as:
1624:Quantum mechanics
1587:quantum supremacy
1531:superdense coding
1503:unitary operators
1459:no-hiding theorem
1333:, it is given by
1200:Quantum algorithm
1196:Quantum supremacy
1171:and computation:
1109:
1108:
1082:
1016:
929:
872:
784:
727:
643:
592:
475:
351:
203:Quantum mechanics
189:classical physics
168:quantum computing
145:digital computers
126:quantum mechanics
110:quantum mechanics
106:scientific method
94:cognitive science
74:quantum mechanics
16:(Redirected from
6626:
6594:
6593:
6471:machine learning
6446:key distribution
6431:image processing
6421:error correction
6347:
6329:
6322:
6315:
6306:
6292:
6291:
6003:Quantum geometry
5998:Quantum dynamics
5855:Superdeterminism
5751:Matrix mechanics
5606:Bra–ket notation
5537:
5530:
5523:
5514:
5500:
5499:
5490:
5489:
5296:
5226:error correction
5155:computing models
5121:Relaxation times
5011:Quantum counting
4900:
4848:quantum capacity
4795:No-teleportation
4780:No-communication
4652:
4645:
4638:
4629:
4623:
4622:
4604:
4598:
4597:
4550:
4544:
4543:
4507:
4496:
4495:
4475:
4459:
4453:
4452:
4416:
4407:
4406:
4380:
4349:Cerf, Nicolas J.
4344:
4338:
4337:
4306:
4300:
4299:
4297:
4295:
4281:
4275:
4274:
4256:
4250:
4249:
4226:
4220:
4219:
4196:
4190:
4189:
4158:
4152:
4151:
4126:(1818): 97–117.
4112:
4106:
4105:
4103:
4102:
4087:
4081:
4080:
4062:
4043:Brassard, Gilles
4035:
4029:
4028:
3993:
3987:
3986:
3951:
3945:
3944:
3923:2060/19690016211
3902:
3896:
3895:
3885:
3883:2060/19690016211
3849:
3843:
3842:
3825:(9): 1898–1908.
3814:
3805:
3804:
3784:
3778:
3777:
3766:10.1038/204771b0
3741:
3735:
3734:
3700:
3694:
3693:
3681:
3675:
3674:
3664:
3655:(6): 2724–2742.
3637:
3631:
3630:
3586:
3577:
3576:
3555:
3549:
3548:
3541:Preskill, John.
3538:
3532:
3531:
3503:
3497:
3496:
3470:
3450:
3444:
3443:
3399:
3393:
3392:
3366:
3351:
3350:
3316:
3310:
3309:
3283:
3277:
3276:
3248:
3242:
3241:
3213:
3207:
3206:
3172:
3166:
3165:
3123:
3094:
3093:
3056:
3040:Typical subspace
2704:
2702:
2701:
2696:
2684:
2682:
2681:
2676:
2674:
2673:
2655:
2653:
2652:
2647:
2642:
2641:
2630:
2626:
2625:
2614:
2613:
2612:
2602:
2579:
2575:
2574:
2563:
2556:
2521:
2519:
2518:
2513:
2475:
2473:
2472:
2467:
2445:
2443:
2442:
2437:
2415:
2413:
2412:
2407:
2386:
2384:
2383:
2378:
2360:
2358:
2357:
2352:
2326:
2324:
2323:
2318:
2313:
2312:
2300:
2299:
2289:
2284:
2266:
2265:
2256:
2254:
2240:
2226:
2225:
2207:
2205:
2204:
2199:
2197:
2196:
2172:
2171:
2155:
2153:
2152:
2147:
2142:
2141:
2108:
2107:
2086:
2085:
2042:
2040:
2039:
2034:
2029:
2028:
2010:
2009:
1997:
1996:
1980:
1975:
1948:
1947:
1914:
1913:
1892:
1891:
1846:
1844:
1843:
1838:
1836:
1835:
1811:
1810:
1794:
1792:
1791:
1786:
1781:
1780:
1747:
1746:
1725:
1724:
1645:density matrices
1599:quantum computer
1595:complexity class
1575:Shor's algorithm
1550:channel capacity
1527:Holevo's theorem
1471:Leonard Susskind
1394:
1392:
1391:
1386:
1332:
1330:
1329:
1324:
1127:Alexander Holevo
1105:
1103:
1102:
1097:
1095:
1094:
1090:
1080:
1079:
1078:
1077:
1041:
1040:
1039:
1029:
1028:
1024:
1014:
1013:
1012:
1011:
975:
974:
973:
952:
950:
949:
944:
942:
941:
937:
927:
926:
925:
924:
897:
896:
895:
885:
884:
880:
870:
869:
868:
867:
834:
833:
832:
807:
805:
804:
799:
797:
796:
792:
782:
781:
780:
779:
752:
751:
750:
740:
739:
735:
725:
724:
723:
722:
686:
685:
684:
666:
664:
663:
658:
656:
655:
651:
641:
640:
639:
638:
617:
616:
615:
605:
604:
600:
590:
589:
588:
587:
557:
556:
555:
527:
525:
524:
519:
505:
504:
503:
493:
488:
487:
483:
473:
472:
471:
470:
431:
430:
429:
419:
403:
401:
400:
395:
381:
380:
379:
369:
364:
363:
359:
349:
348:
347:
346:
310:
309:
308:
298:
253:
246:
239:
223:
116:, non-commuting
78:computer science
21:
6634:
6633:
6629:
6628:
6627:
6625:
6624:
6623:
6609:
6608:
6607:
6602:
6578:
6515:
6426:finite automata
6338:
6333:
6303:
6298:
6280:
6266:Wigner's friend
6242:
6233:Quantum gravity
6194:
6180:Quantum sensing
6160:Quantum network
6140:Quantum machine
6110:Quantum imaging
6073:Quantum circuit
6068:Quantum channel
6027:
5973:Quantum biology
5959:
5935:Elitzur–Vaidman
5910:Davisson–Germer
5893:
5845:Hidden-variable
5835:de Broglie–Bohm
5812:Interpretations
5806:
5770:
5724:
5611:Complementarity
5589:
5546:
5541:
5511:
5506:
5478:
5428:
5417:
5390:Superconducting
5384:
5350:
5341:Neutral atom QC
5333:Ultracold atoms
5327:
5292:implementations
5291:
5285:
5225:
5218:
5185:Quantum circuit
5153:
5147:
5141:
5131:
5091:
5085:
5052:
5045:
5001:Hidden subgroup
4953:
4942:other protocols
4898:
4875:quantum network
4870:Quantum channel
4830:
4824:
4770:No-broadcasting
4760:Gottesman–Knill
4733:
4661:
4656:
4626:
4606:
4605:
4601:
4552:
4551:
4547:
4532:
4509:
4508:
4499:
4492:
4461:
4460:
4456:
4441:
4418:
4417:
4410:
4346:
4345:
4341:
4326:
4308:
4307:
4303:
4293:
4291:
4283:
4282:
4278:
4258:
4257:
4253:
4228:
4227:
4223:
4198:
4197:
4193:
4160:
4159:
4155:
4114:
4113:
4109:
4100:
4098:
4089:
4088:
4084:
4037:
4036:
4032:
3995:
3994:
3990:
3953:
3952:
3948:
3933:
3904:
3903:
3899:
3851:
3850:
3846:
3816:
3815:
3808:
3801:
3786:
3785:
3781:
3743:
3742:
3738:
3723:
3702:
3701:
3697:
3683:
3682:
3678:
3639:
3638:
3634:
3619:
3588:
3587:
3580:
3557:
3556:
3552:
3540:
3539:
3535:
3505:
3504:
3500:
3452:
3451:
3447:
3424:
3401:
3400:
3396:
3389:
3368:
3367:
3354:
3347:
3318:
3317:
3313:
3306:
3285:
3284:
3280:
3273:
3250:
3249:
3245:
3238:
3215:
3214:
3210:
3195:
3174:
3173:
3169:
3146:
3125:
3124:
3097:
3082:
3058:
3057:
3053:
3049:
3044:
2980:
2924:
2887:
2881:
2872:
2866:
2827:quantum circuit
2823:
2817:
2794:
2770:
2758:
2745:
2739:
2719:
2714:
2687:
2686:
2665:
2660:
2659:
2633:
2617:
2604:
2566:
2526:
2525:
2504:
2503:
2496:
2490:
2482:
2452:
2451:
2422:
2421:
2418:Hartley entropy
2392:
2391:
2363:
2362:
2331:
2330:
2304:
2291:
2257:
2244:
2217:
2212:
2211:
2188:
2163:
2158:
2157:
2133:
2099:
2077:
2066:
2065:
2058:
2052:
2020:
2001:
1988:
1939:
1905:
1883:
1851:
1850:
1827:
1802:
1797:
1796:
1772:
1738:
1716:
1705:
1704:
1697:
1691:
1685:
1683:Shannon entropy
1673:
1665:
1649:complex numbers
1621:
1569:In some cases,
1542:quantum channel
1529:. However, in
1493:to them. These
1479:
1335:
1334:
1315:
1314:
1300:Shannon entropy
1284:
1261:
1239:, known as the
1202:
1192:
1168:
1162:
1135:
1114:
1062:
1051:
1030:
996:
988:
964:
959:
958:
915:
907:
886:
858:
847:
823:
818:
817:
764:
753:
741:
707:
699:
675:
670:
669:
629:
618:
606:
578:
570:
546:
541:
540:
494:
455:
447:
420:
410:
409:
370:
337:
326:
299:
289:
288:
257:
193:quantum physics
185:
180:
35:
28:
23:
22:
15:
12:
11:
5:
6632:
6630:
6622:
6621:
6611:
6610:
6604:
6603:
6601:
6600:
6587:
6584:
6583:
6580:
6579:
6577:
6576:
6571:
6566:
6561:
6556:
6555:
6554:
6544:
6539:
6534:
6529:
6523:
6521:
6517:
6516:
6514:
6513:
6508:
6503:
6498:
6493:
6488:
6486:neural network
6483:
6478:
6473:
6468:
6463:
6458:
6453:
6448:
6443:
6438:
6433:
6428:
6423:
6418:
6413:
6408:
6407:
6406:
6396:
6391:
6386:
6381:
6376:
6371:
6366:
6361:
6355:
6353:
6344:
6340:
6339:
6334:
6332:
6331:
6324:
6317:
6309:
6300:
6299:
6297:
6296:
6285:
6282:
6281:
6279:
6278:
6273:
6268:
6263:
6262:
6261:
6250:
6248:
6244:
6243:
6241:
6240:
6235:
6230:
6229:
6228:
6218:
6213:
6211:Casimir effect
6208:
6202:
6200:
6196:
6195:
6193:
6192:
6187:
6182:
6177:
6172:
6170:Quantum optics
6167:
6162:
6157:
6152:
6147:
6142:
6137:
6132:
6127:
6122:
6117:
6112:
6107:
6102:
6097:
6092:
6091:
6090:
6080:
6075:
6070:
6065:
6064:
6063:
6053:
6048:
6043:
6037:
6035:
6029:
6028:
6026:
6025:
6020:
6015:
6010:
6005:
6000:
5995:
5990:
5985:
5980:
5975:
5969:
5967:
5961:
5960:
5958:
5957:
5952:
5947:
5945:Quantum eraser
5942:
5937:
5932:
5927:
5922:
5917:
5912:
5907:
5901:
5899:
5895:
5894:
5892:
5891:
5886:
5881:
5876:
5871:
5866:
5861:
5860:
5859:
5858:
5857:
5842:
5837:
5832:
5827:
5822:
5816:
5814:
5808:
5807:
5805:
5804:
5799:
5794:
5789:
5784:
5778:
5776:
5772:
5771:
5769:
5768:
5763:
5758:
5753:
5748:
5743:
5738:
5732:
5730:
5726:
5725:
5723:
5722:
5721:
5720:
5715:
5705:
5700:
5695:
5690:
5685:
5680:
5675:
5670:
5665:
5660:
5655:
5650:
5645:
5644:
5643:
5638:
5633:
5628:
5618:
5616:Density matrix
5613:
5608:
5603:
5597:
5595:
5591:
5590:
5588:
5587:
5582:
5577:
5572:
5571:
5570:
5560:
5554:
5552:
5548:
5547:
5542:
5540:
5539:
5532:
5525:
5517:
5508:
5507:
5505:
5504:
5494:
5483:
5480:
5479:
5477:
5476:
5474:many others...
5471:
5466:
5461:
5456:
5447:
5433:
5431:
5423:
5422:
5419:
5418:
5416:
5415:
5410:
5405:
5400:
5394:
5392:
5386:
5385:
5383:
5382:
5377:
5372:
5367:
5361:
5359:
5352:
5351:
5349:
5348:
5346:Trapped-ion QC
5343:
5337:
5335:
5329:
5328:
5326:
5325:
5320:
5315:
5310:
5304:
5302:
5300:Quantum optics
5293:
5287:
5286:
5284:
5283:
5278:
5277:
5276:
5269:
5264:
5259:
5254:
5249:
5244:
5239:
5230:
5228:
5220:
5219:
5217:
5216:
5211:
5206:
5205:
5204:
5194:
5193:
5192:
5182:
5181:
5180:
5170:
5165:
5159:
5157:
5149:
5148:
5146:
5145:
5144:
5143:
5139:
5133:
5129:
5118:
5117:
5116:
5106:
5104:Quantum volume
5101:
5095:
5093:
5087:
5086:
5084:
5083:
5078:
5073:
5068:
5063:
5057:
5055:
5047:
5046:
5044:
5043:
5038:
5033:
5028:
5023:
5018:
5013:
5008:
5003:
4998:
4993:
4988:
4983:
4981:Boson sampling
4978:
4973:
4967:
4965:
4959:
4958:
4955:
4954:
4952:
4951:
4946:
4945:
4944:
4939:
4934:
4924:
4919:
4914:
4908:
4906:
4897:
4896:
4891:
4890:
4889:
4879:
4878:
4877:
4867:
4862:
4857:
4852:
4851:
4850:
4845:
4834:
4832:
4826:
4825:
4823:
4822:
4817:
4815:Solovay–Kitaev
4812:
4807:
4802:
4797:
4792:
4787:
4782:
4777:
4772:
4767:
4762:
4757:
4752:
4747:
4741:
4739:
4735:
4734:
4732:
4731:
4730:
4729:
4719:
4718:
4717:
4707:
4702:
4697:
4692:
4691:
4690:
4680:
4675:
4669:
4667:
4663:
4662:
4657:
4655:
4654:
4647:
4640:
4632:
4625:
4624:
4599:
4545:
4530:
4497:
4490:
4454:
4439:
4408:
4371:(2): 621–669.
4339:
4324:
4301:
4276:
4251:
4240:(4): 623–656.
4221:
4210:(3): 379–423.
4191:
4153:
4116:Deutsch, David
4107:
4082:
4030:
4011:(4): 247–253.
3988:
3969:(3): 177–183.
3946:
3931:
3897:
3860:(2): 231–252.
3844:
3806:
3799:
3779:
3736:
3721:
3695:
3676:
3662:10.1.1.89.1572
3632:
3617:
3578:
3550:
3533:
3498:
3445:
3422:
3394:
3387:
3352:
3345:
3311:
3304:
3278:
3271:
3243:
3236:
3208:
3193:
3167:
3144:
3095:
3080:
3060:Vedral, Vlatko
3050:
3048:
3045:
3043:
3042:
3037:
3032:
3027:
3022:
3017:
3012:
3007:
3002:
2997:
2992:
2987:
2981:
2979:
2976:
2975:
2974:
2967:
2960:
2953:
2946:
2939:
2923:
2920:
2916:ancilla qubits
2883:Main article:
2880:
2877:
2868:Main article:
2865:
2862:
2819:Main article:
2816:
2813:
2808:
2807:
2804:
2793:
2790:
2789:
2788:
2785:
2769:
2766:
2757:
2754:
2741:Main article:
2738:
2735:
2718:
2715:
2713:
2710:
2694:
2672:
2668:
2645:
2640:
2636:
2629:
2624:
2620:
2611:
2607:
2601:
2597:
2593:
2590:
2587:
2584:
2578:
2573:
2569:
2562:
2559:
2555:
2552:
2548:
2545:
2542:
2539:
2536:
2533:
2511:
2492:Main article:
2489:
2486:
2481:
2478:
2465:
2462:
2459:
2435:
2432:
2429:
2405:
2402:
2399:
2376:
2373:
2370:
2350:
2347:
2344:
2341:
2338:
2316:
2311:
2307:
2303:
2298:
2294:
2288:
2283:
2280:
2277:
2273:
2269:
2264:
2260:
2253:
2250:
2247:
2243:
2238:
2235:
2232:
2229:
2224:
2220:
2195:
2191:
2187:
2184:
2181:
2178:
2175:
2170:
2166:
2145:
2140:
2136:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2106:
2102:
2098:
2095:
2092:
2089:
2084:
2080:
2076:
2073:
2054:Main article:
2051:
2048:
2032:
2027:
2023:
2019:
2016:
2013:
2008:
2004:
2000:
1995:
1991:
1987:
1984:
1979:
1974:
1971:
1968:
1964:
1960:
1957:
1954:
1951:
1946:
1942:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1912:
1908:
1904:
1901:
1898:
1895:
1890:
1886:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1834:
1830:
1826:
1823:
1820:
1817:
1814:
1809:
1805:
1784:
1779:
1775:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1745:
1741:
1737:
1734:
1731:
1728:
1723:
1719:
1715:
1712:
1687:Main article:
1684:
1681:
1677:Claude Shannon
1672:
1669:
1664:
1661:
1635:, an ion in a
1620:
1617:
1613:
1612:
1606:
1579:RSA encryption
1567:
1553:
1538:
1522:
1516:fault-tolerant
1478:
1475:
1463:
1462:
1456:
1442:
1436:
1430:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1322:
1312:density matrix
1283:
1280:
1265:Claude Shannon
1260:
1257:
1210:Turing machine
1191:
1188:
1161:
1158:
1134:
1131:
1113:
1110:
1107:
1106:
1093:
1089:
1085:
1076:
1072:
1069:
1065:
1061:
1058:
1054:
1050:
1047:
1044:
1038:
1033:
1027:
1023:
1019:
1010:
1006:
1003:
999:
995:
991:
987:
984:
981:
978:
972:
967:
956:
953:
940:
936:
932:
923:
918:
914:
910:
906:
903:
900:
894:
889:
883:
879:
875:
866:
861:
857:
854:
850:
846:
843:
840:
837:
831:
826:
815:
813:Density matrix
809:
808:
795:
791:
787:
778:
774:
771:
767:
763:
760:
756:
749:
744:
738:
734:
730:
721:
717:
714:
710:
706:
702:
698:
695:
692:
689:
683:
678:
667:
654:
650:
646:
637:
632:
628:
625:
621:
614:
609:
603:
599:
595:
586:
581:
577:
573:
569:
566:
563:
560:
554:
549:
538:
535:
529:
528:
517:
514:
511:
508:
502:
497:
492:
486:
482:
478:
469:
465:
462:
458:
454:
450:
446:
443:
440:
437:
434:
428:
423:
418:
407:
404:
393:
390:
387:
384:
378:
373:
368:
362:
358:
354:
345:
340:
336:
333:
329:
325:
322:
319:
316:
313:
307:
302:
297:
286:
280:
279:
273:
267:
260:
259:
256:
255:
248:
241:
233:
227:
226:Evolution of:
184:
181:
179:
176:
55:quantum system
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6631:
6620:
6617:
6616:
6614:
6599:
6598:
6589:
6588:
6585:
6575:
6572:
6570:
6567:
6565:
6562:
6560:
6557:
6553:
6552:Plasma window
6550:
6549:
6548:
6545:
6543:
6540:
6538:
6535:
6533:
6530:
6528:
6525:
6524:
6522:
6518:
6512:
6511:teleportation
6509:
6507:
6504:
6502:
6499:
6497:
6494:
6492:
6489:
6487:
6484:
6482:
6479:
6477:
6474:
6472:
6469:
6467:
6464:
6462:
6459:
6457:
6454:
6452:
6449:
6447:
6444:
6442:
6439:
6437:
6434:
6432:
6429:
6427:
6424:
6422:
6419:
6417:
6414:
6412:
6409:
6405:
6402:
6401:
6400:
6397:
6395:
6392:
6390:
6387:
6385:
6382:
6380:
6377:
6375:
6372:
6370:
6367:
6365:
6362:
6360:
6357:
6356:
6354:
6352:
6348:
6345:
6341:
6337:
6330:
6325:
6323:
6318:
6316:
6311:
6310:
6307:
6295:
6287:
6286:
6283:
6277:
6274:
6272:
6269:
6267:
6264:
6260:
6257:
6256:
6255:
6252:
6251:
6249:
6245:
6239:
6236:
6234:
6231:
6227:
6224:
6223:
6222:
6219:
6217:
6214:
6212:
6209:
6207:
6204:
6203:
6201:
6197:
6191:
6188:
6186:
6183:
6181:
6178:
6176:
6173:
6171:
6168:
6166:
6163:
6161:
6158:
6156:
6153:
6151:
6148:
6146:
6143:
6141:
6138:
6136:
6133:
6131:
6130:Quantum logic
6128:
6126:
6123:
6121:
6118:
6116:
6113:
6111:
6108:
6106:
6103:
6101:
6098:
6096:
6093:
6089:
6086:
6085:
6084:
6081:
6079:
6076:
6074:
6071:
6069:
6066:
6062:
6059:
6058:
6057:
6054:
6052:
6049:
6047:
6044:
6042:
6039:
6038:
6036:
6034:
6030:
6024:
6021:
6019:
6016:
6014:
6011:
6009:
6006:
6004:
6001:
5999:
5996:
5994:
5991:
5989:
5986:
5984:
5983:Quantum chaos
5981:
5979:
5976:
5974:
5971:
5970:
5968:
5966:
5962:
5956:
5953:
5951:
5950:Stern–Gerlach
5948:
5946:
5943:
5941:
5938:
5936:
5933:
5931:
5928:
5926:
5923:
5921:
5918:
5916:
5913:
5911:
5908:
5906:
5903:
5902:
5900:
5896:
5890:
5887:
5885:
5884:Transactional
5882:
5880:
5877:
5875:
5874:Quantum logic
5872:
5870:
5867:
5865:
5862:
5856:
5853:
5852:
5851:
5848:
5847:
5846:
5843:
5841:
5838:
5836:
5833:
5831:
5828:
5826:
5823:
5821:
5818:
5817:
5815:
5813:
5809:
5803:
5800:
5798:
5795:
5793:
5790:
5788:
5785:
5783:
5780:
5779:
5777:
5773:
5767:
5764:
5762:
5759:
5757:
5754:
5752:
5749:
5747:
5744:
5742:
5739:
5737:
5734:
5733:
5731:
5727:
5719:
5716:
5714:
5711:
5710:
5709:
5708:Wave function
5706:
5704:
5701:
5699:
5696:
5694:
5691:
5689:
5686:
5684:
5683:Superposition
5681:
5679:
5678:Quantum state
5676:
5674:
5671:
5669:
5666:
5664:
5661:
5659:
5656:
5654:
5651:
5649:
5646:
5642:
5639:
5637:
5634:
5632:
5631:Excited state
5629:
5627:
5624:
5623:
5622:
5619:
5617:
5614:
5612:
5609:
5607:
5604:
5602:
5599:
5598:
5596:
5592:
5586:
5583:
5581:
5578:
5576:
5573:
5569:
5566:
5565:
5564:
5561:
5559:
5556:
5555:
5553:
5549:
5545:
5538:
5533:
5531:
5526:
5524:
5519:
5518:
5515:
5503:
5495:
5493:
5485:
5484:
5481:
5475:
5472:
5470:
5467:
5465:
5462:
5460:
5457:
5455:
5451:
5448:
5446:
5442:
5438:
5435:
5434:
5432:
5430:
5424:
5414:
5411:
5409:
5406:
5404:
5401:
5399:
5396:
5395:
5393:
5391:
5387:
5381:
5378:
5376:
5373:
5371:
5370:Spin qubit QC
5368:
5366:
5363:
5362:
5360:
5357:
5353:
5347:
5344:
5342:
5339:
5338:
5336:
5334:
5330:
5324:
5321:
5319:
5316:
5314:
5311:
5309:
5306:
5305:
5303:
5301:
5297:
5294:
5288:
5282:
5279:
5275:
5274:
5270:
5268:
5265:
5263:
5260:
5258:
5255:
5253:
5250:
5248:
5245:
5243:
5240:
5238:
5235:
5234:
5232:
5231:
5229:
5227:
5221:
5215:
5212:
5210:
5207:
5203:
5200:
5199:
5198:
5195:
5191:
5188:
5187:
5186:
5183:
5179:
5178:cluster state
5176:
5175:
5174:
5171:
5169:
5166:
5164:
5161:
5160:
5158:
5156:
5150:
5142:
5138:
5134:
5132:
5128:
5124:
5123:
5122:
5119:
5115:
5112:
5111:
5110:
5107:
5105:
5102:
5100:
5097:
5096:
5094:
5088:
5082:
5079:
5077:
5074:
5072:
5069:
5067:
5064:
5062:
5059:
5058:
5056:
5054:
5048:
5042:
5039:
5037:
5034:
5032:
5029:
5027:
5024:
5022:
5019:
5017:
5014:
5012:
5009:
5007:
5004:
5002:
4999:
4997:
4994:
4992:
4989:
4987:
4986:Deutsch–Jozsa
4984:
4982:
4979:
4977:
4974:
4972:
4969:
4968:
4966:
4964:
4960:
4950:
4947:
4943:
4940:
4938:
4935:
4933:
4930:
4929:
4928:
4925:
4923:
4922:Quantum money
4920:
4918:
4915:
4913:
4910:
4909:
4907:
4905:
4901:
4895:
4892:
4888:
4885:
4884:
4883:
4880:
4876:
4873:
4872:
4871:
4868:
4866:
4863:
4861:
4858:
4856:
4853:
4849:
4846:
4844:
4841:
4840:
4839:
4836:
4835:
4833:
4831:communication
4827:
4821:
4818:
4816:
4813:
4811:
4808:
4806:
4803:
4801:
4798:
4796:
4793:
4791:
4788:
4786:
4783:
4781:
4778:
4776:
4773:
4771:
4768:
4766:
4763:
4761:
4758:
4756:
4753:
4751:
4748:
4746:
4743:
4742:
4740:
4736:
4728:
4725:
4724:
4723:
4720:
4716:
4713:
4712:
4711:
4708:
4706:
4703:
4701:
4698:
4696:
4693:
4689:
4686:
4685:
4684:
4681:
4679:
4676:
4674:
4671:
4670:
4668:
4664:
4660:
4653:
4648:
4646:
4641:
4639:
4634:
4633:
4630:
4620:
4616:
4612:
4611:
4603:
4600:
4595:
4591:
4587:
4583:
4579:
4575:
4571:
4567:
4563:
4559:
4555:
4549:
4546:
4541:
4537:
4533:
4527:
4523:
4519:
4515:
4514:
4506:
4504:
4502:
4498:
4493:
4491:9781316809976
4487:
4483:
4479:
4474:
4469:
4465:
4458:
4455:
4450:
4446:
4442:
4440:9781316848142
4436:
4432:
4428:
4424:
4423:
4415:
4413:
4409:
4404:
4400:
4396:
4392:
4388:
4384:
4379:
4374:
4370:
4366:
4362:
4358:
4354:
4350:
4343:
4340:
4335:
4331:
4327:
4321:
4317:
4316:
4311:
4305:
4302:
4290:
4286:
4280:
4277:
4272:
4268:
4264:
4263:
4255:
4252:
4247:
4243:
4239:
4235:
4231:
4225:
4222:
4217:
4213:
4209:
4205:
4201:
4195:
4192:
4187:
4183:
4179:
4175:
4171:
4167:
4163:
4157:
4154:
4149:
4145:
4141:
4137:
4133:
4129:
4125:
4121:
4117:
4111:
4108:
4097:
4093:
4086:
4083:
4078:
4074:
4070:
4066:
4061:
4056:
4052:
4048:
4044:
4040:
4034:
4031:
4026:
4022:
4018:
4014:
4010:
4006:
4002:
3998:
3992:
3989:
3984:
3980:
3976:
3972:
3968:
3964:
3960:
3956:
3950:
3947:
3942:
3938:
3934:
3932:9780080956329
3928:
3924:
3920:
3916:
3912:
3908:
3901:
3898:
3893:
3889:
3884:
3879:
3875:
3871:
3867:
3863:
3859:
3855:
3848:
3845:
3840:
3836:
3832:
3828:
3824:
3820:
3813:
3811:
3807:
3802:
3796:
3792:
3791:
3783:
3780:
3775:
3771:
3767:
3763:
3759:
3755:
3751:
3747:
3740:
3737:
3732:
3728:
3724:
3718:
3714:
3710:
3706:
3699:
3696:
3691:
3687:
3680:
3677:
3672:
3668:
3663:
3658:
3654:
3650:
3646:
3642:
3636:
3633:
3628:
3624:
3620:
3614:
3610:
3606:
3602:
3598:
3594:
3593:
3585:
3583:
3579:
3574:
3573:
3568:
3564:
3560:
3554:
3551:
3546:
3545:
3537:
3534:
3529:
3525:
3521:
3517:
3513:
3509:
3502:
3499:
3494:
3490:
3486:
3482:
3478:
3474:
3469:
3464:
3460:
3456:
3449:
3446:
3441:
3437:
3433:
3429:
3425:
3423:9781138104488
3419:
3415:
3411:
3407:
3406:
3398:
3395:
3390:
3384:
3380:
3376:
3372:
3365:
3363:
3361:
3359:
3357:
3353:
3348:
3342:
3338:
3334:
3330:
3326:
3322:
3315:
3312:
3307:
3301:
3297:
3293:
3289:
3282:
3279:
3274:
3268:
3264:
3260:
3256:
3255:
3247:
3244:
3239:
3237:9780511676550
3233:
3229:
3225:
3221:
3220:
3212:
3209:
3204:
3200:
3196:
3190:
3186:
3182:
3178:
3171:
3168:
3163:
3159:
3155:
3151:
3147:
3145:9780511976667
3141:
3137:
3133:
3129:
3122:
3120:
3118:
3116:
3114:
3112:
3110:
3108:
3106:
3104:
3102:
3100:
3096:
3091:
3087:
3083:
3081:9780199215706
3077:
3073:
3069:
3065:
3061:
3055:
3052:
3046:
3041:
3038:
3036:
3033:
3031:
3028:
3026:
3023:
3021:
3018:
3016:
3013:
3011:
3008:
3006:
3005:Quantum clock
3003:
3001:
2998:
2996:
2993:
2991:
2988:
2986:
2983:
2982:
2977:
2973:
2972:
2968:
2966:
2965:
2961:
2959:
2958:
2954:
2952:
2951:
2947:
2945:
2944:
2940:
2938:
2937:
2933:
2932:
2931:
2929:
2921:
2919:
2917:
2913:
2909:
2905:
2903:
2902:quantum noise
2899:
2895:
2891:
2886:
2878:
2876:
2871:
2863:
2861:
2859:
2854:
2852:
2851:quantum state
2848:
2847:probabilities
2844:
2843:superposition
2840:
2839:quantum state
2836:
2832:
2828:
2822:
2814:
2812:
2805:
2802:
2801:
2800:
2797:
2791:
2786:
2782:
2781:
2780:
2777:
2774:
2767:
2765:
2763:
2755:
2753:
2751:
2744:
2736:
2734:
2731:
2727:
2723:
2716:
2711:
2709:
2706:
2692:
2670:
2666:
2656:
2643:
2638:
2634:
2627:
2622:
2618:
2609:
2605:
2599:
2595:
2591:
2588:
2582:
2576:
2571:
2567:
2560:
2546:
2543:
2537:
2531:
2523:
2509:
2502:
2495:
2487:
2485:
2479:
2477:
2457:
2449:
2433:
2427:
2419:
2403:
2397:
2388:
2374:
2371:
2368:
2345:
2342:
2339:
2336:
2327:
2309:
2305:
2296:
2292:
2286:
2281:
2278:
2275:
2271:
2267:
2262:
2258:
2251:
2248:
2245:
2241:
2236:
2230:
2222:
2218:
2209:
2193:
2189:
2185:
2182:
2179:
2176:
2173:
2168:
2164:
2138:
2134:
2127:
2124:
2121:
2118:
2115:
2112:
2104:
2100:
2093:
2090:
2082:
2078:
2071:
2063:
2062:RĂ©nyi entropy
2057:
2056:RĂ©nyi entropy
2050:RĂ©nyi entropy
2049:
2047:
2043:
2025:
2021:
2014:
2011:
2006:
2002:
1993:
1989:
1982:
1977:
1972:
1969:
1966:
1962:
1958:
1955:
1944:
1940:
1933:
1930:
1927:
1924:
1921:
1918:
1910:
1906:
1899:
1896:
1888:
1884:
1877:
1871:
1868:
1862:
1856:
1848:
1832:
1828:
1824:
1821:
1818:
1815:
1812:
1807:
1803:
1777:
1773:
1766:
1763:
1760:
1757:
1754:
1751:
1743:
1739:
1732:
1729:
1721:
1717:
1710:
1701:
1696:
1690:
1682:
1680:
1678:
1670:
1668:
1662:
1660:
1658:
1654:
1650:
1646:
1642:
1638:
1634:
1630:
1625:
1618:
1616:
1610:
1607:
1604:
1600:
1596:
1592:
1588:
1584:
1580:
1576:
1572:
1568:
1565:
1562:, called the
1561:
1558:
1554:
1551:
1547:
1543:
1539:
1536:
1532:
1528:
1523:
1520:
1517:
1513:
1508:
1507:
1506:
1504:
1500:
1499:Boolean logic
1496:
1492:
1491:quantum gates
1488:
1484:
1476:
1474:
1472:
1468:
1460:
1457:
1454:
1450:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1424:
1423:
1421:
1417:
1413:
1409:
1404:
1402:
1398:
1382:
1376:
1373:
1370:
1367:
1361:
1358:
1355:
1352:
1346:
1340:
1320:
1313:
1309:
1305:
1301:
1297:
1293:
1289:
1281:
1279:
1277:
1272:
1270:
1266:
1258:
1256:
1254:
1250:
1246:
1242:
1238:
1237:Richard Jozsa
1234:
1233:David Deutsch
1229:
1226:
1222:
1217:
1215:
1211:
1207:
1201:
1197:
1189:
1187:
1184:
1179:
1178:
1174:
1167:
1159:
1157:
1155:
1151:
1146:
1144:
1140:
1132:
1130:
1128:
1123:
1119:
1111:
1087:
1083:
1070:
1067:
1063:
1059:
1056:
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112:, due to the
111:
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70:
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60:
56:
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48:
39:
33:
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6595:
6532:Anti-gravity
6476:metamaterial
6440:
6404:post-quantum
6399:cryptography
6119:
6013:Quantum mind
5925:Franck–Hertz
5787:Klein–Gordon
5736:Formulations
5729:Formulations
5658:Interference
5648:Entanglement
5626:Ground state
5621:Energy level
5594:Fundamentals
5558:Introduction
5398:Charge qubit
5323:KLM protocol
5272:
5136:
5126:
4820:Purification
4750:Eastin–Knill
4694:
4609:
4602:
4561:
4557:
4548:
4512:
4463:
4457:
4421:
4368:
4364:
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4304:
4292:. Retrieved
4288:
4279:
4261:
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4237:
4233:
4224:
4207:
4203:
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4172:(1): 82–85.
4169:
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4119:
4110:
4099:. Retrieved
4095:
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2746:
2726:Dense coding
2720:
2712:Applications
2707:
2657:
2524:
2497:
2483:
2389:
2328:
2210:
2059:
2044:
1849:
1702:
1698:
1674:
1666:
1622:
1614:
1601:is known as
1483:Bloch sphere
1480:
1464:
1448:
1411:
1408:Bloch sphere
1405:
1285:
1273:
1262:
1230:
1228:algorithms.
1218:
1203:
1180:
1176:
1173:Cryptography
1169:
1147:
1136:
1118:Stratonovich
1115:
216:
201:
186:
172:cryptography
165:
142:
140:techniques.
123:
102:neuroscience
90:cryptography
71:
58:
46:
45:
6547:Force field
6496:programming
6456:logic clock
6441:information
6416:electronics
6271:EPR paradox
6051:Quantum bus
5920:Double-slit
5898:Experiments
5864:Many-worlds
5802:Schrödinger
5766:Phase space
5756:Schrödinger
5746:Interaction
5703:Uncertainty
5673:Nonlocality
5668:Measurement
5663:Decoherence
5653:Hamiltonian
5429:programming
5408:Phase qubit
5313:Circuit QED
4785:No-deleting
4727:cloud-based
4361:Lloyd, Seth
4053:(1): 7–11.
2898:decoherence
2892:is used in
2448:min-entropy
1306:. Given a
1221:Moore's law
1206:Alan Turing
276:Interaction
264:Schrödinger
207:Schrödinger
157:mathematics
149:transmitted
118:observables
6461:logic gate
6359:algorithms
6199:Extensions
6033:Technology
5879:Relational
5830:Copenhagen
5741:Heisenberg
5688:Tunnelling
5551:Background
5469:libquantum
5403:Flux qubit
5308:Cavity QED
5257:Bacon–Shor
5247:stabilizer
4775:No-cloning
4449:1034577167
4334:1038428525
4294:4 December
4101:2020-11-13
4060:2003.06557
4025:0433.94008
3983:0317.94003
3731:j.ctt7s8nw
3514:: 92–100.
3468:2012.07836
3440:1390.81001
3432:1003273264
3047:References
2908:Peter Shor
2900:and other
1693:See also:
1416:impossible
1245:Peter Shor
1194:See also:
1164:See also:
1116:In 1960s,
533:Observable
270:Heisenberg
211:Heisenberg
153:algorithms
98:psychology
86:philosophy
42:computers.
6506:simulator
6394:computing
6364:amplifier
5905:Bell test
5775:Equations
5601:Born rule
5375:NV center
4810:Threshold
4790:No-hiding
4755:Gleason's
4540:255569451
4473:1106.1445
4403:119250535
4378:1110.3234
4289:Mi.ras.ru
3892:121571330
3657:CiteSeerX
3493:221237249
3461:: 16–32.
3154:665137861
3090:822959053
2693:ρ
2667:λ
2635:λ
2628:
2606:λ
2596:∑
2592:−
2583:ρ
2577:
2561:ρ
2547:−
2544:≡
2538:ρ
2510:ρ
2464:∞
2461:→
2431:→
2401:→
2372:≠
2349:∞
2272:∑
2268:
2249:−
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1963:∑
1959:−
1647:over the
1535:entangled
1467:unitarity
1377:ρ
1374:
1368:ρ
1362:
1356:−
1347:ρ
1321:ρ
1139:atom trap
1092:ℏ
1057:−
1032:ρ
1026:ℏ
966:ρ
955:constant
939:ℏ
888:ρ
882:ℏ
853:−
825:ρ
794:ℏ
759:−
737:ℏ
653:ℏ
624:−
602:ℏ
537:constant
516:⟩
496:ψ
485:ℏ
442:⟩
422:ψ
406:constant
392:⟩
372:ψ
361:ℏ
332:−
321:⟩
301:ψ
284:Ket state
6613:Category
6411:dynamics
6294:Category
6088:Timeline
5840:Ensemble
5820:Bayesian
5713:Collapse
5585:Glossary
5568:Timeline
5437:OpenQASM
5413:Transmon
5290:Physical
5090:Quantum
4991:Grover's
4765:Holevo's
4738:Theorems
4688:timeline
4678:NISQ era
4619:18346434
4594:19708593
4586:10045619
4077:27022972
3999:(1979).
3957:(1973).
3839:51631629
3692:: 12–17.
3627:52859247
3485:32822698
3203:68629072
3162:59717455
3062:(2006).
2978:See also
2922:Journals
1589:include
1412:smallest
1141:and the
1122:Helstrom
6501:sensing
6481:network
6466:machine
6436:imaging
6384:circuit
6379:channel
6351:Quantum
6247:Related
6226:History
5965:Science
5797:Rydberg
5563:History
5427:Quantum
5365:Kane QC
5224:Quantum
5152:Quantum
5081:PostBQP
5051:Quantum
5036:Simon's
4829:Quantum
4666:General
4566:Bibcode
4383:Bibcode
4186:6519532
4148:1438116
4128:Bibcode
4017:0581651
3975:0456936
3941:2020051
3862:Bibcode
3774:4194913
3754:Bibcode
3597:Bibcode
3516:Bibcode
3325:Bibcode
2950:Quantum
1560:entropy
1557:Shannon
1420:measure
1249:problem
230:Picture
6491:optics
6343:Fields
5940:Popper
5445:IBM QX
5441:Qiskit
5380:NMR QC
5358:-based
5262:Steane
5233:Codes
5031:Shor's
4937:SARG04
4745:Bell's
4617:
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161:qubits
6520:Other
6451:logic
5850:Local
5792:Pauli
5782:Dirac
5267:Toric
4710:Qubit
4615:S2CID
4590:S2CID
4468:arXiv
4399:S2CID
4373:arXiv
4182:S2CID
4144:S2CID
4073:S2CID
4055:arXiv
3888:S2CID
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3770:S2CID
3727:JSTOR
3489:S2CID
3463:arXiv
3158:S2CID
3030:Qubit
2831:qubit
2450:when
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1296:qubit
1276:qubit
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6597:List
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5450:Quil
5356:Spin
5252:Shor
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4865:LOCC
4582:PMID
4536:OCLC
4526:ISBN
4486:ISBN
4445:OCLC
4435:ISBN
4330:OCLC
4320:ISBN
4296:2018
3937:OCLC
3927:ISBN
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3613:ISBN
3481:PMID
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3199:OCLC
3189:ISBN
3150:OCLC
3140:ISBN
3086:OCLC
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