1540:
the 810 nm photons come back to the crystal, they are annihilated by interaction with the returning pump beam. In any case, the beam of photons that make it through the crystal and the beam of photons that pass through the beam splitter are both separated into "vertically polarized" and "horizontally polarized" beams, which correspond to the "electrons" and the "positrons" of Hardy's scheme. The two "electron" beams (the photons with one kind of polarization) are united at a beam splitter and go to one or two detectors, and the same for the "positrons" (the other photons). Classically, no photons should be detected at what the authors call the "dark ports" because if they take both directions from the first beam splitter, they will interfere with themselves, whereas if they take only one path, then one cannot detect them both at the dark ports because of the paradox. By introducing a 20° rotation in polarization and using
50:, and these have demonstrated that the phenomenon does occur. However, the consequence of these experiments is only that past events can be inferred after their occurrence as a probabilistic wave collapse. These weak measurements are considered to be an observation themselves, and therefore part of the causation of wave collapse, making the objective results only a probabilistic function rather than a fixed reality. However, a careful analysis of the experiment shows that Hardy's paradox only proves that a
1014:
762:
1352:
569:
1009:{\displaystyle \left|e^{+}e^{-}\right\rangle \to {\frac {1}{4}}\left(3\left|c^{+}\right\rangle \left|c^{-}\right\rangle +\left|c^{+}\right\rangle \left|d^{-}\right\rangle +\left|d^{+}\right\rangle \left|c^{-}\right\rangle -\left|d^{+}\right\rangle \left|d^{-}\right\rangle -2\left|\gamma \right\rangle \left|\gamma \right\rangle \right).}
1049:
389:
1539:
crystal to produce pairs of 810 nm photons with polarizations orthogonal to each other. These then hit a beam splitter, which sends photons back to the barium borate crystal with 50% probability. The 405 nm pumping beam also bounces from a mirror and comes back to the barium borate. If both
1548:
that allowed them to calculate the "occupation" of different arms (paths) and combinations. As predicted by
Aharonov and colleagues, they found a negative value for the combination in which both photons take the outer (no-annihilation) route. The results were not exactly as predicted, and they
250:
381:
1347:{\displaystyle {\frac {|v^{+}\rangle -i|w^{+}\rangle }{\sqrt {2}}}{\frac {|v^{-}\rangle -i|w^{-}\rangle }{\sqrt {2}}}={\frac {1}{2}}\left(|v^{+}\rangle |v^{-}\rangle -i|v^{+}\rangle |w^{-}\rangle -i|w^{+}\rangle |v^{-}\rangle -|w^{+}\rangle |w^{-}\rangle \right).}
54:
cannot exist, as there cannot be a theory that assumes that the system meets the states of reality regardless of the interaction with the measuring apparatus. This confirms that a quantum theory, to be consistent with the experiments, must be non-local (in the
564:{\displaystyle {\frac {1}{2}}\left(\left|v^{+}\right\rangle \left|v^{-}\right\rangle +i\left|v^{+}\right\rangle \left|w^{-}\right\rangle +i\left|w^{+}\right\rangle \left|v^{-}\right\rangle -\left|\gamma \right\rangle \left|\gamma \right\rangle \right).}
72:
137:
1479:
1357:
This is not orthogonal to the expression above for the state before the final beam splitters. The scalar product between them is 1/4, showing that there is a 1 in 16 chance of this happening, paradoxically.
754:
663:
308:
1849:; Botero, Alonso; Popescu, Sandu; Reznik, Benni; Tollaksen, Jeff (2002). "Revisiting Hardy's paradox: counterfactual statements, real measurements, entanglement and weak values".
83:
for quantum particles and antiparticles. We will describe the case using electrons and positrons. Each interferometer consists of bent paths and two beam splitters (labeled
1946:
1795:
Aharonov, Yakir; Albert, David Z.; Vaidman, Lev (1988-04-04). "How the result of a measurement of a component of the spin of a spin-1/2particle can turn out to be 100".
691:
600:
266:
In the actual experiment the interferometers are arranged so that part of their paths overlap as shown in the diagram. If the amplitude for the particle in one arm, say
1019:
Since the probabilities are the squares of the absolute values of these amplitudes, this means a 9 in 16 chance of each particle being detected in its respective
1361:
The situation can be analyzed in terms of two simultaneous interaction-free measurements: from the point of view of the interferometer on the left, a click at
1742:
Yokota, K.; Yamamoto, T.; Koashi, M.; Imoto, N. (2009). "Direct observation of Hardy's paradox by joint weak measurement with an entangled photon pair".
1524:
paths in boxes and noting the effect of their mutual electrostatic attraction. They stated that one would actually find a repulsion between the boxes.
245:{\displaystyle \left|e^{-}\right\rangle \to {\frac {\left|v^{-}\right\rangle +i\left|w^{-}\right\rangle }{\sqrt {2}}}\to i\left|c^{-}\right\rangle .}
91:
in the accompanying diagram) and is tuned so that when operating individually, particles always exit to the same particle detector (the ones labeled
1039:
detectors; and a 4 in 16 (1 in 4) chance that the electron and positron annihilate, so neither is detected. Notice that a detection in both
103:
is for "destructive interference"). For example, for the right-hand side interferometer, when operating alone, entering electrons (labeled
1488:
and colleagues in 2001 pointed out that the number of electrons or positrons in each branch is theoretically observable and is 0 in the
1431:
1729:
1916:
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605:
80:
1676:
Lundeen, J. S.; Steinberg, A. M. (2009). "Experimental Joint Weak
Measurement on a Photon Pair as a Probe of Hardy's Paradox".
376:{\displaystyle \left|w^{+}\right\rangle \left|w^{-}\right\rangle \to \left|\gamma \right\rangle \left|\gamma \right\rangle }
1520:! They proposed a way that this could be observed physically by temporarily trapping the electron and the positron in the
383:. There is a 1 in 4 chance of this happening. We can express the state of the system, before the final beam splitters, as
1936:
1550:
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51:
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published work in which they set up a "Hardy's paradox" system using photons. A 405 nm laser goes through a
1481:
term arises, in fact, from the nonmaximally entangled nature of the state just before the final beam splitters.
1587:
Hardy, Lucien (1992). "Quantum mechanics, local realistic theories, and
Lorentz-invariant realistic theories".
1630:
Hardy, Lucien (1993). "Nonlocality for two particles without inequalities for almost all entangled states".
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observable and is not given by the product of the single-particle values. So we find that the number of
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paths in their respective interferometers, they will annihilate to produce two gamma rays:
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A paradox then arises because sometimes the particles do emerge simultaneously at
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amplitude would reach the second beam splitter and would split into arms
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respectively). If we assume the particles are independent (described by
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If (classically speaking) both the electron and the positron take the
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an annihilation taking place. For this reason, this scheme was named
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1474:{\displaystyle \left|d^{+}\right\rangle \left|d^{-}\right\rangle }
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would thus indicate the presence of the obstructing particle, but
127:), but these constructively interfere and thus always exit in arm
70:
79:
The basic building block of Hardy’s thought experiment are two
1369:. Similarly, for the interferometer on the right, a click at
1397:), we conclude that they can never emerge simultaneously in
749:{\textstyle {\frac {1}{\sqrt {2}}}(|v\rangle -i|w\rangle )}
658:{\textstyle {\frac {1}{\sqrt {2}}}(|v\rangle +i|w\rangle )}
1413:, which cannot occur because of the annihilation process.
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attribute this to imperfect switching (annihilation) and
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with equal amplitudes. The detection of a particle in
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implies the presence of the obstructing electron in
1496:branches. And yet, the number of electron–positron
1803:(14). American Physical Society (APS): 1351–1354.
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270:, were to be obstructed by a second particle in
1428: = 1/16). Quantum mechanically, the
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1377:. Indeed, every time a click is recorded at
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31:in 1992–1993 in which a particle and its
1947:Thought experiments in quantum mechanics
1373:implies the presence of the positron in
119:(in the diagram, the latter part of the
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1027:for one particle being detected in its
99:is for "constructive interference" and
1518:the number in the vv combination is −1
1405:. This would imply that they were in
7:
75:Setup for Hardy's thought experiment
42:Experiments using the technique of
1385:), the other particle is found in
14:
67:Setup description and the results
274:that collides with it, only the
1508:pairs (both particles in their
46:have studied an interaction of
1708:10.1103/PhysRevLett.102.020404
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255:Similarly, positrons (labeled
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1:
1914:Lecture by Aephraim Steinberg
1881:10.1016/s0375-9601(02)00986-6
1857:(3–4). Elsevier BV: 130–138.
1774:10.1088/1367-2630/11/3/033011
1551:interaction-free measurements
111:of electrons taking the path
1043:detectors is represented by
296:interaction-free measurement
81:Mach–Zehnder interferometers
52:local hidden-variable theory
1817:10.1103/physrevlett.60.1351
1652:10.1103/PhysRevLett.71.1665
1609:10.1103/PhysRevLett.68.2981
1023:detector; a 1 in 16 chance
1963:
686:{\displaystyle |d\rangle }
595:{\displaystyle |c\rangle }
115:and electrons taking path
259:) are always detected at
1797:Physical Review Letters
1678:Physical Review Letters
1632:Physical Review Letters
1589:Physical Review Letters
1744:New Journal of Physics
1568:Wave function collapse
1500:in any combination is
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109:quantum superposition
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35:may interact without
1684:(2): 020404–000001.
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602:detectors click for
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1937:Quantum measurement
1873:2002PhLA..301..130A
1809:1988PhRvL..60.1351A
1766:2009NJPh...11c3011Y
1700:2009PhRvL.102b0404L
1644:1993PhRvL..71.1665H
1601:1992PhRvL..68.2981H
1919:2015-10-02 at the
1471:
1424:(with probability
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21:thought experiment
1851:Physics Letters A
1728:. Also available
1638:(11): 1665–1668.
1595:(20): 2981–2984.
1546:weak measurements
1512:path) is 0, each
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