Quantum nonlocality

In theoretical physics, quantum nonlocality is a characteristic of some measurements made at a microscopic level that contradict the assumptions of local realism found in classical mechanics. Despite consideration of hidden variables as a possible resolution of this contradiction, some aspects of entangled quantum states have been demonstrated irreproducible by any local hidden variable theory. Bell's theorem is one such demonstration which has been verified by experiment.[1]

Experiments have generally favoured quantum mechanics as a description of nature, over local hidden variable theories.[2][3] Any physical theory that supersedes or replaces quantum theory must make similar experimental predictions and must therefore also be nonlocal in this sense; quantum nonlocality is a property of the universe that is independent of our description of nature.

Quantum nonlocality does not allow for faster-than-light communication,[4] and hence is compatible with special relativity. However, it prompts many of the foundational discussions concerning quantum theory.

In October 2018, physicists reported that quantum behavior can be explained with classical physics for a single particle, but not for multiple particles as in quantum entanglement and related nonlocality phenomena.[5][6]

History

Einstein, Podolsky and Rosen

In 1935, Einstein, Podolsky and Rosen published a thought experiment with which they hoped to expose the incompleteness of the Copenhagen interpretation of quantum mechanics in relation to the violation of local causality at the microscopic scale that it described.[7] Afterwards, Einstein presented a variant of these ideas in a letter to Erwin Schrödinger,[8] which is the version that is presented here. The state and notation used here are more modern, and akin to Bohm's take on EPR.[9] The quantum state of the two particles prior to measurement can be written as

${\displaystyle \left|\psi _{AB}\right\rangle ={\frac {1}{\sqrt {2}}}{\bigg (}\left|0\right\rangle _{A}\left|1\right\rangle _{B}-\left|1\right\rangle _{A}\left|0\right\rangle _{B}{\bigg )}={\frac {1}{\sqrt {2}}}{\bigg (}\left|+\right\rangle _{A}\left|-\right\rangle _{B}-\left|-\right\rangle _{A}\left|+\right\rangle _{B}{\bigg )}}$

Where ${\displaystyle \left|\pm \right\rangle ={\frac {1}{\sqrt {2}}}\left(\left|0\right\rangle \pm \left|1\right\rangle \right)}$.[10]

Here, subscripts A and B distinguish the two particles, though it is more convenient and usual to refer to these particles as being in the possession of two experimentalists called Alice and Bob. The rules of quantum theory give predictions for the outcomes of measurements performed by the experimentalists. Alice, for example, will measure her particle to be spin-up in an average of fifty percent of measurements. However, according to the Copenhagen interpretation, Alice's measurement causes the state of the two particles to collapse, so that if Alice performs a measurement of spin in the z-direction, that is with respect to the basis ${\displaystyle \{\left|0\right\rangle _{A},\left|1\right\rangle _{A}\}}$, then Bob's system will be left in one of the states ${\displaystyle \{\left|0\right\rangle _{B},\left|1\right\rangle _{B}\}}$. Likewise, if Alice performs a measurement of spin in the x-direction, that is with respect to the basis ${\displaystyle \{\left|+\right\rangle _{A},\left|-\right\rangle _{A}\}}$, then Bob's system will be left in one of the states ${\displaystyle \{\left|+\right\rangle _{B},\left|-\right\rangle _{B}\}}$. Schrödinger referred to this phenomenon as "steering".[11] This steering occurs in such a way that no signal can be sent by performing such a state update; quantum nonlocality cannot be used to send messages instantaneously and is therefore not in direct conflict with causality concerns in Special Relativity.[10]

In the Copenhagen view of this experiment, Alice's measurement—and particularly her measurement choice—have a direct effect on Bob's state. However, under the assumption of locality, actions on Alice's system do not affect the "true", or "ontic" state of Bob's system. We see that the ontic state of Bob's system must be compatible with one of the quantum states ${\displaystyle \left|\uparrow \right\rangle _{B}}$ or ${\displaystyle \left|\downarrow \right\rangle _{B}}$, since Alice can make a measurement that concludes with one of those states being the quantum description of his system. At the same time, it must also be compatible with one of the quantum states ${\displaystyle \left|\leftarrow \right\rangle _{B}}$ or ${\displaystyle \left|\rightarrow \right\rangle _{B}}$ for the same reason. Therefore, the ontic state of Bob's system must be compatible with at least two quantum states; the quantum state is therefore not a complete descriptor of his system. Einstein, Podolsky and Rosen saw this as evidence of the incompleteness of the Copenhagen interpretation of quantum theory, since the wavefunction is explicitly not a complete description of a quantum system under this assumption of locality. Their paper concludes:[7]

While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.

Although various authors (most notably Niels Bohr) criticised the ambiguous terminology of the EPR paper,[12][13] the thought experiment nevertheless generated a great deal of interest. Their notion of a "complete description" was later formalised by the suggestion of hidden variables that determine the statistics of measurement results, but to which an observer does not have access.[14] Bohmian mechanics provides such a completion of quantum mechanics, with introduction of hidden variables; however the theory is explicitly nonlocal.[15] The interpretation therefore does not give an answer to Einstein's question, which was whether or not a complete description of quantum mechanics could be given in terms of local hidden variables in keeping with the "Principle of Local Action".[16]

Probabilistic Nonlocality

In 1964 John Bell answered Einstein's question by showing that such local hidden variables can never reproduce the full range of statistical outcomes predicted by quantum theory.[17] Bell showed that a local hidden variable hypothesis leads to restrictions on the strength of correlations of measurement results. If the Bell inequalities are violated experimentally as predicted by quantum mechanics, then reality cannot be described by local hidden variables and the mystery of quantum nonlocal causation remains. According to Bell:[17]

This [grossly nonlocal structure] is characteristic ... of any such theory which reproduces exactly the quantum mechanical predictions.

Clauser, Horne, Shimony and Holt (CHSH) reformulated these inequalities in a manner that was more conducive to experimental testing (see CHSH inequality).[18] They proposed a scheme whereby two experimentalists, Alice and Bob, make separate measurements of photon polarization in two carefully chosen directions, and derived a simple inequality that is obeyed by all local hidden variable theories, but violated by certain measurements on quantum states.

Bell formalized the idea of a hidden variable by introducing the parameter λ to locally characterize measurement results on each system:[17] "It is a matter of indifference ... whether λ denotes a single variable or a set ... and whether the variables are discrete or continuous". However, it is equivalent (and more intuitive) to think of λ as a local "strategy" or "message" that occurs with some probability ρ(${\displaystyle \lambda }$) when an entangled pair of states is created. EPR's criteria of local separability then stipulates that each local strategy defines the distributions of independent outcomes if Alice measures in direction A and Bob measures in direction B:[17]

${\displaystyle P\left({a,b}{|}{A,B,\lambda }\right)=P\left({a}{|}{A,\lambda }\right)P\left({b}{|}{B,\lambda }\right)}$

where, for instance, ${\displaystyle \scriptstyle P\left({a}{|}{A,\lambda }\right)}$ denotes the probability of Alice getting the outcome a given λ, and that she measured A.

Suppose that λ can take values from some set ${\displaystyle \lambda _{i}}$, where 1 ≤ i ≤ k. If each ${\displaystyle \lambda _{i}}$ has an associated probability ρ(${\displaystyle \lambda _{i}}$) of being selected (such that the probabilities sum to unity) we can average over this distribution to obtain a formula for the joint probability of each measurement result:

${\displaystyle P\left({a,b}{|}{A,B}\right)=\sum _{i=1}^{k}P\left({a,b}{|}{A,B,\lambda _{i}}\right)\rho \left(\lambda _{i}\right)\,.}$

In the CHSH scheme, the measurement result for the polarization of a photon can take one of two values (informally, whether the photon is polarized in that direction, or in the orthogonal direction). We encode this by allowing a and b to take on values ±1. For arbitrary measurements A and B, their correlator E(A, B) is then defined as

${\displaystyle E\left({A,B}\right)=\sum _{a,b}abP\left({a,b}{|}{A,B}\right)\,.}$

Note that the product ab is equal to 1 if Alice and Bob get the same outcome, and −1 if they get different outcomes. The correlator E(A,B) can therefore be seen as the expectation that Alice's and Bob's outcomes are correlated. In the case that Alice chooses from one of two measurements ${\displaystyle A_{0}}$ or ${\displaystyle A_{1}}$, and Bob chooses from ${\displaystyle B_{0}}$ or ${\displaystyle B_{1}}$, the CHSH value for this joint probability distribution is defined as

${\displaystyle S_{CHSH}=E\left({A_{0},B_{0}}\right)+E\left({A_{0},B_{1}}\right)+E\left({A_{1},B_{0}}\right)-E\left({A_{1},B_{1}}\right)\,.}$

Compare this with the expression ${\displaystyle \scriptstyle x\oplus y=XY}$ and the discussion in the above example. The CHSH value ${\displaystyle S_{CHSH}}$ includes a negative contribution of the correlator whenever ${\displaystyle A_{1}}$ and ${\displaystyle B_{1}}$ are chosen (${\displaystyle x=y}$ when ${\displaystyle XY=1}$), and a positive contribution in all other cases (${\displaystyle x}$${\displaystyle y}$ when ${\displaystyle XY=0}$). If the joint probability distribution can be described with local strategies as above, it can be shown that the correlation function always obeys the following CHSH inequality:[10]

${\displaystyle -2\leq S_{CHSH}\leq 2\,.}$

However, if instead of local hidden variables we adopt the rules of quantum theory, it is possible to construct an entangled pair of particles (one each for Alice and Bob) and a set of measurements ${\displaystyle \scriptstyle A_{0},A_{1},B_{0},B_{1}}$ such that ${\displaystyle \scriptstyle S_{CHSH}\;=\;2{\sqrt {2}}}$. This demonstrates an explicit way in which a theory with ontological states that are local, with local measurements and only local actions cannot match the probabilistic predictions of quantum theory, disproving Einstein's hypothesis. Experimentalists such as Alain Aspect have verified the quantum violation of the CHSH inequality,[2] as well as other formulations of Bell's inequality, to invalidate the local hidden variables hypothesis and confirm that reality is indeed nonlocal in the EPR sense.

Possibilistic Nonlocality

The demonstration of nonlocality due to Bell is probabilistic in the sense that it shows that the precise probabilities predicted by quantum mechanics for some entangled scenarios cannot be met by a local theory. (For short, here and henceforth "local theory" means "local hidden variables theory".) However, quantum mechanics permits an even stronger violation of local theories: a possibilistic one, in which we find that local theories cannot agree with quantum mechanics on which events are possible or impossible in an entangled scenario. The first proof of this kind was due to Greenberger, Horne and Zeilinger in 1993.[19] The state involved is often called the GHZ state.

In 1993, Lucien Hardy demonstrated a logical proof of quantum nonlocality that, like the GHZ proof is a possibilistic proof.[20][21][22] We note that the same state ${\displaystyle \left|\psi \right\rangle }$ defined below can be written in a few suggestive ways:

${\displaystyle \left|\psi \right\rangle ={\frac {1}{\sqrt {3}}}\left(\left|00\right\rangle +\left|01\right\rangle +\left|10\right\rangle \right)={\frac {1}{\sqrt {3}}}\left({\sqrt {2}}\left|+0\right\rangle +{\frac {1}{\sqrt {2}}}\left(\left|+1\right\rangle +\left|-1\right\rangle \right)\right)={\frac {1}{\sqrt {3}}}\left({\sqrt {2}}\left|0+\right\rangle +{\frac {1}{\sqrt {2}}}\left(\left|1+\right\rangle +\left|1-\right\rangle \right)\right)}$

where, as above, ${\displaystyle |\pm \rangle ={\tfrac {1}{\sqrt {2}}}(\left|0\right\rangle \pm \left|1\right\rangle )}$.

The experiment consists of this entangled state being shared between two experimenters, each of whom has the ability to measure either with respect to the basis ${\displaystyle \{\left|0\right\rangle ,\left|1\right\rangle \}}$ or ${\displaystyle \{\left|+\right\rangle ,\left|-\right\rangle \}}$. We see that if they each measure with respect to ${\displaystyle \{\left|0\right\rangle ,\left|1\right\rangle \}}$, then they never see the outcome ${\displaystyle \left|11\right\rangle }$. If one measures with respect to ${\displaystyle \{\left|0\right\rangle ,\left|1\right\rangle \}}$ and the other ${\displaystyle \{\left|+\right\rangle ,\left|-\right\rangle \}}$, they never see the outcomes ${\displaystyle \left|-0\right\rangle ,}$ ${\displaystyle \left|0-\right\rangle .}$ However, sometimes they see the outcome ${\displaystyle \left|--\right\rangle }$ when measuring with respect to ${\displaystyle \{\left|+\right\rangle ,\left|-\right\rangle \}}$, since ${\displaystyle \langle --|\psi \rangle =-{\tfrac {1}{2{\sqrt {3}}}}\neq 0.}$

This leads us to the paradox: having the outcome ${\displaystyle |--\rangle }$ we conclude that if one of the experimenters had measured with respect to the ${\displaystyle \{\left|0\right\rangle ,\left|1\right\rangle \}}$ basis instead, the outcome must have been ${\displaystyle |{-}1\rangle }$ or ${\displaystyle |1-\rangle }$, since ${\displaystyle |{-}0\rangle }$ and ${\displaystyle |0-\rangle }$ are impossible. But then, if they had both measured with respect to the ${\displaystyle \{\left|0\right\rangle ,\left|1\right\rangle \}}$ basis, by locality the result must have been ${\displaystyle \left|11\right\rangle }$, which is also impossible.

Differences between nonlocality and entanglement

In the media and popular science, quantum nonlocality is often portrayed as being equivalent to entanglement. While it is true that a pure bipartite quantum state must be entangled in order for it to produce nonlocal correlations, there exist entangled (mixed) states which do not produce such correlations,[23] and there exist non-entangled (namely, separable) states that do produce some type of non-local behavior.[24] For the former, a well-known example is constituted by a subset of Werner states that are entangled but whose correlations can always be described using local hidden variables. On the other hand, reasonably simple examples of Bell inequalities have been found for which the quantum state giving the largest violation is never a maximally entangled state, showing that entanglement is, in some sense, not even proportional to nonlocality.[25][26][27]

In short, entanglement of a two-party state is necessary but not sufficient for that state to be nonlocal. It is important to recognise that entanglement is more commonly viewed as an algebraic concept, noted for being a precedent to nonlocality as well as quantum teleportation and superdense coding, whereas nonlocality is interpreted according to experimental statistics and is much more involved with the foundations and interpretations of quantum mechanics.

Superquantum nonlocality

Whilst the CHSH inequality gives restrictions on the CHSH value attainable by local hidden variable theories, the rules of quantum theory do not allow us to violate Tsirelson's bound of ${\displaystyle \scriptstyle 2{\sqrt {2}}}$, even if we exploit measurements of entangled particles.[28] The question remained whether this was the maximum CHSH value that can be attained without explicitly allowing instantaneous signaling. In 1994 two physicists, Sandu Popescu and Daniel Rohrlich, formulated an explicit set of correlated measurements that respect the "non-signalling" principle, yet give ${\displaystyle S_{CHSH}=4}$: the algebraic maximum.[29] The maximal violation of CHSH consistent with no signalling was also found, earlier, by Rastall and Khalfin and Tsirelson.[30][31] This demonstrated that there exist formulatable theories that are "non-signalling", yet drastically violate the joint probability constraints of quantum theory. The attempt to understand what distinguishes quantum theory from such general theories motivated an abstraction from physical measurements of nonlocality, to the study of nonlocal boxes.[32]

Nonlocal boxes generalize the concept of experimentalists making joint measurements from separate locations. As in the discussion above, the choice of measurement is encoded by the input to the box. A two-party nonlocal box takes an input A from Alice and an input B from Bob, and outputs two values a and b for Alice and Bob respectively and separately, where a, b, A and B take values from some finite alphabet (normally ${\displaystyle \{0,1\}}$). The box is characterized by the probability of outputting pair a, b, given the inputs A, B. This probability is denoted ${\displaystyle \scriptstyle P\left({{a,b}{|}{A,B}}\right)}$ and obeys the normal probabilistic conditions of positivity and normalisation:[32]

${\displaystyle P\left({{a,b}{|}{A,B}}\right)\geq 0\quad \forall {a,b,A,B}}$

and

${\displaystyle \sum _{a,b}P\left({{a,b}{|}{A,B}}\right)=1\quad \forall {A,B}}$

A box is local, or admits a local hidden variable model, if its output probabilities can be characterized in the following way:[23]

${\displaystyle P\left({{a,b}{|}{A,B}}\right)=\sum _{\lambda }p(\lambda )\;P\left({{a}{|}{A,\lambda }}\right)\;P\left({{b}{|}{B,\lambda }}\right)}$

where ${\displaystyle \scriptstyle P\left({{a}{|}{A,\lambda }}\right)}$ and ${\displaystyle \scriptstyle P\left({{b}{|}{B,\lambda }}\right)}$ describe single input/output probabilities at Alice's or Bob's system alone, and the value of ${\displaystyle \lambda }$ is chosen at random according to some fixed probability distribution given by ${\displaystyle p(\lambda )}$. Intuitively, ${\displaystyle \lambda }$ corresponds to a hidden variable, or to a shared randomness between Alice and Bob. If a box violates this condition, it is explicitly called nonlocal. However, the study of nonlocal boxes often encompasses both local and nonlocal boxes.

The set of nonlocal boxes most commonly studied are the so-called non-signalling boxes,[32] for which neither Alice nor Bob can signal their choice of input to the other. Physically, this is a reasonable restriction: setting the input is physically analogous to making a measurement, which should effectively provide a result immediately. Since there may be a large spatial separation between the parties, signalling to Bob would potentially require considerable time to elapse between measurement and result, which is a physically unrealistic scenario.

The non-signalling requirement imposes further conditions on the joint probability, in that the probability of a particular output a or b should depend only on its associated input. This allows for the notion of a reduced or marginal probability on both Alice and Bob's measurements, and is formalised by the conditions:

${\displaystyle \sum _{b}P\left({a,b}{|}{A,B}\right)=\sum _{b}P\left({a,b}{|}{A,B'}\right)\equiv P\left({a}{|}{A}\right)\quad \forall {a,A,B,B'}}$

and

${\displaystyle \sum _{a}P\left({a,b}{|}{A,B}\right)=\sum _{a}P\left({a,b}{|}{A',B}\right)\equiv P\left({b}{|}{B}\right)\quad \forall {b,B,A,A'}}$

The constraints above are all linear, and so define a polytope representing the set of all non-signalling boxes with a given number of inputs and outputs. Moreover, the polytope is convex because any two boxes that exist in the polytope can be mixed (as above, according to some variable ${\displaystyle \lambda }$ with probabilities ${\displaystyle p(\lambda )}$) to produce another box that also exists within the polytope.

Local boxes are clearly non-signalling, however nonlocal boxes may or may not be non-signalling. Since this polytope contains all possible non-signalling boxes of a given number of inputs and outputs, it has as subsets both local boxes and those boxes which can achieve Tsirelson's bound in accord with quantum mechanical correlations. Indeed, the set of local boxes form a convex sub-polytope of the non-signalling polytope.

Popescu and Rohrlich's maximum algebraic violation of the CHSH inequality can be reached by a non-signalling box, referred to as a standard PR box after these authors, with joint probability given by:

${\displaystyle P\left({a,b}{|}{A,B}\right)={\begin{cases}{\frac {1}{2}},&{\mbox{if }}a\oplus b=AB\\0,&{\mbox{otherwise}}\end{cases}}}$

where ${\displaystyle \oplus }$ denotes addition modulo two.[33]

Various attempts have been made to argue why Nature does not (or should not) allow for stronger nonlocality than quantum theory is already known to permit. For example, it was found in the late 2000's that quantum mechanics cannot be more nonlocal without violating the Heisenberg uncertainty principle.[34][35] Strikingly, it was also discovered that if PR boxes did exist, any distributed computation could be performed with only one bit of communication.[36] An even stronger result is that for any nonlocal box theory which violates Tsirelson's bound, there cannot be a sensible measure of mutual information between pairs of systems.[37] This suggests a deep link between nonlocality and the information-theoretic properties of quantum mechanics. Nevertheless, the PR-box is ruled out by a plausible postulate of information theory.[38]

Non-signaling adversaries have recently been considered in quantum cryptography.[39] Such an adversary is constrained only by the non-signaling principle, and so may potentially be more powerful than a quantum adversary.

Blasiak's model

Similarly to de Broglie–Bohm pilot wave theory, Blasiak's model stipulates in the ontology a particle and a pilot wave, and the pilot wave ensures predictions indistinguishable from the quantum case.[6] The main distinction is wave function collapse. De Broglie–Bohm theory avoids collapse by including the whole universe (with all particle detectors, other devices, observers etc.) into the system. Blasiak's model describes a single particle in a quantum circuit (called "interferometric circuit") that contains devices such as phase shifters, beam splitters and particle detectors. Usually, a detector causes collapse (non-local, discontinuous change) of a wave function. But Blasiak's model treats detection without violating the locality principle, via the following ideas.

The wave function consists of fragments. Each fragment emerges from a detection, and propagates from the point of detection, gradually crowding out older fragments. The particle moves inside the expanding region occupied by the most recent fragment. Thus, obsolete fragments are harmless, they never pilot the particle. Gradual elimination of older fragments is implemented via the third ontological component, a field that carries information on the time of the most recent detection in the causal past of a given space-time point.

No information can travel faster-than-light

Nonlocality does not mean that information can travel faster-than-light. Indeed, quantum field theory preserves causality, meaning that no influence can be projected between two points faster than the speed-of-light.

This can be shown by noting that the commutator of two spacelike local quantum operators is always zero (or anti-commutator for fermionic operators).[40]

References

1. ^ Hensen, B, et al. (October 2015). "Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres". Nature. 526 (7575): 682–686. arXiv:1508.05949. Bibcode:2015Natur.526..682H. doi:10.1038/nature15759. PMID 26503041.
2. ^ a b Aspect, Alain; Dalibard, Jean; Roger, Gérard (December 1982). "Experimental Test of Bell's Inequalities Using Time- Varying Analyzers". Physical Review Letters. 49 (25): 1804–1807. Bibcode:1982PhRvL..49.1804A. doi:10.1103/PhysRevLett.49.1804.
3. ^ Rowe MA, et al. (February 2001). "Experimental violation of a Bell's Inequality with efficient detection". Nature. 409 (6822): 791–794. Bibcode:2001Natur.409..791K. doi:10.1038/35057215. PMID 11236986.
4. ^ Ghirardi, G.C.; Rimini, A.; Weber, T. (March 1980). "A general argument against superluminal transmission through the quantum mechanical measurement process". Lettere al Nuovo Cimento. 27 (10): 293–298. doi:10.1007/BF02817189.
5. ^ Staff (11 October 2018). "Public Release: 11-OCT-2018 - Where is it, the foundation of quantum reality?". EurekAlert!. Retrieved 13 October 2018.
6. ^ a b Blasiak, Pawel (13 July 2018). "Local model of a qudit: Single particle in optical circuits". Physical Review. 98 (012118) (1). doi:10.1103/PhysRevA.98.012118.
7. ^ a b Einstein, Albert; Podolsky, Boris; Rosen, Nathan (May 1935). "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?". Physical Review. 47 (10): 777–780. Bibcode:1935PhRv...47..777E. doi:10.1103/PhysRev.47.777.
8. ^ Einstein, Albert. "Letter to E. Schrödinger" [Letter]. Einstein Archives, ID: Call Number 22-47. Hebrew University of Jerusalem.
9. ^ Jevtic, S.; Rudolph, T (2015). "How Einstein and/or Schrödinger should have discovered Bell's theorem in 1936". Journal of the Optical Society of America B. 32 (4). pp. 50–55.
10. ^ a b c Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge University Press. pp. 112–113. ISBN 978-0-521-63503-5.
11. ^ Wiseman, H.M.; Jones, S.J.; Doherty, A.C. (April 2007). "Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox". Physical Review Letters. 98.
12. ^ Bohr, N (July 1935). "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?". Physical Review. 48 (8): 696–702. Bibcode:1935PhRv...48..696B. doi:10.1103/PhysRev.48.696.
13. ^ Furry, W.H. (March 1936). "Remarks on Measurements in Quantum Theory". Physical Review. 49 (6): 476. Bibcode:1936PhRv...49..476F. doi:10.1103/PhysRev.49.476.
14. ^ von Neumann, J. (1932/1955). In Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, translated into English by Beyer, R.T., Princeton University Press, Princeton, cited by Baggott, J. (2004) Beyond Measure: Modern physics, philosophy, and the meaning of quantum theory, Oxford University Press, Oxford, ISBN 0-19-852927-9, pages 144–145.
15. ^ Maudlin, Tim (2011). Quantum Non-Locality and Relativity : Metaphysical Intimations of Modern Physics (3rd ed.). John Wiley & Sons. p. 111. ISBN 9781444331264.
16. ^ Fine, Arthur (Winter 2017). "The Einstein-Podolsky-Rosen Argument in Quantum Theory". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 6 December 2018.
17. ^ a b c d Bell, John (1964). "On the Einstein Podolsky Rosen paradox". Physics. 1 (3): 195–200. doi:10.1103/PhysicsPhysiqueFizika.1.195.
18. ^ Clauser, John F.; Horne, Michael A.; Shimony, Abner; Holt, Richard A. (October 1969). "Proposed Experiment to Test Local Hidden-Variable Theories". Physical Review Letters. 23 (15): 880–884. Bibcode:1969PhRvL..23..880C. doi:10.1103/PhysRevLett.23.880.
19. ^ Daniel M. Greenberger; Michael A. Horne; Anton Zeilinger (2007), Going beyond Bell's Theorem, arXiv:0712.0921, Bibcode:2007arXiv0712.0921G
20. ^ Hardy, Lucien (1993). "Nonlocality for two particles without inequalities for almost all entangled states". Physical Review Letters. 71 (11): 1665–1668. Bibcode:1993PhRvL..71.1665H. doi:10.1103/PhysRevLett.71.1665. PMID 10054467.
21. ^ Braun, D.; Choi, M.-S. (2008). "Hardy's test versus the Clauser-Horne-Shimony-Holt test of quantum nonlocality: Fundamental and practical aspects". Phys. Rev. A. 78 (3): 032114. arXiv:0808.0052. Bibcode:2008PhRvA..78c2114B. doi:10.1103/physreva.78.032114.
22. ^ Nikolić, Hrvoje (2007). "Quantum Mechanics: Myths and Facts". Foundations of Physics. 37 (11): 1563–1611. arXiv:quant-ph/0609163. Bibcode:2007FoPh...37.1563N. doi:10.1007/s10701-007-9176-y.
23. ^ a b Werner, R.F. (1989). "Quantum States with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model". Physical Review A. 40 (8): 4277–4281. Bibcode:1989PhRvA..40.4277W. doi:10.1103/PhysRevA.40.4277. PMID 9902666.
24. ^ Bennett, C.H.; DiVincenzo, D.P.; Fuchs, C.A.; Mor, T.; Rains, E.; Shor, P.W.; Smolin, J.A.; Wootters, W.K. (1999). "Quantum nonlocality without entanglement". Physical Review A. 59 (2): 1070–1091. arXiv:quant-ph/9804053. Bibcode:1999PhRvA..59.1070B. doi:10.1103/PhysRevA.59.1070.
25. ^ Marius Junge; Carlos Palazuelos (2011). "Large violation of Bell inequalities with low entanglement". Communications in Mathematical Physics. 306 (3): 695–746. arXiv:1007.3043. Bibcode:2011CMaPh.306..695J. CiteSeerX 10.1.1.752.4896. doi:10.1007/s00220-011-1296-8.
26. ^ Thomas Vidick; Stephanie Wehner (2011). "More Non-locality with less Entanglement". Physical Review A. 83 (5): 052310. arXiv:1011.5206. Bibcode:2011PhRvA..83e2310V. doi:10.1103/PhysRevA.83.052310.
27. ^ Yeong-Cherng Liang; Tamas Vertesi; Nicolas Brunner (2010). "Semi-device-independent bounds on entanglement". Physical Review A. 83 (2): 022108. arXiv:1012.1513. Bibcode:2011PhRvA..83b2108L. doi:10.1103/PhysRevA.83.022108.
28. ^ Cirel'son, B. S. (1980). "Quantum generalizations of Bell's inequality". Letters in Mathematical Physics. 4 (2): 93–100. Bibcode:1980LMaPh...4...93C. doi:10.1007/BF00417500.
29. ^ Popescu, Sandu; Rohrlich, Daniel (1994). "Nonlocality as an axiom". Foundations of Physics. 24 (3): 379–385. Bibcode:1994FoPh...24..379P. CiteSeerX 10.1.1.508.4193. doi:10.1007/BF02058098.
30. ^ Rastall, Peter (1985). "Locality, Bell's theorem, and quantum mechanics". Foundations of Physics. 15 (9): 963–972. Bibcode:1985FoPh...15..963R. doi:10.1007/bf00739036.
31. ^ Khalfin, L.A.; Tsirelson, B.S. (1985). Lahti; et al. (eds.). Quantum and quasi-classical analogs of Bell inequalities. Symposium on the Foundations of Modern Physics. World Sci. Publ. pp. 441–460.
32. ^ a b c Barrett, J.; Linden, N.; Massar, S.; Pironio, S.; Popescu, S.; Roberts, D. (2005). "Non-local correlations as an information theoretic resource". Physical Review A. 71 (2): 022101. arXiv:quant-ph/0404097. Bibcode:2005PhRvA..71b2101B. doi:10.1103/PhysRevA.71.022101.
33. ^ Barrett, Jonathan; Pironio, Stefano (September 2005). "Popescu-Rohrlich Correlations as a Unit of Nonlocality". Physical Review Letters. 95 (14): 140401. arXiv:quant-ph/0506180. Bibcode:2005PhRvL..95n0401B. doi:10.1103/PhysRevLett.95.140401. PMID 16241631.
34. ^ Michael M. Wolf; David Perez-Garcia; Carlos Fernandez (2009). "Measurements Incompatible in Quantum Theory Cannot Be Measured Jointly in Any Other No-Signaling Theory". Physical Review Letters. 103 (23): 230402. arXiv:0905.2998. Bibcode:2009PhRvL.103w0402W. doi:10.1103/PhysRevLett.103.230402. PMID 20366131.
35. ^ Jonathan Oppenheim; Stephanie Wehner (2010). "The uncertainty principle determines the non-locality of quantum mechanics". Science. 330 (6007): 1072–1074. arXiv:1004.2507. Bibcode:2010Sci...330.1072O. doi:10.1126/science.1192065. PMID 21097930.
36. ^ van Dam, Wim (2005). "Implausible Consequences of Superstrong Nonlocality". arXiv:quant-ph/0501159.
37. ^ Pawlowski, M.; Paterek, T.; Kaszlikowski, D.; Scarani, V.; Winter, A.; Zukowski, M. (October 2009). "Information Causality as a Physical Principle". Nature. 461 (7267): 1101–1104. arXiv:0905.2292. Bibcode:2009Natur.461.1101P. doi:10.1038/nature08400. PMID 19847260.
38. ^ Pfister, Corsin; Wehner, Stephanie (2013). "If no information gain implies no disturbance, then any discrete physical theory is classical". Nature Communications. 4 (1851): 1851. arXiv:1210.0194. doi:10.1038/ncomms2821. PMID 23673636.
39. ^ Barrett, Jonathan; Hardy, Lucien; Kent, Adrian (2005). "No Signalling and Quantum Key Distribution". Physical Review Letters. 95 (1): 010503. arXiv:quant-ph/0405101. Bibcode:2005PhRvL..95a0503B. doi:10.1103/PhysRevLett.95.010503. PMID 16090597.
40. ^ Peskin, M. E., & Schroeder, D. V. (1995). An Introduction To Quantum Field Theory (Frontiers in Physics). Westview Press Incorporated. pp. 25-29