利用者:原子力の熊/アインシュタイン＝ポドルスキー＝ローゼンのパラドックス

アインシュタイン＝ポドルスキー＝ローゼンのパラドックス(EPRのパラドックス)は、量子力学が完全な理論であるか否かを巡る思考実験であり、1935年に発表された。アルベルト・アインシュタイン、ボリス・ポドルスキーならびにネイサン・ローゼンは、例えばニュートン力学は物体の運動が十分に低速であるという近似の下にのみ成立するように、量子力学もまた何らかの近似の下でのみ成立する理論であると考えた。その論拠として、もし量子力学が完全に、つまり近似なしに、正しいならば、特殊相対性理論との間に矛盾が生じることを示したのである。ただし、アインシュタインらの論理は、物理量は常に物理的実在と結びつけられていなければならない、という仮定に基づいている。この「物理的実在」という語は、厳密には定義されていない。

量子力学とコペンハーゲン解釈

20世紀に提唱された量子力学は、それまでの理論では説明できなかった様々な現象に対し十分な説明を加えることができた。現在に至るまで、量子力学と矛盾する実験事実は報告されていない。このことから、大半の物理学者は量子力学は正しいと考えており、アインシュタインらも量子力学が間違っているとは述べていない。しかし、量子力学の解釈については様々な意見がある。

量子力学では、日常的な感覚では理解できない事象が、数式の上では明確に説明される。この典型例が二重スリット実験である。この実験では、電子がある点から発射され、二本のスリットを通り抜け、スクリーン上に投影される。日常感覚でいえば、電子は二本のスリットの「どちらか一方」を通過するはずである。しかし、それではスクリーン上に干渉縞が現れる理由を説明できない。この現象は、量子力学では波動関数が干渉を起こすためだと数学的には説明される。これに物理的解釈を加えるのが、量子力学の解釈問題である。

広く受け入れられているコペンハーゲン解釈では、次のように説明される。電子が存在している位置は、測定が行われるまでは定まっていない。これは「我々には分からない」という意味ではなく、本当に定まっていない。しかし、測定を行なうと、直ちに決まる。二重スリット実験でいえば、「電子がどちらのスリットを通ったか」ということは決まっておらず、スクリーン上に到達する時に、初めて電子の位置は定まる。

アインシュタインは、コペンハーゲン解釈に強く反対した。彼は、スクリーンに到達するまで電子の位置が定まっていないように見えるのは、電子の運動に影響を与える未知の変数、すなわち隠れた変数が存在するからだと考えた。

完全な理論と物理的実在

アインシュタインらは、1935年の 論文で「完全な理論には、全ての物理的実在に対応する要素が含まれていなければならない」と仮定した。これは、例えば、熱力学には個々の分子という物理的実在に相当する要素を含んでいないから不完全である、ということである。また、原理として「もし、体系をかき乱すことなく、ある物理量を確実に予言することが可能であるならば、その物理量に対応する物理的実在が存在する」とした。つまり、コペンハーゲン解釈が正しいならば、電子の位置に対応する物理的実在は存在しないことになる。電子の位置を測定すれば波束が収縮し、体系がかき乱されるからである。

この考え方には異論もある。実証主義の立場からは、そもそも「物理的実在」という概念事態が曖昧で無意味である。あるいは、そもそも量子力学と矛盾する実験事実が存在しないのだから、量子力学が完全か不完全かという議論は無益である、とも批判される。ただし、後者の批判に対してはジェームズ・クラーク・マクスウェルの例が反論として挙げられる。マクスウェルの時代には電磁波の存在は知られていなかったが、それまでに知られていた理論から、実験的には確認されていない「電場と磁場が作り出す波」の存在可能性を指摘したのである。これと同様のことは、量子力学についても起こり得ると考えられる。

なお、この論文はしばしばアインシュタインの考えと見なされているが、実際にはプリンストン高等研究所におけるアインシュタインおよびローゼンとの議論をポドルスキーがまとめたものである。後にアインシュタインがエルヴィン・シュレーディンガーに述べたところによれば、アインシュタインが本当に言いたかったこととは、少しばかり違うとのことである。

Description of the paradox

The EPR paradox draws on a phenomenon predicted by quantum mechanics, known as quantum entanglement, to show that measurements performed on spatially separated parts of a quantum system can apparently have an instantaneous influence on one another.
This effect is now known as "nonlocal behavior" (or colloquially as "quantum weirdness" or "spooky action at a distance"). In order to illustrate this, let us consider a simplified version of the EPR thought experiment put forth by David Bohm.

Measurements on an entangled state

We have a source that emits pairs of electrons, with one electron sent to destination A, where there is an observer named Alice, and another is sent to destination B, where there is an observer named Bob. According to quantum mechanics, we can arrange our source so that each emitted electron pair occupies a quantum state called a spin singlet. This can be viewed as a quantum superposition of two states, which we call state I and state II. In state I, electron A has spin pointing upward along the z-axis (+z) and electron B has spin pointing downward along the z-axis (-z). In state II, electron A has spin -z and electron B has spin +z. Therefore, it is impossible to associate either electron in the spin singlet with a state of definite spin. The electrons are thus said to be entangled.

The EPR thought experiment, performed with electrons. A source (center) sends electrons toward two observers, Alice (left) and Bob (right), who can perform spin measurements.

Alice now measures the spin along the z-axis. She can obtain one of two possible outcomes: +z or -z. Suppose she gets +z. According to quantum mechanics, the quantum state of the system collapses into state I. (Different interpretations of quantum mechanics have different ways of saying this, but the basic result is the same.) The quantum state determines the probable outcomes of any measurement performed on the system. In this case, if Bob subsequently measures spin along the z-axis, he will obtain -z with 100% probability. Similarly, if Alice gets -z, Bob will get +z.

There is, of course, nothing special about our choice of the z-axis. For instance, suppose that Alice and Bob now decide to measure spin along the x-axis, according to quantum mechanics, the spin singlet state may equally well be expressed as a superposition of spin states pointing in the x direction. We'll call these states Ia and IIa. In state Ia, Alice's electron has spin +x and Bob's electron has spin -x. In state IIa, Alice's electron has spin -x and Bob's electron has spin +x. Therefore, if Alice measures +x, the system collapses into Ia, and Bob will get -x. If Alice measures -x, the system collapses into IIa, and Bob will get +x.

In quantum mechanics, the x-spin and z-spin are "incompatible observables", which means that there is a Heisenberg uncertainty principle operating between them: a quantum state cannot possess a definite value for both variables. Suppose Alice measures the z-spin and obtains +z, so that the quantum state collapses into state I. Now, instead of measuring the z-spin as well, Bob measures the x-spin. According to quantum mechanics, when the system is in state I, Bob's x-spin measurement will have a 50% probability of producing +x and a 50% probability of -x. Furthermore, it is fundamentally impossible to predict which outcome will appear until Bob actually performs the measurement.

So how does Bob's electron know, at the same time, which way to point if Alice decides (based on information unavailable to Bob) to measure x and also how to point if Alice measures z? Using the usual Copenhagen interpretation rules that say the wave function "collapses" at the time of measurement, there must be action at a distance or the electron must know more than it is supposed to. To make the mixed part quantum and part classical descriptions of this experiment local, we have to say that the notebooks (and experimenters) are entangled and have linear combinations of + and – written in them, like Schrödinger's Cat.

Incidentally, although we have used spin as an example, many types of physical quantities — what quantum mechanics refers to as "observables" — can be used to produce quantum entanglement. The original EPR paper used momentum for the observable. Experimental realizations of the EPR scenario often use photon polarization, because polarized photons are easy to prepare and measure.

Reality and completeness

We will now introduce two concepts used by Einstein, Podolsky, and Rosen (EPR), which are crucial to their attack on quantum mechanics: (i) the elements of physical reality and (ii) the completeness of a physical theory.

The authors (EPR) did not directly address the philosophical meaning of an "element of physical reality". Instead, they made the assumption that if the value of any physical quantity of a system can be predicted with absolute certainty prior to performing a measurement or otherwise disturbing it, then that quantity corresponds to an element of physical reality. Note that the converse is not assumed to be true; there may be other ways for elements of physical reality to exist, but this will not affect the argument.

Next, EPR defined a "complete physical theory" as one in which every element of physical reality is accounted for. The aim of their paper was to show, using these two definitions, that quantum mechanics is not a complete physical theory.

Let us see how these concepts apply to the above thought experiment. Suppose Alice decides to measure the value of spin along the z-axis (we'll call this the z-spin.) After Alice performs her measurement, the z-spin of Bob's electron is definitely known, so it is an element of physical reality. Similarly, if Alice decides to measure spin along the x-axis, the x-spin of Bob's electron is an element of physical reality after her measurement.

We have seen that a quantum state cannot possess a definite value for both x-spin and z-spin. If quantum mechanics is a complete physical theory in the sense given above, x-spin and z-spin cannot be elements of reality at the same time. This means that Alice's decision — whether to perform her measurement along the x- or z-axis — has an instantaneous effect on the elements of physical reality at Bob's location. However, this violates another principle, that of locality.

Locality in the EPR experiment

The principle of locality states that physical processes occurring at one place should have no immediate effect on the elements of reality at another location. At first sight, this appears to be a reasonable assumption to make, as it seems to be a consequence of special relativity, which states that information can never be transmitted faster than the speed of light without violating causality. It is generally believed that any theory which violates causality would also be internally inconsistent, and thus deeply unsatisfactory.

It turns out that the usual rules for combining quantum mechanical and classical descriptions violate the principle of locality without violating causality. Causality is preserved because there is no way for Alice to transmit messages (i.e. information) to Bob by manipulating her measurement axis. Whichever axis she uses, she has a 50% probability of obtaining "+" and 50% probability of obtaining "-", completely at random; according to quantum mechanics, it is fundamentally impossible for her to influence what result she gets. Furthermore, Bob is only able to perform his measurement once: there is a fundamental property of quantum mechanics, known as the "no cloning theorem", which makes it impossible for him to make a million copies of the electron he receives, perform a spin measurement on each, and look at the statistical distribution of the results. Therefore, in the one measurement he is allowed to make, there is a 50% probability of getting "+" and 50% of getting "-", regardless of whether or not his axis is aligned with Alice's.

However, the principle of locality appeals powerfully to physical intuition, and Einstein, Podolsky and Rosen were unwilling to abandon it. Einstein derided the quantum mechanical predictions as "spooky action at a distance". The conclusion they drew was that quantum mechanics is not a complete theory.

In recent years, however, doubt has been cast on EPR's conclusion due to developments in understanding locality and especially quantum decoherence. The word locality has several different meanings in physics. For example, in quantum field theory "locality" means that quantum fields at different points of space do not interact with one another. However, quantum field theories that are "local" in this sense appear to violate the principle of locality as defined by EPR, but they nevertheless do not violate locality in a more general sense. Wavefunction collapse can be viewed as an epiphenomenon of quantum decoherence, which in turn is nothing more than an effect of the underlying local time evolution of the wavefunction of a system and all of its environment. Since the underlying behaviour doesn't violate local causality, it follows that neither does the additional effect of wavefunction collapse, whether real or apparent. Therefore, as outlined in the example above, neither the EPR experiment nor any quantum experiment demonstrates that faster-than-light signaling is possible.

Resolving the paradox

Hidden variables

There are several ways to resolve the EPR paradox. The one suggested by EPR is that quantum mechanics, despite its success in a wide variety of experimental scenarios, is actually an incomplete theory. In other words, there is some yet undiscovered theory of nature to which quantum mechanics acts as a kind of statistical approximation (albeit an exceedingly successful one). Unlike quantum mechanics, the more complete theory contains variables corresponding to all the "elements of reality". There must be some unknown mechanism acting on these variables to give rise to the observed effects of "non-commuting quantum observables", i.e. the Heisenberg uncertainty principle. Such a theory is called a hidden variable theory.

To illustrate this idea, we can formulate a very simple hidden variable theory for the above thought experiment. One supposes that the quantum spin-singlet states emitted by the source are actually approximate descriptions for "true" physical states possessing definite values for the z-spin and x-spin. In these "true" states, the electron going to Bob always has spin values opposite to the electron going to Alice, but the values are otherwise completely random. For example, the first pair emitted by the source might be "(+z, -x) to Alice and (-z, +x) to Bob", the next pair "(-z, -x) to Alice and (+z, +x) to Bob", and so forth. Therefore, if Bob's measurement axis is aligned with Alice's, he will necessarily get the opposite of whatever Alice gets; otherwise, he will get "+" and "-" with equal probability.

Assuming we restrict our measurements to the z and x axes, such a hidden variable theory is experimentally indistinguishable from quantum mechanics. In reality, of course, there is an (uncountably) infinite number of axes along which Alice and Bob can perform their measurements, so there has to be an infinite number of independent hidden variables. However, this is not a serious problem; we have formulated a very simplistic hidden variable theory, and a more sophisticated theory might be able to patch it up. It turns out that there is a much more serious challenge to the idea of hidden variables.

Bell's inequality

→詳細は「Bell's theorem」を参照

In 1964, John Bell showed that the predictions of quantum mechanics in the EPR thought experiment are significantly different from the predictions of a very broad class of hidden variable theories (the local hidden variable theories). Roughly speaking, quantum mechanics predicts much stronger statistical correlations between the measurement results performed on different axes than the hidden variable theories. These differences, expressed using inequality relations known as "Bell's inequalities", are in principle experimentally detectable. Later work by Eberhard showed that the key properties of local hidden variable theories that lead to Bell's inequalities are locality and counter-factual definiteness. Any theory in which these principles hold produces the inequalities. A. Fine subsequently showed that any theory satisfying the inequalities can be modeled by a local hidden variable theory.

After the publication of Bell's paper, a variety of experiments were devised to test Bell's inequalities. (As mentioned above, these experiments generally rely on photon polarization measurements.) All the experiments conducted to date have found behavior in line with the predictions of standard quantum mechanics.

However, Bell's theorem does not apply to all possible philosophically realist theories, although a common misconception touted by new agers is that quantum mechanics is inconsistent with all notions of philosophical realism. Realist interpretations of quantum mechanics are possible, although as discussed above, such interpretations must reject either locality or counter-factual definiteness. Mainstream physics prefers to keep locality while still maintaining a notion of realism that nevertheless rejects counter-factual definiteness. Examples of such mainstream realist interpretations are the consistent histories interepretation and the transactional interpretation. Fine's work showed that taking locality as a given there exist scenarios in which two statistical variables are correlated in a manner inconsistent with counter-factual definiteness and that such scenarios are no more mysterious than any other despite the inconsistency with counter-factual definiteness seeming 'counter-intuitive'. Violation of locality however is difficult to reconcile with special relativity and is thought to be incompatible with the principle of causality. On the other hand the Bohm interpretation of quantum mechanics instead keeps counter-factual definiteness while introducing a conjectured non-local mechanism called the 'quantum potential'. Some workers in the field have also attempted to formulate hidden variable theories that exploit loopholes in actual experiments, such as the assumptions made in interpreting experimental data although no such theory has been produced that can reproduce all the results of quantum mechanics.

There are also individual EPR-like experiments that have no local hidden variables explanation. Examples have been suggested by David Bohm and by Lucien Hardy.

"Acceptable theories", and the experiment

According to the present view of the situation, quantum mechanics simply contradicts Einstein's philosophical postulate that any acceptable physical theory should fulfill "local realism".

In the EPR paper (1935) the authors realized that quantum mechanics was non-acceptable in the sense of their above-mentioned assumptions, and Einstein thought erroneously that it could simply be augmented by 'hidden variables', without any further change, to get an acceptable theory. He pursued these ideas until the end of his life (1955), i.e. over twenty years.

In contrast, John Bell, in his 1964 paper, showed "once and for all" that quantum mechanics and Einstein's assumptions lead to different results, different by a factor of ${\sqrt {2}}$, for certain correlations. So the issue of "acceptability", up to this time mainly concerning theory (even philosophy), finally became experimentally decisable.

There are many Bell test experiments hitherto, e.g. those of Alain Aspect and others. They all show that pure quantum mechanics, and not  Einstein's "local realism", is acceptable. Thus, according to Karl Popper these experiments falsify Einstein's philosophical assumptions, especially the ideas on "hidden variables", whereas quantum mechanics itself remains a good candidate for a theory, which is acceptable in a wider context.

But apparently an experiment, which would also classify Bohm's non-local quasi-classical theory as non-acceptable, is still lacking.

Implications for quantum mechanics

Most physicists today believe that quantum mechanics is correct, and that the EPR paradox is only a "paradox" because classical intuitions do not correspond to physical reality. How EPR is interpreted regarding locality depends on the interpretation of quantum mechanics one uses. In the Copenhagen interpretation, it is usually understood that instantaneous wavefunction collapse does occur. However, the view that there is no causal instantaneous effect has also been proposed within the Copenhagen interpretation: in this alternate view, measurement affects our ability to define (and measure) quantities in the physical system, not the system itself. In the many-worlds interpretation, a kind of locality is preserved, since the effects of irreversible operations such as measurement arise from the relativization of a global state to a subsystem such as that of an observer.

The EPR paradox has deepened our understanding of quantum mechanics by exposing the fundamentally non-classical characteristics of the measurement process. Prior to the publication of the EPR paper, a measurement was often visualized as a physical disturbance inflicted directly upon the measured system. For instance, when measuring the position of an electron, one imagines shining a light on it, thus disturbing the electron and producing the quantum mechanical uncertainties in its position. Such explanations, which are still encountered in popular expositions of quantum mechanics, are debunked by the EPR paradox, which shows that a "measurement" can be performed on a particle without disturbing it directly, by performing a measurement on a distant entangled particle.

Technologies relying on quantum entanglement are now being developed. In quantum cryptography, entangled particles are used to transmit signals that cannot be eavesdropped upon without leaving a trace. In quantum computation, entangled quantum states are used to perform computations in parallel, which may allow certain calculations to be performed much more quickly than they ever could be with classical computers.

Mathematical formulation

The above discussion can be expressed mathematically using the quantum mechanical formulation of spin. The spin degree of freedom for an electron is associated with a two-dimensional Hilbert space H, with each quantum state corresponding to a vector in that space. The operators corresponding to the spin along the x, y, and z direction, denoted S_x, S_y, and S_z respectively, can be represented using the Pauli matrices:

$S_{x}={\frac {\hbar }{2}}{\begin{bmatrix}0&1\\1&0\end{bmatrix}},\quad S_{y}={\frac {\hbar }{2}}{\begin{bmatrix}0&-i\\i&0\end{bmatrix}},\quad S_{z}={\frac {\hbar }{2}}{\begin{bmatrix}1&0\\0&-1\end{bmatrix}}$

where $\hbar$ stands for Planck's constant divided by 2π.

The eigenstates of S_z are represented as

$\left|+z\right\rangle \leftrightarrow {\begin{bmatrix}1\\0\end{bmatrix}},\quad \left|-z\right\rangle \leftrightarrow {\begin{bmatrix}0\\1\end{bmatrix}}$

With qubits it looks:

$|0\rangle ={\begin{bmatrix}1\\0\end{bmatrix}},\quad |1\rangle ={\begin{bmatrix}0\\1\end{bmatrix}}$

and the eigenstates of S_x are represented as

$\left|+x\right\rangle \leftrightarrow {\frac {1}{\sqrt {2}}}{\begin{bmatrix}1\\1\end{bmatrix}},\quad \left|-x\right\rangle \leftrightarrow {\frac {1}{\sqrt {2}}}{\begin{bmatrix}1\\-1\end{bmatrix}}$

With qubits it looks:

$|+\rangle ={1 \over {\sqrt {2}}}(|0\rangle +|1\rangle )={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1\\1\end{bmatrix}},\quad |-\rangle ={1 \over {\sqrt {2}}}(|0\rangle -|1\rangle )={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1\\-1\end{bmatrix}}$

The Hilbert space of the electron pair is $H\otimes H$, the tensor product of the two electrons' Hilbert spaces. The spin singlet state is

$\left|\psi \right\rangle ={\frac {1}{\sqrt {2}}}{\bigg (}\left|+z\right\rangle \otimes \left|-z\right\rangle -\left|-z\right\rangle \otimes \left|+z\right\rangle {\bigg )}$

With qubits it looks:

$|\psi \rangle ={\frac {1}{\sqrt {2}}}(|01\rangle -|10\rangle )$

where the two terms on the right hand side are what we have referred to as state I and state II above. This is also commonly written as

$\left|\psi \right\rangle ={\frac {1}{\sqrt {2}}}{\bigg (}\left|+-\right\rangle -\left|-+\right\rangle {\bigg )}$

With qubits it looks:

$\left|\psi \right\rangle ={\frac {1}{\sqrt {2}}}{\bigg (}\left|+-\right\rangle -\left|-+\right\rangle {\bigg )}={\frac {1}{\sqrt {2}}}{\bigg (}{1 \over {\sqrt {2}}}(|0\rangle +|1\rangle ){1 \over {\sqrt {2}}}(|0\rangle -|1\rangle )-{1 \over 2}(|0\rangle -|1\rangle )(|0\rangle +|1\rangle ){\bigg )}=$

$={\frac {1}{\sqrt {2}}}({1 \over 2}(|00\rangle -|01\rangle +|10\rangle -|11\rangle )-{1 \over 2}(|00\rangle +|01\rangle -|10\rangle -|11\rangle ))={\frac {1}{\sqrt {2}}}(|10\rangle -|01\rangle )$

From the above equations, it can be shown that the spin singlet can also be written as

$\left|\psi \right\rangle ={\frac {-1}{\sqrt {2}}}{\bigg (}\left|+x\right\rangle \otimes \left|-x\right\rangle -\left|-x\right\rangle \otimes \left|+x\right\rangle {\bigg )}$

With qubits it looks:

$|\psi \rangle =-{\frac {1}{\sqrt {2}}}{\bigg (}\left|+\right\rangle \otimes \left|-\right\rangle -\left|-\right\rangle \otimes \left|+\right\rangle {\bigg )}={\frac {-1}{\sqrt {2}}}(|10\rangle -|01\rangle )={\frac {1}{\sqrt {2}}}(|01\rangle -|10\rangle )$

where the terms on the right hand side are what we have referred to as state Ia and state IIa.

To illustrate how this leads to the violation of local realism, we need to show that after Alice's measurement of S_z (or S_x), Bob's value of S_z (or S_x) is uniquely determined, and therefore corresponds to an "element of physical reality". This follows from the principles of measurement in quantum mechanics. When S_z is measured, the system state ψ collapses into an eigenvector of S_z. If the measurement result is +z, this means that immediately after measurement the system state undergoes an orthogonal projection of ψ onto the space of states of the form

$\left|+z\right\rangle \otimes \left|\phi \right\rangle \quad \phi \in H$

With qubits it looks:

$\left|0\right\rangle \otimes \left|\phi \right\rangle \quad \phi \in H$

For the spin singlet, the new state is

$\left|+z\right\rangle \otimes \left|-z\right\rangle .$

With qubits it looks:

$\left|0\right\rangle \otimes \left|1\right\rangle .$

Similarly, if Alice's measurement result is -z, a system undergoes an orthogonal projection onto

$\left|-z\right\rangle \otimes \left|\phi \right\rangle \quad \phi \in H$

With qubits it looks:

$\left|1\right\rangle \otimes \left|\phi \right\rangle \quad \phi \in H$

which means that the new state is

$\left|-z\right\rangle \otimes \left|+z\right\rangle$

With qubits it looks:

$\left|1\right\rangle \otimes \left|0\right\rangle$

This implies that the measurement for S_z for Bob's electron is now determined. It will be -z in the first case or +z in the second case.

It remains only to show that S_x and S_z cannot simultaneously possess definite values in quantum mechanics. One may show in a straightforward manner that no possible vector can be an eigenvector of both matrices. More generally, one may use the fact that the operators do not commute,

$\left[S_{x},S_{z}\right]=-i\hbar S_{y}\neq 0$

along with the Heisenberg uncertainty relation

$\langle (\Delta S_{x})^{2}\rangle \langle (\Delta S_{z})^{2}\rangle \geq {\frac {1}{4}}\left|\langle \left[S_{x},S_{z}\right]\rangle \right|^{2}$

See also

• Bell test experiments
• Bell state
• Bell's theorem
• CHSH Bell test
• Counter-factual definiteness
• Local hidden variable theory
• Quantum teleportation
• Synchronicity
• Popper's experiment

References

Selected papers

• A. Aspect, Bell's inequality test: more ideal than ever, Nature 398 189 (1999). [2]
• J.S. Bell, On the Einstein-Poldolsky-Rosen paradox, Physics 1 195 (1964).
• J.S. Bell, Bertlmann's Socks and the Nature of Reality. Journal de Physique 42 (1981).
• N. Bohr, Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev. 48, 696 (1935) [3]
• P.H. Eberhard, Bell's theorem without hidden variables. Nuovo Cimento 38B1 75 (1977).
• P.H. Eberhard, Bell's theorem and the different concepts of locality. Nuovo Cimento 46B 392 (1978).
• A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47 777 (1935). [4]

• A. Fine, Hidden Variables, Joint Probability, and the Bell Inequalities. Phys. Rev. Lett. 48, 291 (1982).[5]
• A. Fine, Do Correlations need to be explained?, in Philosophical Consequences of Quantum Theory: Reflections on Bell's Theorem, edited by Cushing & McMullin (University of Notre Dame Press, 1986).
• L. Hardy, Nonlocality for two particles without inequalities for almost all entangled states. Phys. Rev. Lett. 71 1665 (1993).[6]
• M. Mizuki, A classical interpretation of Bell's inequality. Annales de la Fondation Louis de Broglie 26 683 (2001).
• P. Pluch, "Theory for Quantum Probability", PhD Thesis University of Klagenfurt (2006)
• M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe and D. J. Wineland, Experimental violation of a Bell's inequality with efficient detection, Nature 409, 791-794 (15 February 2001). [7]
• M. Smerlak, C. Rovelli, Relational EPR [8]

Books

• J.S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, 1987). ISBN 0-521-36869-3
• J.J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994), pp. 174-187, 223-232. ISBN 0-201-53929-2
• F. Selleri, Quantum Mechanics Versus Local Realism: The Einstein-Podolsky-Rosen Paradox (Plenum Press, New York, 1988) ISBN 0-306-42739-7
• Roger Penrose, The Road to Reality (Alfred A. Knopf, 2005; Vintage Books, 2006 )

External links

• The original EPR paper
• A. Fine, The Einstein-Podolsky-Rosen Argument in Quantum Theory
• Abner Shimony, Bell’s Theorem (2004)
• EPR, Bell & Aspect: The Original References
• Does Bell's Inequality Principle rule out local theories of quantum mechanics? From the Usenet Physics FAQ.
• Theoretical use of EPR in teleportation
• Effective use of EPR in cryptography