In quantum mechanics, the EPR paradox is a thought experiment which demonstrates that the result of a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts. This runs counter to the intuition of special relativity, which states that information cannot be transmitted faster than the speed of light. "EPR" stands for Albert Einstein, Boris Podolsky, and Nathan Rosen, who introduced the thought experiment in a 1935 paper to argue that quantum mechanics is not a complete physical theory. It is sometimes referred to as the EPRB paradox for David Bohm, who converted the original thought experiment into something closer to being experimentally testable.

The EPR paradox is a paradox in the following sense: if one takes quantum mechanics and adds some seemingly reasonable conditions (referred to as "locality", "realism", and "completeness"), then one obtains a contradiction. However, quantum mechanics by itself does not appear to be internally inconsistent, nor -- as it turns out -- does it contradict relativity. As a result of further theoretical and experimental developments since the original EPR paper, most physicists today regard the EPR paradox as an illustration of how quantum mechanics violates classical intuitions, and not as an indication that quantum mechanics is fundamentally flawed.

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## 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 behaviour" (or pejoratively as "quantum weirdness"). In order to illustrate this, let us consider a simplified version of the EPR thought experiment due to 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 I and 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.

Missing image
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.) Therefore, 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.

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. Actual experimental realizations of the EPR scenario often use the polarization of photons, because it is easy to prepare and to measure.

### Reality and completeness

We will now introduce two concepts used by Einstein, Podolsky, and Rosen, 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 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.

In quantum mechanics, the x-spin and z-spin of an electron are "incompatible observables", which means that there is a Heisenberg uncertainty principle operating between them, and a quantum state cannot possess a definite value for both variables. Therefore, 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 quantum mechanics violates 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% 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.

It should be noted that 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 violate the principle of locality as defined by EPR.

## Resolving the paradox

### Hidden variables

There are several possible 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 as-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

In 1964, John Bell showed that the predictions of quantum mechanics in the EPR thought experiment are actually slightly different from the predictions of a very broad class of 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. For a detailed derivation of this result, see the article on Bell's theorem.

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, the book is not completely closed on this issue. First of all, Bell's theorem does not apply to all possible "realist" theories. It is possible to construct theories that escape its implications, and are therefore indistinguishable from quantum mechanics, though these theories are generally non-local -- they are believed to violate both causality and the rules of special relativity. 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. However, no one has ever been able to formulate a local realist theory that can reproduce all the results of quantum mechanics.

### 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. Several different conclusions can be drawn from this, depending on which interpretation of quantum mechanics one uses. In the old Copenhagen interpretation, one concludes that the principle of locality does not hold, and that instantaneous wavefunction collapse really does occur. In the many-worlds interpretation, locality is preserved, and the effects of the measurements arise from the splitting of the observers into different "histories".

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 on 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 Sx, Sy and Sz respectively, can be represented using the Pauli matrices:

[itex] 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} [itex]

where [itex]\hbar[itex] stands for Planck's constant divided by .

The eigenstates of Sz are represented as

[itex]

\left|+z\right\rang \leftrightarrow \begin{bmatrix}1\\0\end{bmatrix}, \quad \left|-z\right\rang \leftrightarrow \begin{bmatrix}0\\1\end{bmatrix} [itex]

and the eigenstates of Sx are represented as

[itex]

\left|+x\right\rang \leftrightarrow \frac{1}{\sqrt{2}} \begin{bmatrix}1\\1\end{bmatrix}, \quad \left|-x\right\rang \leftrightarrow \frac{1}{\sqrt{2}} \begin{bmatrix}1\\-1\end{bmatrix} [itex]

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

[itex]

\left|\psi\right\rang = \frac{1}{\sqrt{2}} \bigg(\left|+z\right\rang \otimes \left|-z\right\rang - \left|-z\right\rang \otimes \left|+z\right\rang \bigg) [itex]

where the two terms on the right hand side are what we have referred to as state I and state II above. From the above equations, it can be shown that the spin singlet can also be written as

[itex]

\left|\psi\right\rang = \frac{-1}{\sqrt{2}} \bigg(\left|+x\right\rang \otimes \left|-x\right\rang - \left|-x\right\rang \otimes \left|+x\right\rang \bigg) [itex]

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 the violation of local realism, we need to show that after Alice's measurement of Sz (or Sx), Bob's value of Sz (or Sx) is uniquely determined, and therefore corresponds to an "element of physical reality". This follows from the principles of measurement in quantum mechanics. When Sz is measured, the system state ψ collapses into an eigenvector of Sz. 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

[itex] \left| +z \right\rangle \otimes \left| \phi\right\rangle \quad \phi \in H [itex]

For the spin singlet, the new state is

[itex] \left| +z \right\rangle \otimes \left| -z \right\rangle. [itex]

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

[itex] \left| -z \right\rangle \otimes \left| \phi\right\rangle \quad \phi \in H [itex]

which means that the new state is

[itex] \left|-z\right\rangle \otimes \left|+z\right\rangle [itex]

This implies that the measurement for Sz 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 Sx and Sz 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,

[itex]

\left[ S_x, S_z\right] = - i\hbar S_y \ne 0 [itex]

along with the Heisenberg uncertainty relation

[itex]

\lang (\Delta S_x)^2 \rang \lang (\Delta S_z)^2 \rang \ge \frac{1}{4} \left|\lang \left[ S_x, S_z\right] \rang \right|^2 [itex]

## References

### Selected papers

• A. Aspect, Bell's inequality test: more ideal than ever, Nature 398 189 (1999). [1] (http://www-ece.rice.edu/~kono/ELEC565/Aspect_Nature.pdf)
• 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).
• 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? (http://www.drchinese.com/David/EPR.pdf) Phys. Rev. 47 777 (1935). [2] (http://prola.aps.org/abstract/PR/v47/i10/p777_1)
• A. Fine, Hidden Variables, Joint Probability, and the Bell Inequalities. Phys. Rev. Lett 48, 291 (1982).
• 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 2 particles without inequalities for almost all entangled states. Phys. Rev. Lett. 71 1665 (1993).
• M. Mizuki, A classical interpretation of Bell's inequality. Annales de la Fondation Louis de Broglie 26 683 (2001).

### Books

• J.S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, 1987). ISBN 0521368693
• J.J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994), pp. 174-187, 223-232. ISBN 0201539292
• F. Selleri, Quantum Mechanics Versus Local Realism: The Einstein-Podolsky-Rosen Paradox (Plenum Press, New York, 1988)

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