EPR paradox

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In quantum mechanics, the EPR paradox (or Einstein–Podolsky–Rosen paradox) is a thought experiment which challenged long-held ideas about the relation between the observed values of physical quantities and the values that can be accounted for by a physical theory. "EPR" stands for Einstein, Podolsky, and Rosen, who introduced the thought experiment in a 1935 paper to argue that quantum mechanics is not a complete physical theory.

According to its authors the EPR experiment yields a dichotomy. Either

  1. The result of a measurement performed on one part A of a quantum system has a non-local effect on the physical reality of another distant part B, in the sense that quantum mechanics can predict outcomes of some measurements carried out at B; or...
  2. Quantum mechanics is incomplete in the sense that some element of physical reality corresponding to B cannot be accounted for by quantum mechanics (that is, some extra variable is needed to account for it.)

As it was shown later by Bell one cannot introduce the notion of "elements of reality" without affecting the predictions of the theory. That is, one cannot complete quantum mechanics with these "elements", because this automatically leads to some logical contradictions (of the type 1=-1).

Einstein never accepted quantum mechanics as a "real" and complete theory, struggling to the end of his life for an interpretation that could comply with relativity without complying with the Heisenberg Uncertainty Principle. As he once said: "God does not play dice", skeptically referring to the Copenhagen Interpretation of quantum mechanics which says there exists no objective physical reality other than that which is revealed through measurement and observation.

The EPR paradox is a paradox in the following sense: if one adds to quantum mechanics some seemingly reasonable (but actually wrong, or questionable as a whole) conditions (referred to as locality, realism (not to be confused with philosophical realism), counterfactual definiteness, and completeness; see Bell inequality and Bell test experiments), 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.

Contents

[edit] Quantum mechanics and its interpretation

During the twentieth century, quantum theory proved to be a successful theory, which describes the physical reality of the mesoscopic and microscopic world. Up to now, no method has been found to contradict the predictions made by quantum theory. This is remarkable, since measurement accuracy has increased, and the size of the systems under consideration has decreased at a fast pace.

Quantum mechanics was developed with the aim of describing atoms and to explain the observed spectral lines in a measurement apparatus. The fact that quantum theory allows for an accurate description of reality is clear from many physical experiments and has probably never been seriously disputed. Interpretations of quantum phenomena are another story.

The question of how to interpret the mathematical formulation of quantum mechanics has given rise to a variety of different answers from people of different philosophical backgrounds.

Quantum theory and quantum mechanics do not account for single measurement outcomes in a deterministic way. According to an accepted interpretation of quantum mechanics known as the Copenhagen interpretation, a measurement causes an instantaneous collapse of the wave function describing the quantum system, and the system after the collapse is random.

The most prominent opponent of the Copenhagen interpretation was Albert Einstein. Einstein did not believe in the idea of genuine randomness in nature, the main argument in the Copenhagen interpretation. In his view, quantum mechanics is incomplete and suggests that there had to be 'hidden' variables responsible for random measurement results.

The famous paper "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?"[1], authored by Einstein, Podolsky and Rosen in 1935, condensed the philosophical discussion into a physical argument. They claim that given a specific experiment, in which the outcome of a measurement could be known before the measurement takes place, there must exist something in the real world, an "element of reality", which determines the measurement outcome. They postulate that these elements of reality are local, in the sense that they belong to a certain point in spacetime. This element may only be influenced by events which are located in the backward light cone of this point in spacetime. Even though these claims sound reasonable and convincing, they are founded on assumptions about nature which constitute what is now known as local realism.

Though the EPR paper has often been taken as an exact expression of Einstein's views, it was primarily authored by Podolsky, based on discussions at the Institute for Advanced Study with Einstein and Rosen. Einstein later expressed to Erwin Schrödinger that "It did not come out as well as I had originally wanted; rather, the essential thing was, so to speak, smothered by the formalism."[1]

[edit] 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.

[edit] Simple version

Before delving into the complicated logic that leads to the 'paradox', it is perhaps worth mentioning the simple version of the argument, as described by Greene and others, which Einstein used to show that 'hidden variables' must exist.

Two electrons are emitted from a source, by pion decay, so that their spins are opposite; one electron’s spin about any axis is the negative of the other's. Also, due to uncertainty, making a measurement of a particle’s spin about one axis disturbs the particle so you now can’t measure its spin about any other axis.

Now say you measure one electron’s spin about the x-axis. This automatically tells you the other electron’s spin about the x-axis. Since you’ve done the measurement without disturbing the other electron in any way, it can’t be that the other electron "only came to have that state when you measured it", because you didn’t measure it! It must have had that spin all along. Also (although you can’t actually do it now you’ve disturbed the electron), you could have taken the measurement about any other axis. So it follows that the other electron also had a definite spin about any other axis – much more information than the particle is capable of holding, and a "hidden variable" according to EPR.

[edit] 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 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.

Here is the crux of the matter. You might imagine that, when Bob measures the x-spin of his particle, he would get an answer with absolute certainty, since prior to this he hasn't disturbed his electron at all. But, as described above, Bob's electron has a 50% probability of producing +x and a 50% probability of -x - random behaviour, not certain. Bob's electron knows that Alice's electron has been measured, and its z-spin detected, and hence B's z-spin calculated, so its x-spin is 'out of bounds'.

Put another way, 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 (i.e. be the opposite of Alice's electron's spin about the x-axis) and also how to point if Alice measures z (i.e. behave randomly), since it is only supposed to know one thing at a time? Using the usual Copenhagen interpretation rules that say the wave function "collapses" at the time of measurement, there must be action at a distance (entanglement) or the electron must know more than it is supposed to (hidden variables).

In case the explanation above is confusing, here is the paradox summed up;

Two photons are emitted, shoot off and are measured later. Whatever axis their spins are measured along, they are always found to be opposite. This can only be explained if the photons are linked in some way. Either they were created with a definite (opposite) spin about every axis - a "hidden variable" argument - or they are linked so that one electron knows what axis the other is having its spin measured along, and becomes its opposite about that one axis - an "entanglement" argument. Moreover, if the two photons have their spins measured about different axes, once A's spin has been measured about the x-axis (and B's spin about the x-axis deduced), B's spin about the y-axis will no longer be certain, as if it knows that the measurement has taken place. Either that, or it has a definite spin already, which gives it a spin about a second axis - a hidden variable.

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.

[edit] 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; even if there are some "elements of physical reality" whose value cannot be predicted, 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 Bob decides to measure spin of his electron along the x-axis, the x-spin of Alice's electron becomes an element of physical reality after the measurement. After such measurements, the conclusion that Alice's and Bob's electrons now have definite values of spin along both the X and Z axis simultaneously is inevitable.

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.

[edit] 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.

[edit] Resolving the paradox

[edit] 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.

[edit] Bell's inequality

Main article: Bell's theorem

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 interpretation 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.

[edit] "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 \frac{3}{2}, for certain correlations. So the issue of "acceptability", up to this time mainly concerning theory (even philosophy), finally became experimentally decidable.

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.

[edit] Implications for quantum mechanics

Most physicists today believe that quantum mechanics is correct, and that the EPR paradox is a "paradox" only 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.

[edit] 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:

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 Sz are represented as

\left|+z\right\rang \leftrightarrow \begin{bmatrix}1\\0\end{bmatrix}, \quad \left|-z\right\rang \leftrightarrow \begin{bmatrix}0\\1\end{bmatrix}
With qubits it looks:
|0\rang=\begin{bmatrix}1\\0\end{bmatrix},\quad |1\rang=\begin{bmatrix}0\\1\end{bmatrix}

and the eigenstates of Sx are represented as

\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}
With qubits it looks:
|+\rang={1\over\sqrt{2}}(|0\rang+|1\rang)=\frac{1}{\sqrt{2}}\begin{bmatrix}1\\1\end{bmatrix},\quad |-\rang={1\over\sqrt{2}}(|0\rang-|1\rang)=\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\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)
With qubits it looks:
|\psi\rang=\frac{1}{\sqrt{2}}(|01\rang-|10\rang)

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\rang = \frac{1}{\sqrt{2}} \bigg(\left|+ -\right\rang - \left|- +\right\rang \bigg)
With qubits it looks:
\left|\psi\right\rang = \frac{1}{\sqrt{2}} \bigg(\left|+ -\right\rang - \left|- +\right\rang \bigg)=\frac{1}{\sqrt{2}}\bigg({1\over\sqrt{2}}(|0\rang+|1\rang){1\over\sqrt{2}}(|0\rang-|1\rang)-{1\over 2}(|0\rang-|1\rang)(|0\rang+|1\rang)\bigg)=

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

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

\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)
With qubits it looks:
|\psi\rang=-\frac{1}{\sqrt{2}}\bigg(\left|+\right\rang \otimes \left|-\right\rang - \left|-\right\rang \otimes \left|+\right\rang \bigg)=\frac{-1}{\sqrt{2}}(|10\rang-|01\rang)=\frac{1}{\sqrt{2}}(|01\rang-|10\rang)

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 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

\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 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,

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

along with the Heisenberg uncertainty relation

\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

[edit] See also

[edit] References

[edit] Selected papers

  • A. Fine, Hidden Variables, Joint Probability, and the Bell Inequalities. Phys. Rev. Lett. 48, 291 (1982).[4]
  • 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).[5]
  • 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). [6]
  • M. Smerlak, C. Rovelli, Relational EPR [7]

[edit] Notes

  1. ^ Quoted in Kaiser, David. "Bringing the human actors back on stage: the personal context of the Einstein-Bohr debate," British Journal for the History of Science 27 (1994): 129-152, on page 147.

[edit] Books

[edit] External links

Retrieved from "http://en.wikipedia.org/wiki/EPR_paradox"
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