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The EPR paradox is an early and influential critique leveled against quantum mechanics. Albert Einstein and his colleagues Boris Podolsky and Nathan Rosen (known collectively as EPR) designed a thought experiment intended to reveal what they believed to be inadequacies of quantum mechanics. To that end they pointed to a consequence of quantum mechanics that its supporters had not noticed.

According to quantum mechanics, a single system has its own wave function, its own unitary quantum-theoretical description. If such a single system can be transformed into two individual systems, doing so does not create two wave functions. Instead, theory indicates that each system shares a single wave function.

It was known from experiments that the outcome of an experiment sometimes cannot be uniquely predicted. An example of such indeterminacy can be seen when a beam of light is incident on a half-silvered mirror. One half of the beam will reflect, the other will pass. But what happens when we keep decreasing the intensity of the beam, so that only one photon is in transit at any time? Half of the photons will pass and another half will be reflected. Even if we 'prepare' the photons by passing them through a polarizer, there will always be an experiment of which the result could not be predicted with certainty.

The routine explanation of this effect was, at that time, provided by Heisenberg's uncertainty principle. Physical quantities come in pairs which are called Conjugate quantities. Example of such a conjugate pair are position and momentum of a particle, or components of spin measured around different axes. When one quantity was measured, and became determined, the conjugated quantity became indeterminate. Heisenberg explained this as a disturbance caused by measurement.

The EPR paper written in 1936 has shown that this explanation is inadequate. It considered two entangled particles, let's call them A and B, and pointed out measuring a quantity of a particle A will cause the conjugated quantity of particle B to become undetermined, even if there was no contact, no classical disturbance.

Heisenberg's principle was an attempt to provide a classical explanation of a quantum effect we call non-locality. There were two possible explanations. Either there was some interaction between the particles, even though they were separated, or the information about the outcome of all possible measurements was already present in both particles.

The EPR authors preferred the second explanation according to which that information was encoded in some 'hidden parameters'. The first explanation, that an effect propagated instantly, across a distance, was (and is) in conflict with the theory of relativity. However, as later experiments and Bell's theorem demonstrated, their preferred explanation was not viable.

They then concluded that quantum mechanics was incomplete since, in its formalism, there was no space for such hidden parameters. They would both be determinate values, not just one of them as indicated by quantum mechanics. If the two values of the remote, undisturbed, system were real, then they must have been real all along and not determined by the act of measurement. The act of measurement might well disturb and change subsequent values of the system measured, but that fact did not deny that there must have been something real there to be measured all along. In short, they gave reason to believe that the second, undisturbed, system had a real and definite position, and a real and definite momentum, and that therefore the first system must also have had a real and definite position, and a real and definite momentum waiting there for the experimenter to disturb and change. However, quantum mechanics could not provide a theoretical description or prediction of these values, and so must be held to be incomplete.

History of EPR developments

The article that first brought forth these matters, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" was published in 1935.[1] Einstein struggled to the end of his life for a theory that could better comply with his idea of causality, protesting against the view that there exists no objective physical reality other than that which is revealed through measurement interpreted in terms of quantum mechanical formalism. However, since Einstein's death, experiments analogous to the one described in the EPR paper have been carried out, starting in 1976 by French scientists Lamehi-Rachti and Mittig[2] at the Saclay Nuclear Research Centre. These experiments appear to show that the local realism idea is false,[3] thereby supporting the position of Bohr et al., against the challenge from Einstein and his group.
Quantum mechanics and its interpretation
Main article: Interpretations of quantum mechanics

Since the early twentieth century, quantum theory has proved to be successful in describing accurately the physical reality of the mesoscopic and microscopic world, in multiple reproducible physics experiments.

Quantum mechanics was developed with the aim of describing atoms and explaining the observed spectral lines in a measurement apparatus. Although disputed, it has yet to be seriously challenged. Philosophical interpretations of quantum phenomena, however, are another matter: 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 persuasions (see Interpretation of quantum mechanics).

Quantum theory and quantum mechanics do not provide single measurement outcomes in a deterministic way. According to the understanding of quantum mechanics known as the Copenhagen interpretation, measurement causes an instantaneous collapse of the wave function describing the quantum system into an eigenstate of the observable state that was measured. Einstein characterized this imagined collapse in the 1927 Solvay Conference. He presented a thought experiment in which electrons are introduced through a small hole in a sphere whose inner surface serves as a detection screen. The electrons will contact the spherical detection screen in a widely dispersed manner. Those electrons, however, are all individually described by wave fronts that expand in all directions from the point of entry. A wave as it is understood in everyday life would paint a large area of the detection screen, but the electrons would be found to impact the screen at single points and would eventually form a pattern in keeping with the probabilities described by their identical wave functions. Einstein asks what makes each electron's wave front "collapse" at its respective location. Why do the electrons appear as single bright scintillations rather than as dim washes of energy across the surface? Why does any single electron appear at one point rather than some alternative point? The behavior of the electrons gives the impression of some signal having been sent to all possible points of contact that would have nullified all but one of them, or, in other words, would have preferentially selected a single point to the exclusion of all others.[4]

The most prominent opponent of the Copenhagen interpretation is Albert Einstein. In his view, quantum mechanics is incomplete. Commenting on this, other writers (such as John von Neumann[5] and David Bohm[6]) have suggested that consequently there would have to be 'hidden' variables responsible for random measurement results, something which was not expressly claimed in the original paper.

That 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 is known before the measurement takes place, there must exist something in the real world, an "element of reality", that determines the measurement outcome. They postulate that these elements of reality are local, in the sense that each belongs to a certain point in spacetime. Each element may only be influenced by events which are located in the backward light cone of its point in spacetime (i.e. the past). These claims are founded on assumptions about nature that 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."[7] In 1936 Einstein presented an individual account of his local realist ideas.[8]
Description of the paradox

The original EPR paradox challenges the prediction of quantum mechanics that it is impossible to know both the position and the momentum of a quantum particle. This challenge can be extended to other pairs of physical properties.
EPR paper

The original paper purports to describe what must happen to "two systems I and II, which we permit to interact ...", and, after some time, "we suppose that there is no longer any interaction between the two parts." In the words of Kumar (2009), the EPR description involves "two particles, A and B, [which] interact briefly and then move off in opposite directions."[9] According to Heisenberg's uncertainty principle, it is impossible to measure both the momentum and the position of particle B exactly. However, according to Kumar, it is possible to measure the exact position of particle A. By calculation, therefore, with the exact position of particle A known, the exact position of particle B can be known. Also, the exact momentum of particle A can be measured, so the exact momentum of particle B can be worked out. Kumar writes: "EPR argued that they had proved that ... [particle] B can have simultaneously exact values of position and momentum. ... Particle B has a position that is real and a momentum that is real."

EPR appeared to have contrived a means to establish the exact values of either the momentum or the position of B due to measurements made on particle A, without the slightest possibility of particle B being physically disturbed.[10]

EPR tried to set up a paradox to question the range of true application of Quantum Mechanics: Quantum theory predicts that both values cannot be known for a particle, and yet the EPR thought experiment purports to show that they must all have determinate values. The EPR paper says: "We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete."[11]

The EPR paper ends by saying:

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.

Measurements on an entangled state

We have a source that emits electron-positron pairs, with the electron sent to destination A, where there is an observer named Alice, and the positron sent to destination B, where there is an observer named Bob. According to quantum mechanics, we can arrange our source so that each emitted pair occupies a quantum state called a spin singlet. The particles are thus said to be entangled. This can be viewed as a quantum superposition of two states, which we call state I and state II. In state I, the electron has spin pointing upward along the z-axis (+z) and the positron has spin pointing downward along the z-axis (-z). In state II, the electron has spin -z and the positron has spin +z. Therefore, it is impossible (without measuring) to know the definite state of spin of either particle in the spin singlet.
The EPR thought experiment, performed with electron-positron pairs. A source (center) sends particles toward two observers, electrons to Alice (left) and positrons to 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 the Copenhagen interpretation of quantum mechanics, the quantum state of the system collapses into state I. 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, there is 100% probability that he will obtain -z. Similarly, if Alice gets -z, Bob will get +z.

There is, of course, nothing special about choosing the z-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. Suppose that Alice and Bob had decided to measure spin along the x-axis. We'll call these states Ia and IIa. In state Ia, Alice's electron has spin +x and Bob's positron has spin -x. In state IIa, Alice's electron has spin -x and Bob's positron has spin +x. Therefore, if Alice measures +x, the system 'collapses' into state Ia, and Bob will get -x. If Alice measures -x, the system collapses into state IIa, and Bob will get +x.

Whatever axis their spins are measured along, they are always found to be opposite. This can only be explained if the particles 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 "feels" which axis the other is having its spin measured along, and becomes its opposite about that one axis—an "entanglement" argument. Moreover, if the two particles have their spins measured about different axes, once the electron's spin has been measured about the x-axis (and the positron's spin about the x-axis deduced), the positron's spin about the y-axis will no longer be certain, as if (a) it knows that the measurement has taken place, or (b) it has a definite spin already, about a second axis—a hidden variable. However, it turns out that the predictions of Quantum Mechanics, which have been confirmed by experiment, cannot be explained by any hidden variable theory. This is demonstrated in Bell's theorem.[12]

In quantum mechanics, the x-spin and z-spin are "incompatible observables", meaning there is a Heisenberg uncertainty principle operating between them: a quantum state cannot possess a definite value for both of these 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. It is 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 positron, he would get an answer with absolute certainty, since prior to this he hasn't disturbed his particle at all. But Bob's positron has a 50% probability of producing +x and a 50% probability of -x—so the outcome is not certain. Bob's positron "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 uncertain.

Put another way, how does Bob's positron know which way to point if Alice decides (based on information unavailable to Bob) to measure x (i.e. to be the opposite of Alice's electron's spin about the x-axis) and also how to point if Alice measures z, since it is only supposed to know one thing at a time? The Copenhagen interpretation rules that say the wave function "collapses" at the time of measurement, so there must be action at a distance (entanglement) or the positron must know more than it's supposed to (hidden variables).

Here is the paradox summed up:

It is one thing to say that physical measurement of the first particle's momentum affects uncertainty in its own position, but to say that measuring the first particle's momentum affects the uncertainty in the position of the other is another thing altogether. Einstein, Podolsky and Rosen asked how can the second particle "know" to have precisely defined momentum but uncertain position? Since this implies that one particle is communicating with the other instantaneously across space, i.e. faster than light, this is the "paradox".

Incidentally, Bell used spin as his example, but many types of physical quantities—referred to as "observables" in quantum mechanics—can be used. The EPR paper used momentum for the observable. Experimental realisations of the EPR scenario often use photon polarization, because polarized photons are easy to prepare and measure.

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, there may be an infinite number of axes along which Alice and Bob can perform their measurements, so there would have 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
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 particular class of hidden variable theories (the local hidden variable theories). Roughly speaking, quantum mechanics has a much stronger statistical correlation with measurement results performed on different axes than do these 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 which lead to Bell's inequalities are locality and counter-factual definiteness. Any theory in which these principles apply produces the inequalities. Arthur 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 (experiments which generally rely on photon polarization measurement). All the experiments conducted to date have found behavior in line with the predictions of standard quantum mechanics theory.

However, Bell's theorem does not apply to all possible philosophically realist theories. It is a common misconception that quantum mechanics is inconsistent with all notions of philosophical realism, but 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 striving also to maintain 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 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 keeps counter-factual definiteness while introducing a conjectured non-local mechanism in form of the 'quantum potential', defined as one of the terms of the Schrödinger equation. 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 theory has been proposed 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.
Einstein's hope for a purely algebraic theory

The Bohm interpretation of quantum mechanics hypothesizes that the state of the universe evolves smoothly through time with no collapsing of quantum wavefunctions. One problem for the Copenhagen interpretation is to precisely define wavefunction collapse. Einstein maintained that quantum mechanics is physically incomplete and logically unsatisfactory. In "The Meaning of Relativity," Einstein wrote, "One can give good reasons why reality cannot at all be represented by a continuous field. From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers (quantum numbers). This does not seem to be in accordance with a continuum theory and must lead to an attempt to find a purely algebraic theory for the representation of reality. But nobody knows how to find the basis for such a theory." If time, space, and energy are secondary features derived from a substrate below the Planck scale, then Einstein's hypothetical algebraic system might resolve the EPR paradox (although Bell's theorem would still be valid). Edward Fredkin in the Fredkin Finite Nature Hypothesis has suggested an informational basis for Einstein's hypothetical algebraic system. If physical reality is totally finite, then the Copenhagen interpretation might be an approximation to an information processing system below the Planck scale.
"Acceptable theories" and the experiment

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

In the EPR paper (1935) the authors realised that quantum mechanics was inconsistent with their assumptions, but Einstein nevertheless thought that quantum mechanics might simply be augmented by hidden variables (i.e. variables which were, at that point, still obscure to him), without any other change, to achieve an acceptable theory. He pursued these ideas for over twenty years until the end of his life, in 1955.

In contrast, John Bell, in his 1964 paper, showed that quantum mechanics and the class of hidden variable theories Bell investigated[13] would lead to different experimental results: different by a factor of 3⁄2 for certain correlations. So the issue of "acceptability", up to that time mainly concerning theory, finally became experimentally decidable.

There are many Bell test experiments, e.g. those of Alain Aspect and others. They support the predictions of quantum mechanics rather than the class of hidden variable theories Bell investigated.[14] According to Karl Popper these experiments showed that the class of "hidden variables" Bell investigated is erroneous.
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 locality is strictly preserved, since the effects of operations such as measurement affect only the state of the particle that is measured. However, the results of the measurement are not unique—every possible result is obtained.

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. In fact, Yakir Aharonov and his collaborators have developed a whole theory of so-called Weak measurement.[citation needed]

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 complex 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 \( S_z \) are represented as

\( \left|+z\right\rang \leftrightarrow \begin{bmatrix}1\\0\end{bmatrix}, \quad \left|-z\right\rang \leftrightarrow \begin{bmatrix}0\\1\end{bmatrix} \)

and the eigenstates of Sx are represented as

\( \left|+x\right\rang \leftrightarrow \frac{1}{\sqrt{2}} \begin{bmatrix}1\\1\end{bmatrix}, \quad \left|-x\right\rang \leftrightarrow \frac{1}{\sqrt{2}} \begin{bmatrix}1\\-1\end{bmatrix}. \)

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

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

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

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

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

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

For the spin singlet, the new state is

\( \left| +z \right\rangle \otimes \left| -z \right\rangle. \)

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

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

which means that the new state is

\( \left|-z\right\rangle \otimes \left|+z\right\rangle \)

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 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,

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

along with the Heisenberg uncertainty relation

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