Bell's theorem

Bell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with local hidden-variable theories, given some basic assumptions about the nature of measurement. The first such result was introduced by John Stewart Bell in 1964, building upon the Einstein–Podolsky–Rosen paradox, which had called attention to the phenomenon of quantum entanglement.

In the context of Bell's theorem, "local" refers to the principle of locality, the idea that a particle can only be influenced by its immediate surroundings, and that interactions mediated by physical fields cannot propagate faster than the speed of light. "Hidden variables" are supposed properties of quantum particles that are not included in quantum theory but nevertheless affect the outcome of experiments. In the words of Bell, "If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."

In his original paper, Bell analyzed independent measurements on two spatially separated particles of an entangled pair. More advanced experiments, known collectively as Bell tests, have been performed many times since. Often, these experiments have had the goal of "closing loopholes", that is, ameliorating problems of experimental design or set-up that could in principle affect the validity of the findings of earlier Bell tests. Bell tests have consistently found that physical systems obey quantum mechanics and violate Bell inequalities; which is to say that the results of these experiments are incompatible with local hidden-variable theories.

The exact nature of the assumptions required to prove a Bell-type constraint on correlations has been debated by physicists and by philosophers. While the significance of Bell's theorem is not in doubt, different interpretations of quantum mechanics disagree about what exactly it implies.

Theorem

There are many variations on the basic idea, some employing stronger mathematical assumptions than others. Significantly, Bell-type theorems do not refer to any particular theory of local hidden variables, but instead show that quantum physics violates general assumptions behind classical pictures of nature. The original theorem proved by Bell in 1964 is not the most amenable to experiment, and it is convenient to introduce the genre of Bell-type inequalities with a later example. Consider the combination<math display="block">a_0b_0 + a_0b_1 + a_1b_0-a_1b_1 = (a_0+a_1)b_0 + (a_0-a_1)b_1 \, .</math>Because both <math>a_0</math> and <math>a_1</math> take the values <math>\pm 1</math>, then either <math>a_0 = a_1</math> or <math>a_0 = -a_1</math>. In the former case, the quantity <math>(a_0-a_1)b_1</math> must equal 0, while in the latter case, <math>(a_0+a_1)b_0 = 0</math>. So, one of the terms on the right-hand side of the above expression will vanish, and the other will equal <math>\pm 2</math>. Consequently, if the experiment is repeated over many trials, with Victor preparing new pairs of particles, the absolute value of the average of the combination <math>a_0b_0 + a_0b_1 + a_1b_0-a_1b_1</math> across all the trials will be less than or equal to 2. No single trial can measure this quantity, because Alice and Bob can only choose one measurement each, but on the assumption that the underlying properties exist, the average value of the sum is just the sum of the averages for each term. Using angle brackets to denote averages<math display="block">| \langle A_0B_0 \rangle + \langle A_0B_1 \rangle + \langle A_1B_0 \rangle - \langle A_1B_1 \rangle | \leq 2 \, .</math>

This is a Bell inequality, specifically, the CHSH inequality. Its derivation here depends upon two assumptions: first, that the underlying physical properties <math>a_0, a_1, b_0,</math> and <math>b_1</math> exist independently of being observed or measured (sometimes called the assumption of realism); and second, that Alice's choice of action cannot influence Bob's result or vice versa (often called the assumption of locality).

thumb|An illustration of the CHSH game: the referee, Victor, sends a bit each to Alice and to Bob, and Alice and Bob each send a bit back to the referee.

The CHSH inequality can also be thought of as a game in which Alice and Bob try to coordinate their actions. Victor prepares two bits, <math>x</math> and <math>y</math>, independently and at random. He sends bit <math>x</math> to Alice and bit <math>y</math> to Bob. Alice and Bob win if they return answer bits <math>a</math> and <math>b</math> to Victor, satisfying

Or, equivalently, Alice and Bob win if the logical AND of <math>x</math> and <math>y</math> is the logical XOR of <math>a</math> and <math>b</math>. Alice and Bob can agree upon any strategy they desire before the game, but they cannot communicate once the game begins. In any theory based on local hidden variables, Alice and Bob's probability of winning is no greater than <math>3/4</math>, regardless of what strategy they agree upon beforehand. However, if they share an entangled quantum state, their probability of winning can be as large as<math display="block">\frac{2+\sqrt{2{4} \approx 0.85 \, .</math>

Bell (1964)

Bell's 1964 paper shows that a very simple local hidden-variable model can in restricted circumstances reproduce the predictions of quantum mechanics, but then he demonstrates that, in general, such models give different predictions. For example, let the vectors <math>\vec{a}</math> and <math>\vec{b}</math> be orthogonal, and let <math>\vec{c}</math> lie in their plane at a 45° angle from both of them. Then<math display="block">P(\vec{a}, \vec{b}) = 0,</math>

while

but

Therefore, there is no local hidden-variable model that can reproduce the predictions of quantum mechanics for all choices of <math>\vec{a}</math>, <math>\vec{b}</math>, and <math>\vec{c}.</math> Experimental results contradict the classical curves and match the curve predicted by quantum mechanics as long as experimental shortcomings are accounted for. Bell noted that this applies when the two detectors are oriented in the same direction (<math>\vec{a} = \vec{b}</math>), and so the EPR criterion would imply that some element of reality must predetermine the measurement result. Because the quantum description of a particle does not include any such element, the quantum description would have to be incomplete. In other words, Bell's 1964 paper shows that, assuming locality, the EPR criterion implies hidden variables and then he demonstrates that local hidden variables are incompatible with quantum mechanics. Because experiments cannot achieve perfect correlations or anti-correlations in practice, Bell-type inequalities based on derivations that relax this assumption are tested instead. In this thought experiment, Victor generates a set of three spin-1/2 particles described by the quantum state<math display="block">|\psi\rangle = \frac{1}{\sqrt{2(|000\rangle - |111\rangle) \, , </math>

where as above, <math>|0\rangle</math> and <math>|1\rangle</math> are the eigenvectors of the Pauli matrix <math>\sigma_z</math>. Victor then sends a particle each to Alice, Bob, and Charlie, who wait at widely separated locations. Alice measures either <math>\sigma_x</math> or <math>\sigma_y</math> on her particle, and so do Bob and Charlie. The result of each measurement is either <math>+1</math> or <math>-1</math>. Applying the Born rule to the three-qubit state <math>|\psi\rangle</math>, Victor predicts that whenever the three measurements include one <math>\sigma_x</math> and two <math>\sigma_y</math>'s, the product of the outcomes will always be <math>+1</math>. This follows because <math>|\psi\rangle</math> is an eigenvector of <math>\sigma_x \otimes \sigma_y \otimes \sigma_y</math> with eigenvalue <math>+1</math>, and likewise for <math>\sigma_y \otimes \sigma_x \otimes \sigma_y</math> and <math>\sigma_y \otimes \sigma_y \otimes \sigma_x</math>. Therefore, knowing Alice's result for a <math>\sigma_x</math> measurement and Bob's result for a <math>\sigma_y</math> measurement, Victor can predict with probability 1 what result Charlie will return for a <math>\sigma_y</math> measurement. According to the EPR criterion of reality, there would be an "element of reality" corresponding to the outcome of a <math>\sigma_y</math> measurement upon Charlie's qubit. Indeed, this same logic applies to both measurements and all three qubits. Per the EPR criterion of reality, then, each particle contains an "instruction set" that determines the outcome of a <math>\sigma_x</math> or <math>\sigma_y</math> measurement upon it. The set of all three particles would then be described by the instruction set<math display="block">(a_x,a_y,b_x,b_y,c_x,c_y) \, , </math>

with each entry being either <math>-1</math> or <math>+1</math>, and each <math>\sigma_x</math> or <math>\sigma_y</math> measurement simply returning the appropriate value.

If Alice, Bob, and Charlie all perform the <math>\sigma_x</math> measurement, then the product of their results would be <math>a_x b_x c_x</math>. This value can be deduced from<math display="block">(a_x b_y c_y) (a_y b_x c_y) (a_y b_y c_x) = a_x b_x c_x a_y^2 b_y^2 c_y^2 = a_x b_x c_x \, , </math>

because the square of either <math>-1</math> or <math>+1</math> is <math>1</math>. Each factor in parentheses equals <math>+1</math>, so<math display="block">a_x b_x c_x = +1 \, , </math>

and the product of Alice, Bob, and Charlie's results will be <math>+1</math> with probability unity. But this is inconsistent with quantum physics: Victor can predict using the state <math>|\psi\rangle</math> that the measurement <math>\sigma_x \otimes \sigma_x \otimes \sigma_x</math> will instead yield <math>-1</math> with probability unity.

This thought experiment can also be recast as a traditional Bell inequality or, equivalently, as a nonlocal game in the same spirit as the CHSH game. In it, Alice, Bob, and Charlie receive bits <math>x,y,z</math> from Victor, promised to always have an even number of ones, that is, <math>x\oplus y\oplus z = 0</math>, and send him back bits <math>a,b,c</math>. They win the game if <math>a,b,c</math> have an odd number of ones for all inputs except <math>x=y=z=0</math>, when they need to have an even number of ones. That is, they win the game if and only if <math>a \oplus b \oplus c = x \lor y \lor z</math>. With local hidden variables the highest probability of victory they can have is 3/4, whereas using the quantum strategy above they win it with certainty. This is an example of quantum pseudo-telepathy.

Kochen–Specker theorem (1967)

In quantum theory, orthonormal bases for a Hilbert space represent measurements that can be performed upon a system having that Hilbert space. Each vector in a basis represents a possible outcome of that measurement. Suppose that a hidden variable <math>\lambda</math> exists, so that knowing the value of <math>\lambda</math> would imply certainty about the outcome of any measurement. Given a value of <math>\lambda</math>, each measurement outcome – that is, each vector in the Hilbert space – is either impossible or guaranteed. A Kochen–Specker configuration is a finite set of vectors made of multiple interlocking bases, with the property that a vector in it will always be impossible when considered as belonging to one basis and guaranteed when taken as belonging to another. In other words, a Kochen–Specker configuration is an "uncolorable set" that demonstrates the inconsistency of assuming a hidden variable <math>\lambda</math> can be controlling the measurement outcomes.

Free will theorem

The Kochen–Specker type of argument, using configurations of interlocking bases, can be combined with the idea of measuring entangled pairs that underlies Bell-type inequalities. This was noted beginning in the 1970s by Kochen, Heywood and Redhead, Stairs, and Brown and Svetlichny. As EPR pointed out, obtaining a measurement outcome on one half of an entangled pair implies certainty about the outcome of a corresponding measurement on the other half. The "EPR criterion of reality" posits that because the second half of the pair was not disturbed, that certainty must be due to a physical property belonging to it. In other words, by this criterion, a hidden variable <math>\lambda</math> must exist within the second, as-yet unmeasured half of the pair. No contradiction arises if only one measurement on the first half is considered. However, if the observer has a choice of multiple possible measurements, and the vectors defining those measurements form a Kochen–Specker configuration, then some outcome on the second half will be simultaneously impossible and guaranteed.

This type of argument gained attention when an instance of it was advanced by John Conway and Simon Kochen under the name of the free will theorem. The Conway–Kochen theorem uses a pair of entangled qutrits and a Kochen–Specker configuration discovered by Asher Peres.

Quasiclassical entanglement

As Bell pointed out, some predictions of quantum mechanics can be replicated in local hidden-variable models, including special cases of correlations produced from entanglement. This topic has been studied systematically in the years since Bell's theorem. In 1989, Reinhard Werner introduced what are now called Werner states, joint quantum states for a pair of systems that yield EPR-type correlations but also admit a hidden-variable model. Werner states are bipartite quantum states that are invariant under unitaries of symmetric tensor-product form: <math display="block">\rho_{AB} = (U \otimes U) \rho_{AB} (U^\dagger \otimes U^\dagger).</math>

In 2004, Robert Spekkens introduced a toy model that starts with the premise of local, discretized degrees of freedom and then imposes a "knowledge balance principle" that restricts how much an observer can know about those degrees of freedom, thereby making them into hidden variables. The allowed states of knowledge ("epistemic states") about the underlying variables ("ontic states") mimic some features of quantum states. Correlations in the toy model can emulate some aspects of entanglement, like monogamy, but by construction, the toy model can never violate a Bell inequality.

History

Background

The question of whether quantum mechanics can be "completed" by hidden variables dates to the early years of quantum theory. In his 1932 textbook on quantum mechanics, the Hungarian-born polymath John von Neumann presented what he claimed to be a proof that there could be no "hidden parameters". The validity and definitiveness of von Neumann's proof were questioned by Hans Reichenbach, in more detail by Grete Hermann, and possibly in conversation though not in print by Albert Einstein.