User:Tal Mor/sandbox/quantum nonlocality

In theoretical physics, quantum nonlocality most commonly refers to the phenomenon by which measurements made at a microscopic level contradict a collection of notions known as local realism that are regarded as intuitively true in classical mechanics. Rigorously, quantum nonlocality refers to quantum mechanical predictions of many-system measurement correlations that cannot be simulated by any local hidden variable theory. Many entangled quantum states exhibit such correlations, as demonstrated by Bell's theorem, and as verified by experiment.

Experiments have generally favoured quantum mechanics as a description of nature, over local hidden variable theories. Any physical theory that supersedes or replaces quantum theory must make similar experimental predictions and must therefore also be nonlocal in this sense; quantum nonlocality is a property of the universe that is independent of our description of nature.

Whilst quantum nonlocality improves the efficiency of various computational tasks, it does not allow for faster-than-light communication, and hence is compatible with special relativity. However, it prompts many of the foundational discussions concerning quantum theory.

Example
Imagine two experimentalists, Alice and Bob, situated in separate laboratories. They conduct a simple experiment in which Alice chooses and pushes one of two buttons, A0 and A1, on her apparatus, and Bob observes on his apparatus one of two indicating lamps, b0 and b1, lighting. In this case there are four possible events that could occur in the experiment: (A0,b0), (A0,b1), (A1,b0) and (A1,b1). Suppose that after many runs of the experiment, only the events (A0,b0) and (A1,b1) occur; this is good evidence that A has an influence on b. Indeed, Alice could easily send messages to Bob by encoding those messages into sequences of 0s and 1s, and causing the b0 or b1 lamp to light up respectively.

More realistically, suppose that the four events occur with conditional probabilities P(b0|A0), P(b1|A0) = 1 - P(b0|A0), P(b0|A1) and P(b1|A1) = 1 - P(b0|A1). Here P(b0|A0) is the probability that Bob's b0 lamp lit up, given that Alice pushed the button A0. We can still rigorize the notion that A has an influence on B in this setting: if P(b0|A0) differs from P(b0|A1) then Alice's choice of button still affects the probabilistic outcome on Bob's side, and it is still possible for Alice to send Bob messages with low probability of error. For example, if P(b0|A0) = $$\scriptstyle \frac{1}{2}$$ and P(b0|A1) = $$\scriptstyle \frac{2}{3}$$, then after 100 runs of the experiment in which Alice pushed the same button, Bob can tell with high probability which button it was by looking at how often b0 occurred.

Here is a more complicated scenario: Alice pushes one of two buttons, A0 and A1, and Bob also pushes one of two buttons, B0 and B1. Alice observes one of two outcomes, a0 and a1, and Bob also observes one of two outcomes, b0 and b1. There are 24 = 16 possible combinations of these 4 events:
 * $$\textstyle (AX, BY, ax, by) $$

where $X, Y$ each indicate whether the corresponding button is pressed and $x, y$ each indicate whether the corresponding lamp is lit (each of X,Y,x,y is 0 or 1). Suppose that of these 16, only 8 combinations actually occur, with the following (conditional) probabilities:
 * $$ P( {ax,by}{|}{AX,BY} ) =

\begin{cases} \frac{1}{2}, & \mbox{if } x \oplus y = XY \\ 0, & \mbox{otherwise} \end{cases} $$ where $$\scriptstyle \oplus$$ denotes addition modulo 2 (XOR) and the juxtaposition "XY" denotes multiplication (AND).

Then if A1 and B1 are both pressed ($XY = 1$) the outcomes with nonzero probability (note that $x ⊕ y = 1 ⇒ x ≠ y$) are perfectly anticorrelated, either (a0,b1) or (a1,b0), with an equal probability for both occurrences. In all other cases ($XY = 0)$ the two outcomes with nonzero probability (note that $x ⊕ y = 0 ⇒ x = y$) are perfectly correlated (either (a0,b0) or (a1,b1), again, equiprobably.

Do these outcomes imply that some influence exists (A on b, or B on a), or not? The question is important, since the answer depends on our fundamental assumptions about how mathematical theories describe physical reality.

On the one hand, Alice cannot send a message to Bob, using her buttons A0, A1 and his indicators b0, b1 (nor Bob to Alice), since it is easily checked that P(bx|A0) = P(bx|A1) for both x = 0 and x = 1 in the above example. That is to say, this particular set of probabilities is non-signalling. In this sense, there is no influence of A on b, or of B on a.

On the other hand, it is provably impossible for two separated parties to simulate this outcome without any kind of interaction or communication between them. Thorough logical analysis reveals that the above outcome can only occur if there is some direct influence between A and B, if we assume local realism and, arguably, counterfactual definiteness. These fundamental assumptions, deeply rooted in our physical intuition, are incompatible with quantum theory. Different interpretations of quantum mechanics reject different parts of local realism and/or counterfactual definiteness (for detail, see Principle of locality). A classical definition of nonlocality, i.e. direct influence of one object on another, distant object, normally takes local realism and counterfactual definiteness for granted.

Einstein, Podolsky and Rosen
In 1935, Einstein, Podolsky and Rosen published a thought experiment with which they hoped to expose the incompleteness of the Copenhagen interpretation of quantum mechanics in relation to the violation of local causality at the microscopic scale that it described. David Bohm later modified the original EPR thought experiment, simplifying the mathematics and highlighting assumptions like reality (which Einstein et al had tacitly assumed). In Bohm's version of the experiment, a spin-zero particle decays into two spin-half particles such that there is no interaction between the two particles after decay. The quantum state of the two particles prior to measurement can be written as


 * $$\left|\psi_{AB}\right\rang =\frac{1}{\sqrt{2}} \bigg(\left|\uparrow\right\rang_A \left|\downarrow\right\rang_B -

\left|\downarrow\right\rang_A \left|\uparrow\right\rang_B \bigg) $$

Here, subscripts A and B distinguish the two particles, though it is more convenient and usual to refer to these particles as being in the possession of two experimentalists called Alice and Bob. The rules of quantum theory give predictions for the outcomes of measurements performed by the experimentalists. Alice, for example, will measure her particle to be spin-up in an average of fifty percent of measurements. However, according to the Copenhagen interpretation, Alice's measurement causes the state of the two particles to collapse, so that if Alice measures spin-up in some direction n, the quantum state after measurement is the corresponding eigenstate


 * $$\left|\psi_{AB}\right\rang = \left|\uparrow_{\mathbf{n}}\right\rang_A \left|\downarrow_{\mathbf{n}}\right\rang_B $$

If Bob also measures spin in direction n, he must get a spin-down result. Hence, spin measurements in the same direction are always anti-correlated, although the particles are assumed at this stage to be non-interacting. Einstein, Podolsky and Rosen saw this as evidence of the incompleteness of the Copenhagen interpretation of quantum theory: if there is no particle interaction then the only explanation for this anti-correlation between measurement outcomes is that each particle carries a pre-existing determinate value (appropriately anti-correlated with the value carried by the other particle) for that measurement. Such a property is unaccounted for by the quantum mechanical state description, and their paper concludes:


 * 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.

Although various authors (most notably Niels Bohr) criticised the ambiguous terminology of the EPR paper, the thought-experiment nevertheless generated a great deal of interest. Their notion of a "complete description" was later formalised by the suggestion of hidden variables that determine the statistics of measurement results, but to which an observer does not have access. Bohmian mechanics provides such a completion of quantum mechanics, with introduction of hidden variables; however the theory is explicitly nonlocal (positions of particles are non contextual observables). Bell's Theorem refers to local hidden variables in keeping with Einstein's "Principle of Local Action".

Demonstration
In 1964 John Bell showed that such local hidden variables could never reproduce the statistical outcomes of individual measurements, as predicted by quantum theory. Bell showed that a local hidden variable hypothesis leads to restrictions on the strength of correlations of measurement results. If the Bell inequalities are violated experimentally as predicted by quantum mechanics, then reality cannot be described by such local hidden variables and the mystery of quantum nonlocal causation remains. According to Bell:


 * This [grossly nonlocal structure] is characteristic... of any such theory which reproduces exactly the quantum mechanical predictions.

Clauser, Horne, Shimony and Holt (CHSH) reformulated these inequalities in a manner that was more conducive to experimental testing (see CHSH inequality). They proposed a scheme whereby two experimentalists, Alice and Bob, make separate measurements of photon polarization in two carefully chosen directions, and derived a simple inequality that is obeyed by all local hidden variable theories, but violated by certain measurements on quantum states.

Bell formalized the idea of a hidden variable by introducing the parameter λ to locally characterize measurement results on each system: "It is a matter of indifference... whether λ denotes a single variable or a set... and whether the variables are discrete or continuous". However, it is equivalent (and more intuitive) to think of λ as a local "strategy" or "message" that occurs with some probability ρ($$\lambda$$) when an entangled pair of states is created. EPR's criteria of local separability then stipulates that each local strategy defines independent distributions for the outcome probabilities if Alice measures in direction A and Bob measures in direction B:


 * $$ P \left ( {a, b}{|}{A, B, \lambda } \right ) = P \left ( {a}{|}{A, \lambda } \right ) P \left ( {b}{|}{B, \lambda } \right )$$

where, for instance, $$\scriptstyle P \left ( {a}{|}{A, \lambda } \right )$$ denotes the probability of Alice getting the outcome a given λ, and that she measured A.

Suppose that λ can take values from some set $$\lambda_i$$, where 1 ≤ i ≤ k. If each $$\lambda_i$$ has an associated probability ρ($$\lambda_i$$) of being selected (such that the probabilities sum to unity) we can average over this distribution to obtain a formula for the joint probability of each measurement result:


 * $$ P \left ( {a, b}{|}{A, B} \right ) = \sum_{i=1}^k P \left ( {a, b} {|} {A, B, \lambda_i } \right ) \rho \left (\lambda_i \right )$$

In the CHSH scheme, the measurement result for the polarization of a photon can take one of two values (informally, whether or not the photon is polarized in that direction). We encode this by allowing a and b to take on values ±1. For each measurement A and B, the correlator E(A, B) is then defined as:


 * $$ E \left ( {A, B} \right ) = \sum_{a,b} ab P\left ( {a, b}{|}{A, B} \right )$$

Note that the product ab is equal to 1 if Alice and Bob get the same outcome, and -1 if they get different outcomes. E(A,B) can therefore be seen as the expectation that Alice's and Bob's outcomes are correlated. In the case that Alice chooses from one of two measurements $$A_0$$ or $$A_1$$, and Bob chooses from $$B_0$$ or $$B_1$$, the CHSH value for this joint probability distribution is defined as:


 * $$ S_{CHSH} = E \left ( {A_0, B_0} \right ) + E \left ( {A_0, B_1} \right ) + E \left ( {A_1, B_0} \right ) - E \left ( {A_1, B_1} \right ) $$

Compare this with the expression $$\scriptstyle x \oplus y = XY$$ and the discussion in the above example. The CHSH value $$S_{CHSH}$$ includes a negative contribution of the correlator whenever $$A_1$$ and $$B_1$$ are chosen ($$x=y$$ when $$XY=1$$), and a positive contribution in all other cases ($$x$$≠$$y$$ when $$XY=0$$). If the joint probability distribution can be described with local strategies as above, it can be shown that the correlation function always obeys the following CHSH inequality:


 * $$ -2 \le S_{CHSH} \le 2$$

However, if instead of local hidden variables we adopt the rules of quantum theory, it is possible to construct an entangled pair of particles (one each for Alice and Bob) and a set of measurements $$\scriptstyle A_0, A_1, B_0, B_1$$ such that $$\scriptstyle S_{CHSH} \;=\; 2\sqrt{2}$$. Experimentalists such as Alain Aspect have verified the quantum violation of the CHSH inequality, as well as other formulations of Bell's inequality, to invalidate the local hidden variables hypothesis and confirm that reality is indeed nonlocal in the EPR sense.

Hardy's proof of quantum nonlocality
In 1993, Lucien Hardy demonstrated a logical proof of quantum nonlocality that, like the GHZ proof, does not require probabilistic inequalities; it is a possibilistic proof.

Consider a pair of entangled photons, and two experimenters who each have the ability to measure either one of two photonic properties: its polarisation, or its colour. The polarisation can be either ↑ or ←, and the colour can be either red or green. The experimenters repeatedly set up this experiment and perform it, each randomly picking whether they should measure colour or polarisation. When they've finished, they reveal some interesting results: whenever one of them measured colour, and the other measured polarisation, they never got a measurement result of (green,↑). They also found that whenever they measured colour, they often found that both photons were green.

An interesting question to ask these researchers, then, is whether or not when they both measured polarisation, they ever got the result (←,←). This is because an absence of such a result demonstrates a failure of local realism, by the following logic: a result of (green, green) is possible, so it's possible for the hidden states of both of the photons to be such that looking at their colour would yield "green"; if the hidden states of the photons were both compatible with green, and one of the researchers measured polarisation, they never saw the (green, ↑) result, which means that whenever a such photon would yield green in a measurement of colour, it must yield ← in a measurement of polarisation; then, if the photons were in a state compatible with (green, green), and both of the researchers measured polarisation, they would have to observe (←,←). Therefore, if in such an experiment it were the case that (←,←), (green,↑) and (↑, green) were all never seen, then that would constitute a violation of local realism. Hardy showed that such a measurement scenario is possible with any entangled resource other than, interestingly, the maximally-entangled Bell state. This is because the argument hinges on an asymmetry in the quantum state space not present in the highly symmetric Bell pairs.

Superquantum nonlocality
Whilst the CHSH inequality gives restrictions on the CHSH value attainable by local hidden variable theories, the rules of quantum theory do not allow us to violate Tsirelson's bound of $$\scriptstyle 2 \sqrt{2}$$, even if we exploit measurements of entangled particles. The question remained whether this was the maximum CHSH value that can be attained without explicitly allowing instantaneous signaling. In 1994 two physicists, Sandu Popescu and Daniel Rohrlich, formulated an explicit set of correlated measurements that respect the "non-signalling" principle, yet give $$S_{CHSH} = 4$$: the algebraic maximum. This demonstrated that there exist formulatable theories that are "non-signalling", yet drastically violate the joint probability constraints of quantum theory. The attempt to understand what distinguishes quantum theory from such general theories motivated an abstraction from physical measurements of nonlocality, to the study of nonlocal boxes.

Nonlocal boxes generalize the concept of experimentalists making joint measurements from separate locations. As in the discussion above, the choice of measurement is encoded by the input to the box. A two-party nonlocal box takes an input A from Alice and an input B from Bob, and outputs two values a and b for Alice and Bob respectively and separately, where a, b, A and B take values from some finite alphabet (normally $$\scriptstyle \{0,1\}$$). The box is characterized by the probability of outputting pair a, b, given the inputs A, B. This probability is denoted $$\scriptstyle P \left ( { {a, b}{|}{A, B} } \right )$$ and obeys the normal probabilistic conditions of positivity and normalisation:


 * $$P \left ( { {a, b}{|}{A, B} } \right ) \ge 0 \quad \forall {a,b,A,B}$$

and


 * $$\sum_{a,b} P \left ( { {a, b}{|}{A, B} } \right ) = 1 \quad \forall {A,B}$$

A box is local, or admits a local hidden variable model, if its output probabilities can be characterized in the following way:


 * $$P \left ( { {a,b}{|}{A,B} } \right ) = \sum_{\lambda} p(\lambda) \; P \left ( { {a}{|}{A,\lambda} } \right ) \; P \left ( { {b}{|}{B,\lambda} } \right )$$

where $$\scriptstyle P \left ( { {a}{|}{A,\lambda} } \right )$$ and $$\scriptstyle P \left ( { {b}{|}{B,\lambda} } \right )$$ describe single input/output probabilities at Alice's or Bob's system alone, and the value of $$\lambda$$ is chosen at random according to some fixed probability distribution given by $$p(\lambda)$$. Intuitively, $$\lambda$$ corresponds to a hidden variable, or to a shared randomness between Alice and Bob. If a box violates this condition, it is explicitly called nonlocal. However, the study of nonlocal boxes often encompasses both local and nonlocal boxes.

The set of nonlocal boxes most commonly studied are the so-called non-signalling boxes, for which neither Alice nor Bob can signal their choice of input to the other. Physically, this is a reasonable restriction: setting the input is physically analogous to making a measurement, which should effectively provide a result immediately. Since there may be a large spatial separation between the parties, signalling to Bob would potentially require considerable time to elapse between measurement and result, which is a physically unrealistic scenario.

The non-signalling requirement imposes further conditions on the joint probability, in that the probability of a particular output a or b should depend only on its associated input. This allows for the notion of a reduced or marginal probability on both Alice and Bob's measurements, and is formalised by the conditions:


 * $$\sum_{b} P \left ( {a,b}{|}{A,B} \right ) = \sum_{b} P \left ( {a,b}{|}{A,B'} \right ) \equiv P \left ( {a}{|}{A} \right ) \quad \forall {a,A,B,B'}$$

and


 * $$\sum_{a} P \left ( {a,b}{|}{A,B} \right ) = \sum_{a} P \left ( {a,b}{|}{A',B} \right ) \equiv P \left ( {b}{|}{B} \right ) \quad \forall {b,B,A,A'}$$

The constraints above are all linear, and so define a polytope representing the set of all non-signalling boxes with a given number of inputs and outputs. Moreover, the polytope is convex because any two boxes that exist in the polytope can be mixed (as above, according to some variable $$\lambda$$ with probabilities $$p(\lambda)$$) to produce another box that also exists within the polytope.

Local boxes are clearly non-signalling, however nonlocal boxes may or may not be non-signalling. Since this polytope contains all possible non-signalling boxes of a given number of inputs and outputs, it has as subsets both local boxes and those boxes which can achieve Tsirelson's bound in accord with quantum mechanical correlations. Indeed, the set of local boxes form a convex sub-polytope of the non-signalling polytope.

Popescu and Rohrlich's maximum algebraic violation of the CHSH inequality can be reached by a non-signalling box, referred to as a standard PR box after these authors, with joint probability given by:


 * $$ P \left ( {a,b}{|}{A,B} \right ) =

\begin{cases} \frac{1}{2}, & \mbox{if } a \oplus b = AB \\ 0, & \mbox{otherwise} \end{cases} $$

where $$\oplus$$ denotes addition modulo two.

Various attempts have been made to argue why Nature does not (or should not) allow for stronger nonlocality than quantum theory is already known to permit. For example, in recent publications it was found that quantum mechanics cannot be more nonlocal without violating the Heisenberg uncertainty principle. Strikingly, it has been discovered that if PR boxes did exist, any distributed computation could be performed with only one bit of communication. An even stronger result is that for any nonlocal box theory which violates Tsirelson's bound, there cannot be a sensible measure of mutual information between pairs of systems. This suggests a deep link between nonlocality and the information-theoretic properties of quantum mechanics. Nevertheless, the PR-box is ruled out by a plausible postulate of information theory.

Non-signaling adversaries have recently been considered in quantum cryptography. Such an adversary is constrained only by the non-signaling principle, thus may potentially be more powerful than a quantum adversary.

Nonlocality vs entanglement
In the media and popular science, quantum nonlocality is often portrayed as being equivalent to entanglement. While it is true that a pure bipartite quantum state must be entangled in order for it to produce nonlocal correlations, there exist entangled states which do not produce such correlations, and there exist non-entangled (separable) states that do produce some type of non-local behaviour. A well-known example of the first case is constituted by a subset of Werner states that are entangled but whose correlations can always be described using local hidden variables. On the other hand, reasonably simple examples of Bell inequalities have been found for which the quantum state giving the largest violation is never a maximally entangled state, showing that entanglement is, in some sense, not even proportional to nonlocality.

In short, entanglement of a two-party state is necessary but not sufficient for that state to be nonlocal. It is important to recognise that entanglement is more commonly viewed as an algebraic concept, noted for being a precedent to nonlocality as well as quantum teleportation and superdense coding, whereas nonlocality is interpreted according to experimental statistics and is much more involved with the foundations and interpretations of quantum mechanics.