As shown, the statistics achievable by two or more parties conducting experiments in a classical system are constrained in a non-trivial way. Analogously, the statistics achievable by separate observers in a quantum theory also happen to be restricted. The first derivation of a non-trivial statistical limit on the set of quantum correlations, due to
B. Tsirelson, is known as
Tsirelson's bound. Consider the CHSH Bell scenario detailed before, but this time assume that, in their experiments, Alice and Bob are preparing and measuring quantum systems. In that case, the CHSH parameter can be shown to be bounded by :-2\sqrt{2}\leq \mathrm{CHSH}\leq 2\sqrt{2}.
The sets of quantum correlations and Tsirelson's problem Mathematically, a box P(a,b|x,y) admits a quantum realization if and only if there exists a pair of Hilbert spaces H_A, H_B, a normalized vector \left|\psi\right\rangle\in H_A\otimes H_B and projection operators E^x_a:H_A\to H_A, F^y_b:H_B\to H_B such that • For all x,y, the sets \{E^x_a\}_a,\{F^y_b\}_b represent complete measurements. Namely, \sum_aE^x_a={\mathbb I}_A, \sum_bF^y_b={\mathbb I}_B. • P(a,b|x,y) =\left\langle\psi\right|E^x_a\otimes F^y_b\left|\psi\right\rangle, for all a,b,x,y. In the following, the set of such boxes will be called Q. Contrary to the classical set of correlations, when viewed in probability space, Q is not a polytope. On the contrary, it contains both straight and curved boundaries. In addition, Q is not closed: this means that there exist boxes P(a,b|x,y) which can be arbitrarily well approximated by quantum systems but are themselves not quantum. In the above definition, the space-like separation of the two parties conducting the Bell experiment was modeled by imposing that their associated operator algebras act on different factors H_A, H_B of the overall
Hilbert space H=H_A\otimes H_B describing the experiment. Alternatively, one could model space-like separation by imposing that these two algebras commute. This leads to a different definition: P(a,b|x,y) admits a field quantum realization if and only if there exists a Hilbert space H, a normalized vector \left|\psi\right\rangle\in H and projection operators E^x_a:H\to H, F^y_b:H\to H such that • For all x,y, the sets \{E^x_a\}_a,\{F^y_b\}_b represent complete measurements. Namely, \sum_aE^x_a={\mathbb I}, \sum_bF^y_b={\mathbb I} . • P(a,b|x,y) =\left\langle\psi\right|E^x_a F^y_b\left|\psi\right\rangle, for all a,b,x,y. • [E^x_a, F^y_b]=0, for all a,b,x,y. Call Q_c the set of all such correlations P(a,b|x,y). How does this new set relate to the more conventional Q defined above? It can be proven that Q_c is closed. Moreover, \bar{Q} \subseteq Q_c, where \bar{Q} denotes the closure of Q. Tsirelson's problem consists in deciding whether the inclusion relation \bar{Q} \subseteq Q_c is strict, i.e., whether or not \bar{Q} = Q_c. This problem only appears in infinite dimensions: when the Hilbert space H in the definition of Q_c is constrained to be finite-dimensional, the closure of the corresponding set equals \bar{Q}. that would imply that \bar{Q} \neq Q_c , thus solving Tsirelson's problem. Tsirelson's problem can be shown equivalent to
Connes embedding problem, a famous conjecture in the theory of operator algebras.
Characterization of quantum correlations Since the dimensions of H_A and H_B are, in principle, unbounded, determining whether a given box P(a,b|x,y) admits a quantum realization is a complicated problem. In fact, the dual problem of establishing whether a quantum box can have a perfect score at a non-local game is known to be undecidable. Characterizing quantum boxes is equivalent to characterizing the cone of completely positive semidefinite matrices under a set of linear constraints. For small fixed dimensions d_A, d_B, one can explore, using variational methods, whether P(a,b|x,y) can be realized in a bipartite quantum system H_A\otimes H_B, with \dim(H_A)=d_A, \dim(H_B)=d_B. That method, however, can just be used to prove the realizability of P(a,b|x,y), and not its unrealizability with quantum systems. To prove unrealizability, the most known method is the Navascués–Pironio–Acín (NPA) hierarchy. This is an infinite decreasing sequence of sets of correlations Q^1\supset Q^2\supset Q^3\supset... with the properties: • If P(a,b|x,y)\in Q_c, then P(a,b|x,y)\in Q^k for all k. • If P(a,b|x,y)\not\in Q_c, then there exists k such that P(a,b|x,y)\not\in Q^k. • For any k, deciding whether P(a,b|x,y)\in Q^k can be cast as a
semidefinite program. The NPA hierarchy thus provides a computational characterization, not of Q, but of Q_c. If \bar{Q}\not=Q_c, (as claimed by Ji, Natarajan, Vidick, Wright, and Yuen) then a new method to detect the non-realizability of the correlations in Q_c- \bar{Q} is needed. If Tsirelson's problem was solved in the affirmative, namely, \bar{Q}=Q_c, then the above two methods would provide a practical characterization of \bar{Q}.
The physics of supra-quantum correlations The works listed above describe what the quantum set of correlations looks like, but they do not explain why. Are quantum correlations unavoidable, even in post-quantum physical theories, or on the contrary, could there exist correlations outside \bar{Q} which nonetheless do not lead to any unphysical operational behavior? In their seminal 1994 paper,
Popescu and Rohrlich explore whether quantum correlations can be explained by appealing to relativistic causality alone. Namely, whether any hypothetical box P(a,b|x,y)\not\in\bar{Q} would allow building a device capable of transmitting information faster than the speed of light. At the level of correlations between two parties, Einstein's causality translates in the requirement that Alice's measurement choice should not affect Bob's statistics, and vice versa. Otherwise, Alice (Bob) could signal Bob (Alice) instantaneously by choosing her (his) measurement setting x (y) appropriately. Mathematically, Popescu and Rohrlich's no-signalling conditions are: \sum_a P(a,b|x,y)= \sum_a P(a,b|x^\prime,y)=:P_B(b|y), \sum_b P(a,b|x,y)= \sum_b P(a,b|x,y^\prime)=:P_A(a|x). Like the set of classical boxes, when represented in probability space, the set of no-signalling boxes forms a
polytope. Popescu and Rohrlich identified a box P(a,b|x,y) that, while complying with the no-signalling conditions, violates Tsirelson's bound, and is thus unrealizable in quantum physics. Dubbed the PR-box, it can be written as: P(a,b|x,y)=\frac{1}{2}\delta_{xy,a\oplus b}. Here a,b,x,y take values in {0,1}, and a\oplus b denotes the sum modulo two. It can be verified that the CHSH value of this box is 4 (as opposed to the Tsirelson bound of 2\sqrt{2}\approx 2.828). This box had been identified earlier, by Rastall and Khalfin and
Tsirelson. In view of this mismatch, Popescu and Rohrlich pose the problem of identifying a physical principle, stronger than the no-signalling conditions, that allows deriving the set of quantum correlations. Several proposals followed: • Non-trivial
communication complexity (NTCC). This principle stipulates that nonlocal correlations should not be so strong as to allow two parties to solve all 1-way communication problems with some probability p>1/2 using just one bit of communication. It can be proven that any box violating Tsirelson's bound by more than 2\sqrt{2}\left(\frac{2}{\sqrt{3}}-1\right)\approx 0.4377 is incompatible with NTCC. • No Advantage for Nonlocal Computation (NANLC). The following scenario is considered: given a function f_{0,1}^n\to 1, two parties are distributed the strings of n bits x,y and asked to output the bits a,b so that a\oplus b is a good guess for f(x\oplus y). The principle of NANLC states that non-local boxes should not give the two parties any advantage to play this game. It is proven that any box violating Tsirelson's bound would provide such an advantage. •
Information Causality (IC). The starting point is a bipartite communication scenario where one of the parts (Alice) is handed a random string x of n bits. The second part, Bob, receives a random number k\in\{1,...,n\}. Their goal is to transmit Bob the bit x_k, for which purpose Alice is allowed to transmit Bob s bits. The principle of IC states that the sum over k of the mutual information between Alice's bit and Bob's guess cannot exceed the number s of bits transmitted by Alice. It is shown that any box violating Tsirelson's bound would allow two parties to violate IC. • Macroscopic Locality (ML). In the considered setup, two separate parties conduct extensive low-resolution measurements over a large number of independently prepared pairs of correlated particles. ML states that any such "macroscopic" experiment must admit a local hidden variable model. It is proven that any microscopic experiment capable of violating Tsirelson's bound would also violate standard Bell nonlocality when brought to the macroscopic scale. Besides Tsirelson's bound, the principle of ML fully recovers the set of all two-point quantum correlators. • Local Orthogonality (LO). This principle applies to multipartite Bell scenarios, where n parties respectively conduct experiments x_1,...,x_n in their local labs. They respectively obtain the outcomes a_1,...,a_n. The pair of vectors (\bar{a}|\bar{x}) is called an event. Two events (\bar{a}|\bar{x}), (\bar{a}^\prime|\bar{x}^\prime) are said to be locally orthogonal if there exists k such that x_k=x_k^\prime and a_k\not=a_k^\prime . The principle of LO states that, for any multipartite box, the sum of the probabilities of any set of pair-wise locally orthogonal events cannot exceed 1. It is proven that any bipartite box violating Tsirelson's bound by an amount of 0.052 violates LO. All these principles can be experimentally falsified under the assumption that we can decide if two or more events are space-like separated. This sets this research program aside from the axiomatic reconstruction of quantum mechanics via
Generalized Probabilistic Theories. The works above rely on the implicit assumption that any physical set of correlations must be closed under wirings. This means that any effective box built by combining the inputs and outputs of a number of boxes within the considered set must also belong to the set. Closure under wirings does not seem to enforce any limit on the maximum value of CHSH. However, it is not a void principle: on the contrary, in \tilde{Q} is a set of correlations that is closed under wirings and can be characterized via semidefinite programming. It contains all correlations in Q_c\supset \bar{Q}, but also some non-quantum boxes P(a,b|x,y)\not\in Q_c. Remarkably, all boxes within the almost quantum set are shown to be compatible with the principles of NTCC, NANLC, ML and LO. There is also numerical evidence that almost-quantum boxes also comply with IC. It seems, therefore, that, even when the above principles are taken together, they do not suffice to single out the quantum set in the simplest Bell scenario of two parties, two inputs and two outputs. ==Device independent protocols==