We study the properties of no-signaling correlations that cannot be reproduced by local measurements on entangled quantum states. We say that such correlations violate Tsirelson bounds. We show that if these correlations are obtained by some reversible unitary quantum evolution $U$, then $U$ cannot be written in the product form ${U}_{A}\ensuremath{\bigotimes}{U}_{B}$. This implies that $U$ can be used for signaling and for entanglement generation. This result is completely general and in fact can be viewed as a characterization of Tsirelson bounds. We then show how this result can be used as a tool to study Tsirelson bounds and we illustrate this by rederiving the Tsirelson bound of $2\sqrt{2}$ for the Clauser-Horn-Shimony-Holt inequality, and by deriving a new Tsirelson bound for qutrits.
We study the properties of no-signaling correlations that cannot be reproduced by local measurements on entangled quantum states. We say that such correlations violate Tsirelson bounds. We show that if these correlations are obtained by some reversible unitary quantum evolution U, then U cannot be written in the product form UA\ensuremath⨂UB. This implies that U can be used for signaling and for entanglement generation. This result is completely general and in fact can be viewed as a characterization of Tsirelson bounds. We then show how this result can be used as a tool to study Tsirelson bounds and we illustrate this by rederiving the Tsirelson bound of 22 for the Clauser-Horn-Shimony-Holt inequality, and by deriving a new Tsirelson bound for qutrits.
We describe a new technique for obtaining Tsirelson bounds, which are upper bounds on the quantum value of a Bell inequality. Since quantum correlations do not allow signalling, we obtain a Tsirelson bound by maximizing over all no-signalling probability distributions. This maximization can be cast as a linear programme. In a setting where three parties, A, B and C, share an entangled quantum state of arbitrary dimension, we (i) bound the trade-off between AB's and AC's violation of the Clauser–Horne–Shimony–Holt inequality and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities, relevant for interactive proof systems and cryptography.
Abstract In this article, we investigate classical simulations of quantum-type probabilities and correlations that defy Boole’s conditions of possible experience, specifically the Clauser-Horne-Shimony-Holt inequality, and even surpass the Tsirelson bound. We demonstrate that such violations can be achieved by encoding a single bit to communicate the measurement context. Our findings highlight the crucial role of context communication in reproducing quantum correlations that are stronger than classical limits, providing insights into the fundamental principles underlying entangled systems.
Quantum theory imposes a strict limit on the strength of nonlocal correlations. It only allows for a violation of the Clauser, Horne, Shimony, and Holt (CHSH) inequality up to the value 22, known as Tsirelson's bound. In this paper, we consider generalized CHSH inequalities based on many measurement settings with two possible measurement outcomes each. We demonstrate how to prove Tsirelson bounds for any such generalized CHSH inequality using semidefinite programming. As an example, we show that for any shared entangled state and observables X1,…,Xn and Y1,…,Yn with eigenvalues $\ifmmode\pm\else\textpm\fi{}1$ we have $\ensuremath{\mid}⟨{X}_{1}{Y}_{1}⟩+⟨{X}_{2}{Y}_{1}⟩+⟨{X}_{2}{Y}_{2}⟩+⟨{X}_{3}{Y}_{2}⟩+\ensuremath{\cdots}+⟨{X}_{n}{Y}_{n}⟩\ensuremath{-}⟨{X}_{1}{Y}_{n}⟩\ensuremath{\mid}\ensuremath{\leqslant}2n\phantom{\rule{0.2em}{0ex}}\mathrm{cos}[\ensuremath{\pi}∕(2n)]$. It is well known that there exist observables such that equality can be achieved. However, we show that these are indeed optimal. Our approach can easily be generalized to other inequalities for such observables.
A no-signaling theory may, in principle, be more nonlocal than quantum mechanics: the Popescu--Rohrlich box reaches the algebraic Clauser--Horne--Shimony--Holt value 4, whereas nature stops at the Tsirelson value 22. Absolute Frame Theory (AFT) produces nonlocal correlations---two systems entangled in M share a single point of the substratum A (Axiom II)---and forbids superluminal signaling (the channel is Bremermann-bounded). We ask where those correlations fall, and we establish, by two independent routes, that they cannot exceed the Tsirelson bound: AFT admits no super-quantum correlations. The operator route proves, inside the reconstructed Hilbert theory, the Tsirelson operator identity, with the local commutativity [Ai,Bj]=0 itself derived from AFT structure---the tensor factorization that local tomography supplies and the field microcausality of the constrained embedding---rather than assumed. The information route reduces the bound to a physical principle: AFT's shared resource is read out through a completely positive, trace-preserving channel of finite classical capacity (a theorem of the channel, founded on action quantization and the trace over the inaccessible fiber), this channel obeys the data-processing inequality, and the combination places AFT in the class that satisfies Information Causality, which singles out 22 among the no-signaling theories. We state plainly what this does and does not achieve. It does not close the question of why the bound is 22 rather than some other value: both routes, chased to their foundations, terminate at the same posited fact---the M--A signature transition (the Wick rotation) that installs the complex-Hilbert structure---and finite capacity alone, stripped of that structure, does not exclude a Popescu--Rohrlich box. The two routes are thus the operator-algebraic and information-theoretic faces of one ceiling, and they reduce the selection of 22 to a single substratum fact, with the residue an instance of the deepest open question of the programme rather than a problem specific to nonlocal correlations.