A Physical Approach to Tsirelson’s Problem

Abstract

Tsirelson's problem deals with how to model separate measurements in quantum mechanics. In addition to its theoretical importance, the resolution of Tsirelson's problem could have great consequences for device independent quantum key distribution and certified randomness. Unfortunately, understanding present literature on the subject requires a heavy mathematical background. In this paper, we introduce quansality, a new theoretical concept that allows to reinterpret Tsirelson's problem from a foundational point of view. Using quansality as a guide, we recover all known results on Tsirelson's problem in a clear and intuitive way.

Physical characterization of quantum devices from nonlocal correlations

“…Note that as long as robust self-testing is possible the guess does not need to be exact, i.e., it is enough that p(a, b|x, y) = ψ|E Let us now discuss the conditions under which our method will certify that p(a, b|x, y) self-tests the mathematical guess (|ψ , {E x a , F y b }) in a robust way. Following the notation of [35], we say that p(a, b|x, y) is a 'relativistic' quantum distribution if there exist a state |ψ ∈ H and projection operators E…”

Quantum bounds for inequalities involving marginal expectation values

“…commute with each other. We choose to put the operators in separate Hilbert spaces [30,31]. The ±1 possible measurements are embedded in the operators' eigenvalues, e.g., spin projectors to the x and y spin axes.…”

Quantum Steering and Spacelike Separation

“…Tsirelson's non-locality problem consists, precisely, in deciding if both descriptions of separate measurements are equivalent at the level of correlations, i.e., if any distribution of the form P (a, b|x, y) = tr(E a x F b y ρ), with [E a x , F b y ] = 0, can be approximated by distributions of the formP (a, b) = tr(Ẽ a x ⊗F b yρ ). It can be proven that both models of space separation lead to the same set of correlations if all measurement operators involved are assumed to act over finite-dimensional Hilbert spaces [10,13]. Tsirelson's problem thus reduces to finding out if both sets of correlations also remain the same in infinite dimensions.…”

“…In other words: can all bipartite distributions of the form P (a, b|x, y) = tr{σE a x F b y }, with [E a x , F b y ] = 0, be approximated by distributions of the formP (a, b|x, y) = tr{σẼ a x ⊗ F b y }? In [13] it was shown that the answer to this question is positive if either s = d = 2, or the initial set of measurements {E a x } s x=1 does not allow Alice to induce heat vision in the joint system [15]. The heat vision effect refers to the phenomenon that certain collections of von Neumann measurements can bring the system to a non-convergent dynamics when applied randomly and sequentially.…”