Tsirelson’s Bound

Introduction

Tsirelson’s bound is a fundamental concept in the theory of quantum mechanics and quantum information theory. Named after the mathematician Boris Tsirelson, this bound provides a limit to the quantum mechanical correlation that can be achieved between two separated systems, playing a key role in distinguishing quantum theory from other possible physical theories.

Mathematical Statement

Consider the scenario of a Bell-type experiment, where two parties (often referred to as Alice and Bob) each have a quantum system and can choose to measure their system in one of two ways, denoted for Alice and for Bob. The measurements are assumed to be dichotomic, meaning they yield either +1 or -1 as the result. In quantum mechanics, these measurements are represented by Hermitian operators acting on a shared quantum state .

In this context, Tsirelson’s bound refers to the maximal quantum violation of the CHSH inequality, a specific Bell inequality. Mathematically, this inequality can be written as:

where:

.

In quantum mechanics, however, correlations can be stronger and can reach up to a value of , which is Tsirelson’s bound.

Significance and Implications

Tsirelson’s bound has profound implications for the foundations of quantum theory. It helps to delineate the region of “quantum correlations” within the larger “no-signalling” polytope, which contains correlations that do not allow for faster-than-light communication.

This bound also has practical implications for quantum cryptography and quantum computing, playing a significant role in the security of quantum key distribution protocols and in the study of quantum computational complexity.

Tsirelson’s Problem

An interesting question, known as Tsirelson’s problem, concerns the set of quantum correlations in the multipartite scenario. Specifically, the question is whether the set of multipartite quantum correlations is convex or whether there exist some “extremal” quantum correlations that cannot be achieved by a convex combination of others. This question was resolved by Ji, Natarajan, Vidick, Wright and Yuen, who proved that the set is not convex.

Conclusion

Tsirelson’s bound serves as a cornerstone in the mathematical structure of quantum mechanics, distinguishing quantum correlations from those of any non-quantum theory. Its relevance extends beyond foundational aspects of quantum theory to the burgeoning fields of quantum information and quantum computation.

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