Optimal non-signalling violations via tensor norms
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Optimal non-signalling violations via tensor norms Abderramán Amr Rey1
· Carlos Palazuelos1,2 · Ignacio Villanueva1,2,3
Received: 13 May 2019 / Accepted: 16 September 2019 © Universidad Complutense de Madrid 2019
Abstract In this paper we characterize the set of bipartite non-signalling probability distributions in terms of tensor norms. Using this characterization we give optimal upper and lower bounds on Bell inequality violations when non-signalling distributions are considered. Interestingly, our upper bounds show that non-signalling Bell inequality violations cannot be significantly larger than quantum Bell inequality violations. Keywords Quantum information · Bell inequalities · Non-signalling distributions · Tensor norms Mathematics Subject Classification 81P45 · 46B28
1 Introduction A very remarkable feature of quantum mechanics is that it predicts the existence of experimental data which cannot be reproduced within any local and realistic physical theory, even in the presence of hidden variables. This idea was first formalized by Bell [3] and has played a major role in the recent development of quantum information science (see the survey [5]).
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Abderramán Amr Rey [email protected] Carlos Palazuelos [email protected] Ignacio Villanueva [email protected]
1
Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain
2
Instituto de Ciencias Matemáticas (ICMAT), C/ Nicolás Cabrera, Campus de Cantoblanco, 28049 Madrid, Spain
3
Instituto Matemáticas Interdisciplinar (IMI), Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain
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A. Amr et al.
One of the main ideas in Bell’s work, non-locality, can be studied in itself, independently from quantum mechanics. Bell’s scenario is usually described by two parties spatially separated, typically named Alice and Bob, who perform different measurements to obtain certain outputs. If we label Alice’s and Bob’s measurement devices by x and y respectively so that x, y = 1, . . . , N and Alice’s and Bob’s possible outputs by a and b respectively so that a, b = 1, . . . , K , the main object of study is the tensor N ,K , P = {P(a, b|x, y)}x,y;a,b=1
where P(a, b|x, y) denotes the probability that Alice and Bob obtain the pair of outputs (a, b) when they measure with the inputs x and y respectively. Note that, 2 2 from an algebraic point of view, each P is just an element in R N K . Moreover, the fact that P describes a measurement scenario implies that certain restrictions must be fulfilled; namely, P(a, b|x, y) ≥ 0 and a,b P(a, b|x, y) = 1 for every a, b, x, y. 2
2
Let us denote by C the subset of R N K given by such elements. We will refer to them as probability distributions. The main point in Bell’s work was to understand that the assumption of a physical theory to explain the experiment (and, more generally, Nature) leads to a subset of C which will be formed by those probability distributions which are compatible with such a theory. A minimal requirement for a t
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