Shared randomness and device-independent dimension witnessing

Abstract

It has been shown that the conditional probability distributions obtained by performing measurements on an uncharacterized physical system can be used to infer its underlying dimension in a device-independent way both in the classical and quantum setting. We analyze several aspects of the structure of the sets of probability distributions corresponding to a certain dimension taking into account whether shared randomness is available as a resource or not. We first consider the so-called prepare-and-measure scenario. We show that quantumness and shared randomness are not comparable resources. That is, on the one hand, there exist behaviours that require a quantum system of arbitrarily large dimension in order to be observed while they can be reproduced with a classical physical system of minimal dimension together with shared randomness. On the other hand, there exist behaviours which require exponentially larger dimensions classically than quantumly even if the former is supplemented with shared randomness. We also show that in the absence of shared randomness, the sets corresponding to a sufficiently small dimension are negligible (zero-measure and nowhere dense) both classically and quantumly. This is in sharp contrast to the situation in which this resource is available and explains the exceptional robustness of dimension witnesses in the setting in which devices can be taken to be uncorrelated. We finally consider the Bell scenario in the absence of shared randomness and prove some non-convexity and negligibility properties of these sets for sufficiently small dimensions. This shows again the enormous difference induced by the availability or not of this resource.

A general bound for the dimension of quantum behaviours in the prepare-and-measure scenario

“…Another example to consider this scenario is when the parties have to use their devices to implement a particular task and the constraints of the problem allow them to use shared randomness as a resource. In fact, it is known that the availability or not of shared randomness can have drastic consequences in what comes to the necessary dimension underlying a given observed behaviour; for instance, without this resource at disposal almost all behaviours are high-dimensional while, when it is given, low-dimension behaviours are no longer negligible [13]. The main goal of the present work is to close this gap and provide general analytical conditions to test the dimension of quantum behaviours in the prepare-andmeasure scenario even when devices might share randomness.…”

Witnessing Irreducible Dimension

“…DOI: 10.1103/PhysRevLett.119.080401 Introduction.-The Hilbert space dimension of a quantum system limits the amount of information that can be stored in it. The study of the power of fixed-dimensional systems is still topical today [1][2][3], and several experimental groups are implementing high-dimensional encoding and decoding of information [4][5][6]. Thus, for the purposes of quantum information processing, a proper certification of dimension should capture the users' capacity of exploiting that dimensionality, not just the dimension that "is there"-after all, the simplest particle or a single mode of any field are already infinite-dimensional.…”

“…The parties label each of the four-valued input and outcomes a; b; x; y ∈ f0; 1; 2; 3g as two bits: c ¼ 2c 1 [24]. Thus, (4) certifies that there are indeed two subsystems in a separable state, both on Alice's side (denoted A 1 and A 2 ) and on Bob's (B 1 and B 2 ).…”

Covariance Bell inequalities

“…We showed explicitly, on our simplest example, how the value of the Bell expression allows one to place device-independent lower bounds on the amount of shared randomness in a local model (both in terms of its dimension and its entropy). Such witnesses may help addressing certain problems in quantum nonlocality related to shared randomness [35][36][37], in particular finding what is the minimal amount of shared randomness necessary to simulate the correlations of entangled states admitting a local model [38]. Our example with binary inputs allows one to certify the use of relatively little shared randomness (e.g.…”