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arXiv:1809.02414 · Journal of Physics A Mathematical and Theoretical

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

Julio I. de Vicente

Published Jan 11, 2019• 8 citations
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Abstract

Abstract The prepare-and-measure scenario offers the possibility to infer the dimension of an unknown physical system in a device-independent way, i.e. using only raw measurement data with apparatuses regarded as black boxes. We provide here a general lower bound on the dimension necessary to observe arbitrary quantum behaviours in this scenario based on simple matrix analysis. This bound holds even if the preparer and measurer share randomness. This is relevant in scenarios where the parties are free to access this resource or it is not safe to assume that the devices are not correlated. We further use this result to bound the success probability of random access codes in general as a function of the dimension of the quantum systems sent from one party to another and we provide constructions of dimension witnesses.

DOI
10.1088/1751-8121/aafde7

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Abstract The prepare-and-measure scenario offers the possibility to infer the dimension of an unknown physical system in a device-independent way, i.e. using only raw measurement data with apparatuses regarded as black boxes. We provide here a general lower bound on the dimension necessary to observe arbitrary quantum behaviours in this scenario based on simple matrix analysis. This bound holds even if the preparer and measurer share randomness. This is relevant in scenarios where the parties are free to access this resource or it is not safe to assume that the devices are not correlated. We further use this result to bound the success probability of random access codes in general as a function of the dimension of the quantum systems sent from one party to another and we provide constructions of dimension witnesses.
We address the problem of testing the dimensionality of classical and quantum systems in a "black-box" scenario. We develop a general formalism for tackling this problem. This allows us to derive lower bounds on the classical dimension necessary to reproduce given measurement data. Furthermore, we generalize the concept of quantum dimension witnesses to arbitrary quantum systems, allowing one to place a lower bound on the Hilbert space dimension necessary to reproduce certain data. Illustrating these ideas, we provide simple examples of classical and quantum dimension witnesses.
Device-independent dimension witnesses provide a lower bound on the dimensionality of classical and quantum systems in a ``black box''scenario where only correlations between preparations, measurements, and outcomes are considered. We address the problem of the robustness of dimension witnesses, namely that to witness the dimension of a system or to discriminate between its quantum or classical nature, even in the presence of loss. We consider the case when shared randomness is allowed between preparations and measurements and provide a threshold in the detection efficiency such that dimension witnessing can still be performed.
Analyzing the dimension of an unknown quantum system in a device-independent manner, i.e., using only the measurement statistics, is a fundamental task in quantum physics and quantum information theory. In this paper, we consider this problem in the prepare-and-measure scenario. Specifically, we provide a lower bound on the dimension of the prepared quantum systems, which is a function that only depends on the measurement statistics. Furthermore, we show that our bound performs well in several examples. In particular, we show that our bound provides insights into the notion of dimension witness, and we also use it to show that the sets of restricted-dimensional prepare-and-measure correlations are not always convex.
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 the 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. 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 behaviors 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 behaviors that 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 it 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 we prove some nonconvexity and negligibility properties of these sets for sufficiently small dimensions. This shows again the enormous difference induced by the availability (or lack thereof) of this resource.
We investigate whether one can detect the presence of a quantum resource in some operational task or equivalently whether every quantum resource provides an advantage over its free counterpart in some black box scenarios where one does not have much information about the devices. For any dimension d , we find that for any resource theory with less than d 2 number of linearly independent free states or free operations, there exist correlations that can detect the presence of a quantum resource. For this purpose, we introduce the framework for detecting quantum resources semidevice independently by considering the prepare-and-measure scenario with the restriction on the dimension of the quantum channel connecting the preparation box with the measurement box. We then explicitly construct witnesses to observe the presence of various quantum resources. We expect these results will open avenues for detecting and finding uses of quantum resources in general operational tasks.
We imagine an experiment on an unknown quantum mechanical system in which the system is prepared in various ways and a range of measurements are performed. For each measurement MMM and preparation $\ensuremath{\rho}$ the experimenter can determine, given enough time, the probability of a given outcome aaa: $p(a\ensuremath{\mid}M,\ensuremath{\rho})$. How large does the Hilbert space of the quantum system have to be in order to allow us to find density matrices and measurement operators that will reproduce the given probability distribution? In this paper, we prove a simple lower bound for the dimension of the Hilbert space. The main insight is to relate this problem to the construction of quantum random access codes, for which interesting bounds on the Hilbert space dimension already exist. We discuss several applications of our result to hidden-variable or ontological models, to Bell inequalities, and to properties of the smooth min-entropy.