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arXiv:1505.01736 · Physical Review A

Testing dimension and nonclassicality in communication networks

Joseph Bowles, Nicolas Brunner, Marcin Pawłowski

Published Aug 24, 2015• 35 citations
arXivHTMLPDF

Abstract

We consider networks featuring preparation, transformation, and measurement devices, in which devices exchange communication via mediating physical systems. We investigate the problem of testing the dimension of the mediating systems in the device-independent scenario, that is, based on observable data alone. A general framework for tackling this problem is presented, considering both classical and quantum systems. These methods can then also be used to certify the nonclassicality of the mediating systems, given an upper bound on their dimension. Several case studies are reported, which illustrate the relevance of the framework. These examples also show that, for fixed dimension, quantum systems largely outperform classical ones. Moreover, the use of a transformation device considerably improves noise tolerance when compared to simple prepare-and-measure networks. These results suggest that the classical simulation of quantum systems becomes costly in terms of dimension, even for simple networks.

DOI
10.1103/physreva.92.022351

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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.
We consider the problem of testing the dimension of uncharacterized classical and quantum systems in a prepare-and-measure setup. Here we assume the preparation and measurement devices to be independent, thereby making the problem nonconvex. We present a simple method for generating nonlinear dimension witnesses for systems of arbitrary dimension. The simplest of our witnesses is highly robust to technical imperfections, and can certify the use of qubits in the presence of arbitrary noise and arbitrarily low detection efficiency. Finally, we show that this witness can be used to certify the presence of randomness, suggesting applications in quantum information processing.
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.
We show that the entropy of a message can be tested in a device-independent way. Specifically, we consider a prepare-and-measure scenario with classical or quantum communication, and develop two different methods for placing lower bounds on the communication entropy, given observable data. The first method is based on the framework of causal inference networks. The second technique, based on convex optimization, shows that quantum communication provides an advantage over classical communication, in the sense of requiring a lower entropy to reproduce given data. These ideas may serve as a basis for novel applications in device-independent quantum information processing.
Preparing and measuring physical systems are the operational building blocks of any physical experiment, and to describe them is the first purpose of any physical theory.Remarkably, even when only uncharacterized preparation and measurement devices are present, it is sometimes possible to distinguish between the behaviors of quantum and classical systems from only observational data.Certifying the physical origin of measurement statistics in the prepare and measure scenario is of primal importance for developing quantum networks, distributing quantum keys, and certifying randomness, to mention a few applications, but, surprisingly, no general methods to do so are known.We progress on this problem by crafting a general, sufficient condition to certify that a given set of preparations can only generate classical statistics, for any number of generalized measurements.As an application, we employ the method to demonstrate nonclassicality activation in the prepare and measure scenario, also considering its application in random access codes.Following that, we adapt our method to certify, again through a sufficient condition, whether a given set of measurements can never give rise to nonclassical behaviors, irrespective of what preparations they may act upon.This, in turn, allows us to find a large set of incompatible measurements that cannot be used to demonstrate nonclassicality, thus showing incompatibility is not sufficient for nonclassicality in the prepare and measure scenario.
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.