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arXiv:2007.16145 · Quantum

Informationally restricted correlations: a general framework for classical and quantum systems

Armin Tavakoli, Emmanuel Zambrini Cruzeiro, Erik Woodhead, Stefano Pironio

Published Jan 5, 2022• 27 citations
arXivHTMLPDF

Abstract

We introduce new methods and tools to study and characterise classical and quantum correlations emerging from prepare-and-measure experiments with informationally restricted communication. We consider the most general kind of informationally restricted correlations, namely the ones formed when the sender is allowed to prepare statistical mixtures of mixed states, showing that contrary to what happens in Bell nonlocality, mixed states can outperform pure ones. We then leverage these tools to derive device-independent witnesses of the information content of quantum communication, witnesses for different quantum information resources, and demonstrate that these methods can be used to develop a new avenue for semi-device independent random number generators.

DOI
10.22331/q-2022-01-05-620

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We introduce new methods and tools to study and characterise classical and quantum correlations emerging from prepare-and-measure experiments with informationally restricted communication. We consider the most general kind of informationally restricted correlations, namely the ones formed when the sender is allowed to prepare statistical mixtures of mixed states, showing that contrary to what happens in Bell nonlocality, mixed states can outperform pure ones. We then leverage these tools to derive device-independent witnesses of the information content of quantum communication, witnesses for different quantum information resources, and demonstrate that these methods can be used to develop a new avenue for semi-device independent random number generators.
A key aspect in quantum information is to understand the advantage offered by quantum systems over classical ones in communication tasks. In recent years, a fundamental approach to this problem has been developed, focusing on quantum correlations in prepare-and-measure scenarios. Inspired by the developments in Bell nonlocality and device-independent information processing, this line of research aims to characterize the possibilities and limits of quantum systems for communication, in particular to precisely capture the advantage they offer over classical systems. In addition to fundamental insights, these ideas also underpin the concept of semi-device-independent quantum information processing. Exploring trade-offs between security, performance and ease-of-implementation, this approach opens promising directions for novel quantum information processing technologies and devices. A number of protocols and proof-of-principle demonstrations have been reported in recent years, in particular for quantum randomness certification and key distribution. Here, we provide a comprehensive introduction to quantum prepare-and-measure correlations and semi-device independent applications.
Quantum communication leads to strong correlations, that can outperform classical ones. Complementary to previous works in this area, we investigate correlations in prepare-and-measure scenarios assuming a bound on the information content of the quantum communication, rather than on its Hilbert-space dimension. Specifically, we explore the extent of classical and quantum correlations given an upper bound on the one-shot accessible information. We provide a characterisation of the set of classical correlations and show that quantum correlations are stronger than classical ones. We also show that limiting information rather than dimension leads to stronger quantum correlations. Moreover, we present device-independent tests for placing lower bounds on the information given observed correlations. Finally, we show that quantum communication carrying log⁡d bits of information is at least as strong a resource as d -dimensional classical communication assisted by pre-shared entanglement.
Prepare-and-measure scenarios are among the simplest and most widely used settings for quantum communication. A sender prepares a physical system in one of several possible states, which a receiver then measures. Despite this simple structure, these scenarios offer a rich framework for exploring the differences between classical and quantum communication, revealing quantum advantages, and developing protocols for quantum cryptography and device certification. In this tutorial, we give a ground-up introduction to prepare-and-measure scenarios and the methods used to study them. We introduce different frameworks for constraining and characterising communication, with an emphasis on semi-device-independent approaches to quantum information. We also cover modern convex-optimisation techniques for characterising the correlations achievable in these scenarios and illustrate how these tools can be applied in practice. Our aim is to give newcomers the conceptual and technical foundations needed to approach the current literature and apply prepare-and-measure methods in their own research. We conclude with a discussion of open problems and directions for future work.
Quantum theory admits correlations between distant parties that are stronger than those permitted by classical physics. Such correlations, which are termed nonlocal, have been studied extensively since their discovery by John Bell in 1964. This thesis makes progress on various aspects of quantum nonlocality both from a foundational and practical perspective. We begin by studying a foundational question on the relationship between nonlocality and quantum entanglement. We show that the strongest form of nonlocality, also called pseudo-telepathy or full nonlocality, can be demonstrated by non-maximally entangled states and prove that all pure entangled states can demonstrate full nonlocality in the many-copy scenario. We also provide non-trivial examples of fully nonlocal mixed states. We then turn to the study of another phenomenon in quantum theory which is essential in almost all demonstrations of non-classicality—measurement incompatibility. We generalise the notion of measurement incompatibility and show that our notion has strong operational relevance: it captures the randomness in the outcomes of an untrusted measurement device against a classical adversary. We use our formalism to derive new bounds on the experimental losses and noise that can be tolerated in quantum communication protocols before they become insecure. We then begin a systematic study of routed Bell experiments, a variant of standard Bell experiments that was introduced to demonstrate nonlocality over long distances in the presence of losses and noise. We formalize the notion of genuine long-range quantum correlations in routed Bell experiments and develop numerical and analytical tools to characterize them. We show that routed Bell experiments allow for the demonstration of long-range quantum correlations at arbitrarily large distances using only entangled qubit states. We then prove the security of fully device-independent (DI) quantum key distribution (QKD) protocols based on routed Bell tests and quantify the resulting improvement in key rate and distance compared to standard DI QKD protocols.
Abstract A promising platform for semi-device-independent quantum information is prepare-and-measure experiments restricted only by a bound on the energy of the communication. Here, we investigate the role of shared entanglement in such scenarios. For classical communication, we derive a general correlation criterion for nonlocal resources and use it to show that entanglement can fail to be a resource in standard tasks. For quantum communication, we consider the basic primitive for energy-constrained communication, namely the probabilistic transmission of a bit, and show that the advantages of entanglement only can be unlocked by non-unitary encoding schemes that purposefully decohere the entangled state. We also find that these advantages can be increased by using entanglement of higher dimension than qubit. We leverage these insights to investigate the impact of entanglement for quantum random number generation, which is a standard application of these systems but whose security so far only has been established against classical side information. In the low-energy regime, our attacks on the protocol indicate that the security remains largely intact, thereby paving the way for strengthened security without more complex setups and with negligible performance reductions.
Quantum correlations in Bell and prepare-and-measure experiments are central resources for probing nonclassicality and enabling device-based quantum information protocols. In the absence of shared public randomness (i.e., without run-to-run mixing), even qubit correlation sets are typically nonconvex, making standard convex characterizations inadequate. Here we derive qubit-specific constraints from uncertainty relations, yielding a state-independent consistency test for observed statistics in both prepare-and-measure and Bell scenarios. The test captures explicit nonconvex boundaries in representative correlation families and enables correlation-based device inference by constraining (and sometimes uniquely determining) unitary-invariant measurement parameters even away from extreme points. Moreover, incorporating the inferred qubit constraints as additional conditions in a moment-matrix relaxation strengthens separability tests and can certify entanglement even for Bell-local correlations within the independent-device model. These tools provide a practical route to characterize and leverage low-dimensional quantum devices, including certification, randomness generation, and entanglement verification.