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Quantum Information and Computation

A non-distillability criterion for secret correlations

Lluís Masanes, Andreas Winter

Published Jan 1, 2010• 0 citations
Publisher

Abstract

Within entanglement theory there are criteria which certify that some quantum states cannot be distilled into pure entanglement. An example is the positive partial transposition criterion. Here we present, for the first time, the analogous thing for secret correlations. We introduce a computable criterion which certifies that a probability distribution between two honest parties and an eavesdropper cannot be (asymptotically) distilled into a secret key. The existence of non-distillable correlations with positive secrecy cost, also known as bound information, is an open question. This criterion may be the key for finding bound information. However, if it turns out that this criterion does not detect bound information, then, a very interesting consequence follows: any distribution with positive secrecy cost can increase the secrecy content of another distribution. In other words, all correlations with positive secrecy cost constitute a useful resource.

DOI
10.26421/qic10.1-2-11

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Within entanglement theory there are criteria which certify that some quantum states cannot be distilled into pure entanglement. An example is the positive partial transposition criterion. Here we present, for the first time, the analogous thing for secret correlations. We introduce a computable criterion which certifies that a probability distribution between two honest parties and an eavesdropper cannot be (asymptotically) distilled into a secret key. The existence of non-distillable correlations with positive secrecy cost, also known as bound information, is an open question. This criterion may be the key for finding bound information. However, if it turns out that this criterion does not detect bound information, then, a very interesting consequence follows: any distribution with positive secrecy cost can increase the secrecy content of another distribution. In other words, all correlations with positive secrecy cost constitute a useful resource.
Within entanglement theory there are criteria which certify that some quantum states cannot be distilled into pure entanglement. An example is the positive partial transposition criterion. Here we present, for the first time, the analogous thing for secret correlations. We introduce a computable criterion which certifies that a probability distribution between two honest parties and an eavesdropper cannot be (asymptotically) distilled into a secret key. The existence of non-distillable correlations with positive secrecy cost, also known as bound information, is an open question. This criterion may be the key for finding bound information. However, if it turns out that this criterion does not detect bound information, then, a very interesting consequence follows: any distribution with positive secrecy cost can increase the secrecy content of another distribution. In other words, all correlations with positive secrecy cost constitute a useful resource.
There exist bipartite quantum states that cost entanglement to create, yet from which no singlet can be distilled. This extreme irreversibility is known as bound entanglement. A quarter-century ago, Gisin and Wolf asked whether classical information theory admits the same phenomenon. Are there correlations, shared by two parties and an eavesdropper, that cost secret bits to create, yet from which none can be distilled? This question is part of a broader program exploring the relation between quantum and classical information theory. We answer it in the affirmative, providing a simple example, a distribution of two bits and a trit. Bound entanglement thus has a classical counterpart, bound information. An even more direct correspondence was originally conjectured, namely that measuring purifications of bound-entangled states yields distributions with bound information. The very states that motivated the conjecture, however, yield no bound information when measured in the standard basis, whereas suitable measurements of purifications of separable states do. The analogy between bound entanglement and bound information thus lies in the accounting of resources, not in a correspondence between individual quantum states and the classical probability distributions induced by measuring them.
One big problem of the communication between two parties is the secrecy. That means how\nmuch information a third party can obtain by intercepting the messages transmitted from\none honest party to the other one. Therefore cryptography offers a wide range of protocols\nto ensure security with assumptions on the eavesdropper. So one was looking for an\ninformation-theoretical description of the scenario to get unconditional secure communication.\nIn this scenario we are considering two honest parties that want to communicate over\nan authenticated channel that the eavesdropper is wiretapping. This scenario introduced the definition of the intrinsic information and the secret-key rate which are a measure of the secrecy in this setting. Later because of strong analogies to quantum mechanics it turned out that this description was lacking a phenomena called bound information which is the disability of a probability distribution to create a secret-key even though it has predicted secrecy.\nNearly ten years of research have shown the existence of bound information for the multipartite case where several parties are communicating but not yet for the bipartite case.\nHence the approach of non-distillability seems a very promising one to find this conjecture.\nMotivated by this the approach we implemented this tool and simulated some distributions that have conjectured bound information. Thereby we improved the tool to reduce its calculationtime and to get closer to the aim.
Non-additivity is one of the distinctive traits of Quantum Information Theory: the combined use of quantum objects may be more advantageous than the sum of their individual uses. Non-additivity effects have been proven, for example, for quantum channel capacities, entanglement distillation or state estimation. In this work, we consider whether non-additivity effects can be found in Classical Information Theory. We work in the secret-key agreement scenario in which two honest parties, having access to correlated classical data that are also correlated to an eavesdropper, aim at distilling a secret key. Exploiting the analogies between the entanglement and the secret-key agreement scenario, we provide some evidence that the secret-key rate may be a non-additive quantity. In particular, we show that correlations with conjectured bound information become secret-key distillable when combined. Our results constitute a new instance of the subtle relation between the entanglement and secret-key agreement scenario.
Non-additivity is one of the distinctive traits of Quantum Information Theory: the combined use of quantum objects may be more advantageous than the sum of their individual uses. Non-additivity effects have been proven, for example, for quantum channel capacities, entanglement distillation or state estimation. In this work, we consider whether non-additivity effects can be found in Classical Information Theory. We work in the secret-key agreement scenario in which two honest parties, having access to correlated classical data that are also correlated to an eavesdropper, aim at distilling a secret key. Exploiting the analogies between the entanglement and the secret-key agreement scenario, we provide some evidence that the secret-key rate may be a non-additive quantity. In particular, we show that correlations with conjectured bound information become secret-key distillable when combined. Our results constitute a new instance of the subtle relation between the entanglement and secret-key agreement scenario.