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IEEE International Symposium on Information Theory (ISIT)

Secret, public and quantum correlation cost of triples of random variables

Andreas Winter

Published Jan 1, 2005• 39 citations
Publisher

Abstract

The inverse of Maurer's secret key distillation problem from (many independent realisations of) a triple of random variables X, Y, Z by two players (Alice and Bob) against an eavesdropper (Eve) is considered: the formation of the joint distribution (up to local degrading of Z) from secret key and public communication. We determine the asymptotically minimal amount of secret key for this task, and indeed the full trade-off of secret (between Alice and Bob) vs. public (shared between Alice, Bob and Eve) correlation for this problem. Our result generalises a theorem of Wyner on the "common information of a pair of random variables", which is recovered as the special case of Z being independent of XY. We investigate the secret key required as a function of the probability distribution and compare to an analogous notion based on prior shared entanglement

DOI
10.1109/isit.2005.1523752

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The inverse of Maurer's secret key distillation problem from (many independent realisations of) a triple of random variables X, Y, Z by two players (Alice and Bob) against an eavesdropper (Eve) is considered: the formation of the joint distribution (up to local degrading of Z) from secret key and public communication. We determine the asymptotically minimal amount of secret key for this task, and indeed the full trade-off of secret (between Alice and Bob) vs. public (shared between Alice, Bob and Eve) correlation for this problem. Our result generalises a theorem of Wyner on the "common information of a pair of random variables", which is recovered as the special case of Z being independent of XY. We investigate the secret key required as a function of the probability distribution and compare to an analogous notion based on prior shared entanglement
We study and solve the problem of distilling a secret key from quantum states representing correlation between two parties (Alice and Bob) and an eavesdropper (Eve) via one–way public discussion: we prove a coding theorem to achieve the ‘wire–tapper’ bound, the difference of the mutual information Alice–Bob and that of Alice–Eve, for so–called classical–quantum–quantum–correlations, via one–way public communication. This result yields information–theoretic formulae for the distillable secret key, giving ‘ultimate’ key rate bounds if Eve is assumed to possess a purification of Alice and Bob's joint state. Specializing our protocol somewhat and making it coherent leads us to a protocol of entanglement distillation via one–way LOCC (local operations and classical communication) which is asymptotically optimal: in fact we prove the so–called ‘hashing inequality’, which says that the coherent information (i.e. the negative conditional von Neumann entropy) is an achievable Einstein–Podolsky–Rosen rate. This result is known to imply a whole set of distillation and capacity formulae, which we briefly review.
In this paper, we consider the problem of generating arbitrary three-party correlations from a combination of public and secret correlations. Two parties-called Alice and Bob-share perfectly correlated bits that are secret from a collaborating third party, Charlie. At the same time, all three parties have access to a separate source of correlated bits, and their goal is to convert these two resources into multiple copies of some given tripartite distribution P(XYZ). We obtain a single-letter characterization of the tradeoff between public and private bits that are needed to achieve this task. The rate of private bits is shown to generalize Wyner's classic notion of common information held between a pair of random variables. The problem we consider can be contrasted fruitfully with the task of secrecy formation, in which P(XYZ) is generated using public communication and local randomness but with Charlie functioning as an adversary instead of a collaborator. We describe in detail the differences between the collaborative and adversarial scenarios.
We consider the extraction of shared secret key from correlations that are generated by either a classical or quantum source. In the classical setting, two honest parties (Alice and Bob) use public discussion and local randomness to distill secret key from some distribution pXYZp_{XYZ}pXYZ​ that is shared with an unwanted eavesdropper (Eve). In the quantum settings, the correlations pXYZp_{XYZ}pXYZ​ are delivered to the parties as either an \textit{incoherent} mixture of orthogonal quantum states or as \textit{coherent} superposition of such states; in both cases, Alice and Bob use public discussion and local quantum operations to distill secret key. While the power of quantum mechanics increases Alice and Bob's ability to generate shared randomness, it also equips Eve with a greater arsenal of eavesdropping attacks. Therefore, it is not obvious who gains the greatest advantage for distilling secret key when replacing a classical source with a quantum one. In this paper we first demonstrate that the classical key rate is equivalent to the quantum key rate when the correlations are generated incoherently in the quantum setting. For coherent sources, we next show that the rates are incomparable, and in fact, their difference can be arbitrarily large in either direction. However, we identify a large class of non-trivial distributions pXYZp_{XYZ}pXYZ​ that possess the following properties: (i) Eve's advantage is always greater in the quantum source than in its classical counterpart, and (ii) for the quantum entanglement shared between Alice and Bob in the coherent source, the so-called entanglement cost/squashed entanglement/relative entropy of entanglement can all be computed. With property (ii), we thus present a rare instance in which the various entropic entanglement measures of a quantum state can be explicitly calculated.
Alice and Bob want to share a secret key and to communicate an independent message, both of which they desire to be kept secret from an eavesdropper Eve. We study this problem of secret communication and secret key generation when two resources are available – correlated sources at Alice, Bob, and Eve, and a noisy broadcast channel from Alice to Bob and Eve which is independent of the sources. We are interested in characterizing the fundamental trade-off between the rates of the secret message and secret key. We present an achievable solution and prove its optimality for the parallel channels and sources case when each sub-channel and source component satisfies a degradation order (either in favor of the legitimate receiver or the eavesdropper). This includes the case of jointly Gaussian sources and an additive Gaussian channel, for which the secrecy region is evaluated. I.
Consider a scenario in which two parties Alice and Bob as well as an opponent Eve receive the output of a binary symmetric source (e,g. installed in a satellite) over individual, not necessarily independent binary symmetric channels. Alice and Bob share no secret key initially and can only communicate over a public channel completely accessible to Eve. The authors derive a lower bound on the rate at which Alice and Bob can generate secret-key bits about which Eve has arbitrarily little information, This lower bound is strictly positive as long as Eve's binary symmetric channel is not perfect, even if Alice's and Bob's channels are by orders of magnitude less reliable than Eve's channel.>
We consider distillation of secret bits from partially secret noisy correlations PABE, shared between two honest parties and an eavesdropper. The most studied distillation scenario consists of joint operations on a large number of copies of the distribution (PABE)N, assisted with public communication. Here we consider distillation with only one copy of the distribution, and instead of rates, the ¿quality¿ of the distilled secret bits is optimized, where the ¿quality¿ is quantified by the secret-bit fraction of the result. The secret-bit fraction of a binary distribution is the proportion which constitutes a secret bit between Alice and Bob. With local operations and public communication the maximal extractable secret-bit fraction from a distribution PABEis found, and is denoted by ¿[PABE]. This quantity is shown to be nonincreasing under local operations and public communication, and nondecreasing under eavesdropper's local operations: ¿ is a secrecy monotone. It is shown that if ¿ [PABE] > 1/2 then PABEis distillable, thus providing a sufficient condition for distillability. A simple expression for ¿[PABE] is found when the eavesdropper is decoupled, and when the honest parties' information is binary and the local operations are reversible. Intriguingly, for general distributions the (optimal) operation requires local degradation of the data.