In this paper we derive a new upper bound on the quantum key distillation capacity. This upper bound is an extension of the classical bound of Gohari and Anantharam on the source model problem. Our bound strictly improves the quantum extension of reduced intrinsic information bound of Christandl et al. Although this bound is proposed for quantum settings, it also serves as an upper bound for the special case of classical source model, and may improve the bound of Gohari and Anantharam. The problem of quantum key distillation is one in which two distant parties, Alice and Bob, and an adversary, Eve, have access to copies of quantum systems A, B, E respectively, prepared jointly according to an arbitrary state ρABE. Alice and Bob desire to distill secret key bits that are secure from Eve, using only local quantum operations and authenticated public classical communication (LOPC).
In this paper we derive a new upper bound on the quantum key distillation capacity. This upper bound is an extension of the classical bound of Gohari and Anantharam on the source model problem. Our bound strictly improves the quantum extension of reduced intrinsic information bound of Christandl et al. Although this bound is proposed for quantum settings, it also serves as an upper bound for the special case of classical source model, and may improve the bound of Gohari and Anantharam. The problem of quantum key distillation is one in which two distant parties, Alice and Bob, and an adversary, Eve, have access to copies of quantum systems A, B, E respectively, prepared jointly according to an arbitrary state ρABE. Alice and Bob desire to distill secret key bits that are secure from Eve, using only local quantum operations and authenticated public classical communication (LOPC).
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.
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 pXYZ that is shared with an unwanted eavesdropper (Eve). In the quantum settings, the correlations 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 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.
We consider the task of distilling local purity from a noisy quantum state ρABC, wherein we provide a protocol for three parties, Alice, Bob and Charlie, to distill local purity (at a rate P) from many independent copies of a given quantum state ρABC. The three parties have access to their respective subsystems of ρABC, and are only allowed to use local unitary operations. In addition, Alice and Bob can communicate with Charlie using a one-way multiple-access dephasing channel of link rates R1and R2, respectively. The objective of the protocol is to minimize the usage of the dephasing channel (in terms of rates R1and R2) while maximizing the asymptotic purity that can be jointly distilled from ρABC. To achieve this, we employ ideas from distributed measurement compression protocols, and in turn, characterize a set of sufficient conditions on (P, R1,R2) in terms of quantum information theoretic quantities such that P amount of purity can be distilled using rates R1and R2.
The theory of quantum cryptography aims to guarantee unconditional information-theoretic security against an omnipotent eavesdropper. In many practical scenarios, however, the assumption of an all-powerful adversary is excessive and can be relaxed considerably. In this paper we study secret-key distillation across a lossy and noisy quantum wiretap channel between Alice and Bob, with a separately parameterized realistically lossy quantum channel to the eavesdropper Eve. We show that under such restricted eavesdropping, the key rates achievable can exceed the secret-key-distillation capacity against an unrestricted eavesdropper in the quantum wiretap channel. Furthermore, we show upper bounds on the key rates based on the relative entropy of entanglement. This simple restricted eavesdropping model is widely applicable, for example, to free-space quantum optical communication, where realistic collection of light by Eve is limited by the finite size of her optical aperture. Future work will include calculating bounds on the amount of light Eve can collect under various realistic scenarios.
Local pure states are an important resource for quantum computing. The problem of distilling local pure states from mixed ones can be cast in an information theoretic paradigm. The bipartite version of this problem where local purity must be distilled from an arbitrary quantum state shared between two parties, Alice and Bob, is closely related to the problem of separating quantum and classical correlations in the state and in particular, to a measure of classical correlations called the one-way distillable common randomness. In Phys. Rev. A 71, 062303 (2005), the optimal rate of local purity distillation is derived when many copies of a bipartite quantum state are shared between Alice and Bob, and the parties are allowed unlimited use of a unidirectional dephasing channel. In the present paper, we extend this result to the setting in which the use of the channel is bounded. We demonstrate that in the case of a classical-quantum system, the expression for the local purity distilled is efficiently computable and provide examples with their tradeoff curves.
After carrying out a protocol for quantum key agreement over a noisy quantum channel, the parties Alice and Bob must process the raw key in order to end up with identical keys about which the adversary has virtually no information. In principle, both classical and quantum protocols can be used for this processing. It is a natural question which type of protocols is more powerful. We prove for general states but under the assumption of incoherent eavesdropping that Alice and Bob share some so-called intrinsic information in their classical random variables, resulting from optimal measurements, if and only if the parties' quantum systems are entangled. In addition, we provide evidence that the potentials of classical and of quantum protocols are equal in every situation. Consequently, many techniques and results from quantum information theory directly apply to problems in classical information theory, and vice versa. For instance, it was previously believed that two parties can carry out unconditionally secure key agreement as long as they share some intrinsic information in the adversary's view. The analysis of this purely classical problem from the quantum information-theoretic viewpoint shows that this is true in the binary case, but false in general. More explicitly, bound entanglement, i.e., entanglement that cannot be purified by any quantum protocol, has a classical counterpart. This "bound intrinsic information" cannot be distilled to a secret key by any classical protocol. As another application we propose a measure for entanglement based on classical information-theoretic quantities.