We develop the resource theory of private randomness extraction in the distributed and device-dependent scenario. We begin by introducing the notion of independent random bits, which are bipartite states containing ideal private randomness for each party, and motivate the natural set of free operations. As a conceptual tool, we introduce virtual quantum state merging, which is essentially the flip side of quantum state merging, without communication. We focus on the bipartite case and find the rate regions achievable in different settings. Surprisingly, it turns out that local noise can boost randomness extraction. As a consequence of our analysis, we resolve a long-standing problem by giving an operational interpretation for the reverse coherent information (up to a constant term logd) as the number of private random bits obtained by sending quantum states from one honest party (server) to another one (client) via the eavesdropped quantum channel.
We develop the resource theory of private randomness extraction in the distributed and device-dependent scenario. We begin by introducing the notion of independent random bits, which are bipartite states containing ideal private randomness for each party, and motivate the natural set of free operations. As a conceptual tool, we introduce virtual quantum state merging, which is essentially the flip side of quantum state merging, without communication. We focus on the bipartite case and find the rate regions achievable in different settings. Surprisingly, it turns out that local noise can boost randomness extraction. As a consequence of our analysis, we resolve a long-standing problem by giving an operational interpretation for the reverse coherent information (up to a constant term logd) as the number of private random bits obtained by sending quantum states from one honest party (server) to another one (client) via the eavesdropped quantum channel.
We investigate the problem of producing local pure states by performing local noisy operations assisted by one-way classical communication on a given bipartite mixed state in the one-shot setting. We consider the following two scenarios:1)Scenario I: A party, say Alice, is provided with a single copy of some quantum state ρAon system A. The task for Alice is to extract pure qubit states using only noisy operations on A. We call this task purity concentration.2)Scenario II: Two parties, Alice and Bob possess the A and B sub-systems, respectively, of a given bipartite quantum state ρAB. They are allowed to perform any local noisy operations and communicate via a one-way dephasing (i.e., classical) channel. The task for them is to design a protocol using these resources such that together they can extract pure local qubit states from the shared state ρAB. We call this task local purity distillation.
Secret-key distillation from quantum states and channels is a central task of interest in quantum information theory, as it facilitates private communication over a quantum network. Here, we study the task of secret-key distillation from bipartite states and point-to-point quantum channels using local operations and one-way classical communication (one-way LOCC). We employ the resource theory of unextendible entanglement to study the transformation of a bipartite state under one-way LOCC, and we obtain several efficiently computable upper bounds on the number of secret bits that can be distilled from a bipartite state using one-way LOCC channels; these findings apply not only in the one-shot setting but also in some restricted asymptotic settings. We extend our formalism to private communication over a quantum channel assisted by forward classical communication. We obtain efficiently computable upper bounds on the one-shot forward-assisted private capacity of a channel, thus addressing a question in the theory of quantum-secured communication that has been open for some time now. Our formalism also provides upper bounds on the rate of private communication when using a large number of channels in such a way that the error in the transmitted private data decreases exponentially with the number of channel uses. Moreover, our bounds can be computed using semidefinite programs, thus providing a computationally feasible method to understand the limits of private communication over a quantum network.
We give the first systematic study on the second order asymptotics of the operational task of coherence distillation with and without assistance. In the unassisted setting, we introduce a variant of randomness extraction framework where free incoherent operations are allowed before the incoherent measurement and the randomness extractors. We then show that the maximum number of random bits extractable from a given quantum state is precisely equal to the maximum number of coherent bits distillable from the same state. This relation enables us to derive tight second order expansions of both tasks in the independent and identically distributed setting. Remarkably, the incoherent operation classes that can empower coherence distillation for generic states all admit the same second order expansions, indicating their operational equivalence for coherence distillation in both asymptotic and large block length regimes. We then generalize the above line of research to the assisted setting, arising naturally in bipartite quantum systems where Bob distills coherence from the state at hand, aided by the benevolent Alice possessing the other system. More precisely, we introduce a new assisted incoherent randomness extraction task and establish an exact relation between this task and the assisted coherence distillation. It strengthens the one-shot relation in the unassisted setting and confirms that this cryptographic framework offers a new perspective to the study of quantum coherence distillation. Likewise, this relation yields second order characterizations to the assisted tasks. As by-products, we show the strong converse property of the tasks above from their second order expansions.
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.
Quantum communication relies on the existence of high quality quantum channels to exchange information. In practice, however, all communication links are affected by noise from the environment. Here we investigate the ability of quantum channels to perform quantum communication tasks by restricting the participants to use only local operations and one-way classical communication (one-way LOCC) along with the available quantum channel. In particular, a channel can be used to distill a highly entangled state between two parties, which further enables quantum or private communication. In this work, we invoke the framework of superchannels to study the distillation of a resourceful quantum state, such as a maximally entangled state or a private state, using multiple instances of a point-to-point quantum channel. We use the idea ofk-extendibility to obtain a semidefinite relaxation of the set of one-way LOCC superchannels and define a class of entanglement measures for quantum channels that decrease monotonically under such superchannels; therefore these measures, dubbed collectively the “unextendible entanglement of a channel”, yield upper bounds on several communication-theoretic quantities of interest in the regimes of resource distillation and zero error.We then generalize the formalism ofk-extendibility to bipartite superchannels, thus obtaining functions that are monotone under two-extendible superchannels. This allows us to analyze probabilistic distillation of ebits or secret key bits from a bipartite state when using a resourceful quantum channel. Moreover, we propose semidefinite programs to evaluate several of these quantities, providing a computationally feasible method of comparison between quantum channels for resource distillation.
Given a bipartite quantum system, correlations between its subsystems can be understood as information that each one carries about the other. In order to give a model-independent description of secure information disposal, we propose the paradigm of private quantum decoupling, corresponding to locally reducing correlations in a given bipartite state without transferring them to the environment. In this framework, the concept of private local randomness naturally arises as a resource, and total correlations get divided into eliminable and ineliminable ones. We prove upper and lower bounds on the amount of ineliminable correlations present in an arbitrary bipartite state, and show that, in tripartite pure states, ineliminable correlations satisfy a monogamy constraint, making apparent their quantum nature. A relation with entanglement theory is provided by showing that ineliminable correlations constitute an entanglement parameter. In the limit of infinitely many copies of the initial state provided, we compute the regularized ineliminable correlations to be measured by the coherent information, which is thus equipped with a new operational interpretation. In particular, our results imply that two subsystems can be privately decoupled if their joint state is separable.