In this paper, we develop the resource theory of quantum secret key. Operating under the assumption that entangled states with zero distillable key do not exist, we define the key cost of a quantum state, and device. We study its properties through the lens of a quantity that we call the key of formation. The main result of our paper is that the regularized key of formation is an upper bound on the key cost of a quantum state. The core protocol underlying this result is privacy dilution, which converts states containing ideal privacy into ones with diluted privacy. Next, we show that the key cost is bounded from below by the regularized relative entropy of entanglement, which implies the irreversibility of the privacy creation-distillation process for a specific class of states. We further focus on mixed-state analogues of pure quantum states in the domain of privacy, and we prove that a number of entanglement measures are equal to each other for these states, similar to the case of pure entangled states. The privacy cost and distillable key in the single-shot regime exhibit a yield-cost relation, and basic consequences for quantum devices are also provided. Importantly, our results presented here will remain valid even if entangled states with zero distillable key were shown to exist.
In this paper, we develop the resource theory of quantum secret key. Operating under the assumption that entangled states with zero distillable key do not exist, we define the key cost of a quantum state, and device. We study its properties through the lens of a quantity that we call the key of formation. The main result of our paper is that the regularized key of formation is an upper bound on the key cost of a quantum state. The core protocol underlying this result is privacy dilution, which converts states containing ideal privacy into ones with diluted privacy. Next, we show that the key cost is bounded from below by the regularized relative entropy of entanglement, which implies the irreversibility of the privacy creation-distillation process for a specific class of states. We further focus on mixed-state analogues of pure quantum states in the domain of privacy, and we prove that a number of entanglement measures are equal to each other for these states, similar to the case of pure entangled states. The privacy cost and distillable key in the single-shot regime exhibit a yield-cost relation, and basic consequences for quantum devices are also provided. Importantly, our results presented here will remain valid even if entangled states with zero distillable key were shown to exist.
The notions of privacy tests and k-extendible states have both been instrumental in quantum information theory, particularly in understanding the limits of secure communication. In this paper, we determine the maximum probability with which an arbitrary k-extendible state can pass a privacy test, and we prove that it is equal to the maximum fidelity between an arbitrary k-extendible state and the standard maximally entangled state. Our findings, coupled with the resource theory of k-unextendibility, lead to an efficiently computable upper bound on the one-shot, one-way distillable key of a bipartite state, and we prove that it is equal to the best-known efficiently computable upper bound on the one-shot, one-way distillable entanglement. We also establish efficiently computable upper bounds on the one-shot, forward-assisted private capacity of channels. Extending our formalism to the independent and identically distributed setting, we obtain single-letter efficiently computable bounds on the n-shot, one-way distillable key of a state and the n-shot, forward-assisted private capacity of a channel. For some key examples of interest, our bounds are significantly tighter than other known efficiently computable bounds.
Although it is well known that the amount of resources that can be asymptotically distilled from a quantum state or channel does not exceed the resource cost needed to produce it, the corresponding relation in the nonasymptotic regime hitherto has not been well understood.Here, we establish a quantitative relation between the one-shot distillable resource yield and dilution cost in terms of transformation errors involved in these processes.Notably, our bound is applicable to quantum state and channel manipulation with respect to any type of quantum resource and any class of free transformations thereof, encompassing broad types of settings, including entanglement, quantum thermodynamics, and quantum communication.We also show that our techniques provide strong converse bounds relating the distillable resource and the resource dilution cost in the asymptotic regime.Moreover, we introduce a class of channels that generalize the twirling maps encountered in many resource theories, and by directly connecting it with resource quantification, we compute analytically several smoothed resource measures and improve our one-shot yield-cost bound in relevant theories.We use these operational insights to exactly evaluate important measures for various resource states in the resource theory of magic states.
We study entanglement distillation and dilution of states and channels in the single-shot regime. With the help of a recently introduced conversion distance, we provide compact closed-form expressions for the dilution and distillation of pure states and show how this can be used to efficiently calculate these quantities on multiple copies of pure states. These closed-form expressions also allow us to obtain second-order asymptotics. We then prove that the ε-single-shot entanglement cost of mixed states is given exactly in terms of an expression containing a suitably smoothed version of the conditional max-entropy. For pure states, this expression reduces to the smoothed max-entropy of the reduced state, for which we provide a closed-form expression. Analogously, we provide a closed-form expression for the smoothed min-entropy and connect it to the ε-single-shot distillable entanglement. Based on these results, we bound the single-shot entanglement cost of channels. We then turn to the one-way entanglement distillation of states and channels and provide bounds in terms of a quantity we denote coherent information of entanglement.
Hereunder, we study the class of irreducible private states that are private states from which all the secret content is accessible via measuring their key part. We provide the first protocol which distills key not only from the key part, but also from the shield if only the state is reducible. We prove also a tighter upper bound on the performance of that protocol, given in terms of regularized relative entropy of entanglement instead of relative entropy of entanglement previously known. This implies in particular that the irreducible private states are all strictly irreducible if and only if the entangled but key-undistillable states ('entangled key-undistilable states') exist. In turn, all the irreducible private states of the dimension 4 ⨂ 4 are strictly irreducible, that is, after an attack on the key part they become separable. Provided the bound key states exist, we consider different subclasses of the irreducible private states and their properties. Finally we provide a lower bound on the trace norm distance between key-undistillable states and private states, in sufficiently high dimensions.
We establish an operational theory of coherence (or of superposition) in quantum systems, by focusing on the optimal rate of performance of certain tasks. Namely, we introduce the two basic concepts-"coherence distillation" and "coherence cost"-in the processing quantum states under so-called incoherent operations [Baumgratz, Cramer, and Plenio, Phys. Rev. Lett. 113, 140401 (2014)]. We, then, show that, in the asymptotic limit of many copies of a state, both are given by simple single-letter formulas: the distillable coherence is given by the relative entropy of coherence (in other words, we give the relative entropy of coherence its operational interpretation), and the coherence cost by the coherence of formation, which is an optimization over convex decompositions of the state. An immediate corollary is that there exists no bound coherent state in the sense that one would need to consume coherence to create the state, but no coherence could be distilled from it. Further, we demonstrate that the coherence theory is generically an irreversible theory by a simple criterion that completely characterizes all reversible states.
The traditional perspective in quantum resource theories concerns how to use free operations to convert one resourceful quantum state to another one. For example, a fundamental and well known question in entanglement theory is to determine the distillable entanglement of a bipartite state, which is equal to the maximum rate at which fresh Bell states can be distilled from many copies of a given bipartite state by employing local operations and classical communication for free. It is the aim of this paper to take this kind of question to the next level, with the main question being: What is the best way of using free channels to convert one resourceful quantum channel to another? Here we focus on the the resource theory of entanglement for bipartite channels and establish several fundamental tasks and results regarding it. In particular, we establish bounds on several pertinent information processing tasks in channel entanglement theory, and we define several entanglement measures for bipartite channels, including the logarithmic negativity and the κ-entanglement. We also show that the max-Rains information of [Bäuml et al., Physical Review Letters, 121, 250504 (2018)] has a divergence interpretation, which is helpful for simplifying the results of this earlier work.