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Register a new userQuantum entropy-typical subspace and universal data compression
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The quantum entropy-typical subspace theory is specified. It is shown that any mixed state with von Neumann entropy less than h can be preserved approximately by the entropy-typical subspace with entropy= h. This result implies an universal compression scheme for the case that the von Neumann entropy of the source does not exceed h.
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We present a new scheme for the compression of one-way quantum messages, in the setting of coherent entanglement assisted quantum communication. For this, we present a new technical tool that we call the convex split lemma, which is inspired by the classical compression schemes that use rejection sampling procedure. As a consequence, we show new bounds on the quantum communication cost of single-shot entanglement-assisted one-way quantum state redistribution task and for the sub-tasks quantum state splitting and quantum state merging. Our upper and lower bounds are tight up to a constant and hence stronger than previously known best bounds for above tasks. Our protocols use explicit quantum operations on the sides of Alice and Bob, which are different from the decoupling by random unitaries approach used in previous works. As another application, we present a port-based teleportation scheme which works when the set of input states is restricted to a known ensemble, hence potentially saving the number of required ports. Furthermore, in case of no prior knowledge about the set of input states, our average success fidelity matches the known average success fidelity, providing a new port-based teleportation scheme with similar performance as appears in literature.
The existence of the typical set is key for the consistence of the ensemble formalization of statistical mechanics. We demonstrate here that the typical set can be defined and characterized for a general class of stochastic processes. This includes processes showing arbitrary path dependence, long range correlations or dynamic sampling spaces. We show how the typical set is characterized from general forms of entropy and how one can transform these general entropic forms into extensive functionals and, in some cases, to Shannon path entropy. The definition of the typical set and generalized forms of entropy for systems with arbitrary phase space growth may help to provide an ensemble picture for the thermodynamic paths of many systems away from equilibrium. In particular, we argue that a theory of expanding/shrinking phase spaces in processes displaying an intrinsic degree of stochasticity may lead to new frameworks for exploring the emergence of complexity and robust properties in open ended evolutionary systems and, in particular, of biological systems.
Recently a new quantum generalization of the Renyi divergence and the corresponding conditional Renyi entropies was proposed. Here we report on a surprising relation between conditional Renyi entropies based on this new generalization and conditional Renyi entropies based on the quantum relative Renyi entropy that was used in previous literature. Our result generalizes the well-known duality relation H(A|B) + H(A|C) = 0 of the conditional von Neumann entropy for tripartite pure states to Renyi entropies of two different kinds. As a direct application, we prove a collection of inequalities that relate different conditional Renyi entropies and derive a new entropic uncertainty relation.
Given two pairs of quantum states, a fundamental question in the resource theory of asymmetric distinguishability is to determine whether there exists a quantum channel converting one pair to the other. In this work, we reframe this question in such a way that a catalyst can be used to help perform the transformation, with the only constraint on the catalyst being that its reduced state is returned unchanged, so that it can be used again to assist a future transformation. What we find here, for the special case in which the states in a given pair are commuting, and thus quasi-classical, is that this catalytic transformation can be performed if and only if the relative entropy of one pair of states is larger than that of the other pair. This result endows the relative entropy with a fundamental operational meaning that goes beyond its traditional interpretation in the setting of independent and identical resources. Our finding thus has an immediate application and interpretation in the resource theory of asymmetric distinguishability, and we expect it to find application in other domains.
The existence of a positive log-Sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the Markov approximation. For classical spin systems, such constant was proven to exist, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, via a quasi-factorization of the entropy in terms of the conditional entropy in some sub-$sigma$-algebras. In this work we analyze analogous quasi-factorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasi-factorization results for it. As an illustration of their potential, we use one of them to obtain a positive log-Sobolev constant for the heat-bath dynamics with product fixed point.
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