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arXiv:2210.11462 · Lobachevskii Journal of Mathematics

Local Lower Bounds on Characteristics of Quantum and Classical Systems

Максим Евгеньевич Широков

Published Jun 1, 2023• 1 citation
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Abstract

We consider methods for obtaining local lower bounds on characteristics of quantum (resp. classical) systems, i.e., lower bounds valid in the trace norm $$\epsilon$$ -neighborhood of a given state (resp. probability distribution). Our basic tool is the quasi-classical modification of the Alicki–Fannes–Winter technique which gives faithful one-side continuity bounds with the rank/energy constraint imposed on only one of the two quantum states (resp. probability distributions). We also propose a new universal method that allows us to obtain easy computable faithful local lower bounds on many important characteristics of quantum (resp. classical) systems. The main attention is paid to infinite-dimensional systems.

DOI
10.1134/s1995080223060422

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We consider methods for obtaining local lower bounds on characteristics of quantum (resp. classical) systems, i.e., lower bounds valid in the trace norm ϵ\epsilonϵ -neighborhood of a given state (resp. probability distribution). Our basic tool is the quasi-classical modification of the Alicki–Fannes–Winter technique which gives faithful one-side continuity bounds with the rank/energy constraint imposed on only one of the two quantum states (resp. probability distributions). We also propose a new universal method that allows us to obtain easy computable faithful local lower bounds on many important characteristics of quantum (resp. classical) systems. The main attention is paid to infinite-dimensional systems.
Abstract We describe the most general form of the Alicki–Fannes–Winter method for obtaining uniform continuity bounds for characteristics of composite quantum systems and consider several modifications of this method, which make it applicable to a wide class of characteristics in both finite-dimensional and infinite-dimensional cases. We present uniform continuity bounds for the most important characteristics of composite quantum systems. Along with the results obtained earlier by various authors, we describe a number of new results proved by the proposed methods. In particular, we obtain new continuity bounds for the quantum discord, one-way classical correlation and its regularization in finite-dimensional and infinite-dimensional bipartite quantum systems.
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Abstract General methods of local continuity analysis of characteristics of infinite-dimensional composite quantum systems are considered. A new approximation technique for obtaining local continuity conditions for various characteristics of quantum systems is proposed and described in detail. This technique is used to prove several general results (a Simon-type dominated convergence theorem, a theorem on the preservation of continuity under convex mixtures, etc.). Local continuity conditions are derived for the following characteristics of composite quantum systems: the quantum conditional entropy, the quantum (conditional) mutual information, the one-way classical correlation and its regularization, the quantum discord and its regularization, the entanglement of formation and its regularization, and the constrained Holevo capacity of a partial trace and its regularization.
The approximate coherent state rank is the minimal number of (classical) coherent states required to approximate a continuous-variable bosonic quantum state and directly relates to the classical complexity of simulating bosonic computations. Despite its importance, little is known about lower bounds on this quantity, even for basic families of states. In this work, we initiate a systematic study of lower bounds on the approximate coherent state rank. Our contributions are as follows. (i) We introduce a technique based on low-rank approximation theory yielding generic lower bounds on the approximate coherent state rank of arbitrary single-mode states. (ii) Using this technique, we find a complete characterization of all single-mode states of finite approximate coherent state rank, and we obtain in particular analytical expressions for the approximate coherent state rank of squeezed states and of finite superpositions of Fock states. (iii) We further show that our single-mode lower bounds can be lifted to multimode lower bounds for finite superpositions of multimode Fock states. (iv) Finally, we prove a super-polynomial lower bound on the approximate coherent state rank of the nnn-mode Fock state ∣1⟩⊗n|1\rangle^{\otimes n}∣1⟩⊗n, by exploiting a connection to the permanent. To do so, we show that the algebraic complexity of approximate multi-linear formulas for the permanent is super-polynomial, building upon the proof of a lower bound for exact formulas due to [Raz, JACM 2009]. Our results establish an unconditional barrier to efficient classical simulation of Boson Sampling via coherent state decompositions and connect non-classicality of bosonic quantum systems to central questions in algebraic complexity.
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The quantum relative entropy is frequently used as a distance, or distinguishability measure between two quantum states. In this article we study the relation between this measure and a number of other measures used for that purpose, including the trace norm distance. More specifically, we derive lower and upper bounds on the relative entropy in terms of various distance measures for the difference of the states based on unitarily invariant norms. The upper bounds can be considered as statements of continuity of the relative entropy distance in the sense of Fannes. We employ methods from optimisation theory to obtain bounds that are as sharp as possible.