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Classical no-cloning under Liouville dynamics by non-Csisz{\' a}r f-divergence

Takuya Yamano, Osamu Iguchi

Published Mar 7, 2008• 0 citations
OpenAlex

Abstract

The Csiszar f-divergence, which is a class of information distances, is known to offer a useful tool for analysing the classical counterpart of the cloning operations that are quantum mechanically impossible for the factorized and marginality classical probability distributions under Liouville dynamics. We show that a class of information distances that does not belong to this divergence class also allows for the formulation of a classical analogue of the quantum no-cloning theorem. We address a family of nonlinear Liouville-like equations, and generic distances, to obtain constraints on the corresponding functional forms, associated with the formulation of classical analogue of the no-cloning principle.

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Classical no-cloning theorem under Liouville dynamics by non-Csiszár f-divergence

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The Csiszár f -divergence, which is a class of information distances, is known to offer a useful tool for analysing the classical counterpart of the cloning operations that are quantum-mechanically impossible for the factorized and marginality classical probability distributions under Liouville dynamics. We show that a class of information distances that does not belong to this divergence class also allows for the formulation of a classical analogue of the quantum no-cloning theorem. We address a family of nonlinear Liouville-like equations, and generic distances, to obtain constraints on the corresponding functional forms, associated with the formulation of a classical analogue of the no-cloning principle.

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Classical No-Cloning Theorem

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A classical version of the no-cloning theorem is discussed. We show that an arbitrary probability distribution associated with a (source) system cannot be copied onto another (target) system while leaving the original distribution of the source system unperturbed. For classical dynamical systems such a perfect cloning process is not permitted by the Liouvillian (ensemble) evolution associated with the joint probability distribution of the composite source-target-copying machine system.

Physical Review Letters•May 9, 2002•60 citations
The Csiszar f-divergence, which is a class of information distances, is known to offer a useful tool for analysing the classical counterpart of the cloning operations that are quantum mechanically impossible for the factorized and marginality classical probability distributions under Liouville dynamics. We show that a class of information distances that does not belong to this divergence class also allows for the formulation of a classical analogue of the quantum no-cloning theorem. We address a family of nonlinear Liouville-like equations, and generic distances, to obtain constraints on the corresponding functional forms, associated with the formulation of classical analogue of the no-cloning principle.
The Csiszár f -divergence, which is a class of information distances, is known to offer a useful tool for analysing the classical counterpart of the cloning operations that are quantum-mechanically impossible for the factorized and marginality classical probability distributions under Liouville dynamics. We show that a class of information distances that does not belong to this divergence class also allows for the formulation of a classical analogue of the quantum no-cloning theorem. We address a family of nonlinear Liouville-like equations, and generic distances, to obtain constraints on the corresponding functional forms, associated with the formulation of a classical analogue of the no-cloning principle.
We consider the possibility of performing, in the context of Liouville dynamics, certain information related, universal processes on classical ensemble probability distributions. This problem constitutes a classical counterpart of the study of universal processes in quantum mechanics, which is nowadays the focus of intensive research. The time independence of appropriate entropic distances between time dependent solutions of Liouville equation allows to establish the impossibility of some of the alluded processes. The present problem is also considered within the framework of discrete classical dynamics.
It is well known that the statistics of closed classical systems evolves according to the Liouville theorem. Here we study the dynamics of the marginal statistics of classical systems coupled to external degrees of freedom, by developing a time-local equation of motion using Green's functions and a series expansion method. We also compare this equation of motion with its supposed quantum counterpart, namely the quantum master equation, which we hope could shed some light on quantum-classical analogy (QCA) from the perspective of "open system dynamics". We notice an apparent exception to QCA in this case, as the first-order classical equation of motion derived herein contains a term that does not appear to have a quantum analogue. We also propose possible ways of getting around this tension, which may help re-establish QCA (in first perturbative order). We do not draw a definitive conclusion about QCA in the context of open system dynamics but hope to provide a starting point for investigations along this line.
We introduce a new family of quantum distances based on symmetric Csiszár divergences, a class of distinguishability measures that encompass the main dissimilarity measures between probability distributions. We prove that these quantum distances can be obtained by optimizing over a set of quantum measurements followed by a purification process. Specifically, we address in the first place the case of distinguishing pure quantum states, solving an optimization of the symmetric Csiszár divergences over von Neumann measurements. In the second place, by making use of the concept of purification of quantum states, we arrive at a new set of distinguishability measures, which we call extended quantum Csiszár distances. In addition, as it has been demonstrated that a purification process can be physically implemented, the proposed distinguishability measures for quantum states could be endowed with an operational interpretation. Finally, by taking advantage of a well-known result for classical Csiszár divergences, we show how to build quantum Csiszár true distances. Thus, our main contribution is the development and analysis of a method for obtaining quantum distances satisfying the triangle inequality in the space of quantum states for Hilbert spaces of arbitrary dimension.