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arXiv:0803.1063 · Europhysics Letters (EPL)

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

Takuya Yamano, Osamu Iguchi

Published Sep 1, 2008• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1209/0295-5075/83/50007

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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.
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.
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.
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.