arXiv:1303.1999 · Physical Review A

Purification-based metric to measure the distance between quantum states and processes

Tristán M. Osán, Pedro W. Lamberti

Published Jun 17, 2013• 6 citations

Abstract

In this work we study the properties of a purification-based entropic metric for measuring the distance between both quantum states and quantum processes. This metric is defined as the square root of the entropy of the average of two purifications of mixed quantum states which maximize the overlap between the purified states. We analyze this metric and show that it satisfies many appealing properties, which suggest this metric is an interesting proposal for theoretical and experimental applications of quantum information.

DOI: 10.1103/physreva.87.062319

Related papers

Natural metric for quantum information theory

Lamberti, Pedro W., Portesi, Mariela Adelina, Javier Sparacino

We study in detail a very natural metric for quantum states. This new proposal has two basic ingredients: entropy and purification. The metric for two mixed states is defined as the square root of the entropy of the average of representative purifications of those states. Some basic properties are analyzed and its relation to other distances is investigated. As an illustrative application, the proposed metric is evaluated for one-qubit mixed states.

Jan 1, 200915 citations

NATURAL METRIC FOR QUANTUM INFORMATION THEORY

P. W. LAMBERTI, M. PORTESI, J. SPARACINO

arXiv:0807.0583International Journal of Quantum InformationAug 1, 200912 citations

Purification complexity without purifications

Shan-Ming Ruan

Abstract We generalize the Fubini-Study method for pure-state complexity to generic quantum states by taking Bures metric or quantum Fisher information metric (QFIM) on the space of density matrices as the complexity measure. Due to Uhlmann’s theorem, we show that the mixed-state complexity exactly equals the purification complexity measured by the Fubini-Study metric for purified states but without explicitly applying any purification. We also find the purification complexity is non-increasing under any trace-preserving quantum operations. We also study the mixed Gaussian states as an example to explicitly illustrate our conclusions for purification complexity.

arXiv:2006.01088Journal of High Energy PhysicsJan 18, 202119 citations

Metric character of the quantum Jensen-Shannon divergence

Pedro W. Lamberti, A. P. Majtey, Antoni Borràs +2 more

In a recent paper, the generalization of the Jensen-Shannon divergence in the context of quantum theory has been studied [Majtey et al., Phys. Rev. A 72, 052310 (2005)]. This distance between quantum states has shown to verify several of the properties required for a good distinguishability measure. Here we investigate the metric character of this distance. More precisely we show, formally for pure states and by means of a numerical procedure for mixed states, that its square root verifies the triangle inequality.

arXiv:0801.1586Physical Review AMay 9, 2008150 citations

Entanglement measures and purification procedures

Vlatko Vedral, Martin Bodo Plenio

We improve previously proposed conditions each measure of entanglement has to satisfy. We present a class of entanglement measures that satisfy these conditions and show that the quantum relative entropy and Bures metric generate two measures of this class. We calculate the measures of entanglement for a number of mixed two spin-1/2 systems using the quantum relative entropy, and provide an efficient numerical method to obtain the measures of entanglement in this case. In addition, we prove a number of properties of our entanglement measure that have important physical implications. We briefly explain the statistical basis of our measure of entanglement in the case of the quantum relative entropy. We then argue that our entanglement measure determines an upper bound to the number of singlets that can be obtained by any purification procedure.

arXiv:quant-ph/9707035Physical Review AMar 1, 19981,417 citations

A Framework for Non-Asymptotic Quantum Information Theory

Marco Tomamichel

This thesis consolidates, improves and extends the smooth entropy framework for non-asymptotic information theory and cryptography. We investigate the conditional min- and max-entropy for quantum states, generalizations of classical Rényi entropies. We introduce the purified distance, a novel metric for unnormalized quantum states and use it to define smooth entropies as optimizations of the min- and max-entropies over a ball of close states. We explore various properties of these entropies, including data-processing inequalities, chain rules and their classical limits. The most important property is an entropic formulation of the asymptotic equipartition property, which implies that the smooth entropies converge to the von Neumann entropy in the limit of many independent copies. The smooth entropies also satisfy duality and entropic uncertainty relations that provide limits on the power of two different observers to predict the outcome of a measurement on a quantum system. Finally, we discuss three example applications of the smooth entropy framework. We show a strong converse statement for source coding with quantum side information, characterize randomness extraction against quantum side information and prove information theoretic security of quantum key distribution using an intuitive argument based on the entropic uncertainty relation.

arXiv:1203.2142Jan 1, 2012225 citations