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Physical Review Letters

Uniqueness of the Entanglement Measure for Bipartite Pure States and Thermodynamics

Vlatko Vedral, Elham Kashefi

Published Jun 27, 2002• 43 citations
Publisher

Abstract

We apply the axiomatic approach to thermodynamics presented by Giles to derive a unique measure of entanglement for bipartite pure states. This implies that local manipulations of entanglement in quantum information theory and adiabatic transformations of states in thermodynamics have the same underlying mathematical structure. We discuss possible extensions of our results to mixed and multipartite states.

DOI
10.1103/physrevlett.89.037903

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We apply the axiomatic approach to thermodynamics presented by Giles to derive a unique measure of entanglement for bipartite pure states. This implies that local manipulations of entanglement in quantum information theory and adiabatic transformations of states in thermodynamics have the same underlying mathematical structure. We discuss possible extensions of our results to mixed and multipartite states.
A review is given on the thermodynamical structure of bipartite entanglement. By comparing it to the axiomatic formulation of thermodynamics presented by Giles it is shown that for finite dimensional systems the two theories are formally inequivalent. The same approach is used to demonstrate the full equivalence in the asymptotic limit for pure quantum states. For mixed states a different method for obtaining the second law is described applied to two different classes of operations, PPT-preserving and asymptotically non-entangling operations.
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We show that quantum entanglement can be understood and quantified using the same axiomatic approach to thermodynamics pioneered by Caratheodory and later extended by Buchdahl and Giles. This leads to the existence of a unique measure of entanglement for pure states of systems consisting of two subsystems, analogous to the thermodynamical entropy. Thermodynamics and (quantum) information are seen to be just two different manifestations of ordering of (abstract) states connected by (abstract) processes.
Entanglement of pure states of bipartite quantum systems has been shown to have a unique measure in terms of the von Neumann entropy of the reduced states of either of its subsystems. The measure is established under entanglement manipulation of an asymptotically large number of copies of the bipartite pure state. In this paper, two different asymptotic measures of entanglement for arbitrary sequences of bipartite pure states are established. These are shown to coincideonly whenthe sequence isinformation stable, in terms of the quantum spectral information rates of its corresponding sequence of subsystem states. Additional bounds on the optimal rates of entanglement manipulation protocols in quantum information theory are also presented. These include bounds given by generalizations of the coherent information bounds, Rains' bound, and the relative entropy of entanglement.