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Physical Review Letters

Disentangling Quantum States while Preserving All Local Properties

Tal Mor

Published Aug 16, 1999• 27 citations
Publisher

Abstract

We consider here a disentanglement process which transforms a state $\ensuremath{\rho}$ of two subsystems into an unentangled (i.e., separable) state, while not affecting the reduced density matrix of either subsystem. Recently, Terno [Phys. Rev. A 59, 3320 (1999)] showed that an arbitrary state cannot be disentangled, by a physically allowable process, into a tensor product of its reduced density matrices. In this Letter we show that there are sets of states which can be disentangled, but only into separable states other than the product of the reduced density matrices, and other sets of states which cannot be disentangled at all. Thus, we prove that a universal disentangling machine cannot exist.

DOI
10.1103/physrevlett.83.1451

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We consider here a disentanglement process which transforms a state \ensuremathρ\ensuremath{\rho}\ensuremathρ of two subsystems into an unentangled (i.e., separable) state, while not affecting the reduced density matrix of either subsystem. Recently, Terno [Phys. Rev. A 59, 3320 (1999)] showed that an arbitrary state cannot be disentangled, by a physically allowable process, into a tensor product of its reduced density matrices. In this Letter we show that there are sets of states which can be disentangled, but only into separable states other than the product of the reduced density matrices, and other sets of states which cannot be disentangled at all. Thus, we prove that a universal disentangling machine cannot exist.
Disentanglement is the process which transforms a state ρρρ of two subsystems into an unentangled state, while not effecting the reduced density matrices of each of the two subsystems. Recently Terno showed that an arbitrary state cannot be disentangled into a tensor product of its reduced density matrices. In this letter we present various novel results regarding disentanglement of states. Our main result is that there are sets of states which cannot be successfuly disentangled (not even into a separable state). Thus, we prove that a universal disentangling machine cannot exist.
We discuss optimal disentanglement processes of states of two two-level systems that belong to (i) the universal set, (ii) the set in which the states of one party lie on a single great circle of the Bloch sphere, and (iii) the set in which the states of one party commute with each other, by teleporting the states of one party (on which the disentangling machine is acting) through three particular types of separable channels, each of which is a mixture of Bell states. In the general scenario, by teleporting one party's state of an arbitrary entangled state of two two-level parties through some mixture of Bell states, we have shown that this entangled state can be made separable by using a physically realizable map $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ acting on one party's states, if $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(I)=I,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}({\ensuremath{\sigma}}_{j})={\ensuremath{\lambda}}_{j}{\ensuremath{\sigma}}_{j},$ where ${\ensuremath{\lambda}}_{j}>~0$ (for j=1,2,3)j=1,2,3)j=1,2,3) and ${\ensuremath{\lambda}}_{1}+{\ensuremath{\lambda}}_{2}+{\ensuremath{\lambda}}_{3}<~1.$
In early days of quantum theory it was believed that the results of measurements performed on two distant physical systems should be uncorrelated thus their quantum state should be separable it means described by a simple tensor product of the individual local state vectors or a tensor product of individual local density operators. It was shown many years ago by EPR that two systems which interacted in the past and separated afterwards had to be described in most cases by particular non-separable states which are called entangled. It was noticed by Zanardi et al. that a Hilbert space of possible state vectors of compound physical system could be partitioned in different way by introducing various tensor product structures induced by the experimentally accessible observables (interactions and measurements). Therefore a separable state in one partition could become entangled in different partition. In this sense the entanglement was relative to a particular set of experimental capabilities. Continuing this line of thought Torre et al. claimed to prove that for any separable state there existed strongly correlated physical observables therefore all quantum states were entangled. The same claims one may find in several recent papers. In this paper we will discuss in operational way the differences existing between separable, non-separable and entangled states and we will show that the conclusions of Torre et al. were unjustified. A sufficient condition for entanglement is the violation of BI-CHSH and/or steering inequalities. Since there exist experiments outside the quantum physics violating these inequalities therefore in the operational approach one cannot say that the entanglement is an exclusive quantum phenomenon.
Three kinds of definitions of disentanglement of 3-particle systems are given. Three basic regulations of a 3-particle system pure-state disentanglement are completely analyzed. It is shown that there are some sets of 3-particle entangled states that can be disentangled into product states or 1-particle product and 2-particle separable states or fully separable states, but a universal disentangling machine into product states or 1-particle product and 2-particle separable states or fully separable states cannot exist.