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arXiv:quant-ph/9812020

On the Disentanglement of States

Tal Mor

Published Dec 9, 1998• 4 citations
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Abstract

Disentanglement is the process which transforms a state $ρ$ 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.

DOI
10.48550/arxiv.quant-ph/9812020

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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 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.
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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.
It is not possible to disentangle a qubit in an unknown state $|\ensuremath{\psi}〉$ from a set of $N\ensuremath{-}1$ ancilla qubits prepared in a specific reference state ∣0〉.|0〉.∣0〉. That is, it is not possible to perfectly perform the transformation $(|\ensuremath{\psi},0,\dots{},0〉+|0,\ensuremath{\psi},\dots{},0〉+\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}+|0,0,\dots{},\ensuremath{\psi}〉)\ensuremath{\rightarrow}|0,\dots{},0〉\ensuremath{\bigotimes}|\ensuremath{\psi}〉.$ The question is then how well we can do? We consider a number of different methods of extracting an unknown state from an entangled state formed from that qubit and a set of ancilla qubits in a known state. Measuring the whole system is, as expected, the least effective method. We present various quantum ``devices'' which disentangle the unknown qubit from the set of ancilla qubits. In particular, we present the optimal universal disentangler which disentangles the unknown qubit with a fidelity that does not depend on the state of the qubit, and a probabilistic disentangler which performs the perfect disentangling transformation, but with a probability less than 1.
As a contribution to the field of quantum mereology, we study how a change of tensor product structure in a finite-dimensional Hilbert space affects its entanglement properties. In particular, we ask whether, given a time-evolving state, there exists a tensor product structure in which no entanglement is generated. We give a concrete, constructive example of disentangling tensor product structure in the case of a C-NOT gate evolution between two qbits, before showing that this cannot be achieved for most time-evolving quantum states.