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Acta Physica Sinica

DISENTANGLEMENT OF 3-PAETICLE SYSTEMS

Feng Jian, Ji‐Suo Wang, Gao Yun-feng, Mingsheng Zhan, (1)聊城师范学院光通信研究所,聊城252059; (2)中国科学院武汉物理与数学研究所波谱与原子分子物理国家重点实验室,武汉430071; (3)中国科学院武汉物理与数学研究所波谱与原子分子物理国家重点实验室,武汉430071;中国科学院安徽光学精密机械研究所激光光谱学开放研究实验室,合肥230031;聊城师范学院光通信研究所,聊城252059

Published Jan 1, 2001• 3 citations

Abstract

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.

DOI: 10.7498/aps.50.2083

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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.

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.

Entanglement is known to be a relative notion, defined with respect to the choice of physical observables to be measured (i.e., the measurement setup used). This implies that, in general, the same state can be both separable and entangled for different measurement setups, but this does not exclude the existence of states which are separable (or entangled) for all possible setups. We show that for systems of bosonic particles there indeed exist such universally separable states: They are independently and identically distributed (i.i.d.) pure states. In contrast, there is no such state for fermionic systems with a few exceptional cases. We also find that none of the fermionic and bosonic systems admits universally entangled states.

By properties of independence state vector in quantum mechanics, a theory to produce entanglement is put forward for the first time and the definition of entangled state and the basic forms of entangledment for multiparticle system are given. Finally the definition is given of entangled degree for basic form entangled state and the formula is derived of entangled degree for two and three_particles system. Furthermore, the criterion of disentangled state for three particles system is described.

We propose a measure of entanglement between two subsystems of arbitrary (but finite) number of levels. This measure is invariant under unitary transformations of the subsystems and varies between 0 (product states) and 1 (maximum entangled states). The case of two two-level systems is worked out explicitly, where the known complementarity between one- and two-particle interference results in a very natural way from a sum rule. Generalizations to more than two subsystems are discussed. Here we show that for three particles there cannot exist a pure state that is completely characterized by three-particle entanglement alone. For the example of the Greenberger-Horne-Zeilinger (GHZ) state, an experimental setup is proposed, in which its corresponding two-particle entanglement would show up.

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.