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Journal of Physics A Mathematical and General

Multi-particle entanglement via two-party entanglement

Gilles Brassard A, Tal Mor

Published Aug 24, 2001• 4 citations
Publisher

Abstract

Entanglement between n particles is a generalization of the entanglement between two particles, and a state is considered entangled if it cannot be written as a mixture of tensor products of the n particles' states. We present the key notion of semi-separability , used to investigate n -particle entanglement by looking at two-party entanglement between its various subsystems. We provide necessary conditions for n -particle separability (that is, sufficient conditions for n -particle entanglement). We also provide necessary and sufficient conditions in the case of pure states. By surprising examples, we show that such conditions are not sufficient for separability in the case of mixed states, suggesting entanglement of a strange type.

DOI
10.1088/0305-4470/34/35/306

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Entanglement between n particles is a generalization of the entanglement between two particles, and a state is considered entangled if it cannot be written as a mixture of tensor products of the n particles' states. We present the key notion of semi-separability , used to investigate n -particle entanglement by looking at two-party entanglement between its various subsystems. We provide necessary conditions for n -particle separability (that is, sufficient conditions for n -particle entanglement). We also provide necessary and sufficient conditions in the case of pure states. By surprising examples, we show that such conditions are not sufficient for separability in the case of mixed states, suggesting entanglement of a strange type.
Given a pure state |ψN>∈ℋN of a quantum system composed of n qubits, where ℋN is the Hilbert space of dimension N=2n, this paper answers two questions: what conditions should the amplitudes in |ψN> satisfy for this state to be separable (i) into a tensor product of n qubit states |ψ2>0⊗ |ψ2>1 ⊗⋯⊗ |ψ2>n-1, and (ii), into a tensor product of two subsystems states |ψP> ⊗ |ψQ> with P=2p and Q=2q such that p+q=n? For both questions, necessary and sufficient conditions are proved, thus characterizing at the same time families of separable and entangled states of n qubit systems. A number of more refined questions about separability in n qubit systems can be studied on the basis of these results.
We provide necessary and sufficient conditions for separability of mixed states of n-particle systems. The conditions are formulated in terms of maps which are positive on product states of n−1n-1n−1 particles. The method of providing of the maps on the basis of unextendable product bases is provided. The three qubit state problem is reformulated in the form suggesting possibility of explicite characterisation of all maps needed for separability condition.
The purpose of this paper is to obtain a sufficient and necessary condition as a criteria to test whether an arbitrary multipartite state is entangled or not. Based on the tensor expression of a multipartite pure state, the paper shows that a state is separable iff $| \mathbf{C}(ρ)| $ =0 for pure states and iff C(ρ)C(ρ)C(ρ) vanishes for mixed states.
In the quest to completely describe entanglement in the general case of a finite number of parties sharing a physical system of finite-dimensional Hilbert space an entanglement magnitude is introduced for its pure and mixed states: robustness. It corresponds to the minimal amount of mixing with locally prepared states which washes out all entanglement. It quantifies in a sense the endurance of entanglement against noise and jamming. Its properties are studied comprehensively. Analytical expressions for the robustness are given for pure states of two-party systems, and analytical bounds for mixed states of two-party systems. Specific results are obtained mainly for the qubit-qubit system (qubit denotes quantum bit). As by-products local pseudomixtures are generalized, a lower bound for the relative volume of separable states is deduced, and arguments for considering convexity a necessary condition of any entanglement measure are put forward.
Unlike for bipartite states consisting of distinguishable particles, in the case of identical parties the notion of entanglement is still under debate. In the following, we review two different approaches to the entanglement of systems consisting of two bosons or fermions; the first approach is based on the particle aspect typical of first quantization and identifies separable pure states as those that allow to assign two orthogonal single particle vector states to both parties. The second approach makes full use of the mode aspect of second quantization whereby separability can be formulated as absence of nonlocal correlation among two different sets of modes. While the first approach applies to pure states only, the second one is more general and characterizes generic entangled states. In the following, we shall show that the mode-based approach indeed contains the particle-based one.