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Pairwise entanglement in symmetric multi-qubit systems

Wang, X, Klaus Mølmer

Published Jun 25, 2001• 82 citations
OpenAlex

Abstract

The concurrence, a quantitative measure of the entanglement between a pair of particles, is determined for the case where the pair is extracted from a symmetric state of N two-level systems. Examples are given for both pure and mixed states of the N-particle system, and for a pair extracted from two ensembles with correlated collective spins.

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The concurrence, a quantitative measure of the entanglement between a pair of particles, is determined for the case where the pair is extracted from a symmetric state of N two-level systems. Examples are given for both pure and mixed states of the N-particle system, and for a pair extracted from two ensembles with correlated collective spins.
Interaction with environment may lead to the transition of quantum system from pure state to the mixed one. In this case, the problem of definition of entanglement may arise. In particular, quantitative measure of entanglement concurrence is introduced differently for pure and mixed states. To solve this problem, alternative definition of concurrence of mixed states is introduced in present work. In particular, concurrence of mixed states is determined as an average of pure states concurrences. Averaging is carried out over the quantum ensemble, corresponding to all possible realization of random variable.
We consider an assembly of N indistinguishable two-level atoms, for which a concept of spin- squeezing and entanglement measures can be well defined. Permutation symmetry of such a state allows one to use an entanglement measure called the concurrence as the measures of pairwise and also N-partite entanglement. Dy using these measures, we show that negative pair-spin correlations of the state are quantum ones. It implies the quantumness of a spin-squeezing, which also pos­sesses pair-spin negative correlations. We also provide the entanglement measures of a macroscopic entangled state and an atomic Dicke state for super-radiant and spontaneous emissions.
Quantum entanglement reflects itself through non-local correlations among the subsystems of a quantum system. This thesis focuses on constructing a complete set of local invariants characterizing symmetric two qubit systems and analyzing the pairwise entanglement properties like collective spin squeezing - exhibited by multiqubits - in terms of two qubit entanglement invariants. Further, the collective pairwise entanglement properties of symmetric multiqubit states are shown to be captured entirely by the off diagonal block of the covariance matrix.
We propose a direct measure of tripartite quantum entanglement in an arbitrary symmetric pure state of N correlated two-level atoms (qubits). We compute the third order moments of the collective pseudo-spin operators in terms of the individual atomic operators of the atoms in the assembly and find that all the bipartite quantum correlation terms among the atoms cancel out leaving only the all possible tripartite quantum correlation terms among them. We observe that the third order moments are made up of solely the tripartite quantum correlations among the atoms. This helps to extract out only the tripartite quantum correlations among the N atoms and propose a measure and quantification of these correlations. We also propose the necessary and sufficient condition for the presence of tripartite quantum entanglement in such multi-atomic systems. We conjecture the way of determining the third order moments of the pseudo-spin operators of the two-level atoms in the assembly, experimentally.
Pairwise entanglement properties of a symmetric multi-qubit system are analyzed through a complete set of two-qubit local invariants. Collective features of entanglement, such as spin squeezing, are expressed in terms of invariants and a classification scheme for pairwise entanglement is proposed. The invariant criteria given here are shown to be related to the recently proposed (Phys. Rev. Lett.95, 120502 (2005)) generalized spin squeezing inequalities for pairwise entanglement in symmetric multi-qubit states.
Given a multipartite quantum system that consists of two-level particles (qubits), one may or may not have access to all the subsystems. What can we know about the entanglement of the multiqubit system and residual correlations beyond two-tangle if we have access only to two-qubits at a time? Algebraic analysis of two-qubit states yields monogamy constraints on distribution of entanglement between sub-systems of an N-qubit state and criterion to determine if the state has multipartite entanglement. Monogamy constraints, reported in this letter, are relations between well known entanglement measures such as one-tangle, two-tangles and three-tangles of an N-qubit pure state.