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The Rochester Conferences on Coherence and Quantum Optics and the Quantum Information and Measurement meeting

X-matrices as a platform for studying multipartite entanglement

Seyed Mohammad Hashemi Rafsanjani, Shantanu Agarwal, Curtis J. Broadbent, Joseph H. Eberly

Published Jan 1, 2013• 0 citations
Publisher

Abstract

We derive an algebraic formula for the genuinely multipartite entanglement of a N-qubit X state. The entanglement dynamics of a N-qubit GHZ-state is a decoherence scenario. We also show that the mixedness of a state limits its entanglement, and the critical mixedness, beyond which no X-state is entangled, increases with the number of qubits.

DOI
10.1364/qim.2013.w6.51

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We derive an algebraic formula for the genuinely multipartite entanglement of a N-qubit X state. The entanglement dynamics of a N-qubit GHZ-state is a decoherence scenario. We also show that the mixedness of a state limits its entanglement, and the critical mixedness, beyond which no X-state is entangled, increases with the number of qubits.
Beyond the simplest case of bipartite qubits, the composite Hilbert space of multipartite systems is largely unexplored. In order to explore such systems, it is important to derive analytic expressions for parameters which characterize the system's state space. Two such parameters are the degree of genuine multipartite entanglement and the degree of mixedness of the system's state. We explore these two parameters for an N-qubit system whose density matrix has an X form. We derive the class of states that has the maximum amount of genuine multipartite entanglement for a given amount of mixedness. We compare our results with the existing results for the N = 2 case. The critical amount of mixedness above which no N-qubit X-state possesses genuine multipartite entanglement is derived. It is found that as N increases, states with higher mixedness can still be entangled.
We find an algebraic formula for the NNN-partite concurrence of NNN qubits in an XXX matrix. XXX matrices are density matrices whose only nonzero elements are diagonal or antidiagonal when written in an orthonormal basis. We use our formula to study the dynamics of the NNN-partite entanglement of NNN remote qubits in generalized NNN-party Greenberger-Horne-Zeilinger (GHZ) states. We study the case in which each qubit interacts with a local amplitude damping channel. It is shown that only one type of GHZ state loses its entanglement in finite time; for the rest, NNN-partite entanglement dies out asymptotically. Algebraic formulas for the entanglement dynamics are given in both cases. We directly confirm that the half-life of the entanglement is proportional to the inverse of NNN. When entanglement vanishes in finite time, the time at which entanglement vanishes can decrease or increase with NNN depending on the initial state. In the macroscopic limit, this time is independent of the initial entanglement.
More than a century after the seminal work of Schmidt and with all the enthusiasm that have surrounded entanglement ever since the controversial EPR paper, it remains an open challenge to determine whether a given state possesses entanglement or not. The problem is even more difficult if one considers the entanglement among more than two parties, i.e. multipartite entanglement. In the following we first introduce the concept of multipartite entanglement and discuss what it means to quantify the entanglement of a given state. We then introduce a class of multiqubit states, called X-states, and find an algebraic formula for the multipartite entanglement of such states. We show that using this formula one can find a lower bound for the entanglement of any multiqubit state. We then explore the connection between the entanglement and purity in multiqubit states. In the fourth chapter, we introduce a geometrical measure of entanglement and quantify it for the set of GHZ-diagonal states. These are states that can be written as a convex sum of completely bit-flipped states. Using these results we can develop an upper and a lower bound for the entanglement of any density matrix. In the final chapter we survey some of the insights that can be developed using the results of the preceding chapters. We first explore the decay of entanglement in a decoherence scenario where each qubit is experiences decay through an amplitude damping channel, and finally we make a proposal to preserve and control multipartite entanglement through the phenomenon of collapse and revival.
We derive the entangled thermal mixed states by using the formalism of phase states for a finite-dimensional algebra of a multi-qubit system in contact with an independent thermal environment of absolute temperature [Formula: see text]. Thermal entangled states describing the multi-qubit system in equilibrium with the thermal bath are a special kind of mixed states that exhibit genuine multipartite correlation. We define the unitary phase operators for a multipartite system of non-interacting qubits. Entangled density matrices are derived for qubits interacting via an Hermitian Hamiltonian of Heisenberg type [Formula: see text]. By assuming that the noisy interaction of the entangled qubit ensemble with the bath is governed by a local Hamiltonian [Formula: see text], we show that the entangled phase states can be decohered. When the multi-qubits entangled system reaches the equilibrium with the thermal bath, the decohered mixed states are identified with entangled thermal states. The thermal mixed states for bipartite and multipartite systems are explicitly expressed and their bipartite entanglement properties are investigated.
We construct a simplex for multipartite qubit states of even number nnn of qubits, which has the same geometry concerning separability, mixedness, kind of entanglement, amount of entanglement, and nonlocality as the bipartite qubit states. We derive the entanglement of the class of states which can be described by only three real parameters with the help of a multipartite measure for all discrete systems. We prove that the bounds on this measure are optimal for the whole class of states and that it reveals that the states possess only nnn-partite entanglement and not, e.g., bipartite entanglement. We then show that this nnn-partite entanglement can be increased by stochastic local operations and classical communication to the purest maximal entangled states. However, pure nnn-partite entanglement cannot be distilled; consequently all entangled states in the simplex are nnn-partite bound entangled. We study also Bell inequalities and find the same geometry as for bipartite qubits. Moreover, we show how the (hidden) nonlocality for all nnn-partite bound entangled states can be revealed.