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Physical Review A

Simulated annealing and entanglement of formation for (n⊗m) -dimensional mixed states

S. Allende, D. Altbir, J. C. Retamal

Published Aug 24, 2015• 7 citations
Publisher

Abstract

The simulated annealing algorithm, a method commonly used to calculate properties of condensed-matter systems, is used to calculate entanglement of formation for higher-dimensional mixed states. The method relies on representing a mixed state as a pure state in an enlarged Hilbert space, and on the search for the extension minimizing the convex roof of entanglement of formation. Exact results are found, and predictions for system reservoir dynamics in higher dimensions are confirmed. These findings open the way for new applications of the method in quantum information.

DOI
10.1103/physreva.92.022348

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The simulated annealing algorithm, a method commonly used to calculate properties of condensed-matter systems, is used to calculate entanglement of formation for higher-dimensional mixed states. The method relies on representing a mixed state as a pure state in an enlarged Hilbert space, and on the search for the extension minimizing the convex roof of entanglement of formation. Exact results are found, and predictions for system reservoir dynamics in higher dimensions are confirmed. These findings open the way for new applications of the method in quantum information.
We present a conjugate gradient method for calculating the entanglement of formation of arbitrary mixed quantum states of any dimension and with any bipartite division of the Hilbert space. The development of the gradient used by the algorithm, its implications for the number of states required in the optimal decomposition, and the way that conjugate gradient minimization has been adapted for this particular problem are outlined. We have found that the algorithm exhibits linear convergence for general mixed states, and that it correctly reproduces the known results for pairs of qubits and for isotropic states. The results of an example application of the code are discussed: calculating the entanglement of formation of a Psi+ Bell state of two qutrits when one of those qutrits is subject to various decoherence channels. The results for qutrits are contrasted with those for qubits: for the types of decoherence considered here, qutrit entanglement appears to be more robust than qubit entanglement.
Entanglement of formation for arbitrary mixed states in a ($2\ensuremath{\bigotimes}2$)-dimensional Hilbert space is well known. For higher-dimensional mixed states analytical expressions for quantifying entanglement exist for a particular family of states. In this work, we calculate analytically the entanglement of formation for a family of bipartite ($2\ensuremath{\bigotimes}d$)-dimensional mixed states which can be obtained from tripartite $2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}d$ pure states following the entanglement relations obtained by Koashi and Winter [Phys. Rev. A 69, 022309 (2004)].
We develop a fast algorithm to calculate the entanglement of formation of a mixed state, which is defined as the minimum average entanglement of the pure states that form the mixed state. The algorithm combines conjugat-gradient and steepest-descent algorithms and outperforms both. Using this new algorithm, we obtain the statistics of the entanglement of formation on ensembles of random density matrices of higher dimensions than possible before. The correlation between the entanglement of formation and other quantities that are easier to compute, such as participation ratio and negativity are presented. Our results suggest a higher percentage of zero-entanglement states among zero-negativity states than previously reported.
The entanglement of a pure state of a pair of quantum systems is defined as the entropy of either member of the pair. The entanglement of formation of a mixed state $\ensuremath{\rho}$ is the minimum average entanglement of an ensemble of pure states that represents \ensuremath{\rho}. An earlier paper conjectured an explicit formula for the entanglement of formation of a pair of binary quantum objects (qubits) as a function of their density matrix, and proved the formula for special states. The present paper extends the proof to arbitrary states of this system and shows how to construct entanglement-minimizing decompositions.