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arXiv:1001.0067 · Physical Review A

Efficient numerical method to calculate the three-tangle of mixed states

Kun Cao, Zheng-Wei Zhou, Guang‐Can Guo, Lixin He

Published Mar 25, 2010• 24 citations
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Abstract

We demonstrate an efficient numerical method to calculate the three-tangle of general mixed states. We construct an ``energy function'' (target function) for the three-tangle of the mixed state under certain constraints. The energy function (target function) is then optimized via the replica exchange Monte Carlo method. We have extensively tested the method for the examples with known analytical results, showing remarkable agreement. The method can be applied to other optimization problems in quantum information theory.

DOI
10.1103/physreva.81.034302

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We demonstrate an efficient numerical method to calculate the three-tangle of general mixed states. We construct an ``energy function'' (target function) for the three-tangle of the mixed state under certain constraints. The energy function (target function) is then optimized via the replica exchange Monte Carlo method. We have extensively tested the method for the examples with known analytical results, showing remarkable agreement. The method can be applied to other optimization problems in quantum information theory.
This thesis presents a numerical method for computing the two-tangle and three-tangle of mixed states.A target function is constructed for both the two-tangle and three-tangle of mixed states, where a number of constraints have been introduced in order to make sure that the whole construct makes physical sense.The target function is then minimized by means of the COBYLA algorithm, and tested by comparison with known analytical results.The numerical computation of the twotangle shows good agreement with the analytical results.As for the three-tangle, it can be reasonably approximated by the COBYLA algorithm, given that the algorithm is allowed to run for some time.Presumably, an increasingly better approximation to the analytical results will be obtained by increasing the allowed computation time.
The quantitative assessment of the entanglement in multipartite quantum states is, apart from its fundamental importance, a practical problem. Recently there has been significant progress in developing new methods to determine certain entanglement measures. In particular, there is a method---in principle, analytical---to compute a certified lower bound for the three-tangle. The purpose of this work is to provide a manual for the implementation of this approach and to explicitly discuss several analytically solvable cases in order to gauge the numerical tools. Moreover, we derive a simple analytical bound for the mixed-state three-tangle.
The quantitative assessment of the entanglement in multipartite quantum states is, apart from its fundamental importance, a practical problem. Recently there has been significant progress in developing new methods to determine certain entanglement measures. In particular, there is an - in principle analytical - method to compute a certified lower bound for the three-tangle. The purpose of this work is to provide a manual for the implementation of this approach and to explicitly discuss several analytically solvable cases in order to gauge the numerical tools. Moreover, we derive a simple analytical bound for the mixed state three-tangle.
We give a complete solution for the three-tangle of mixed three-qubit states composed of a generalized Greenberger–Horne–Zeilinger (GHZ) state, a |000⟩+ b |111⟩, and a generalized W state, c |001⟩+ d |010⟩+ f |100⟩. Using the methods introduced by Lohmayer et al (2006 Phys. Rev. Lett. 97 260502), we provide explicit expressions for the mixed-state three-tangle and the corresponding optimal decompositions for this more general case. Moreover, as a special case, we obtain a general solution for a family of states consisting of a generalized GHZ state and an orthogonal product state.
Tensor network states (TNS) methods combined with the Monte Carlo (MC) technique have been proven a powerful algorithm for simulating quantum many-body systems. However, because the ground state energy is a highly non-linear function of the tensors, it is easy to get stuck in local minima when optimizing the TNS of the simulated physical systems. To overcome this difficulty, we introduce a replica-exchange molecular dynamics optimization algorithm to obtain the TNS ground state, based on the MC sampling technique, by mapping the energy function of the TNS to that of a classical mechanical system. The method is expected to effectively avoid local minima. We make benchmark tests on a 1D Hubbard model based on matrix product states (MPS) and a Heisenberg J1-J2 model on square lattice based on string bond states (SBS). The results show that the optimization method is robust and efficient compared to the existing results.