Numerical Determination of the Three-Tangle of Mixed States

Sameh M. Saad

Published Aug 31, 2015• 0 citations

Abstract

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.

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OpenAlex

Abstract

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We present a new tripartite entanglement measure for three-qubit mixed states. The new measure

t_{\mathrm{r}}(ρ)

, which we refer to as the r-tangle, is given as a kind of the tangle, but has a feature which the tangle does not have; if we can derive an analytical form of $ t_{\mathrm{r}}(ρ)$ for a three-qubit mixed state

ρ

, we can also derive

t_{\mathrm{r}}(ρ')

analytically for any states

ρ'

which are SLOCC-equivalent to the state

ρ

. The concurrence of two-qubit states also satisfies the feature, but the tangle does not. These facts imply that the r-tangle

t_{\mathrm{r}}

is the appropriate three-partite counterpart of the concurrence. We also derive an analytical form of the r-tangle

t_{\mathrm{r}}

for mixtures of a generalized GHZ state and a generalized W state, and hence for all states which are SLOCC-equivalent to them.

Three-tangle for a rank-3 mixture composed of Greenberger-Horne-Zeilinger,

W

, and flipped-

W

states is analytically calculated. The optimal decompositions in the full range of parameter space are constructed by making use of the convex-roof extension. We also provide an analytical technique, which determines whether or not an arbitrary rank-3 state has vanishing three-tangle. This technique is developed by making use of the Bloch sphere

{S}^{8}

of the qutrit system. The Coffman-Kundu-Wootters inequality is discussed by computing one-tangle and concurrences. It is shown that the one-tangle is always larger than the sum of squared concurrences and three-tangle. The physical implication of three-tangle is briefly discussed.