How to practically obtain a lower bound to the 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 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 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.

arXiv:1310.8311Physical Review AFeb 12, 201420 citations

Estimating multipartite entanglement measures

Andreas Osterloh, Philipp Hyllus

We investigate the lower bound obtained from experimental data of a quantum state $\ensuremath{\rho}$, as proposed independently by O. G\"uhne et al. [Phys. Rev. Lett. 98, 110502 (2007)] and J. Eisert et al. [New J. Phys. 9, 46 (2007)], and apply it to mixed states of three qubits. The measure we consider is the convex-roof extended three-tangle. Our findings highlight an intimate relation to lower bounds obtained recently from so-called characteristic curves of a given entanglement measure. We apply the bounds to estimate the three-tangle present in recently performed experiments aimed at producing a three-qubit Greenberger-Horne-Zeilinger (GHZ) state. A nonvanishing lower bound is obtained if the GHZ fidelity of the produced states is larger than $3/4$ .We investigate the lower bound obtained from experimental data of a quantum state $\ensuremath{\rho}$, as proposed independently by O. G\"uhne et al. [Phys. Rev. Lett. 98, 110502 (2007)] and J. Eisert et al. [New J. Phys. 9, 46 (2007)], and apply it to mixed states of three qubits. The measure we consider is the convex-roof extended three-tangle. Our findings highlight an intimate relation to lower bounds obtained recently from so-called characteristic curves of a given entanglement measure. We apply the bounds to estimate the three-tangle present in recently performed experiments aimed at producing a three-qubit Greenberger-Horne-Zeilinger (GHZ) state. A nonvanishing lower bound is obtained if the GHZ fidelity of the produced states is larger than $3/4$.

arXiv:0908.3467Physical Review AFeb 9, 201019 citations

Analytically computable tangle for three-qubit mixed states

Hiroyasu Tajima

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.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.

arXiv:1308.5488Aug 26, 20130 citations

Numerical Determination of the Three-Tangle of Mixed States

Sameh M. Saad

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.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.

Aug 31, 20150 citations

Typical features of the Mintert-Buchleitner lower bound for concurrence

Antoni Borràs, A. P. Majtey, Angel Ricardo Plastino +2 more

Despite the great importance of quantum entanglement, a computationally practical, physically motivated, and experimentally implementable way to estimate the amount of entanglement associated with a general (mixed) state of a bipartite quantum system is still lacking. An entanglement lower bound recently advanced by Mintert and Buchleitner [Phys. Rev. Lett. 98, 140505 (2007)] was proposed as a possible tool for the aforementioned task. Mintert and Buchleitner presented evidence indicating that, due to its tightness, in some important cases the alluded bound can be used as an entanglement measure. From the physical point of view, what makes this bound particularly appealing as an entanglement measure is that it is based on an observable quantity. For this last reason, and also because of its other attractive feature, mathematical simplicity, this promising bound and its possible use as an entanglement estimator certainly deserve to be the focus of careful attention. The usefulness of the bound as an entanglement measure, however, depends on its degree of tightness. Here we perform a systematic survey of the behavior of the alluded bound in state space, in order to determine its typical properties (in particular its tightness) and thus assess its value as an entanglement estimator.Despite the great importance of quantum entanglement, a computationally practical, physically motivated, and experimentally implementable way to estimate the amount of entanglement associated with a general (mixed) state of a bipartite quantum system is still lacking. An entanglement lower bound recently advanced by Mintert and Buchleitner [Phys. Rev. Lett. 98, 140505 (2007)] was proposed as a possible tool for the aforementioned task. Mintert and Buchleitner presented evidence indicating that, due to its tightness, in some important cases the alluded bound can be used as an entanglement measure. From the physical point of view, what makes this bound particularly appealing as an entanglement measure is that it is based on an observable quantity. For this last reason, and also because of its other attractive feature, mathematical simplicity, this promising bound and its possible use as an entanglement estimator certainly deserve to be the focus of careful attention. The usefulness of the bound as an entanglement measure, however, depends on its degree of tightness. Here we perform a systematic survey of the behavior of the alluded bound in state space, in order to determine its typical properties (in particular its tightness) and thus assess its value as an entanglement estimator.

Physical Review AFeb 13, 200912 citations

Quantifying Entanglement of Maximal Dimension in Bipartite Mixed States

Gael Sentís, Christopher Eltschka, Otfried Gühne +2 more

The Schmidt coefficients capture all entanglement properties of a pure bipartite state and therefore determine its usefulness for quantum information processing. While the quantification of the corresponding properties in mixed states is important both from a theoretical and a practical point of view, it is considerably more difficult, and methods beyond estimates for the concurrence are elusive. In particular this holds for a quantitative assessment of the most valuable resource, the forms of entanglement that can only exist in high-dimensional systems. We derive a framework for lower bounding the appropriate measure of entanglement, the so-called G-concurrence, through few local measurements. Moreover, we show that these bounds have relevant applications also for multipartite states.The Schmidt coefficients capture all entanglement properties of a pure bipartite state and therefore determine its usefulness for quantum information processing. While the quantification of the corresponding properties in mixed states is important both from a theoretical and a practical point of view, it is considerably more difficult, and methods beyond estimates for the concurrence are elusive. In particular this holds for a quantitative assessment of the most valuable resource, the forms of entanglement that can only exist in high-dimensional systems. We derive a framework for lower bounding the appropriate measure of entanglement, the so-called G-concurrence, through few local measurements. Moreover, we show that these bounds have relevant applications also for multipartite states.

arXiv:1605.09783Physical Review LettersNov 4, 201625 citations

3/4

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.