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arXiv:0806.2598 · Physical Review A

Observable estimation of entanglement for arbitrary finite-dimensional mixed states

Chengjie Zhang, Yan-Xiao Gong, Yong-Sheng Zhang, Guang‐Can Guo

Published Oct 9, 2008• 59 citations
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Abstract

We present observable upper bounds for the squared concurrence, which are the dual inequalities of the observable lower bounds introduced by Mintert and Buchleitner [Phys. Rev. Lett. 98, 140505 (2007)] and Aolita et al. [Phys. Rev. A 78, 022308 (2008)]. These bounds can be used to estimate entanglement for arbitrary experimental unknown finite-dimensional states by a few experimental measurements on a twofold copy $\ensuremath{\rho}\ensuremath{\bigotimes}\ensuremath{\rho}$ of the mixed states. Furthermore, the degree of mixing for a mixed state and some properties of the linear entropy also have certain relations with the upper and lower bounds of its squared concurrence.

DOI
10.1103/physreva.78.042308

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We present observable upper bounds for the squared concurrence, which are the dual inequalities of the observable lower bounds introduced by Mintert and Buchleitner [Phys. Rev. Lett. 98, 140505 (2007)] and Aolita et al. [Phys. Rev. A 78, 022308 (2008)]. These bounds can be used to estimate entanglement for arbitrary experimental unknown finite-dimensional states by a few experimental measurements on a twofold copy \ensuremathρ\ensuremath⨂\ensuremathρ\ensuremath{\rho}\ensuremath{\bigotimes}\ensuremath{\rho}\ensuremathρ\ensuremath⨂\ensuremathρ of the mixed states. Furthermore, the degree of mixing for a mixed state and some properties of the linear entropy also have certain relations with the upper and lower bounds of its squared concurrence.
We derive measurable lower bounds on concurrence of arbitrary mixed states, for both bipartite and multipartite cases. First, we construct measurable lower bonds on the purely algebraic bounds of concurrence [F. Mintert et al. (2004), Phys. Rev. lett., 92, 167902]. Then, using the fact that the sum of the square of the algebraic bounds is a lower bound of the squared concurrence, we sum over our measurable bounds to achieve a measurable lower bound on concurrence. With two typical examples, we show that our method can detect more entangled states and also can give sharper lower bonds than the similar ones.
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We derive measurable lower bounds on concurrence of arbitrary mixed states, for both bipartite and multipartite cases. First, we construct measurable lower bonds on the \textit{purely algebraic} bounds of concurrence [F. Mintert \textit{et al.} (2004), Phys. Rev. lett., 92, 167902]. Then, using the fact that the sum of the square of the algebraic bounds is a lower bound of the squared concurrence, we sum over our measurable bounds to achieve a measurable lower bound on concurrence. With two typical examples, we show that our method can detect more entangled states and also can give sharper lower bonds than the similar ones.
We present observable lower bounds for several bipartite entanglement measures including entanglement of formation, geometric measure of entanglement, concurrence, convex-roof extended negativity, and GGG concurrence. The lower bounds facilitate estimates of these entanglement measures for arbitrary finite-dimensional bipartite states. Moreover, these lower bounds can be calculated analytically from the expectation value of a single observable. Based on our results, we use several real experimental measurement data to get lower bounds of entanglement measures for these experimentally realized states. In addition, we also study the relations between entanglement measures.
We present observable lower and upper bounds for the entanglement of formation (EOF) and quantum discord (QD), which facilitates estimates of EOF and QD for arbitrary experimental unknown states in finite-dimensional bipartite systems. These bounds can be easily obtained by a few experimental measurements on a twofold copy $\ensuremath{\varrho}\ensuremath{\bigotimes}\ensuremath{\varrho}$ of the mixed states. Based on our results, we use the experimental measurement data of the real experiment given by Schmid et al. [Phys. Rev. Lett. 101, 260505 (2008)] to obtain the lower and upper bounds of EOF and QD for the experimental unknown state.
The dynamics of the quantum entanglement is a fundamental characteristic for various quantum systems. Since the computable entanglement measure for higher dimensional quantum states itself is absent, the dynamics of the entanglement expressed in an operational method will be of interest. We study the dynamics of $\ensuremath{\tau}$, an analytical lower bound of squared concurrence, of a bipartite $d\ensuremath{\bigotimes}d$ quantum state when one party goes through an arbitrary noisy channel. For a pure input state, the range of $\ensuremath{\tau}$ is obtained explicitly. For a mixed input state, an upper bound of $\ensuremath{\tau}$ is found. Interestingly, the tangle ${\ensuremath{\tau}}^{\ensuremath{'}}$, as an upper bound of squared concurrence, also has a similar dynamical property. Our results are similar to that of Konrad et al. and can help the estimation of high-dimension bipartite entanglement in experiments.