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arXiv:1107.5458 · Physical Review A

Observable estimation of entanglement of formation and quantum discord for bipartite mixed quantum states

Chengjie Zhang, Sixia Yu, Qing Chen, C. H. Oh

Published Nov 21, 2011• 24 citations
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Abstract

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.

DOI
10.1103/physreva.84.052112

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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 \ensuremathϱ\ensuremath⨂\ensuremathϱ\ensuremath{\varrho}\ensuremath{\bigotimes}\ensuremath{\varrho}\ensuremathϱ\ensuremath⨂\ensuremathϱ 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.
We present observable lower and upper bounds of the quantum discord, which facilitates estimates of quantum discord 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 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 for the experimental unknown state. Furthermore, lower and upper bounds for entanglement of formation can also be obtained as a corollary.
Quantum correlations play vital roles in the quantum features in quantum information processing tasks. Among the measures of quantum correlations, quantum discord (QD) and entanglement of formation (EOF) are two significant ones. Recent research has shown that there exists a relation between QD and EOF, which makes QD more significant in quantum information theory. However, until now, there exists no general method of characterizing quantum discord in high-dimensional quantum systems. In this paper, we have proposed a general method for calculating quantum discord in arbitrary-dimensional bipartite quantum systems in terms of Hurwitz's theory. Applications including the Werner state, the spin-1 XXZ model thermal equilibrium state, the Horodecki state, and the separable-bound-free entanglement state are investigated. We present the method of obtaining the EOF of arbitrary-dimensional bipartite quantum states via purification, and the relationship between QD and EOF.
The common use in the literature of orthogonal measurements in obtaining quantum discord for two-qubit states is discussed and compared with more general measurements. We prove the optimality of orthogonal measurements for rank-2 states. While for rank-3 and -4 mixed states they are not optimal, we present strong numerical evidence showing that they give the correct quantum discord up to minimal corrections. Based on the connection, through purification with an ancilla, between discord and entanglement of formation (EoF), we give a tight upper bound for the EoF of a 2⊗ N mixed state of rank 2, given by an optimal decomposition of 2 elements. We also provide an alternative way to compute the quantum discord for two qubits based on the Bloch vectors of the state.
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.