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arXiv:1110.3499

Measurement-induced nonlocality for an arbitrary bipartite state

Sayyed Yahya Mirafzali, Iman Sargolzahi, Ali Ahanj, Kurosh Javidan, Mohsen Sarbishaei

Published Oct 16, 2011• 3 citations
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Abstract

Measurement-induced nonlocality is a measure of nonlocalty introduced by Luo and Fu [Phys. Rev. Lett \textbf{106}, 120401 (2011)]. In this paper, we study the problem of evaluation of Measurement-induced nonlocality (MIN) for an arbitrary $m\times n$ dimensional bipartite density matrix $ρ$ for the case where one of its reduced density matrix, $ρ^{a}$, is degenerate (the nondegenerate case was explained in the preceding reference). Suppose that, in general, $ρ^{a}$ has $d$ degenerate subspaces with dimension $m_{i} (m_{i} \leq m, i=1, 2, ..., d)$. We show that according to the degeneracy of $ρ^{a}$, if we expand $ρ$ in a suitable basis, the evaluation of MIN for an $m\times n$ dimensional state $ρ$, is degraded to finding the MIN in the $m_{i}\times n$ dimensional subspaces of state $ρ$. This method can reduce the calculations in the evaluation of MIN. Moreover, for an arbitrary $m\times n$ state $ρ$ for which $m_{i}\leq 2$, our method leads to the exact value of the MIN. Also, we obtain an upper bound for MIN which can improve the ones introduced in the above mentioned reference. In the final, we explain the evaluation of MIN for $3\times n$ dimensional states in details.

DOI
10.48550/arxiv.1110.3499

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Measurement-induced nonlocality is a measure of nonlocalty introduced by Luo and Fu [Phys. Rev. Lett \textbf{106}, 120401 (2011)]. In this paper, we study the problem of evaluation of Measurement-induced nonlocality (MIN) for an arbitrary m×nm\times nm×n dimensional bipartite density matrix ρρρ for the case where one of its reduced density matrix, ρaρ^{a}ρa, is degenerate (the nondegenerate case was explained in the preceding reference). Suppose that, in general, ρaρ^{a}ρa has ddd degenerate subspaces with dimension mi(mi≤m,i=1,2,...,d)m_{i} (m_{i} \leq m, i=1, 2, ..., d)mi​(mi​≤m,i=1,2,...,d). We show that according to the degeneracy of ρaρ^{a}ρa, if we expand ρρρ in a suitable basis, the evaluation of MIN for an m×nm\times nm×n dimensional state ρρρ, is degraded to finding the MIN in the mi×nm_{i}\times nmi​×n dimensional subspaces of state ρρρ. This method can reduce the calculations in the evaluation of MIN. Moreover, for an arbitrary m×nm\times nm×n state ρρρ for which mi≤2m_{i}\leq 2mi​≤2, our method leads to the exact value of the MIN. Also, we obtain an upper bound for MIN which can improve the ones introduced in the above mentioned reference. In the final, we explain the evaluation of MIN for 3×n3\times n3×n dimensional states in details.
Measurement-induced nonlocality is a measure of nonlocalty introduced by Luo and Fu [Phys. Rev. Lett \textbf{106}, 120401 (2011)]. In this paper, we study the problem of evaluation of Measurement-induced nonlocality (MIN) for an arbitrary m×nm\times nm×n dimensional bipartite density matrix ρ\rhoρ for the case where one of its reduced density matrix, ρa\rho^{a}ρa, is degenerate (the nondegenerate case was explained in the preceding reference). Suppose that, in general, ρa\rho^{a}ρa has ddd degenerate subspaces with dimension mi(mi≤m,i=1,2,...,d)m_{i} (m_{i} \leq m , i=1, 2, ..., d)mi​(mi​≤m,i=1,2,...,d). We show that according to the degeneracy of ρa\rho^{a}ρa, if we expand ρ\rhoρ in a suitable basis, the evaluation of MIN for an m×nm\times nm×n dimensional state ρ\rhoρ, is degraded to finding the MIN in the mi×nm_{i}\times nmi​×n dimensional subspaces of state ρ\rhoρ. This method can reduce the calculations in the evaluation of MIN. Moreover, for an arbitrary m×nm\times nm×n state ρ\rhoρ for which mi≤2m_{i}\leq 2mi​≤2, our method leads to the exact value of the MIN. Also, we obtain an upper bound for MIN which can improve the ones introduced in the above mentioned reference. Finally, we explain the evaluation of MIN for 3×n3\times n3×n dimensional states in details.
Measurement-induced nonlocality (MIN), introduced by Luo and Fu [Phys. Rev. Lett.106, 120401 (2011)], is a kind of quantum correlation which is different from entanglement and quantum discord (QD). MIN is defined over one-sided projective measurements. In this paper, we introduce a MIN over two-sided projective measurements. The nullity of this two-sided MIN is characterized, a formula for calculating two-sided MIN for pure states is proposed, and a lower bound of (two-sided) MIN for maximally entangled mixed states is given. In addition, we find that (two-sided) MIN is not continuous. Both finite- and infinite-dimensional cases are considered.
Measurement-induced nonlocality (MIN), which describes the maximum global effect caused by locally invariant measurements, was introduced by Luo and Fu ( Phys. Rev. Lett. , 106 (2011) 120401). In this paper, a new measure of MIN based on Wigner-Yanase skew information is proposed. It is shown that this measure not only has good computability but also eliminates the noncontractivity problem appearing in the original measure of MIN defined by the Hilbert-Schmidt norm. The analytical formulas of MIN based on Wigner-Yanase skew information for any pure states, ( 2 × n ) -dimensional mixed states, and some higher-dimensional symmetric states are presented. Furthermore, the tight upper bound to MIN based on Wigner-Yanase skew information in the general case is also derived.
Nonlocality is one unique property of quantum mechanics that differs from the classical world. One of its quantifications can be properly described as the maximum global effect caused by locally invariant measurements, known as measurement-induced nonlocality (MIN) (2011 Phys. Rev. Lett. 106 120401). Here, we propose quantifying the MIN by the trace norm. We show explicitly that this measure is monotonically decreasing under the action of a completely positive trace-preserving map, which is the general local quantum operation, on the unmeasured party for the bipartite state. This property avoids the undesirable characteristic appearing in the known measure of MIN defined by the Hilbert–Schmidt norm which may be increased or decreased by trivial local reversible operations on the unmeasured party. We obtain analytical formulas of the trace-norm MIN for any -dimensional pure state, two-qubit state, and certain high-dimensional states. As with other quantum correlation measures, the newly defined MIN can be directly applied to various models for physical interpretations.