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Unitary invariant discord as a measures of bipartite quantum correlations in an N-qubit quantum system

Alexander Ivanovich Zenchuk

Published Oct 18, 2011• 6 citations
Publisher
DOI
10.1007/s11128-011-0319-x

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We introduce a measure of quantum correlations in the $N$-qubit quantum system which is invariant with respect to the $SU(2^N)$ group of transformations of this system. This measure is a modification of the quantum discord introduced earlier and is referred to as the unitary or $SU(2^N)$-invariant discord. Since the evolution of a quantum system is equivalent to the proper unitary transformation, the introduced measure is an integral of motion and is completely defined by eigenvalues of the density matrix. As far as the calculation of the unitary invariant discord is rather complicated computational problem, we propose its modification which may be found in a simpler way. The case N=2 is considered in details. In particular, it is shown that the modified SU(4)-invariant discord reaches the maximum value for a pure state. {A geometric measure of the unitary invariant discord of an $N$-qubit state is introduced and a simple formula for this measure is derived, which allows one to consider this measure as a witness of quantum correlations.} The relation of the unitary invariant discord with the quantum state transfer along the spin chain is considered. We also compare the modified SU(4)-invariant discord with the geometric measure of SU(4)-invariant discord of the two-qubit systems in the thermal equilibrium states governed by the different Hamiltonians.

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A computable multipartite multimode Gaussian correlation measure and the monogamy relation for continuous-variable systems

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In this paper, a computable multipartite multimode Gaussian quantum correlation measure ${\mathcal M}^{(k)}$ is proposed for any $k$-partite continuous-variable (CV) systems with $k\geq 2$. ${\mathcal M}^{(k)}$ depends only on the covariance matrix of CV states, is invariant under any permutation of subsystems, is a quantification without ancilla problem, nonincreasing under $k$-partite local Gaussian channels (particularly, invariant under $k$-partite local Gaussian unitary operations), vanishes on $k$-partite product states. For a $k$-partite Gaussian state $ρ$, ${\mathcal M}^{(k)}(ρ)=0$ if and only if $ρ$ is a $k$-partite product state. Thus, for the bipartite case, ${\mathcal M}={\mathcal M}^{(2)}$ is an accessible replacement of the Gaussian quantum discord and Gaussian geometric discord. Moreover, ${\mathcal M}^{(k)}$ satisfies the unification condition, hierarchy condition that a multipartite quantum correlation measure should obey. ${\mathcal M}^{(k)}$ is not bipartite like monogamous, but, ${\mathcal M}^{(k)}$ is complete monogamous and tight complete monogamous.

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Unitary invariant discord as a measure of bipartite quantum correlations in an NNN-qubit quantum system
We introduce a measure of quantum correlations in the NNN-qubit quantum system which is invariant with respect to the SU(2N)SU(2^N)SU(2N) group of transformations of this system. This measure is a modification of the quantum discord introduced earlier and is referred to as the unitary or SU(2N)SU(2^N)SU(2N)-invariant discord. Since the evolution of a quantum system is equivalent to the proper unitary transformation, the introduced measure is an integral of motion and is completely defined by eigenvalues of the density matrix. As far as the calculation of the unitary invariant discord is rather complicated computational problem, we propose its modification which may be found in a simpler way. The case N=2 is considered in details. In particular, it is shown that the modified SU(4)-invariant discord reaches the maximum value for a pure state. {A geometric measure of the unitary invariant discord of an NNN-qubit state is introduced and a simple formula for this measure is derived, which allows one to consider this measure as a witness of quantum correlations.} The relation of the unitary invariant discord with the quantum state transfer along the spin chain is considered. We also compare the modified SU(4)-invariant discord with the geometric measure of SU(4)-invariant discord of the two-qubit systems in the thermal equilibrium states governed by the different Hamiltonians.
In this paper, a computable multipartite multimode Gaussian quantum correlation measure M(k){\mathcal M}^{(k)}M(k) is proposed for any kkk-partite continuous-variable (CV) systems with k≥2k\geq 2k≥2. M(k){\mathcal M}^{(k)}M(k) depends only on the covariance matrix of CV states, is invariant under any permutation of subsystems, is a quantification without ancilla problem, nonincreasing under kkk-partite local Gaussian channels (particularly, invariant under kkk-partite local Gaussian unitary operations), vanishes on kkk-partite product states. For a kkk-partite Gaussian state ρρρ, M(k)(ρ)=0{\mathcal M}^{(k)}(ρ)=0M(k)(ρ)=0 if and only if ρρρ is a kkk-partite product state. Thus, for the bipartite case, M=M(2){\mathcal M}={\mathcal M}^{(2)}M=M(2) is an accessible replacement of the Gaussian quantum discord and Gaussian geometric discord. Moreover, M(k){\mathcal M}^{(k)}M(k) satisfies the unification condition, hierarchy condition that a multipartite quantum correlation measure should obey. M(k){\mathcal M}^{(k)}M(k) is not bipartite like monogamous, but, M(k){\mathcal M}^{(k)}M(k) is complete monogamous and tight complete monogamous.