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arXiv:1302.5239

Quantum discord for two-qubit CS states: Analytical solution

M. A. Yurishchev

Published Feb 21, 2013• 3 citations
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Abstract

We found an universal local unitary (orthogonal) transformation which transforms any centrosymmetric (CS) density matrix ρ_{CS} into the X density matrix ρ_X: H\otimes Hρ_{CS}H\otimes H=ρ_X, where H is the Hadamard transformation. Because quantum discord is invariant under the local unitary transformations, this remarkable property allows to get the discord of general two-qubit CS states in the analytical form using the corresponding well-known formulas for the X states. Examples of systems with CS density matrices provided from the recent literature, including XXZ spin model with the Dzyaloshinsky-Moriya interaction, a gas of spin-carrying particles in closed nanopore, and a family of pseudopure states.

DOI
10.48550/arxiv.1302.5239

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We found an universal local unitary (orthogonal) transformation which transforms any centrosymmetric (CS) density matrix ρ_{CS} into the X density matrix ρ_X: H\otimes Hρ_{CS}H\otimes H=ρ_X, where H is the Hadamard transformation. Because quantum discord is invariant under the local unitary transformations, this remarkable property allows to get the discord of general two-qubit CS states in the analytical form using the corresponding well-known formulas for the X states. Examples of systems with CS density matrices provided from the recent literature, including XXZ spin model with the Dzyaloshinsky-Moriya interaction, a gas of spin-carrying particles in closed nanopore, and a family of pseudopure states.
Quantum discord is a function of density-matrix elements (and through them, e.~g., of temperature, applied fields, time, and so forth). The domain of such a function in the case of two-qubit system with X or centrosymmetric (CS) density matrix can consist at most of three subdomains: two ones, where the quantum discord is expressed in closed analytical forms (Q_0 and Q_{π/2}), and an intermediate subdomain in which for determining the quantum discord Q_θit is required to solve numerically a one-dimensional minimization problem to find the optimal measurement angle θ\in(0,π/2). Exact equations for determining the boundaries between these subdomains are obtained and solved for a number of models. The Q_θsubdomains are discovered in the anisotropic spin dimers in external field. On the other hand, coinciding boundaries and therefore sudden transitions between optimal measurement angles θ=π/2 and θ=0 are observed in dynamics of spin currying particles in closed nanopore and also in phase flip channels. In latter cases the solutions are entirely analytical.
A local orthogonal transformation that transforms any centrosymmetric density matrix of a two-qubit system to the X form has been found. A piecewise-analytic-numerical formula Q = min{Q π/2, Q θ, Q 0}, where Q π/2 and Q 0 are analytical expressions and the branch Q 0 θ can be obtained only by numerically searching for the optimal measurement angle θ ∈ (0, π/2), is proposed to calculate the quantum discord Q of a general X state. The developed approaches have been applied for a quantitative description of the recently predicted flickering (periodic disappearance and reappearance) of the quantum-information pair correlation between nuclear 1/2 spins of atoms or molecules of a gas (for example, 129Xe) in a bounded volume in the presence of a strong magnetic field.
Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. It has been discussed so far for small systems with only a few independent parameters. We extend here to a much broader class of states when the second party is of arbitrary dimension d, so long as the first, measured, party is a qubit. We present two formulae to calculate quantum discord, the first relating to the original entropic definition and the second to a recently proposed geometric distance measure which leads to an analytical formulation. The tracing over the qubit in the entropic calculation is reduced to a very simple prescription. And, when the d-dimensional system is a so-called X state, the density matrix having non-zero elements only along the diagonal and anti-diagonal so as to appear visually like the letter X, the entropic calculation can be carried out analytically. Such states of the full bipartite qubit-qudit system may be named "extended X states", whose density matrix is built of four block matrices, each visually appearing as an X. The optimization involved in the entropic calculation is generally over two parameters, reducing to one for many cases, and avoided altogether for an overwhelmingly large set of density matrices as our numerical investigations demonstrate. Our results also apply to states of a N-qubit system, where "extended X states" consist of (2^(N+2) - 1) states, larger in number than the (2^(N+1) - 1) of X states of N qubits. While these are still smaller than the total number (2^(2N) - 1) of states of N qubits, the number of parameters involved is nevertheless large. In the case of N = 2, they encompass the entire 15-dimensional parameter space, that is, the extended X states for N = 2 represent the full qubit-qubit system.
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.
The geometric quantum discord of a two-qudit state has been studied in many papers; however, its exact analytical value in the explicit form is known only for a general two-qubit state, a general qubit-qudit state, and some special families of two-qudit states. Based on the general Bloch vectors formalism [E. R. Loubenets et al., J. Phys. A: Math. Theor. 54, 195301 (2021)], we find the explicit exact analytical value of the geometric quantum discord for an arbitrary two-qudit state of any dimension via the parameters of its correlation matrix and the Bloch vectors of its reduced states. This general analytical result includes all the known exact results on the geometric quantum discord only as particular cases and proves rigorously that the lower bound on the geometric discord presented in [S. Rana et al., Phys. Rev. A 85, 024102 (2012)] constitutes its exact value for each two-qudit state. Moreover, our general result allows us to find for an arbitrary two-qudit state, pure or mixed, the upper and lower bounds on its geometric quantum discord, expressed via the Hilbert space characteristics of this state.