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arXiv:1306.0228 · Physical Review A

Quantum discord for two-qubit X states: Analytical formula with very small worst-case error

Yichen Huang

Published Jul 18, 2013• 135 citations
arXivHTMLPDF

Abstract

Quantum discord is a measure of quantum correlation beyond entanglement. Computing quantum discord for simple quantum states is a basic problem. An analytical formula of quantum discord for two-qubit $X$ states is first claimed in [Ali, Rau, and Alber, Phys. Rev. A 81, 042105 (2010)], but later found to be not always correct. I observe numerically that the formula is valid with worst-case absolute error 0.0021. For symmetric two-qubit $X$ states, I give a counterexample to the analytical formula derived in [F. F. Fanchini et al., Phys. Rev. A 81, 052107 (2010)], but observe that the formula is valid with worst-case absolute error 0.0006. The formula has been used in many research papers. The results in all these works are approximately correct, even if they may not be exactly correct.

DOI
10.1103/physreva.88.014302

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Quantum discord is a measure of quantum correlation beyond entanglement. Computing quantum discord for simple quantum states is a basic problem. An analytical formula of quantum discord for two-qubit XXX states is first claimed in [Ali, Rau, and Alber, Phys. Rev. A 81, 042105 (2010)], but later found to be not always correct. I observe numerically that the formula is valid with worst-case absolute error 0.0021. For symmetric two-qubit XXX states, I give a counterexample to the analytical formula derived in [F. F. Fanchini et al., Phys. Rev. A 81, 052107 (2010)], but observe that the formula is valid with worst-case absolute error 0.0006. The formula has been used in many research papers. The results in all these works are approximately correct, even if they may not be exactly correct.
Quantum discord provides a measure for quantifying quantum correlations beyond entanglement and is very hard to compute even for two-qubit states because of the minimization over all possible measurements. Recently a simple algorithm to evaluate the quantum discord for two-qubit XXX states was proposed by Ali, Rau, and Alber [Phys. Rev. A 81, 042105 (2010)] with minimization taken over only a few cases. Here we shall at first identify a class of XXX states, whose quantum discord can be evaluated analytically without any minimization, for which their algorithm is valid, and also identify a family of XXX states for which their algorithm fails. And then we demonstrate that this special family of XXX states provides furthermore an explicit example for the inequivalence between the minimization over positive operator-valued measures and that over von Neumann measurements.
Quantum discord for two-qubit XXX-states: A comprehensive approach inspired by classical polarization optics
Classical correlation and quantum discord are computed for two-qubit XXX-states. Our approach, which is inspired by the methods of classical polarization optics, is geometric in the sense that the entire analysis is tied to the correlation ellipsoid of all normalized conditional states of the A-qubit with measurement elements applied to the B-qubit. Aspects of the computation which depend on the location of the reduced state of A inside the ellipsoid get clearly separated from those which do not. Our treatment is comprehensive\,: all known results are reproduced, often more economically, and several new insights and results emerge. Detailed reexamination of the famous work of Ali, Rau, and Alber [Phys. Rev. A {\bf 81}, 042105 (2010)], in the light of ours and a counterexample we manufacture against their principal theorem points to an uncommon situation in respect of their principal result\,: their theorem turns out to be `numerically correct' in all but a very tiny region in the space of XXX-states, notwithstanding the fact that their proof of the theorem seems to make, in the disguise of an unusual group theoretic argument, an {\em a priori} assumption equivalent to the theorem itself.
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.
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.
Quantum discord in any mixture of two bi-qubit arbitrary product states is studied with two different approaches. In the first approach the maximal classical correlations are obtained via numerical computations, while in the second approach they are analytically derived. Quantum correlations captured with both approaches completely coincide, as is in accord with the conclusion of Cen et al. [Phys. Rev. A 83 (2011) 054101]. The symmetry reduction of the concerned states concerning quantum correlations is accomplished. The captured discords are amply analyzed so that some inherent distinct properties are revealed.