The counterintuitive effect of non-unique ordering of two-qubit states with quantum entanglement measures was discovered over ten years ago. More precisely, it was shown by Monte Carlo simulations that there exist states for which the entanglement of formation and the negativity do not impose the same ordering of states, i.e. E F (ρ AB )⩽(⩾)E F (ρ' AB ) is not equivalent to N (ρ AB )⩽(⩾) N (ρ' AB ). Recently, it was discovered that quantum discord and the geometric quantum discord do not necessarily imply the same ordering of two-qubit X -states, which means that the lack of the unique ordering of states with quantum entanglement measures goes beyond entanglement. Inspired by this observation, we study the problem of the states ordering with quantum discords, considering two-qubit Bell-diagonal states for analytical simplicity. In particular, we identify some classes of states for which the states ordering with quantum discords is preserved as long as the states belong to the same class and give a few illustrative examples.
The counterintuitive effect of non-unique ordering of two-qubit states with quantum entanglement measures was discovered over ten years ago. More precisely, it was shown by Monte Carlo simulations that there exist states for which the entanglement of formation and the negativity do not impose the same ordering of states, i.e. E F (ρ AB )⩽(⩾)E F (ρ' AB ) is not equivalent to N (ρ AB )⩽(⩾) N (ρ' AB ). Recently, it was discovered that quantum discord and the geometric quantum discord do not necessarily imply the same ordering of two-qubit X -states, which means that the lack of the unique ordering of states with quantum entanglement measures goes beyond entanglement. Inspired by this observation, we study the problem of the states ordering with quantum discords, considering two-qubit Bell-diagonal states for analytical simplicity. In particular, we identify some classes of states for which the states ordering with quantum discords is preserved as long as the states belong to the same class and give a few illustrative examples.
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 X 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 X states, whose quantum discord can be evaluated analytically without any minimization, for which their algorithm is valid, and also identify a family of X states for which their algorithm fails. And then we demonstrate that this special family of X states provides furthermore an explicit example for the inequivalence between the minimization over positive operator-valued measures and that over von Neumann measurements.
Abstract One of the primary goal of quantum information theory is to understand and quantify the various correlations that exist in quantum systems. Quantum correlation is broader than quantum entanglement, and quantum discord has always attracted more attention. In this paper, we investigate the hidden properties of quantum discord. In particular, for a class of two-qubit states, we prove the existence of hidden quantum discord under local filtering operations and give sufficient conditions for the absence of hidden quantum discord. For a class of three-qubit states, we investigate hidden geometric discord and analyze its dynamic behavior under several conditions. Furthermore, by comparing two- and three-qubit systems, we discover the inconsistency of hidden discord for ρ = 1 2 N ( I + ∑ j = 1 3 c j σ j ⊗ … ⊗ σ j ) ( c 3 = 1 , N = 2 , 3 ) .
Quantum discord, as introduced by Olliver and Zurek [Phys. Rev. Lett. 88, 017901 (2001)], is a measure of the discrepancy between two natural yet different quantum analogs of the classical mutual information. This notion characterizes and quantifies quantumness of correlations in bipartite states from a measurement perspective, and is fundamentally different from the various entanglement measures in the entanglement vs separability paradigm. The phenomenon of nonzero quantum discord is a manifestation of quantum correlations due to noncommutativity rather than due to entanglement, and has interesting and significant applications in revealing the advantage of certain quantum tasks. We will evaluate analytically the quantum discord for a large family of two-qubit states, and make a comparative study of the relationships between classical and quantum correlations in terms of the quantum discord. We furthermore compare the quantum discord with the entanglement of formation, and illustrate that the latter may be larger than the former, although for separable states, the entanglement of formation always vanishes and thus is less than the quantum discord.
We present a reliable algorithm to evaluate quantum discord for general two-qubit states, amending and extending an approach recently put forward for the subclass of X states. A closed expression for the discord of arbitrary states of two qubits cannot be obtained, as the optimization problem for the conditional entropy requires the solution to a pair of transcendental equations in the state parameters. We apply our algorithm to run a numerical comparison between quantum discord and an alternative, computable measure of nonclassical correlations, namely, the geometric discord. We identify the extremally nonclassically correlated two-qubit states according to the (normalized) geometric discord, at a fixed value of the conventional quantum discord. The latter cannot exceed the square root of the former for systems of two qubits.
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.
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.