Quantum discord goes beyond entanglement and exists in a wide range of quantum states that may be separable, playing a crucial role in quantum information tasks. In this paper, we firstly proposed a zero-discord criterion for two-qubit system based on the partial transposition of density matrix, and then extended it to the qubit-qudit system. By detailed examples we demonstrate the effectiveness of these criteria in detecting discord. Moreover, we provide an analytical lower bound of geometric quantum discord(GQD) using eigenvalue vectors of density matrix. Finally, we presented a one-way work deficit lower bound based on our lower bound of GQD.
Quantum discord goes beyond entanglement and exists in a wide range of quantum states that may be separable, playing a crucial role in quantum information tasks. In this paper, we firstly proposed a zero-discord criterion for two-qubit system based on the partial transposition of density matrix, and then extended it to the qubit-qudit system. By detailed examples we demonstrate the effectiveness of these criteria in detecting discord. Moreover, we provide an analytical lower bound of geometric quantum discord(GQD) using eigenvalue vectors of density matrix. Finally, we presented a one-way work deficit lower bound based on our lower bound of GQD.
Quantum discord goes beyond entanglement and exists in a wide range of quantum states that may be separable, playing a crucial role in quantum information tasks. In this paper, we firstly proposed a zero-discord criterion for two-qubit system based on the partial transposition of density matrix, and then extended it to the qubit–qudit system. By detailed examples we demonstrate the effectiveness of these criteria in detecting discord. Moreover, we provide an analytical lower bound of geometric quantum discord(GQD) using eigenvalue vectors of density matrix. Finally, we presented a one-way work deficit lower bound based on our lower bound of GQD.
Here, we show that partial transposition, which is initially introduced to study entanglement, can also inspire many results on quantum discord including: (I) a discord criterion of spectrum invariant under partial transposition, stating that one state must contain discord if its spectrum is changed by the action of partial transposition, (II) an approach to estimate the geometric quantum discord and the one-way deficit based on the change of spectrum. To compare with entanglement theory, we also lower-bound the geometric quantum entanglement and the entanglement of relative entropy. Thus, on one hand, we illustrate an approach to specify and estimate discord based on partial transposition. On the other hand, we show that, entanglement and discord, two basic notions of nonclassical correlations, can be placed on the same ground such that their interplay and distinction can be illustrated in within a universal framework.
Here, we show that partial transposition, which is initially introduced to study entanglement, can also inspire many results on quantum discord, including (I) a discord criterion of a spectrum invariant under partial transposition, stating that one state must contain discord if its spectrum is changed by the action of partial transposition, and (II) an approach to estimate the geometric quantum discord and the one-way deficit based on the change of the spectrum. To compare with entanglement theory, we also lower bound the geometric quantum entanglement and the entanglement of relative entropy. Thus, on one hand, we illustrate an approach to specify and estimate discord based on partial transposition. On the other hand, we show that, entanglement and discord, two basic notions of nonclassical correlations, can be placed on the same ground such that their interplay and distinction can be illustrated within a universal framework.
Quantum discord quantifies quantum correlations beyond entanglement and assumes nonzero values, which are notoriously hard to compute, for almost all quantum states. Here we provide computable tight bounds for the quantum discord for qubit-qudit states. In the case of two qubits our lower and upper bounds coincide for a 7-parameter family of filtered X-states, whose quantum discords can therefore be evaluated analytically. An application to the accessible information of the binary qubit channel is also presented. For the qubit-qudit state output by the circuit of deterministic computation with one qubit, nontrivial lower and upper bounds that respect the zero-discord conditions are obtained for its quantum discord.
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, 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.