We exploit quantum discord (and geometric discord) to detect quantum correlations present in a well-known communication model called quantum random access codes (QRACs), which has a variety of applications in quantum information theory. In spite of the fact that there is no entanglement between the two parts involved in this model, analytical derivation shows that the quantum discord is nonzero and highlights that quantum discord might be regarded as a figure of merit to characterize the quantum feature of QRACs, since this model has no classical counterparts. To gain further insight, we also investigate the dynamical behavior of quantum discord under some specific state rotations. In the two-state case, the connection between quantum discord and dimension witness is graphically discussed and intriguingly, our results illustrate that these two quantities are monotonically related to each other. For state encodings in the real $|0\ensuremath{\rangle}\ensuremath{-}|1\ensuremath{\rangle}$ plane, we derive an explicit analytical expression of the geometric discord and find that geometric discord reaches the maximal value for the optimal encoding strategy. However, for arbitrary state encodings in the Bloch sphere, our numerical simulations reveal that maximal geometric discord could not coincide with optimal $2\ensuremath{\rightarrow}1$ QRAC.
We exploit quantum discord (and geometric discord) to detect quantum correlations present in a well-known communication model called quantum random access codes (QRACs), which has a variety of applications in quantum information theory. In spite of the fact that there is no entanglement between the two parts involved in this model, analytical derivation shows that the quantum discord is nonzero and highlights that quantum discord might be regarded as a figure of merit to characterize the quantum feature of QRACs, since this model has no classical counterparts. To gain further insight, we also investigate the dynamical behavior of quantum discord under some specific state rotations. In the two-state case, the connection between quantum discord and dimension witness is graphically discussed and intriguingly, our results illustrate that these two quantities are monotonically related to each other. For state encodings in the real ∣0\ensuremath⟩\ensuremath−∣1\ensuremath⟩ plane, we derive an explicit analytical expression of the geometric discord and find that geometric discord reaches the maximal value for the optimal encoding strategy. However, for arbitrary state encodings in the Bloch sphere, our numerical simulations reveal that maximal geometric discord could not coincide with optimal 2\ensuremath→1 QRAC.
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 entanglement plays a key role in quantum information and quantum computation and thus attracts much attention in many branches of physics both in theory and in experiment. But recent studies revealed that some separable states (non-entangled state) may speed up certain tasks over their classical counterparts and may also possess certain kinds of quantum correlations. For example, geometric quantum discord, which is a more general quantum correlation measure than entanglement, can be nonzero for some separable states. From a practical point of view, it is proposed that the geometric quantum discord be responsible for the power of many quantum information processing tasks. In order to capture such correlations, Ollivier and Zurek introduced quantum discord, which measures the discrepancy between two natural yet different quantum analogues of two classically equivalent expressions of mutual information. However, the calculation of quantum discord is based on numerical maximization procedure, and there are few analytical expressions even for a two-qubit state. In order to obtain the analytical results of quantum discord, a geometric measure of quantum discord which measures the quantum correlations through the minimum Hilbert-Schmidt distance between the given state and zero discord state is introduced. Geometric quantum discord is defined as an effective measure of quantum correlation, and the geometric quantum discord through the minimal distance between the quantum state and the set of zero-discord states in a bipartite quantum system can be worked out. In this paper, by using the geometric quantum discord measurement method, the geometric quantum discord in Tavis-Cummings model is investigated, and the influences of the initial state purity, entanglement degree, dipole-dipole coupling intensity between two atoms, and field in the Fock state on the evolution characteristic of geometric quantum discord are analyzed. The results show that the geometric quantum discord appears periodically. It initially decreases to a minimum value, and then turns out to be increased for different initial states. The rigorous analysis and numerical results reveal that when we take a suitable initial state, the geometric quantum discord of two atoms can be kept in correlation. When the atoms are in the different initial states, the quantum properties of the system are significant. The photon number of the field can lead the quantum discord to be weakened. Geometric quantum discord can be increased by increasing the cavity photon number and the dipole-dipole coupling intensity. Geometric quantum discord can be enhanced obviously by increasing the strength of the dipole-dipole coupling interaction. The conclusions may conduce to the understanding of quantum correlation for the other systems from the view of geometric quantum discord.
Among various definitions of quantum correlations, quantum discord has attracted considerable attention. To find an analytical expression for quantum discord is an intractable task. Exact results are known only for very special states, namely two-qubit X-shaped states. We present in this paper a geometric viewpoint, from which two-qubit quantum discord can be described clearly. The known results on X state discord are restated in the directly perceivable geometric language. As a consequence, the dynamics of classical correlations and quantum discord for an X state in the presence of decoherence is endowed with geometric interpretation. More importantly, we extend the geometric method to the case of more general states, for which numerical as well as analytical results on quantum discord have not yet been obtained. Based on the support of numerical computations, some conjectures are proposed to help us establish the geometric picture. We find that the geometric picture for these states has an intimate relationship with that for X states. Thereby, in some cases, analytical expressions for classical correlations and quantum discord can be obtained.
Abstract Entangled quantum states are regarded as a key resource in quantum key distribution (QKD) protocols. However, quantum correlations, other than entanglement can also play a significant role in the QKD protocols. In this work, we will focus on one such measure of quantum correlation, known as geometric quantum discord (GQD). Firstly, we derive an analytical expression of GQD for two-qutrit quantum systems and further generalize it for d 1 ⨂ d 2 dimensional systems. Next, we apply the concept of GQD in studying QKD. In particular, if the shared resource state is an entangled state constructed with the linear combination of the tensor product of the Bell pair and the state σ i ’s, i = 0, 1, 2, 3, then we have shown that under some assumption on σ i ’s, the lower bound for a distillable secret key rate K D can be expressed in terms of GQD of σ 0 + σ 1 2 and σ 2 + σ 3 2 . Thus, the distillable key rate depends upon the GQD of σ 0 + σ 1 2 and σ 2 + σ 3 2 , when the communicating parties uses private states for generating a secret key in presence of an eavesdropper. Further, for a certain range of GQD of σ 0 + σ 1 2 and σ 2 + σ 3 2 <
Dakic, Vedral, and Brukner [arXiv:1004.0190 (2010)] introduced a geometric measure of quantum discord and derived an explicit formula for any two-qubit state. This measure is significant in capturing quantum correlations from a geometric perspective. In this brief report, we evaluate the geometric measure of quantum discord for an arbitrary state and obtain an explicit and tight lower bound. Furthermore, we reveal an intrinsic feature of geometric measure of quantum discord by showing that it actually coincides with a simpler quantity based on von Neumann measurements.
Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. In general, this correlation is different from entanglement, and quantum discord may be nonzero even for certain separable states. Even in the simple case of bipartite quantum systems, this different kind of quantum correlation has interesting and significant applications in quantum information processing. So far, quantum discord has been calculated explicitly only for a rather limited set of two-qubit quantum states and expressions for more general quantum states are not known. In this article, we derive explicit expressions for quantum discord for a larger class of two-qubit states, namely, a seven-parameter family of so called X states that have been of interest in a variety of contexts in the field. We also study the relation between quantum discord, classical correlation, and entanglement for a number of two-qubit states to demonstrate that they are independent measures of correlation with no simple relative ordering between them.