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arXiv:1608.08136

Zero discord implies classicality

Michel Boyer

Published Aug 29, 2016• 1 citation
arXivHTMLPDF

Abstract

The "classical-quantum" ($\mathit{cq}$) discord of a bipartite state $ρ^{AB}$ is the smallest difference between the mutual information $S(ρ^{A:B})$ of $ρ$ and that of $ρ$ after a measurement channel is applied on the $A$ system. Relating zero discord to the strong subadditivity of the Von Neumann entropy, Datta proved that a state has zero $\mathit{cq}$ discord iff and only if it can be written in the form $\sum_i p_i\, |i\rangle\langle i|\otimes ρ^B_i$ for $p_i$ a probability distribution, $|i\rangle$ a basis of the $A$ system and $ρ^B_i$ states of the $B$ system. We provide a simple proof of that same result using directly a theorem of Petz on channels that leave unchanged the relative entropy of two given states.

DOI
10.48550/arxiv.1608.08136

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The "classical-quantum" (cq\mathit{cq}cq) discord of a bipartite state ρABρ^{AB}ρAB is the smallest difference between the mutual information S(ρA:B)S(ρ^{A:B})S(ρA:B) of ρρρ and that of ρρρ after a measurement channel is applied on the AAA system. Relating zero discord to the strong subadditivity of the Von Neumann entropy, Datta proved that a state has zero cq\mathit{cq}cq discord iff and only if it can be written in the form ∑ipi ∣i⟩⟨i∣⊗ρiB\sum_i p_i\, |i\rangle\langle i|\otimes ρ^B_i∑i​pi​∣i⟩⟨i∣⊗ρiB​ for pip_ipi​ a probability distribution, ∣i⟩|i\rangle∣i⟩ a basis of the AAA system and ρiBρ^B_iρiB​ states of the BBB system. We provide a simple proof of that same result using directly a theorem of Petz on channels that leave unchanged the relative entropy of two given states.
The strong subadditivity inequality of von Neumann entropy relates the entropy of subsystems of a tripartite state ${\ensuremath{\rho}}_{ABC}$ to that of the composite system. Here, we define ${\mathbit{T}}^{(a)}({\ensuremath{\rho}}_{ABC})$ as the extent to which ${\ensuremath{\rho}}_{ABC}$ fails to satisfy the strong subadditivity inequality $S({\ensuremath{\rho}}_{B})+S({\ensuremath{\rho}}_{C})\ensuremath{\le}S({\ensuremath{\rho}}_{AB})+S({\ensuremath{\rho}}_{AC})$ with equality and investigate its properties. In particular, by introducing auxiliary subsystem EEE, we consider any purification $|{\ensuremath{\psi}}_{ABCE}\ensuremath{\rangle}$ of ${\ensuremath{\rho}}_{ABC}$ and formulate ${\mathbit{T}}^{(a)}({\ensuremath{\rho}}_{ABC})$ as the extent to which the bipartite quantum correlations of ${\ensuremath{\rho}}_{AB}$ and ${\ensuremath{\rho}}_{AC}$, measured by entanglement of formation and quantum discord, change under the transformation $B\ensuremath{\rightarrow}BE$ and $C\ensuremath{\rightarrow}CE$. Invariance of quantum correlations of ${\ensuremath{\rho}}_{AB}$ and ${\ensuremath{\rho}}_{AC}$ under such transformation is shown to be a necessary and sufficient condition for vanishing ${\mathbit{T}}^{(a)}({\ensuremath{\rho}}_{ABC})$. Our approach allows one to characterize, intuitively, the structure of states for which the strong subadditivity is saturated. Moreover, along with providing a conservation law for quantum correlations of states for which the strong subadditivity inequality is satisfied with equality, we find that such states coincide with those that the Koashi-Winter monogamy relation is saturated.
Within the unified framework of exploiting the relative entropy as a distance measure of quantum correlations, we make explicit the hierarchical structure of quantum coherence, quantum discord, and quantum entanglement in multipartite systems. On this basis, we define a basis-independent measure of quantum coherence and prove that it is exactly equivalent to quantum discord. Furthermore, since the original relative entropy of coherence is a basis-dependent quantity, we investigate the local and nonlocal unitary creation of quantum coherence, focusing on the two-qubit unitary gates. Intriguingly, our results demonstrate that nonlocal unitary gates do not necessarily outperform the local unitary gates. Finally, the additivity relationship of quantum coherence in tripartite systems is discussed in detail, where the strong subadditivity of von Neumann entropy plays an essential role.
Measurements of Quantum Systems disturb their states. To quantify this non-classical characteristic, Zurek and Ollivier introduced the quantum discord, a quantum correlation which can be nonzero even when entanglement in the system is zero. Discord has aroused great interest as a resource that is more robust against the effects of decoherence and offers exponential speed up of certain computational algorithms. Here, we study general two-level bipartite systems and give general results on the relationship between discord, entanglement, and linear entropy, and identify the states for which discord takes a maximal value for a given entropy or entanglement, thus placing strong bounds on entanglement-discord and entropy-discord relations. We find out that although discord and entanglement are identical for pure states, they differ when generalized to mixed states as a result of the difference in the method of generalization.
Quantum discord is the quantitative difference between two alternative expressions for bipartite mutual information, given respectively in terms of two distinct definitions for the conditional entropy. By constructing a stochastic model of shared states, classical discord can be similarly defined, quantifying the presence of some stochasticity in the measurement process. Therefore, discord can generally be understood as a quantification of the system's state disturbance due to local measurements, be it quantum or classical. We establish an operational meaning of classical discord in the context of state merging with noisy measurement and thereby show the quantum-classical separation in terms of a negative conditional entropy.
Recent observation that almost all quantum states bear non-classical correlations [A. Ferraro et al, Phys. Rev. A 81, 052328 (2010)] may seem to imply that the Markovian bipartite systems are practically deprived of zero discord states. Nevertheless, complementary to the result of Ferraro et al, we construct a model of a Markovian bipartite system providing zero discord for arbitrary long time interval, that we term 'Markovian classicality'. Our model represents a matter-of-principle formal proof, i.e. a sufficient condition for the, otherwise not obvious, existence of Markovian classicality. Interestingly enough, we are not able to offer any alternative to the model. Physical relevance of the model is twofold. First, the model is in intimate relation to the topics of quantum information locality, quantum discord saturation and quantum decorrelation. Second, the model is of the general physical interest. It pertains to a specific structure (decomposition into parts/subsystems) of a composite system, not to a special physical kind of composite systems. Being a characteristic of a structure, by definition, the model of Markovian classicality is not a model of sudden death of discord. We emphasize wide-range implications of our results.