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arXiv:1101.1717

Non-negative discord strengthens the subadditivity of quantum entropy functions

Patrick J. Coles

Published Jan 10, 2011• 6 citations
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Abstract

The definition of quantum discord is generalized to allow for any concave entropy function, and its non-negativity strengthens the subadditivity condition for that entropy function. In a sense, this condition is intermediate in strength between subadditivity and strong subadditivity, hence called firm subadditivity, allowing one to further classify entropy functions based on whether they satisfy this intermediate condition. It is proven that the quadratic entropy $1-\Tr(ρ^2)$ satisfies the firm subadditivity condition, whereas some other subadditive Tsallis entropies are not firmly subadditive.

DOI
10.48550/arxiv.1101.1717

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The definition of quantum discord is generalized to allow for any concave entropy function, and its non-negativity strengthens the subadditivity condition for that entropy function. In a sense, this condition is intermediate in strength between subadditivity and strong subadditivity, hence called firm subadditivity, allowing one to further classify entropy functions based on whether they satisfy this intermediate condition. It is proven that the quadratic entropy 1−\Tr(ρ2)1-\Tr(ρ^2)1−\Tr(ρ2) satisfies the firm subadditivity condition, whereas some other subadditive Tsallis entropies are not firmly subadditive.
Due to some ambiguity in the definition of mutual Tsallis entropy in classical probability theory, its generalization to quantum theory is discussed and, as a consequence, two types of generalized quantum discords, called q-discords, are defined in terms of quantum Tsallis entropy. Both q-discords for two-qubit Werner and isotropic states are determined and compared and it is shown that one of them is non-negative, at least for states under investigation, for all q > 0. Finally, an analytical expression for q-discord of certain family of two-qubit X-states is presented. Using this example, we show that both types of q-discords can take negative values for some q > 1, hence their use as correlations measures is rather limited.
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.
Several ways have been proposed in the literature to define a coherence measure based on Tsallis relative entropy. One of them is defined as a distance between a state and a set of incoherent states with Tsallis relative entropy taken as a distance measure. Unfortunately, this measure does not satisfy the required strong monotonicity, but a modification of this coherence has been proposed that does. We introduce three new Tsallis coherence measures coming from a more general definition that also satisfy the strong monotonicity, and compare all five definitions between each other. Using three coherence measures that we discuss, one can also define a discord. Two of these have been used in the literature, and another one is new. We also discuss two correlation measures based on Tsallis relative entropy. We provide explicit expressions for all three discord and two correlation measure on pure states. Lastly, we provide tight upper and lower bounds on two discord and correlations measures on any quantum state, with the condition for equality.
Due to some ambiguity in defining mutual Tsallis entropy in the classical probability theory, its generalization to quantum theory is discussed and, as a consequence, two types of generalized quantum discord, called qqq-discords, are defined in terms of quantum Tsallis entropy. qqq-discords for two-qubit Werner and isotropic states are calculated and it is shown that one of them is positive, at least for states under investigation, for all q>0q>0q>0. Finally, an analytical expression for qqq-discord of certain family of two-qubit X states is presented.
The strong subadditivity of entropy plays a key role in several areas of physics and mathematics. It states that the entropy $S[\ensuremath{\varrho}]=\ensuremath{-}\mathrm{Tr}(\ensuremath{\varrho}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varrho})$ of a density matrix ${\ensuremath{\varrho}}_{123}$ on the product of three Hilbert spaces satisfies $S[{\ensuremath{\varrho}}_{123}]\ensuremath{-}S[{\ensuremath{\varrho}}_{12}]\ensuremath{\leqslant}S[{\ensuremath{\varrho}}_{23}]\ensuremath{-}S[{\ensuremath{\varrho}}_{2}]$. We strengthen this to $S[{\ensuremath{\varrho}}_{123}]\ensuremath{-}S[{\ensuremath{\varrho}}_{12}]\ensuremath{\leqslant}{\ensuremath{\sum}}_{\ensuremath{\alpha}}{n}^{\ensuremath{\alpha}}(S[{\ensuremath{\varrho}}_{23}^{\ensuremath{\alpha}}]\ensuremath{-}S[{\ensuremath{\varrho}}_{2}^{\ensuremath{\alpha}}])$, where the ${n}^{\ensuremath{\alpha}}$ are weights and the ${\ensuremath{\varrho}}_{23}^{\ensuremath{\alpha}}$ are partitions of ${\ensuremath{\varrho}}_{23}$. Correspondingly, there is a strengthening of the theorem that the map $A\ensuremath{\mapsto}\mathrm{Tr}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[L+\mathrm{ln}\phantom{\rule{0.2em}{0ex}}A]$ is concave. As applications we prove some monotonicity and convexity properties of the Wehrl coherent state entropy and entropy inequalities for quantum gases.