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Register a new userQuantum coherence quantifiers based on the R{e}nyi $alpha$-relative entropy
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The resource theories of quantum coherence attract a lot of attention in recent years. Especially, the monotonicity property plays a crucial role here. In this paper we investigate the monotonicity property for the coherence measures induced by the R{e}nyi $alpha$-relative entropy which present in [Phys. Rev. A 94, 052336, 2016]. We show that the R{e}nyi $alpha$-relative entropy of coherence does not in general satisfy the monotonicity requirement under the subselection of measurements condition and it also does not satisfy the extension of monotonicity requirement which presents in [Phys. Rev. A 93, 032136, 2016]. Due to the R{e}nyi $alpha$-relative entropy of coherence can act as a coherence monotone quantifier, we examine the trade-off relations between coherence and mixedness. Finally, some properties for the single qubit of R{e}nyi $2$-relative entropy of coherence are derived.
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We investigate monogamy relations related to the R{e}nyi-$alpha$ entanglement and polygamy relations related to the R{e}nyi-$alpha$ entanglement of assistance. We present new entanglement monogamy relations satisfied by the $mu$-th power of R{e}nyi-$alpha$ entanglement with $alphain[sqrt{7}-1)/2,(sqrt{13}-1)/2]$ for $mugeqslant2$, and polygamy relations satisfied by the $mu$-th power of R{e}nyi-$alpha$ entanglement of assistance with $alphain[sqrt{7}-1)/2,(sqrt{13}-1)/2]$ for $0leqmuleq1$. These relations are shown to be tighter than the existing ones.
We investigate monogamy relations and upper bounds for generalized $W$-class states related to the R{e}nyi-$alpha$ entropy. First, we present an analytical formula on R{e}nyi-$alpha$ entanglement (R$alpha$E) and R{e}nyi-$alpha$ entanglement of assistance (REoA) of a reduced density matrix for a generalized $W$-class states. According to the analytical formula, we show monogamy and polygamy relations for generalized $W$-class states in terms of R$alpha$E and REoA. Then we give the upper bounds for generalized $W$-class states in terms of R$alpha$E. Next, we provide tighter monogamy relations for generalized $W$-class states in terms of concurrence and convex-roof extended negativity and obtain the monogamy relations for R$alpha$E by the analytical expression between R$alpha$E and concurrence. Finally, we apply our results into quantum games and present a new bound of the nonclassicality of quantum games restricting to generalized $W$-class states.
Based on the resource theory for quantifying the coherence of quantum channels, we introduce a new coherence quantifier for quantum channels via maximum relative entropy. We prove that the maximum relative entropy for coherence of quantum channels is directly related to the maximally coherent channels under a particular class of superoperations, which results in an operational interpretation of the maximum relative entropy for coherence of quantum channels. We also introduce the conception of sub-superchannels and sub-superchannel discrimination. For any quantum channels, we show that the advantage of quantum channels in sub-superchannel discrimination can be exactly characterized by the maximum relative entropy of coherence for quantum channels. Similar to the maximum relative entropy of coherence for channels, the robustness of coherence for quantum channels has also been investigated. We show that the maximum relative entropy of coherence for channels provides new operational interpretations of robustness of coherence for quantum channels and illustrates the equivalence of the dephasing-covariant superchannels, incoherent superchannels, and strictly incoherent superchannels in these two operational tasks.
In this work, we establish an exact relation which connects the heat exchange between two systems initialized in their thermodynamic equilibrium states at different temperatures and the R{e}nyi divergences between the initial thermodynamic equilibrium state and the final non-equilibrium state of the total system. The relation tells us that the various moments of the heat statistics are determined by the Renyi divergences between the initial equilibrium state and the final non-equilibrium state of the global system. In particular the average heat exchange is quantified by the relative entropy between the initial equilibrium state and the final non-equilibrium state of the global system. The relation is applicable to both finite classical systems and finite quantum systems.
The existence of a positive log-Sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the Markov approximation. For classical spin systems, such constant was proven to exist, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, via a quasi-factorization of the entropy in terms of the conditional entropy in some sub-$sigma$-algebras. In this work we analyze analogous quasi-factorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasi-factorization results for it. As an illustration of their potential, we use one of them to obtain a positive log-Sobolev constant for the heat-bath dynamics with product fixed point.
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