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Register a new userApproximate tensorization of the relative entropy for noncommuting conditional expectations
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In this paper, we derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. The latter inequality, which we call approximate tensorization of the relative entropy, can be expressed as a lower bound for the sum of relative entropies between a given density and its respective projections onto two intersecting von Neumann algebras in terms of the relative entropy between the same density and its projection onto an algebra in the intersection, up to multiplicative and additive constants. In particular, our inequality reduces to the so-called quasi-factorization of the entropy for commuting algebras, which is a key step in modern proofs of the logarithmic Sobolev inequality for classical lattice spin systems. We also provide estimates on the constants in terms of conditions of clustering of correlations in the setting of quantum lattice spin systems. Along the way, we show the equivalence between conditional expectations arising from Petz recovery maps and those of general Davies semigroups.
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The property of superadditivity of the quantum relative entropy states that, in a bipartite system $mathcal{H}_{AB}=mathcal{H}_A otimes mathcal{H}_B$, for every density operator $rho_{AB}$ one has $ D( rho_{AB} || sigma_A otimes sigma_B ) ge D( rho_A || sigma_A ) +D( rho_B || sigma_B) $. In this work, we provide an extension of this inequality for arbitrary density operators $ sigma_{AB} $. More specifically, we prove that $ alpha (sigma_{AB})cdot D({rho_{AB}}||{sigma_{AB}}) ge D({rho_A}||{sigma_A})+D({rho_B}||{sigma_B})$ holds for all bipartite states $rho_{AB}$ and $sigma_{AB}$, where $alpha(sigma_{AB})= 1+2 || sigma_A^{-1/2} otimes sigma_B^{-1/2} , sigma_{AB} , sigma_A^{-1/2} otimes sigma_B^{-1/2} - mathbb{1}_{AB} ||_infty$.
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.
Noncommuting observables cannot be simultaneously measured, however, under local hidden variable models, they must simultaneously hold premeasurement values, implying the existence of a joint probability distribution. We study the joint distributions of noncommuting observables on qubits, with possible criteria of positivity and the Frechet bounds limiting the joint probabilities, concluding that the latter may be negative. We use symmetrization, justified heuristically and then more carefully via the Moyal characteristic function, to find the quantum operator corresponding to the product of noncommuting observables. This is then used to construct Quasi-Bell inequalities, Bell inequalities containing products of noncommuting observables, on two qubits. These inequalities place limits on local hidden variable models that define joint probabilities for noncommuting observables. We find Quasi-Bell inequalities have a quantum to classical violation as high as $frac{3}{2}$, higher than conventional Bell inequalities. The result demonstrates the theoretical importance of noncommutativity in the nonlocality of quantum mechanics, and provides an insightful generalization of Bell inequalities.
Many thermodynamic relations involve inequalities, with equality if a process does not involve dissipation. In this article we provide equalities in which the dissipative contribution is shown to involve the relative entropy (a.k.a. Kullback-Leibler divergence). The processes considered are general time evolutions both in classical and quantum mechanics, and the initial state is sometimes thermal, sometimes partially so. By calculating a transport coefficient we show that indeed---at least in this case---the source of dissipation in that coefficient is the relative entropy.
We prove the existence of a universal recovery channel that approximately recovers states on a v. Neumann subalgebra when the change in relative entropy, with respect to a fixed reference state, is small. Our result is a generalization of previous results that applied to type-I v. Neumann algebras by Junge at al. [arXiv:1509.07127]. We broadly follow their proof strategy but consider here arbitrary v. Neumann algebras, where qualitatively new issues arise. Our results hinge on the construction of certain analytic vectors and computations/estimations of their Araki-Masuda $L_p$ norms. We comment on applications to the quantum null energy condition.
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