Quantum functional inequalities (e.g., the logarithmic Sobolev and Poincaré inequalities) have found widespread application in the study of the behavior of primitive quantum Markov semigroups. The classical counterparts of these inequalities are related to each other via a so-called transportation cost inequality of order 2 (TC2). The latter inequality relies on the notion of a metric on the set of probability distributions called the Wasserstein distance of order 2. (TC2) in turn implies a transportation cost inequality of order 1 (TC1). In this paper, we introduce quantum generalizations of the inequalities (TC1) and (TC2), making use of appropriate quantum versions of the Wasserstein distances, one recently defined by Carlen and Maas and the other defined by us. We establish that these inequalities are related to each other, and to the quantum modified logarithmic Sobolev- and Poincaré inequalities, as in the classical case. We also show that these inequalities imply certain concentration-type results for the invariant state of the underlying semigroup. We consider the example of the depolarizing semigroup to derive concentration inequalities for any finite dimensional full-rank quantum state. These inequalities are then applied to derive upper bounds on the error probabilities occurring in the setting of finite blocklength quantum parameter estimation.
Quantum Markov semigroups characterize the time evolution of an important class of open quantum systems. Studying convergence properties of such a semigroup and determining concentration properties of its invariant state have been the focus of much research. Quantum versions of functional inequalities (like the modified logarithmic Sobolev and Poincaré inequalities) and the so-called transportation cost inequalities have proved to be essential for this purpose. Classical functional and transportation cost inequalities are seen to arise from a single geometric inequality, called the Ricci lower bound, via an inequality which interpolates between them. The latter is called the HWI inequality, where the letters I, W and H are, respectively, acronyms for the Fisher information (arising in the modified logarithmic Sobolev inequality), the so-called Wasserstein distance (arising in the transportation cost inequality) and the relative entropy (or Boltzmann H function) arising in both. Hence, classically, the above inequalities and the implications between them form a remarkable picture which relates elements from diverse mathematical fields, such as Riemannian geometry, information theory, optimal transport theory, Markov processes, concentration of measure and convexity theory. Here, we consider a quantum version of the Ricci lower bound introduced by Carlen and Maas and prove that it implies a quantum HWI inequality from which the quantum functional and transportation cost inequalities follow. Our results hence establish that the unifying picture of the classical setting carries over to the quantum one. 1 Here we assume that the QMS is primitive, i.e. it has a unique invariant state. 2 Given a metric space (X , d), a probability measure µ is said to satisfy Gaussian (resp. exponential) concentration on it if there exist positive constants a, b such that for any A ⊆ X , and r > 0, µ(A) ≥ 1/2 =⇒ µ(Ar) ≥ 1 − ae −bf (r) .
We generalize the concepts of weak quantum logarithmic Sobolev inequality (LSI) and weak hypercontractivity (HC), introduced in the quantum setting by Olkiewicz and Zegarlinski, to the case of non-primitive quantum Markov semigroups (QMS). The originality of this work resides in that this new notion of hypercontractivity is given in terms of the so-called amalgamated Lp norms introduced recently by Junge and Parcet in the context of operator spaces theory. We make three main contributions. The first one is a version of Gross' integration lemma: we prove that (weak) HC implies (weak) LSI. Surprisingly, the converse implication differs from the primitive case as we show that LSI implies HC but with a weak constant equal to the cardinal of the center of the decoherence-free algebra. Building on the first implication, our second contribution is the fact that strong LSI and therefore strong HC do not hold for non-trivially primitive QMS. This implies that the amalgamated Lp norms are not uniformly convex for 1 ≤ p ≤ 2. As a third contribution, we derive universal bounds on the (weak) logarithmic Sobolev constants for a QMS on a finite dimensional Hilbert space, using a similar method as Diaconis and Saloff-Coste in the case of classical primitive Markov chains, and Temme, Pastawski and Kastoryano in the case of primitive QMS. This leads to new bounds on the decoherence rates of decohering QMS. Additionally, we apply our results to the study of the tensorization of HC in non-commutative spaces in terms of the completely bounded norms (CB norms) recently introduced by Beigi and King for unital and trace preserving QMS. We generalize their results to the case of a general primitive QMS and provide estimates on the (weak) constants.This part is organised as follows: in Section 2.1 we introduce our notations and recall the definitions of quantum Markov semigroups, their decoherence-free algebra and the notion of environment-induced decoherence. Section 2.2 is devoted to the exposition of the weighted L p norms and the L p Dirichlet forms associated to a quantum Markov semigroup. The main results of this article are presented in Section 2.3, namely the equivalence between hypercontractivity and logarithmic Sobolev inequality in the context of amalgamated L p spaces, and the existence of universal constants. In Section 2.4 we apply our framework to the estimation of decoherence rates. Finally, the study of hypercontractivity for the CB-norms is presented in Section 2.5. Quantum Markov semigroups and environment-induced decoherenceLet (H, .|. ) be a finite dimensional Hilbert space of dimension d H . We denote by B(H) the Banach space of bounded operators on H, by B sa (H) the subspace of self-adjoint operators on H, i.e. B sa (H) = {X = B(H); X = X * }, and by B + sa (H) the cone of positive semidefinite operators on H, where the adjoint of an operator Y is written as Y * . The identity operator on H is denoted by I H , dropping the index H when it is unnecessary. In the case when H ≡ C k , we will also use the notation I k ...
The α-sandwiched Rényi divergence satisfies the data processing inequality, i.e. monotonicity under quantum operations, for α ≥ 1/2. In this article, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality for the α-sandwiched Rényi divergence for all α ≥ 1/2. For the range α ∈ [1/2, 1), our result provides the only condition for equality obtained thus far. To prove our result, we first consider the special case of partial trace and derive a condition for equality based on the original proof of the data processing inequality by Frank and Lieb (J Math Phys 54(12):122201, 2013) using a strict convexity/concavity argument. We then generalize to arbitrary quantum operations via the Stinespring Representation Theorem. As applications of our condition for equality in the data processing inequality, we deduce conditions for equality in various entropic inequalities. We formulate a Rényi version of the Araki-Lieb inequality and analyze the case of equality, generalizing a result by Carlen and Lieb (Lett Math Phys 101(1):1-11, 2012) about equality in the original Araki-Lieb inequality. Furthermore, we prove a general lower bound on a Rényi version of the entanglement of formation and observe that it is attained by states saturating the Rényi version of the Araki-Lieb inequality. Finally, we prove that the known upper bound on the entanglement fidelity in terms of the usual fidelity is saturated only by pure states.
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