We establish a quantum version of the classical isoperimetric inequality relating the Fisher information and the entropy power of a quantum state. The key tool is a Fisher information inequality for a state which results from a certain convolution operation: the latter maps a classical probability distribution on phase space and a quantum state to a quantum state. We show that this inequality also gives rise to several related inequalities whose counterparts are well-known in the classical setting: in particular, it implies an entropy power inequality for the mentioned convolution operation as well as the isoperimetric inequality and establishes concavity of the entropy power along trajectories of the quantum heat diffusion semigroup. As an application, we derive a Log-Sobolev inequality for the quantum Ornstein-Uhlenbeck semigroup and argue that it implies fast convergence towards the fixed point for a large class of initial states.
We establish a quantum version of the classical isoperimetric inequality relating the Fisher information and the entropy power of a quantum state. The key tool is a Fisher information inequality for a state which results from a certain convolution operation: the latter maps a classical probability distribution on phase space and a quantum state to a quantum state. We show that this inequality also gives rise to several related inequalities whose counterparts are well-known in the classical setting: in particular, it implies an entropy power inequality for the mentioned convolution operation as well as the isoperimetric inequality and establishes concavity of the entropy power along trajectories of the quantum heat diffusion semigroup. As an application, we derive a Log-Sobolev inequality for the quantum Ornstein-Uhlenbeck semigroup and argue that it implies fast convergence towards the fixed point for a large class of initial states.
We establish a quantum version of the classical isoperimetric inequality relating the Fisher information and the entropy power of a quantum state. The key tool is a Fisher information inequality for a state which results from a certain convolution operation: the latter maps a classical probability distribution on phase space and a quantum state to a quantum state. We show that this inequality also gives rise to several related inequalities whose counterparts are well-known in the classical setting: in particular, it implies an entropy power inequality for the mentioned convolution operation as well as the isoperimetric inequality, and establishes concavity of the entropy power along trajectories of the quantum heat diffusion semigroup. As an application, we derive a Log-Sobolev inequality for the quantum Ornstein-Uhlenbeck semigroup, and argue that it implies fast convergence towards the fixed point for a large class of initial states.
Abstract 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.
Functional inequalities constitute a very powerful toolkit in studying various problems arising in classical information theory, statistics and many-body systems. Extensions of these tools to the noncommutative setting have been introduced in the beginning of the 90's in order to study the asymptotic properties of certain quantum Markovian evolutions. In this thesis, we study various extensions and problems arising from the specific noncommutative nature of such processes. The first logarithmic Sobolev inequality to be proved, due to Gross, was for the Ornstein Uhlenbeck semigroup, that is the Brownian motion with friction on the real line. The generalization of this result to the quantum Ornstein Uhlenbeck semigroup was found very recently by Carlen and Maas, and de Palma and Huber by means of different techniques. The latter proof consists of a quantum generalization of the so-called entropy power inequality. Here, we consider another possible version of the entropy power inequality and use it to derive asymptotic properties of the frictionless quantum Brownian motion. The proof of Carlen and Maas discussed in the previous paragraph relies on their new quantum extension of the classical notion of displacement convexity. This is classically known to imply most of the usual functional inequalities such as the modified logarithmic Sobolev inequality and Poincaré's inequality. Here, we further study the framework introduced by Carlen and Maas. In particular, we show how displacement convexity implies quantum functional and transportation cost inequalities. The latter are then used to derive certain concentration inequalities of quantum states in the spirit of Bobkov and Goetze. These concentration inequalities are used in order to derive finite sample size bounds for the task of quantum parameter estimation. The main advantage of classical logarithmic Sobolev inequalities over other methods resides in their tensorization property: the strong log-Sobolev constant of the product of independent Markovian evolutions is equal to the maximum over the set of strong log-Sobolev constants of the individual evolutions. However, this property is strongly believed to fail in the non-commutative case, due to the non-multiplicativity of noncommutative Lp to Lq norms. In this thesis, we show tensorization of the logarithmic Sobolev constants for the simplest quantum Markov semigroup, namely the generalized depolarizing semigroup. Moreover, we consider a new general method to overcome the issue of tensorization for general primitive quantum Markov semigroups by looking at their contractivity properties under the completely bounded Lp to Lq norms. This method was first investigated in the restricted case of unital semigroups by Beigi and King. Noncommutative functional inequalities considered in the present literature only deal with primitive quantum Markovian semigroups which model memoryless irreversible dynamics converging to a specific faithful state. However, quantum Markov semigroups can in general display a much richer behavior referred to as decoherence: In particular, under some mild conditions, any such semigroup is known to converge to an algebra of observables which effectively evolve unitarily. Here, we introduce the concept of a decoherence-free logarithmic Sobolev inequality, and the related notion of hypercontractivity of the associated evolution, to study the decoherence rate of non-primitive quantum Markov semigroups. Moreover, we utilize the transference method recently introduced by Gao, Junge and LaRacuente, in order to find decoherence times associated to a class of decoherent Markovian evolutions of great importance in the field of quantum error protection, namely collective decoherence semigroups. Finally, we develop the notion of quantum reverse hypercontractivity, first introduced by Cubitt, Kastoryano, Montanaro and Temme in the unital case, and apply it in conjunction with the tensorization of the modified logarithmic Sobolev inequality for the generalized depolarizing semigroup in order to find strong converse rates in quantum hypothesis testing and for the classical capacity of classical-quantum channels. Moreover, the transference method also allows us to find strong converse bounds on the various capacities of quantum Markovian evolutions.
We develop a method for deriving Gaussian optimality results for bosonic quantum systems from inequalities for qubit systems. Inspired by Gross's derivation of the Gaussian logarithmic Sobolev inequality from an inequality on the two-point space, we use qubits as quantum two-point spaces and lift bosonic Weyl operators to pseudo-Weyl operators on large qubit systems. By controlling weighted norms and relative entropy in the fluctuation limit, we establish two transfer theorems that convert tensorized qubit inequalities into inequalities for bosonic Gaussian channels. As a first application, we prove that the sharp 2-logarithmic Sobolev inequality for the bosonic Ornstein--Uhlenbeck semigroup tensorizes without loss in the constant. We also establish a sharp output entropy inequality for tensor powers of a class of qubit channels. This includes a quantum Mrs.~Gerber's lemma determining the minimum output entropy of tensor powers of the qubit depolarizing channel at fixed input entropy. Applying our relative entropy transfer theorem, we obtain multimode constrained minimum output entropy bounds for Gaussian additive-noise channels and thermal attenuators in the heating range.
We propose a generalization of the quantum entropy power inequality involving condi-tional entropies. For the special case of Gaussian states, we give a proof based on perturba-tion theory for symplectic spectra. We discuss some implications for entanglement-assisted classical communication over additive bosonic noise channels. 1 Classical and quantum entropy-power inequalities The entropy power inequality, proposed by Shannon [27] and later established with increasing rigor by Stam [29] and Blachman [5], has become a fundamental tool in classical informa-tion theory. Shannon’s original application of the entropy power inequality is a lower bound on the capacity of an additive (but potentially non-Gaussian) noise channel [27, Theorem 18]. However, the usefulness of the entropy power inequality is especially evident in multi-terminal information theory. Among the most well-known applications are the characterization of the Gaussian broadcast channel by Bergman [4], the Gaussian two-description problem by Ozarow [25] and the quadratic Gaussian CEO problem by Oohama [24]. In these multi-user settings, Fano’s inequality by itself is insufficient to characterize different tradeoffs. A more recent application proposed by Liu and Viswanath [23] uses entropy power inequalities to solve certain optimization problems. The entropy power inequality lower bounds the differential entropy of the convolution of two independent random variables X, Y taking values in Rn. Its covariance-preserving version states that H( λX + 1 − λY) ≥ λH(X) + (1 − λ)H(Y) for all 0 ≤ λ ≤ 1. (1) Eq. (1) can be shown [22, 32] to be equivalent to the more commonly used statement e2H(X+Y)/n ≥ e2H(X)/n + e2H(Y)/n. The latter explains the terminology as e2H(X)/n is the power, i.e., variance of a Gaussian ran-dom variable with identical entropy as H(X). Inequalities such as (1) are closely related to Log-Sobolev inequalities (see e.g., [31]) as well as Brunn-Minkowski-type inequalities [6]. Gen-eralizations to free probability [30] and quantum states have been considered. 1 ar
We prove the quantum conditional entropy power inequality for quantum additive noise channels. This inequality lower bounds the quantum conditional entropy of the output of an additive noise channel in terms of the quantum conditional entropies of the input state and the noise when they are conditionally independent given the memory. We also show that this conditional entropy power inequality is optimal in the sense that we can achieve equality asymptotically by choosing a suitable sequence of Gaussian input states. We apply the conditional entropy power inequality to find an array of information-theoretic inequalities for conditional entropies which are the analogs of inequalities which have already been established in the unconditioned setting. Furthermore, we give a simple proof of the convergence rate of the quantum Ornstein-Uhlenbeck semigroup based on entropy power inequalities.