Schedule for: 24w5240 - Quantum Markov Semigroups and Channels: Special Classes and Applications

Probabilistic quantization is a deeper level of classical probability showing how, any classical random variable or random field with all moments, naturally generates (i.e. in a purely deductive way and without any artificial construction) all the structures that we are used to attribute to usual (boson or fermion) quantum theory. In this lecture there will be no time to illustrate these structures and our attention will be concentrated only on one of them: a natural generalization of the combinatorics describing the moments of classical, boson and fermion Random fields called ‘generalized gaussianity’. As in the known cases, this combinatorics is a consequence of a set of commutation relations which, are associated to any random variable with all moments through its canonical quantum decomposition. In the lecture only real valued random variables will be discussed. These results arise from joint work with Yun Gang Lu.

Gaussian (quasi-free) QMS describe the evolution of open quantum systems of bosons interaction with the surrounding environment. They also generalize bosonic quadratic Hamiltonians. In this talk we first introduce QMS and describe the Gorini-Kossakowski-Lindblad-Sudharshan (GKLS) structure of their generators. Next, we illustrate the construction of Gaussian QMS by the minimal semigroup method and discuss some properties of the Markovian dynamics such as irreducibility, ergodicity, existence of invariant states and the structure of the decoherence-free subalgebra in which the reduced evolution is homomorphic.

Quantum channels and quantum Markov semigroups (QMS) describe the evolution of an open quantum system interacting with the surrounding environment. Gaussian quantum channels and Markov semigroups play a key role because several models are based on linear couplings of bosonic systems to other bosonic systems with quadratic Hamiltonians. Recall that a QMS is called Gaussian when the algebra is $\mathcal B(\Gamma(\mathbb C^d))$ of all bounded operators on the Fock space $\Gamma(\mathbb C^d)$ and if the predual semigroup acting on trace class operators on $\Gamma(\mathbb C^d)$ preserves Gaussian states. If there exits a faithful invariant density we explicitly compute the optimal exponential convergence rate, namely the spectral gap of the generator, in non-commutative $L^2$-spaces determined by the invariant density showing that the exact value is the lowest eigenvalue of a certain matrix determined by the diffusion and drift matrices. The spectral gap turns out to depend on the non-commutative $L^2$-space considered, whether the one determined by the so-called GNS or KMS multiplication by the square root of the invariant density. In the first case, it is strictly positive if and only if there is the maximum number of linearly independent noises. While, we exhibit explicit examples in which it is strictly positive only with KMS multiplication. We do not assume any symmetry or quantum detailed balance condition with respect to the invariant density.

Gaussian quantum markov semigroups are an interesting class of quantum Markov semigroups both in the way they appear in physical models and in the way they can be treated mathematically, albeit their inherent problem of not being uniformly continuous. Indeed many of the domain problems that arise, in general, when dealing with such semigroups can be solved and many problems one can study on them have a linear algebra reduction, allowing for a simple solution. In this talk we will tackle the symmetry problem with respect to the GNS embedding. Starting from a gaussian faithful invariant state, we consider the embedding of bounded operators onto Hilbert-Schmidt operators and study self-adjointness of the generator it induces on the latter space, from the generator of the original gaussian quantum Markov semigroup. To this end, in true gaussian QMS fashion, we recover some classical result that held for uniformly continuous semigroups and further specify them as algebraic properties of the parameters. Eventually, we present a structure result for the generator of GNS-symmetric semigroups, finally shedding some light on the true amount of degrees of freedom one has available when constructing gaussian QMSs.

Gaussian quantum Markov semigroups (GQMS) are a fundamental class of Markovian evolutions of Bosonic systems which can be characterized in different equivalent ways: they preserve the set of quantum Gaussian states, and their generator has a quadratic Hamiltonian part and linear jump operators (where quadratic and linear are meant with respect to creation and annihilation operators). Moreover, they can be seen as a natural generalization of the classical Ornstein-Uhlenbeck semigroup to the Bosonic setting. In our talk we will present some recent results characterizing the set of invariant states of GQMS; moreover, we will show what restrictions on the generator are imposed by the existence of an invariant state and what consequences it has on the dynamics (convergence to equilibrium and environment-induced decoherence). Our research builds on some recent results obtained in https://doi.org/10.1007/s00032-022-00355-0 and https://doi.org/10.1142/S0219025724400046.

In this talk I will talk about Unitarily Invariant Random Matrices which are modified by a block linear transformation. The motivation which will be explained comes from Quantum Information and the PPT criterion in the framework of Random Quantum Channels. First, we solve the problem of finding the asymptotic spectral distribution by using Operator Valued Free Probability. Second, we find some situations where one can show that such modifications can be written as sums of asymptotically free random matrices, thus explaining the appearance of some of the formulas. This talk is based on joint work with Ion Nechita and Carlos Vargas.

We discuss some fundamental ideas in quantum probability analogous to the classical probability theory and also talk about some key differences in this theory. Background: Linear algebra

The close correspondence between the eigenvalues of the graph and the energy levels of the electrons in hydrocarbon molecules motivated Gutman in 1978 to define the energy of a graph. On the other hand, inspired by Non-Commutative Probability Arizmendi and Juarez-Romero introduced the concept of energy a vertex. Since we can recover the energy of a graph by summing the individual energies of its vertices, it follows that the energy of a vertex should be understood as its contribution to the energy of the graph. In the present exposition, the main objective is to analyze how the fusion of a tree with a bipartite graph affects the energy of the tree vertices. The results reveal an alternating pattern with respect to the coalescence vertex: the energy decreases for vertices located at odd distances and increases for those located at even distances. In addition, an analysis of the long-term effects indicates that, in some cases, the edges incident on the coalescing vertices tend to disappear. Finally, we will discuss some advance made for general bipartite graphs.

Relaxation rates provide important characteristics both for classical and quantum processes. They do control how fast the system thermalizes, equilibrates, decoheres, and/or dissipate. Moreover, very often they are directly accessible to be measured in the laboratory and hence they define key physical properties of the system. Experimentally measured relaxation rates can be used to test validity of particular theoretical models. In my talk I analyze a fundamental question: does quantum mechanics provide any nontrivial constraint for relaxation rates? To answer this question I prove the conjecture formulated few years ago that any quantum dynamical semigroup implies that a maximal rate is bounded from above by the sum of all relaxation rates divided by the dimension of the Hilbert space. It should be stressed that this constraint is universal (it is valid for all quantum systems with finite number of energy levels) and it is tight (cannot be improved). In addition, the constraint plays a role analogous role to the seminal Bell inequalities and the well known Leggett-Garg inequalities (sometimes called temporal Bell inequalities). Violations of Bell inequalities rule out local hidden variable models, and violations of Leggett-Garg inequalities rule out macrorealism. Similarly, violations of the relaxation bound rule out Markovian (meaning CP-divisible) evolution.

Given a uniformly continuous Quantum Markov Semigroup (QMS) $\mathcal T = (\mathcal T_t)_{t\geq 0}$ on the algebra of all linear and bounded operators acting on a complex separable Hilbert space, we study the decoherence-free subalgebra $\mathcal N (\mathcal T )$, where maps $\mathcal T_t$ act as automorphisms. By using the direct integral decomposition on factors of this algebra, we determine the general structure of the infinitesimal generator of $\mathcal T$ and we specialise it in the atomic case. In particular, in this situation, we prove that the structure of $\mathcal N (\mathcal T )$ induces a decomposition of the system into its noiseless and purely dissipative parts [2], determining the structure of invariant states, as well as decoherence-free subsystems and subspaces [4]. Moreover, we show that, when there is a faithful invariant state, the decoherence-free subalgebra has to be atomic and decoherence takes place [5].

In the context of the classical central limit theorem, let $Y_n$ denote the $n$-th normalized sums of i.i.d copies of a random variable $X$ with mean 0 and variance 1. Following Shannon's work in information theory, Lieb conjectured in 1978 that the differential entropy of $Y_n$ increases monotonically in $n$. This conjecture was finally settled by Artstein, Ball, Barthe and Naor (ABBN) in 2004. In fact, the ABBN article proved more general results and tied the so-called entropy power inequalities into this framework. These inequalities are extremely useful in proving several coding theorems in information theory. On the non-commutative side of the story, Cushen and Hudson, in 1971, proved a quantum probability analog of the classical central limit theorem. The monotonicity of von Neumann entropy under the Cushen-Hudson central limit theorem remains an open problem in this area. Guha in 2008 showed that certain quantum analogs of entropy power inequalities, if proved, will produce several coding theorems in quantum information theory, but these problems also remain open to this day. In this talk, we discuss an overview of this area of research and show a new result on the monotonicity of the entropy of the distribution of observables.

Quantum graphs are an operator space generalization of classical graphs, which serve the role of confusability graphs associated with quantum channels. In this talk, I will provide an introduction to quantum graphs arising in the context of operator systems theory, non-commutative topology and quantum information theory. I will then present a notion of quantum graph homomorphisms, which are basically quantum channels satisfying certain conditions generalizing homomorphisms between classical graphs. Different characterizations of quantum graph homomorphisms will be presented using algebraic conditions and winning strategies of nonlocal games.

In this talk, we will introduce quantum walks, which are the quantum generalization of random walks. We will present some differences between quantum walks and random walks. We will also discuss the properties of quantum walks on graphs. Finally, we will explore the phenomenon of localization and how it manifests in specific infinite graphs called spidernets, using the method of quantum decomposition for spectral analysis of graphs.

In this talk we will describe some of the major research developments in stochastic (Ito-Stratonovich and white noise calculus) analysis, control and nonlinear filtering of classical fields such as fluid dynamics, and their connection to infinite dimensional partial differential equations. We will then outline ways of extending such studies to quantum nonlinear evolutions such as Yang-Mills-Higgs-Spinor systems and also operator theoretic evolutions such as Hudson-Parthasarathy and GKLS evolutions.

The super version of imprimitivity theorem is available now to describe global supersymmetry of systems using the representations of super Lie groups (SLG). This result uses the equivalence between super Harish- Chandra pairs and super Lie groups at the categorigal level and is applicable to super Poincaré group and generalizes a smooth SI to super context. We apply the result to build supersymmetric quantum fields. Towards this end, we set up a super fock space of a disjoint union of super Hilbert spaces which is equivalent to super tensoring of boson (even) part symmetrically and that of fermion (odd) part antisymmetrically of the super particle Hilbert space. This leads to a super fock space that is disjoint union of bosonic and fermionic spaces, that is $\mathbb Z_2$-graded. We derive covariant Weyl operators for light-like fields, with the massless super spinorial multiplet as an illustrative example. First, we build a representation of a light-like little group in terms of Weyl operators. We then use this construction to induce a representation of Poincar´e group to construct the fields via super version of imprimitivity theorem.

I will discuss the operator space fragmentation. It is analogous to the Hilbert space fragmentation, which is actively discussed in the recent works in many-body physics. From the mathematical point of view, operator space fragmentation means that for a quantum Markov semigroup for a many-body system one has at least exponentially (in particle number) many operator subspaces that are invariant with respect to such semigroup and these subspaces have linear (in particle number) dimension and have trivial intersection. I will consider several examples of quantum Markov semigroups such that operator space fragmentation occurs very naturally due to invariant dynamics of moments (of some fixed order) of bosonic or fermionic creation and annihilation operators. These quantum Markov semigroups can themselves be built from unitary dynamics with the same property by averaging it with resect to classical Levy fields. I will also discuss that a wide class of dissipative XX models manifest operator space fragmentation, as Jordan products of so-called Onsager strings span invariant spaces in this case. Finally, I will show that if one averages a quantum Markov semigroup with a quadratic (in creation and annihilation operators) generator with respect to classical subordinators then a generalization of operator space fragmentation arises. Namely, instead of invariant subspaces with trivial intersection, in this case one has a hierarchy of nested invariant subspaces.

We review first the definition of the creation, preservation, annihilation, and number operators generated by a random variable having finite moments of all orders. Then we compute these operators for the random variables of the Meixner class. We close the talk by describing the random variables whose annihilation operators are linear combinations of translation operators. This is a joint work with N. Obata and H. Yoshida.

In this talk I will review an optimal-transport inspired approach to define Ricci curvature bounds for quantum Markov semigroups. I will focus on intertwining techniques for Ricci curvature bounds and show how appropriately chosen intertwining semigroups lead to improved Ricci curvature bounds. As an application I will discuss depolarizing semigroups and show that the Ricci curvature bound implies the optimal relative entropy decay rate for qubits. This is joint work with Florentin Münch and Haonan Zhang.

Recently, several notions of Ricci curvature lower bounds have been introduced for quantum Markov semigroups to establish functional inequalities. In proving curvature lower bounds for specific examples, an intertwining technique has proven to be a useful tool. In this talk, I will discuss related notions of Ricci curvature lower bounds for quantum Markov semigroups and explain how we benefit from developing the intertwining idea. This is based on joint work with Melchior Wirth and Florentin Münch.

The main result of this talk is the proof of the boundedness of the Gaussian Riesz potentials $I_{\beta}$, for $\beta\geq 1$ on $L^{p(\cdot)}(\gamma_d)$, the Gaussian variable Lebesgue spaces under a certain additional condition of regularity on $p(\cdot)$. This result trivially gives us an alternative and direct proof of the boundedness of $I_\beta$ on Gaussian Lebesgue spaces $L^p(\gamma_d)$.

Gaussian QMS's is an interesting class of semigroups widely studied in the literature. The natural question arises of whether or not the Gaussian QMS's can be deduced from fundamental laws of physics (first principles) so that they describe real physical phenomena. Here we approach this problem by studying those Gaussian QMS's that arise in the weak coupling limit of a system interacting with a reservoir.

Given a Markov generator of weak coupling limit type $\mathcal{L}$, there exists a special class of invariant states associated, the so called uniform and completely non equilibrium states, which were introduced and characterized in \cite{Marco-Fer-Julio}. From the point of view of its interaction graph $G_{\mathcal{L}}$, see \cite{AccardiFQ}, each energy level $\epsilon_{n}$ is represented by a vertex and the Bohr frequency by edges, under certain conditions, these states are associated with Eulerian cycles. We shall talk about some configurations of this graph, the called k-jump interaction graph $G_{\mathcal{L}_{k}}$, where $k$ corresponds to a Bohr frequency and we shall show their associated non equilibrium invariant states.

We deal with the non-generic simplest case: a two generic three level system with degeneracity, in order to get the inversion population phenomena.

We will present in this talk a rigorous definition of moments of an unbounded observable concerning a quantum state in terms of the so-called Yosida’s approximations for unbounded generators of contractions semigroups. We use this notion to characterize Gaussian states in terms of the moments of the field operator. The above analysis permits to obtain rigorous formulae for the mean value vector and the covariance matrix of a Gaussian state.