Related papers: Iterates of quantum operations
Let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual and let $(\Omega, \mathbb{P})$ be a probability space. A bounded positive random linear operator on $L^1(M,\tau)$ is a map $\gamma : \Omega \times L^1(M,\tau) \to…
We study mean ergodicity in amenable operator semigroups and establish the connection to the convergence of strong and weak ergodic nets. We then use these results in order to show the convergence of uniform families of ergodic nets that…
Recent results of M.Junge and Q.Xu on the ergodic properties of the averages of kernels in noncommutative L^p-spaces are applied to the analysis of the almost uniform convergence of operators induced by the convolutions on compact quantum…
The entangled ergodic theorem concerns the study of the convergence in the strong, or merely weak operator topology, of the multiple Cesaro mean $$\frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1} U^{n_{\a(1)}}A_{1}U^{n_{\a(2)}}...…
The complex Hilbert space of standard quantum mechanics may be treated as a real Hilbert space. The pure states of the complex theory become mixed states in the real formulation. It is then possible to generalize standard quantum mechanics,…
We study how iterated and composed completely positive maps act on operator-valued kernels. Each kernel is realized inside a single Hilbert space where composition corresponds to applying bounded creation operators to feature vectors. This…
Entanglement in quantum many-body systems is the key concept for future technology and science, opening up a possibility to explore uncharted realms in an enormously large Hilbert space. The hybrid quantum-classical algorithms have been…
The random matrix ensembles are applied to the quantum chaotic systems. The quantum systems are studied using the finite dimensional real, complex and quaternion Hilbert spaces of the eigenfunctions. The linear operators describing the…
Any discrete quantum process is represented by a sequence of quantum channels. We consider ergodic quantum processes obtained by a map that takes the points along the trajectory of a discrete ergodic dynamical system to the space of quantum…
The notion of compact quantum subgroup is revisited and an alternative definition is given. Induced representations are considered and a Frobenius reciprocity theorem is obtained. A relationship between ergodic actions of compact quantum…
The quantum geometric tensor (QGT) characterizes the Hilbert space geometry of the eigenstates of a parameter-dependent Hamiltonian. In recent years, the QGT and related quantities have found extensive theoretical and experimental utility,…
We use techniques of proof mining to obtain a computable and uniform rate of metastability (in the sense of Tao) for the mean ergodic theorem for a finite number of commuting linear contractive operators on a uniformly convex Banach space.
Quantum supermaps are higher-order maps transforming quantum operations into quantum operations. Here we extend the theory of quantum supermaps, originally formulated in the finite dimensional setting, to the case of higher-order maps…
Let $U$ be a unitary operator acting on the Hilbert space $\ch$, and $\a:\{1,..., 2k\}\mapsto\{1,..., k\}$ a pair partition. Then the ergodic average $$ \frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1} U^{n_{\a(1)}}A_{1}U^{n_{\a(2)}}...…
We develop a Perron-Frobenius type theory for products of random quantum channels acting on finite-dimensional matrix algebras sampled from a stationary and ergodic stochastic process, which, in keeping with the literature, we call ergodic…
In this paper, among other things, we state and prove the mean ergodic theorem for amenable semigroup algebras.
Usually models for quantum computations deal with unitary gates on pure states. In this paper we generalize the usual model. We consider a model of quantum computations in which the state is an operator of density matrix and the gates are…
Accurately describing work extraction from a quantum system is a central objective for the extension of thermodynamics to individual quantum systems. The concepts of work and heat are surprisingly subtle when generalizations are made to…
Quantum operations (QO) describe any state change allowed in quantum mechanics, such as the evolution of an open system or the state change due to a measurement. We address the problem of which unitary transformations and which observables…
For a Dunford-Schwartz operator in a fully symmetric space of measurable functions of an arbitrary measure space, we prove pointwise convergence of the conventional and weighted ergodic averages.