Normal completely positive maps on the space of quantum operations

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 transforming quantum operations with input in a separable von Neumann algebra and output in the algebra of the bounded operators on a given separable Hilbert space. In this setting we prove two dilation theorems for quantum supermaps that are the analogues of the Stinespring and Radon-Nikodym theorems for quantum operations. Finally, we consider the case of quantum superinstruments, namely measures with values in the set of quantum supermaps, and derive a dilation theorem for them that is analogue to Ozawa's theorem for quantum instruments. The three dilation theorems presented here show that all the supermaps defined in this paper can be implemented by connecting devices in quantum circuits.Comment: 47 pages (in one-column format), including new results about quantum operations on separable von Neumann algebra

Generators of detailed balance quantum Markov semigroups

For a quantum Markov semigroup T on the algebra B(h) with a faithful invariant state, we defne an adjoint T' with respect to the scalar product determined by the invariant state. In this paper, we solve the open problems of characterizing adjoints T' that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators in the Gorini-Kossakowski-Sudarshan-Lindblad representation of the generator of T

On two quantum versions of the detailed balance condition.

Quantum detailed balance conditions are often formulated as relationships between the generator of a quantum Markov semigroup and the generator of a dual semigroup with respect to a certain scalar product defined by an invariant state. In this paper we survey some results describing the structure of norm continuous quantum Markov semigroups on B(h) satisfying a quantum detailed balance condition when the duality is defined by means of pre-scalar products on B(h) of the form tr(\rho^{1-s}x*\rho^sy) (s\in [0, 1]) in order to compare the resulting quantum versions of the classical detailed balance condition. Moreover, we discuss the structure of generators of a quantum Markov semigroup which commute with the modular automorphism because this condition appears when we consider pre-scalar products with s\not= 1/2

On the asymptotic behavior of generic quantum Markov semigroups

We study ergodicity and decoherence for generic quantum Markov semigroups: in particular, we highlight the strong relationships among these properties, the convergence in law of the classical Markov process associated with the diagonal part of the semigroup, and the structure of its communication classes

Decoherence for quantum Markov semi-groups on matrix algebras

In this work we describe a necessary and sufficient condition for decoherence of quantum Markov evolutions acting on matrix spaces (according to the definition introduced by Blanchard and Olkiewicz). This condition is related to the spectral analysis of the generator L of the semi-group and is easily stated: the evolution displays decoherence if and only if the maximal algebra N (T ) on which the semigroup is ∗-automorphic contains all the eigenvalues of L associated with eigenvectors with null real part. Moreover, this condition is surely verified when the semigroup admits a faithful invariant state

Vector valued reproducing kernel Hilbert spaces and universality

This paper is devoted to the study of vector valued reproducing kernel Hilbert spaces. We focus on two aspects: vector valued feature maps and universal kernels. In particular, we characterize the structure of translation invariant kernels on abelian groups and we relate it to the universality problem