Franco Ventriglia

Homodyne estimation of quantum state purity by exploiting the covariant uncertainty relation

We experimentally verify the uncertainty relations for the mixed states in tomographic representation by measuring the radiation field tomograms, i.e. homodyne distributions. Thermal states of a single-mode radiation field are discussed in detail as a paradigm of the mixed quantum state. On considering the connection between generalized uncertainty relations and optical tomograms, it is seen that the purity of the states can be retrieved by statistical analysis of the homodyne data. The purity parameter assumes a relevant role in quantum information where the effective fidelities of protocols depend critically on the purity of the information carrier states. In this context, the homodyne detector becomes an easy-to-handle purity-meter for the state on line with a running quantum information protocol.

Alternative structures and bi-Hamiltonian systems

In the study of bi-Hamiltonian systems (both classical and quantum) one starts with a given dynamics and looks for all alternative Hamiltonian descriptions it admits.In this paper we start with two compatible Hermitian structures (the quantum analog of two compatible classical Poisson brackets) and look for all the dynamical systems which turn out to be bi-Hamiltonian with respect to them.In the study of bi-Hamiltonian systems (both classical and quantum) one starts with a given dynamics and looks for all alternative Hamiltonian descriptions it admits. In this paper, we start with two compatible Hermitian structures (the quantum analogue of two compatible classical Poisson brackets) and look for all the dynamical systems which turn out to be bi-Hamiltonian with respect to them.

Semigroup of positive maps for qudit states and entanglement in tomographic probability representation

Stochastic and bistochastic matrices providing positive maps for spin states (for qudits) are shown to form semigroups with dense intersection with the Lie groups

A pedagogical presentation of a Csstarf-algebraic approach to quantum tomography

IGL(n, \mathbb{R})

Tomography in Abstract Hilbert Spaces

and

Brownian motion on Lie groups and open quantum systems

GL(n, \mathbb{R})

Quantum Tomography twenty years later

respectively. The density matrix of a qudit sta

Groupoids and the tomographic picture of quantum mechanics

It is now well established that quantum tomography provides an alternative picture of quantum mechanics. It is common to introduce tomographic concepts starting with the Schrodinger–Dirac picture of quantum mechanics on Hilbert spaces. In this picture, states are a primary concept and observables are derived from them. On the other hand, the Heisenberg picture, which has evolved in the C-algebraic approach to quantum mechanics, starts with the algebra of observables and introduces states as a derived concept. The equivalence between these two pictures amounts, essentially, to the Gelfand–Naimark–Segal construction. In this construction, the abstract C-algebra is realized as an algebra of operators acting on a constructed Hilbert space. The representation that is defined may be reducible or irreducible, but in either case it allows us to identify a unitary group associated with the C-algebra by means of its invertible elements. In this picture both states and observables are appropriate functions on the group; it also follows that quantum tomograms are strictly related with appropriate functions (positive-type) on the group. In this paper we present, using very simple examples, a tomographic description emerging from the set of ideas connected with the C-algebra picture of quantum mechanics. In particular, we introduce the tomographic probability distributions for finite and compact groups, and formulate an autonomous criterion to recognize a given probability distribution as a tomogram of quantum state.

On the tomographic picture of quantum mechanics

The tomographic description of a quantum state is formulated in an abstract infinite-dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product. Resolutions of the unity, written in terms of over-complete sets of rank-one projectors and of associated Gram-Schmidt operators taking into account their non-orthogonality, are then used to reconstruct a quantum state from its tomograms. Examples of well known tomographic descriptions illustrate the exposed theory.

Alternative structures and bi-Hamiltonian systems on a Hilbert space

We study the twirling semigroups of (super) operators, namely certain quantum dynamical semigroups that are associated, in a natural way, with the pairs formed by a projective representation of a locally compact group and a convolution semigroup of probability measures on this group. The link connecting this class of semigroups of operators with (classical) Brownian motion is clarified. It turns out that every twirling semigroup associated with a finite-dimensional representation is a random unitary semigroup, and, conversely, every random unitary semigroup arises as a twirling semigroup. Using standard tools of the theory of convolution semigroups of measures and of convex analysis, we provide a complete characterization of the infinitesimal generator of a twirling semigroup associated with a finite-dimensional unitary representation of a Lie group.