Metric on the space of quantum states from relative entropy. Tomographic reconstruction

Abstract

In the framework of quantum information geometry, we derive, from quantum relative Tsallis entropy, a family of quantum metrics on the space of full rank, N level quantum states, by means of a suitably defined coordinate free differential calculus. The cases N = 2, N = 3 are discussed in detail and notable limits are analyzed. The radial limit procedure has been used to recover quantum metrics for lower rank states, such as pure states.By using the tomographic picture of quantum mechanics we have obtained the Fisher-Rao metric for the space of quantum tomograms and derived a reconstruction formula of the quantum metric of density states out of the tomographic one. A new inequality obtained for probabilities of three spin-1/2 projections in three perpendicular directions is proposed to be checked in experiments with superconducting circuits.

Geometry from divergence functions and complex structures

“…A coordinate-free formulation of this procedure is given in [14,25], and a general formalism based on the Lie groupoid structure of M × M is presented in [22]. Here, on the other hand, we would like to give an alternative (but equivalent), coordinate-free formulation of the extraction procedure which is inspired by Kähler geometry and thus will lead us to define also a presymplectic form on M × M starting from a divergence function.…”

“…where we used equation (25), and we set θ j ⊗ s θ k = θ j ⊗ θ k + θ k ⊗ θ j . According to the results in section 2 of [14], F is a divergence function if we have…”

“…Note that, however, a similar procedure may be also considered for the family of Tsallis entropies (see [25]), and for the family of (α − z)-Renyi relative entropies [14]. For the sake of simplicity, we omit the explicit expression of the pre-symplectic form and the symmetric (0, 2) tensor on M × M. Indeed, the explicit expressions of these objects turn out to be particularly long, and the computations that we are about to perform in order to compute the metric tensor on M already give enough details to obtain the tensors on M × M by means of equations (29) and (31).…”

“…Furthermore, in the same paper it is shown also that the metric tensor derived from Von Neumann-Umegaki relative entropy is monotone with respect to quantum stochastic maps (completely positive and trace preserving maps on the C * -algebra associated to the quantum system). According to Petz theorem, monotone metric tensor can be decomposed into the sum of two terms, a classical part which is the Fisher-Rao metric tensor, and a quantum term which is coupled to the classical one via a monotone function f (a relation between this monotone function and the tomographic procedure to reconstruct a quantum state from the knowledge of different associated probability densities, has been proposed in [24,25]). As already pointed out, Eq.…”

Spin Kinetic Equations in the Probability Representation of Quantum Mechanics

“…There exist other probability distributions for the density matrix [23,25], which depend linearly on the probabilities p…”