A pedagogical intrinsic approach to relative entropies as potential functions of quantum metrics: Theq–zfamily

Abstract

The so-called q-z-Rényi Relative Entropies provide a huge two-parameter family of relative entropies which includes almost all well-known examples of quantum relative entropies for suitable values of the parameters. In this paper we consider a log-regularized version of this family and use it as a family of potential functions to generate covariant (0, 2) symmetric tensors on the space of invertible quantum states in finite dimensions. The geometric formalism developed here allows us to obtain the explicit expressions of such tensor fields in terms of a basis of globally defined differential forms on a suitable unfolding space without the need to introduce a specific set of coordinates. To make the reader acquainted with the intrinsic formalism introduced, we first perform the computation for the qubit case, and then, we extend the computation of the metric-like tensors to a generic n-level system. By suitably varying the parameters q and z, we are able to recover well-known examples of quantum metric tensors that, in our treatment, appear written in terms of globally 1 arXiv:1711.09769v2 [quant-ph]

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.…”

“…A global basis {α j , β k } of 1-forms is obtained as the dually-related basis of {X j , Y k }. If i d : M −→ M × M is the diagonal immersion given by m → i d (m) = (m, m), it follows from proposition 2 in [14] that X j is i d -related with X j + Y j , and thus we have…”

“…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…”

“…where we used proposition 5 in [14]. By comparing equation (37) with the results of proposition 5 in [14], we obtain that the metric-like tensor on M extracted from the function F by means of the almost complex structure J as explained in this section coincides with the metric-like tensor on M extracted from the function F by means of the procedure outlined in [14]. At this point, if J is the almost complex structure associated with another choice of global bases on M × M , a direct computation shows that 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).…”