Group Actions and Monotone Metric Tensors: The Qubit Case

Abstract

In recent works, a link between group actions and information metrics on the space of faithful quantum states has been highlighted in particular cases. In this contribution, we give a complete discussion of this instance for the particular case of the qubit.

Group actions and Monotone Quantum Metric Tensors

“…From the results in [17] we know that there are at least 3 monotone metric tensors for which this construction is possible for any finite-level quantum system. Moreover, from the results in [21] we know that in the case of a two-level quantum systems, the Lie groups GL(H) and T * U(H) are the only Lie groups for which the construction described above is actually possible. Here, we want to understand if the group actions of GL(H) and T * U(H) found in [21] can be extended from a 2-level quantum system to a system with an arbitrary, albeit finite, number of levels.…”

“…At this purpose, it is important to recall all those properties, shared by GL(H) and T * U(H) and by their actions, that are at the heart of the results of [17,21]. First of all, both GL(H) and T * U(H) contain the Lie group U(H) as a Lie subgroup, and contain the elements λI with λ > 0 and I the identity operator on H. Then, all the (transitive) actions of both GL(H) and T * U(H) on S (H) appearing in the analysis of [17,21] arise as a sort of normalization of suitable (transitive) actions on P(H). Specifically, if G denotes either GL(H) or T * U(H), then every G-action δ on S (H) can be written as…”

“…Following [17,21], when considering the general linear group GL(H), the reference action δ0 appearing in equation ( 57) is the action β in equation (10). Therefore, denoting with Ẑa b a fundamental vector field of δ0 and with Ẑφ a b a fundamental vector field of δφ , from equation ( 14), equation (60), and [10, Thm.…”

“…In this work we further investigate the problem by showing that all the group actions and metric tensors found in [21] for a two-level system actually appear also for a quantum system with an arbitrary, albeit finite, number of levels. Moreover, we characterize all the values of κ for which the Riemannian metric tensor associated with the action β κ in equation ( 5) is actually a quantum monotone metric tensor (cfr.…”

“…Once we have these three "isolated" instances, it is only natural to wonder if they are truly isolated cases, or if there are other monotone metric tensors for which a similar construction is possible. In [21] this problem is completely solved in the two-level case in which a direct, coordinate-based solution is possible. The result is that the only two groups for which the aforementioned construction works are precisely the general linear group GL(H) and the cotangent group T * U(H).…”