5 research outputs found
Canonical divergence for measuring classical and quantum complexity
A new canonical divergence is put forward for generalizing an
information-geometric measure of complexity for both, classical and quantum
systems. On the simplex of probability measures it is proved that the new
divergence coincides with the Kullback-Leibler divergence, which is used to
quantify how much a probability measure deviates from the non-interacting
states that are modeled by exponential families of probabilities. On the space
of positive density operators, we prove that the same divergence reduces to the
quantum relative entropy, which quantifies many-party correlations of a quantum
state from a Gibbs family.Comment: 17 page
Group actions and monotone metric tensors: The qubit case
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
Quantum States, Groups and Monotone Metric Tensors
A novel link between monotone metric tensors and actions of suitable
extensions of the unitary group on the manifold of faithful quantum states is
presented here by means of three illustrative examples related with the
Bures-Helstrom metric tensor, the Wigner-Yanase metric tensor, and the
Bogoliubov-Kubo-Mori metric tensor.Comment: 16 pages. Minor adjustments. Comments are welcome
From the Jordan product to Riemannian geometries on classical and quantum states
The Jordan product on the self-adjoint part of a finite-dimensional
-algebra is shown to give rise to Riemannian metric
tensors on suitable manifolds of states on , and the covariant
derivative, the geodesics, the Riemann tensor, and the sectional curvature of
all these metric tensors are explicitly computed. In particular, it is proved
that the Fisher--Rao metric tensor is recovered in the Abelian case, that the
Fubini--Study metric tensor is recovered when we consider pure states on the
algebra of linear operators on a finite-dimensional
Hilbert space , and that the Bures--Helstrom metric tensors is
recovered when we consider faithful states on .
Moreover, an alternative derivation of these Riemannian metric tensors in terms
of the GNS construction associated to a state is presented. In the case of pure
and faithful states on , this alternative geometrical
description clarifies the analogy between the Fubini--Study and the
Bures--Helstrom metric tensor.Comment: 32 pages. Minor improvements. References added. Comments are welcome