Generalization of quantum monotonic metric with duality via a purification

Anna Jenčová

We investigate relations between Uhlmann’s parallelism, monotone Riemannian metrics and dual affine connections on the space of density matrices.

Renato Ascoli, Ettore Minardi

Jun Suzuki

Shan-Ming Ruan

Abstract We generalize the Fubini-Study method for pure-state complexity to generic quantum states by taking Bures metric or quantum Fisher information metric (QFIM) on the space of density matrices as the complexity measure. Due to Uhlmann’s theorem, we show that the mixed-state complexity exactly equals the purification complexity measured by the Fubini-Study metric for purified states but without explicitly applying any purification. We also find the purification complexity is non-increasing under any trace-preserving quantum operations. We also study the mixed Gaussian states as an example to explicitly illustrate our conclusions for purification complexity.

Greg Kuperberg, Nik Weaver

Keiji Matsumoto

Many problems in quantum information theory can be vied as interconversion between resources. In this talk, we apply this view point to state estimation theory, motivated by the following observations. First, a monotone metric takes value between SLD and RLD Fisher metric. This is quite analogous to the fact that entanglement measures are sandwiched by distillable entanglement and entanglement cost. Second, SLD add RLD are mutually complement via purification of density matrices, but its operational meaning was not clear. To find a link between these observations, we define reverse estimation problem, or simulation of quantum state family by probability distribution family, proving that RLD Fisher metric is a solution to local reverse estimation problem of quantum state family with 1-dim parameter. This result gives new proofs of some known facts and proves one new fact about monotone distances. We also investigate information geometry of RLD, and reverse estimation theory of a multi-dimensional parameter family.