A fibre bundle over manifolds of quantum channels and its application to quantum statistics
Akio Fujiwara, Hiroshi Imai · Journal of Physics A Mathematical and Theoretical · 2008
A fibre bundle structure is introduced over manifolds of quantum channels. This structure has a close connection with the problem of estimating an unknown quantum channel Γ θ specified by a parameter θ. It is shown that the quantum Fisher information of the family of output states maximized over all input states , which quantifies the ultimate statistical distinguishability of the parameter θ, is expressed in terms of a geometrical quantity on the fibre bundle. Using this formula, a criterion for the maximum quantum Fisher information of the n th extended channel (id ⊗ Γ θ ) ⊗ n to be O ( n ) is derived. This criterion further proves that for almost all quantum channels, the maximum quantum Fisher information increases in the order of O ( n ).