Fidelity Preserving Maps on Density Operators

We prove that any bijective fidelity preserving transformation on the set of all density operators on a Hilbert space is implemented by an either unitary or antiunitary operator on the underlying Hilbert space. Let H be a Hilbert space. The set of all density operators on H, that is, the set of all positive self-adjoint operators on H with finite trace is denoted by C 1 (H). (We note that one may prefer normalized density operators; see the first remark at the end of the paper.) According to Uhlmann [6, 7], for any A,B ∈ C 1 (H) we define the fidelity of A and B by