State fidelity between two mixed states

ITensor Julia Questions
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1

Hello,

I want to calculate the state fidelity between two mixed states which are given in MPO forms.

F(\rho,\sigma):=\text{tr}\sqrt{\rho^{1/2} \sigma \rho^{1/2}} (=F(\sigma,\rho))

This definition is seen in section 9.2.2 of Nielsen & Chuang’ s textbook “Quantum Computation and Quantum Information”.

How can I get the state fidelity?

I think I can calculate this if I can get the square root of an MPO, but I don’t know the way.

Thank you very much in advance.

2

I’m not an expert on MPS and MPO myself, but I am afraid this is not a trivial thing to do with computational efficiency.

However, for specific classes of mixed states that arise from partitions of a spin chain which is globally described by a matrix product state, this reference [1807.01640] Uhlmann fidelities from tensor networks presents some efficient methods.

3

I agree there probably isn’t an efficient way to compute this type of fidelity, given \rho and \sigma as MPOs. Is there a different type of fidelity that could work for the purpose you need?

I do know of one special set of cases where you can get the square root of an MPO, and that is where the MPO is the exponential of an operator H and you can make it by time evolving (real or imaginary) by H. Then you can just time evolve for half the time to get the square root.