How to measure fidelity between two mixed quantum states?

          @misc{ pirsa_PIRSA:08080050,
            doi = {10.48660/08080050},
            url = {https://pirsa.org/08080050},
            author = {Zyczkowski, Karol},
            keywords = {},
            language = {en},
            title = {How to measure fidelity between two mixed quantum states?},
            publisher = {Perimeter Institute},
            year = {2008},
            month = {aug},
            note = {PIRSA:08080050 see, \url{https://pirsa.org}}
          }
          

Abstract

Assume one laboratory designed a technique to produce quantum states in a given state $ ho$. The other lab wants to generate exactly the same state and they produce states $sigma$. If we want to know how well the second lab is doing we need to characterize the distance between $sigma$ and $ ho$ by some means,e.g. by trying to measure their fidelity, which allows us to find the Bures distance between them. The task is simple if the given state is pure, $ ho=|psi angle langle psi|$, since then fidelity reduces to the expectation value, $F=langlepsi| sigma| psi angle$. If $ ho$ is mixed the explicit formula for fidelity contains the trace of an absolute value of an operator which is not simple to compute nor to measure. Therefore we provide lower and upper bounds for fidelity and propose schemes to measure them. These experimental schemes require much less effort than the full quantum tomography of both states in question. The bounds for fidelity are called {sl sub-} and {sl super-fidelity}, respectively, due to their properties: as fidelity is multiplicative with respect to the tensor product, the sub-fidelity is sub-multiplicative, while super-fidelity is shown to be super-multiplicative. In the case of any two states of a one qubit system the bounds are strict and all three quantities coincide. The super-fidelity allowes us to define a modified Bures distance which for larger systems induces an alternative geometry of the space of quantum states.