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Generalized Hofmann quantum process fidelity bounds for quantum filters

We propose and investigate bounds on quantum process fidelity of quantum filters, i.e. probabilistic quantum operations represented by a single Kraus operator K. These bounds generalize the Hofmann bounds on quantum process fidelity of unitary operations [H.F. Hofmann, Phys. Rev. Lett. 94, 160504 (2005)], and are based on probing the quantum filter by pure states forming two mutually unbiased bases. Determination of these bounds therefore requires much less measurements than full quantum process tomography. We find that it is particularly suitable to construct one of the probe basis from the right eigenstates of K, because in this case the bounds are tight in the sense that if the actual filter coincides with the ideal one then both the lower and upper bounds are equal to one. We theoretically investigate application of these bounds to a two-qubit optical quantum filter formed by interference of two photons on a partially polarizing beam splitter. For experimentally convenient choice of factorized input states and measurements we study the tightness of the bounds. We show that more stringent bounds can be obtained by more sophisticated processing of the data using convex optimization and we compare our methods for different choice of the input probe states.