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Determining an unknown quantum state from an ensemble of identical systems isa fundamental, yet experimentally demanding, task in quantum science. Here westudy the number of measurement bases needed to fully characterize an arbitrarymulti-mode state containing a definite number of photons, or an arbitrarymixture of such states. We show this task can be achieved using only linearoptics and photon counting, which yield a practical though non-universal set ofprojective measurements. We derive the minimum number of measurement settingsrequired and numerically show that this lower bound is saturated with randomlinear optics configurations, such as when the corresponding unitarytransformation is Haar-random. Furthermore, we show that for N photons, anyunitary 2N-design can be used to derive an analytical, though non-optimal,state reconstruction protocol.

Multiphoton Tomography with Linear Optics and Photon Counting