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Description
Bell-state measurements are central to photonic quantum information with applications in quantum teleportation, entanglement swapping, quantum repeaters, dense coding, and fusion-based photonic quantum computation. A passive linear optical interferometer with vacuum auxiliary modes and photon number resolving detectors can unambiguously identify at most two of the four equiprobable Bell states with an average success probability that cannot exceed 1/2. Ancillary photons provide a direct route beyond the one-half limit while retaining passive linear optics.
For a fixed circuit, the computational task is to determine which photon-counting patterns occur for exactly one of the four Bell inputs and to sum their probabilities. Individual passive-linear-optical amplitudes can be evaluated from matrix permanents, but unambiguous discrimination requires a simultaneous four-label classification measurement patterns. At the largest size considered here, M = 32 modes and Q = 30 photons give approximately 2.33 × 10^17 ideal full detector patterns. This rules out naive enumeration of the complete outcome space, but it does not imply that all structured evaluations are intractable.
In this research we sucessfully adapt a tree tensor network (TTN), whose bipartitions follow the recursive optical circuit of beam splitters, to ancilla-assisted linear-optical Bell-state discrimination. Photon-number U(1) symmetry resolves each virtual bond into fixed-charge sectors that significantly reduces complexity of computations and allows for finding probabilities of unambiguous state discrimination with no other than machine precision related approximations. To our knowledge, this is the first treatment to combine a U(1)-resolved TTN with simultaneous four-label detector-support classification for ancilla-assisted passive-linear-optical Bell-state discrimination.
The method reproduces known probabilities of Bell-state discrimination and additionally provides photon number resolving detector patterns uniquelly associated with each Bell state in terms of photon number prefix cylinders, which avoids enumeration of all possible patterns. Scalling is discussed.