Group-theoretic Algorithms for Multiphoton Interferometery

Introduction and Background WignerD-functions are the matrix elements of the representations of finite group SU(n) of n × n special unitary matrices. Wigner D-functions of SU(2) group elements are used in nuclear, atomic and molecular physics [1–3]. In the Standard Model of particle physics, SU(2), SU(3) and SU(6) D-functions are used to describe transformations that preserve global or local symmetries [4–6]. Recently, SU(n) transformations have been the subject of considerable interest because of the BosonSampling problem. The output from an n-channel passive optical interferometer affecting a SU(n) transformation on indistinguishable single-photon pulse inputs is computationally hard classically subject to conjectures [7–9]. The action of three-channel optical interferometers on partially distinguishable single-photon inputs is best described by SU(3) D-functions [10–12]. D-functions of SU(2) group elements are well studied and tabulated [13]. SU(3) D-functions in a weight basis can be calculated as products of SU(2) Wigner Dfunctions [14, 15]. D-functions in the weight basis, which connect eigenstates of the weight-basis elements of su(n), su(n − 1) . . . su(2) Lie algebras, are especially suitable for studying permutational symmetric Bosonic systems. D-functions in the GelfandTsetlin basis for su(n) algebras can be constructed [16,17] but these functions lack the manifest permutational symmetries that arise in physical systems like BosonSampling interferometers.