Retrodictive quantum state engineering by Kenneth Lyell Pregnell; Griffith University. School of

By Kenneth Lyell Pregnell; Griffith University. School of Science

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It should be mentioned that this transformation is not unique, indeed Reck et al. select a different one in [62], however so long as the transformation matrix is unitary the net results will be equivalent. The above transformation is chosen for two reasons: firstly, in the limit where the beam-splitter becomes totally transmissive the transformation should approach the identity operator, and secondly, any phase change occurring upon reflection should be identical on both sides of the beam-splitter.

7) gives a factorised expression for U(N + 1) in terms of rotational matrices and a diagonal matrix as U(N + 1) = D · R(2) . . R(N ) · R(N + 1). 8) The achievement of Reck et al. was to realise that the necessary rotational matrices R(m), m = 2, 3, . . , N + 1, and the diagonal matrix D can be constructed from selected arrays of beam-splitters and phase-shifters. 8). To achieve this physical realisation we first need to consider the effect a lossless beam-splitter has on the input and output modes of a travelling optical field.

35) is identical to that which would be obtained if the dynamics were decomposed into a single preparation event at time ˆ b and a measurement event associated with the MDO Φ ˆ b at t0 associated with the PDO Λ i jk time t1 . The measurement event is conditioned on the combined outcome (j, k). As such, we interpret the second preparation event occurring at t1 to be part of the measurement event. 37) where the mode label b is dropped for notational convenience. Such a definition is consistent with all the probability formulae and thus provides a valid alternative interpretation to the arrangement proposed above.

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