Here are my notes on Scott’s third lecture, delivered this Wednesday, 18/12/2013.

**Linear Optics: review**

A set of n bosons can be described by what mathematicians call a symmetric product. An easier way of putting it is to think of a finite number n of photons, which can be found in m different modes. These modes can describe directions of propagation, position, or some other suitable property that can be used to discriminate them. If we have just a single photon, its state will be an superposition of all m modes with arbitrary complex amplitudes: $latex |\psi\rangle = \sum_{i=1}^m \alpha_i |i\rangle$.

Transformations on this single-photon state are described by a $latex m \times m$ unitary U. The role of circuit gates is played here by beam splitters and phase shifters, which implement an arbitrary unitary transformation on a dual-rail qubit (consisting of a single photon in a superposition of two…

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