Categorical traces from single-photon linear optics (Q2918876)

In the book under review the authors formally analyse a modified version of the Sagnac interferometer. In this setup, a photon may repeatedly enter an optical loop dependent on its polarisation. Since the number of times that a photon traverses the optical loop is not fixed, the device inherently implements a delay, and for single photons with a defined initial state entering the system, the output will be complicated to describe in the conventional circuit picture. The authors introduce a formal language for such kind of setups and analyse the properties of the resulting linear maps from input to output states.NEWLINENEWLINEThe paper starts with a short historical background about the Sagnac interferometer and then provides basic formal definitions of Hilbert spaces, inner product, states, unitary maps and measurements. In a small section the optics toolkit is introduced, which includes standard unitary operations on one or two qubits. These are encoded in the photon polarisation (horizontal or vertical) and the photon transmission channel. The discussed elements are the beamsplitter, the polarising beamsplitter, the phase plate, and the half-wave plate. After a discussion of the Sagnac interferometer, a modified version is introduced, where the original beamsplitter is modified by a polarising version and where inside the optical loop a unitary map may be applied. It is ensured that all incoming photons are horizontally polarised. The presence of a polarising beamsplitter now enables photons to traverse the loop multiple times, and it is not clear at what time the photons will exit the beamsplitter. Formally, the input and output states are then discretized in time and space, which gives rise to an infinite-dimensional Hilbert space. The setup then acts in a way on this Hilbert space which is formalised by the twisted dagger construction: the repeated application of the conditioned optical loop. The authors analyse its properties and connect it for a special case to linear maps from input to output states. They also address how one would describe combined modified Sagnac interferometers and introduce partial traces and weak partial traces. In their conclusion, the authors conjecture that the description of many quantum-optical experiments might be simpler using their twisted dagger construction than with unitary circuits. They propose to extend these mathematical concepts much further.NEWLINENEWLINEThe paper is written in a clear but very formal style and relatively self-contained, such that it might be suitable for computer scientists and mathematicians. It provides an efficient description of specific setups. It would have profited from a stronger motivation in terms of realistic applications of the modified Sagnac interferometer and from a more extensive exposition of connections with the unitary circuit picture.NEWLINENEWLINEFor the entire collection see [Zbl 1245.00037].