Single plane minimal tomography of double slit qubits

The determination of the density matrix of an ensemble of identically prepared quantum systems by performing a series of measurements, known as quantum tomography, is minimal when the number of outcomes is minimal. The most accurate minimal quantum tomography of qubits, sometimes called a tetrahedron measurement, corresponds to projections over four states which can be represented on the Bloch sphere as the vertices of a regular tetrahedron. We investigate whether it is possible to implement the tetrahedron measurement of double slit qubits of light, using measurements performed on a single plane. Assuming Gaussian slits and free propagation, we demonstrate that a judicious choice of the detection plane and the double slit geometry allows the implementation of a tetrahedron measurement. Finally, we consider possible sets of values which could be used in actual experiments.