Experimental characterization of quantum processes: A selective and efficient method in arbitrary finite dimensions

The temporal evolution of a quantum system can be characterized by quantum process tomography, a complex task that consumes a number of physical resources scaling exponentially with the number of subsystems. An alternative approach to the full reconstruction of a quantum channel allows selecting which coefficient from its matrix description to measure, and how accurately, reducing the amount of resources to be polynomial. The possibility of implementing this method is closely related to the possibility of building a complete set of mutually unbiased bases (MUBs) whose existence is known only when the dimension of the Hilbert space is the power of a prime number. However, an extension of the method that uses tensor products of maximal sets of MUBs, has been introduced recently. Here we explicitly describe how to implement this algorithm to selectively and efficiently estimate any parameter characterizing a quantum process in a non-prime power dimension, and we conducted for the first time an experimental verification of the method in a Hilbert space of dimension $d=6$. That is the small space for which there is no known a complete set of MUBs but it can be decomposed as a tensor product of two other Hilbert spaces of dimensions $D_1=2$ and $D_2=3$, for which a complete set of MUBs is known. The $6$-dimensional states were codified in the discretized transverse momentum of the photon wavefront. The state preparation and detection stages are dynamically programmed with the use of only-phase spatial light modulators, in a versatile experimental setup that allows to implement the algorithm in any finite dimension.