Coherent probes suffice to efficiently characterize all bosonic Gaussian processes

Characterizing quantum processes is an indispensable requirement for a wide variety of quantum information processing tasks. This task can be challenging in continuous-variable systems since the Hilbert space of such systems is infinite-dimensional. Simplifications can be made when we restrict our attention to linear optical processes which can be efficiently described by a unitary transformation of the mode operators. Several methods have already been developed to efficiently characterize such a unitary transformation.

Beyond the linear optical regime, efficient characterization remains possible for Gaussian systems, as such systems can be described by an affine transformation of the mode operators. Thus, O(N²) parameters fully describe an N-mode Gaussian process. The task then is to develop a method to efficiently find these parameters using experimentally available resources. Several methods have been developed for this task, though they differ in experimental resource requirements, scalability, and applicability to the full class of Gaussian processes.

In this work, I show that all bosonic Gaussian processes can be efficiently characterized with coherent probes and quadrature detection alone. In particular, a choice of 2N+1 coherent probes of equal energy, followed by 2N quadrature detections suffice to characterize a N-mode bosonic Gaussian process. In total, O(N²) measurements are employed, thereby demonstrating that this method is asymptotically optimal. Finally, I show evidence of the applicability of this method in characterizing quantum processes beyond the Gaussian regime.