Crystallizing highly-likely subspaces that contain an unknown quantum state of light

Abstract

In continuous-variable tomography, with finite data and limited computation resources, reconstruction of a quantum state of light is performed on a finite-dimensional subspace. In principle, the data themselves encode all information about the relevant subspace that physically contains the state. We provide a straightforward and numerically feasible procedure to uniquely determine the appropriate reconstruction subspace by extracting this information directly from the data for any given unknown quantum state of light and measurement scheme. This procedure makes use of the celebrated statistical principle of maximum likelihood, along with other validation tools, to grow an appropriate seed subspace into the optimal reconstruction subspace, much like the nucleation of a seed into a crystal. Apart from using the available measurement data, no other assumptions about the source or preconceived parametric model subspaces are invoked. This ensures that no spurious reconstruction artifacts are present in state reconstruction as a result of inappropriate choices of the reconstruction subspace. The procedure can be understood as the maximum-likelihood reconstruction for quantum subspaces, which is an analog to, and fully compatible with that for quantum states.

Extracting the physical sector of quantum states

“…This procedure requires nothing more than data obtained from a set of commuting measurements. As in [25], the extraction of the physical sector does not depend on any other assumptions or calibration details about the source. By construction, this procedure has a linear complexity in the dimension of the physical sector.…”

“…In reference [25], we showed that, when the measurement device is calibrated, one can systematically extract the physical sector (that is, both the Hilbert-space support and dimension) and simultaneously reconstruct any unknown state directly from the measurement data without any assumption about the state. In this paper, we introduce an even more efficient procedure that extracts the physical sector of any state from the data without state reconstruction and provide the pseudocode.…”