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In continuous-variable tomography, with finite data and limited computationresources, reconstruction of a quantum state of light is performed on afinite-dimensional subspace. No systematic method was ever developed to assignsuch a reconstruction subspace---only ad hoc methods that rely onhard-to-certify assumptions about the source and strategies. We provide astraightforward and numerically feasible procedure to uniquely determine theappropriate reconstruction subspace for any given unknown quantum state oflight and measurement scheme. This procedure makes use of the celebratedstatistical 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 theavailable measurement data, no other spurious assumptions about the source orad hoc strategies are invoked. As a result, there will no longer bereconstruction artifacts present in state reconstruction, which is a usualconsequence of a bad choice of reconstruction subspace. The procedure can beunderstood as the maximum-likelihood reconstruction for quantum subspaces,which is an analog to, and fully compatible with that for quantum states.

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