For quantum state Procrustes optimization, restricting projective measurements to an r-dimensional subspace costs at most c log(r)^4 in objective value, independent of the ambient dimension.
Orthogonalization of Positive Operator Valued Measures
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We show that a partition of the unity (or POVM) on a Hilbert space that is almost orthogonal is close to an orthogonal POVM in the same von Neumann algebra. This generalizes to infinite dimension previous results in matrix algebras by Kempe-Vidick and Ji-Natarajan-Vidick-Wright-Yuen. Quantitatively, our result are also finer, as we obtain a linear dependance, which is optimal. We also generalize to infinite dimension a duality result between POVMs and minimal majorants of finite subsets in the predual of a von Neumann algebra.
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The suboptimality ratio of projective measurements restricted to low-rank subspaces
For quantum state Procrustes optimization, restricting projective measurements to an r-dimensional subspace costs at most c log(r)^4 in objective value, independent of the ambient dimension.