The top-left N by N submatrix of an M by M circular orthogonal ensemble random matrix, scaled by sqrt(M), converges to a complex symmetric Gaussian matrix in total variation distance for N much smaller than sqrt(M).
Singular value bounds In this section we prove the lemmas used in the proof of Proposition 3.2
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Proof of Hiding Conjecture in Gaussian Boson Sampling
The top-left N by N submatrix of an M by M circular orthogonal ensemble random matrix, scaled by sqrt(M), converges to a complex symmetric Gaussian matrix in total variation distance for N much smaller than sqrt(M).