violates_bapat_sunder(A,B, tol = ATOL)

Checks if matrices A and B violate the Bapat-Sunder conjecture, see Boson bunching is not maximized by indistinguishable particles

add_columns_to_make_square_unitary(M_dagger)

Makes a square matrix U of which the first columns are M_dagger, which has to be unitary by itself by a random choice of vectors that are then orthonormalized.

cholesky_semi_definite_positive(A)

cholesky decomposition (A = R' * R) for a sdp but not strictly positive definite matrix

incorporate_in_a_unitary(X)

incorporates the renormalized matrix X in a double sized unitary through the proof of Lemma 29 of Aaronson Arkipov seminal The Computational Complexity of Linear Optics

incorporate_in_a_unitary_non_square(X)

same as incorporate_in_a_unitary but for a matrix renormalized X of type (m,n) with m >= n generates a minimally sized unitary (ex 99 interferometer for the 72 M' of the first counter example of drury)

random_search_counter_example_bapat_sunder(;m,n,r, physical_H = true)

Brute-force search of counter-examples of rank r.

search_until_user_stop(search_function)

Runs search_function until user-stop (Ctrl+C).

J_array(theta, n)

returns the J as defined in in Eq.10 of Universality of Generalized Bunching and Efficient Assessment of Boson Sampling, with the permutations coming in the order given by permutations(collect(1:n))

density_matrix_from_J(J,n)

density matrix associated to a J function as defined in Universality of Generalized Bunching and Efficient Assessment of Boson Sampling, computed through Eq. 46.

schur_matrix(H)

computes the Schur matrix as defined in Eq. 1 of Linear Algebra and its Applications 490 (2016) 196–201