schmidt_decomposition_bitpartite_states

function [ W ] = Schmidt( v,m,n );
% Input column vector bipartite density matrix, v and desired m,n such that 
% m*n = dim(v). The density matrix is the tensor of two hilbert spaces 
% by definition. This function returns the Schmidt number and associated 
% bases separated by a line with the index of the schmidt number and finally
% the same bipartite density matrix in the dimensions mxn. This new bipartite 
% density matrix is the product of the Schmidt number and associated bases.

e_m = eye(m);
    e_n = eye(n);

    h = 0;
    for i = 1:m;
        
        for j = 1:n;
            
            h = h+1;
            e_i = e_m(:,i);
            e_j = e_n(:,j);
            K = kron(e_i, e_j);
            s = ctranspose(K)*v;
            x = ctranspose(K)*v;
            W(i,j) = x;
            Q = {e_i,e_j,s};
            ei = Q{1};
            ej = Q{2};
            ex = Q{3};
            m = {ei,ej,ex}; 
            
        end
    end

[r,f,o] = svd(W);
if size(e_n) > size(e_m);
    ms = size(e_m);
min_dim = ms(1,1);
end
if size(e_m)>=size(e_n);
    ns = size(e_n);
    min_dim = ns(1,1);
end
fprintf('v is the sum of s_i*k, where k is the tensor of the ith row in sigma and ith column in V of SVD of input vector%g.\n')

for i = 1:min_dim;
    
    s_i = f(i,i)
    u_i = r(:,i);
    v_i = o(:,i);%returns unitary matrices such that x = u*s*v' so we take column instead of row
    k = kron(u_i,v_i)
    
end