Bartosz ReguĊ‚a

Upper bounds on the distillable randomness of bipartite quantum states

Code accompanying the manuscript arXiv:2212.09073.

The implementation requires CVX, QETLAB, and (for the $\Upsilon$ measure) CVXQUAD.

$\beta$ measure

The function takes two arguments, a bipartite input state $\sigma_{AB}$ and (optionally) a single-partite state $\rho_{A}$. If the second argument is not provided, $\rho_A$ is taken to be $\sigma_A$. For simplicity, the local systems of $\sigma_{AB}$ are assumed to be of equal dimension.

function [cvx_optval,L,V] = beta(sigm,varargin)

    d = sqrt(max(size(sigm)));

    if nargin<2
        rho = PartialTrace(sigm);
    else
        rho = varargin{1};
    end
    
    cvx_begin sdp quiet 
    
    variable L(d,d) hermitian 
    variable V(d^2,d^2) hermitian 
    
    PartialTranspose(V+sigm)>=0
    PartialTranspose(V-sigm)>=0
    
    kron(rho,L)+V >= 0
    kron(rho,L)-V >= 0
    
    minimise trace(L)
    
    cvx_end

end

$\Upsilon$ measure

function [c,L,V] = upsilon(rho)

    d = sqrt(max(size(rho)));
    
    cvx_begin sdp quiet 
    
    variable sigm(d^2,d^2) hermitian semidefinite
    variable L(d,d) hermitian 
    variable V(d^2,d^2) hermitian 
    
    PartialTranspose(V+sigm)>=0
    PartialTranspose(V-sigm)>=0
    
    kron(PartialTrace(rho),L)+V >= 0
    kron(PartialTrace(rho),L)-V >= 0
    
    trace(L) <= 1
    
    minimise quantum_rel_entr(rho,sigm)
    
    cvx_end

    c = cvx_optval/log(2);

end