One-shot coherence distillation

Phys. Rev. Lett 121, 010401 (2018)

The measures $T^{(m)}_{\mathcal{I}}$ and $T^{(m)}_{\mathcal{J}}$

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The computation of the quantities $T^{(m)}_{\mathcal{I}}$ for pure states reduces to a semi-analytical formula, as described in the manuscript. For general mixed states, each measure can be computed with a semidefinite program. The below Matlab implementation takes as input either pure state vectors or general density matrices, and for states of rank 2 or more also returns the optimal witness W. The semidefinite program is implemented using CVX.

function [val,W] = TI(rho,m)
% [val,W] TI(rho,m) computes the family of coherence measures as defined in PRL 121, 010401 (2018)
% rho: dxd density matrix or normalised state vector
% m: non-negative integer coefficient indicating the chosen measure
% the optimal witness W s.t. val=trace(rho*W) is returned only for non-pure rho

dim = size(rho);
d = max(dim);

if m>=d; m = d-1; end

if rank(rho) == 1 % pure state case
    if min(dim) ~= 1
        [~,~,v] = svd(rho);
        rho = v(:,1);
    m = m+1; % to compute the (m+1)-distillation norm
    psi = sort(abs(rho),'descend');
    k = 1;
    minsum = sum(psi(m:d).^2);

    for j=2:m
        jsum = sum(psi(m-j+1:d).^2)/j;
        if jsum < minsum
            minsum = jsum;
            k = j;
    val = sum(psi(1:m-k))+k*sqrt(minsum);
    val = val^2-1;
    W = 0;
else % general case
    rho = (rho+rho')/2; % to avoid numerical problems
    cvx_begin quiet sdp
        variable W(d,d) hermitian
        -eye(d) <= W <= m*eye(d)
        diag(W) <= 0
        maximise trace(rho*W);
    val = cvx_optval;


Notice that a simple change of the constraint diag(W) <= 0 into diag(W) == 0 gives instead the family of measures $T^{(m)}_{\mathcal{J}}$, which characterise the one-shot fidelity of coherence distillation.

One-shot distillable coherence

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The one-shot distillable coherence under MIO and DIO (and for pure states also SIO and IO) is computed through a semidefinite program. An example implementation with CVX is given below.

function dis = distillable_coherence(rho,tol)
% distillable_coherence(rho,tol) computes the one-shot distillable coherence under MIO/DIO
% rho: dxd density matrix or normalised state vector
% tol: error tolerance 0<=tol<=1

dim = size(rho);
d = max(dim);
if dim(1)==1; rho=rho'*rho; 
elseif dim(2)==1; rho=rho*rho'; end
rho = (rho+rho')/2; % to avoid numerical problems

cvx_begin sdp quiet

    variable G(d,d) hermitian
    variable k nonnegative
    0 <= G <= eye(d)
    trace(rho*G) >= 1-tol;
    diag(G) == k*ones(d,1)

    minimise k


dis = log2(floor(1/cvx_optval));