No second law of entanglement manipulation after all

Code accompanying the manuscript Nat. Phys. 19, 184–189 (2023) (arXiv:2111.02438).

Tempered negativity

The code below shows how to implement the tempered negativity $N_\tau(\rho)$, a lower bound on entanglement cost of any state. It requires CVX for MATLAB and additionally uses the helper function PartialTranspose from QETLAB.

It takes three arguments:

As the last two arguments are both optional, function calls of the form TempN(rho), TempN(rho, [dA, dB]), TempN(rho, omega), TempN(rho, omega, [dA, dB]) are all valid.

The function returns the value of $N_\tau$ and an optimal dual operator $X$.

function [cvx_optval, X] = TempN(rho, varargin)

% the ugly block of code below handles the various optional arguments
if nargin<2
    omega = rho;
    d = sqrt(max(size(rho)))*[1,1];
elseif nargin==2
    if min(size(varargin{1})) == 1
        omega = rho;
        d = varargin{1} * ones(1,3-max(size(varargin{1})));
        omega = varargin{1};
        d = sqrt(max(size(rho)))*[1,1];
elseif nargin==3
    omega = varargin{1};
    d = varargin{2} * ones(1,3-max(size(varargin{2})));;

D = prod(d);

rho = (rho+rho')/2; % to avoid numerical issues
omega = (omega+omega')/2;

% main SDP
cvx_begin sdp quiet

    variable X(D,D) hermitian

    maximize trace(X*rho)

    -eye(D) <= PartialTranspose(X,2,d) <= eye(D)

    -trace(X*omega)*eye(D) <= X <= trace(X*omega)*eye(D)