Issues calculating quantum relative entropy of 2 cvx matricies

1

Hi all,
I’m currently trying to calculate quantum relative entropy to solve a minimization problem to obtain a sub-optimal quantum state. specifically I’m attempting to utilize CVX to implement equation (16) from this paper : https://journals.aps.org/prx/pdf/10.1103/PhysRevX.9.041064
My CVX code is as follows:

cvx_begin sdp 
    
   %rho_AB is 12x 12 due to Nc = 2 and alic's subsystem being 4d, thus 4×(Nc​+1)
    variable rho_AB(matrix_size,matrix_size) 

    %implement the R subsytem into rho_AB
    rho_AB_exp= kron(rho_AB,eye(2));

    G_Rho_AB = G(rho_AB_exp,K);

    Z_G_Rho_AB = (Z0 * G_Rho_AB * Z0) + (Z1 * G_Rho_AB * Z1);
    disp(['Z_G_Rho_AB size =' mat2str(size(Z_G_Rho_AB))])
    


    D_Rho_AB = trace(G_Rho_AB*log(G_Rho_AB)) - trace(G_Rho_AB-log(Z_G_Rho_AB));
    minimize(rel_entr(G_Rho_AB,Z_G_Rho_AB));

This is not working, giving me the error of:

 > disciplined convex programming error:
    Illegal operation: log( {zero} )

This is my first time using Matlab and CVX so I’m really confused as to what specifically this error means, and in general how to calculate quantum relative entropy. I attempted to use CVXQUAD quantum_rel_entr but due to my matrices being 24x24 the memory usage is way too high. Does anyone know how i can solve this issue, or how i can go about calculating quantum relative entropy for the 2 matrices G_Rho_AB and Z_G_Rho_AB?

Thanks in advanced :slight_smile:

2

CVXQUAD uses a very computationally demanding method to handle quantum_rel_entr. n by n matrix input (in your case, 24 by 24), turns into several 2n^2 by 2n^2 “SDP” matrices, which in your case are 1152 by 1152 (and some smaller, but still very large SDP matrices). That “squaring” of the dimension is a “killer”. See table 1 of Semidefinite approximations of the matrix logarithm, written by the CVXQUAD author and co-authors. .The CVX processing time to formulate that can be quite large, not to mention getting an SDP solver to solve it.

DDS (Domain-Driven Solver) version 2.2 runs under MATLAB, and handles quantum relative entropy using an interior point method barrier function which is much less computationally demanding than CVXQUAD, but was yet known (but is now, thanks to the CVXQUAD author and co-author) to be self-concordant when CVXQUAD was written, and therefore "guaranteed (in some sense) to work correctly.

However, DDS is not available as a solver under CVX, and likely never will be.

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3

Really appreciate the feedback Mark. So if I wanted to approach solving:

trace(G_Rho_AB*log(G_Rho_AB)) - trace(G_Rho_AB-log(Z_G_Rho_AB));

There’s no feasible way to do this within cvx itself? If so does the issue stem from the log() needed within the formula?

4

That log is a a matrix log. That can’t be handled with CVX’s “own” functions, but can be handled with CVXQUAD, which adds to CVX’s functions.

I think DDS is your best bet for solving via MATLAB, unless you get a monster machine on which to run CVX/CVXQUAD/SDP solver called by CVX,

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5

Understood, much appreciated Mark!