MathJax example

DDS Users' Guide

Quantum Entropy

Let \(qe(X):=\text{TR}(X\ln(X))\) and consider \(\ell\) quantum entropy constraints of the form \begin{eqnarray} \label{eq:QE-1} % &\min& c^\top x \nonumber \\ qe(F^i_0+x_1 F^i_1+ \cdots+x_n F^i_n) \leq g_i^\top x+d_i, \ \ \ i\in\{1,\ldots,\ell\}. \end{eqnarray} \(F^i_j\)'s are \(n_i\)-by-\(n_i\) symmetric matrices. To input these constraints to DDS as the \(k\)th block, we define: \begin{eqnarray} \label{eq:QE-4} &&\text{cons\{k,1\}='QE'}, \ \ \text{cons\{k,2\}}=[n_1, \ldots,n_\ell], \nonumber \\ &&\text{A\{k,1\}}:=\left [\begin{array}{c} g_1^\top \\ \text{sm2vec}(F^1_1), \cdots, \text{sm2vec}(F^1_n) \\ \vdots \\ g_\ell^\top \\ \text{sm2vec}(F^\ell_1), \cdots, \text{sm2vec}(F^\ell_n)\end{array} \right ], \ \ \ \text{b\{k,1\}}:=\left [ \begin{array}{c} d_1 \\ \text{sm2vec}(F^1_0)\\ \vdots \\ d_\ell \\ \text{sm2vec}(F^\ell_0) \end{array} \right]. \end{eqnarray}