Lieb's concavity theorem, matrix geometric means and semidefinite optimization
Hamza Fawzi, James Saunderson
The zip file contains CVX definitions for the convex sets as well as convex/concave functions listed below.
matrix_geo_mean_hypo_cone | $\{(A,B,T) : A\#_t B \succeq T\}$ | $t \in [0,1]$ |
matrix_geo_mean_epi_cone | $\{(A,B,T) : A\#_t B \preceq T\}$ | $t \in [-1,0]\cup [1,2]$ |
lieb_hypo_cone | $\{(A,B,T) : A^{1-t} \otimes B^t \succeq T\}$ | $t \in [0,1]$ |
ando_epi_cone | $\{(A,B,T) : A^{1-t} \otimes B^t \preceq T\}$ | $t \in [-1,0]\cup [1,2]$ |
% The following line constrains matrices A,B,T to satisfy A #_{3/4} B >= T {A,B,T} == matrix_geo_mean_hypo_cone(4,3/4);
lieb_ando | $(A,B)\mapsto \tr\left[K^* A^{1-t} K B^t\right]$ | Concave for $t \in [0,1]$ Convex for $t \in [-1,0]\cup [1,2]$ |
tsallis_entr | $A \mapsto \frac{1}{t} \tr\left[A^{1-t} - A\right]$ | Concave Converges to von Neumann entropy $S(A) = -\tr[A \log A]$ when $t\rightarrow 0$ |
tsallis_rel_entr | $(A,B) \mapsto \frac{1}{t} \tr\left[A - A^{1-t} B^t\right]$ | Convex Converges to relative entropy $S(A,B) = \tr[A (\log A - \log B)]$ when $t\rightarrow 0$ |
trace_mpower | $A\mapsto \tr\left[A^t\right]$ | Convex for $t \in [0,1]$ Concave for $t \in [-1,0]\cup [1,2]$ |
trace_KAK | $A\mapsto \tr\left[(K^* A^t K)^{1/t}\right]$ | Concave for $t \in [-1,1]\setminus \{0\}$ Convex for $t \in [1,2]$ |
Note on approximating entropy functions using tsallis_entr and tsallis_rel_entr:
The functions tsallis_entr and tsallis_rel_entr can be used to approximate the von Neumann entropy and the relative entropy respectively by choosing $t$ close to 0. However it is important to keep in mind that choosing $t$ very small may result in a semidefinite program that is very ill-conditioned. We encourage the user to try different values of $t$ and check the resulting approximation (note that our constructions are optimized for powers of two, $t = 2^{-k}$). A value of $t$ that seemed to work in our examples is $t=2^{-6}$.
We are currently working on more sophisticated methods to approximate von Neumann entropy and quantum relative entropy. See the paper for more information.
The CVX code below solves the following optimization problem ($\rho_1$ and $\rho_2$ are fixed positive semidefinite matrices) \[ \begin{array}{ll} \underset{p}{\text{maximize}} & S_t(p_1 \rho_1 + p_2 \rho_2) - p_1 S(\rho_1) - p_2 S(\rho_2)\\ \text{subject to} & p \geq 0, p_1 + p_2 = 1 \end{array} \] where $S_t(A) = \frac{1}{t} \tr\left[A^{1-t} - A\right]$ is the Tsallis entropy. When $t$ is small, the solution of this optimization problem approximates the capacity of a classical-quantum channel (see e.g., the paper Efficient Approximation of Quantum Channel Capacities by Sutter et al.). We use the value $t = 1/2^6$ in the code below.
% Example 2.16 in % "Efficient Approximation of Quantum Channel Capacities" by Sutter et al. (arXiv:1407.8202) rho1 = [1 0; 0 0]; rho2 = (1/2) * [1 1; 1 1]; Srho1 = sum(entr(eig(rho1))); Srho2 = sum(entr(eig(rho2))); cvx_begin variable p(2,1) maximize ((tsallis_entr(p(1)*rho1 + p(2)*rho2,2^(-6)) - p(1)*Srho1 - p(2)*Srho2)/log(2)); subject to p >= 0; sum(p) == 1; cvx_end