Quantum relative Lorenz curves

The theory of majorization and its variants, including thermomajorization, have been found to play a central role in the formulation of many physical resource theories, ranging from entanglement theory to quantum thermodynamics. Here we formulate the framework of quantum relative Lorenz curves, and show how it is able to unify majorization, thermomajorization, and their noncommutative analogs. In doing so, we define the family of Hilbert $\alpha$ divergences and show how it relates with other divergences used in quantum information theory. We then apply these tools to the problem of deciding the existence of a suitable transformation from an initial pair of quantum states to a final one, focusing in particular on applications to the resource theory of athermality, a precursor of quantum thermodynamics.

Figure 1

Example of classical testing region in dimension $n=4$ with ${\vec{p}}_1={(1/2,1/4,1/4,0)}^T$ and ${\vec{p}}_2={(1/6,1/6,1/3,1/3)}^T$. The Lorenz curve (i.e., the upper boundary) is determined by the vertices at which the Lorenz curve changes slope.

Figure 2

Numerical example of the quantum testing region for two random four-dimensional density matrices. Notice that the curve is only roughly approximated as the sampled measurements are not enough to determine it neatly. Quantum Lorenz curves are efficiently obtained by semidefinite linear programming, e.g., using Eq. (2) in the main text.

Figure 3

Typical behavior of ${‖r{\rho }_1-{\rho }_2‖}_1$, for two random density matrices ${\rho }_1$ and ${\rho }_2$ on ${\mathbb{C}}^3$, as a function of $r\in \mathbb{R}$ (continuous line). For $r\le r_{*}\equiv inf({\rho }_2/{\rho }_1)=sup\{\lambda :\lambda {\rho }_1-{\rho }_2\le 0\}$, the curve becomes equal to $1-r$ (dashed line). For $r\ge r^{*}\equiv sup({\rho }_2/{\rho }_1)=inf\{\lambda :\lambda {\rho }_1-{\rho }_2\ge 0\}$, the curve becomes equal to $r-1$ (dotted line).