Maximally and Minimally Correlated States Attainable within a Closed Evolving System

The amount of correlation attainable between the components of a quantum system is constrained if the system is closed. We provide some examples, largely from the field of quantum thermodynamics, where knowing the maximal possible variation in correlations is useful. The optimization problem it raises requires us to search for the maximally and minimally correlated states on a unitary orbit, with and without energy conservation. This is fully solvable for the smallest system of two qubits. For larger systems, the problem is reduced to a manageable, classical optimization.

Figure 1

Regions $\mathcal{R}$ (shaded) of allowed ${\lambda }_A$ ($y$ axis) and ${\lambda }_B$ ($x$ axis) when the joint state of two qubits, with spectrum $\Lambda =\{{\lambda }_i{\}}_{i=1}^4$, has various ranks: $\Lambda =$ (a) $\{1,0,0,0\}$, (b) $\{0.8,0.2,0,0\}$, (c) $\{0.5,0.5,0,0\}$, and (d) $\{0.6,0.3,0.1,0\}$. ${\lambda }_A,{\lambda }_B\in [0,\frac{1}{2}]$. For each spectrum, the hollow circles correspond to ${\rho }_{\min }$ and the large filled ones to ${\rho }_{\max }$. In (d), a state with energy $E$ defines the set ${\mathcal{R}}_E$ of states which could also have energy $E$. It is bounded from “above” by the solid-and-dotted line, on which the state itself is situated. The maximally correlated state in ${\mathcal{R}}_E$ is at $q$.