Maximally coherent mixed states: Complementarity between maximal coherence and mixedness

Quantum coherence is a key element in topical research on quantum resource theories and a primary facilitator for design and implementation of quantum technologies. However, the resourcefulness of quantum coherence is severely restricted by environmental noise, which is indicated by the loss of information in a quantum system, measured in terms of its purity. In this work, we derive the limits imposed by the mixedness of a quantum system on the amount of quantum coherence that it can possess. We obtain an analytical trade-off between the two quantities that upperbound the maximum quantum coherence for fixed mixedness in a system. This gives rise to a class of quantum states, “maximally coherent mixed states,” whose coherence cannot be increased further under any purity-preserving operation. For the above class of states, quantum coherence and mixedness satisfy a complementarity relation, which is crucial to understand the interplay between a resource and noise in open quantum systems.

Figure 1

Plot showing the trade-off between the (scaled) coherence, $C_{l_1}\left(\rho \right)/(d-1)$, and mixedness $M_l\left(\rho \right)$ as obtained from Eq. (3). The red line represents the extremal parabola in Eq. (19), which corresponds to the MCMS class that satisfies a complementarity relation between coherence and mixedness. The figure plots the (scaled) coherence, along the $Y$ axis, and mixedness, along the $X$ axis, for $1\times {10}^5$ randomly generated states in $d=2$, 3, 4, and 5 dimensions, using a specific Mathematica package [54].