Distinguishability times and asymmetry monotone-based quantum speed limits in the Bloch ball

Abstract

For both unitary and open qubit dynamics, we compare asymmetry monotone-based bounds on the minimal time required for an initial qubit state to evolve to a final qubit state from which it is probabilistically distinguishable with fixed minimal error probability (i.e., the minimal error distinguishability time). For the case of unitary dynamics generated by a time-independent Hamiltonian, we derive a necessary and sufficient condition on two asymmetry monotones that guarantees that an arbitrary state of a two-level quantum system or a separable state of $N$ two-level quantum systems will unitarily evolve to another state from which it can be distinguished with a fixed minimal error probability $\delta \in [0,1/2]$. This condition is used to order the set of qubit states based on their distinguishability time, and to derive an optimal release time for driven two-level systems such as those that occur, e.g., in the Landau-Zener problem. For the case of non-unitary dynamics, we compare three lower bounds to the distinguishability time, including a new type of lower bound which is formulated in terms of the asymmetry of the uniformly time-twirled initial system-plus-environment state with respect to the generator $H_{SE}$ of the Stinespring isometry corresponding to the dynamics, specifically, in terms of $\Vert [H_{SE},\rho_{\text{av}}(\tau)]\Vert_{1}$, where $\rho_{\text{av}}(\tau):={1\over \tau}\int_{0}^{\tau}dt\, e^{-iH_{SE}t}\rho \otimes \vert 0\rangle_{E}\langle 0\vert_{E} e^{iH_{SE}t}$.