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 ${\rho }_{\text{av}}(\tau ):=\frac{1}{\tau }{\int }_0^{\tau }dt{e}^{-i{H}_{SE}t}\rho \otimes |0{⟩}_E⟨0{|}_E{e}^{i{H}_{SE}t}$.

13 pages, 4 figures

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• Purity Speed Limit of Open Quantum Systems from Magic Subspaces
• Saturating the one-axis twisting quantum Cramér-Rao bound with a total spin readout