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

T.J. Volkoff1 and K.B. Whaley2

1Department of Physics, Konkuk University, Seoul 05029, Korea
2Berkeley Quantum Information and Computation Center and Department of Chemistry, UC Berkeley, Berkeley, California 94720, USA

### 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 $\rho_{\text{av}}(\tau):={1\over \tau}\int_{0}^{\tau}dt\, e^{-iH_{SE}t}\rho \otimes \vert 0\rangle_{E}\langle 0\vert_{E} e^{iH_{SE}t}$.

The ability to reliably distinguish the states of a quantum system at different points in time is central to the performance of all high-precision quantum technologies, from atomic clocks to quantum sensors. However, the minimal time required to reliably distinguish such states, i.e., the "distinguishability time", depends sensitively on the quantum states involved, on their dynamics, on the method that is used to discriminate those states, and on what level of reliability is considered acceptable. In this work, we develop a framework for calculating distinguishability times that takes all these aspects of the problem into account, providing exact results or bounding the distinguishability time from below using quantities called "asymmetry monotones". Such lower bounds are referred to as quantum speed limits. Our framework leads to a completely new method for deriving tight quantum speed limits that approximate the true distinguishability time We apply our results to qubits, the basic building blocks of quantum clocks and quantum processors. Because of the simple description of qubit quantum states as points in a closed three-dimensional ball, we can directly compare our new quantum speed limits to established bounds for complex dynamical models that arise from coupling qubits to a realistic noisy environment.

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[2] Jie Sun and Songfeng Lu, "On the time–energy uncertainty relation of Kieu applied to quantum evolutions of Grover’s search", International Journal of Quantum Information 18 05, 2050021 (2020).

[3] A. A. Dìaz V., V. Martikyan, S. J. Glaser, and D. Sugny, "Purity speed limit of open quantum systems from magic subspaces", Physical Review A 102 3, 033104 (2020).

[4] Francesco Campaioli, Felix A. Pollock, and Kavan Modi, "Tight, robust, and feasible quantum speed limits for open dynamics", Quantum 3, 168 (2019).

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