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

T.J. Volkoff; K.B. Whaley

doi:10.22331/q-2018-10-01-96

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

Published:	2018-10-01, volume 2, page 96
Eprint:	arXiv:1508.04181v4
Doi:	https://doi.org/10.22331/q-2018-10-01-96
Citation:	Quantum 2, 96 (2018).

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

Featured image: Oscillations in the minimal error probability of distinguishing a pure qubit state from its initial state as a function of time and polar angle on the Bloch sphere for a model that interpolates between Markovian and non-Markovian dynamics (intermediate parameter regime shown). Blue indicates higher distinguishability, whereas yellow indicates lower distinguishability. The distinguishability time decreases as the polar angle increases from $\theta =0$, corresponding to initial state $\vert 0 \rangle \langle 0 \vert$, to $\theta =\pi$, corresponding to initial state $\vert 1 \rangle \langle 1 \vert$. The figure also shows that only pure states with polar angle near $\pi over 2$ orthogonalize, i.e., reach states which can be distinguished from the initial state with zero probability of error.

Popular summary

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.

► BibTeX data

@article{Volkoff2018distinguishability,
  doi = {10.22331/q-2018-10-01-96},
  url = {https://doi.org/10.22331/q-2018-10-01-96},
  title = {Distinguishability times and asymmetry monotone-based quantum speed limits in the {B}loch ball},
  author = {Volkoff, T.J. and Whaley, K.B.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {2},
  pages = {96},
  month = oct,
  year = {2018}
}

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