Discriminating Lindbladian Dynamics
Abstract
The fundamental task of quantum hypothesis testing, which consists in discriminating quantum objects, has been extensively studied for quantum states and channels. Here, we investigate the fundamental limits of discriminating an arbitrary pair of quantum Markovian dynamics generated by Lindbladians, focusing on the roles of total exposure time and temporal resolution. We allow adaptive choices of evolution durations and arbitrary instantaneous controls between them. We characterise the optimal type-II error exponent (Stein exponent) as the largest instantaneous increase of quantum relative entropy, and give an algebraic criterion for its finiteness in terms of the Lindbladians. Remarkably, an infinite exponent is equivalent to a vanishing type-II error probability at sufficiently large finite total time, for every positive type-I tolerance. While some dynamics admit a finite protocol with zero type-II error, others require increasingly fine temporal control. For the latter, the error vanishes as the minimum allowed evolution duration, or resolution time, tends to zero at fixed sufficiently large total time, but remains bounded away from zero at every fixed positive resolution time. We achieve this using quantum Zeno protocols, establishing temporal resolution as a resource for discrimination. When the null hypothesis is the identity semigroup, an infinite asymptotic error exponent has a stronger consequence: at a fixed finite total exposure time, the type-II error can be made arbitrarily small while keeping the type-I error exactly zero. This is impossible in the corresponding classical setting. Discriminating the identity semigroup from generalised amplitude damping on a qubit provides examples of all three regimes: a finite asymptotic error exponent, zero type-II error attainable at positive resolution time, and vanishing type-II error requiring the resolution time to tend to zero.
Authors 4
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Lyon 1 Université · École Normale Supérieure de Lyon · RWTH Aachen University
Affiliation as printed
Inria , ENS de Lyon , Université Claude Bernard Lyon 1 , LIP , 69342 Lyon cedex 07 , France
Institute for Quantum Information , RWTH Aachen University , Germany
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Affiliation as printed
Scuola Normale Superiore , Piazza dei Cavalieri 7 , 56126 Pisa , Italy
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Affiliation as printed
Institut für Theoretische Physik & IQST , Albert-Einstein-Allee 11 , Universität Ulm , D-89081 Ulm , Germany
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Affiliation as printed
Institut für Theoretische Physik & IQST , Albert-Einstein-Allee 11 , Universität Ulm , D-89081 Ulm , Germany