Relevance of classical solutions to quantum theories

Artem Alexandrov

We bridge two sides of singular perturbation theory: the classical theory of slow-fast systems and the semi-classical approach to quantum mechanical systems. For a specific but physically important class of dynamical systems, we show that purely classical and exotic objects, so-called canard solutions, are shadows of instantons in the corresponding quantum system. We demonstrate that canard solutions exist in a domain of parameter space whose boundaries are determined by an instanton action. We illustrate our statements analytically for the relevant example, the overdamped Josephson junction, and confirm them numerically. For the Josephson junction, the canard window is the exponentially narrow gap between consecutive Shapiro steps.

Jack Cohn

Sina Salek, Daniela Cadamuro, Philipp Kammerlander +1 more

Rate-distortion theory provides bounds for compressing data produced by an information source to a specified encoding rate that is strictly less than the source's entropy. This necessarily entails some loss, or distortion, between the original source data and the best approximation after decompression. The so-called Information Bottleneck Method is designed to compress only “relevant” information. Which information is relevant is determined by the correlation between the data being compressed and a variable of interest, so-called side information. In this paper, an Information Bottleneck Method is introduced for the compression of quantum data. The channel communication picture is used for compression and decompression. The rate of compression is derived using an entanglement-assisted protocol with classical communication, and under an unproved conjecture that the rate function is convex in the distortion parameter. The optimum channel achieving this rate for a given input state is characterized. The conceptual difficulties arising due to differences in the mathematical formalism between quantum and classical probability theory are discussed and solutions are presented.

Theoretical Physics

Iwo Białynicki‐Birula

Davit Aghamalyan, Aleek Maity, Varun Narasimhachar +1 more

Abstract We present a novel framework for optimal control in both classical and quantum systems. Our approach leverages the Dirac–Bergmann algorithm: a systematic method for formulating and solving constrained dynamical systems. In contrast to the standard Pontryagin principle, which is used in control theory, our approach bypasses the need to perform a variation to obtain the optimal solution. Instead, the Dirac–Bergmann algorithm generates the optimal solution automatically, through the closure of the Poisson bracket algebra of the full set of constraints and the Hamiltonian. The efficacy of our framework is demonstrated through two quintessential examples: (1) the classical brachistochrone problem and (2) the time-optimal control of a generic quantum system, relevant for quantum technological applications. In the latter example, both closed and open quantum systems are discussed.