Preservers on Quantum Structures

Anatolij Dvureĕenskij

Robin Blume-Kohout, Hui Khoon Ng, David Poulin +1 more

Quantum systems carry information. Quantum theory supports at least two distinct kinds of information (classical and quantum), and a variety of different ways to encode and preserve information in physical systems. A system's ability to carry information is constrained and defined by the noise in its dynamics. This paper introduces an operational framework, using information-preserving structures, to classify all the kinds of information that can be perfectly (i.e., with zero error) preserved by quantum dynamics. We prove that every perfectly preserved code has the same structure as a matrix algebra, and that preserved information can always be corrected. We also classify distinct operational criteria for preservation (e.g., ``noiseless,'' ``unitarily correctible,'' etc.) and introduce two natural criteria for measurement-stabilized and unconditionally preserved codes. Finally, for several of these operational criteria, we present efficient (polynomial in the state-space dimension) algorithms to find all of a channel's information-preserving structures.

Mirko Navara

Benjamin H. Feintzeig

Abstract This article presents a novel sense in which theoretical structure has been preserved across the transition from classical to quantum physics. I import mathematical tools from category theory that have been used for structural comparisons in the context of theoretical equivalence and apply these tools to new situations involving theory change. The structural preservation takes the form of a categorical equivalence between categories of models of classical and quantum physics. I situate the significance of this structural preservation in terms of prospects for theory construction in quantum physics.

L. A. Lugiato, A. Gatti, K.I. Petsas

Jacob J. Lavin

This work investigates structural constraints imposed on linear dynamical systems by the preservation of inner-product geometry. In particular, we analyze transformations that preserve transition probabilities and show that continuous isometry-preserving flows must be unitary. Closely related to Wigner’s theorem, the paper emphasizes the rigidity of Hilbert-space geometry and the restrictive consequences that follow from requiring distinguishability-preserving evolution. The results are intended as a structural analysis rather than a reconstruction of quantum theory, highlighting how modest geometric assumptions constrain admissible dynamics.