Instantaneous Sobolev Regularization for Dissipative Bosonic Dynamics

TL;DR

This paper demonstrates that certain dissipative quantum evolutions on bosonic systems instantly regularize states, providing new analytic tools for stability and error correction in quantum computation.

Contribution

It introduces a class of Lindblad operators with polynomial creation and annihilation terms that induce immediate Sobolev regularization in bosonic quantum dynamics.

Findings

States become infinitely regular in all moments instantly under the dynamics.

Explicit trace norm convergence estimates for bosonic cat codes.

Improved bounds on exponential convergence to fixed points.

Abstract

We investigate quantum Markov semigroups on bosonic Fock space and identify a broad class of infinite-dimensional dissipative evolutions that exhibit instantaneous Sobolev-regularization. Motivated by stability problems in quantum computation, we show that for certain Lindblad operators that are polynomials of creation and annihilation operators, the resulting dynamics immediately transform any initial state into one with finite expectation in all powers of the number operator. A key application is in the bosonic cat code, where we obtain explicit estimates in the trace norm for the speed of convergence. These estimates sharpen existing perturbative bounds at both short and long times, offering new analytic tools for assessing stability and error suppression in bosonic quantum information processing. For example, we improve the strong exponential convergence of the (shifted) $2$-photon dissipation to its fixed point to the uniform topology.