Symmetry of bilinear master equations for a quantum oscillator

TL;DR

This paper explores the full symmetry transformations of bilinear master equations for a quantum oscillator, revealing that more general adjoint-symmetric operators can preserve the physical properties of the system.

Contribution

It introduces a broader class of transformation operators beyond unitary products, expanding the understanding of symmetries in reduced quantum dynamics.

Findings

Transformation operators must be adjoint-symmetric to preserve hermiticity and probability.

More general operators, including mixtures, are necessary to fully describe symmetries.

The valid parameter range is constrained by the positive semidefinite condition of the density operator.

Abstract

We study the most general continuous transformation on the generators of bilinear master equations of a quantum oscillator. We find that transformation operators that preserve the hermiticity of density operators and conserve the probability of reduced dynamics should be adjoint-symmetric, and they are not limited to the pure product of unitary operators in the bra and ket space but could be a mixture of them. We need to include the more general transformation operators to explore the full symmetry of generic reduced dynamics. We discuss how the operators are related to those considered in previous works, and illustrate how they leave the reduced dynamics form invariant, or map one into the other. The positive semidefinite requirement on the density operator can be imposed to give a valid range of transformation parameters.