In many derivations of the Lindblad equation, the authors say something like “There is a gauge freedomA gauge freedom of the Lindblad equation means a transformation we can to both the Lindblad operators and (possibly) the system’s self-Hamiltonian, without changing the reduced dynamics.^{a } in our choice of Lindblad (“jump”) operators that we can use to make those operators traceless for convenience”. However, the nature of this freedom and convenience is often obscure to non-experts.

While reading Hayden & Sorce’s nice recent paper [arXiv:2108.08316] motivating the choice of traceless Lindblad operators, I noticed for the first time that the trace-ful parts of Lindblad operators are just the contributions to Hamiltonian part of the reduced dynamics that arise *at first order* in the system-environment interaction. In contrast, the so-called “Lamb shift” Hamiltonian is second order.

Consider a system-environment decomposition of Hilbert space with a global Hamiltonian , where , , and are the system’s self Hamiltonian, the environment’s self-Hamiltonian, and the interaction, respectively. Here, we have (without loss of generality) decomposed the interaction Hamiltonian into a tensor product of Hilbert-Schmidt-orthogonal sets of operators and , with a real parameter that control the strength of the interaction.

This Hamiltonian decomposition is not unique in the sense that we can alwaysThere is also a similar freedom with the environment in the sense that we can send and .^{b } send and , where is any Hermitian operator acting only on the system. When reading popular derivations of the Lindblad equation

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like in the textbook by Breuer & Petruccione, one could be forgivenSpecifically, I have forgiven myself for doing this…^{c } for thinking that this freedom is eliminated by the necessity of satisfying the assumption that , which is crucially deployed in the “microscopic” derivation of the Lindblad equation operators and from the global dynamics generated by .… [continue reading]