Specified finite GNS detailed balance

Let $d\ge 1$ be finite, let $\mathcal{H}={\mathbb{C}}^d$ with its standard inner product, and write $M_d(\mathbb{C})$ for its matrices. Let $H_{\mathrm{s}}=H_{\mathrm{s}}^{\ast }\in M_d(\mathbb{C})$ be a Hamiltonian in joules, let $\mathrm{\hslash }>0$ have units joule-seconds, and let $m\ge 0$ be an integer with $L_1,\dots ,L_m\in M_d(\mathbb{C})$ a finite family of jump matrices in seconds${}^{-1\mathrm{/}2}$. Matrices on which the generators act are dimensionless. The Schrödinger generator $\mathcal{L}$ is the finite GKSL generator of Finite GKSL generators and units.

Its trace-dual observable generator is

$G(A)=\frac{i}{\mathrm{\hslash }}[H_{\mathrm{s}},A]+D(A),D(A)={\sum }_{j=1}^m(L_j^{\ast }A{L}_j-\frac{1}{2}\{L_j^{\ast }{L}_j,A\})\mathrm{.}$

Here $[U,V]=UV-VU$, $\{U,V\}=UV+VU$, and $A^{\ast }$ is the conjugate transpose. The units of $G$, $D$, and $\mathcal{L}$ are seconds${}^{-1}$. The generator preserves the identity, $D(I)=0$, so $e^{tD}(I)=I$.

Let $\sigma ={\sigma }^{\ast }>0$ be a dimensionless density matrix on $\mathcal{H}$, so $\mathrm{Tr}\sigma =1$. Define

$⟨A,B{⟩}_{\sigma }:=\mathrm{Tr}(\sigma B^{\ast }A),\mathrm{\parallel }A{\mathrm{\parallel }}_{\sigma }:=⟨A,A{⟩}_{\sigma }^{1\mathrm{/}2}\mathrm{.}$

This inner product is linear in its first variable. It is positive definite because $\sigma >0$. The data satisfy the specified finite GNS detailed-balance convention when $[H_{\mathrm{s}},\sigma ]=0$ and $D$ is self-adjoint for this inner product, meaning $⟨D(A),B{⟩}_{\sigma }=⟨A,D(B){⟩}_{\sigma }$ for all $A,B\in M_d(\mathbb{C})$. The Hamiltonian part of $G$ is kept separate and is anti-self-adjoint. Quantum detailed-balance conventions vary; in particular, this definition does not identify GNS symmetry with KMS or Kubo–Mori symmetry.

The cited Alicki–Lendi convention uses the same dissipative self-adjointness condition with the Hamiltonian part separated. Its inner product is written conjugate-linear in the first variable; swapping the two slots and conjugating gives the first-linear convention used here, without changing self-adjointness. Carlen–Maas explicitly distinguish the GNS inner product from the KMS inner product. No equivalence with another detailed-balance convention is assumed.