Finite system-environment and Markov model assignments

Work with finite-dimensional system and environment spaces $H_S={\mathbb{C}}^d$ and $H_E={\mathbb{C}}^m$, where $d,m\ge 1$, using the ordered tensor product $H_S\otimes H_E$ with the system factor first. A system or environment state is a positive semidefinite Hermitian matrix of trace one; these state matrices are dimensionless. At the initial time, for every allowed system state $\rho$, adopt the joint preparation $\rho \otimes \eta$, where the fixed environment state $\eta$ does not depend on $\rho$. For each $t\ge 0$, choose a declared unitary $U_t$ on $H_S\otimes H_E$, meaning $U_t^{\ast }{U}_t=U_t{U}_t^{\ast }=I$, with $U_0=I$. The reduced system dynamics is the map ${\mathrm{\Phi }}_t(\rho )={\mathrm{Tr}}_E\text{\hspace{0.17em}⁣}(U_t(\rho \otimes \eta )U_t^{\ast })$, where for the fixed product basis the finite partial trace has entries $({\mathrm{Tr}}_{E}R{)}_{\alpha \beta }={\sum }_{j=1}^m{R}_{(\alpha ,j),(\beta ,j)}$. This assignment excludes initially correlated joint states. It imposes no one-parameter group, continuity, or semigroup law on $U_t$.

A time-homogeneous finite GKSL model may instead be selected as a separate approximation: choose a fixed finite-dimensional Hamiltonian matrix $H=H^{\ast }$ in joules, a constant $\mathrm{\hslash }>0$ in joule-seconds, and finitely many jump matrices $L_a$ in ${\mathrm{s}}^{-1\mathrm{/}2}$, and use $T_t=e^{t\mathcal{L}}$ for $t\ge 0$ with $\mathcal{L}(X)=-\frac{i}{\mathrm{\hslash }}[H,X]+{\sum }_a(L_{a}X{L}_a^{\ast }-\frac{1}{2}\{L_a^{\ast }{L}_a,X\})$. Here $\mathcal{L}:M_d(\mathbb{C})\to M_d(\mathbb{C})$ is complex-linear, $[A,B]=AB-BA$, $\{A,B\}=AB+BA$, and $A^{\ast }$ is conjugate transpose. The exponential means $e^{t\mathcal{L}}:={\sum }_{n=0}^{\mathrm{\infty }}{t}^n{\mathcal{L}}^n\mathrm{/}n!$, with ${\mathcal{L}}^0=\mathrm{id}$ and powers given by composition. This series converges absolutely in the operator norm induced by the finite matrix trace norm $\mathrm{\parallel }X{\mathrm{\parallel }}_1=\mathrm{Tr}\sqrt{X^{\ast }X}$, as in the finite GKSL semigroup theorem cited below; its hypotheses are exactly the finite Hermitian $H$, positive $\mathrm{\hslash }$, and finite jump family specified here. Each application must state its rates and physical regime. Selecting this approximation is not a consequence of the finite unitary reduction, and this postulate claims no weak-coupling or reservoir limit.

In both alternatives, the parameter $t$ indexing $U_t$ or $T_t$ denotes elapsed physical time measured in SI seconds. Thus $t\mathcal{L}$ in the GKSL exponential is dimensionless, since the adopted finite GKSL generator has units ${\mathrm{s}}^{-1}$.

Partial trace over a finite tensor factor

Finite completely positive trace-preserving maps

Finite GKSL generators and units

Complete finite Choi-Kraus equivalence

Finite GKSL generates CPTP semigroups

The diagram records only the defined finite-map composition. It supplies no proof of complete positivity or trace preservation.

diagram failed to render:
\rho \arrow[r, "{X\mapsto X\otimes\eta}"] \arrow[drr, "\Phi_t"'] &
\rho\otimes\eta \arrow[r, "{R\mapsto U_t R U_t^*}"] &
U_t(\rho\otimes\eta)U_t^* \arrow[d, "\operatorname{Tr}_E"] \\
& & \Phi_t(\rho)