Specified effective GKSL bath and instrument assumption

Let $d\ge 1$ be finite and let the modeled subsystem Hilbert space be $\mathcal{H}={\mathbb{C}}^d$. A state is a dimensionless matrix $\rho ={\rho }^{\ast }\in M_d(\mathbb{C})$ such that $x^{\ast }\rho x\ge 0$ for every $x\in {\mathbb{C}}^d$ and $\mathrm{Tr}\rho =1$. An initial preparation is a chosen state ${\rho }_0$. Let $H_{\mathrm{s}}=H_{\mathrm{s}}^{\ast }\in M_d(\mathbb{C})$ have units joules, let $\mathrm{\hslash }>0$ have units joule-seconds, and let $L_1,\dots ,L_m\in M_d(\mathbb{C})$, $m\ge 0$ finite, have units seconds${}^{-1\mathrm{/}2}$. We adopt the time-independent finite GKSL generator of Finite GKSL generators and units,

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

where $[U,V]=UV-VU$ and $\{U,V\}=UV+VU$. The exact state law is

$\frac{d}{dt}{\rho }_t=\mathcal{L}({\rho }_t),{\rho }_{t=0}={\rho }_0,{\rho }_t=e^{t\mathcal{L}}{\rho }_0,0\le t\le t_{\mathrm{r}\mathrm{e}\mathrm{s}}\mathrm{.}$

Here $t_{\mathrm{r}\mathrm{e}\mathrm{s}}\in (0,\mathrm{\infty })$ is the finite resolved duration, in seconds, adopted for this model. For mathematical long-time calculations, the same constant generator defines the continuation $e^{t\mathcal{L}}{\rho }_0$ for every $t\ge 0$; this continuation makes no physical-applicability claim for $t>t_{\mathrm{r}\mathrm{e}\mathrm{s}}$. By Finite GKSL generates CPTP semigroups it is a density-state evolution on that mathematical continuation as well.

Let $k_{\mathrm{B}}>0$ denote Boltzmann's constant, with SI units joules per kelvin; it converts absolute temperature to an inverse-energy parameter. Let $T_{\mathrm{b}\mathrm{a}\mathrm{t}\mathrm{h}}>0$ be the adopted absolute temperature parameter in kelvins and set $\beta :=(k_{\mathrm{B}}{T}_{\mathrm{b}\mathrm{a}\mathrm{t}\mathrm{h}}{)}^{-1}$, with units joules${}^{-1}$. The temperature and rates are model inputs here; no measured calibration is asserted. The temperature parameter alone imposes no detailed-balance condition on an arbitrary choice of $L_j$. When the generator is specialized to the connected Gibbs-jump model, its energies, $\beta$, and rates must satisfy the hypotheses in Connected Gibbs jump generators satisfy detailed balance.

Within this effective formulation, the dimension $d$, preparation ${\rho }_0$, general coefficient matrices $H_{\mathrm{s}}$ and $L_j$, positive constants $\mathrm{\hslash }$ and $k_{\mathrm{B}}$, temperature $T_{\mathrm{b}\mathrm{a}\mathrm{t}\mathrm{h}}$, and resolved duration $t_{\mathrm{r}\mathrm{e}\mathrm{s}}$ are primitive, adopted model inputs with the types and units stated above; none is derived here from a microscopic reservoir or inferred from a measurement. This status is local to this formulation, not a claim that each is fundamental in every physical theory. The space ${\mathbb{C}}^d$, inverse-energy parameter $\beta$, generator $\mathcal{L}$, and state family ${\rho }_t$ are derived by their displayed constructions once those inputs and the physical state-law postulate are fixed. The selected Hermitian observable $A$ and its unit assignment are additional adopted observable data; its expectation is derived by the trace formula below.

In the connected Gibbs-jump parameterization, the orthonormal energy basis, real level energies $E_a$ in joules, and rates $w_{ab}\ge 0$ in seconds${}^{-1}$ are instead the adopted primitive spectral and rate data, subject to the Gibbs-balance and connectivity hypotheses of Connected Gibbs jump generators satisfy detailed balance. In that parameterization the operator coefficients are derived as $H_{\mathrm{s}}={\sum }_a{E}_a\mathrm{\mid }a⟩⟨a\mathrm{\mid }$ and $L_{ab}=\sqrt{w_{ab}}\mathrm{\mid }b⟩⟨a\mathrm{\mid }$; they replace, rather than supplement, the independent general matrix choices above. The Gibbs weights ${\pi }_a=e^{-\beta E_a}\mathrm{/}{\sum }_b{e}^{-\beta E_b}$ are derived, while the energy-resolved instrument remains a separate adopted assumption. In particular, neither the temperature nor the Gibbs-balance relation derives the numerical rates.

For that Gibbs-jump specialization, record predictions additionally adopt the particular ideal energy-resolved instrument defined in Specified energy-jump instrument realizes a finite chain, and only on $0\le t\le t_{\mathrm{r}\mathrm{e}\mathrm{s}}$ as a physical model. For every allowed initial density ${\rho }_0$, define $q_a:=⟨a\mathrm{\mid }{\rho }_0\mathrm{\mid }a⟩\ge 0$, so ${\sum }_a{q}_a=1$. The full label path is initialized by $\mathrm{Pr}(a(0)=a)=q_a$ and, conditional on this label, follows the supplier's holding and destination laws. Before the first click this label is an auxiliary variable for the record law, not a claim that the conditional quantum state is an energy eigenstate or that an additional projective measurement has replaced ${\rho }_0$ by its diagonal part. More precisely, with the supplier's diagonal no-click matrix $K$, the unnormalized no-click density is $N_t=e^{tK}{\rho }_0{e}^{t{K}^{\ast }}$ and its trace is $S(t)={\sum }_a{q}_a{e}^{-r_{a}t}>0$; conditional on no click the quantum state is $N_t\mathrm{/}S(t)$. The first click $a\to b$ has time density $w_{ab}{q}_a{e}^{-r_{a}t}$ and, when this density is positive, resets the quantum state to $\mathrm{\mid }b⟩⟨b\mathrm{\mid }$, because $L_{ab}{N}_t{L}_{ab}^{\ast }=w_{ab}{q}_a{e}^{-r_{a}t}\mathrm{\mid }b⟩⟨b\mathrm{\mid }$. Zero-density branches are not normalized. Thus the no-click survival and every labelled click law agree with the stated initial-label mixture and its subsequent supplier chains, while the no-click quantum state retains any initial coherences. For a diagonal Gibbs preparation, $q=\pi$. Its law for all $t\ge 0$ is used only as a mathematical continuation when deriving a long-time coefficient. A one-time channel family alone does not select this record law. Full The Axiom of Choice is retained as the formal set-theoretic premise in the monitored-chain supplier's construction of the joint path law; it is not a physical postulate about nature. A selected observable is represented by a Hermitian matrix $A\in M_d(\mathbb{C})$ with its assigned physical units, and its model expectation $\mathrm{Tr}({\rho }_{t}A)$ has those same units; no measurement outcome is reported.

This postulate distinguishes the adopted effective bath model from derived finite-dimensional consequences. It does not provide measured rates or uncertainties from an experiment. If rates later come from a measurement, its apparatus, calibration, sampling conditions, and uncertainty must be stated as empirical premises in a separate appropriately scoped item.