Continuous monitoring versus pulsed syndrome protocols

Fix a nonzero finite-dimensional complex inner-product system space $H$ and a finite time horizon $[0,T]$, where $T>0$. A pulsed syndrome protocol specifies separated checking times. At checking time $t_j$, for every earlier outcome history $h$, it specifies a finite-dimensional output space $K_{j,h}$, a finite instrument with outcome maps ${\mathcal{I}}_{j,h,a}:\mathcal{L}(H)\to \mathcal{L}(K_{j,h})$, and decoder channels ${\mathcal{D}}_{j,h,a}:\mathcal{L}(K_{j,h})\to \mathcal{L}(H)$ selected using $h$ and the newly observed outcome $a$. Thus the branch operation ${\mathcal{D}}_{j,h,a}\circ {\mathcal{I}}_{j,h,a}$ returns an operator on $H$, the input space for the next checking operation; any intervening evolution in the protocol must also accept and return operators on $H$. The instrument outcome is called a syndrome record; a zero-probability outcome has no assigned conditional state.

For this interface, a continuous-monitoring protocol specifies an effective conditional evolution together with a time-indexed measurement record. For a specified initial density operator, the model supplies a probability space $(\mathrm{\Omega },\mathcal{F},P)$ and, at each $t\in [0,T]$, a measurable record value space $(E_t,{\mathcal{E}}_t)$ and a random element $Y_t:(\mathrm{\Omega },\mathcal{F})\to (E_t,{\mathcal{E}}_t)$. The record may be accumulated continuous-current values or a jump record listing detected event times and their labels up to $t$; the chosen record value spaces and measurable structures are part of the model. Define the available record history by

${\mathcal{G}}_t={\sigma }_{\mathrm{\Omega }}(\{Y_s^{-1}(B):0\le s\le t,B\in {\mathcal{E}}_s\})\mathrm{.}$

This increasing family of sub-sigma-algebras of $\mathcal{F}$ is the history used here, without completion or right-continuous augmentation. Write $\mathcal{D}(H)$ for the density operators on $H$ and equip it with the sigma-algebra ${\mathrm{\Sigma }}_D$ generated by preimages of real Borel sets under the real and imaginary matrix-entry maps in a fixed basis of $H$. The conditional-state rule supplies maps ${\rho }_t:\mathrm{\Omega }\to \mathcal{D}(H)$ measurable from ${\mathcal{G}}_t$ to ${\mathrm{\Sigma }}_D$. Feedback, when present, supplies measurable control value spaces $(U_t,{\mathcal{U}}_t)$ and maps $u_t:\mathrm{\Omega }\to U_t$ measurable from ${\mathcal{G}}_t$ to ${\mathcal{U}}_t$. These measurability requirements mean that states and controls are adapted to the available record history, rather than to unobserved future outcomes. The model specifies the law, initial state and evolution rule linking these objects, the measured coupling, its strength, and the detection efficiency (the modeled fraction of the monitored signal registered). Any additional regularity or stochastic-integration convention needed by that evolution or feedback law is also a model input. A no-feedback model specifies the absence of feedback. This definition asserts neither existence nor uniqueness of a solution of any proposed evolution rule.

Engineered dissipative stabilization is a separate specification of additional generator terms intended to influence the evolution without requiring a measurement record. In the finite GKSL model these terms are specified by their additional coupling matrices and the resulting dissipators. A model may combine stabilization with either protocol.

A comparison as the pulsed interval tends to zero must separately specify a family of pulsed models, the proposed continuous model, a common set $X$ of comparison objects with assignments $x_{\delta },x_{\mathrm{c}}\in X$ representing the pulsed model at interval $\delta >0$ and the continuous model, a topology $\mathcal{T}$ on $X$, and an error observable or performance functional with its domain and codomain specified. No convergence of $x_{\delta }$ to $x_{\mathrm{c}}$ or equality of performance is part of this definition.

Ahn, Doherty and Landahl describe weak-measurement unravellings of a master equation, a conditioned state and measurement record, and record-dependent feedback in their stated Born–Markov setting. Their diffusive example uses one coupling operator. This is an input to the terminology and model distinction above, not a proof that a particular pulsed protocol has a continuous limit.

Given: The stated model data and the mathematical interfaces in F1–F7. The independent monitoring citation supplies terminology, not a general stochastic existence theorem.

Goal: Justify the compatibility and information structure of the specifications without asserting a physical realization or a limiting theorem.

1.1 The output operator space of each ${\mathcal{I}}_{j,h,a}$ is exactly the input operator space of ${\mathcal{D}}_{j,h,a}$, whose output is $\mathcal{L}(H)$. Their composition therefore has the required input for the next operation. Decoder selection uses only recorded outcomes, and conditional states retain the instrument's positive-weight convention. [given, F1, F2]

1.2 Every generating set $Y_s^{-1}(B)$ belongs to $\mathcal{F}$ by record measurability, so F6 gives ${\mathcal{G}}_t\subseteq \mathcal{F}$. For $s\le t$ the generators at $s$ are included among those at $t$, and minimality gives ${\mathcal{G}}_s\subseteq {\mathcal{G}}_t$. The same generated-sigma-algebra construction defines ${\mathrm{\Sigma }}_D$ on the density-operator set using real Borel coordinate tests. Measurability of the supplied ${\rho }_t$ and $u_t$ relative to ${\mathcal{G}}_t$ is the stipulated information constraint; no conditional-expectation construction or solution theorem is inferred. [given, F4, F5, F6, F7]

2.1 The stabilization clause names additional terms within the finite GKSL definition; it does not claim that an arbitrary added linear map is a physical generator. The comparison data put both represented objects in the same set $X$ with a specified topology, so the proposed limiting claim has a defined setting. Neither this specification nor the monitoring citation proves convergence or equality of performance. [given, F3, F7, step 1.1, step 1.2] ∎