An absorbing two-outcome instrument: amplitude suppression and a branch without an outgoing polarization state

Adopt the following finite detector state and instrument assignments as additional assumptions of this example, using the preparation and instrument formulas of P1-P3 on this explicitly declared effective model; their massive-matter formulation alone does not supply this extension. Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the hypothesis of the mathematical suppliers. Fix a propagation label $k_0\in {\mathbb{R}}^3\setminus \{0\}$ in ${\mathrm{m}}^{-1}$ relative to fixed Euclidean spatial coordinates, and a real orthonormal transverse pair $v_1,v_2$ with $k_0\cdot v_1=k_0\cdot v_2=0$. As an additional finite detector-model assumption, assign $H=\mathbb{C}{e}_0\oplus V\cong {\mathbb{C}}^3$, where $V=\mathbb{C}{v}_1\oplus \mathbb{C}{v}_2$, with orthonormal coordinates $(e_0,e_1,e_2)$, $e_j\leftrightarrow v_j$ for $j=1,2$, and state space $D(H)$ (The state space: density operators, pure and mixed states, and purity, Real and complex inner product spaces, with the inner product linear in the first argument, Hilbert space). A unit vector $\xi$ represents the preparation $T_{\xi ,\xi }$. The level $e_0$ is the declared absorbed/no-photon marker, orthogonal to the polarization plane $V=\mathrm{span}(e_1,e_2)$; it is a bookkeeping level rather than an assigned Fock vacuum. The extension to $H$ and the instrument rule below are adopted assumptions of this effective detector model, not consequences of the massive-matter state assignment or a full radiation dynamics. Angles and all amplitude coordinates are dimensionless, with angles read in radians. Fix a unit vector $v=\cos \gamma e_1+\sin \gamma e_2,v^{\perp }:=-\sin \gamma e_1+\cos \gamma e_2,$ so that $(e_0,v,v^{\perp })$ is an orthonormal basis, and fix a detection parameter $p\in (0,1)$, a dimensionless model parameter. Declare the two-outcome instrument with Kraus family (Declared ideal measurements and instruments of the framework clause (d)) $M_{\mathrm{d}\mathrm{e}\mathrm{t}}:=\sqrt{p}{T}_{e_0,v},M_{\mathrm{n}\mathrm{o}}:=T_{e_0,e_0}+\sqrt{1-p}{T}_{v,v}+T_{v^{\perp },v^{\perp }},$ where $T_{x,y}z=⟨z,y⟩x$ (Rank-one operator calculus), with outcomes $\mathrm{d}\mathrm{e}\mathrm{t}$ and $\mathrm{n}\mathrm{o}$, effects $F_{\mathrm{d}\mathrm{e}\mathrm{t}}=M_{\mathrm{d}\mathrm{e}\mathrm{t}}^{\ast }{M}_{\mathrm{d}\mathrm{e}\mathrm{t}}$, $F_{\mathrm{n}\mathrm{o}}=M_{\mathrm{n}\mathrm{o}}^{\ast }{M}_{\mathrm{n}\mathrm{o}}$, probabilities $p_{\mathrm{d}\mathrm{e}\mathrm{t}}=\mathrm{tr}(\rho F_{\mathrm{d}\mathrm{e}\mathrm{t}})$, $p_{\mathrm{n}\mathrm{o}}=\mathrm{tr}(\rho F_{\mathrm{n}\mathrm{o}})$ and updates $\rho \mapsto M_a\rho M_a^{\ast }\mathrm{/}{p}_a$ for positive probabilities. The example considers preparations $\rho =T_{\psi ,\psi }$ with $\psi =\alpha v+\beta v^{\perp },\mathrm{\mid }\alpha {\mathrm{\mid }}^2+\mathrm{\mid }\beta {\mathrm{\mid }}^2=1,$ so that the preparation lies in the polarization plane, and computes:

(i) The family is normalized, $M_{\mathrm{d}\mathrm{e}\mathrm{t}}^{\ast }{M}_{\mathrm{d}\mathrm{e}\mathrm{t}}+M_{\mathrm{n}\mathrm{o}}^{\ast }{M}_{\mathrm{n}\mathrm{o}}=I$, with effects $F_{\mathrm{d}\mathrm{e}\mathrm{t}}=p{T}_{v,v}$ and $F_{\mathrm{n}\mathrm{o}}=T_{e_0,e_0}+(1-p)T_{v,v}+T_{v^{\perp },v^{\perp }}$; hence $p_{\mathrm{d}\mathrm{e}\mathrm{t}}=p\mathrm{\mid }\alpha {\mathrm{\mid }}^2,p_{\mathrm{n}\mathrm{o}}=1-p\mathrm{\mid }\alpha {\mathrm{\mid }}^2\mathrm{.}$

(ii) When $\alpha \mathrm{\ne }0$, conditioned on the outcome $\mathrm{d}\mathrm{e}\mathrm{t}$, the conditional state is $T_{e_0,e_0}$: the branch carries the absorbed/no-photon record and leaves no outgoing state in the polarization plane. In particular the model does not produce an outgoing one-photon polarization state for the detected branch.

(iii) Conditioned on the outcome $\mathrm{n}\mathrm{o}$, the conditional state is the pure state of the normalized vector $\frac{\sqrt{1-p}\alpha v+\beta v^{\perp }}{\sqrt{(1-p)\mathrm{\mid }\alpha {\mathrm{\mid }}^2+\mathrm{\mid }\beta {\mathrm{\mid }}^2}},$ that is, the $v$-component is amplitude-suppressed by $\sqrt{1-p}$ rather than projected away. For the fully detectable preparation $\psi =v$ ($\alpha =1$) and $p=\frac{1}{2}$ one has $p_{\mathrm{d}\mathrm{e}\mathrm{t}}=p_{\mathrm{n}\mathrm{o}}=\frac{1}{2}$ and the no-detection conditional state is again $T_{v,v}$; a repetition of the instrument on that branch again detects with probability $\frac{1}{2}$, so the no-detection outcome is not repeatable.

(iv) When $\alpha \mathrm{\ne }0$, after the detection branch the state is $T_{e_0,e_0}$, and a repetition gives $p_{\mathrm{d}\mathrm{e}\mathrm{t}}=\mathrm{tr}(T_{e_0,e_0}{F}_{\mathrm{d}\mathrm{e}\mathrm{t}})=p\mathrm{\mid }⟨e_0,v⟩{\mathrm{\mid }}^2=0$: the detection outcome is not repeatable either. The effects $F_{\mathrm{d}\mathrm{e}\mathrm{t}}$ and $F_{\mathrm{n}\mathrm{o}}$ are not projections for $0<p<1$, so the instrument is not a declared ideal sharp measurement; it is a declared general instrument whose update is part of its declaration (Finite effects and instruments: normalization and non-determination of the update clause (d), Declared ideal measurements and instruments of the framework clauses (d)-(f)).

No real detector, efficiency, dark count, dead time, photon number or measured rate is described; the computation is a model computation in the declared three-level space.

• Interpretation of the marker. The level $e_0$ is a bookkeeping record of the branch in which the model detector has absorbed the excitation of the polarization mode. The example is honest about being a declared three-level idealisation: it asserts no photon-number content, no outgoing vacuum field, no detector dynamics and no real absorption mechanism.
• Why this is not a projection. The effects $F_{\mathrm{d}\mathrm{e}\mathrm{t}}$ and $F_{\mathrm{n}\mathrm{o}}$ sum to $I$ and have eigenvalues in $[0,1]$, so they are legitimate effects of a finite-resolution apparatus model; the principal contrast with the projective case is that the update of the no-detection branch suppresses the detected component by the amplitude $\sqrt{1-p}$ rather than projecting it away, and that neither outcome is repeatable.