T101 applies the conjugate-half mixing framework of T100 to a two-output interference experiment and derives the resulting structure of observable probability corrections. While T100 established that weak mixing between the observable sector and its conjugate counterpart produces first-order corrections constrained by total-probability conservation, T101 specializes those corrections to interference measurements and determines the form that any physically admissible deviation must take. The starting point is the standard two-output Born readout \[P_+^{(0)}=\cos^2\theta,\qquadP_-^{(0)}=\sin^2\theta,\] where \(\theta\) parameterizes the interferometer configuration and \(\phi\) denotes the relative phase. From T100, observable probabilities in the presence of weak conjugate-half mixing satisfy \[P_\pm^{\mathrm{obs}}=P_\pm^{(0)}+\eta \Delta_\pm+O(\eta^2),\] together with the conservation law \[\Delta_++\Delta_-=0.\] The theorem shows that, in a two-output experiment, conservation immediately forces an antisymmetric correction structure: \[\Delta_+=-\Delta_-=f(\theta,\phi),\] yielding \[P_+^{\mathrm{obs}}=\cos^2\theta+\eta f(\theta,\phi)+O(\eta^2),\] \[P_-^{\mathrm{obs}}=\sin^2\theta-\eta f(\theta,\phi)+O(\eta^2).\] Thus, conjugate-half mixing cannot appear as an unconstrained additive deviation. Any increase in one output channel must be accompanied by an equal decrease in the other. The theorem next derives positivity constraints on the correction function. Since probabilities must remain between zero and one, the correction cannot remain finite at the interference endpoints \[\theta=0,\qquad\theta=\frac{\pi}{2},\] where one channel has unit probability and the other has zero probability. Consequently, \[f(0,\phi)=0,\qquadf\!\left(\frac{\pi}{2},\phi\right)=0,\] or must be suppressed sufficiently rapidly near those points. A minimal admissible envelope satisfying these constraints is \[\alpha(\theta)=\sin(2\theta),\] which vanishes at both endpoints while remaining positive throughout the interior of the interference interval. Incorporating the leading phase dependence yields the simplest structured correction \[f(\theta,\phi)=c\,\sin(2\theta)\cos\phi,\] where \(c\) is a coupling parameter describing the strength of the projected conjugate-half contribution. Substitution gives the minimal interference-correction model \[P_+^{\mathrm{obs}}=\cos^2\theta+\eta c\,\sin(2\theta)\cos\phi+O(\eta^2),\] \[P_-^{\mathrm{obs}}=\sin^2\theta-\eta c\,\sin(2\theta)\cos\phi+O(\eta^2).\] The theorem therefore predicts that any observable signature of conjugate-half mixing must appear as antisymmetric redistribution between interference outputs rather than as excess total probability. The characteristic experimental signal is \[\delta P_+=-\delta P_-,\] with the correction vanishing near the interference endpoints and reaching maximal magnitude in the interior of the interference pattern. T101 serves as the interference-level realization of the conjugate-half mixing framework. T100 established the general conservation law governing first-order Born corrections. T101 translates that abstract constraint into a concrete interference prediction, showing that weak conjugate-half mixing manifests as differential redistribution of probability across interference channels while preserving total probability to first order.
T101 applies the conjugate-half mixing framework of T100 to a two-output interference experiment and derives the resulting structure of observable probability corrections. While T100 established that weak mixing between the observable sector and its conjugate counterpart produces first-order corrections constrained by total-probability conservation, T101 specializes those corrections to interference measurements and determines the form that any physically admissible deviation must take. The starting point is the standard two-output Born readout [P_+^{(0)}=\cos^2\theta,\qquadP_-^{(0)}=\sin^2\theta,] where (\theta) parameterizes the interferometer configuration and (\phi) denotes the relative phase. From T100, observable probabilities in the presence of weak conjugate-half mixing satisfy [P_\pm^{\mathrm{obs}}=P_\pm^{(0)}+\eta \Delta_\pm+O(\eta^2),] together with the conservation law [\Delta_++\Delta_-=0.] The theorem shows that, in a two-output experiment, conservation immediately forces an antisymmetric correction structure: [\Delta_+=-\Delta_-=f(\theta,\phi),] yielding [P_+^{\mathrm{obs}}=\cos^2\theta+\eta f(\theta,\phi)+O(\eta^2),] [P_-^{\mathrm{obs}}=\sin^2\theta-\eta f(\theta,\phi)+O(\eta^2).] Thus, conjugate-half mixing cannot appear as an unconstrained additive deviation. Any increase in one output channel must be accompanied by an equal decrease in the other. The theorem next derives positivity constraints on the correction function. Since probabilities must remain between zero and one, the correction cannot remain finite at the interference endpoints [\theta=0,\qquad\theta=\frac{\pi}{2},] where one channel has unit probability and the other has zero probability. Consequently, [f(0,\phi)=0,\qquadf!\left(\frac{\pi}{2},\phi\right)=0,] or must be suppressed sufficiently rapidly near those points. A minimal admissible envelope satisfying these constraints is [\alpha(\theta)=\sin(2\theta),] which vanishes at both endpoints while remaining positive throughout the interior of the interference interval. Incorporating the leading phase dependence yields the simplest structured correction [f(\theta,\phi)=c,\sin(2\theta)\cos\phi,] where (c) is a coupling parameter describing the strength of the projected conjugate-half contribution. Substitution gives the minimal interference-correction model [P_+^{\mathrm{obs}}=\cos^2\theta+\eta c,\sin(2\theta)\cos\phi+O(\eta^2),] [P_-^{\mathrm{obs}}=\sin^2\theta-\eta c,\sin(2\theta)\cos\phi+O(\eta^2).] The theorem therefore predicts that any observable signature of conjugate-half mixing must appear as antisymmetric redistribution between interference outputs rather than as excess total probability. The characteristic experimental signal is [\delta P_+=-\delta P_-,] with the correction vanishing near the interference endpoints and reaching maximal magnitude in the interior of the interference pattern. T101 serves as the interference-level realization of the conjugate-half mixing framework. T100 established the general conservation law governing first-order Born corrections. T101 translates that abstract constraint into a concrete interference prediction, showing that weak conjugate-half mixing manifests as differential redistribution of probability across interference channels while preserving total probability to first order.
T104 computes the explicit observable interference correction generated by Q5 leakage-return transport. T101 established the admissible structure of a hidden-half correction term but left the correction function unspecified. T102 derived the leakage-return operator C=L†BL, and T103 identified the leakage generator B=iAL from the explicit Q5 leakage adjacency. T104 completes the chain by directly computing the resulting interference correction. The central structural result is that first-order mixing vanishes identically: L†BL=0. This occurs because the leakage generator transports amplitude through the helical chain e3↔e4↔e5, while the observable return map extracts only the boundary modes e3 and e5. A single application of B moves amplitude into the middle gate-null mode e4, preventing any first-order return contribution. Observable mixing therefore begins only at second order. The first nontrivial return operator is C2=L†B2L=−801(I−iσy), which is Hermitian and generates the leading observable correction. Applying the T100 renormalization procedure yields normalized probability shifts Δ+=−160sinϕsin(4θ),Δ−=+160sinϕsin(4θ), with Δ++Δ−=0. The resulting observable probabilities are P+obs=cos2θ−2η2160sinϕsin(4θ)+O(η3),P−obs=sin2θ+2η2160sinϕsin(4θ)+O(η3). The correction is therefore second order in the leakage coupling and possesses a characteristic angular envelope sin(4θ). This replaces the earlier minimal placeholder structure of T101 and introduces a richer node pattern, with nulls at θ=0,4π,2π, together with maxima at θ=8π,83π. The phase dependence is proportional to sinϕ, arising from the e±iπ/4 phases contained in the leakage generator AL. This shifts the correction into quadrature relative to the leading Born interference fringe. T104 establishes that the Q5 leakage-return mechanism produces a fully explicit interference correction that is antisymmetric between output channels, conserves total probability, vanishes at the required endpoints, and emerges naturally from the two-step transport structure of the leakage chain. The theorem replaces the phenomenological correction function of T101 with a direct Q5-derived prediction and identifies sinϕsin(4θ) as the characteristic observable signature of leakage-return transport.
This note proposes a simple, falsifiable interferometric test for structured deviations from standard two-path interference. In conventional interference, the observed intensity is described by a single effective phase parameter: I(ϕ)=P1+P2+2P1P2cosϕ, and deviations are typically attributed to visibility loss or phase offset. The present protocol tests for a different class of behaviour: phase-locked deviations that depend on the internal structure of the phase profile applied to one arm of the interferometer. A controlled phase modulation of the form ϕ(x)=n⋅f(x)\phi(x) = n \cdot f(x) ϕ(x)=n⋅f(x) is introduced in one arm, where nn n indexes the complexity of the imposed phase structure. The resulting interference signal is measured as a function of the global phase and decomposed into quadrature components: In(ϕ)=Cn+Vn[Ancosϕ+Bnsinϕ]. A deviation metric is defined relative to the baseline (flat-phase) case: Δn=(An−A0)2+(Bn−B0)2. The testable prediction is that, if structured phase-dependent effects are present, then: the baseline satisfies Δ0≈0\Delta_0 \approx 0 Δ0≈0; the deviation Δn\Delta_n Δn increases systematically with the imposed phase structure, and the deviation remains phase-locked in the quadrature basis and cannot be absorbed into a simple phase offset. The protocol includes explicit control conditions to distinguish such behaviour from conventional effects: symmetric modulation of both arms should eliminate the deviation; random phase noise should reduce visibility but not produce systematic scaling in Δn\Delta_n Δn; and phase referencing is required to resolve the quadrature structure. The outcome is binary. If no structured deviation is observed, the hypothesis is falsified. If a phase-locked, structure-dependent deviation is observed, then the standard interference model is incomplete under these conditions. This proposal is intentionally model-independent. While motivated by broader work on structured phase transport in discrete systems, the experimental test does not rely on that framework and can be evaluated directly.
T102 derives the first-principles origin of the conjugate-half mixing operator introduced abstractly in T100–T101. While the earlier theorems established that weak mixing between the observable sector and its conjugate counterpart produces first-order corrections to Born probabilities, the mixing map itself remained an assumed object. T102 removes that assumption by deriving the operator directly from the leakage-return dynamics of the extended Q5 transport system. The starting point is the leakage architecture developed in T93–T96. The extended state space is Hext=C2⊕HL, with a coupled generator G=(AL−L†B), where K⊥=I−ΠY is the leakage-complement projector introduced in T96, while L:C2→HL is the leakage injection map into the leakage sector. The operator B governs circulation within the leakage sector. From T94, leakage-return dynamics are described by R(t)=L†etBLψ. Expanding the leakage evolution operator, etB=I+tB+O(t2), gives R(t)=L†Lψ+tL†BLψ+O(t2). The theorem identifies the first nontrivial mixing contribution as the circulation-induced term tL†BLψ. Introducing the coupling scale η∼t, The effective observable state becomes ψeff=ψ+ηCψ+O(η2), with the derived mixing operator C=L†BL. The theorem, therefore, demonstrates that conjugate-half mixing is not an independent postulate. It emerges naturally from the sequence: leakageLsurface circulationBreturnL†observable correction. A key structural property follows immediately. Since B†=−B, The mixing operator satisfies C†=−C. The operator is therefore anti-Hermitian, implying ℜ⟨ψ,Cψ⟩=0 and ensuring preservation of normalization to first order: ∥ψ+ηCψ∥2=∥ψ∥2+O(η2). The theorem next establishes compatibility with the Born-correction framework of T100. Substituting ψeff=ψ+ηCψ+O(η2) into the observable readout yields Δi=2ℜ⟨Πiψ,ΠiCψ⟩, which reproduces the first-order correction structure previously introduced in the conjugate-half mixing program. The abstract operator C of T100–T101 is therefore identified explicitly with the leakage-return operator L†BL. An important stability result also follows. If a state is perfectly gate-aligned, ΠYψ=ψ, then K⊥ψ=0. Since L injects only the leakage component, \[K_\perp\psi=0\quad\Rightarrow\quadL\psi=0.\] Therefore Cψ=L†BLψ=0. Perfectly admitted states receive no conjugate-half mixing correction. Mixing arises only from leakage away from the gate-admissible mode. Several conceptual consequences follow. The mixing operator is revealed as a composite object consisting of three distinct stages: leakage injection (L), leakage-sector circulation (B), and return transport (L†). No new free parameter is introduced at this stage. All structures arise from previously established operators within the leakage architecture. T102 serves as the derivational bridge between the leakage program and the Born-correction program. T94 introduced residual return dynamics. T100–T101 introduced observable mixing corrections. T102 demonstrates that both descriptions arise from the same underlying mechanism, establishing C=L†BL as the first-principles origin of conjugate-half mixing within the extended Q5 framework.
In a two-particle interferometer, one can study the variation of both single- and joint-detection probabilities as functions of the phase shifts of the beams. By combining the usual definition for one-particle fringe visibility vi (i=1,2) with a reasonable proposed definition for two-particle fringe visibility v12, we show that vi2+v122\ensuremath{\le}1 or, equivalently, viv12\ensuremath{\le}1/2. Some extensions are discussed.
We begin with a study of operations and the effects they measure. We define the probability that an effect a occurs when the system is in a state ρ by Pρ(a)=Tr(ρa). If Pρ(a)=0 and I is an operation that measures a, we define the conditional probability of an effect b given a relative to I by Pρ(b∣a)=Tr[I(ρ)b]/Pρ(a). We characterize when Bayes' quantum second rule Pρ(b∣a)=Pρ(a)Pρ(b)Pρ(a∣b) holds. We then consider Lüders and Holevo operations. We next discuss instruments and the observables they measure. If A and B are observables and an instrument I measures A, we define the observable B conditioned on A relative to I and denote it by (B∣A). Using these concepts, we introduce Bayes' quantum first rule. We observe that this is the same as the classical Bayes' first rule, except it depends on the instrument used to measure A. We then extend this to Bayes' quantum first rule for expectations. We show that two observables B and C are jointly commuting if and only if there exists an atomic observable A such that B=(B∣A) and C=(C∣A). We next obtain a general uncertainty principle for conditioned observables. Finally, we discuss observable conditioned quantum entropies. The theory is illustrated with many examples. Quanta 2024; 13: 1–10.
T103 identifies the explicit local leakage-sector circulation generator required by the leakage-return architecture developed in T93–T102. T102 established that observable conjugate-half mixing arises through the factorization C=L†BL, where L injects non-admitted crossing amplitude into the leakage sector, B governs leakage-sector circulation, and L† returns circulated amplitude to the observable sector. However, T102 left the circulation generator B unspecified. T103 supplies its local Q5 form. The theorem draws upon the leakage-generator development established later in the theorem chain. The local leakage sector is modelled by a three-mode basis (e3,e4,e5), where e3 and e5 are return-capable boundary modes and e4 is a balancing intermediary mode. The resulting local leakage adjacency has chain structure e3↔e4↔e5, with no direct coupling between e3 and e5. The corresponding local leakage adjacency operator is AL=8010e−iπ/40eiπ/40e−iπ/40eiπ/40. The theorem verifies that AL is Hermitian: AL†=AL. The leakage-sector circulation generator is then defined by B=iAL. Since AL is Hermitian, the circulation generator satisfies B†=−B, and is therefore anti-Hermitian. A direct consequence is that leakage evolution etB is unitary: (etB)†etB=I, showing that leakage circulation preserves norm. Leakage is therefore interpreted as conservative circulation rather than dissipative loss. The theorem next examines the transport structure encoded by AL. Because the matrix entries (AL)13=(AL)31=0, there is no direct coupling between the two boundary modes e3 and e5. All transport between them must pass through the middle mode e4. The middle mode, therefore, acts as a balancing mediator within the leakage sector. This structural restriction has dynamical consequences. The first-order leakage evolution etB=I+tB+O(t2) contains no direct boundary-to-boundary transfer because B13=B31=0. Consequently, transfer between e3 and e5 first appears through the second-order pathway \[e_3\rightarrowe_4\rightarrowe_5.\] Boundary return is therefore intrinsically mediated and delayed relative to local circulation. Substituting the derived generator into the T102 mixing formula yields C=L†(iAL)L. This provides the first explicit local realization of the conjugate-half mixing operator. The mixing process is no longer governed by an unspecified circulation generator but by a concrete leakage adjacency derived from the Q5 leakage architecture. Several conceptual consequences follow. Leakage transport is mediated rather than direct. Circulation preserves norm and therefore represents redistribution rather than loss. The middle leakage mode plays an essential balancing role. Most importantly, the observable mixing operator inherits its structure from a specific leakage-sector geometry rather than from an arbitrary correction term. T103 serves as the bridge between the abstract leakage-return factorization of T102 and the explicit interference calculations that follow. T102 established the form C=L†BL. T103 identifies the local generator B=iAL. Together, these results provide the explicit operator C=L†(iAL)L, which forms the starting point for the observable interference-correction analysis developed in T104. Epistemic status: Solid: The Hermiticity of AL, anti-Hermiticity of B=iAL, norm preservation of etB, absence of direct e3↔e5 coupling, and second-order nature of boundary-to-boundary transfer follow directly from the displayed local leakage matrix. Conditional: The normalization factor 1/80, the identification of (e3,e4,e5) with the physical leakage sector, and the embedding of the local leakage block into the full Q5 architecture depend upon the later global embedding results and remain conditional pending completion of the full leakage-sector construction.