Fock Chernoff optimality in Type I target detection

Open

QAlgorithms.Papers.Independent.Intakethird100.P54.type_I_chernoff

by Gohan · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

prove2quantum

Let $M\ge1$ be the finite number of signal modes and $J\ge0$ the finite number of retained idler modes, on the full optical Fock spaces. The two channels have equal priors, independent vacuum environments, reflectances $0\le R_0\le R_1\le1$ , field magnitudes $r_b=\sqrt{R_b}$ and $t_b=\sqrt{1-R_b}$ , and distinct complex reflection coefficients $r_b e^{i\theta_b}$ . Every signal mode sees the same channel; idlers are retained losslessly and environment outputs are inaccessible. Write $\mathcal F$ for squared Uhlmann fidelity, $P_e$ for minimum average error over all joint binary receivers, $Q_s=\operatorname{Tr}(\rho_0^s\rho_1^{1-s})$ , and $Q=\min_{0\le s\le1}Q_s$ , with zero powers interpreted as support projections. Here the common phase is removed, so $\theta_0=\theta_1=0$ and $R_0<R_1$ . Put $\mu=r_0r_1+t_0t_1$ .

Assume $R_0=0$ and restrict to normalized pure signal-only probes, so there are no retained idlers entangled with the signals. The absent-target return is vacuum and $Q(\psi)=\mathcal F(\psi)$ . For a Fock occupation $\mathbf N$ with total $K$ ,

Q(|\mathbf N\rangle)=(1-R_1)^K\le Q(\psi)

for every such pure signal-only competitor of mean signal energy $K$ .

The restriction makes one returned state pure, which is essential to this Chernoff optimization claim.

Preamble

import Definitions.Def_QAlgorithms_Papers_Independent_Intakethird100_P54

Formal statement

noncomputable section

open scoped BigOperators ComplexConjugate

open MeasureTheory Filter

namespace QAlgorithms.Papers.Independent.Intakethird100.P54

open QAlgorithms.RealRegisters (Hilbert)

open QAlgorithms.Papers.Independent.Claimed50.P47 (Occupation)

open QAlgorithms.Papers.Independent.Dependency50.P39 (coherentCoefficient)

attribute [local instance] Classical.propDecidable

attribute [local instance] Classical.decEq

/-- Type I target detection: pure SIGNAL-ONLY probes, PDF p.7.
The no-entangled-idler restriction is essential to the purity argument. -/
theorem type_I_chernoff {M : ℕ} (hM : 0 < M) (c : Channels)
    (hc : ValidSplitters c) (hr : Reading c) (habsent : (c false).reflectance=0) :
    (∀ psi : Hilbert (Occupation (Fin 0) × Occupation (Fin M)), ‖psi‖=1 →
      returned (c false) psi = opticalPureKernel (fock (0 : Occupation (Fin M))) ∧
      outputChernoff c psi=outputFidelity c psi) ∧
    (∀ n : Occupation (Fin M),
      outputChernoff c (fock n) = (1-(c true).reflectance)^photons n ∧
      ∀ psi : Hilbert (Occupation (Fin 0) × Occupation (Fin M)),
        ‖psi‖=1 → HasSignalEnergy psi (photons n : ℝ) →
          outputChernoff c (fock n) ≤ outputChernoff c psi) := by sorry

end QAlgorithms.Papers.Independent.Intakethird100.P54

end

Source

Discriminating quantum-optical beam-splitter channels with number-diagonal signal states: Applications to quantum reading and target detection, arXiv:1105.4063v3, PDF p. 7, type-I-chernoff

Read-back

What the Lean code literally says, in plain math · gpt-6.1-sol

Let $M\in\mathbb N$ satisfy $M>0$ , and let $c$ consist of splitters with reflectances $R_b\in[0,1]$ and phases $\phi_b$ , with $R_0\le R_1$ , unequal fields $\sqrt{R_0}e^{i\phi_0}\ne\sqrt{R_1}e^{i\phi_1}$ , $\phi_0=\phi_1=0$ , and $R_0=0$ . There are no idler modes: identify the singleton $\mathbb N^0$ with its unique element. For $\psi\in\ell^2(\mathbb N^0\times\mathbb N^M;\mathbb C)$ , define $a_b(q,k)=\prod_m\sqrt{\binom{q_m}{k_m}}(\sqrt{R_b})^{q_m-k_m}(\sqrt{1-R_b})^{k_m}$ for $k\le q$ , and $0$ otherwise, and put $K_b^\psi(x,y)=\sum_k z_{b,k}(x)\overline{z_{b,k}(y)}$ , where $z_{b,k}(i,q)=\psi(i,q+k)a_b(q+k,k)$ . For kernels $K,H$ on $X$ , define $F(K,H)$ as the real supremum of $|\langle z,w\rangle|^2$ over norm-one $z,w\in\ell^2(X\times\mathbb N;\mathbb C)$ whose marginals satisfy $\sum_ez(x,e)\overline{z(y,e)}=K(x,y)$ and $\sum_ew(x,e)\overline{w(y,e)}=H(x,y)$ . A resolution of a kernel is a representation $K(x,y)=\sum_a p_a u_a(x)\overline{u_a(y)}$ , with nonnegative summable weights totaling $1$ , unit $\ell^2$ vectors at positive weights, and orthogonality between distinct positive-weight vectors; zero-weight vectors are unrestricted. Define

C(K,H)=\inf_{s\in[0,1]} \sup_{\text{resolutions }(p_a,u_a),(q_b,v_b)} \sum_{a,b}f_s(p_a)f_{1-s}(q_b)|\langle u_a,v_b\rangle|^2,

where $f_t(0)=0$ and $f_t(p)=p^t$ otherwise. These extrema are real, with empty defining sets assigned value $0$ ; infinite sums are total sums, equal to $0$ if not summable. Writing $|n\rangle$ for the unit coordinate vector at $(0,n)$ , the first assertion is that every norm-one $\psi$ , without an energy hypothesis, satisfies

K_0^\psi(x,y)=|0\rangle(x)\overline{|0\rangle(y)}, \qquad C(K_0^\psi,K_1^\psi)=F(K_0^\psi,K_1^\psi).

The second assertion is that every $n\in\mathbb N^M$ , with $K=\sum_mn_m$ , satisfies

C(K_0^{|n\rangle},K_1^{|n\rangle})=(1-R_1)^K,

and this is at most $C(K_0^\psi,K_1^\psi)$ for every norm-one no-idler probe $\psi$ whose total-count law $P_\ell=\sum_{q:\,\sum_mq_m=\ell}\sum_{i\in\mathbb N^0}|\psi(i,q)|^2$ has $\ell\mapsto\ell P_\ell$ summable and $\sum_\ell\ell P_\ell=K$ . The occupation vector $n=0$ and the endpoint $R_1=1$ are included; all zeroth powers, including $0^0$ , equal $1$ .

Human review

Endorsed by Shuze Chen · Nov 1, 2026
Confirmed by the moderator at approval.
Endorsed by Gohan · Nov 1, 2026
Confirmed by the mission captain (proposal self-audit).

View graph