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Coherent and incoherent two-path bin laws and the marker overlap

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $(D, A, d x)$ be a $σ$ -finite measure space, let $L^{2} (D)$ be the complex Hilbert space of The space $L^{p} (μ)$ as the quotient by null functions and Complex Lp classes and Euclidean test-function conventions, and let $a, b \in L^{2} (D)$ .

(a) Coherent law. If $a + b \neq 0$ as a class, i.e. $∥ a + b ∥ > 0$ , then $p_{c o h} (B) : = \frac{\int_{B} ∣ a + b ∣^{2} d x}{∥ a + b ∥^{2}}, B \in A,$ defines a probability measure on $(D, A)$ (Measures on sigma-algebras, Probability measures and probability spaces), and for every $B$ $\int_{B} ∣ a + b ∣^{2} d x = \int_{B} ∣ a ∣^{2} d x + \int_{B} ∣ b ∣^{2} d x + 2 Re \int_{B} a \overline{b} d x .$

(b) Incoherent law. If $a \neq 0$ , $b \neq 0$ and $w \in [0, 1]$ , then $p_{m i x} (B) : = w \frac{\int_{B} ∣ a ∣^{2} d x}{∥ a ∥^{2}} + (1 - w) \frac{\int_{B} ∣ b ∣^{2} d x}{∥ b ∥^{2}}$ defines a probability measure on $(D, A)$ .

(c) Signed difference. Let $a, b$ be unit vectors with $Re ⟨ a, b ⟩ \neq - 1$ , and put $N (B) : = \int_{B} (∣ a ∣^{2} + ∣ b ∣^{2}) d x, c (B) : = Re \int_{B} a \overline{b} d x, p_{c o h} (B) : = \frac{N (B) + 2 c (B)}{2 + 2 c (D)}, p_{m i x} (B) : = \frac{N (B)}{2} .$ Then for every $B$ , $N (B) - 2 c (B) = \int_{B} ∣ a - b ∣^{2} d x \geq 0$ , and $p_{c o h} (B) - p_{m i x} (B) = \frac{2 c (B) - c (D) N (B)}{2 (1 + c (D))} .$ Equality $p_{c o h} (B) = p_{m i x} (B)$ for every measurable bin holds exactly when $2 Re (a \overline{b}) = c (D) (∣ a ∣^{2} + ∣ b ∣^{2})$ almost everywhere. In particular, if $a ⊥ b$ then $c (D) = 0$ and the signed difference is $c (B)$ , so orthogonality alone does not imply equality on each bin. If $b = a$ then equality holds on every bin. Coherence of the preparation changes the bin law according to the amplitudes and their actual normalization.

(d) Marker overlap. Let $K$ be a complex Hilbert space (Hilbert space) with unit vectors $e_{1}, e_{2}$ , let $α, β \in C$ with $∣ α ∣^{2} + ∣ β ∣^{2} = 1$ , let $a, b$ be unit vectors and put $Ψ : = α a \otimes e_{1} + β b \otimes e_{2}$ in the completed tensor product $L^{2} (D) \hat{\otimes} K$ . The reduced weight $ρ : = {Tr}_{K} T_{Ψ, Ψ}$ is defined even when $Ψ = 0$ , with $tr ρ = ∥ Ψ ∥^{2}$ . If $Ψ \neq 0$ , $B \in A$ has finite measure and $P_{B}$ is multiplication by $1_{B}$ , then the normalized bin law of the reduced weight $ρ$ is $\frac{tr (ρ P_{B})}{tr ρ} = \frac{1}{∥ Ψ ∥^{2}} \int_{B} (∣ α ∣^{2} ∣ a ∣^{2} + ∣ β ∣^{2} ∣ b ∣^{2} + 2 Re (α \overline{β} ⟨ e_{1}, e_{2} ⟩ a (x) \overline{b (x)})) d x,$ and $∣ ⟨ e_{1}, e_{2} ⟩ ∣ \leq 1$ : the marker overlap bounds the complex cross coefficient in modulus (its phase may rotate the real interference contribution); that contribution vanishes identically when $e_{1} ⊥ e_{2}$ , leaving the incoherent weight. If $Ψ = 0$ , no normalized law is defined.

(e) Finite sampling. If a detector model produces, for a fixed bin $B$ , a positive integer $N$ of independent trials each with success probability $p = p_{c o h} (B)$ or $p = p_{m i x} (B)$ , and $S_{N}$ is the number of successes, then $S_{N}$ is binomial $(N, p)$ and $E [S_{N} / N] = p$ , $Var (S_{N} / N) = p (1 - p) / N$ (The CHSH bound and the finite-sampling law for independent trials, clause (c)).

All assertions are identities and inequalities for the specified preparation vectors; no detector law, no measured counting rate and no empirical claim is asserted, and (e) is conditional on the stated independent-trials model.

Facts & Assumptions

Given: a $σ$ -finite measure space $(D, A, d x)$ , classes $a, b \in L^{2} (D)$ ; in (d) a Hilbert space $K$ with unit vectors $e_{1}, e_{2}$ and coefficients $α, β$ .

[F1]

For measurable $B$ the set function $B \mapsto \int_{B} g d x$ is countably additive for $g \in L^{1}$ : for pairwise disjoint $B_{n}$ with union $B$ , the partial sums of $\int ∣ g ∣ 1_{B_{n}}$ increase to $\int ∣ g ∣ 1_{B} \leq \int ∣ g ∣$ , and monotone convergence applied to the positive and negative parts gives $\int_{B} g = \sum_{n} \int_{B_{n}} g$ (Monotone convergence for the integral).

[F2]

For $a, b \in L^{2} (D)$ and measurable $B$ , $\int_{B} ∣ a + b ∣^{2} = \int_{B} ∣ a ∣^{2} + \int_{B} ∣ b ∣^{2} + 2 Re \int_{B} a \overline{b}$ , and $∣ \int_{B} a \overline{b} ∣ \leq ∥ a ∥_{L^{2} (B)} ∥ b ∥_{L^{2} (B)} \leq ∥ a ∥ ∥ b ∥$ (Cauchy-Schwarz inequality for $L^{2}$ , Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors, $L^{2}$ with the integral pairing is a Hilbert space).

[F3]

Marker overlap contraction of Marker overlap contraction of the two-path cross term: with $Ψ = α a \otimes e_{1} + β b \otimes e_{2}$ , $ρ = {Tr}_{K} T_{Ψ, Ψ}$ is a positive trace-class weight with $tr ρ = ∥ Ψ ∥^{2}$ and, for every measurable $B$ of finite measure, $tr (ρ P_{B}) = \int_{B} (∣ α ∣^{2} ∣ a ∣^{2} + ∣ β ∣^{2} ∣ b ∣^{2} + 2 Re (α \overline{β} ⟨ e_{1}, e_{2} ⟩ a \overline{b})) d x$ ; also $∣ ⟨ e_{1}, e_{2} ⟩ ∣ \leq 1$ .

[F4]

Binomial frequencies: if $S_{N}$ is binomial $(N, p)$ with $N \geq 1$ , $p \in [0, 1]$ , then $E [S_{N} / N] = p$ and $Var (S_{N} / N) = p (1 - p) / N$ (The CHSH bound and the finite-sampling law for independent trials, clause (c)).

Proof

1.1givenF1F2algebra

[Coherent law] Assume $∥ a + b ∥ > 0$ ; then $∥ a + b ∥^{2} = \int_{D} ∣ a + b ∣^{2} > 0$ is finite, and $p_{c o h} (D) = 1$ . The set function $B \mapsto \int_{B} ∣ a + b ∣^{2}$ is countably additive by [F1] applied to $g = ∣ a + b ∣^{2} \in L^{1}$ , and nonnegative; dividing by the finite positive number $∥ a + b ∥^{2}$ preserves countable additivity and total mass one, so $p_{c o h}$ is a probability measure. The expansion is the pointwise identity $∣ a + b ∣^{2} = ∣ a ∣^{2} + ∣ b ∣^{2} + 2 Re (a \overline{b})$ integrated over $B$ , legitimate for $L^{1}$ classes. This is (a).

1.2givenF1algebra

[Incoherent law] If $a \neq 0$ and $b \neq 0$ , the two normalized set functions $B \mapsto ∥ a ∥^{- 2} \int_{B} ∣ a ∣^{2}$ and $B \mapsto ∥ b ∥^{- 2} \int_{B} ∣ b ∣^{2}$ are probability measures by [F1] applied to $∣ a ∣^{2}$ and $∣ b ∣^{2}$ , so any convex combination with $w \in [0, 1]$ is a probability measure. This is (b).

1.3givenF2algebra

[Signed difference] Let $a, b$ be unit vectors; then $N (D) = 2$ and, Cauchy–Schwarz gives $c (D) \geq - 1$ , and the hypothesis $c (D) \neq - 1$ therefore gives $∥ a + b ∥^{2} = 2 + 2 c (D) > 0$ . Expanding, $p_{c o h} (B) - p_{m i x} (B) = \frac{N (B) + 2 c (B)}{2 + 2 c (D)} - \frac{N (B)}{2} = \frac{2 N (B) + 4 c (B) - 2 N (B) - 2 c (D) N (B)}{4 (1 + c (D))} = \frac{2 c (B) - c (D) N (B)}{2 (1 + c (D))}$ , and $N (B) - 2 c (B) = \int_{B} (∣ a ∣^{2} + ∣ b ∣^{2} - 2 Re a \overline{b}) = \int_{B} ∣ a - b ∣^{2} \geq 0$ . The signed difference is the integral over $B$ of the integrable real function $[2 Re (a \overline{b}) - c (D) (∣ a ∣^{2} + ∣ b ∣^{2})] / [2 (1 + c (D))]$ . Its integral vanishes on every measurable bin exactly when this density is zero almost everywhere: otherwise one of its positive or negative level sets has a strictly signed integral. This proves the stated all-bin criterion. If $a ⊥ b$ , only $c (D) = 0$ follows, giving difference $c (B)$ . For an explicit counterexample, take $D = {1, 2}$ with $d x ({1}) = d x ({2}) = 1$ , $a = (1, 1) / \sqrt{2}$ and $b = (1, - 1) / \sqrt{2}$ . Their inner product is zero, but $c ({1}) = 1 / 2$ , so their coherent and balanced mixed laws differ on that bin. If $b = a$ (unit), then $N (B) = 2 \int_{B} ∣ a ∣^{2}$ and $c (B) = \int_{B} ∣ a ∣^{2}$ , so $2 c (B) - c (D) N (B) = 2 \int_{B} ∣ a ∣^{2} - 2 \int_{B} ∣ a ∣^{2} = 0$ ; the difference vanishes for every $B$ . This is (c).

1.4givenF3algebra

[Marker law] Under the hypotheses of (d), $tr ρ = ∥ Ψ ∥^{2}$ by [F3]; if $Ψ \neq 0$ the normalized law is $tr (ρ P_{B}) / tr ρ$ , and substituting [F3]'s formula gives the displayed identity. The bound $∣ ⟨ e_{1}, e_{2} ⟩ ∣ \leq 1$ bounds the complex cross coefficient by $∣ α β ∣$ , while its phase need not contract the real interference part; this is [F3]. When $Ψ = 0$ , the raw weight remains defined and has zero trace, so normalization is omitted; if $e_{1} ⊥ e_{2}$ the cross term $2 Re (α \overline{β} ⟨ e_{1}, e_{2} ⟩ a \overline{b})$ vanishes pointwise, leaving the incoherent weight $∣ α ∣^{2} ∣ a ∣^{2} + ∣ β ∣^{2} ∣ b ∣^{2}$ . This is (d).

2.1givenF4algebra∎

[Finite sampling] Under the stated independent-trials model, the number of successes $S_{N}$ is by definition binomial $(N, p)$ , and the mean and variance of $S_{N} / N$ are those of [F4]. This is (e), and it is conditional on that model.

Coherent and incoherent two-path bin laws and the marker overlap

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources