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Conditional jump-time phases form a positive Gram matrix

Statement

Let $t \geq 0$ , let $μ$ be a probability measure on the Borel sets of $[0, t]$ , and let $θ_{1}, \dots, θ_{r}$ be bounded Borel-measurable real functions, where $r \geq 1$ . Define $G_{j k} = \int_{[0, t]} e^{i (θ_{j} (s) - θ_{k} (s))} d μ (s)$ . Then $G$ is Hermitian positive semidefinite, $G_{j j} = 1$ , and $∣ G_{j k} ∣ \leq 1$ .

For the following conditional branch, let $ρ$ be a density operator on $C^{r}$ and let $D (s)$ be diagonal with entries $e^{i θ_{j} (s)}$ . If its normalized output is the entrywise average $σ = \int D (s) ρ D (s)^{†} d μ (s)$ , then $σ_{j k} = G_{j k} ρ_{j k}$ . Thus $G_{j k}$ is the relative-coherence multiplier for this branch. A common input-independent waiting law supplies $μ$ only when a model also establishes this diagonal phase action and the stated averaging formula.

Facts & Assumptions

[F1]

A probability measure has total mass one. Probability measures and probability spaces

[F2]

On complex $L^{2} (μ)$ , the pairing is $⟨ f, g ⟩ = \int f \overline{g} d μ$ , linear in the first variable. The complex L² pairing on equivalence classes

[F3]

This pairing is an inner product and satisfies $∣ ⟨ f, g ⟩ ∣ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ . The complex L² pairing is well-defined and satisfies Cauchy–Schwarz

[F4]

A Gram matrix has entries given by pairwise inner products. The Gram matrix G(v₀,…,vᵣ₋₁)=(⟨ vᵢ,vⱼ⟩)_i,j<r and Gram determinant, with empty value 1

[F5]

A density operator is a non-negative trace-one endomorphism; matrix entries and adjoints use the stated finite coordinate conventions. Density operator Finite operator and ket conventions

[F6]

The Borel sigma-algebra of a topological space is generated by its open sets. A function between measurable spaces is measurable when inverse images of all measurable sets are measurable. The Borel sigma-algebra of a topological space A measurable function between measurable spaces

[F7]

The complex exponential is complex differentiable everywhere, and complex differentiability implies continuity. The complex exponential is entire and its complex derivative is itself Complex differentiability at a point implies continuity there

[F8]

A continuous map between topological spaces has Borel inverse images of Borel sets. A continuous map has Borel preimages of Borel sets

[F9]

For real $u$ , $∣ e^{i u} ∣ = 1$ ; for complex $z, w$ , $e^{z + w} = e^{z} e^{w}$ . Also $z \overline{z} = ∣ z ∣^{2}$ , and conjugation and modulus obey their usual sum and product laws. exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0 exp(z+w)=exp z exp w, and the complex exponential extends the real exponential Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive

[F10]

Complex functions are measurable when their real and imaginary components are measurable; products and conjugation preserve this property. The space $L^{2} (μ; C)$ consists of measurable functions with finite squared-modulus integral, modulo equality almost everywhere, and complex integration is componentwise. Complex Lp classes and Euclidean test-function conventions

[F11]

The integral of complex integrable functions is complex-linear. The Lebesgue integral is linear on L¹(μ)

[F12]

On a finite-dimensional inner-product space, a non-negative operator is a self-adjoint endomorphism $T$ satisfying $⟨ T v, v ⟩ \geq 0$ for every $v$ ; here positive semidefinite means non-negative. Non-negative and positive operators

Proof

Given: The probability measure, phases, and, for the conditional assertion, $ρ$ , $D (s)$ and the averaging formula in the statement.

Goal: Prove the matrix properties and the conditional coherence formula.

1.1givenF6F7F8F10

Put $E (u) = e^{i u}$ for real $u$ and $f_{j} = E \circ θ_{j}$ . By F7, $E$ is continuous: the map $u \mapsto i u$ preserves distances, and composing it with the continuous exponential preserves continuity. The coordinate projections $z \mapsto Re z$ and $z \mapsto Im z$ are continuous, since each coordinate difference is bounded in absolute value by the modulus of the complex difference. For every Borel $B \subseteq C$ , $f_{j}^{- 1} (B) = θ_{j}^{- 1} (E^{- 1} (B))$ is Borel by F6 and F8. The same inverse-image argument for the continuous real and imaginary components of $E$ shows that $f_{j}$ is measurable in the componentwise convention of F10.

2.1F1F2F4F9F10step 1.1algebra

F9 gives $∣ f_{j} (s) ∣ = 1$ . F1 and F10 give $\int ∣ f_{j} ∣^{2} d μ = μ ([0, t]) = 1$ , so each $f_{j}$ represents an $L^{2} (μ)$ vector of norm one. For real $a, b$ , exponential addition gives $E (a) = E (a - b) E (b)$ . Multiplying by $\overline{E (b)}$ and using $E (b) \overline{E (b)} = ∣ E (b) ∣^{2} = 1$ gives $E (a - b) = E (a) \overline{E (b)}$ . Taking $a = θ_{j} (s)$ and $b = θ_{k} (s)$ identifies the stated phase-difference integrand with $f_{j} (s) \overline{f_{k} (s)}$ ; F2 makes this product integrable. The stated entries satisfy $G_{j k} = ⟨ f_{j}, f_{k} ⟩$ ; thus $G$ is their Gram matrix.

3.1F3F5F12step 2.1algebra

Regard $G$ as the coordinate endomorphism $c \mapsto G c$ of $C^{r}$ with its standard first-variable-linear inner product, as in F5. Conjugate symmetry gives $G_{k j} = \overline{G_{j k}}$ , so its conjugate transpose equals $G$ and this endomorphism is self-adjoint. For any $c \in C^{r}$ , first-variable linearity gives $⟨ G c, c ⟩_{r o o t} = c^{†} G c = \sum_{j, k} \overline{c_{j}} c_{k} ⟨ f_{j}, f_{k} ⟩ = ∥ \sum_{j} \overline{c_{j}} f_{j} ∥_{2}^{2} \geq 0$ . F12 therefore identifies this endomorphism as non-negative. Hence $G$ is Hermitian positive semidefinite. Also $G_{j j} = ∥ f_{j} ∥_{2}^{2} = 1$ , and Cauchy–Schwarz gives $∣ G_{j k} ∣ \leq 1$ .

4.1givenF1F5F9F10F11step 1.1step 2.1∎

By diagonal matrix multiplication and the phase identity in step 2.1, $(D (s) ρ D (s)^{†})_{j k} = f_{j} (s) \overline{f_{k} (s)} ρ_{j k} = e^{i (θ_{j} (s) - θ_{k} (s))} ρ_{j k}$ . The product $f_{j} \overline{f_{k}}$ is measurable by F10 and the entry is integrable, since its modulus is the constant $∣ ρ_{j k} ∣$ and $μ ([0, t]) = 1$ . Entrywise integration and scalar linearity therefore give $σ_{j k} = ρ_{j k} \int e^{i (θ_{j} - θ_{k})} d μ = G_{j k} ρ_{j k}$ . This uses the assumed averaging formula; it does not derive a waiting law or phase action from a physical model.

Depends on

Used by · 4

Dependency tree · 2 levels

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