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Causal propagators, time slice and algebraic locality — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Bell-pair postselection changes a conditional remote expectation

Statement

Assume Countable Choice as in The Axiom of Countable Choice ( ${A C}_{ω}$ ). On $H = C^{2} \otimes C^{2}$ , let $ψ = \frac{∣ 00 ⟩ + ∣ 11 ⟩}{\sqrt{2}},$ where $e_{0} = (1, 0)$ and $e_{1} = (0, 1)$ are the standard orthonormal basis of $C^{2}$ , $j, a, b, c, d \in {0, 1}$ , $∣ j ⟩ : = e_{j}$ , and $∣ a b ⟩ : = e_{a} \otimes e_{b}$ . Give $H$ the inner product linear in the first argument and conjugate-linear in the second determined by $⟨ e_{a} \otimes e_{b}, e_{c} \otimes e_{d} ⟩ = 1$ if $a = c$ and $b = d$ , and $0$ otherwise, extended by these sesquilinearity rules. Let $I_{H}$ be the identity on $H$ . For either $K = H$ or $K = C^{2}$ , write $B (K)$ for the bounded complex-linear maps $K \to K$ , with boundedness measured by the induced norm. Define the initial vector functional $ω : B (H) \to C$ by $ω (C) : = ⟨ C ψ, ψ ⟩$ . For $j \in {0, 1}$ let $P_{j} x : = ⟨ x, e_{j} ⟩ e_{j}$ for $x \in C^{2}$ ; Alice uses the local projections $M_{0} = P_{0} \otimes I_{2}, M_{1} = P_{1} \otimes I_{2},$ where $I_{2}$ is the identity on $C^{2}$ . Bob's observable is $Z_{B} = I_{2} \otimes Z$ , with $Z$ the linear operator determined by $Z ∣ 0 ⟩ = ∣ 0 ⟩$ and $Z ∣ 1 ⟩ = - ∣ 1 ⟩$ . Define the branch weight $p_{j} : = ∥ M_{j} ψ ∥^{2}$ and, when $p_{j} > 0$ , the normalized branch vector $ψ_{j} : = M_{j} ψ / \sqrt{p_{j}}$ . Each branch weight is $1 / 2$ ; conditional on branch $0$ Bob's $Z_{B}$ expectation $ω_{j} (Z_{B}) : = ⟨ Z_{B} ψ_{j}, ψ_{j} ⟩$ is $+ 1$ , and conditional on branch $1$ it is $- 1$ . The initial and nonselective $Z_{B}$ expectations are both zero. These are finite-dimensional mathematical state calculations, not claims about an apparatus or observed data.

Statement references

Local trace-preserving operations preserve commuting expectations

Facts & Assumptions

Given: The two-factor Hilbert space with the displayed orthonormal basis and inner-product convention, the vector $ψ$ , the projections $M_{0}, M_{1}$ , and the operator $Z_{B}$ .

[A1]

Assume Countable Choice as stated in The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F1]

Real and complex inner-product spaces and their induced length: the inner product is linear in its first argument, conjugate-linear in its second, and induces the norm $∥ x ∥ = \sqrt{⟨ x, x ⟩}$ .

[F2]

Local trace-preserving operations preserve commuting expectations: for a nonzero complex Hilbert space $H$ , a state on $B (H)$ is a complex-linear functional $ω$ with $ω (I_{H}) = 1$ and $ω (X^{*} X) \in [0, \infty)$ for every bounded $X$ ; for commuting unital $*$ -subalgebras and a finite normalized family in one of them, the nonselective expectation of every element of the other equals its initial expectation.

[F3]

Hilbert space and Every finite-dimensional normed space is Banach: a complex Hilbert space is a complex inner-product space complete in its induced norm; the four displayed orthonormal basis vectors make $H$ finite-dimensional, so the induced norm is complete and $H$ is a nonzero Hilbert space. The same argument with $e_{0}, e_{1}$ makes $C^{2}$ a Hilbert space.

[F4]

The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators: $B (K)$ is the set of bounded complex-linear operators $K \to K$ for each of the finite-dimensional Hilbert spaces $K = H, C^{2}$ used here.

[F5]

The Hilbert-space adjoint of a bounded operator: for bounded $X$ on a Hilbert space with the linear-first convention, $⟨ X x, y ⟩ = ⟨ x, X^{*} y ⟩$ .

Proof

1.1F1F2F3F4F5givenalgebra

The four vectors $∣ a b ⟩$ are orthonormal, so $∥ ψ ∥^{2} = (1 + 1) / 2 = 1$ . By [F1], $ω$ is complex-linear and $ω (I_{H}) = 1$ . For every $X \in B (H)$ , adjointness [F5] and conjugate symmetry give $ω (X^{*} X) = ⟨ X^{*} X ψ, ψ ⟩ = \overline{⟨ ψ, X^{*} X ψ ⟩} = \overline{⟨ X ψ, X ψ ⟩} = ∥ X ψ ∥^{2} \geq 0$ , so $ω$ is a state in the sense of [F2]. The projections satisfy $M_{0}^{*} = M_{0}$ , $M_{1}^{*} = M_{1}$ , $M_{0} M_{1} = 0$ , and $M_{0} + M_{1} = I_{H}$ . Hence $\sum_{j = 0}^{1} M_{j}^{*} M_{j} = I_{H}$ , and $M_{0} ψ = \frac{∣ 00 ⟩}{\sqrt{2}}, M_{1} ψ = \frac{∣ 11 ⟩}{\sqrt{2}} .$ The outcome probabilities are $p_{j} = ∥ M_{j} ψ ∥^{2} = 1 / 2$ , so both normalized conditional vectors are defined.

2.1F1step 1.1givenalgebra

Dividing the branch vectors in step 1.1 by $\sqrt{p_{j}} = 1 / \sqrt{2}$ gives $ψ_{0} = ∣ 00 ⟩$ and $ψ_{1} = ∣ 11 ⟩$ . Since $Z_{B} ∣ 00 ⟩ = ∣ 00 ⟩$ and $Z_{B} ∣ 11 ⟩ = - ∣ 11 ⟩$ , their conditional expectations are respectively $ω_{0} (Z_{B}) = ⟨ Z_{B} ψ_{0}, ψ_{0} ⟩ = + 1$ and $ω_{1} (Z_{B}) = ⟨ Z_{B} ψ_{1}, ψ_{1} ⟩ = - 1$ . Also $Z_{B} ψ = \frac{∣ 00 ⟩ - ∣ 11 ⟩}{\sqrt{2}}, ω (Z_{B}) = ⟨ Z_{B} ψ, ψ ⟩ = ⟨ \frac{∣ 00 ⟩ - ∣ 11 ⟩}{\sqrt{2}}, \frac{∣ 00 ⟩ + ∣ 11 ⟩}{\sqrt{2}} ⟩ = 0,$ because the cross terms vanish by orthogonality.

3.1A1F1F2F4step 1.1step 2.1givenalgebra∎

Let $A : = {A \otimes I_{2} : A \in B (C^{2})}$ and $B : = {I_{2} \otimes B : B \in B (C^{2})}$ . For every elementary tensor $u \otimes v$ , $(A \otimes I_{2}) (I_{2} \otimes B) (u \otimes v) = A u \otimes B v = (I_{2} \otimes B) (A \otimes I_{2}) (u \otimes v)$ ; elementary tensors span $H$ , so these unital $*$ -algebras commute. The two projections lie in $A$ and the normalization was checked in step 1.1. Applying [F2] to the state proved in step 1.1 and $C = Z_{B}$ gives the nonselective expectation $\sum_{j} ω (M_{j}^{*} Z_{B} M_{j}) = \sum_{j} ⟨ M_{j}^{*} Z_{B} M_{j} ψ, ψ ⟩ = ω (Z_{B}) = ⟨ Z_{B} ψ, ψ ⟩ = 0$ . Equivalently, the probability-weighted conditional values from step 2.1 average to $(1 / 2) (+ 1) + (1 / 2) (- 1) = 0$ . Thus post-selection changes the conditional remote expectation in either branch, while retaining and averaging the outcomes leaves the original remote marginal unchanged.

Remarks

The branch states are conditional on the reported local outcome. Their different remote expectations do not give a nonselective signal; the outcome-averaged identity is the algebraic conclusion of Local trace-preserving operations preserve commuting expectations.

5 · Examples, experiments and counterexamples

None yet.