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Factorization of local statistics for product preparations, and its failure for non-product pure preparations

Physical scope

The local statistics of a pure preparation of a declared composite of two distinguishable subsystems: for a product preparation every pair of local finite-outcome measurements has a factorized joint law, and for a non-product pure preparation there are explicit local rank-one effects whose joint probability differs from the product of the local probabilities. The statement asserts no dynamics, no interaction, no apparatus realization, no measured correlation and no empirical fact; it is a conditional computation in the declared model.

Statement

Adopt the framework's composition and state readings for a declared composite of two distinguishable subsystems $A, B$ with assigned spaces $H, K$ (Composite systems of the framework: product bases, local readouts, local effects and local instruments, Reduced preparations of a declared composite: the subsystem states of the framework), and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice), as required by the cited spectral-law supplier. Let $Ψ \in H \hat{\otimes} K$ be a unit vector, $ρ = T_{Ψ, Ψ}$ the associated pure preparation, and $ρ_{A} = {Tr}_{K} ρ$ , $ρ_{B} = {Tr}_{H} ρ$ its reduced preparations. Then:

(a) Product preparations factorize. If $Ψ = κ (u \otimes v)$ with $∥ u ∥ = ∥ v ∥ = 1$ , then for every bounded effect $F$ on $H$ and every bounded effect $F^{'}$ on $K$ $tr (ρ (F \otimes F^{'})) = tr (ρ_{A} F) tr (ρ_{B} F^{'}) .$ In particular, for a declared finite-outcome measurement of $A$ with effect family $(F_{i})$ and one of $B$ with $(F_{j}^{'})$ , the joint outcome probabilities factorize, $p (i, j) = p_{A} (i) p_{B} (j)$ : in a product preparation the two subsystems have independent local statistics.

(b) Non-product pure preparations are correlated. If $Ψ$ is not a product vector and $Ψ = \sum_{r \in J} s_{r} κ (v_{r} \otimes u_{r})$ is its Schmidt expansion with $s_{1} \geq s_{2} > 0$ , then the rank-one effects $P : = T_{v_{1}, v_{1}}$ on $H$ and $P^{'} : = T_{u_{2}, u_{2}}$ on $K$ satisfy $tr (ρ (P \otimes P^{'})) = 0 \neq s_{1}^{2} s_{2}^{2} = tr (ρ_{A} P) tr (ρ_{B} P^{'}),$ so the joint law of the corresponding two-outcome local measurements does not factorize; the product of the marginal probabilities exceeds the joint probability by $s_{1}^{2} s_{2}^{2} > 0$ .

(c) Scope. The statement is a conditional computation for pure preparations of the declared composite; it asserts no dynamics, no interaction, no apparatus realization, no measured correlation and no empirical fact, and it does not claim that a joint law failing to factorize establishes any particular physical mechanism.

Facts & Assumptions

Given: The declared composite with assigned spaces $H, K$ ; a unit vector $Ψ \in H \hat{\otimes} K$ ; the preparations $ρ, ρ_{A}, ρ_{B}$ ; bounded effects $F, F^{'}$ ; the Schmidt expansion of $Ψ$ when it exists; the Axiom of Countable Choice.

[F1]

Schmidt reduction. If $Ψ = \sum_{r} s_{r} κ (v_{r} \otimes u_{r})$ is the Schmidt expansion of Infinite Schmidt decomposition and the pure product criterion with orthonormal families $(v_{r})$ , $(u_{r})$ and $s_{r} > 0$ , $\sum_{r} s_{r}^{2} = 1$ , then $ρ_{A} = \sum_{r} s_{r}^{2} T_{v_{r}, v_{r}}$ and $ρ_{B} = \sum_{r} s_{r}^{2} T_{u_{r}, u_{r}}$ ; for a product vector $Ψ = κ (u \otimes v)$ one has $ρ_{A} = T_{u, u}$ and $ρ_{B} = T_{v, v}$ (Schmidt reduction of a pure composite preparation, purity of the subsystems, and the product criterion clauses (a), (b), Separable partial trace and its basic properties clause (c)).

[F2]

Rank-one pairing. For bounded $S$ and vectors $x, y$ in any of the factor or composite Hilbert spaces, $T_{x, y} S = T_{x, S^{*} y}$ by Rank-one operator calculus clause (d), and clause (b) gives $tr (T_{x, y} S) = ⟨ x, S^{*} y ⟩ = ⟨ S x, y ⟩$ ; the product is trace class also by Trace class is a two sided Banach operator ideal clause 2. For unit $w$ , clauses (c), (d) give $T_{w, w}^{*} = T_{w, w}$ and $T_{w, w}^{2} = T_{w, w}$ , so it is an orthogonal projection. If $F$ is bounded and positive, the square root in [F5] further gives $tr (T_{Ψ, Ψ} F) = ⟨ F Ψ, Ψ ⟩ = ∥ F^{1 / 2} Ψ ∥^{2} \geq 0$ for unit $Ψ$ ; this positivity assertion requires positivity of $F$ .

[F3]

Tensor calculus. $(S \otimes S^{'}) κ (x \otimes y) = κ (S x \otimes S^{'} y)$ and $(S \otimes S^{'})^{*} = S^{*} \otimes S^{' *}$ , so $(F \otimes F^{'}) κ (x \otimes y) = κ (F x \otimes F^{'} y)$ for bounded $F, F^{'}$ (Bounded tensor maps: extension, norm estimate, adjoint and commutation clauses (a), (c)); product vectors and the embedding $κ$ are those of Hilbert tensor construction: inner product, completion and product bases.

[F4]

Finite-outcome probability rule. P2 permits a finite family of bounded effects $G_{k}$ on a system with $0 \leq G_{k} \leq I$ and $\sum_{k} G_{k} = I$ ; the outcome probabilities in $ρ$ are $tr (ρ G_{k})$ (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes clause (iv)). A product family of local effects is used as a joint measurement only after these conditions are checked below.

[F5]

Positive square roots. Every bounded positive operator $F$ has a bounded positive square root $F^{1 / 2}$ with $(F^{1 / 2})^{2} = F$ ; if $0 \leq F \leq I$ , then $0 \leq I - F \leq I$ and both have such square roots (Continuous functional calculus for bounded self adjoint operators).

[F6]

Choice hypothesis. The Axiom of Choice is the additional mathematical assumption used by the cited spectral-law supplier's spectral construction (The Axiom of Choice).

Proof

technique · first declare and verify the product POVM for two local effect families, then evaluate the joint pairing on product preparations and project a non-product preparation onto Schmidt vectors from different factors

Given: The data of the statement and [F1]-[F5].

1.1givenF3F4F5algebra

[Declared joint POVM] Let $(F_{i})_{i = 1}^{n}$ and $(F_{j}^{'})_{j = 1}^{m}$ be finite local effect families with $\sum_{i} F_{i} = I_{H}$ and $\sum_{j} F_{j}^{'} = I_{K}$ . Define $G_{i j} : = F_{i} \otimes F_{j}^{'}$ . By [F5] the positive square roots exist, and the tensor adjoint and product rules [F3] give $G_{i j} = (F_{i}^{1 / 2} \otimes (F_{j}^{'})^{1 / 2})^{*} (F_{i}^{1 / 2} \otimes (F_{j}^{'})^{1 / 2}) \geq 0$ . Bilinearity of bounded tensoring gives $\sum_{i, j} G_{i j} = (\sum_{i} F_{i}) \otimes (\sum_{j} F_{j}^{'}) = I_{H} \otimes I_{K}$ . Thus every $G_{i j} \leq I$ and the family $(G_{i j})$ satisfies the finite-outcome requirements of P2 on the product outcome set; [F4] assigns the joint probabilities $p (i, j) = tr (ρ G_{i j})$ . This is the joint law used in (a).

1.2givenF1F2F3F4F6algebra

[Non-product case] Let $Ψ$ be non-product. By [F1] its norm-convergent Schmidt expansion has at least two positive coefficients; label two of them $s_{1} \geq s_{2} > 0$ and put $P = T_{v_{1}, v_{1}}$ , $P^{'} = T_{u_{2}, u_{2}}$ . Since $P v_{r} = δ_{r, 1} v_{1}$ and $P^{'} u_{r} = δ_{r, 2} u_{2}$ by the rank-one definition and orthonormality, [F3] gives $(P \otimes P^{'}) κ (v_{r} \otimes u_{r}) = δ_{r, 1} δ_{r, 2} κ (v_{1} \otimes u_{2}) = 0$ for every $r$ . Boundedness of $P \otimes P^{'}$ passes this identity through the norm-convergent Schmidt expansion, so $(P \otimes P^{'}) Ψ = 0$ and $tr (ρ (P \otimes P^{'})) = 0$ by [F2]. The trace-norm expansions of the reduced preparations [F1], paired with the bounded projections, give $tr (ρ_{A} P) = s_{1}^{2}$ and $tr (ρ_{B} P^{'}) = s_{2}^{2}$ by [F2]. Thus the marginal product exceeds the joint probability by $s_{1}^{2} s_{2}^{2} > 0$ . The two local measurements $(P, I - P)$ and $(P^{'}, I - P^{'})$ are effect families by [F2], [F4], and their $(+, +)$ entry already proves nonfactorization.

2.1givenF1F2F3step 1.1algebra

[Product case] Let $Ψ = κ (u \otimes v)$ with $∥ u ∥ = ∥ v ∥ = 1$ and let $F, F^{'}$ be bounded effects. By [F5] and the tensor calculus [F3], the same square-root factorization as in step 1.1 shows $F \otimes F^{'} \geq 0$ , and $I_{H} \otimes I_{K} - F \otimes F^{'} = (I_{H} - F) \otimes I_{K} + F \otimes (I_{K} - F^{'}) \geq 0$ , since each term on the right factors as $R^{*} R$ using [F5] and [F3]. Thus $F \otimes F^{'}$ is an effect. By [F2] and [F3], $tr (ρ (F \otimes F^{'})) = ⟨ (F \otimes F^{'}) Ψ, Ψ ⟩_{\hat{\otimes}} = ⟨ F u, u ⟩_{H} ⟨ F^{'} v, v ⟩_{K} = tr (T_{u, u} F) tr (T_{v, v} F^{'}) = tr (ρ_{A} F) tr (ρ_{B} F^{'})$ by [F1]. For arbitrary $F, F^{'}$ , apply step 1.1 to the binary local families $(F, I_{H} - F)$ and $(F^{'}, I_{K} - F^{'})$ ; it declares $F \otimes F^{'}$ as their joint $(+, +)$ effect. For general finite outcome families, step 1.1 gives the corresponding declared joint probabilities and marginals, so they factorize. This is (a).

3.1step 1.1step 2.1step 1.2∎

[Assembly and scope] Steps 2.1 and 1.2 prove (a) and (b); only the declared composite, its pure preparations and the cited mathematics were used, and no dynamics, apparatus or empirical input entered, which is the scope statement (c).

Remarks

One pair of effects suffices. Clause (b) exhibits a single pair of local rank-one effects that detects the correlation; the two-outcome measurements with effects $(P, I - P)$ and $(P^{'}, I - P^{'})$ have joint law $(0, s_{1}^{2}, s_{2}^{2}, 1 - s_{1}^{2} - s_{2}^{2})$ in the ordering $(+, +), (+, -), (-, +), (-, -)$ for the unit vector $Ψ$ . All entries are nonnegative because $\sum_{r} s_{r}^{2} = 1$ , and the first entry differs from $s_{1}^{2} s_{2}^{2}$ .
No converse beyond pure states. The theorem concerns pure preparations; for mixed preparations no analogous factorization criterion is asserted.

Factorization of local statistics for product preparations, and its failure for non-product pure preparations

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

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