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Schmidt reduction of a pure composite preparation, purity of the subsystems, and the product criterion

Physical scope

The pure preparations of a declared composite of two distinguishable subsystems: the Schmidt expansion of the composite vector, the reduced preparations of the factors, the purity criterion for the reduced states, and the product-vector characterization, including pure-state separability equivalence for the trace-norm closed convex hull. The statement is conditional on the framework's composition reading and the cited mathematics; it asserts no dynamics, no measurement, no interaction, no apparatus realization and no empirical fact, and it makes no claim about mixed states beyond what is stated.

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, The state space of the framework: preparations, pure rays and density operators, Reduced preparations of a declared composite: the subsystem states of the framework), and assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). The adopted composite reading also carries the Axiom of Choice (The Axiom of Choice) for its local spectral readouts; the Schmidt and partial-trace deductions below use only Countable Choice. Let $Ψ \in H \hat{\otimes} K$ satisfy $∥ Ψ ∥ = 1$ , let $ρ = T_{Ψ, Ψ} \in D (H \hat{\otimes} K), ρ_{A} = {Tr}_{K} ρ \in D (H), ρ_{B} = {Tr}_{H} ρ \in D (K)$ be the pure preparation of the declared composite and its two reduced preparations. Then:

(a) Schmidt expansion and reductions. There are an at most countable index set $J$ , orthonormal families $(v_{r})_{r \in J} \subseteq H$ , $(u_{r})_{r \in J} \subseteq K$ and numbers $s_{r} > 0$ with $Ψ = \sum_{r \in J} s_{r} κ (v_{r} \otimes u_{r}) (norm convergence), \sum_{r \in J} s_{r}^{2} = 1,$ and the reduced preparations are given by the trace-norm convergent series $ρ_{A} = \sum_{r \in J} s_{r}^{2} T_{v_{r}, v_{r}}, ρ_{B} = \sum_{r \in J} s_{r}^{2} T_{u_{r}, u_{r}} .$ Moreover $tr ρ_{A}^{2} = tr ρ_{B}^{2} = \sum_{r \in J} s_{r}^{4}$ .

(b) Purity and product criterion. The following are equivalent: (i) exactly one $s_{r}$ is nonzero; (ii) $ρ_{A}$ is pure, that is $tr ρ_{A}^{2} = 1$ ; (iii) $ρ_{B}$ is pure; (iv) $Ψ$ is a product vector, $Ψ = κ (u \otimes v)$ with $∥ u ∥ = ∥ v ∥ = 1$ ; (v) $ρ = ρ_{A} \otimes ρ_{B}$ ; (vi) $ρ$ is separable in the trace-norm closed-convex-hull sense of Product vectors, separable states and entangled states of a two-factor composite clause (b). In that case $s_{r_{0}} = 1$ at the unique nonzero index and $Ψ = κ (v_{r_{0}} \otimes u_{r_{0}})$ , $ρ_{A} = T_{u, u}$ , $ρ_{B} = T_{v, v}$ for $u = v_{r_{0}}$ , $v = u_{r_{0}}$ .

(c) Non-product pure preparations. If $Ψ$ is not a product vector, then at least two coefficients $s_{r}$ are nonzero, $ρ_{A}$ and $ρ_{B}$ are mixed, $tr ρ_{A}^{2} < 1$ , and $ρ \neq ρ_{A} \otimes ρ_{B}$ ; the pure preparation is entangled in the full trace-norm closed-convex-hull sense of Product vectors, separable states and entangled states of a two-factor composite clause (b). More precisely, if $m = \max_{r} s_{r}^{2} < 1$ , every separable $σ$ satisfies $tr (σ ρ) \leq m$ and $∥ ρ - σ ∥_{1} \geq 1 - m > 0$ .

(d) Scope. The statement concerns the declared composite, its pure preparations and the reduced preparations of the factors. It asserts no dynamics, no interaction, no measurement, no apparatus realization and no empirical fact; and it concerns pure composite preparations only, mixed-state separability being outside its scope.

Facts & Assumptions

Given: The declared composite with assigned spaces $H, K$ ; a unit vector $Ψ \in H \hat{\otimes} K$ ; the preparations $ρ, ρ_{A}, ρ_{B}$ ; the Axiom of Countable Choice and the inherited Axiom of Choice.

[F1]

Schmidt expansion. For $Ψ \in H \hat{\otimes} K$ there are an at most countable $J$ , orthonormal families $(v_{r}) \subseteq H$ , $(u_{r}) \subseteq K$ and $s_{r} > 0$ with $Ψ = \sum_{r} s_{r} κ (v_{r} \otimes u_{r})$ in norm, $∥ Ψ ∥^{2} = \sum_{r} s_{r}^{2}$ ; the $s_{r}$ are the positive square roots of the nonzero eigenvalues of the positive operator $C C^{*} : H_{0} \to H_{0}$ , where $C : \overline{K_{0}} \to H_{0}$ is the linear coefficient map on countable closed factor supports of $Ψ$ from the cited Schmidt theorem (for an orthonormal basis $(f_{j})$ of $K_{0}$ and $Ψ = \sum_{j} h_{j} \otimes f_{j}$ , it is defined by $C {\overset{ˉ}{f}}_{j} = h_{j}$ and continuous linear extension); and for $∥ Ψ ∥ = 1$ the reduced density $ρ = {Tr}_{K} T_{Ψ, Ψ}$ satisfies $tr ρ^{2} = \sum_{r} s_{r}^{4}$ , with $tr ρ^{2} = 1$ exactly when exactly one $s_{r}$ is nonzero, i.e. exactly when $Ψ$ is a product vector (Infinite Schmidt decomposition and the pure product criterion; the completed tensor product and the embedding $κ$ are those of Hilbert tensor construction: inner product, completion and product bases). For bounded $X \in B (H)$ , $x \in H$ and $y \in K$ the tensor identity $(X \otimes I_{K}) κ (x \otimes y) = κ (X x \otimes y)$ holds (Bounded tensor maps: extension, norm estimate, adjoint and commutation clause (a)).

[F2]

Partial trace. There is a linear positive trace-preserving map ${Tr}_{K}$ on trace-class operators with $tr (({Tr}_{K} T) X) = tr (T (X \otimes I_{K}))$ for every trace-class $T$ and every bounded $X \in B (H)$ and $∥ {Tr}_{K} T ∥_{1} \leq ∥ T ∥_{1}$ ; it maps density operators to density operators (Separable partial trace and its basic properties clauses (a), (b)); on rank-one operators ${Tr}_{K} T_{u \otimes v, u^{'} \otimes v^{'}} = ⟨ v, v^{'} ⟩ T_{u, u^{'}}$ (clause (c)).

[F3]

Rank-one trace pairing. For trace-class $S$ and $x, y \in H$ one has $tr (S T_{x, y}) = ⟨ S x, y ⟩$ , and $T_{x, x} \geq 0$ with $tr T_{x, x} = ∥ x ∥^{2}$ , $T_{x, y}$ being bounded with $∥ T_{x, y} ∥ = ∥ x ∥ ∥ y ∥$ ; the map $(x, y) \mapsto T_{x, y}$ is trace-norm continuous (Rank-one operator calculus clauses (a), (b), (c), (d), (e), (f), Trace class is a two sided Banach operator ideal). For every bounded $X$ , clauses (d) and (b) of that rank-one supplier give $T_{x, y} X = T_{x, X^{*} y}$ and $tr (T_{x, y} X) = ⟨ x, X^{*} y ⟩ = ⟨ X x, y ⟩$ ; thus the bounded pairings used below do not require $X$ to be trace class.

[F4]

Purity criterion. A state is pure exactly when its purity $tr ρ^{2}$ equals one (Purity, spectral weights and the criterion for a pure state, Density operators and scalar spectral probability clause (a)).

[F5]

Reduced preparations and product preparations. $ρ_{A} = {Tr}_{K} ρ$ and $ρ_{B} = {Tr}_{H} ρ$ are the reduced preparations of the declared composite, and for a product preparation $T_{κ (u \otimes v), κ (u \otimes v)} = T_{u, u} \otimes T_{v, v}$ , so that $ρ_{A} = T_{u, u}$ and $ρ_{B} = T_{v, v}$ (Reduced preparations of a declared composite: the subsystem states of the framework clauses (a), (c), Composite systems of the framework: product bases, local readouts, local effects and local instruments clause (b)).

[F6]

The mathematical equivalence of a pure projector being separable with its vector being product, and the strict Schmidt-overlap/trace-norm witness for non-product vectors, are Pure-state separability is equivalent to being a product: a Schmidt overlap witness. Its closed convex hull is exactly the one declared in the definition.

Proof

Proof technique: direct; obtain the Schmidt expansion from the infinite Schmidt theorem, pass the partial trace identity to the trace-norm limit through finite truncations to identify the two reduced preparations, and read off the purity and product criteria from the supplier's purity statement and the purity criterion for states.

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

1.1givenF1

[Expansion and normalization] By [F1] there are $J$ , $(v_{r})$ , $(u_{r})$ and $s_{r} > 0$ with $Ψ = \sum_{r} s_{r} κ (v_{r} \otimes u_{r})$ in norm and $\sum_{r} s_{r}^{2} = ∥ Ψ ∥^{2} = 1$ ; this is the expansion of (a).

1.2givenF1F3algebra

[Finite truncations] Put $Ψ_{N} : = \sum_{r \in J_{N}} s_{r} κ (v_{r} \otimes u_{r})$ for an increasing sequence of finite subsets $J_{N} ↑ J$ . For bounded $X \in B (H)$ and $N$ one has, using the tensor identity of [F1] on elementary tensors, $⟨ (X \otimes I_{K}) Ψ_{N}, Ψ_{N} ⟩ = \sum_{r, r^{'} \in J_{N}} s_{r} s_{r^{'}} ⟨ X v_{r}, v_{r^{'}} ⟩ ⟨ u_{r}, u_{r^{'}} ⟩ = \sum_{r \in J_{N}} s_{r}^{2} ⟨ X v_{r}, v_{r} ⟩$ by orthonormality of $(u_{r})$ ; since $Ψ_{N} \to Ψ$ , $∥ X \otimes I_{K} ∥ \leq ∥ X ∥$ and $\sum_{r} s_{r}^{2} < \infty$ , the sesquilinear form is continuous and the series converges absolutely, so $tr (ρ (X \otimes I_{K})) = ⟨ (X \otimes I_{K}) Ψ, Ψ ⟩ = \sum_{r} s_{r}^{2} ⟨ X v_{r}, v_{r} ⟩$ .

2.1givenF2F3step 1.2algebra

[Reduction identity] Let $S : = \sum_{r} s_{r}^{2} T_{v_{r}, v_{r}}$ , a trace-class operator with $S \geq 0$ and $tr S = 1$ by [F3] and absolute convergence. For bounded $X$ , step 1.2 and [F2] give $tr (ρ_{A} X) = tr (ρ (X \otimes I_{K})) = \sum_{r} s_{r}^{2} ⟨ X v_{r}, v_{r} ⟩$ ; taking $X = T_{w, w^{'}}$ and using [F3] twice, $⟨ ρ_{A} w, w^{'} ⟩ = tr (ρ_{A} T_{w, w^{'}}) = \sum_{r} s_{r}^{2} ⟨ T_{w, w^{'}} v_{r}, v_{r} ⟩ = \sum_{r} s_{r}^{2} ⟨ v_{r}, w^{'} ⟩ ⟨ w, v_{r} ⟩ = ⟨ S w, w^{'} ⟩$ for all $w, w^{'} \in H$ , so $ρ_{A} = S$ by taking $w^{'} = (ρ_{A} - S) w$ and using $∥ (ρ_{A} - S) w ∥^{2} = 0$ ; interchanging the factors gives $ρ_{B} = \sum_{r} s_{r}^{2} T_{u_{r}, u_{r}}$ with the same argument.

3.1givenF1F4F5F6step 2.1algebra

[Purity and product criterion] By [F1] and step 2.1, $tr ρ_{A}^{2} = \sum_{r} s_{r}^{4} = tr ρ_{B}^{2}$ and $tr ρ_{A}^{2} = 1$ holds exactly when exactly one $s_{r}$ is nonzero and exactly when $Ψ$ is a product vector; purity of $ρ_{A}$ is equivalent to $tr ρ_{A}^{2} = 1$ by [F4], and likewise for $ρ_{B}$ . If exactly one $s_{r_{0}}$ is nonzero, then $s_{r_{0}}^{2} = 1$ and $Ψ = κ (v_{r_{0}} \otimes u_{r_{0}})$ is a product vector, so $ρ = ρ_{A} \otimes ρ_{B}$ by [F5]; conversely for a product vector [F5] gives $ρ_{A} = T_{u, u}$ , $ρ_{B} = T_{v, v}$ , which are pure by [F3] and [F4]. This is (b); its failure case, in which at least two $s_{r}$ are nonzero and both reduced preparations are mixed with $tr ρ_{A}^{2} < 1$ and $ρ \neq ρ_{A} \otimes ρ_{B}$ , is (c). Conversely, if $ρ = ρ_{A} \otimes ρ_{B}$ , pair both operators with the unit product vector $q_{r} = κ (v_{r} \otimes u_{r})$ . The expansion gives $⟨ ρ q_{r}, q_{r} ⟩ = s_{r}^{2}$ , while step 2.1 and the tensor action in [F1] give $⟨ (ρ_{A} \otimes ρ_{B}) q_{r}, q_{r} ⟩ = s_{r}^{4}$ . Thus every positive $s_{r}^{2}$ equals its square and is one; their sum one forces exactly one coefficient. Finally [F6] identifies this product case exactly with separability, proving (vi); in the non-product case it gives entanglement and the stated strict trace-norm witness.

4.1step 1.1step 2.1step 3.1∎

[Assembly and scope] Step 1.1 gives the expansion of (a) and step 2.1 its reduction formulae; step 3.1 establishes (b) and (c); only the declared composite, its pure preparations and the cited mathematics were used, which is the scope statement (d).

Remarks

Pure versus mixed scope. Clause (c) proves non-product pure preparations entangled for the full trace-norm closed convex hull. A general classification or criterion for mixed-state separability is not claimed.
Choice. The adopted composite interface carries the Axiom of Choice for local spectral readouts. The deductions in this proof use only Countable Choice through the tensor construction and Schmidt theorem; no spectral theorem for unbounded operators enters the deduction.

Schmidt reduction of a pure composite preparation, purity of the subsystems, and the product criterion

Statement

Facts & Assumptions

Proof

Remarks

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