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Statistical mixtures, spectral decompositions and the extremal role of pure preparations

Physical scope

Model-internal statements about preparations of the non-relativistic quantum framework under the state postulate: statistical mixtures of preparations with probability weights are preparations; every preparation has a spectral decomposition as a countable mixture of pairwise orthogonal pure preparations; and the pure preparations are exactly the extreme points of the preparation set, so every mixed preparation is a nontrivial mixture of two preparations. The weights are classical preparation probabilities, not amplitudes. The item asserts no apparatus realization of any mixture, no measurement or probability law, no dynamics and no empirical result.

Statement

Adopt the state and composition postulate P1 of the framework (State and composition postulate P1: Hilbert spaces, preparations and composite systems) and use the state-space vocabulary of The state space of the framework: preparations, pure rays and density operators. Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), which is the choice hypothesis of the spectral and mixture suppliers cited below. Let $H \neq {0}$ be the Hilbert space assigned to a declared system. Then:

(a) Mixtures. Let $(ρ_{n})$ be a finite or countable family of preparations with real weights $w_{n} \geq 0$ satisfying $\sum_{n} w_{n} = 1$ . Then the statistical mixture $\sum_{n} w_{n} ρ_{n}$ of clause (iii) of P1 converges in trace norm to a preparation, namely to the density operator $T = \sum_{n} w_{n} ρ_{n}$ of The state space is convex, closed under countable mixtures, and complete in the trace norm. The mixture is affine in its data: a term of zero weight may be added or removed without changing $T$ , and splitting one weight between two copies of the same preparation, $w ρ = w_{1} ρ + w_{2} ρ$ with $w_{1} + w_{2} = w$ , leaves $T$ unchanged. If the index set is partitioned into consecutive finite blocks with total weights $u_{k} > 0$ and block mixtures $σ_{k}$ , then $\sum_{k} u_{k} σ_{k} = \sum_{n} w_{n} ρ_{n}$ in trace norm, so mixtures may be regrouped.

(b) Spectral decomposition. Every preparation $ρ$ is a countable statistical mixture of pairwise orthogonal pure preparations: there exist a finite or countable orthonormal family $(u_{n})$ of unit vectors and positive weights $λ_{n}$ with $\sum_{n} λ_{n} = 1$ such that $ρ = \sum_{n} λ_{n} T_{u_{n}, u_{n}},$ the sum converging in trace norm, and the weights $λ_{n}$ are exactly the positive eigenvalues of $ρ$ counted with multiplicity.

(c) Extremality and nontrivial decompositions. A preparation is pure if and only if its density operator is an extreme point of the state space $D (H)$ in the sense of the condition displayed in The extreme points of the state space are exactly the pure states; consequently every mixed preparation $ρ$ admits a representation as a statistical mixture of two preparations in a nontrivial way: if $λ_{1} \geq λ_{2} > 0$ are two positive eigenvalues of $ρ$ with orthonormal eigenvectors $u_{1}$ and $u_{2}$ , then with $ε : = λ_{2} / 2$ the operators $σ_{\pm} : = ρ \pm ε (T_{u_{1}, u_{1}} - T_{u_{2}, u_{2}})$ are preparations with $ρ = \frac{1}{2} σ_{+} + \frac{1}{2} σ_{-}$ and $σ_{+} \neq σ_{-}$ .

Facts & Assumptions

Given: The adopted state postulate P1, the assigned Hilbert space $H \neq {0}$ , preparations $ρ, ρ_{n} \in D (H)$ , weights $w_{n} \geq 0$ with $\sum_{n} w_{n} = 1$ , and where required two positive eigenvalues $λ_{1} \geq λ_{2} > 0$ of $ρ$ with orthonormal eigenvectors $u_{1}, u_{2}$ .

[F1]

The state postulate (State and composition postulate P1: Hilbert spaces, preparations and composite systems): pure preparations are the unit rays of $H$ ; general preparations are the density operators on $H$ ; a statistical mixture of preparations $ρ_{n}$ with weights $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ , is the trace-norm convergent sum $ρ = \sum_{n} w_{n} ρ_{n}$ , whose weights are a classical preparation uncertainty and not amplitudes; and the density operator associated with a unit ray $[ψ]$ is $T_{ψ, ψ}$ .

[F2]

State-space vocabulary and mixtures (The state space of the framework: preparations, pure rays and density operators, The state space is convex, closed under countable mixtures, and complete in the trace norm): preparations are the states $D (H)$ ; countable probability-weighted sums of states converge absolutely in the trace norm to a state, finite convex combinations of states are states, and consecutive finite blocks of a weighted family may be grouped with the grouped weights producing the same sum.

[F3]

Spectral expansion (Density operators and scalar spectral probability clause (a), Purity, spectral weights and the criterion for a pure state): every density operator $ρ$ admits an expansion $ρ = \sum_{n} λ_{n} T_{u_{n}, u_{n}}$ converging in trace norm, with $λ_{n} > 0$ , $\sum_{n} λ_{n} = 1$ , and $(u_{n})$ an orthonormal family of eigenvectors; the $λ_{n}$ are the positive eigenvalues of $ρ$ counted with multiplicity. The purity criterion gives that $ρ$ is pure exactly when the multiplicity list has one positive eigenvalue equal to $1$ .

[F4]

Pure states are extreme (The extreme points of the state space are exactly the pure states): the extreme points of $D (H)$ are exactly the pure states.

[F5]

Pure states and rays (Rays, unit vectors and rank-one Hilbert projections, The state space: density operators, pure and mixed states, and purity): $T_{u, u}$ with $u \in S (H)$ are exactly the pure states, and $T_{u_{1}, u_{1}} \neq T_{u_{2}, u_{2}}$ for orthonormal unit vectors $u_{1}, u_{2}$ .

[F6]

Extremality condition (The extreme points of the state space are exactly the pure states): a state $x \in D (H)$ is extreme when $x = (1 - t) y + t z$ with $y, z \in D (H)$ and real $0 < t < 1$ implies $y = z = x$ .

Proof

1.1givenF1F2algebra

[(a) mixtures and affinity] By [F2] the sum $T = \sum_{n} w_{n} ρ_{n}$ converges absolutely in trace norm to a state, so it is a preparation in the sense of [F1]; the identity $\sum_{n} w_{n} ρ_{n} = \sum_{n} w_{n} ρ_{n} + 0 \cdot ρ^{'}$ for an extra preparation $ρ^{'}$ and the identity $w ρ = w_{1} ρ + w_{2} ρ$ for $w_{1} + w_{2} = w$ are identities in the real vector space of self-adjoint trace-class operators, so the value of the mixture is unchanged; and the grouping statement for consecutive blocks is clause (c) of [F2].

1.2givenF2F3F5algebra

[(b) the spectral decomposition is a mixture of pure preparations] By [F3] the preparation $ρ$ has an expansion $ρ = \sum_{n} λ_{n} T_{u_{n}, u_{n}}$ with positive weights $λ_{n}$ , $\sum_{n} λ_{n} = 1$ and $(u_{n})$ orthonormal; by [F5] each $T_{u_{n}, u_{n}}$ is a pure state, and pairwise orthogonality of the $u_{n}$ makes the pure preparations pairwise distinct; the convergence in trace norm is the convergence of a statistical mixture as in [F2]. Thus $ρ$ is exhibited as the mixture of the pure preparations $T_{u_{n}, u_{n}}$ with the weights $λ_{n}$ , which by [F3] are its positive eigenvalues counted with multiplicity.

1.3givenF2F4

[(c) extremality] By [F4] the extreme points of $D (H)$ are exactly the pure states; since preparations are the states of $D (H)$ by [F2], a preparation is pure exactly when its density operator is extreme.

2.1givenstep 1.3F1F2F3F5F6algebra

[(c) the nontrivial splitting of a mixed preparation] By [F3], a mixed $ρ$ has at least two positive eigenvalues counted with multiplicity. Choose two with $λ_{1} \geq λ_{2} > 0$ and orthonormal eigenvectors $u_{1}, u_{2}$ , and put $ε = λ_{2} / 2$ . In the spectral expansion of [F3], choose the orthonormal eigenvectors to include $u_{1}, u_{2}$ (within either selected eigenspace, replace its finite orthonormal basis by one containing the specified vectors). The operators $σ_{\pm} = ρ \pm ε (T_{u_{1}, u_{1}} - T_{u_{2}, u_{2}})$ therefore have the same trace-norm convergent expansion with the two selected weights replaced by $λ_{1} \pm ε$ and $λ_{2} \mp ε$ . All weights remain positive, since $λ_{1} - ε \geq λ_{2} / 2 > 0$ and $λ_{2} - ε = λ_{2} / 2 > 0$ , and their sum remains one. By [F2] and [F5] each $σ_{\pm}$ is a density operator and hence a preparation. Their average is $ρ$ , while $σ_{+} - σ_{-} = 2 ε (T_{u_{1}, u_{1}} - T_{u_{2}, u_{2}}) \neq 0$ by [F5]. Thus [F1] reads the identity as a nontrivial statistical mixture with weights $1 / 2$ , so $ρ$ is not extreme by [F6], consistent with step 1.3.

Statistical mixtures, spectral decompositions and the extremal role of pure preparations

Statement

Facts & Assumptions

Proof

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