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Distinct density operators are separated by the law of a bounded rank-one projection

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )); the Axiom of Choice is used only where the spectral theorem for unbounded self-adjoint operators is invoked (The Axiom of Choice). Let $H$ be a nonzero complex Hilbert space (Hilbert space) and let $ρ, σ$ be density operators on $H$ with $ρ \neq σ$ (Density operators and scalar spectral probability clause (a)). Then:

(a) There is a unit vector $w \in H$ with $⟨ ρ w, w ⟩ \neq ⟨ σ w, w ⟩$ .

(b) For the rank-one orthogonal projection $P : = T_{w, w}$ of Rank-one operator calculus (The Hilbert orthogonal projection onto a closed subspace), whose spectral projection-valued measure assigns $P$ to the Borel set ${1}$ , the two scalar laws of Density operators and scalar spectral probability clause (b) satisfy $μ_{ρ}^{P} ({1}) = tr (ρ P) = ⟨ ρ w, w ⟩, μ_{σ}^{P} ({1}) = tr (σ P) = ⟨ σ w, w ⟩,$ so the bounded sharp observable $P$ assigns different outcome probabilities to $ρ$ and $σ$ .

(c) For every unit vector $w$ and every self-adjoint trace-class $T$ one has $∣ tr (T T_{w, w}) ∣ \leq ∥ T ∥_{1}$ ; in particular $∣ ⟨ ρ w, w ⟩ - ⟨ σ w, w ⟩ ∣ \leq ∥ ρ - σ ∥_{1}$ .

Facts & Assumptions

Given: The Axiom of Countable Choice and, for the spectral PVM of clause (b), the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice); a nonzero complex Hilbert space $H$ ; density operators $ρ, σ$ on $H$ ; unit vectors $w \in H$ .

[F1]

Rank-one calculus. $T_{u, v} x = ⟨ x, v ⟩ u$ defines a trace-class operator with $∥ T_{u, v} ∥_{1} = ∥ u ∥ ∥ v ∥$ and $tr (T_{u, v}) = ⟨ u, v ⟩$ ; $T_{u, u}$ is self-adjoint and positive with $⟨ T_{u, u} x, x ⟩ = ∣ ⟨ x, u ⟩ ∣^{2}$ (Rank-one operator calculus).

[F2]

One-dimensional projections. For $u \in S (H)$ the operator $T_{u, u}$ is the Hilbert orthogonal projection onto the one-dimensional subspace $C u$ ; in particular $T_{u, u}^{2} = T_{u, u} = T_{u, u}^{*}$ (Rays, unit vectors and rank-one Hilbert projections, The Hilbert orthogonal projection onto a closed subspace).

[F3]

Density operators and their law. A density operator $τ$ is self-adjoint, positive and trace class with $tr τ = 1$ , and admits an expansion $τ = \sum_{n} t_{n} T_{v_{n}, v_{n}}$ with $t_{n} \geq 0$ , $\sum_{n} t_{n} = 1$ and $(v_{n})$ orthonormal; for every PVM $E$ , $B \mapsto tr (τ E (B))$ is a probability measure with $tr (τ E (B)) = \sum_{n} t_{n} ∥ E (B) v_{n} ∥^{2}$ (Density operators and scalar spectral probability clauses (a) and (b)).

[F4]

Trace ideal and trace bounds. Products of a trace-class operator with a bounded operator are trace class and $∥ A T B ∥_{1} \leq ∥ A ∥ ∥ T ∥_{1} ∥ B ∥$ ; the trace is linear on trace-class operators with $∣ tr S ∣ \leq ∥ S ∥_{1}$ (Trace class is a two sided Banach operator ideal clause 2, Trace is absolutely convergent and basis independent clause 4).

[F5]

Quadratic form and norm. For a bounded self-adjoint $T$ one has $∥ T ∥ = \sup {∣ ⟨ T x, x ⟩ ∣ : ∥ x ∥ = 1}$ , and the value is $0$ exactly when $T = 0$ (Norm of a self adjoint operator from its quadratic form).

[F6]

Spectral PVM of a projection. Every orthogonal projection $P$ is a self-adjoint operator with $0 \leq P \leq I$ ; by the spectral theorem it has a unique regular PVM $E_{P}$ on the Borel sets of $R$ with $P = \int λ d E_{P} (λ)$ (Spectral theorem for unbounded self-adjoint operators (PVM form)).

Proof

1.1givenF3F4F5algebra

[Quadratic forms separate trace-class states] Put $T : = ρ - σ$ , a nonzero self-adjoint trace-class operator by [F3] and [F4]. Suppose $⟨ T w, w ⟩ = 0$ for every unit vector $w$ ; then the supremum of [F5] is $0$ , so $∥ T ∥ = 0$ and $T = 0$ , contradicting $ρ \neq σ$ . Hence there is a unit $w$ with $0 \neq ⟨ (ρ - σ) w, w ⟩ = ⟨ ρ w, w ⟩ - ⟨ σ w, w ⟩$ , proving (a).

1.2givenF1F3F4algebra

[The trace pairing of a rank-one projection] Let $τ = \sum_{n} t_{n} T_{v_{n}, v_{n}}$ be the expansion of a density operator and let $w$ be a unit vector. For each $n$ , the product rule $T_{v_{n}, v_{n}} T_{w, w} = T_{v_{n}, ⟨ v_{n}, w ⟩ w}$ holds by direct evaluation on any $x$ : $T_{v_{n}, v_{n}} T_{w, w} x = ⟨ x, w ⟩ ⟨ w, v_{n} ⟩ v_{n} = T_{v_{n}, ⟨ v_{n}, w ⟩ w} x$ ; hence by [F1] its trace is $⟨ v_{n}, ⟨ v_{n}, w ⟩ w ⟩ = ∣ ⟨ v_{n}, w ⟩ ∣^{2}$ . Multiplying by $t_{n}$ , summing and using the trace ideal and trace bound [F4] together with $\sum_{n} t_{n} ∥ v_{n} ∥^{2} ∥ w ∥^{2} = 1$ gives the absolutely convergent identity $tr (τ T_{w, w}) = \sum_{n} t_{n} ∣ ⟨ v_{n}, w ⟩ ∣^{2}$ . On the other hand $⟨ τ w, w ⟩ = \sum_{n} t_{n} ⟨ T_{v_{n}, v_{n}} w, w ⟩ = \sum_{n} t_{n} ∣ ⟨ w, v_{n} ⟩ ∣^{2}$ by [F1] and continuity of the pairing against the norm convergent series. Since $∣ ⟨ w, v_{n} ⟩ ∣ = ∣ ⟨ v_{n}, w ⟩ ∣$ , the two series agree: $tr (τ T_{w, w}) = ⟨ τ w, w ⟩$ for every density operator $τ$ .

2.1step 1.1step 1.2F2F3F6algebra

[The projection separates the two states] Fix a unit $w$ with $⟨ ρ w, w ⟩ \neq ⟨ σ w, w ⟩$ , furnished by step 1.1, and put $P : = T_{w, w}$ , an orthogonal projection by [F2]. The map $E^{'} (B) : = 1_{1 \in B} P + 1_{0 \in B} (I - P)$ is a regular PVM on the Borel sets of $R$ with $\int λ d E^{'} (λ) = 1 \cdot P + 0 \cdot (I - P) = P$ : the values are orthogonal projections with $E^{'} (\emptyset) = 0$ and $E^{'} (R) = I$ , multiplicativity holds because $P (I - P) = 0$ , and countable additivity holds on the two-point space ${0, 1}$ . By uniqueness in the spectral theorem [F6], $E^{'}$ is the spectral PVM of $P$ , so $E_{P} ({1}) = P$ and $μ_{τ}^{P} ({1}) = tr (τ P)$ for every density operator $τ$ by [F3]. Step 1.2 identifies these numbers as $⟨ ρ w, w ⟩$ and $⟨ σ w, w ⟩$ , which differ by the choice of $w$ ; this proves (b).

3.1step 1.1step 1.2step 2.1F2F4algebra∎

[The uniform bound] Let $w$ be a unit vector and let $T$ be self-adjoint trace class. By [F2] and [F4], $∥ T_{w, w} ∥ \leq ∥ T_{w, w} ∥_{1} = 1$ and $∣ tr (T T_{w, w}) ∣ \leq ∥ T T_{w, w} ∥_{1} \leq ∥ T ∥_{1}$ ; applying this to $T = ρ - σ$ and using step 1.2 gives $∣ ⟨ ρ w, w ⟩ - ⟨ σ w, w ⟩ ∣ = ∣ tr ((ρ - σ) T_{w, w}) ∣ \leq ∥ ρ - σ ∥_{1}$ , which is (c); the existence statement of (a) and the separating probability identity of (b) are steps 1.1 and 2.1.

Distinct density operators are separated by the law of a bounded rank-one projection

Statement

Facts & Assumptions

Proof

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