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The non-selective Lueders map and the discarded-outcome ensemble

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a nonzero complex Hilbert space (Hilbert space), let $(P_{a})_{a \in A}$ be a finite or countably infinite family of pairwise orthogonal orthogonal projections on $H$ (The Hilbert orthogonal projection onto a closed subspace) whose strong sum is the identity, $\sum_{a \in A} P_{a} x = x (x \in H),$ let $ρ$ be a density operator (Density operators and scalar spectral probability clause (a)), and put $p_{a} : = tr (ρ P_{a})$ . Define the non-selective Lueders map $Φ (ρ) : = \sum_{a \in A} P_{a} ρ P_{a} .$ Then:

(a) the series converges in trace norm; $Φ (ρ)$ is a positive trace-class operator with $tr Φ (ρ) = 1$ , hence a density operator, and $Φ (ρ) = \sum_{a} P_{a} Φ (ρ) P_{a}$ ;

(b) for every $b$ , $P_{b} Φ (ρ) = Φ (ρ) P_{b} = P_{b} Φ (ρ) P_{b} = P_{b} ρ P_{b}$ ; in particular $Φ (ρ)$ commutes with every $P_{b}$ ;

(c) discarded-outcome ensemble. $Φ (ρ) = \sum_{a : p_{a} > 0} p_{a} ρ_{a}$ , where $ρ_{a} = P_{a} ρ P_{a} / p_{a}$ is the selective update of The selective Lueders update by an orthogonal projection; the series converges in trace norm;

(d) if $B \in B (H)$ is bounded and $P_{a} B = B P_{a}$ for every $a$ , then $tr (Φ (ρ) B) = tr (ρ B)$ ;

(e) if $ψ = α u + β v$ with $u, v$ orthonormal, $u \in ran P_{1}$ , $v \in ran P_{2}$ for distinct labels $1 \neq 2$ , and $ρ = T_{ψ, ψ}$ , then $Φ (ρ) = ∣ α ∣^{2} T_{u, u} + ∣ β ∣^{2} T_{v, v}$ , and $tr (Φ (ρ) (T_{u, v} + T_{v, u})) = 0, tr (ρ (T_{u, v} + T_{v, u})) = 2 Re (α \overline{β});$

(f) if $ρ = T_{ψ, ψ}$ for a nonzero $ψ$ then $Φ (ρ) = \sum_{a : P_{a} ψ \neq 0} ∥ P_{a} ψ ∥^{2} T_{ν (P_{a} ψ), ν (P_{a} ψ)}$ where $ν : H ∖ {0} \to H$ is the normalization map $ν (x) : = x / ∥ x ∥$ , the series converging in trace norm.

No physical interpretation of the map is asserted.

Facts & Assumptions

Given: A nonzero complex Hilbert space $H$ ; pairwise orthogonal projections $(P_{a})_{a \in A}$ with strong sum $I$ ; a density operator $ρ$ ; probabilities $p_{a} = tr (ρ P_{a})$ ; a bounded operator $B$ commuting with every $P_{a}$ ; a vector $ψ = α u + β v$ .

[F1]

Countable Kraus normalization. If $(M_{a})$ is a finite or countable family of bounded operators with $\sum_{a} M_{a}^{*} M_{a} \to I$ strongly, then $\sum_{a} ∥ M_{a} x ∥^{2} = ∥ x ∥^{2}$ for every $x$ ; for every positive trace-class $ρ$ the series $\sum_{a} M_{a} ρ M_{a}^{*}$ converges in trace norm to a positive trace-class operator with trace $tr ρ$ ; the map extends uniquely to a trace-preserving linear positive map of the trace-class operators, and it maps density operators to density operators (Countable Kraus normalization and trace-norm convergence clauses (a)-(c)).

[F2]

Orthogonal projections and orthogonality. Each $P_{a}$ satisfies $P_{a}^{2} = P_{a} = P_{a}^{*}$ and $P_{a} P_{b} = 0$ for $a \neq b$ ; $∥ P_{a} ∥ \leq 1$ ; the hypothesis of the Statement is that the finite partial sums $\sum_{a \leq N} P_{a}$ converge strongly to $I$ (The Hilbert orthogonal projection onto a closed subspace).

[F3]

Zero-probability summands vanish. If $p_{a} = tr (ρ P_{a}) = 0$ , then $P_{a} ρ P_{a} = 0$ . Indeed, using the expansion [F3'] and trace-norm continuity of left and right multiplication by $P_{a}$ , $P_{a} ρ P_{a} = \sum_{n} w_{n} T_{P_{a} u_{n}, P_{a} u_{n}}$ with trace $\sum_{n} w_{n} ∥ P_{a} u_{n} ∥^{2} = tr (ρ P_{a}) = 0$ (Rank-one operator calculus clauses (b) and (d)); every summand is nonnegative, so each summand vanishes and the series is $0$ .

[F3'] Expansion of a density operator. $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ with $(u_{n})$ a finite or countable orthonormal family and $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ (Density operators and scalar spectral probability clause (a)).

[F4]

Rank-one calculus. $T_{u, v} x = ⟨ x, v ⟩ u$ is trace class with $tr (T_{u, v}) = ⟨ u, v ⟩$ ; for bounded $B$ , $B T_{u, v} = T_{B u, v}$ and $T_{u, v} B = T_{u, B^{*} v}$ (Rank-one operator calculus clauses (b)-(d)). For every $x, y \in H$ , $T_{x, x}^{*} = T_{x, x}$ and $⟨ T_{x, x} y, y ⟩ = ∣ ⟨ y, x ⟩ ∣^{2} \geq 0$ , so $T_{x, x}$ is positive by Self-adjoint, positive, unitary and normal operators.

[F5]

Trace-class ideal and cyclicity. For bounded $B$ and trace-class $T$ , $B T$ and $T B$ are trace class with $tr (B T) = tr (T B)$ (Trace class is a two sided Banach operator ideal, Nuclear bounded cyclicity for trace-class operators clause (b)).

[F6]

Selective update. For a single projection $P$ with $tr (ρ P) > 0$ the operator $ρ_{P} = P ρ P / tr (ρ P)$ is a density operator with $P ρ_{P} P = ρ_{P}$ and $Q ρ_{P} Q = 0$ for every projection $Q$ with $Q P = 0$ (The selective Lueders update by an orthogonal projection clauses (a)-(c)).

Proof

1.1givenF1F2algebra

[The map is the projection Kraus map] For each $a$ , $P_{a}^{*} P_{a} = P_{a}^{2} = P_{a}$ by [F2], so $\sum_{a \leq N} P_{a}^{*} P_{a} = \sum_{a \leq N} P_{a} \to I$ strongly: the hypothesis of [F1] holds with $M_{a} = P_{a}$ . Therefore $Φ (ρ) = \sum_{a} P_{a} ρ P_{a}$ converges in trace norm, is positive trace class with $tr Φ (ρ) = tr ρ = 1$ , and is a density operator. This proves the first part of (a).

2.1step 1.1F1F2F5algebra

[The compressed identity] Fix $b$ . Left and right multiplication by the bounded operator $P_{b}$ are continuous in trace norm, since $∥ P_{b} T ∥_{1} \leq ∥ P_{b} ∥ ∥ T ∥_{1}$ and $∥ T P_{b} ∥_{1} \leq ∥ P_{b} ∥ ∥ T ∥_{1}$ for trace-class $T$ (Trace class is a two sided Banach operator ideal); applying this to the trace-norm convergent series of step 1.1 gives $P_{b} Φ (ρ) = \sum_{a} P_{b} P_{a} ρ P_{a}$ and $Φ (ρ) P_{b} = \sum_{a} P_{a} ρ P_{a} P_{b}$ . Since $P_{b} P_{a} = 0$ for $a \neq b$ and $P_{b}^{2} = P_{b}$ [F2], all summands vanish except $a = b$ , and that summand is $P_{b} ρ P_{b}$ ; hence $P_{b} Φ (ρ) = P_{b} ρ P_{b} = Φ (ρ) P_{b}$ and therefore $P_{b} Φ (ρ) P_{b} = P_{b} ρ P_{b}$ . Summing the identity $P_{b} Φ (ρ) P_{b} = P_{b} ρ P_{b}$ over $b$ and using step 1.1 once more gives $Φ (ρ) = \sum_{b} P_{b} Φ (ρ) P_{b}$ . This proves (a) and (b).

2.2step 1.1F3F6algebra

[The discarded-outcome mixture] For a label $a$ with $p_{a} > 0$ , [F6] gives $P_{a} ρ P_{a} = p_{a} ρ_{a}$ with $ρ_{a}$ a density operator, and [F3] gives $P_{a} ρ P_{a} = 0$ for a label with $p_{a} = 0$ . Hence the summands with $p_{a} = 0$ vanish and, reindexing the trace-norm convergent series of step 1.1 over the subfamily ${a : p_{a} > 0}$ , $Φ (ρ) = \sum_{a : p_{a} > 0} p_{a} ρ_{a}$ . This is (c).

2.3step 1.1F1F5algebra

[Invariance under compatible bounded quantities] Let $B$ be bounded with $P_{a} B = B P_{a}$ for every $a$ . Then $P_{a} B ρ P_{a} = B P_{a} ρ P_{a}$ and, using the trace-norm continuity of left multiplication by $B$ together with step 1.1, $B Φ (ρ) = \sum_{a} P_{a} (B ρ) P_{a}$ . The right-hand side is the Kraus map of [F1] evaluated at the trace-class operator $B ρ$ , so $tr (B Φ (ρ)) = tr Φ (B ρ) = tr (B ρ)$ by trace preservation; since $tr (Φ (ρ) B) = tr (B Φ (ρ))$ [F5], this gives $tr (Φ (ρ) B) = tr (ρ B)$ . This is (d).

3.1step 2.2F2F4algebra

[Coherence destruction] Let $ψ = α u + β v$ with $u, v$ orthonormal, $u \in ran P_{1}$ and $v \in ran P_{2}$ , $1 \neq 2$ , and $ρ = T_{ψ, ψ}$ . For $a \notin {1, 2}$ one has $P_{a} u = P_{a} P_{1} u = 0$ and $P_{a} v = P_{a} P_{2} v = 0$ by [F2], so $P_{a} ψ = 0$ ; and $P_{1} ψ = α u$ , $P_{2} ψ = β v$ . Hence the series has only two nonzero summands, and [F4] gives $P_{1} ρ P_{1} = T_{P_{1} ψ, P_{1} ψ} = ∣ α ∣^{2} T_{u, u}$ and $P_{2} ρ P_{2} = ∣ β ∣^{2} T_{v, v}$ , so $Φ (ρ) = ∣ α ∣^{2} T_{u, u} + ∣ β ∣^{2} T_{v, v}$ . For the pairing with $T_{u, v}$ : $tr (T_{u, u} T_{u, v}) = ⟨ T_{u, u} u, v ⟩ = ⟨ u, v ⟩ = 0$ and $tr (T_{v, v} T_{u, v}) = ⟨ T_{v, v} u, v ⟩ = ⟨ u, v ⟩ ⟨ v, v ⟩ = 0$ , and conjugating gives the same vanishing for $T_{v, u}$ ; hence $tr (Φ (ρ) (T_{u, v} + T_{v, u})) = 0$ . For the coherent state, $tr (ρ T_{u, v}) = ⟨ T_{ψ, ψ} u, v ⟩ = ⟨ u, ψ ⟩ ⟨ ψ, v ⟩ = \overline{α} β$ and $tr (ρ T_{v, u}) = \overline{⟨ u, ψ ⟩ ⟨ ψ, v ⟩} = α \overline{β}$ , so the cross term is $2 Re (α \overline{β})$ . This is (e).

4.1step 1.1F4algebra∎

[The pure-state formula] Let $ρ = T_{ψ, ψ}$ with $ψ \neq 0$ . By step 1.1 with $ρ = T_{ψ, ψ}$ and the rank-one rules [F4], $P_{a} ρ P_{a} = T_{P_{a} ψ, P_{a} ψ}$ , and $T_{P_{a} ψ, P_{a} ψ} = ∥ P_{a} ψ ∥^{2} T_{ν (P_{a} ψ), ν (P_{a} ψ)}$ when $P_{a} ψ \neq 0$ , while both sides vanish when $P_{a} ψ = 0$ ; the trace-norm convergence of the resulting series is that of step 1.1. This is (f).

The non-selective Lueders map and the discarded-outcome ensemble

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

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