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Consequences of a declared ideal sharp measurement: repeatability and discarded ensembles

Physical scope

A conditional consequence of the adopted ideal conditional-measurement postulate P3 for a declared ideal sharp measurement: the Born outcome probabilities, the selective Lueders conditional states, the certainty of an immediately repeated outcome, the discarded-outcome state as the outcome-weighted mixture of the conditional states, invariance of bounded quantities compatible with the declared measurement under discarding, and the loss of coherence between non-selected branches. The theorem asserts nothing about any laboratory apparatus, about outcome uniqueness or about measurements outside the declared class.

Statement

Adopt the state postulate P1, the observable and probability postulate P2 and the ideal conditional-measurement postulate P3 of the framework (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Ideal conditional measurement postulate P3: the projective Lueders update and discarded outcomes), and assume the Axiom of Countable Choice, the hypothesis of the cited mathematics. Let $(P_{a})_{a \in A}$ be a declared ideal sharp measurement of a system with state space $D (H)$ (Declared ideal measurements and instruments of the framework, The state space of the framework: preparations, pure rays and density operators) and let $ρ$ be a state. Then:

(a) Outcome law. For every outcome label $a$ the probability is $p_{a} = tr (ρ P_{a})$ , with $p_{a} \geq 0$ and $\sum_{a} p_{a} = 1$ ; for the pure state of a unit vector $ψ$ it is $p_{a} = ∥ P_{a} ψ ∥^{2}$ .

(b) Conditional state and repeatability. If $p_{a} > 0$ , the conditional state after the outcome $a$ is the state $ρ_{a} = P_{a} ρ P_{a} / p_{a}$ , with $tr (ρ_{a} P_{a}) = 1$ and $tr (ρ_{a} P_{b}) = 0$ for $b \neq a$ . Immediate repetition of the same declared measurement therefore returns the same outcome with certainty: $P (second outcome b ∣ first outcome a) = δ_{a b} .$

(c) Discarded outcomes. The discarded-outcome state $Φ (ρ) = \sum_{a} P_{a} ρ P_{a}$ is a state and coincides with the outcome-weighted mixture of the conditional states, $Φ (ρ) = \sum_{a : p_{a} > 0} p_{a} ρ_{a},$ the series converging in trace norm; moreover $tr (Φ (ρ) P_{a}) = p_{a}$ for every $a$ and $Φ (ρ) = \sum_{a} P_{a} Φ (ρ) P_{a}$ . The conditional-vs-discarded distinction is therefore exactly the distinction between one selected branch $ρ_{a}$ and the mixture of all branches.

(d) Compatible bounded quantities. If $B$ is a bounded operator with $P_{a} B = B P_{a}$ for every $a$ , then $tr (Φ (ρ) B) = tr (ρ B)$ . In particular, if the declared measurement is the spectral measurement of a bounded assigned sharp observable $A = \sum_{a} a_{a} P_{a}$ (a finite or norm-convergent series with real $a_{a}$ ), then $A$ commutes with every $P_{a}$ and the mean of $A$ is unchanged by discarding the outcomes.

(e) Loss of coherence. 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 $Φ (ρ) = ∣ α ∣^{2} T_{u, u} + ∣ β ∣^{2} T_{v, v}, tr (Φ (ρ) (T_{u, v} + T_{v, u})) = 0,$ whereas for the coherent preparation $tr (ρ (T_{u, v} + T_{v, u})) = 2 Re (α \overline{β})$ : the discarded-outcome ensemble is the classical mixture of the branch states and carries no coherence between the ranges of $P_{1}$ and $P_{2}$ .

(f) Scope. Clauses (a)-(e) are conditional on the adopted P3 and on the mathematical items cited; they do not assert that any laboratory apparatus realizes a declared ideal sharp measurement, do not select a unique recorded outcome, and do not describe the disturbance produced by an apparatus outside the declared class (clause (v) of P3). The identification of a device with a declared measurement is a tested identification with uncertainty under P7.

Facts & Assumptions

Given: The adopted postulates P1, P2, P3; a declared ideal sharp measurement $(P_{a})_{a \in A}$ with strong sum $I$ ; a state $ρ$ ; probabilities $p_{a} = tr (ρ P_{a})$ ; the discarded-outcome state $Φ (ρ) = \sum_{a} P_{a} ρ P_{a}$ ; a bounded operator $B$ commuting with every $P_{a}$ ; vectors $ψ, u, v$ and scalars $α, β$ as in clause (e).

[F1]

Declared measurements. The family $(P_{a})$ is a finite or countable family of pairwise orthogonal orthogonal projections with strong sum $I$ , and its Lueders instrument has outcome probabilities $p_{a} = tr (ρ P_{a})$ , conditional states $ρ_{a} = P_{a} ρ P_{a} / p_{a}$ for $p_{a} > 0$ , and discarded-outcome state $Φ (ρ) = \sum_{a} P_{a} ρ P_{a}$ (Declared ideal measurements and instruments of the framework clauses (a)-(b); Ideal conditional measurement postulate P3: the projective Lueders update and discarded outcomes clauses (i)-(iv)).

[F2]

Selective update. For $p_{a} > 0$ the operator $ρ_{a}$ is a state with $tr (ρ_{a} P_{a}) = 1$ , $tr (ρ_{a} Q) = 0$ for every orthogonal projection $Q$ with $Q P_{a} = 0$ , and $P_{a} ρ_{a} P_{a} = ρ_{a}$ (The selective Lueders update by an orthogonal projection clauses (a)-(c)).

[F3]

Non-selective map. The series $Φ (ρ)$ converges in trace norm; $Φ (ρ)$ is a state; $Φ (ρ) = \sum_{a : p_{a} > 0} p_{a} ρ_{a}$ ; $P_{b} Φ (ρ) P_{b} = P_{b} ρ P_{b}$ for every $b$ , so in particular $tr (Φ (ρ) P_{b}) = p_{b}$ ; $Φ (ρ) = \sum_{b} P_{b} Φ (ρ) P_{b}$ ; and for bounded $B$ commuting with every $P_{a}$ one has $tr (Φ (ρ) B) = tr (ρ B)$ (The non-selective Lueders map and the discarded-outcome ensemble clauses (a)-(d)).

[F4]

Rank-one pairing. $T_{u, v} x = ⟨ x, v ⟩ u$ is trace class with $tr (T_{u, v}) = ⟨ u, v ⟩$ , and $tr (S T_{u, v}) = ⟨ S u, v ⟩$ for trace-class $S$ (Rank-one operator calculus clause (b), Nuclear bounded cyclicity for trace-class operators clause (b)).

[F5]

States and mixtures. Under Countable Choice, finite convex combinations of states are states (The state space is convex, closed under countable mixtures, and complete in the trace norm clause (a)); the trace is linear and $∣ tr T ∣ \leq ∥ T ∥_{1}$ , hence trace-norm continuous (Trace is absolutely convergent and basis independent clause 4). The given state implies $H \neq {0}$ , as required by the convexity supplier.

Proof

1.1givenF1F2F3F4algebra

[Clause (a)] By [F1] the outcome probabilities of the declared measurement are $p_{a} = tr (ρ P_{a})$ . Since $0 \leq P_{a} \leq I$ and $ρ \geq 0$ with $tr ρ = 1$ , positivity of $P_{a} ρ P_{a}$ and cyclicity give $p_{a} = tr (P_{a} ρ P_{a}) \geq 0$ [F2, F4]. Trace-norm convergence of $Φ (ρ) = \sum_{a} P_{a} ρ P_{a}$ and continuity of the trace give $\sum_{a} p_{a} = tr Φ (ρ) = 1$ by [F3]. For a pure preparation, [F1] with the rank-one calculus [F4] gives $p_{a} = tr (T_{ψ, ψ} P_{a}) = ⟨ P_{a} ψ, ψ ⟩ = ∥ P_{a} ψ ∥^{2}$ . This is (a).

1.2givenF3F4algebra

[Clause (e)] Under the hypotheses of clause (e), the computation of The non-selective Lueders map and the discarded-outcome ensemble clause (e) applies verbatim to $ρ = T_{ψ, ψ}$ : it gives $Φ (ρ) = ∣ α ∣^{2} T_{u, u} + ∣ β ∣^{2} T_{v, v}$ , $tr (Φ (ρ) (T_{u, v} + T_{v, u})) = 0$ , and $tr (ρ (T_{u, v} + T_{v, u})) = 2 Re (α \overline{β})$ . This is (e).

2.1step 1.1givenF1F2algebra

[Clause (b)] If $p_{a} > 0$ , [F2] gives that $ρ_{a}$ is a state with $tr (ρ_{a} P_{a}) = 1$ ; for $b \neq a$ the projections $P_{b}$ and $P_{a}$ satisfy $P_{b} P_{a} = 0$ , so [F2] gives $tr (ρ_{a} P_{b}) = 0$ . Applying the outcome law of clause (a) to the preparation $ρ_{a}$ , the probability of the outcome $b$ in an immediate repetition of the same declared measurement is $tr (ρ_{a} P_{b}) = δ_{a b}$ . This is (b).

3.1step 2.1givenF3F5algebra

[Clause (c)] By [F3], $Φ (ρ)$ is a state and equals the trace-norm convergent series $\sum_{a : p_{a} > 0} p_{a} ρ_{a}$ ; since $tr ((p_{a} ρ_{a}) P_{b}) = δ_{a b} p_{a}$ by clause (b) and trace-norm continuity of the trace, summing gives $tr (Φ (ρ) P_{b}) = p_{b}$ for every $b$ , and [F3] also gives $Φ (ρ) = \sum_{b} P_{b} Φ (ρ) P_{b}$ . This is (c).

4.1step 3.1givenF3algebra

[Clause (d)] If $B$ is bounded and $P_{a} B = B P_{a}$ for every $a$ , [F3] gives $tr (Φ (ρ) B) = tr (ρ B)$ . If $A = \sum_{a} a_{a} P_{a}$ is a bounded assigned sharp observable with real coefficients $a_{a}$ , then $A P_{b} = \sum_{a} a_{a} P_{a} P_{b} = a_{b} P_{b} = P_{b} A$ , so $A$ commutes with every $P_{a}$ and the identity applies to $B = A$ : the mean $tr (ρ A)$ of the bounded observable $A$ is unchanged by discarding the outcomes. This is (d).

5.1step 1.2givenF1F3algebra∎

[Clause (f) and scope] Clauses (a)-(e) were obtained from the adopted clauses of P3 [F1] and the cited mathematical lemmas [F2, F3, F4]; no clause used a claim about a laboratory apparatus, an apparatus dynamics, or a preferred recorded outcome. Clause (v) of P3 explicitly excludes outcome uniqueness and arbitrary apparatuses from the declared class, and P7 makes the identification of a device with the declared measurement a tested identification with uncertainty. This is (f).

Consequences of a declared ideal sharp measurement: repeatability and discarded ensembles

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources