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Compatible Lüders repeatability

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a nonzero complex Hilbert space, let $E, F$ be regular projection-valued measures on the Borel $σ$ -algebra of $R$ with $E (B) F (C) = F (C) E (B)$ for all Borel $B, C \subseteq R$ , and let $G$ be the joint projection-valued measure on the Borel $σ$ -algebra of $R^{2}$ with $G (B \times C) = E (B) F (C)$ and marginals $E, F$ (Joint PVM construction for strongly commuting regular projection-valued measures). Fix finite families of Borel sets $B_{1}, \dots, B_{n}$ with $B_{i} \cap B_{k} = \emptyset$ for $i \neq k$ and $C_{1}, \dots, C_{m}$ with $C_{j} \cap C_{l} = \emptyset$ for $j \neq l$ , put

$P_{i} : = E (B_{i}), Q_{j} : = F (C_{j}), R_{i j} : = P_{i} Q_{j},$

and let $ρ$ be a density operator on $H$ (Density operators and scalar spectral probability(a)). The listed projections are stipulated to be outcomes of ideal Lüders instruments, with outcome probabilities $tr (ρ P_{i})$ and post-measurement states $ρ_{i} : = P_{i} ρ P_{i} / tr (ρ P_{i})$ when $tr (ρ P_{i}) > 0$ , and likewise for $(Q_{j})$ . If either listed family fails to sum to the identity, include its complementary projection $I - \sum_{i} P_{i}$ or $I - \sum_{j} Q_{j}$ as an additional outcome; only probabilities for the listed outcomes are displayed below. A sequential run measures the first instrument and then the second on the resulting state. Then:

(a) $P_{i}, Q_{j}$ and $R_{i j}$ are orthogonal projections (The Hilbert orthogonal projection onto a closed subspace), $R_{i j} = Q_{j} P_{i} = G (B_{i} \times C_{j})$ , and $P_{i} P_{k} = 0$ for $i \neq k$ , $Q_{j} Q_{l} = 0$ for $j \neq l$ , $R_{i j} R_{k l} = 0$ whenever $(i, j) \neq (k, l)$ .

(b) The probability of the listed ordered outcome pair $(i then j)$ equals, when $tr (ρ P_{i}) > 0$ , $tr (P_{i} ρ P_{i} Q_{j}) / tr (ρ P_{i}) \cdot tr (ρ P_{i}) = tr (ρ R_{i j})$ , and is zero if the first outcome has zero probability; the same number is obtained in the reversed order, so the sequential order of the two compatible sharp measurements gives the same masses to all listed outcome pairs: $tr (P_{i} ρ P_{i} Q_{j}) = tr (Q_{j} ρ Q_{j} P_{i}) = tr (ρ R_{i j}) = tr (ρ G (B_{i} \times C_{j})) .$

(c) If $tr (ρ R_{i j}) > 0$ , the conditional state after the pair is order-independent: $\frac{Q_{j} P_{i} ρ P_{i} Q_{j}}{tr (ρ R_{i j})} = \frac{P_{i} Q_{j} ρ Q_{j} P_{i}}{tr (ρ R_{i j})} = \frac{R_{i j} ρ R_{i j}}{tr (ρ R_{i j})},$ and it is a density operator.

(d) Repeatability. Under the positive-probability hypothesis $tr (ρ R_{i j}) > 0$ of (c), with $σ : = R_{i j} ρ R_{i j} / tr (ρ R_{i j})$ one has $tr (σ R_{i j}) = 1$ and $tr (σ P_{i}) = tr (σ Q_{j}) = 1$ , and the Lüders update by $R_{i j}$ , $P_{i}$ or $Q_{j}$ leaves $σ$ unchanged: $R_{i j} σ R_{i j} / tr (σ R_{i j}) = P_{i} σ P_{i} / tr (σ P_{i}) = Q_{j} σ Q_{j} / tr (σ Q_{j}) = σ$ . Thus immediate repetition of the compatible pair, or of either single sharp measurement, reproduces the same outcomes with certainty.

(e) If $⋃_{j} C_{j} = R$ then $\sum_{j} tr (ρ R_{i j}) = tr (ρ P_{i})$ , and if $⋃_{i} B_{i} = R$ then $\sum_{i} tr (ρ R_{i j}) = tr (ρ Q_{j})$ : the joint law has the sharp-bin marginals.

(f) Scope. All of this is a conditional calculation for the stipulated ideal Lüders instrument and the joint PVM $G$ ; no claim is made that a laboratory apparatus realizes the Lüders instrument or the joint measurement, and no dynamical time development is described.

Facts & Assumptions

Given: A nonzero complex Hilbert space $H$ ; regular PVMs $E, F$ with $E (B) F (C) = F (C) E (B)$ ; the joint PVM $G$ ; Borel sets $B_{1}, \dots, B_{n}$ and $C_{1}, \dots, C_{m}$ pairwise disjoint within each family; a density operator $ρ$ ; the projections $P_{i}, Q_{j}, R_{i j}$ .

[F1]

An orthogonal projection on $H$ is a bounded operator $P$ with $P^{2} = P = P^{*}$ ; for a PVM $E$ every value $E (B)$ is an orthogonal projection, $E (\emptyset) = 0$ , $E (R) = I$ , $E (B \cap C) = E (B) E (C)$ , and $E$ is strongly countably additive (The Hilbert orthogonal projection onto a closed subspace, Joint PVM construction for strongly commuting regular projection-valued measures).

[F2]

$G$ is a regular PVM on the Borel $σ$ -algebra of $R^{2}$ with $G (B \times C) = E (B) F (C)$ for Borel rectangles and marginals $G (B \times R) = E (B)$ , $G (R \times C) = F (C)$ ; it is the unique such PVM and every value $G (S)$ is an orthogonal projection with $0 \leq G (S) \leq I$ (Joint PVM construction for strongly commuting regular projection-valued measures).

[F3]

A density operator is a self-adjoint positive trace-class $ρ$ with $tr ρ = 1$ ; for every PVM $E$ on a measurable space the set function $μ_{ρ} (S) : = tr (ρ E (S))$ is a probability measure, and in every expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ with orthonormal $(u_{n})$ and weights $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ , one has $μ_{ρ} (S) = \sum_{n} w_{n} ∥ E (S) u_{n} ∥^{2}$ (Density operators and scalar spectral probability(a),(b)).

[F4]

For bounded $B$ and trace-class $T$ : $B T$ and $T B$ are trace class, $tr (B T) = tr (T B)$ , $tr$ is linear on trace-class operators, and $∣ tr (B T) ∣ \leq ∥ B ∥ ∥ T ∥_{1}$ (Nuclear bounded cyclicity for trace-class operators(b), Trace is absolutely convergent and basis independent).

[F5]

The Lüders instrument and the sequential protocol of the Statement are stipulated; the conditional state is defined only when the corresponding probability is positive.

Proof

1.1F1F2givenalgebra

[Sharp bins and their products are projections] By [F1], $P_{i} = E (B_{i})$ and $Q_{j} = F (C_{j})$ are orthogonal projections; $P_{i} P_{k} = E (B_{i}) E (B_{k}) = E (B_{i} \cap B_{k}) = E (\emptyset) = 0$ for $i \neq k$ and $Q_{j} Q_{l} = 0$ for $j \neq l$ ; the commutation hypothesis and [F2] give $P_{i} Q_{j} = E (B_{i}) F (C_{j}) = G (B_{i} \times C_{j}) = F (C_{j}) E (B_{i}) = Q_{j} P_{i}$ , so $R_{i j} = P_{i} Q_{j} = Q_{j} P_{i}$ is self-adjoint ( $R_{i j}^{*} = Q_{j}^{*} P_{i}^{*} = Q_{j} P_{i} = R_{i j}$ ) and idempotent ( $R_{i j}^{2} = P_{i} Q_{j} P_{i} Q_{j} = P_{i} (P_{i} Q_{j}) Q_{j} = P_{i}^{2} Q_{j}^{2} = P_{i} Q_{j}$ ), hence an orthogonal projection; also $R_{i j} R_{k l} = P_{i} Q_{j} P_{k} Q_{l} = P_{i} P_{k} Q_{j} Q_{l}$ using commutation, which vanishes unless $i = k$ and $j = l$ .

1.2F3F4algebra

[Trace and Born rules] For every bounded operator $B$ one has $tr (ρ B) = tr (B ρ)$ , and $tr (ρ \cdot)$ is linear in its bounded argument with $∣ tr (ρ B) ∣ \leq ∥ B ∥$ ; for every PVM and Borel set $S$ the number $tr (ρ E (S))$ lies in $[0, 1]$ and equals $\sum_{n} w_{n} ∥ E (S) u_{n} ∥^{2}$ in any expansion of [F3], so $tr (ρ R_{i j}) = tr (ρ G (B_{i} \times C_{j})) = \sum_{n} w_{n} ∥ G (B_{i} \times C_{j}) u_{n} ∥^{2}$ and $tr (ρ P_{i}) = \sum_{n} w_{n} ∥ P_{i} u_{n} ∥^{2}$ are nonnegative and at most $1$ .

2.1step 1.1step 1.2F3F4F5algebra

[Sequential Lüders probability] If $tr (ρ P_{i}) > 0$ , measuring the first instrument gives listed outcome $i$ with probability $tr (ρ P_{i})$ and state $ρ_{i} = P_{i} ρ P_{i} / tr (ρ P_{i})$ (stipulated, [F5]); measuring then the second gives conditional probability $tr (ρ_{i} Q_{j}) = tr (P_{i} ρ P_{i} Q_{j}) / tr (ρ P_{i})$ . Cyclicity and $P_{i} Q_{j} = Q_{j} P_{i}$ , $P_{i}^{2} = P_{i}$ give $tr (P_{i} ρ P_{i} Q_{j}) = tr (ρ P_{i} Q_{j} P_{i}) = tr (ρ P_{i} Q_{j})$ , so the probability of the listed pair is $tr (ρ P_{i} Q_{j}) = tr (ρ R_{i j})$ by step 1.1. When $tr (ρ Q_{j}) > 0$ , the reversed order gives conditional probability $tr (Q_{j} ρ Q_{j} P_{i}) / tr (ρ Q_{j}) = tr (ρ Q_{j} P_{i}) / tr (ρ Q_{j}) = tr (ρ R_{i j}) / tr (ρ Q_{j})$ ; multiplying by the probability of the first outcome gives the same joint probability. If a listed first outcome has zero probability, then $0 \leq R_{i j} \leq P_{i}, Q_{j}$ and the positive density expansion [F3] give $tr (ρ R_{i j}) = 0$ , so the listed pair has probability zero and no division is taken.

3.1step 1.1step 2.1F3F4algebra

[Order-independent conditional state] Assume $tr (ρ R_{i j}) > 0$ ; running $(P_{i})$ then $(Q_{j})$ produces the product of the two stipulated denominator-normalizations, i.e. the state $Q_{j} P_{i} ρ P_{i} Q_{j} / tr (ρ R_{i j})$ (by the computation of step 2.1), while the reversed order produces $P_{i} Q_{j} ρ Q_{j} P_{i} / tr (ρ R_{i j})$ ; by step 1.1 both numerators equal $R_{i j} ρ R_{i j}$ and the denominators are the same positive number, so the two conditional states coincide and equal $R_{i j} ρ R_{i j} / tr (ρ R_{i j})$ , which is a density operator because $R_{i j} ρ R_{i j}$ is positive trace class with trace $tr (ρ R_{i j}) > 0$ [F3, F4].

3.2step 1.1step 2.1F1F4algebra

[Marginals of the joint law] If $⋃_{j} C_{j} = R$ , the finite additivity of a PVM gives $\sum_{j} Q_{j} = F (⋃_{j} C_{j}) = F (R) = I$ [F1], so $\sum_{j} tr (ρ P_{i} Q_{j}) = tr (ρ P_{i} \sum_{j} Q_{j}) = tr (ρ P_{i})$ by linearity of $tr (ρ \cdot)$ and the ideal property [F4]; symmetrically $\sum_{i} tr (ρ P_{i} Q_{j}) = tr (ρ Q_{j})$ when $⋃_{i} B_{i} = R$ , so the joint law has the sharp-bin marginals.

4.1step 1.1step 3.1F4algebra

[Repeatability] Let $σ : = R_{i j} ρ R_{i j} / tr (ρ R_{i j})$ from step 3.1. Then $tr (σ R_{i j}) = tr (R_{i j} ρ R_{i j} R_{i j}) / tr (ρ R_{i j}) = tr (ρ R_{i j}) / tr (ρ R_{i j}) = 1$ since $R_{i j}^{2} = R_{i j}$ and $R_{i j} R_{i j} R_{i j} = R_{i j}$ [step 1.1, F4]; likewise $tr (σ P_{i}) = tr (P_{i} R_{i j} ρ R_{i j} P_{i}) / tr (ρ R_{i j}) = tr (R_{i j} ρ R_{i j}) / tr (ρ R_{i j}) = 1$ because $P_{i} R_{i j} = R_{i j} = R_{i j} P_{i}$ , and $tr (σ Q_{j}) = 1$ the same way. Hence the Lüders updates $R_{i j} σ R_{i j} / tr (σ R_{i j})$ , $P_{i} σ P_{i} / tr (σ P_{i})$ and $Q_{j} σ Q_{j} / tr (σ Q_{j})$ all return the same state: e.g. $R_{i j} σ R_{i j} = R_{i j}^{3} ρ R_{i j}^{3} / tr (ρ R_{i j}) = R_{i j} ρ R_{i j} / tr (ρ R_{i j}) = σ$ and the denominators equal $1$ , and similarly $P_{i} σ P_{i} = P_{i} R_{i j} ρ R_{i j} P_{i} / tr (ρ R_{i j}) = R_{i j} ρ R_{i j} / tr (ρ R_{i j}) = σ$ and $Q_{j} σ Q_{j} = σ$ . Thus an immediate repetition of the pair, or of either single sharp measurement, returns the same outcome with probability $1$ .

5.1step 4.1F2F5algebra∎

[Scope] Steps 2.1-4.1 are conditional computations inside the stipulated Lüders instrument and the joint PVM $G$ of [F2]; they assert joint probabilities, order independence and repeatability for that ideal model only, with no laboratory realization, no nonselective statement beyond step 3.2, and no time development.

Compatible Lüders repeatability

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources