Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Outcome effects do not determine the conditional update, and general instruments need not be repeatable

Physical scope

A declared-instrument statement on the two-level model: two normalized instruments with identical outcome effects have different conditional updates and therefore different predictions for a subsequent measurement, and a third instrument with effects proportional to the identity is not repeatable. The theorem is a statement about declared apparatus models under P2(iv) and P3(v); it asserts no apparatus realization and no empirical fact.

Statement

Adopt the state postulate P1, the observable and probability postulate P2 and the ideal conditional-measurement postulate P3 (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. Let a declared two-level system have state space $D (C^{2})$ with orthonormal basis $(e_{1}, e_{2})$ of $C^{2}$ (Declared ideal measurements and instruments of the framework, The state space of the framework: preparations, pure rays and density operators), write $P_{i} : = T_{e_{i}, e_{i}}$ for the rank-one projections and $W : = T_{e_{2}, e_{1}} + T_{e_{1}, e_{2}}$ for the swap, and declare the dimensionless sharp observable $B : = P_{1} - P_{2}$ (its spectral values have unit $1$ ). Then:

(a) Two normalized instruments with equal effects. The families $(M_{0}, M_{1}) = (P_{1}, P_{2}), (M_{0}^{'}, M_{1}^{'}) = (W P_{1}, P_{2})$ are both normalized declared instruments on $C^{2}$ , and both have the effects $F_{0} = F_{0}^{'} = P_{1}$ , $F_{1} = F_{1}^{'} = P_{2}$ .

(b) Different updates and different predictions. For every state $ρ$ with $p_{0} = tr (ρ P_{1}) > 0$ the conditional state after the outcome $0$ is $T_{e_{1}, e_{1}}$ for the first instrument and $T_{e_{2}, e_{2}}$ for the second. Consequently, for the bounded assigned sharp observable $B = P_{1} - P_{2}$ , the probability of the value $+ 1$ after the outcome $0$ is $1$ under the first instrument and $0$ under the second, although the two instruments have identical outcome effects and identical outcome probabilities $p_{0}, p_{1}$ in every preparation. Hence the effects do not determine the conditional update; the update is part of the declared apparatus model (clause (v) of P3, Finite effects and instruments: normalization and non-determination of the update clause (d)).

(c) A non-repeatable declared instrument. The family $N_{+} : = \frac{1}{\sqrt{2}} I, N_{-} : = \frac{1}{\sqrt{2}} B$ is a normalized declared instrument with effects $F_{+} = F_{-} = \frac{1}{2} I$ . Its outcome probabilities are $\frac{1}{2}, \frac{1}{2}$ in every preparation, and conditioned on either outcome an immediate repetition of the same declared instrument again yields each outcome with probability $\frac{1}{2}$ ; the probability of repeating the same outcome is $\frac{1}{2}$ , not $1$ . So declared instruments need not be repeatable, in contrast to the projective Lueders case of Consequences of a declared ideal sharp measurement: repeatability and discarded ensembles clause (b).

(d) Scope. Clauses (a)-(c) are statements about the declared instruments displayed; they assert no property of any laboratory device, no uniqueness of recorded outcomes and no dynamics of an apparatus. Whether a device is described by one of these instruments is a tested identification with uncertainty under P7.

Facts & Assumptions

Given: The adopted postulates P1, P2, P3; the two-level space $C^{2}$ with orthonormal basis $(e_{1}, e_{2})$ ; the rank-one projections $P_{1} = T_{e_{1}, e_{1}}$ , $P_{2} = T_{e_{2}, e_{2}}$ ; the swap $W = T_{e_{2}, e_{1}} + T_{e_{1}, e_{2}}$ ; the operator $B = P_{1} - P_{2}$ ; the Kraus families of clauses (a) and (c); a state $ρ$ .

[F1]

Rank-one calculus and two-level algebra. $T_{u, v} x = ⟨ x, v ⟩ u$ is bounded with $T_{u, v} T_{x, y} = ⟨ x, v ⟩ T_{u, y}$ and $T_{u, v}^{*} = T_{v, u}$ ; for orthonormal $u, v$ , $T_{u, u}$ and $T_{v, v}$ are orthogonal projections with $T_{u, u} + T_{v, v} = I$ on the span, $T_{u, u} T_{v, v} = 0$ , and $tr (T_{u, u} S) = ⟨ S u, u ⟩$ for trace-class $S$ (Rank-one operator calculus, The Hilbert orthogonal projection onto a closed subspace).

[F2]

Declared instruments. A normalized finite Kraus family $(M_{a, r})$ with $\sum_{a, r} M_{a, r}^{*} M_{a, r} = I$ declares an instrument with effects $F_{a} = \sum_{r} M_{a, r}^{*} M_{a, r}$ , outcome probabilities $p_{a} = tr (ρ F_{a})$ and conditional updates $ρ \mapsto p_{a}^{- 1} \sum_{r} M_{a, r} ρ M_{a, r}^{*}$ for $p_{a} > 0$ ; the update is part of the declaration and is not determined by the effects (Declared ideal measurements and instruments of the framework clause (d), clause (iv) of Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, clause (v) of Ideal conditional measurement postulate P3: the projective Lueders update and discarded outcomes).

[F3]

Abstract non-determination. There are two normalized instruments on $C^{2}$ with the same outcome effects whose updates differ on the same input; effects alone do not determine the update (Finite effects and instruments: normalization and non-determination of the update clause (d)).

[F4]

Projective updates are repeatable. For a declared ideal sharp measurement, the conditional state after outcome $a$ returns $a$ with certainty on immediate repetition (Consequences of a declared ideal sharp measurement: repeatability and discarded ensembles clause (b)).

[F5]

States. $T_{ψ, ψ}$ with $∥ ψ ∥ = 1$ is a state, and $tr (T_{u, u} S) = ⟨ S u, u ⟩$ for bounded self-adjoint $S$ (The state space: density operators, pure and mixed states, and purity, [F1]).

Proof

1.1givenF1F2algebra

[Normalization of the first two instruments] For the first family, $M_{0}^{*} M_{0} = P_{1}^{2} = P_{1}$ and $M_{1}^{*} M_{1} = P_{2}$ , so $M_{0}^{*} M_{0} + M_{1}^{*} M_{1} = P_{1} + P_{2} = I$ [F1]. For the second, $M_{0}^{' *} = P_{1} W^{*}$ gives $M_{0}^{' *} M_{0}^{'} = P_{1} W^{*} W P_{1} = P_{1}^{2} = P_{1}$ because $W^{*} W = I$ , and $M_{1}^{' *} M_{1}^{'} = P_{2}$ ; again the sum is $I$ . Both families are therefore normalized declared instruments [F2] with the displayed effects $F_{0} = P_{1}$ , $F_{1} = P_{2}$ and $F_{0}^{'} = P_{1}$ , $F_{1}^{'} = P_{2}$ . This is (a).

2.1step 1.1F2algebra

[Equal outcome probabilities] Since the effects of the two instruments coincide, for every state $ρ$ and each outcome the probability is $p_{0} = tr (ρ P_{1})$ or $p_{1} = tr (ρ P_{2})$ with the same value under either instrument [F2]; in particular $p_{0} + p_{1} = tr (ρ (P_{1} + P_{2})) = 1$ .

2.2step 1.1F1F2F4algebra

[Non-repeatability] For the family of clause (c), $N_{+}^{*} N_{+} = \frac{1}{2} I$ and $N_{-}^{*} N_{-} = \frac{1}{2} B^{2} = \frac{1}{2} I$ by $B^{2} = P_{1} + P_{2} = I$ [F1], so the family is normalized [F2] with effects $F_{+} = F_{-} = \frac{1}{2} I$ and outcome probabilities $\frac{1}{2}, \frac{1}{2}$ in every preparation by [F2]. If the outcome $+$ occurs, the conditional state is $\frac{1}{2} ρ / \frac{1}{2} = ρ$ , so the probability that an immediate repetition returns $+$ is again $tr (ρ \cdot \frac{1}{2} I) = \frac{1}{2}$ ; if the outcome $-$ occurs, the conditional state is $(\frac{1}{2} B ρ B) / (\frac{1}{2}) = B ρ B$ , and the probability that the repetition returns $-$ is $tr (B ρ B \cdot \frac{1}{2} I) = \frac{1}{2} tr (B^{2} ρ) = \frac{1}{2}$ . Hence the probability of repeating the same outcome is $\frac{1}{2}$ in both cases, not $1$ ; the declared instrument is not repeatable. This is (c), and it is consistent with the repeatability of the projective case [F4] because $N_{\pm}$ are not projections.

3.1step 1.1step 2.1F1F2algebra

[Different conditional updates] Let $ρ$ be a state with $p_{0} > 0$ . For the first instrument, $M_{0} ρ M_{0}^{*} = P_{1} ρ P_{1}$ , and evaluating in the eigen-expansion of $ρ$ (Density operators and scalar spectral probability clause (a)) or directly by [F1] gives $P_{1} ρ P_{1} = tr (ρ P_{1}) P_{1} = p_{0} P_{1}$ ; hence the conditional state is $P_{1} = T_{e_{1}, e_{1}}$ . For the second, $M_{0}^{'} ρ M_{0}^{' *} = W P_{1} ρ P_{1} W^{*} = p_{0} W P_{1} W^{*} = p_{0} T_{e_{2}, e_{2}}$ by [F1] and $W e_{1} = e_{2}$ ; hence the conditional state is $T_{e_{2}, e_{2}}$ . The two updates differ, while step 2.1 gives equal outcome probabilities.

4.1step 3.1F1F5algebra

[Different subsequent predictions] The observable $B = P_{1} - P_{2}$ is bounded and self-adjoint with spectral projections $P_{1}$ at $+ 1$ and $P_{2}$ at $- 1$ , so the probability of the value $+ 1$ in a preparation $σ$ is $tr (σ P_{1})$ . In the conditional state $T_{e_{1}, e_{1}}$ of the first instrument this is $tr (T_{e_{1}, e_{1}} P_{1}) = 1$ by [F1], and in the conditional state $T_{e_{2}, e_{2}}$ of the second it is $tr (T_{e_{2}, e_{2}} P_{1}) = ∣ ⟨ e_{1}, e_{2} ⟩ ∣^{2} = 0$ . Thus instruments with the same effects make different predictions for a subsequent measurement, which proves (b).

5.1step 2.2F2F3algebra∎

[Scope] Steps 1.1-5.1 are computations inside the declared instruments of clauses (a) and (c) using the adopted P2(iv) and P3(v) through [F2]; they assert no device behaviour and no outcome uniqueness. This is (d).

Outcome effects do not determine the conditional update, and general instruments need not be repeatable

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources