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Declared countable instruments: normalization, discarded ensembles and the repeatability criterion

Physical scope

A conditional statement about declared countable instruments of the framework: a normalized countable Kraus family defines outcome effects and probabilities, a trace-preserving discarded-outcome map and conditional updates, and the instrument is repeatable exactly when the Kraus family satisfies the stated algebraic criterion. The theorem asserts no apparatus realization, no preferred outcome and no empirical result; the non-repeatability of general instruments is recorded by an explicit supplied instance.

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, the hypothesis of the cited mathematics. Let $H$ be the nonzero complex Hilbert space assigned to a declared system (The state space of the framework: preparations, pure rays and density operators), and let $(M_{a})_{a \geq 1}$ be a declared normalized countable Kraus family on $H$ : bounded operators with $\sum_{a = 1}^{N} M_{a}^{*} M_{a} ⟶ I strongly as N \to \infty .$ Declare the countable instrument with outcome set ${1, 2, \dots}$ , effects $F_{a} : = M_{a}^{*} M_{a}$ , outcome probabilities $p_{a} : = tr (ρ F_{a})$ , conditional updates $ρ_{a} : = M_{a} ρ M_{a}^{*} / p_{a}$ for $p_{a} > 0$ , and discarded-outcome map $Φ (ρ) : = \sum_{a \geq 1} M_{a} ρ M_{a}^{*}$ (Declared ideal measurements and instruments of the framework clause (d)). Then:

(a) POVM and discarded ensemble. Each $F_{a}$ is an effect, $0 \leq F_{a} \leq I$ ; for every state $ρ$ one has $p_{a} \geq 0$ and $\sum_{a} p_{a} = 1$ ; $Φ$ is a positive trace-preserving linear map of the trace-class operators and maps states to states; and $Φ (ρ) = \sum_{a : p_{a} > 0} p_{a} ρ_{a}$ , a trace-norm convergent mixture of the conditional states.

(b) Repeatability criterion. The declared instrument is repeatable, i.e. for every state $ρ$ and every $a$ with $p_{a} > 0$ an immediate repetition of the same declared instrument returns the outcome $a$ with probability $1$ and every $b \neq a$ with probability $0$ , if and only if $M_{b} M_{a} = 0 for all a, b with b \neq a .$ Equivalently, $M_{a}^{*} F_{a} M_{a} = F_{a}$ for every $a$ . If $H$ is finite-dimensional, this criterion also forces each $F_{a}$ to be an orthogonal projection; the projection condition is not imposed for general $H$ .

(c) Projective case. For a declared ideal sharp measurement $(P_{a})$ the Kraus family $M_{a} = P_{a}$ satisfies the criterion of (b) and is repeatable, in agreement with Consequences of a declared ideal sharp measurement: repeatability and discarded ensembles clause (b); a general normalized Kraus family need not satisfy it (Outcome effects do not determine the conditional update, and general instruments need not be repeatable clause (c)).

(d) Non-determination. The effects $(F_{a})$ determine the outcome probabilities but not the conditional updates; the Kraus family is part of the declared instrument (clause (v) of P3, Outcome effects do not determine the conditional update, and general instruments need not be repeatable clause (b)).

(e) Scope. Clauses (a)-(d) are conditional statements about the declared countable instrument; they assert no apparatus realization, no outcome uniqueness and no empirical fact, and no claim is made that every positive trace-preserving map admits such a Kraus family (Countable Kraus normalization and trace-norm convergence clause (c)).

Facts & Assumptions

Given: The adopted postulates P1, P2, P3; a normalized countable Kraus family $(M_{a})_{a \geq 1}$ on $H$ ; its effects $F_{a} = M_{a}^{*} M_{a}$ , probabilities $p_{a}$ , updates $ρ_{a}$ and discarded map $Φ$ ; a state $ρ$ .

[F1]

Countable Kraus normalization. For a bounded family with $\sum_{a \leq N} M_{a}^{*} M_{a} \to I$ strongly: $\sum_{a} ∥ M_{a} x ∥^{2} = ∥ x ∥^{2}$ 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 linear positive trace-preserving map of the trace-class operators and carries states to states (Countable Kraus normalization and trace-norm convergence clauses (a)-(c)).

[F2]

Density expansions. A state has an expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ with orthonormal $(u_{n})$ , $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ ; then $tr (ρ B) = \sum_{n} w_{n} ⟨ B u_{n}, u_{n} ⟩$ for bounded $B$ (Density operators and scalar spectral probability clause (a), Nuclear bounded cyclicity for trace-class operators clause (b)).

[F3]

Effects of the Kraus family. Each $F_{a} = M_{a}^{*} M_{a}$ is bounded and self-adjoint, with $⟨ F_{a} x, x ⟩ = ∥ M_{a} x ∥^{2} \geq 0$ . The normalization and [F1] give $⟨ (I - F_{a}) x, x ⟩ = \sum_{b \neq a} ∥ M_{b} x ∥^{2} \geq 0$ for every $x$ , so $0 \leq F_{a} \leq I$ in the general-H quadratic-form order used to define effects in Declared ideal measurements and instruments of the framework clause (d).

[F4]

Zero quadratic form for a positive form. If $X$ is bounded and self-adjoint with $q_{X} (x) : = ⟨ X x, x ⟩ \geq 0$ for every $x$ , then $b_{X} (x, y) : = ⟨ X x, y ⟩$ satisfies $∣ b_{X} (x, y) ∣^{2} \leq q_{X} (x) q_{X} (y)$ . If $q_{X} (y) > 0$ , expanding $q_{X} (x - \frac{b_{X} (x, y)}{q_{X} (y)} y) \geq 0$ gives this inequality; if $q_{X} (y) = 0$ , positivity of $q_{X} (x + t y)$ for every $t \in C$ forces $b_{X} (x, y) = 0$ . Consequently, if $q_{X} (x) = 0$ for every $x$ , then all pairings $⟨ X x, y ⟩$ vanish and $X = 0$ .

[F5]

Updates and their formulas. For $p_{a} > 0$ one has $ρ_{a} = M_{a} ρ M_{a}^{*} / p_{a}$ with $tr (M_{a} ρ M_{a}^{*}) = tr (ρ F_{a}) = p_{a}$ (Declared ideal measurements and instruments of the framework clause (d), Rank-one operator calculus clause (e), Nuclear bounded cyclicity for trace-class operators clause (b)).

Proof

1.1givenF1F2F3algebra

[The effects and the probabilities] By [F3] each $F_{a}$ is an effect. For a state $ρ$ with expansion [F2], $\sum_{a} p_{a} = \sum_{a} \sum_{n} w_{n} ∥ M_{a} u_{n} ∥^{2} = \sum_{n} w_{n} ∥ u_{n} ∥^{2} = 1$ by [F1] with $x = u_{n}$ and $\sum_{n} w_{n} = 1$ ; all terms are nonnegative, so the interchange of the sums is legitimate and $p_{a} \geq 0$ for each $a$ .

2.1step 1.1F1F2F5algebra

[The discarded map and the mixture identity] By [F1] the series $\sum_{a} M_{a} ρ M_{a}^{*}$ converges in trace norm, its sum $Φ (ρ)$ is positive trace class with trace $tr ρ = 1$ , and $Φ$ is trace preserving; hence $Φ (ρ)$ is a state [F1, F2]. For a label with $p_{a} = 0$ , [F5] gives $tr (M_{a} ρ M_{a}^{*}) = p_{a} = 0$ , so the positive trace-class summand $M_{a} ρ M_{a}^{*}$ is $0$ ; therefore the nonzero summands are exactly those with $p_{a} > 0$ , and $M_{a} ρ M_{a}^{*} = p_{a} ρ_{a}$ for them [F5]. Reindexing the trace-norm convergent series gives $Φ (ρ) = \sum_{a : p_{a} > 0} p_{a} ρ_{a}$ . This is (a).

3.1step 1.1step 2.1givenF1F5algebra

[Sufficiency of the criterion] Assume $M_{b} M_{a} = 0$ whenever $b \neq a$ . For a state $ρ$ and $p_{a} > 0$ , cyclicity and [F5] give $tr (ρ_{a} F_{b}) = tr (ρ (M_{b} M_{a})^{*} (M_{b} M_{a})) / p_{a} = 0$ for every $b \neq a$ . By clause (a), already proved in steps 1.1-2.1, the outcome probabilities in the state $ρ_{a}$ sum to $1$ , so its probability for $a$ is $1$ . Hence the instrument is repeatable, without any assumption that $F_{a}$ commutes with $M_{a}$ .

4.1step 3.1givenF1F4F5algebra

[Necessity and the equivalent identity] Assume repeatability. For a unit vector $x$ , if $M_{a} x \neq 0$ , the state $T_{x, x}$ has $p_{a} = ∥ M_{a} x ∥^{2} > 0$ , and the probability for $b \neq a$ after outcome $a$ is $∥ M_{b} M_{a} x ∥^{2} / ∥ M_{a} x ∥^{2} = 0$ by the rank-one rules and [F5]. If $M_{a} x = 0$ , then $M_{b} M_{a} x = 0$ directly. Thus $M_{b} M_{a}$ vanishes on every vector, proving necessity. Applying [F1] to $M_{a} x$ shows $\sum_{b} ∥ M_{b} M_{a} x ∥^{2} = ∥ M_{a} x ∥^{2}$ . Under the criterion only $b = a$ remains, so $⟨ M_{a}^{*} F_{a} M_{a} x, x ⟩ = ∥ M_{a}^{2} x ∥^{2} = ∥ M_{a} x ∥^{2} = ⟨ F_{a} x, x ⟩$ . For the self-adjoint difference $D : = M_{a}^{*} F_{a} M_{a} - F_{a}$ , equality of these quadratic forms gives $q_{D} (x) = ⟨ D x, x ⟩ = 0$ for every $x$ ; the complex polarization identity $4 ⟨ D x, y ⟩ = q_{D} (x + y) - q_{D} (x - y) + i q_{D} (x + i y) - i q_{D} (x - i y)$ then gives $⟨ D x, y ⟩ = 0$ for all $x, y$ , hence $D = 0$ . Conversely $M_{a}^{*} F_{a} M_{a} = F_{a}$ gives $∥ M_{a}^{2} x ∥^{2} = ∥ M_{a} x ∥^{2}$ ; by [F1], $\sum_{b} ∥ M_{b} M_{a} x ∥^{2} = ∥ M_{a} x ∥^{2}$ , so all nonnegative terms with $b \neq a$ vanish.

5.1step 4.1F1F3F4algebra

[The finite-dimensional projection consequence] Suppose $H$ is finite-dimensional and the criterion holds. Put $E_{a} : = {y : F_{a} y = y}$ . By [F1], for $y = M_{a} x$ the criterion implies $∥ M_{a} y ∥^{2} = ∥ y ∥^{2}$ , hence $⟨ (I - F_{a}) y, y ⟩ = 0$ . By [F3], $I - F_{a}$ is self-adjoint with nonnegative quadratic form; [F4] gives $⟨ (I - F_{a}) y, z ⟩ = 0$ for every $z$ , so $(I - F_{a}) y = 0$ and $ran M_{a} \subseteq E_{a}$ . On $E_{a}$ , $∥ M_{a} y ∥ = ∥ y ∥$ , so $M_{a} ∣_{E_{a}}$ is injective; finite dimensionality makes its image all of $E_{a}$ . If $x ⊥ E_{a}$ and $e \in E_{a}$ , then $⟨ M_{a} x, M_{a} e ⟩ = ⟨ x, F_{a} e ⟩ = ⟨ x, e ⟩ = 0$ . Thus $M_{a} x$ lies in $E_{a}$ and is orthogonal to all of $E_{a}$ , so is zero. Consequently $F_{a}$ is identity on $E_{a}$ and zero on $E_{a}^{⊥}$ , hence is the orthogonal projection onto $E_{a}$ . This proves the final finite-dimensional assertion of (b); the surjectivity step uses finite dimensionality and is not asserted for general $H$ .

6.1step 4.1step 5.1givenF1F3algebra

[Clauses (c) and (d)] For a declared ideal sharp measurement, $M_{a} = P_{a}$ satisfies $M_{b} M_{a} = P_{b} P_{a} = 0$ for $b \neq a$ and $F_{a} = P_{a}^{2} = P_{a}$ , so the criterion of (b) holds and the instrument is repeatable, in agreement with Consequences of a declared ideal sharp measurement: repeatability and discarded ensembles clause (b); the explicit instrument of Outcome effects do not determine the conditional update, and general instruments need not be repeatable clause (c) violates repeatability, so the criterion is not automatic. Since the outcome probabilities are functions of the effects alone while distinct normalized Kraus families may share effects, the effects do not determine the update; this is the non-determination recorded in Outcome effects do not determine the conditional update, and general instruments need not be repeatable clause (b) and clause (v) of P3. This is (c) and (d).

7.1step 6.1givenF1algebra∎

[Scope] Clauses (a)-(d) were derived from the declared Kraus family and the cited mathematics; no step used a claim about an apparatus, and the converse direction of the criterion did not assume that every positive trace-preserving map has a Kraus family, since the family was declared. This is (e).

Remarks

The finite-dimensional qualification in (b) is essential. For an explicit infinite-dimensional countermodel, take a Hilbert space with orthonormal basis $e_{0}, f_{1}, f_{2}, \dots, g_{1}, g_{2}, \dots$ and fix $0 < t < 1$ . Define bounded operators $M_{1} e_{0} = \sqrt{t} f_{1}$ , $M_{1} f_{j} = f_{j + 1}$ , $M_{1} g_{j} = 0$ , and $M_{2} e_{0} = \sqrt{1 - t} g_{1}$ , $M_{2} g_{j} = g_{j + 1}$ , $M_{2} f_{j} = 0$ ; put $M_{a} = 0$ for $a \geq 3$ . The images within each operator are orthogonal, so these formulas extend continuously from finite sums, with norm at most $1$ . Writing $P_{F}$ and $P_{G}$ for the orthogonal projections onto the closed spans of the $f_{j}$ and $g_{j}$ , their effects are $F_{1} = P_{F} + t T_{e_{0}, e_{0}}$ and $F_{2} = P_{G} + (1 - t) T_{e_{0}, e_{0}}$ , hence $F_{1} + F_{2} = I$ . Since $M_{2} M_{1} = M_{1} M_{2} = 0$ , (b) proves repeatability; however $F_{1} e_{0} = t e_{0}$ and $F_{1}^{2} e_{0} = t^{2} e_{0}$ , so $F_{1}$ is not a projection. This is a conditional operator model, not an apparatus claim.

Declared countable instruments: normalization, discarded ensembles and the repeatability criterion

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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