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A declared finite effect family is realised by a dilated ideal projective measurement

Physical scope

A conditional model statement: a declared finite family of bounded effects on a system is realised as the outcome probabilities of a declared ideal sharp measurement on an enlarged model space carrying a finite ancilla register, and the induced conditional reduced states are given explicitly. The construction is an idealisation within the declared composite model space; it asserts no physical apparatus, no physical ancilla and no preferred outcome.

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 and the Axiom of Choice. Let $H$ be the nonzero complex Hilbert space assigned to a declared system, and let $(F_{a})_{a = 1}^{n}$ be a declared finite family of effects on $H$ in the general-H sense of Declared ideal measurements and instruments of the framework clause (d): each $F_{a}$ is bounded and self-adjoint, $0 \leq ⟨ F_{a} x, x ⟩ \leq ∥ x ∥^{2}$ for every $x \in H$ , and $\sum_{a} F_{a} = I$ . The outcome probabilities are those of clause (iv) of P2. Let $C = C^{n}$ with orthonormal basis $(e_{a})_{a = 1}^{n}$ , form the completed tensor product $H \hat{\otimes} C$ (Hilbert tensor construction: inner product, completion and product bases), and define $W : H \to H \hat{\otimes} C, W ψ : = \sum_{a = 1}^{n} F_{a}^{1 / 2} ψ \otimes e_{a}, P_{a} : = I_{H} \otimes T_{e_{a}, e_{a}},$ where $F_{a}^{1 / 2}$ is the positive square root supplied by the continuous functional calculus (Continuous functional calculus for bounded self adjoint operators) and $T_{x, x}$ is the rank-one projection of Rank-one operator calculus. Then:

(a) The dilation. $W$ is a linear isometry, the operators $P_{a}$ are pairwise orthogonal orthogonal projections on $H \hat{\otimes} C$ with strong sum $I$ , and $W^{*} P_{a} W = F_{a}$ for every $a$ .

(b) Realised outcome probabilities. For every state $ρ$ on $H$ the operator $\tilde{ρ} : = W ρ W^{*}$ is a state on $H \hat{\otimes} C$ , and the declared ideal sharp measurement $(P_{a})_{a = 1}^{n}$ on the composite model space has outcome probabilities $tr (\tilde{ρ} P_{a}) = tr (ρ F_{a}) (a = 1, \dots, n) .$

(c) The induced instrument. Let $Λ$ be the trace-preserving map $Λ (σ) = \sum_{b = 1}^{n} F_{b}^{1 / 2} σ F_{b}^{1 / 2}$ (the square-root Kraus map of the effect family). If $p_{a} : = tr (ρ F_{a}) > 0$ , the conditional composite state after the outcome $a$ is ${\tilde{ρ}}_{a} = P_{a} \tilde{ρ} P_{a} / p_{a}$ , and its reduction to $H$ is the state ${Tr}_{C} {\tilde{ρ}}_{a} = \frac{1}{p_{a}} F_{a}^{1 / 2} ρ F_{a}^{1 / 2} .$ The non-selective composite state reduces to ${Tr}_{C} \sum_{a} P_{a} \tilde{ρ} P_{a} = \sum_{a} F_{a}^{1 / 2} ρ F_{a}^{1 / 2} = Λ (ρ)$ . Thus this dilation induces on $H$ the declared square-root instrument with one Kraus operator $M_{a} = F_{a}^{1 / 2}$ for each outcome $a$ and effects $F_{a}$ (Declared ideal measurements and instruments of the framework clause (d)).

(d) Projective case and scope of the update. If the effects are pairwise orthogonal projections, $F_{a} = P_{a}^{\circ}$ with $P_{a}^{\circ} P_{b}^{\circ} = 0$ for $a \neq b$ , then the induced update of clause (c) is the Lueders update $P_{a}^{\circ} ρ P_{a}^{\circ} / p_{a}$ . For general effects, clause (c) identifies the update selected by this particular dilation; the effects alone do not determine the update among all declared instruments, as shown by Outcome effects do not determine the conditional update, and general instruments need not be repeatable.

(e) Scope. The ancilla register $C$ and the isometry $W$ are mathematical objects of a declared composite model space; the theorem asserts no physical ancilla, no physical realisation of the dilated projective measurement and no preferred outcome. Whether a laboratory device realises the declared effects or either update is a tested identification with uncertainty under P7.

Facts & Assumptions

Given: The adopted postulates P1, P2, P3; a nonzero complex Hilbert space $H$ ; a declared finite effect family $(F_{a})_{a = 1}^{n}$ with $\sum_{a} F_{a} = I$ ; the ancilla $C = C^{n}$ with orthonormal basis $(e_{a})$ ; the isometry $W$ and projections $P_{a}$ of the Statement; a state $ρ$ on $H$ .

[F1]

Functional calculus, tensor maps and adjoints. The spectrum $σ (F)$ consists of those $z \in C$ for which $F - z I$ has no bounded inverse (Spectrum and resolvent of a bounded operator); for bounded self-adjoint $F$ it is real (Spectrum of a self adjoint operator is real). Under AC the continuous functional calculus is an isometric unital star-homomorphism $f \mapsto f (F)$ on the continuous complex functions $C (σ (F))$ with the supremum norm, sends the coordinate function to $F$ , and has range $C^{*} (I, F)$ , the operator-norm closure of polynomials in $F$ (Continuous functional calculus for bounded self adjoint operators). The positive square root used here is justified in step 1.1. The completed tensor product has $⟨ u \otimes c, u^{'} \otimes c^{'} ⟩ = ⟨ u, u^{'} ⟩ ⟨ c, c^{'} ⟩$ , and bounded tensor maps act on elementary tensors by $(S \otimes T) (u \otimes c) = S u \otimes T c$ (Hilbert tensor construction: inner product, completion and product bases, Bounded tensor maps: extension, norm estimate, adjoint and commutation). Every bounded map between Hilbert spaces has a unique bounded Hilbert adjoint satisfying $⟨ T x, y ⟩ = ⟨ x, T^{*} y ⟩$ (The Hilbert-space adjoint of a bounded operator, Hilbert-adjoint identities). The rank-one formula gives $T_{e_{a}, e_{a}} ξ = ⟨ ξ, e_{a} ⟩ e_{a}$ (Rank-one operator calculus).

[F2]

Tensor, trace-ideal and partial-trace rules. On $H \hat{\otimes} C$ the pairing is $⟨ u \otimes c, u^{'} \otimes c^{'} ⟩ = ⟨ u, u^{'} ⟩ ⟨ c, c^{'} ⟩$ , bounded tensor maps preserve the tensor inner product rules, and ${Tr}_{C}$ is the trace-preserving positive linear map with ${Tr}_{C} (S \otimes T_{c, c^{'}}) = ⟨ c, c^{'} ⟩ S$ (Hilbert tensor construction: inner product, completion and product bases, Bounded tensor maps: extension, norm estimate, adjoint and commutation, Separable partial trace and its basic properties). If $X$ is trace class and $A, B$ are bounded, then $A X B$ is trace class and the trace pairing with bounded operators is trace-norm continuous; for trace-class $X$ and bounded $A$ , $tr (A X) = tr (X A)$ (Trace class is a two sided Banach operator ideal, Nuclear bounded cyclicity for trace-class operators). In particular, ${Tr}_{C} (W X W^{*}) = \sum_{b} F_{b}^{1 / 2} X F_{b}^{1 / 2}$ for trace-class $X$ , since $(I \otimes ⟨ e_{b} ∣) W = F_{b}^{1 / 2}$ and $W^{*} (I \otimes ∣ e_{b} ⟩) = F_{b}^{1 / 2}$ .

[F3]

Density expansions and effects. A state has a trace-norm expansion $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ with $(u_{n})$ orthonormal, $w_{n} \geq 0$ and $\sum_{n} w_{n} = 1$ . For effects $0 \leq F_{a} \leq I$ with $\sum_{a} F_{a} = I$ , the general-H normalization calculation gives $tr (ρ F_{a}) \in [0, 1]$ and $\sum_{a} tr (ρ F_{a}) = 1$ (Density operators and scalar spectral probability clause (a), clause (iv) of Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes).

[F4]

Declared instrument. A normalized Kraus family $(M_{a, b})$ with $\sum_{a, b} M_{a, b}^{*} M_{a, b} = I$ declares an instrument with effects $F_{a} = \sum_{b} M_{a, b}^{*} M_{a, b}$ , outcome probabilities $tr (ρ F_{a})$ and conditional updates $ρ \mapsto p_{a}^{- 1} \sum_{b} M_{a, b} ρ M_{a, b}^{*}$ only when $p_{a} > 0$ (Declared ideal measurements and instruments of the framework clause (d)).

[F5]

Projective Lueders case. If the effects are pairwise orthogonal projections, the update $F_{a}^{1 / 2} ρ F_{a}^{1 / 2} / p_{a}$ equals the Lueders update $F_{a} ρ F_{a} / p_{a}$ because $F_{a}^{1 / 2} = F_{a}$ for a projection (The selective Lueders update by an orthogonal projection, The Hilbert orthogonal projection onto a closed subspace).

[F6]

Non-determination by effects. There are two normalized declared instruments with the same effects and different conditional updates (Outcome effects do not determine the conditional update, and general instruments need not be repeatable clauses (a)-(b)).

[F7]

Trace pairing. For every trace-class $X$ on a Hilbert space and bounded $A$ , $tr (A X) = tr (X A)$ , and the pairing is continuous in trace norm (Nuclear bounded cyclicity for trace-class operators clause (b)).

[F8]

Rank-one trace formula. For $u \in H$ , $T_{u, u}$ is trace class, self-adjoint and positive because $T_{u, u}^{*} = T_{u, u}$ and $⟨ T_{u, u} x, x ⟩ = ∣ ⟨ x, u ⟩ ∣^{2} \geq 0$ ; also $tr (T_{u, u} B) = tr (B T_{u, u}) = ⟨ B u, u ⟩$ for bounded $B$ (Rank-one operator calculus clauses (b)-(d), Nuclear bounded cyclicity for trace-class operators clause (b), Self-adjoint, positive, unitary and normal operators).

[F9]

Apparatus identification. P7(iii) requires physical identifications of preparations, dynamics and detector effects to be tested against apparatus records, with uncertainty and calibration qualifications; P7(v) distinguishes that testing from deductive proof of the framework (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions).

Proof

1.1givenF1algebra

[Positive square roots] Fix an effect $F = F_{a}$ . For a real $λ \notin [0, 1]$ , the quadratic-form bounds and Cauchy-Schwarz give $∥ (F - λ I) x ∥ \geq d ∥ x ∥$ , with $d = dist (λ, [0, 1]) > 0$ . This bound makes $F - λ I$ injective with closed range: a convergent sequence of images has Cauchy preimages, which converge in $H$ . Its range is dense, since a vector $y$ orthogonal to it satisfies $(F - λ I)^{*} y = 0$ , hence $y = 0$ ; the orthogonal-decomposition property implies that the closed range is all of $H$ . Its inverse is bounded by $1 / d$ . Together with the real-spectrum result [F1], this proves $σ (F) \subseteq [0, 1]$ . Apply the calculus to $f (t) = \sqrt{t}$ and $g (t) = t^{1 / 4}$ on this spectrum. Star preservation and multiplicativity give $R = f (F) = R^{*}$ , $R^{2} = F$ , and $R = g (F)^{*} g (F) \geq 0$ . If $S$ is another bounded positive square root, $S$ commutes with $F = S^{2}$ , hence with $R \in C^{*} (I, F)$ by norm continuity of multiplication. For $D = S - R$ , $(S + R) D = S^{2} - R^{2} = 0$ . Thus $y = D x$ satisfies $⟨ S y, y ⟩ + ⟨ R y, y ⟩ = 0$ . Each summand is nonnegative; varying $t \in C$ in the nonnegative forms $⟨ S (y + t z), y + t z ⟩$ and $⟨ R (y + t z), y + t z ⟩$ shows $S y = R y = 0$ . Hence $D^{2} x = 0$ and $∥ D x ∥^{2} = ⟨ D^{2} x, x ⟩ = 0$ , proving uniqueness. Write this root as $F_{a}^{1 / 2}$ .

2.1step 1.1givenF1algebra

[Clause (a)] Put $R_{a} : = F_{a}^{1 / 2}$ . For $ψ \in H$ , the tensor inner product and $R_{a}^{*} R_{a} = F_{a}$ give $∥ W ψ ∥^{2} = \sum_{a} ∥ R_{a} ψ ∥^{2} = \sum_{a} ⟨ F_{a} ψ, ψ ⟩ = ∥ ψ ∥^{2}$ , so $W$ is a linear isometry. On elementary tensors, $P_{a} (ξ \otimes e_{b}) = δ_{a b} ξ \otimes e_{a}$ ; boundedness and density therefore show that the $P_{a}$ are orthogonal projections, pairwise orthogonal, and sum to $I$ . Finally, for $ψ, ϕ \in H$ , $⟨ W^{*} P_{a} W ψ, ϕ ⟩ = ⟨ P_{a} W ψ, W ϕ ⟩ = ⟨ R_{a} ψ, R_{a} ϕ ⟩ = ⟨ F_{a} ψ, ϕ ⟩$ , so $W^{*} P_{a} W = F_{a}$ . These are the claims in (a).

3.1step 2.1givenF1F2F3F8algebra

[The dilated state and the outcome probabilities] Write $ρ = \sum_{n} w_{n} T_{u_{n}, u_{n}}$ by [F3]. The rank-one identities give $W T_{u_{n}, u_{n}} W^{*} = T_{W u_{n}, W u_{n}}$ ; bounded left and right multiplication preserve trace class and trace-norm limits by [F2], so $\tilde{ρ}$ is the trace-norm sum of these positive rank-one operators. Its trace is $\sum_{n} w_{n} ∥ W u_{n} ∥^{2} = \sum_{n} w_{n} = 1$ because $W$ is an isometry; positivity follows termwise and is preserved by the operator-norm limit, which follows from trace-norm convergence and [F2]. Thus $\tilde{ρ}$ is a state. For each $a$ , [F1] gives $P_{a} W u_{n} = F_{a}^{1 / 2} u_{n} \otimes e_{a}$ , hence $tr (\tilde{ρ} P_{a}) = \sum_{n} w_{n} tr (T_{W u_{n}, W u_{n}} P_{a}) = \sum_{n} w_{n} ∥ F_{a}^{1 / 2} u_{n} ∥^{2} = \sum_{n} w_{n} ⟨ F_{a} u_{n}, u_{n} ⟩ = tr (ρ F_{a})$ , using the trace formula and trace-norm continuity in [F2], the density expansion [F3], and [F8]. This proves (b).

4.1step 3.1F1F2F3F4F7algebra

[The induced conditional state] Assume $p_{a} > 0$ and define $J_{a} : H \to H \hat{\otimes} C$ by $J_{a} ξ : = ξ \otimes e_{a}$ . The tensor-product inner product makes $J_{a}$ an isometry, and the partial-trace rule [F2] gives ${Tr}_{C} (J_{a} X J_{a}^{*}) = X$ for every trace-class $X$ , since $J_{a} X J_{a}^{*} = X \otimes T_{e_{a}, e_{a}}$ . Directly from the definition of $W$ , $P_{a} W = J_{a} F_{a}^{1 / 2}$ . Hence $P_{a} \tilde{ρ} P_{a} = J_{a} F_{a}^{1 / 2} ρ F_{a}^{1 / 2} J_{a}^{*}$ , so ${Tr}_{C} {\tilde{ρ}}_{a} = F_{a}^{1 / 2} ρ F_{a}^{1 / 2} / p_{a}$ . This is a state because it is the partial trace of the state ${\tilde{ρ}}_{a}$ [F2, F3]. Summing the same identity over $a$ gives ${Tr}_{C} \sum_{a} P_{a} \tilde{ρ} P_{a} = \sum_{a} F_{a}^{1 / 2} ρ F_{a}^{1 / 2} = Λ (ρ)$ . The family $M_{a} : = F_{a}^{1 / 2}$ is normalized because $\sum_{a} M_{a}^{*} M_{a} = \sum_{a} F_{a} = I$ ; its effects are $F_{a}$ and its conditional updates are exactly the displayed reduced states [F4]. By [F7], $tr (Λ (X)) = \sum_{a} tr (F_{a}^{1 / 2} X F_{a}^{1 / 2}) = \sum_{a} tr (X F_{a}) = tr X$ for every trace-class $X$ , so $Λ$ is trace preserving as stated. This is (c).

5.1step 4.1givenF5algebra

[Projective case] If $F_{a} = P_{a}^{\circ}$ is a projection, then $F_{a}^{1 / 2} = F_{a}$ and $F_{b}^{1 / 2} F_{a}^{1 / 2} = P_{b}^{\circ} P_{a}^{\circ} = δ_{a b} P_{a}^{\circ}$ for the pairwise orthogonal family; hence the induced update of step 4.1 reduces to $P_{a}^{\circ} ρ P_{a}^{\circ} / p_{a}$ , the Lueders update [F5].

5.2step 4.1givenF6algebra

[Effects do not determine every instrument's update] The update induced by the displayed dilation is the square-root update from step 4.1. This is a particular choice of instrument associated with the effects; [F6] supplies two normalized declared instruments with the same effects and different updates. Thus the effects alone do not prescribe a unique update among all declared instruments. This is the remaining claim of (d).

6.1step 5.2givenF1F2F9algebra∎

[Scope] The preceding steps were carried out entirely in the declared composite model space $H \hat{\otimes} C$ with the mathematical ancilla $C$ ; no physical ancilla, apparatus or outcome selection was used. The laboratory-identification qualification is precisely the adopted P7 discipline [F9], not a consequence of the dilation. This is (e).

A declared finite effect family is realised by a dilated ideal projective measurement

Statement

Facts & Assumptions

Proof

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