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Finite Naimark compression: a finite POVM as a compression of a PVM

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $H$ be a nonzero complex Hilbert space (Hilbert space) and $(F_{1}, \dots, F_{n})$ a finite POVM on $H$ : a finite family of bounded self-adjoint operators $0 \leq F_{a} \leq I$ with $\sum_{a} F_{a} = I$ (Finite effects and instruments: normalization and non-determination of the update). Let $C = C^{n}$ with standard orthonormal basis $(e_{a})$ and form $H \hat{\otimes} C$ (Hilbert tensor construction: inner product, completion and product bases). Then

$W : H \to H \hat{\otimes} C, W ψ : = \sum_{a = 1}^{n} F_{a}^{1 / 2} ψ \otimes e_{a},$

is a linear isometry, the operators $P_{a} : = I_{H} \otimes T_{e_{a}, e_{a}}$ are orthogonal projections with $\sum_{a} P_{a} = I$ (a finite PVM on ${1, \dots, n}$ , The Hilbert orthogonal projection onto a closed subspace), and

$W^{*} P_{a} W = F_{a} (a = 1, \dots, n) .$

The square roots $F_{a}^{1 / 2}$ are the positive square roots supplied by the bounded functional calculus. The construction asserts no physical realization of the ancilla.

Facts & Assumptions

Given: $H$ ; a finite POVM $(F_{a})$ ; the ancilla $C = C^{n}$ with orthonormal basis $(e_{a})$ .

[F1]

$0 \leq F_{a} \leq I$ ; for the bounded self-adjoint $F_{a}$ the continuous functional calculus provides a unique isometric unital star-homomorphism $f \mapsto f (F_{a})$ ; in particular $\sqrt{F_{a}}$ is bounded, self-adjoint, positive, satisfies ${\sqrt{F_{a}}}^{2} = F_{a}$ and commutes with $F_{a}$ (Continuous functional calculus for bounded self adjoint operators, Borel functional calculus for a bounded normal operator, Finite effects and instruments: normalization and non-determination of the update).

[F2]

The completed tensor product has the inner product $⟨ u \otimes c, u^{'} \otimes c^{'} ⟩ = ⟨ u, u^{'} ⟩ ⟨ c, c^{'} ⟩$ , the elementary tensors span a dense subspace, and bounded operators extend as $S \otimes T$ with adjoint $S^{*} \otimes T^{*}$ (Hilbert tensor construction: inner product, completion and product bases, Bounded tensor maps: extension, norm estimate, adjoint and commutation).

[F3]

$T_{e_{a}, e_{a}}$ is the orthogonal projection of $C$ onto $C e_{a}$ ; $T_{e_{a}, e_{a}} T_{e_{b}, e_{b}} = δ_{a b} T_{e_{a}, e_{a}}$ and $\sum_{a} T_{e_{a}, e_{a}} = I_{C}$ (Finite effects and instruments: normalization and non-determination of the update, The Hilbert orthogonal projection onto a closed subspace, Hilbert projections are linear, self-adjoint and contractive).

Proof

1.1givenF1F2F3algebra

[The isometry] The map $ψ \mapsto \sum_{a} F_{a}^{1 / 2} ψ \otimes e_{a}$ is well defined and linear; for finite sums of elementary tensors the inner product is computed by [F2], and for $ψ \in H$ , $∥ W ψ ∥^{2} = \sum_{a, b} ⟨ F_{a}^{1 / 2} ψ, F_{b}^{1 / 2} ψ ⟩ ⟨ e_{a}, e_{b} ⟩ = \sum_{a} ∥ F_{a}^{1 / 2} ψ ∥^{2} = \sum_{a} ⟨ F_{a} ψ, ψ ⟩ = ⟨ ψ, ψ ⟩$ by [F1] and [F3]. Hence $W$ is an isometry.

2.1step 1.1F2F3algebra

[The projections] The operators $P_{a} = I_{H} \otimes T_{e_{a}, e_{a}}$ are orthogonal projections by [F2] and [F3]; distinct ones are orthogonal and $\sum_{a} P_{a} = I_{H} \otimes I_{C} = I$ by [F3].

3.1step 1.1F1F2algebra∎

[Compression] Compute the adjoint directly: for $ξ \in H$ , vectors $ξ_{a} \in H$ and scalars $z_{a}$ , $⟨ W ξ, \sum_{a} z_{a} ξ_{a} \otimes e_{a} ⟩ = \sum_{a} \overline{z_{a}} ⟨ F_{a}^{1 / 2} ξ, ξ_{a} ⟩ = \sum_{a} \overline{z_{a}} ⟨ ξ, F_{a}^{1 / 2} ξ_{a} ⟩$ by [F2] and [F1], so $W^{*} (\sum_{a} z_{a} ξ_{a} \otimes e_{a}) = \sum_{a} z_{a} F_{a}^{1 / 2} ξ_{a}$ ; in particular $W^{*} (ξ \otimes e_{a}) = F_{a}^{1 / 2} ξ$ . Then $P_{a} W ψ = F_{a}^{1 / 2} ψ \otimes e_{a}$ and $W^{*} P_{a} W ψ = F_{a}^{1 / 2} F_{a}^{1 / 2} ψ = F_{a} ψ$ by [F1].

Finite Naimark compression: a finite POVM as a compression of a PVM

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources