Eigenvalue-1 decomposability and norm-1 informational completeness of quantum observables in infinite dimensions

In the mathematical theory of quantum measurement, an observable is a positive-operator-valued measure (POVM) on a measurable outcome space. Two optimality properties of an observable can be in tension: determining its values (for every outcome set, some state localizes the outcome, exactly or approximately) and determining the state (informational completeness). Haapasalo and Pellonpää [J. Math. Phys. 58, 122104 (2017)] proved that for countably generated outcome spaces on a separable Hilbert space, an observable is pre-processing clean — it cannot be realized by adding quantum noise to any strictly better, statistically equivalent observable — if and only if it decomposes as a direct sum of a projection-valued measure (PVM) on one fixed closed subspace and a POVM on the orthogonal complement, the observable and the PVM sharing the same zero effects (the PVM carries the support structure of the observable). That paper explicitly leaves open (a) whether the eigenvalue-1 property of all nonzero effects alone already forces this fixed-subspace decomposition for non-discrete observables, and (b) whether any norm-1 observable can be informationally complete on an infinite-dimensional separable Hilbert space. In finite dimensions question (b) is settled in the negative direction (norm-1 observables are discrete with at most $d$ outcomes and hence never informationally complete), while question (a) has a positive answer for discrete outcome spaces; but in infinite dimensions the canonical phase observable is norm-1 without the eigenvalue-1 property, breaking the finite-dimensional chain of implications. The problem asks for a rigorous resolution of (a) and (b), each by proof or by explicit counterexample/construction.

Background

Quantum observables. In the Hilbert-space formulation of quantum mechanics, a general measurement is represented by a positive-operator-valued measure (POVM, also called an observable): a map $\mathsf{M}$ from a measurable outcome space $(\Omega,\Sigma)$ to effects on a Hilbert space $\mathcal{H}$ — bounded positive operators between $0$ and the identity — that is normalized ($\mathsf{M}(\Omega)=I$) and countably additive. Measuring $\mathsf{M}$ on a system in a state (density operator) $\rho$ yields outcome statistics $p^{\mathsf{M}}_{\rho}(X)=\mathrm{tr}[\rho\,\mathsf{M}(X)]$ for $X\in\Sigma$. When every effect $\mathsf{M}(X)$ is a projection, $\mathsf{M}$ is a projection-valued measure (PVM), the idealized sharp (von Neumann) measurement; general POVMs model unsharp or noisy measurements.

Two optimality properties. An observable is informationally complete (IC) if its outcome statistics determine the state, i.e., if $\rho\mapsto p^{\mathsf{M}}_{\rho}$ is injective on all states (IC within pure states is the analogous injectivity on rank-one states). In a $d$-dimensional Hilbert space an IC POVM must have at least $d^{2}$ outcomes. Complementarily, $\mathsf{M}$ determines its values if for every outcome set $X$ with $\mathsf{M}(X)\neq 0$ and every $\varepsilon>0$ there is a state $\rho$ with $p^{\mathsf{M}}_{\rho}(X)>1-\varepsilon$; this is equivalent to the norm-1 property, $\|\mathsf{M}(X)\|=1$ for every nonzero effect. The stronger requirement that every nonzero effect $\mathsf{M}(X)$ have eigenvalue 1 — so that some pure state yields an outcome in $X$ with certainty — is the eigenvalue-1 property.

Finite-dimensional picture. If $\mathcal{H}$ is finite-dimensional, every effect has discrete spectrum, so norm-1 implies the eigenvalue-1 property; a counting argument through the minimal Naimark dilation (the standard dilation of a POVM to a PVM on a larger Hilbert space) then shows that a norm-1 observable is discrete with at most $d=\dim\mathcal{H}$ outcomes, and hence cannot be informationally complete. Thus, in finite dimensions, an observable that determines its values cannot also determine the state.

Pre-processing and cleanliness. A channel is a unital completely positive map $\Phi:\mathcal{L}(\mathcal{H})\to\mathcal{L}(\mathcal{H}')$ between bounded-operator algebras. An observable $\mathsf{M}'$ is a pre-processing of $\mathsf{M}$, written $\mathsf{M}'\le_{\mathrm{pre}}\mathsf{M}$, if $\mathsf{M}'(X)=\Phi(\mathsf{M}(X))$ for all $X$; physically, measuring $\mathsf{M}'$ in a state $\rho$ is the same as first transforming $\rho$ with the predual channel $\Phi^{*}$ and then measuring $\mathsf{M}$, so a pre-processed observable is one realized by processing the state before a better measurement. Two observables are statistically equivalent, $\mathsf{M}'\sim\mathsf{M}$, if they have the same zero effects ($\mathsf{M}'(X)=0$ exactly when $\mathsf{M}(X)=0$). $\mathsf{M}$ is pre-processing clean (pre-processing maximal) if $\mathsf{M}\le_{\mathrm{pre}}\mathsf{M}'$ with $\mathsf{M}'\sim\mathsf{M}$ implies $\mathsf{M}'\le_{\mathrm{pre}}\mathsf{M}$: no statistically equivalent observable is strictly more informative, i.e., $\mathsf{M}$ cannot be realized by adding quantum noise to the state before a strictly better equivalent measurement. For finite outcome sets on a fixed finite-dimensional system the notion goes back to Buscemi, Keyl, D'Ariano, Perinotti and Werner [J. Math. Phys. 46, 082109 (2005)], where the clean POVMs are exactly the norm-1 ones (when the outcome number does not exceed the dimension).

What the source paper proved. Haapasalo and Pellonpää [J. Math. Phys. 58, 122104 (2017)] extended this structure theory to general observables. For discrete outcome spaces (finite or countable $\Omega$), pre-processing cleanliness is equivalent to the eigenvalue-1 property (their Theorem 13). For a countably generated $\Sigma$ on a separable $\mathcal{H}$, cleanliness is equivalent to the direct-sum decomposition

$\mathsf{M}(X)=\mathsf{E}(X)\oplus\mathsf{F}(X),\qquad X\in\Sigma,$

with one fixed closed subspace $\mathcal{M}\subseteq\mathcal{H}$, a PVM $\mathsf{E}$ on $\mathcal{M}$, and a POVM $\mathsf{F}$ on $\mathcal{M}^{\perp}$, the PVM $\mathsf{E}$ and the observable $\mathsf{M}$ sharing the same zero effects, $\mathsf{M}\sim\mathsf{E}$ (their Theorem 14; the proof tool — every POVM absolutely continuous with respect to a PVM $\mathsf{P}$ is a pre-processing of $\mathsf{P}$ — is their Theorem 12, due to Pellonpää [J. Phys. A 44, 085304 (2011)]). Clean observables automatically have the eigenvalue-1 property; pre-processing maximal observables determine their values and are never informationally complete, and eigenvalue-1 observables are never IC even within pure states, in arbitrary Hilbert spaces.

Where the finite-dimensional logic breaks. In infinite dimensions the implication norm-1 $\Rightarrow$ eigenvalue-1 fails: the canonical phase observable $\Phi_{\mathrm{can}}$ (the phase POVM built from the phase states associated with the number basis of a harmonic-oscillator mode) has norm 1 but none of its nontrivial effects has eigenvalue 1; it is not clean and not IC (all number states give identical phase statistics). Conversely, the standard informationally complete continuous observable — the covariant phase-space POVM $\mathsf{G}_{S}$ generated by a state $S$, realized as the eight-port (double) homodyne measurement — fails the norm-1 property: $\|\mathsf{G}_{S}(Z)\|\le\pi^{-1}\int_{Z}d^{2}z<1$ for small-area outcome sets $Z$. Its informational completeness is fully characterized: $\mathsf{G}_{S}$ is IC if and only if the function $(q,p)\mapsto\mathrm{tr}[S\,D(q,p)]$ (with $D(q,p)$ the Weyl displacement operators translating canonical position and momentum) has support $\mathbb{R}^{2}$ [Kiukas, Lahti, Schultz and Werner, J. Math. Phys. 53, 102103 (2012)]. Thus the known IC continuous observables are all unsharp, and the known norm-1 non-classical observable fails IC; whether this mutual exclusion is a theorem for infinite-dimensional systems or an artifact of the known examples is exactly what is open.

Problem Statement

Let $\mathcal{H}$ be a separable infinite-dimensional Hilbert space, $(\Omega,\Sigma)$ a countably generated measurable space, and $\mathsf{M}:\Sigma\to\mathcal{L}(\mathcal{H})$ a POVM. Use the definitions of the background: norm-1 ($\|\mathsf{M}(X)\|=1$ for all $X$ with $\mathsf{M}(X)\neq 0$), eigenvalue-1 (every nonzero effect has eigenvalue 1), informational completeness (injectivity of $\rho\mapsto\mathrm{tr}[\rho\,\mathsf{M}(\cdot)]$ on all states), and pre-processing cleanliness ($\mathsf{M}\le_{\mathrm{pre}}\mathsf{M}'$ with $\mathsf{M}'\sim\mathsf{M}$ implies $\mathsf{M}'\le_{\mathrm{pre}}\mathsf{M}$). Resolve the following two linked questions, which together delimit how far determination of values excludes determination of the state for general observables in infinite dimensions.

(a) Eigenvalue-1 decomposability. Suppose every nonzero effect $\mathsf{M}(X)$, $X\in\Sigma$, has eigenvalue 1. Does it follow that there exist a closed subspace $\mathcal{M}\subseteq\mathcal{H}$, a PVM $\mathsf{E}:\Sigma\to\mathcal{L}(\mathcal{M})$, and a POVM $\mathsf{F}:\Sigma\to\mathcal{L}(\mathcal{M}^{\perp})$ such that $\mathsf{M}\sim\mathsf{E}$ (that is, $\mathsf{E}(X)=0$ if and only if $\mathsf{M}(X)=0$, so the PVM part carries the zero-effect structure of $\mathsf{M}$) and $\mathsf{M}(X)=\mathsf{E}(X)\oplus\mathsf{F}(X)$ for all $X\in\Sigma$ — equivalently, that $\mathsf{M}$ is pre-processing clean? Provide a proof or an explicit counterexample. The converse implication is known, as is a positive answer when $\Omega$ is discrete; the canonical phase shows that dropping the eigenvalue-1 hypothesis breaks the link.

(b) Norm-1 versus informational completeness. Does there exist a POVM with a countably generated outcome space on an infinite-dimensional separable Hilbert space that is both norm-1 and informationally complete over all states? Equivalently: can an observable that determines its values (in the approximate-localization sense) also determine every quantum state from its measurement statistics? Provide an explicit construction — which, by the eigenvalue-1 no-go, must be norm-1 without the eigenvalue-1 property — or a no-go proof covering at least countably generated outcome spaces on a separable Hilbert space. Optionally, the analogous question for regular observables (nontrivial effects have spectrum extending both above and below $1/2$; the source proves no such observable is IC in finite dimensions) may be treated together; it is not required for a complete answer.

A complete answer resolves (a) and (b) rigorously, each by proof or by explicit counterexample/construction. The two parts are jointly the two explicitly stated loose ends of one program — the operator-level boundary between value determination and state determination for continuous observables — so an answer to one part alone is a partial result.

References

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