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arXiv:2607.19287

A general estimation framework for continuous-variable systems

Luca Innocenti, Simone Artini, Diana A. Chisholm, Salvatore Lorenzo, Alessandro Ferraro, G. Massimo Palma, Mauro Paternostro, Gabriele Lo Monaco

quant-phSubmitted Jul 21, 2026• Announced Jul 22, 2026• 0 citations
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Abstract

We show that informational completeness, while sufficient to have a bijection between ideal measurement probabilities and quantum states, does not guarantee statistically stable reconstruction from finite measurement data. To address this problem, we develop a general estimation theory for continuous-variable systems in which stable reconstructibility is characterized by the POVM effects forming a measurement frame. Informational completeness is therefore necessary, but not sufficient, for stable reconstruction. Our framework is based on measurement frames in a $σ$-regularized operator geometry, where the reference state $σ$ encodes prior information about relevant features of the measured states. For any fixed measurement scheme, observables may be inaccessible, weakly reconstructible only through estimators with divergent variance, or stably reconstructible by finite-variance unbiased estimators. The relevant regime is determined by the range of the POVM synthesis operator. Our framework provides practical methods for constructing estimators and gives an operational interpretation of singular quasiprobability distributions, including the Glauber-Sudarshan $P$ representation: quasiprobabilities act as unbiased estimators for associated observables, and their singularities reflect a pathological feature of the corresponding measurement: its lack of loewr frame bound. We furthermore show how this formalism naturally provides operational regularization procedures tied to prior information. Overall, our framework provides a unified view of continuous-variable tomography, quasiprobability representations, and classical-shadow estimation.

Comments
60 pages, very few figures; comments welcome!

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We show that informational completeness, while sufficient to have a bijection between ideal measurement probabilities and quantum states, does not guarantee statistically stable reconstruction from finite measurement data. To address this problem, we develop a general estimation theory for continuous-variable systems in which stable reconstructibility is characterized by the POVM effects forming a measurement frame. Informational completeness is therefore necessary, but not sufficient, for stable reconstruction. Our framework is based on measurement frames in a σσσ-regularized operator geometry, where the reference state σσσ encodes prior information about relevant features of the measured states. For any fixed measurement scheme, observables may be inaccessible, weakly reconstructible only through estimators with divergent variance, or stably reconstructible by finite-variance unbiased estimators. The relevant regime is determined by the range of the POVM synthesis operator. Our framework provides practical methods for constructing estimators and gives an operational interpretation of singular quasiprobability distributions, including the Glauber-Sudarshan PPP representation: quasiprobabilities act as unbiased estimators for associated observables, and their singularities reflect a pathological feature of the corresponding measurement: its lack of loewr frame bound. We furthermore show how this formalism naturally provides operational regularization procedures tied to prior information. Overall, our framework provides a unified view of continuous-variable tomography, quasiprobability representations, and classical-shadow estimation.
Quantum measurements lie at the core of quantum information processing, enabling tasks ranging from state reconstruction and parameter estimation to sensing and computation. As quantum technologies scale and become increasingly precise, a central question emerges: what does it mean for a quantum measurement to be good? This thesis addresses this question by developing quantitative, operational, and physically grounded frameworks for assessing the quality and informational power of quantum measurements.The first part of the thesis focuses on the intrinsic quality of quantum measurements independent of any particular input state. Adopting the formalism of quantum resource theories, measurements are treated as resources ordered under classical post-processing, which cannot increase informational content. Restricting attention to informationally complete measurements, we employ frame theory to introduce a novel resource monotone termed completeness stability, defined as the minimum eigenvalue of a suitably scaled frame operator. We establish its operational significance by showing that the average error in quantum state tomography and the average variance in shadow tomography are tightly bounded by the inverse of the completeness stability. Maximizing this monotone identifies weighted complex projective 2-designs, including symmetric informationally complete POVMs and mutually unbiased bases, as optimal measurement configurations.The second part of the thesis extends these ideas to quantum parameter estimation, where measurement performance necessarily depends on correlations between the measurement and the encoded quantum state. By introducing a state-dependent inner product on operator space, we develop an operator-theoretic formulation in which both the classical and quantum Fisher information arise naturally from a single frame operator associated with the measurement and the reference state. Within this framework, the spectral properties of the frame operator provide a complete characterization of estimation performance. The maximum and minimum eigenvalues identify the best- and worst-case parameter-encoding directions, respectively, yielding sharp bounds on the achievable ratio of classical and quantum Fisher information. This establishes a direct connection between informational completeness, estimation robustness, and resource-theoretic monotonicity.The final part of the thesis returns to foundational questions concerning quantum measurement and arrival time. Using Bohmian mechanics, we develop a microscopic, first-principles account of the Hong–Ou–Mandel effect, a paradigmatic two-particle interference phenomenon. By analyzing particle trajectories guided by the wave function, we show how bosonic bunching and fermionic anti-bunching emerge dynamically from the guiding equation without invoking wave-function collapse. Beyond reproducing the Hong–Ou–Mandel dip, we demonstrate how experimental parameters such as barrier strength, phase velocity, and path asymmetry influence both coincidence probabilities and temporal correlations.Taken together, this thesis provides a unified treatment of quantum measurement quality across operational, metrological, and dynamical regimes. By bridging resource theory, frame theory, Fisher information, and trajectory-based quantum dynamics, it offers new insights into the structure, limitations, and physical interpretation of quantum measurements in both applied and foundational contexts.
Informationally complete measurements form the foundation of universal quantum state reconstruction, while quantum parameter estimation is based on the local structure of the manifold of quantum states. Here we establish a general link between these two aspects, in the context of a single informationally complete measurement, by employing a suitably adapted operator frame theory. In particular, we bound the ratio between the classical and quantum Fisher information in terms of the spectral decomposition of the associated frame operator, and connect these bounds to the optimal and least optimal directions for parameter encoding. The geometric and operational characterization of information extraction thus obtained reveals the fundamental tradeoff imposed by informational completeness on local quantum parameter estimation.
Informationally overcomplete POVMs are known to outperform minimally complete measurements in many tomography and estimation tasks, and they also leave a purely classical freedom in shadow tomography: the same observable admits infinitely many unbiased linear reconstructions from identical measurement data. We formulate the choice of reconstruction coefficients as a convex minimax problem and give an algorithm with guaranteed convergence that returns the tightest state-independent variance bound achievable by post-processing for a fixed POVM and observable. Numerical examples show that the resulting estimators can dramatically reduce sampling complexity relative to standard (canonical) reconstructions, and can even improve the qualitative scaling with system size for structured noncommuting targets.
A general symmetric-informationally-complete (GSIC)--positive-operator-valued measure (POVM) is known to provide an optimal quantum state tomography among minimal IC POVMs with a fixed average purity. In this paper we provide a general construction of a GSIC POVM by means of a complete orthogonal basis (COB), also interpreted as a normal quasiprobability representation. A spectral property of a COB is shown to play a key role in the construction of SIC POVMs and also for the bound of the mean-square error of the state tomography. In particular, a necessary and sufficient condition to construct a SIC POVM for any ddd is constructively given by the power of traces of a COB. We give three simple constructions of COBs from which one can systematically obtain GSIC POVMs.
It is well-known in classical frame theory that overcomplete representations of a given vector space provide robustness to additive noise on the frame coefficients of an unknown vector. We describe how the same robustness can be shown to exist in the context of quantum state estimation. A key element of the discussion is the application of classical frame theory to operator-valued vector spaces, or operator spaces, which arise naturally in quantum mechanics. Specifically, in the problem we describe the frame vectors are represented by the elements of an informationally complete or overcomplete (IC or IOC) POVM, the frame coefficients are represented by the outcome probabilities of a quantum measurement made on an unknown state, and the error on the frame coefficients arises from finite sample size estimations of the probabilities. We show that with this formulation of the problem, there is a tradeoff in estimation performance between the number of copies of the unknown system and the number of POVM elements. Lastly, we present evidence through simulation that the same tradeoff is present in the context of quantum binary state detection -- the probability of error can be reduced either by increasing the number of copies of the unknown system or by increasing the number of POVM elements.
We show that the effective resolution of quantum inference is determined by a sampling Gram operator that defines the experimentally accessible degrees of freedom and the reconstruction bandwidth supported by the data. Acting analogously to a transfer function in classical imaging, this operator together with scaling of deviance provides an operational criterion for distinguishing genuinely resolved features from structures arising from incomplete sampling, statistical fluctuations, or reconstruction assumptions. Reconstruction in the Gram eigenbasis emerges naturally as a measurement-adapted compression of the inference problem, retaining only those modes supported by typical measurement outcomes. Within finite frame theory, the same structure appears as the frame operator governing stable reconstruction. The resulting framework connects quantum tomography, inverse reconstruction, and measurement resolution, providing a practical tool for assessing whether fine phase-space structures inferred in contemporary experiments are genuinely resolved by the measurement or merely reconstructed by the model.