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arXiv:2012.05355

Informationally Overcomplete POVMs for Quantum State Estimation and Binary Detection

Catherine A. Medlock, Alan V. Oppenheim, Petros T. Boufounos

Published Dec 9, 2020• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2012.05355

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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.
In recent years, informationally complete measurements have attracted considerable attention, especially in the context of classical shadows. In the particular case of informationally overcomplete measurements, for which the number of possible outcomes exceeds the dimension of the space of linear operators in Hilbert space, the dual positive operator-valued measure operators used to interpret the measurement outcomes are not uniquely defined. In this paper, we propose a method to optimize the dual operators after the measurements have been carried out in order to produce sharper, unbiased estimations of observables of interest. We discuss how this procedure can produce zero-variance estimations in cases where the classical shadows formalism, which relies on so-called canonical duals, incurs exponentially large measurement overheads. We also analyze the algorithm in the context of quantum simulation with randomized Pauli measurements, and show that it can significantly reduce statistical errors with respect to canonical duals on multiple observable estimations.
Accurately estimating expectation values of observables from a finite number of measurement shots is a central challenge in quantum information science. Informationally overcomplete measurements provide a route to reduce estimation variance through optimized classical post-processing. However, the interplay between measurement geometry and dual-frame construction remains largely unexplored. In this work, we study Platonic solid POVMs ---highly symmetric, overcomplete single-qubit measurements whose effects correspond to the vertices of the five Platonic solids on the Bloch sphere--- for the estimation of molecular Hamiltonians. Using kkk-locally optimal dual frames, we show that the geometry of the POVM can have a non-trivial and non-monotonic effect on the estimation variance. We further propose a joint optimization of the POVM orientation and effect weights using a classical proxy state, either a product state or a Matrix Product State (MPS) approximation. We demonstrate that optimized Platonic solid POVMs can outperform standard randomized Pauli measurements, provided the MPS bond dimension is sufficient to faithfully represent the target state. These results reveal a trade-off between classical preprocessing and estimation accuracy, suggesting a practical route to improved observable estimation on near-term quantum hardware.
A quantum injective frame is a frame whose measurements for density operators can be used as a distinguishing feature in a quantum system, and the frame quantum detection problem demands a characterization of all such frames. Very recently, the quantum detection problem for continuous as well as discrete frames in both finite and infinite dimensional Hilbert spaces received significant attention. The quantum detection problem pertaining to the characterization of informationally complete positive operator-valued measures (POVM) can be split into two cases: The quantum injectivity or state separability problem and the rang analysis or quantum state estimation problem. Building upon this notion, this note is aimed at the quantum detection problem for fusion frames. The injectivity of a family of vectors and a family of closed subspaces is characterized in terms of some operator equations in Hilbert–Schmidt and trace classes.
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.
This monograph addresses operating characteristics for binary hypothesis testing in both classical and quantum settingsand overcomplete quantum measurements for quantum binary state discrimination. We specifically explore decision and measurement operating characteristics defined as the tradeoff between probability of detection and probability of false alarm as parameters of the pre-decision operator and the binary decision rule are varied. In the classical case we consider in detail the Neyman-Pearson optimality of the operating characteristics when they are generated using threshold tests on a scalar score variable rather than threshold tests on the likelihood ratio. In the quantum setting, informationally overcomplete POVMs are explored to provide robust quantum binary state discrimination. We focus on equal trace rank one POVMs which can be specified by arrangements of points on a sphere that we refer to as an Etro sphere.
Abstract Informationally overcomplete measurements find important applications in quantum tomography and quantum state estimation. The most popular are maximal sets of mutually unbiased bases, for which trace relations between measurement operators are well known. In this paper, we introduce a more general class of informationally overcomplete positive, operator-valued measure (POVMs) that are generated by equiangular tight frames of arbitrary rank. This class provides a generalization of equiangular measurements to non-projective POVMs, which include rescaled mutually unbiased measurements and bases. We provide a method of their construction, analyze their symmetry properties, and provide examples for highly symmetric cases. In particular, we find a wide class of generalized equiangular measurements that are conical two-designs, which allows us to derive the index of coincidence. Our results show benefits of considering a single informationally overcomplete measurement over informationally complete collections of POVMs.