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arXiv:2604.21323

Sufficient support size of measurements for quantum estimation

Koichi Yamagata

quant-phSubmitted Apr 23, 2026• Announced Apr 24, 2026• 0 citations
v1 · Apr 23, 2026v2 · Aug 12, 2026
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Abstract

In quantum estimation for a $d$-parameter family of density operators on a finite-dimensional Hilbert space $\mathcal{H}$, an estimator is specified by a pair $\left(M,\hatθ\right)$, where $M$ is a POVM with a finite outcome set $Ω$ and $\hatθ:Ω\to\mathbb{R}^{d}$ is a classical estimator map. Since the number of outcomes $\left|Ω\right|$ is a priori unbounded, the space of admissible POVMs is vast, which makes the search for optimal estimators difficult. In this paper, for the minimization of the weighted trace of the mean squared error among locally unbiased estimators, we prove that it suffices to consider POVMs with at most $\left({\rm dim}\,\mathcal{H}\right)^{2}+d(d+1)/2-1$ outcomes, and that an optimal measurement can be chosen to be rank-one. For the minimization of the average weighted trace of the mean squared error in Bayesian estimation, we show that it suffices to consider POVMs with at most $\left( {\rm dim}\, \mathcal{H}\right)^{2}$outcomes, and again an optimal POVM can be taken to be rank-one. Furthermore, when the model admits a real sufficient subalgebra, we show that the $\left( {\rm dim}\, \mathcal{H} \right)^{2}$ term in the above support-size bounds can be reduced in both the locally unbiased and Bayesian settings. These bounds substantially reduce the search space for optimal measurements and justify restricting numerical optimization to rank-one POVMs with finitely many outcomes.

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Abstract. We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify a quantum system which is constrained by prior information? We show that if the prior information restricts the system to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the system. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that 4d − 4 measurement outcomes (POVM elements) is enough in order to identify all pure states in a d-dimensional Hilbert space, and that the minimal number is at most 2 log2(d) smaller than this upper bound. 1.
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The Cramér-Rao bound serves as a crucial lower limit for the mean square error of an estimator in frequentist parameter estimation. Paradoxically, it requires highly accurate prior knowledge of the estimated parameter for constructing the optimal unbiased estimator. In contrast, Bhattacharyya bounds offer a more robust estimation framework with respect to prior accuracy by introducing additional constraints on the estimator. In this work, we examine divergences that arise in the computation of these bounds and establish the conditions under which they remain valid. Notably, we show that when the number of constraints exceeds the number of measurement outcomes, an estimator with finite variance typically does not exist. Furthermore, we systematically investigate the properties of these bounds using paradigmatic examples, comparing them to the Cramér-Rao and Bayesian approaches.