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arXiv:2109.05807 · Physical Review A

Incompatibility measures in multiparameter quantum estimation under hierarchical quantum measurements

Hongzhen Chen, Yu Chen, Haidong Yuan

Published Jun 24, 2022• 17 citations
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Abstract

The incompatibility of the measurements constrains the achievable precisions in multiparameter quantum estimation. Understanding the tradeoff induced by such incompatibility is a central topic in quantum metrology. Here we provide an approach to study the incompatibility under general $p$-local measurements, which are the measurements that can be performed collectively on at most $p$ copies of quantum states. We demonstrate the power of the approach by presenting a hierarchy of analytical bounds on the tradeoff among the precisions of different parameters. These bounds lead to a necessary condition for the saturation of the quantum Cram\'er-Rao bound under $p$-local measurements, which recovers the partial commutative condition at $p=1$ and the weak commutative condition at $p=\ensuremath{\infty}$. As a further demonstration of the power of the framework, we present another set of tradeoff relations with the right logarithmic operators.

DOI
10.1103/physreva.105.062442

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The incompatibility of the measurements constrains the achievable precisions in multiparameter quantum estimation. Understanding the tradeoff induced by such incompatibility is a central topic in quantum metrology. Here we provide an approach to study the incompatibility under general ppp-local measurements, which are the measurements that can be performed collectively on at most ppp copies of quantum states. We demonstrate the power of the approach by presenting a hierarchy of analytical bounds on the tradeoff among the precisions of different parameters. These bounds lead to a necessary condition for the saturation of the quantum Cram'er-Rao bound under ppp-local measurements, which recovers the partial commutative condition at p=1p=1p=1 and the weak commutative condition at p=\ensuremath∞p=\ensuremath{\infty}p=\ensuremath∞. As a further demonstration of the power of the framework, we present another set of tradeoff relations with the right logarithmic operators.
When collective measurements on an infinite number of copies of identical quantum states can be performed, the precision limit of multi-parameter quantum estimation is quantified by the Holevo bound. In practice, however, the collective measurements are always restricted to a finite number of quantum states, under which the precision limit is still poorly understood. Here we provide an approach to study the multi-parameter quantum estimation with general ppp-local measurement where the collective measurements are restricted to at most ppp copies of quantum states. We demonstrate the power of the approach by providing a hierarchy of nontrivial tradeoff relations for multi-parameter quantum estimation which quantify the incompatibilities of general ppp-local measurement. These tradeoff relations also provide a necessary condition for the saturation of the quantum Cramer-Rao bound under ppp-local measurement, which is shown reducing to the weak commutative condition when p→∞p\rightarrow \inftyp→∞. To further demonstrate the versatility of the approach, we also derive another set of tradeoff relations in terms of the right logarithmic operators(RLD).
The quantum Cramér-Rao (QCR) bound sets the ultimate local precision limit for unbiased multiparameter estimation. Unlike in the single-parameter case, however, its saturability is not generally guaranteed and is often analyzed through a hierarchy of commutativity-based conditions. Here, we resolve the logical structure of these conditions for unitary parameter-encoding transformations. We identify strict separations among the conditions, reveal previously overlooked gaps in their implications, and construct explicit counterintuitive examples that expose the boundaries among distinct classes. In particular, we show that commutativity of the parameter-encoding generators alone does not guarantee saturation of the QCR bound when realistic noise leads to mixed probe states. Our results provide a systematic classification of saturation conditions in multiparameter quantum metrology and clarify fundamental precision limits of noisy distributed quantum sensing beyond idealized pure-state regimes.
Multiparameter quantum estimation faces a fundamental challenge due to the inherent incompatibility of optimal measurements for different parameters, a direct consequence of quantum non-commutativity. This incompatibility is quantified by the gap between the symmetric logarithmic derivative (SLD) quantum Cramér-Rao bound, which is not always attainable, and the asymptotically achievable Holevo bound. This work provides a comprehensive analysis of this gap by introducing and contrasting two scalar measures. The first is the weight-independent quantumness measure RRR, which captures the intrinsic incompatibility of the estimation model. The second is a tighter, weight-dependent measure T[W]T[W]T[W] which explicitly incorporates the cost matrix WWW assigning relative importance to different parameters. We establish a hierarchy of bounds based on these two measures and derive necessary and sufficient conditions for their saturation. Through analytical and numerical studies of tunable qubit and qutrit models with SU(2) unitary encoding, we demonstrate that the weight-dependent bound CT[W]C_{T}[W]CT​[W] often provides a significantly tighter approximation to the Holevo bound CH[W]C_{H}[W]CH​[W] than the RRR-dependent bound, especially in higher-dimensional systems. We also develop an approach based on CT[W]C_{T}[W]CT​[W] to compute the Holevo bound CH[W]C_{H}[W]CH​[W] analytically. Our results highlight the critical role of the weight matrix's structure in determining the precision limits of multiparameter quantum metrology.
We present a comprehensive and pedagogical formulation of Bayesian multiparameter quantum estimation. Within this framework, we analyse the role of measurement incompatibility and establish its quantitative effect on attainable precision. We achieve this by deriving upper bounds based on the pretty good measurement -- a notion from hypothesis testing -- combined with the evaluation of the Nagaoka-Hayashi lower bound. In general, we prove that, as in the many-copy regime of local estimation theory, incompatibility can at most double the minimum loss relative to the idealised scenario in which individually optimal measurements are assumed jointly implementable. Therefore, in practical situations, the latter may provide a sufficient and computationally efficient benchmark without solving the full optimisation problem. Our results, which we illustrate through applications of discrete phase imaging, phase and dephasing estimation, and qubit sensing, provide analytical and numerical tools for assessing ultimate precision limits and the role of measurement incompatibility in Bayesian multiparameter quantum metrology, including an open-source package for all the bounds discussed here.
A fundamental challenge in multiparameter quantum estimation arises from the incompatibility of optimal measurements for different parameters, leading to intricate precision trade-offs that obscure the understanding of ultimate quantum limits. Here, we present an approach that precisely quantifies these trade-offs for an arbitrary number of parameters encoded in pure quantum states. Our approach not only derives tight analytical bounds for the trade-offs induced by measurement incompatibility but also provides a systematic methodology to design optimal measurement strategies that saturate these limits. To demonstrate the practical significance of our findings, we apply our framework to quantum radar and obtain a refined Arthurs-Kelly relation that characterizes the ultimate performance for the simultaneous estimation of range and velocity with any given amount of entanglement. This showcases the transformative potential of our findings for a wide range of applications in quantum metrology, sensing, and beyond.
In multiparameter quantum estimation, the optimal measurements for different parameters encoded in a quantum state are in general incompatible, giving rise to nontrivial tradeoffs between their attainable precisions. Understanding and characterizing such tradeoffs is essential for determining the ultimate precision limits in multiparameter quantum estimation and is therefore a central topic in quantum metrology. In this article, we present an approach that precisely quantifies the tradeoff resulting from incompatible optimal measurements in multiparameter estimation. We derive a tight analytical tradeoff relation that determines the ultimate precision limits for estimating an arbitrary number of parameters encoded in pure quantum states. Additionally, we provide a systematic methodology for constructing optimal measurements that saturate this tight bound in an analytical and structured manner. To demonstrate the power of our findings, we apply our methodology to quantum radar, resulting in a refined Arthurs-Kelly relation that characterizes the ultimate performance for the simultaneous estimation of range and velocity.