Masahito Hayashi
The author studies the Cramér-Rao type bound by a linear programming approach. By this approach, he found a necessary and sufficient condition that the Cramér-Rao type bound is attained by a random measurement. In a spin 1/2 system, this condition is satisfied.
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The author studies the Cramer-Rao type bound by a linear programming approach. By this approach, he found a necessary and sufficient condition that the Cramer-Rao type bound is attained by a random measurement. In a spin 1/2 system, this condition is satisfied.
We calculate the Holevo Cramér-Rao bound for estimation of the displacement experienced by one mode of an two-mode squeezed vacuum state with squeezing r and find that it is equal to 4exp(-2r). This equals the sum of the mean squared error obtained from a dual homodyne measurement, indicating that the bound is tight and that the dual homodyne measurement is optimal.
In this letter, we show that for all the so-called path-symmetric states, the measurement of parity of photon number at the output of an optical interferometer achieves maximal phase sensitivity at the quantum Cramer-Rao bound. Such optimal phase sensitivity with parity is attained at a suitable bias phase, which can be determined a priori. Our scheme is applicable for local phase estimation.
In this paper, new methodology -- direct approach -- for the determination of the attainable CR type bound of the pure state model, is proposed and successfully applied to the wide variety of pure state models, for example, the 2-dimensional arbitrary model, the coherent model with arbitrary dimension. When the weight matrix is $SLD$ Fisher information, the bound is determined for arbitrary pure state models. Manifestation of complex structure in the Cramer-Rao type bound is also discussed.
Recently, a widely-used computation expression for quantum Fisher information was shown to be discontinuous at the parameter points where the rank of the parametric density operator changes. The quantum Cramér-Rao bound can be violated on such singular parameter points if one uses this computation expression for quantum Fisher information. We point out that the discontinuity of the computation expression of quantum Fisher information is accompanied with the unboundedness of the symmetric logarithmic derivation operators, based on which the quantum Fisher information is formally defined and the…
We experimentally analyzed the statistical errors in quantum-state estimation and examined whether their lower bound, which is derived from the Cramer-Rao inequality, can be truly attained or not. In the experiments, polarization states of bi-photons produced via spontaneous parametric down-conversion were estimated employing tomographic measurements. Using a new estimation strategy based on Akaike's information criterion, we demonstrated that the errors actually approach the lower bound, while they fail to approach it using the conventional estimation strategy.
The Cramér-Rao approach and global quantum estimation of bosonic states
本文研究玻色态量子估计中Cramér-Rao方法在非独立同分布全局估计下的适用性，揭示其适用与失效情形，并强调外推至全局估计时需谨慎。
In a unified viewpoint in quantum channel estimation, we compare the Cramer-Rao and the mini-max approaches, which gives the Bayesian bound in the group covariant model. For this purpose, we introduce the local asymptotic mini-max bound, whose maximum is shown to be equal to the asymptotic limit of the mini-max bound. It is shown that the local asymptotic mini-max bound is strictly larger than the Cramer-Rao bound in the phase estimation case while the both bounds coincide when the minimum mean square error decreases with the order O(1/n). We also derive a sufficient condition for that the min…
Estimation of parameters is a pivotal task throughout science and technology. Quantum Cramér-Rao bound provides a fundamental limit of precision allowed to achieve under quantum theory. For closed quantum systems, it has been shown how the estimation precision depends on the underlying dynamics. Here, we propose a general formulation for metrology scenarios in open quantum systems, aiming to relate the precision more directly to properties of the underlying dynamics. This feature may be employed to enhance an estimation precision, e.g., by quantum control techniques. Specifically, we derive a …
The quantum Cramér-Rao bound sets a fundamental limit on the accuracy of unbiased parameter estimation in quantum systems, relating the uncertainty in determining a parameter to the inverse of the quantum Fisher information. We experimentally demonstrate near saturation of the quantum Cramér-Rao bound in the phase estimation of a solid-state spin system, provided by a nitrogen-vacancy center in diamond. This is achieved by comparing the experimental uncertainty in phase estimation with an independent measurement of the related quantum Fisher information. The latter is independently extracted f…
The quantum Cramér-Rao bound (QCRB) provides an ultimate precision limit allowed by quantum mechanics in parameter estimation. Given any quantum state dependent on a single parameter, there is always a positive-operator valued measurement (POVM) saturating the QCRB. However, the QCRB-saturating POVM cannot always be implemented efficiently, especially in multipartite systems. In this paper, we show that the POVM based on local operations and classical communication (LOCC) is QCRB-saturating for arbitrary pure states or rank-two mixed states with varying probability distributions over fixed eig…
In quantum metrology, the Holevo Cramér-Rao bound has attracted renewed interest in recent years due to its superiority over the Helstrom Cramér-Rao bound and its asymptotic attainability for multi-parameter estimation. Its evaluation, however, is often much more difficult than that of the Helstrom version, calling into question the actual improvement offered by the Holevo CRB and whether it is worth the trouble. Here I prove that the Holevo bound is at most thrice the Helstrom version, so the improvement must be limited. The result also shows that the Helstrom version remains a pretty good bo…
In multiparameter quantum metrology, the weighted-arithmetic-mean error of estimation is often used as a scalar cost function to be minimized during design optimization. However, other types of mean error can reveal different facets of permissible error combination. By introducing the weighted $f$-mean of estimation error and quantum Fisher information, we derive various quantum Cramér-Rao bounds on mean error in a very general form and also give their refined versions with complex quantum Fisher information matrices. We show that the geometric- and harmonic-mean quantum Cramér-Rao bounds can …
For decoherence processes induced by weak interactions with the environment, a general quantum channel with one noise parameter has been formulated. This channel is called low-noise channel and very useful for investigating the parameter estimation in the leading order. In this paper, we formulate the low-noise channel with multiple unknown parameters in order to address the simultaneous achievability of the Cram{é}r-Rao bound for the parameters estimation. In general, the simultaneous achievement of the Cram{é}r-Rao bound for multi-parameter estimations suffers from non-commutativity of optim…
Many results in the quantum metrology literature use the Cramér-Rao bound and the Fisher information to compare different quantum estimation strategies. However, there are several assumptions that go into the construction of these tools, and these limitations are sometimes not taken into account. While a strategy that utilises this method can considerably simplify the problem and is valid asymptotically, to have a rigorous and fair comparison we need to adopt a more general approach. In this work we use a methodology based on Bayesian inference to understand what happens when the Cramér-Rao bo…
In collisional thermometry, a system in contact with the thermal bath is probed by a stream of ancillas. Coherences and collective measurements were shown to improve the Fisher information in some parameter regimes, for a stream of independent and identically prepared (i.i.d.) ancillas in some specific states [Seah et al., Phys. Rev. Lett., 180602 (2019)]. Here we refine the analysis of this metrological advantage by optimising over the possible input ancilla states, also for block-i.i.d.~states of block size b=2. For both an indirect measurement interaction and a coherent energy exchange chan…
Estimation of high-dimensional unitary transformations saturating the Quantum Cramér-Rao bound
本文提出估计$d$维幺正变换的方法，在$d>2$时对接近恒等的变换饱和量子Cramér-Rao界，$d=2$时借助先验信息达到最优，且无先验信息时精度优于标准量子过程层析。
Quantum metrology derives its capabilities from the careful employ of quantum resources for carrying out measurements. This advantage, however, relies on refined data postprocessing, assessed based on the variance of the estimated parameter. When Bayesian techniques are adopted, more elements become available for assessing the quality of the estimation. Here we adopt generalized classical Cramér-Rao bounds for looking in detail into a phase-estimation experiment performed with quantum light. In particular, we show that the third-order absolute moment can give a superior capability in revealing…
Saturation of the Cramér-Rao Bound for the Atomic Resonance Frequency with Phased Array of Hyperbolic Secant Pulses
本文分析双曲正割形 π 脉冲共振实验的经典与量子 Fisher 信息，并证明交替相位脉冲序列可全局饱和量子 Cramér-Rao 界。
We formulate multiparameter quantum estimation in the parametric and semiparametric setting. While the Holevo Cramér-Rao bound (CRB) requires no substantial modifications in moving from the former to the latter, we generalize the Helstrom CRB appropriately. We show that the Holevo CRB cannot be greater than twice the generalized Helstrom CRB. We also present a tighter, intermediate, bound. Finally, we show that for parameters encoded in the first moments of a Gaussian state there always exists a Gaussian measurement that gives a classical Fisher information matrix that is one-half of the quant…
A usual assumption in quantum estimation is that the unknown parameter labels the possible states of the system, while it influences neither the sample space of outcomes nor the measurement aimed at extracting information on the parameter itself. This assumption is crucial to prove the quantum Cramér-Rao theorem and to introduce the quantum Fisher information as an upper bound to the Fisher information of any possible measurement. However, there are relevant estimation problems where this assumption does not hold and an alternative approach should be developed to find the genuine ultimate boun…
Quantum number-path entanglement is a resource for super-sensitive quantum metrology and in particular provides for sub-shotnoise or even Heisenberg-limited sensitivity. However, such number-path entanglement has thought to have been resource intensive to create in the first place --- typically requiring either very strong nonlinearities, or nondeterministic preparation schemes with feed-forward, which are difficult to implement. Recently in [Phys. Rev. Lett. 114, 170802 (2015)] we showed that number-path entanglement from a BosonSampling inspired interferometer can be used to beat the shot-no…
Achieving the Multi-parameter Quantum Cramér-Rao Bound with Antiunitary Symmetry
本文提出利用反幺正对称性优化参数编码策略，通过两类量子统计模型实验证明多参数估计可同时达到量子Cramér-Rao界且精度至少提升两倍。
Hierarchies of Frequentist Bounds for Quantum Metrology: From Cramér-Rao to Barankin
本文通过选择测试可观测量定义无偏性约束，解析优化量子测量与估计量，得到包含量子Cramér-Rao界和量子Barankin界的层级下界，并推广量子Fisher信息以处理有限数据下的阈值行为。
Quantum Cramér-Rao bound on quantum metric as a multi-observable uncertainty relation
本文证明Uhlmann相对振幅是生成函数，通过它对系统参数微分可得到包含量子Fisher信息矩阵和平均Uhlmann曲率的量子Fisher张量，并给出其Bloch表示，应用于自旋系综展示了恒定Ricci标量、真空Einstein方程和宇宙常数。
Approaching the Multiparameter Quantum Cramér-Rao Bound via Classical Correlation and Entangling Measurements
提出LOEM策略，利用经典关联正交纯态与纠缠测量，实现多参数量子Cramér-Rao界的饱和，并经光子实验验证，达到海森堡精度。
Holevo Cramér-Rao bound for multi-parameter estimation in nonlinear interferometers
该研究针对非线性干涉仪纯高斯态多参数估计问题，推导了双模位移估计的HCRB，对比发现其与双零差测量结果一致且比SLD-CRB更紧，并给出相空间直观分析。
It is challenged only recently that the precision attainable in any measurement of a physical parameter is fundamentally limited by the quantum Cramér-Rao Bound (QCRB). Here, targeting at measuring parameters in strongly dissipative systems, we propose an innovative measurement scheme called {\it dissipative adiabatic measurement} and theoretically show that it can beat the QCRB. Unlike projective measurements, our measurement scheme, though consuming more time, does not collapse the measured state and, more importantly, yields the expectation value of an observable as its measurement outcome,…
Only with the simultaneous estimation of multiple parameters are the quantum aspects of metrology fully revealed. This is due to the incompatibility of observables. The fundamental bound for multi-parameter quantum estimation is the Holevo Cramér-Rao bound (HCRB) whose evaluation has so far remained elusive. For finite-dimensional systems we recast its evaluation as a semidefinite program, with reduced size for rank-deficient states. We use this result to study phase and loss estimation in optical interferometry and three-dimensional magnetometry with noisy multi-qubit systems. For the former,…
Multimode Gaussian quantum light, including multimode squeezed and/or multipartite quadrature entangled light, is a very general and powerful quantum resource with promising applications to quantum information processing and metrology involving continuous variables. In this paper, we determine the ultimate sensitivity in the estimation of any parameter when the information about this parameter is encoded in such Gaussian light, irrespective of the exact information extraction protocol used in the estimation. We then show that, for a given set of available quantum resources, the most economical…
In quantum metrology, it is widely believed that the quantum Cramer-Rao bound is attainable bound while it is not true. In order to clarify this point, we explain why the quantum Cramer-Rao bound cannot be attained geometrically. In this manuscript, we investigate noiseless channel estimation under energy constraint for states, using a physically reasonable error function, and present the optimal state and the attainable bound. We propose the experimental generation of the optimal states for enhanced metrology using squeezing transformations. This makes the estimation of unitary channels physi…
A Precisão da Metrologia Quântica: Limite de Cramér-Rao, Informação de Fisher e possíveis Aplicações Tecnológicas
本文探讨经典与量子计量学基本原理，研究Cramér-Rao界、Fisher信息的物理意义，分析量子态超越经典精度极限的机制，介绍相关技术应用，并回顾量子高斯态压缩参数估计的研究结果。
We describe a compact and reliable method to calculate the Fisher information for the estimation of a dynamical parameter in a continuously measured linear Gaussian quantum system. Unlike previous methods in the literature, which involve the numerical integration of a stochastic master equation for the corresponding density operator in a Hilbert space of infinite dimension, the formulas here derived depends only on the evolution of first and second moments of the quantum states, and thus can be easily evaluated without the need of any approximation. We also present some basic but physically me…
Saturating the one-axis twisting quantum Cramér-Rao bound with a total spin readout
本文证明含一次单轴扭曲、一次时间反演扭曲及总自旋测量的 twist-untwist 协议，可渐近饱和单轴扭曲自旋相干态干涉的最低量子 Cramér-Rao 界，并分析有限程情形。
SU(1,1) interferometers, based on the usage of nonlinear elements, are superior to passive interferometers in phase sensitivity. However, the SU(1,1) interferometer cannot make full use of photons carrying phase information as the second nonlinear element annihilates some of the photons inside. Here, we focus on improving phase sensitivity and propose a new protocol based on a modified SU(1,1) interferometer, where the second nonlinear element is replaced by a beam splitter. We utilize two coherent states as inputs and implement balanced homodyne measurement at the output. Our analysis suggest…
Quantum-enhanced phase estimation paves the way to ultra-precision sensing and is of great realistic significance. In this paper we investigate theoretically the estimation of a second-order nonlinear phase shift using a coherent state and parity measurement. A numerical expression is derived, the resolution and the sensitivity of parity signal are contrasted to linear phase estimation protocol, and the signal visibility is analyzed. Additionally, by virtue of phase-averaging approach to eliminate any hidden resources, we make an attempt at unveiling the low-down on the fundamental sensitivity…
Global Precision Limits in Critical Quantum Metrology: From Cramér-Rao to Ziv-Zakai
研究临界量子计量学精度极限，通过量子齐夫-扎凯界推导均方误差极限，发现二阶量子相变时克拉美-罗奥界精度无显著提升，临界量子传感器精度仅恒定增益，源于先验知识，提供评估框架和研究量子传感的常规方法。
The quantum Fisher information constrains the achievable precision in parameter estimation via the quantum Cramér-Rao bound, which has attracted much attention in Hermitian systems since the 60s of the last century. However, less attention has been paid to non-Hermitian systems. In this Letter, working with different logarithmic operators, we derive two previously unknown expressions for quantum Fisher information, and two Cramér-Rao bounds lower than the well-known one are found for non-Hermitian systems. These lower bounds are due to the merit of non-Hermitian observable and it can be unders…
We study quantum-limited 3D magnetometry using two qubits. Two qubits form the smallest multi-qubit system for 3D magnetometry, the simultaneous estimation of three phases, as it is impossible with a single qubit. We provide an analytical expression for the Holevo-Cramér-Rao bound (HCRB),the fundamental attainable quantum bound of multiparameter estimation, for 3D magnetometry using two-qubit pure states and show its attainability by rank-1 projective measurements. We also examine the attainability of the HCRB in the presence of dephasing noise using numerical methods. While attaining the HCRB…
The Most Informative Cramér--Rao Bound for Quantum Two-Parameter Estimation with Pure State Probes
针对纯态探针量子双参数估计问题，本文提出形式更简化的可达Cramér-Rao界新表达式，确定对应最优测量，并给出网格态位移估计的精度极限。
A superresolution-enhanced spectrometer beyond the Cramer-Rao bound in phase sensitivity
本文提出一种基于高阶强度关联的纯经典超分辨光谱仪，其频率分辨率随强度乘积阶数线性提升，可超越克拉美-罗下界，并借助干涉仪扫描模式实现对环境噪声的鲁棒性。
This is a tutorial aimed at illustrating some recent developments in quantum parameter estimation beyond the Cramèr-Rao bound, as well as their applications in quantum metrology. Our starting point is the observation that there are situations in classical and quantum metrology where the unknown parameter of interest, besides determining the state of the probe, is also influencing the operation of the measuring devices, e.g. the range of possible outcomes. In those cases, non-regular statistical models may appear, for which the Cramèr-Rao theorem does not hold. In turn, the achievable precision…
Saturating the Quantum Cramér--Rao Bound in Prioritised Parameter Estimation
本文研究量子多参数估计中的测量不相容性权衡，发现存在可在最大化某一参数信息的同时不完全丢失其他参数信息的系统，并给出不同于 Heisenberg 型的新权衡关系，且在 Quantinuum 俘获离子量子计算机上通过最优纠缠测量进行了实验验证。
Role of the extended Hilbert space in the attainability of the Quantum Cramér-Rao bound for multiparameter estimation
本文通过显式例子研究扩展希尔伯特空间上SLD算符对易条件，证明支撑空间、支撑-核空间和核空间任意两者上的对易既非必要也非充分，并给出一维核空间下SLDCRB可达到的充要条件。
Hybrid Cramér-Rao bound for Quantum Bayes-Point Estimation with Nuisance Parameters
本文提出一种混合框架，用于在存在干扰参数的情况下进行量子参数估计，引入混合部分量子 Fisher 信息矩阵，并推导出混合风险的Cramér-Rao型下界，通过qubit模型和数值示例展示如何系统利用干扰变量的先验信息。
A geometric perspective: experimental evaluation of the quantum Cramer-Rao bound
该研究从量子几何视角探究多参数估计的精度极限，通过量子几何测量实验评估量子克拉美-罗界，阐明量子不确定性原理对界饱和的阻碍，并提取三参数估计的可达到QCRB。
Using differential geometry, I derive a form of the Bayesian Cramér-Rao bound that remains invariant under reparametrization. With the invariant formulation at hand, I find the optimal and naturally invariant bound among the Gill-Levit family of bounds. By assuming that the prior probability density is the square of a wavefunction, I also express the bounds in terms of functionals that are quadratic with respect to the wavefunction and its gradient. The problem of finding an unfavorable prior to tighten the bound for minimax estimation is shown, in a special case, to be equivalent to finding t…
Enhancing the precision of a measurement requires maximizing the information that can be gained about the quantity of interest from probing a system. For optical based measurements, such an enhancement can be achieved through two approaches, increasing the number of photons used to interrogate the system and using quantum states of light to increase the amount of quantum Fisher information gained per photon. Here we consider the use of quantum states of light with a large number of photons, namely the bright single-mode and two-mode squeezed states, that take advantage of both of these approac…
How precisely can we estimate cosmological parameters by performing a quantum measurement on a cosmological quantum state? In quantum estimation theory the variance of an unbiased parameter estimator is bounded from below by the inverse of measurement-dependent Fisher information and ultimately by quantum Fisher information, which is the maximization of the former over all positive operator valued measurements. Such bound is known as the quantum Cramer-Rao bound. We consider the evolution of a massless scalar field with Bunch-Davies vacuum in a spatially flat FLRW spacetime, which results in a…
The field of quantum metrology seeks to apply quantum techniques and/or resources to classical sensing approaches with the goal of enhancing the precision in the estimation of a parameter beyond what can be achieved with classical resources. Theoretically, the fundamental minimum uncertainty in the estimation of a parameter for a given probing state is bounded by the quantum Cramér-Rao bound. From a practical perspective, it is necessary to find physical measurements that can saturate this fundamental limit and to show experimentally that it is possible to perform measurements with the require…
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