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Questions tagged [quantum-metrology]

The study of making high-resolution and highly sensitive measurements of physical parameters using quantum theory to describe the physical systems, particularly exploiting quantum entanglement and quantum squeezing.

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Where does the "error propagation formula" $(\Delta \theta)^2=(\Delta M)^2/|\partial_\theta\langle M\rangle|^2$ come from, in estimation theory?

Consider the single parameter estimation setting, where we have a distribution depending on $\theta$ and we're looking for a "good" estimator for $\theta$. A commonly mentioned strategy, ...
glS's user avatar
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Generalizing error propagation formula to multi-parameters

For single parameter phase estimation we have the Cramer-Rao bound $$(\Delta \theta)^2 \geq \frac{1}{F_{Q}[\rho, \hat{A}]},$$where $F_{Q}$ is the quantum Fisher information and where instead of an ...
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Fisher information from likelihood function for discrete quantum case

In the context of a single phase estimation problem of a quantum photonics experiment. For example consider a 3-photon quantum circuit (such as the Mach-Zehnder which depends on some phase shift ...
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Calculation of QCRB from QCRB

On page 3 of Zhuang et al. (2018), they found the quantum Cramér-Rao bound (QCRB) of the parameter using the quantum fisher information matrix. See Equations $(15)$ and Eq $(16)$. The problem is ...
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How to compute the SLDs for pure single-qubit states?

In Demkowicz-Dobrzanski et al. (arXiv:2001.11742), the authors mention in Eq. (74), page 22, that the symmetric logarithmic derivatives (SLDs) for pure states parametrised in the usual way via the ...
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Fisher information of parametric channel

Suppose $\Phi_\theta$ is a quantum channel whose action can be written for any state $\rho\in \mathcal S(\mathcal H_S)$ in the Stinespring representation as $\Phi_\theta(\rho)= \text{Tr}_E(U_\theta (\...
Quantastic's user avatar
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How to compute the QFI of a thermal state?

Let $\rho=\frac{1}{Z}\exp(-\beta H)$ be the thermal state associated to the Hamiltonian $$H=\hbar\omega\sum_i\left( a_i^\dagger a_i+\frac12\right).$$ I wonder how the quantum Fisher information of ...
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What are the similarity and difference between quantum fidelity estimation and parameter estimation problem?

Quantum fidelity (1) estimation is to estimate the similarity between two quantum states or process. Could quantum fidelity be viewd as a parameter? And what are the similarity and difference between ...
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In quantum metrology, how do we achieve unbiased estimator in the local unbiased formalism?

In quantum metrology, the main goal is to estimate some unknown parameter to the limit of quantum mechanics. Suppose there's a unitary quantum channel $U_{\theta}=e^{-i\theta \sigma _z}$ with $\theta$ ...
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Paris 2009 paper on Quantum Estimation. From eq. 12 to eq. 16

In the paper "Quantum estimation for quantum technology", by Matteo Paris (2009), one is concerned with estimating a parameter $\lambda$ encoded in a quantum state $\rho_\lambda = \sum_n \...
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Prove that the Bures metric satisfies a contractive property and has unitary invariance

In this paper, the authors assert that the Bures metric satisfies a contractive property and has unitary invariance. These terms aren't defined or proved in the paper, nor is any reference given for a ...
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Is Quantum Cramer-Rao bound for single parameter always attainable?

First I will give some background of Quantum Cramer-Rao bound. There is an amount called Fisher Information:$F(\lambda)=\sum_x{p\left( x|\lambda \right) \left( \partial _{\lambda}\ln p\left( x|\lambda ...
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Period of phase leads the advantage of Heisenberg's Limit disappear?

In Quantum Metrology, the aim is to estimate some unknown parameters(I will talk about one parameter estimation in this post, while multiparameter is also available) as precise as possible. Without ...
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How is quantum metrology realized in experiment?

Are there some lists of the method to realize quantum metrology, or rather, utilizing quantum resources to enhance the precision of parameter estimation. The experimental method I've know is listed as ...
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Are SIC-POVMs optimal for quantum state reconstruction?

Mutually unbiased bases (MUBs) are pairs of orthonormal bases $\{u_j\}_j,\{v_j\}_j\in\mathbb C^N$ such that $$|\langle u_j,v_k\rangle|= \frac{1}{\sqrt N},$$ for all $j,k=1,...,N$. These are useful for ...
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Does the symmetric logarithmic derivative operator have a geometric interpretation?

In the context of Bures metric and quantum Fisher information, an important object is the symmetric logarithmic derivative (SLD). This is usually introduced as a way to express the derivative of a ...
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Biggest variance of $h=\sum_i H_i$?

What's the biggest variance of $h=\sum_i H_i$ where $H_i$ is the hamiltonian act on the ith qubit? If the n qubits state is separable, i.e., the state is $\mid\psi_1\rangle\otimes\mid\psi_2\rangle\...
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Unit vanishes in the Quantum Cramer-Rao Bound?

The Quantum Cramer-Rao Bound states that the precision we can achieve is bounded below by: $$(\Delta \theta)^2\ge\frac{1}{mF_Q[\varrho,H]},$$ where $m$ is the number of independent repetitions, and $...
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Understanding the $M$ upper bound in the paper: "Multipartite entanglement and high-precision metrology"

This paper is a paper in 2012 and cited by a lot of papers. And there does not exist comment in arxiv or error statement in PRA. But when I reading this paper, I think the right part of the eq(23) ...
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What is the difference between "Shot-Noise-Limit" and "Standard Quantum Limit"?

It seems that in a lot of papers in the field of quantum metrology, there are two terms Shot-Noise-Limit and Standard Quantum Limit which are frequently referred to. What's the difference between them,...
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2 votes
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What happens in the Cramer-Rao bound if the quantum Fisher information is zero?

The famous Cramer-Rao bound is $$\Delta\theta\ge\frac{1}{\sqrt {F[\rho,H]}}$$ But what happens if the denominator vanishes, i.e., $F[\rho,H]=0$ ($F[\rho,H]$ here stands for the quantum fisher ...
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How is the connection between Bures fidelity and quantum Fisher information derived?

I recently came to know that there is a connection between Bures Fidelity $(F_B)$ and Quantum Fisher Information $(F_Q)$ given by $$[F_{B}(\rho, \rho_\theta)]^2 = 1 - \frac{\theta^2}{4} F_Q[\rho, A] + ...
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Why is the quantum Fisher information for pure states $F_Q[\rho,A]=4(\Delta A)^2$?

Assume that a density matrix is given in its eigenbasis as $$\rho = \sum_{k}\lambda_k |k \rangle \langle k|.$$ On page 19 of this paper, it states that the Quantum Fisher Information is given as $$F_{...
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Why is the quantum Fisher information $J_f=[f(\frac43-f)]^{-1}$ for maximally entangled qubit pairs?

I am reading paper Channel Identification and its Impact on Quantum LDPC Code Performance (arXiv) where the authors discuss the scenario where the decoder of a Quantum LDPC code uses an estimation of ...
Josu Etxezarreta Martinez's user avatar