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Questions tagged [quantum-fisher-information]

The quantum analog of the classical Fisher information, a way of measuring the amount of information that a random observable A carries about an unknown parameter θ of a distribution that models A. The quantum Fisher information constrains the achievable precision in the statistical estimation of θ via the quantum Cramér–Rao bound.

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How to derive the higher terms in the Taylor expansion of the Bures fidelity?

The Wikipedia article for the quantum Fisher Information mentions that one can expand the Bures fidelity and the quantum Fisher Information will appear as the second-order correction term. However in ...
I1ussion'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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In what limit does the estimator sample variance converge to the Cramer-Rao bound?

In the context of a single phase estimation problem of a quantum photonics experiment (related post). For example consider a 3-photon quantum circuit (such as the Mach-Zehnder which depends on some ...
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Modelling Mach-Zehnder and saturating Cramer-Rao bound

I am simulating (using Mathematica) a Mach-Zehnder interferometer, with photon counting measurements at the end (based on the setup described in the recent post) for the input state $|\psi\rangle:=|0,...
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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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Classical Fisher Information for 1-qubit vs 2-qubit in PennyLane

I'm attempting to examine the Classical Fisher Information (CFI) for a 1-qubit system in comparison to a 2-qubit system.(PennyLane) I anticipated that the CFI for the 2-qubit system would be double(at ...
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Plotting Fisher Information using PennyLane

I made 2-qubit based circuit with post-selection method. Post-selection method is, Let $$ K = \begin{bmatrix} \sqrt{1-\gamma} & 0 \\ 0 & 1 \end{bmatrix}\,. $$ Then, $$ \rho_{\text{post-...
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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 ...
Noobgrammer's user avatar
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Quantum computation of classical Fisher information

Consider a pure $n$-qubit quantum state $|\psi_\theta\rangle$ prepared by some parametrized quantum circuit. There exist well-known algorithms to efficiently estimate the quantum Fisher information ...
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Create qnode with density matrix on pennylane

I'm using pennylane. What I want to do is Create a qnode with the 2*2 density matrix of a single qubit one. It has the parameter as phi Given density matrix: $$\...
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Problems trying to plot the classical Fisher information with Pennylane

I'm working with pennylane. My goal is to plot CFI(Classical Fisher Information)with following quantum state. With the above equation I set gamma as 0. Then It becomes: If gamma is not equal to zero,...
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How is the quantum geometric tensor derived?

In https://arxiv.org/abs/2302.13515 the authors discuss in page 23 the quantum geometric tensor, defined as $$\mathcal Q_{\mu\nu} = \langle\partial_\mu\Psi|(I-|\Psi\rangle\!\langle\Psi|)|\partial_\nu\...
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What are the antisymmetric terms in $\sigma_{mn}$ in the expression for the Fisher information?

Given a parameter-dependent density operator $\hat\rho^\lambda$ and its spectral decomposition $\{\rho_m^\lambda, |\psi_n^\lambda\rangle\}$, Eq. $(17)$ from this review shows that one can compute its ...
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Why can't Quantum Fisher Information be negative?

Quantum Fisher Information is proportional to Fidelity susceptibility. Mathematically the equation is: $QFI=-\frac{\partial^2 d_B(\epsilon) }{\partial \epsilon^2}$ where above equation shows QFI is ...
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How do I find Quantum Fisher Information from an array of fidelity values for various parameter values?

I have an array of Fidelity values corresponding to parameter values. For example, eps=[0,0.1,0.2....,1] where eps is the parameter Fid=[1.0, 0.9, 0.96, 0.91, 0.85, 0.78, 0.71, 0.65, 0.59, 0.54] I ...
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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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How to derive the quantum Fisher information from the relative entropy?

The quantum relative entropy (QRE) between two states $\rho$ and $\sigma$ is given by $$ S(\rho\|\sigma)=\operatorname{Tr}(\rho\ln\rho)-\operatorname{Tr}(\rho\ln\sigma) $$ Now if $\rho$ and $\sigma$ ...
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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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What does $\langle\partial_i\psi(\theta)|\psi(\theta)\rangle$ mean when implementing the Quantum Fisher information matrix?

Following this paper, the quantum Fisher information matrix (QFIM) - $\mathcal{F}$ can be calculated as: $\mathcal{F}_{i, j}(\theta)=4 \operatorname{Re}\left[\left\langle\partial_{i} \psi(\boldsymbol{\...
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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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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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How does the quantum Fisher information provide bounds for the estimation of output states?

Assume you have some quantum process $Q$ (e.g. quantum state tomography) that intakes initialised states $\rho_{i}$, $i=1,\ldots,n$ and gives some output $\rho'_i$. $$ \rho_1 \to Q \to \rho'_1 \\ \...
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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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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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Stabilizer state QFI lower limit query

On page 1 of this paper it states that the QFI (Quantum Fisher Information) for pure states $\psi$ is $$\mathcal{Q}(\psi) = \sum_{i,j=1}^n\text{Tr}(X_iX_j\psi)-\text{Tr}(X_i \psi)\text{Tr}(X_j \psi)~~~...
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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 where the authors discuss the scenario where the decoder of a Quantum LDPC code uses an estimation of the ...
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