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.
Questions tagged [quantum-metrology]
28 questions
12
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1 answer
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, because it seems they all refer to the limit…
narip
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11
votes
2 answers
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 parametrised state as a superoperator acting on the…
glS
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10
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1 answer
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) should be $2M+(N-M)(N-M+2)$ instead of…
narip
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8
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1 answer
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:$F(\lambda)=\sum_x{p\left( x|\lambda \right) \left( \partial _{\lambda}\ln p\left( x|\lambda \right) \right) ^2}$ where $p\left( x|\lambda…
narip
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7
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1 answer
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 $$F_{Q}[\rho,A] = 2…
John Doe
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7
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1 answer
Are SIC-POVMs optimal for quantum state reconstruction?
Mutually unbiased bases (MUBs) are pairs of orthonormal bases $\{u_j\}_j,\{v_j\}_j\in\mathbb C^N$ such that
$$|\langle u_j,v_k\rangle|= \frac{1}{\sqrt N},$$
for all $j,k=1,...,N$.
These are useful for a variety of reasons, e.g. because they provide…
glS
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7
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1 answer
Biggest variance of $h=\sum_i H_i$?
What's the biggest variance of $h=\sum_i H_i$ where $H_i$ is the hamiltonian act on the ith qubit?
If the n qubits state is separable, i.e., the state is $\mid\psi_1\rangle\otimes\mid\psi_2\rangle\otimes\cdots\mid\psi_n\rangle$. Obviously, the…
narip
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6
votes
0 answers
Why is the quantum Fisher information $J_f=[f(\frac43-f)]^{-1}$ for maximally entangled qubit pairs?
I am reading paper Channel Identification and its Impact on Quantum LDPC Code Performance (arXiv) where the authors discuss the scenario where the decoder of a Quantum LDPC code uses an estimation of the depolarization probability of the channel in…
Josu Etxezarreta Martinez
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6
votes
1 answer
Where does the "error propagation formula" $(\Delta \theta)^2=(\Delta M)^2/|\partial_\theta\langle M\rangle|^2$ come from, in estimation theory?
Consider the single parameter estimation setting, where we have a distribution depending on $\theta$ and we're looking for a "good" estimator for $\theta$.
A commonly mentioned strategy, found e.g. in Eq. (7) of [TA2014], is to measure some…
glS
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5
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1 answer
How to compute the SLDs for pure single-qubit states?
In Demkowicz-Dobrzanski et al. (arXiv:2001.11742), the authors mention in Eq. (74), page 22, that the symmetric logarithmic derivatives (SLDs) for pure states parametrised in the usual way via the Bloch sphere are
$$L_\theta =…
glS
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5
votes
1 answer
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
$$[F_{B}(\rho, \rho_\theta)]^2 = 1 - \frac{\theta^2}{4} F_Q[\rho, A] + O(\theta^3),$$
where
$\rho_\theta =…
Mike
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4
votes
1 answer
Why $\delta X = \frac{X_\textrm{est}}{|d\langle X_{\textrm{est}} \rangle_X/dX|}-X$ quantifies the derivation of a parameter estimation?
In a famous paper "Statistical Distance and the Geometry of Quantum States" by Braunstein and Caves, the authors discuss the problem of estimating an unknown parameter $X$. One considers an estimator $X_\textrm{est}$ of $X$, which is a function of…
Laplacian
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4
votes
3 answers
How to compute the energy of a non-eigenstate in a Heisenberg limited form?
Let us have a Hamiltonian $H$ and a state $|\psi\rangle = \sum_i a_i |E_i\rangle$, a linear combination of eigenstates $|E_i\rangle$ of $H$ with eigenvalues $E_i$. What is the best way to achieve a Heisenberg-limited measurement of the energy of…
Pablo
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4
votes
2 answers
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 explicit estimator you can consider the error…
John Doe
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4
votes
1 answer
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 $\rho_\lambda = \sum_n \rho_n |\psi_n\rangle \langle \psi_n|$ - where both eigenvalue…
G Frazao
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