7
votes
Accepted
Does the symmetric logarithmic derivative operator have a geometric interpretation?
First, the classical correspondence, explaining why the SLD should be present. The Fisher information is the expectation value of the score, where the score is the logarithmic derivative of the ...
- 2,959
6
votes
Accepted
Why is the quantum Fisher information for pure states $F_Q[\rho,A]=4(\Delta A)^2$?
Suppose $\lambda_0 = 1$ and the rest are $0$.
$$
F_Q [\rho,A] = 2 \sum_{k,l} \frac{(\lambda_k-\lambda_l)^2}{\lambda_k + \lambda_l} | \langle k |A| l \rangle |^2\\
= 2 \sum_{k=0,l \neq 0} \frac{(1-0)^...
- 3,603
6
votes
Accepted
Unit vanishes in the Quantum Cramer-Rao Bound?
You are correct: the units must indeed match. If we take a standard evolution with unitary $U=\exp(-i H \theta)$, then the units of $H$ and $\theta$ must match such that $H\theta$ is unitless. For a ...
- 2,959
6
votes
Accepted
Understanding the $M$ upper bound in the paper: "Multipartite entanglement and high-precision metrology"
Your conclusion appears correct to me. It seems that Eq.(23), modified with your proposed change to the RHS, can be verified by combining Eq.(3), for the upper bound on the $M$ unentangled particles, ...
- 3,302
5
votes
Accepted
What is the difference between "Shot-Noise-Limit" and "Standard Quantum Limit"?
You are correct that both terms reference the central limit theorem (CLT), which states that[1]
...the average of a large number $n$ of independent measurements (each
having standard deviation $\...
- 2,308
4
votes
How is the connection between Bures fidelity and quantum Fisher information derived?
Let $ \rho = \sum_n \rho_n |\psi_n \rangle \langle \psi_n | $ be the eigendecomposition of $\rho$. We will calculate everything in terms of $ |\psi_n \rangle$ basis.
Note that
$ \frac{d \rho_\theta}{d ...
- 1,386
4
votes
Accepted
Stabilizer state QFI lower limit query
The state $\psi$ (this is denoting the density matrix, even though it's a pure state) can be described as a sum of all the products of the stabilizers. We are promised that $X_i$ is not in the ...
- 52.3k
4
votes
Accepted
How to derive the quantum Fisher information from the relative entropy?
Expressing the derivative $\partial_i\rho$ in terms of its eigenvalues and eigenvectors will show us that these two are not equal. I will assume a full-rank density matrix $\rho$ to streamline the ...
- 2,959
4
votes
Accepted
Is Quantum Cramer-Rao bound for single parameter always attainable?
The answer given in the literature is always yes, this is guaranteed to be possible for single-parameter estimation. What you are noticing eventually gives rise to some cool things that I'll mention ...
- 2,959
3
votes
Accepted
What does $\langle\partial_i\psi(\theta)|\psi(\theta)\rangle$ mean when implementing the Quantum Fisher information matrix?
In fact, $\partial_{k} \psi(\boldsymbol{\theta})$ is a vector. For example, $\frac{\mathrm{d}}{dx} \begin{pmatrix}x\\x^2\end{pmatrix}=\begin{pmatrix}1\\2x\end{pmatrix}$ , i.e., the derivative is ...
- 2,782
3
votes
Accepted
Paris 2009 paper on Quantum Estimation. From eq. 12 to eq. 16
The first question is correct: consider unitary evolution with generator $H_\lambda$ such that $|\partial_\lambda \psi_n\rangle=iH_\lambda |\psi_n\rangle$ for all $n$. The states $|\psi_n\rangle$ need ...
- 2,959
2
votes
Accepted
What are the antisymmetric terms in $\sigma_{mn}$ in the expression for the Fisher information?
Antisymmetric means that $\sigma_{nm}=-\sigma_{mn}$. Since the sum ranges over all values of $m$ and $n$, adding an antisymmetric term adds something proportional to
$$|\langle \psi_m^\lambda|\...
- 2,959
1
vote
Does the symmetric logarithmic derivative operator have a geometric interpretation?
Although, the Bures metric, the Fisher tensor and the symmetric logarithmic derivative appear mainly in quantum estimation theory, and even though the original discovery by Helstrom was in this ...
- 2,455
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