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8 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 $\...
ryanhill1's user avatar
  • 2,503
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 ...
Quantum Mechanic's user avatar
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)^...
AHusain's user avatar
  • 3,663
6 votes
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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 ...
Quantum Mechanic's user avatar
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, ...
Jonathan Trousdale's user avatar
5 votes
Accepted

Biggest variance of $h=\sum_i H_i$?

Here is an approach that requires no specific knowledge about $|\psi\rangle$ whatsoever. In your description you implied that each $H_i$ has the same maximum and minimum eigenvalues $\lambda_m$ and $\...
forky40's user avatar
  • 7,183
5 votes

Are SIC-POVMs optimal for quantum state reconstruction?

First of all, here's a short disclaimer: I'm not an in-depth expert in this field, I'm just currently getting in contact with tomography more and more often :) So take the following with a grain of ...
Markus Heinrich's user avatar
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 ...
tsgeorgios's user avatar
  • 1,416
4 votes
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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 ...
Quantum Mechanic's user avatar
3 votes
Accepted

How to compute the SLDs for pure single-qubit states?

These results are specific to qubits and your direct verification is the best way to do it. You can see this because they required the property of Pauli matrices that $\{\pmb{v}\cdot\pmb{\sigma},\pmb{...
Quantum Mechanic's user avatar
3 votes
Accepted

How to compute the QFI of a thermal state?

QFI must always be computed with respect to a parameter "$\theta$". Perhaps it is the temperature that you want to use here, or $\beta$? Regardless, we can put the state into your desired ...
Quantum Mechanic's user avatar
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 ...
Quantum Mechanic's user avatar
3 votes
Accepted

Period of phase leads the advantage of Heisenberg's Limit disappear?

You are correct, but the SQL is a local limit, when you already have a very good idea what the value of $\theta$ is, so there is no contradiction. Let's work through it. You measure some relative ...
Quantum Mechanic's user avatar
2 votes

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 ...
David Bar Moshe's user avatar
2 votes
Accepted

Fisher information from likelihood function for discrete quantum case

This is correct. If you retained each of the measurement results as opposed to the total number of times each result occured, you would neglect the multinomial factor. In addition, it is worth ...
Quantum Mechanic's user avatar
2 votes

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

The basic starting point will be to distinguish between "a measurement" in quantum theory and "the result of a measurement" or "the measurement of an operator." When one ...
Quantum Mechanic's user avatar
1 vote

Generalizing error propagation formula to multi-parameters

Short answer: It always behooves oneself to learn the classical version of a theory before the quantum one. Many many people (including but certainly not limited to OP) could have their problems ...
Quantum Mechanic's user avatar

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