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6 votes
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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 ...
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5 votes
Accepted

Error in the Toth's 2012 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, ...
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4 votes
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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 $\...
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  • 1,636
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 ...
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  • 1,306
4 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)^...
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  • 3,523
4 votes
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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 $\...
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  • 4,537
4 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 ...
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4 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 ...
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3 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 ...
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3 votes
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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 ...
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1 vote
Accepted

Prove that the Bures metric satisfies a contractive property and has unitary invariance

Contractivity refers to the fact that, under the action of any CPTP map $\mathcal{E}$, a given metric satisfies $M(\rho,\sigma) \geq M(\mathcal{E}(\rho), \mathcal{E}(\sigma))$. Unitary invariance ...
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  • 86
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 ...
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