As of May 31, 2023, we have updated our Code of Conduct.

New answers tagged

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|\...
Quantum Mechanic's user avatar
0 votes

How to show $T(\rho,\sigma)≥\sum_i|r_i − s_i|$ with $r_i,s_i$ eigenvalues of $\rho,\sigma$?

Here's a rephrasing of the same approach to the proof, using a slightly different notation. You want to prove that $$T(\rho,\sigma) \equiv \frac12\|\rho-\sigma\|_1 \ge \frac12\sum_k |\lambda_k(\rho) - ...
glS's user avatar
  • 21.7k
2 votes

Is it true that $|r_i-s_i| \le 1/2$ for all $i$, where $r_i$ and $s_i$ are the eigenvalues of density matrices $\rho$ and $\sigma$?

In this proof, we assume that the trace distance between $\rho$ and $\sigma$ is upper-bounded by $\frac{1}{\mathrm{e}}<\frac12$. As mentioned by glS in the comments, the trace distance is lower-...
Tristan Nemoz's user avatar
1 vote

What is the conditional min-entropy of a pure bipartite state?

Reading again the paper you linked, I think the way the authors were thinking about the result was of showing this via the relations between conditional min- and max-entropies, see discussion at the ...
glS's user avatar
  • 21.7k
3 votes
Accepted

What is the conditional min-entropy of a pure bipartite state?

I'll use an equivalent definition of the min-entropy $$ \begin{aligned} H_{\min}(A|B) = - \log_2 \min& \quad \lambda \\ \mathrm{s.t.}& \quad \rho_{AB} \leq \lambda I_A \otimes \sigma_B \\ &...
Rammus's user avatar
  • 4,906
1 vote

What is the conditional min-entropy of a pure bipartite state?

Remember that the max relative entropy satisfies, in general, $$D_{\rm max}(\rho\|\sigma) = \log \inf\{\eta\ge0:\,\, \rho\le \eta \sigma\}.$$ In particular, $$D_{\rm max}(\rho\|I\otimes \sigma)=\log \...
glS's user avatar
  • 21.7k
2 votes

Clarification about inverses in sandwiched Renyi divergence

Firstly, the sandwiched divergence can be infinite even when $\rho$ and $\sigma$ are not orthogonal. For example, consider $\rho = \frac{|0\rangle \langle 0| + |1\rangle\langle 1|}{2}$ and $\sigma = |...
Rammus's user avatar
  • 4,906

Top 50 recent answers are included