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example of a Schur-concave metric on density matrices

I am looking for an example of a metric/distance function $D(\rho,\sigma)$ which is Schur-concave apart from fidelity. In particular I am interested in the relation $D(\rho, \sum_i p_i \sigma_i) \leq_{...
Ghost-of-PPPF's user avatar
2 votes
1 answer

Why can we write $\rho=\sum_\mu q_\mu|\varphi_\mu\rangle\!\langle\varphi_\mu|$ iff $q\preceq \mathrm{spec}(\rho)$?

Exercise 2.6 in Preskill's notes (chapter 2, around page 48, pdf available here) asks to prove that an arbitrary state $\rho=\sum_i p_i |\alpha_i\rangle\!\langle\alpha_i|$, where $p_i$ and $|\alpha_i\...
user15342's user avatar
  • 155
4 votes
1 answer

Prove that $A\preceq B$ implies $A=\Psi(B)$ for some channel $\Psi$

Define $\newcommand{\PP}{\mathbb{P}}\newcommand{\ket}[1]{\lvert #1\rangle}\newcommand{\tr}{\operatorname{tr}}\newcommand{\ketbra}[1]{\lvert #1\rangle\!\langle #1\rvert}\PP_\psi\equiv\ketbra\psi$, and ...
glS's user avatar
  • 25.4k