Timeline for Formula for a completely positive map

8 events

when toggle format	what		by	license	comment
Dec 13, 2019 at 13:18	comment	added	Arnold Neumaier		@NathanielJohnston: But if you go around a closed loop, do you get the same phases?
Dec 13, 2019 at 12:39	comment	added	Nathaniel Johnston		@ArnoldNeumaier - The eigenvalues of $A$ and $B$ vary continuously with the entries of $A$ and $B$, and if those eigenvalues are distinct then so do the projections onto the eigenspaces, and thus their spectral decompositions, and thus the matrix $C$ (or $U$). Things get murky if you have repeated eigenvalues though.
Dec 13, 2019 at 10:31	comment	added	Arnold Neumaier		@lcv:: It is not clear to me whether $U$ or $C$ can be chosen to be continuous.
Dec 13, 2019 at 10:26	history	edited	Arnold Neumaier	CC BY-SA 4.0	added a qualifying question
Dec 12, 2019 at 19:18	comment	added	lcv		@nathanielJohnston yes I believe so. If he doesn't ask for the map to be trace preserving at least.
Dec 12, 2019 at 16:10	comment	added	Nathaniel Johnston		@Icv - The same idea works even if $A$ and $B$ are just both positive definite with distinct eigenvalues. Then you can construct a matrix $C$ that sends the vectors in a spectral decomposition of $A$ to the vectors in a spectral decomposition of $B$, and the map $x \mapsto CxC^\dagger$ works.
Dec 11, 2019 at 16:48	comment	added	lcv		For once, if $A$ and $B$ are iso-spectral (with distinct eigenvalues) such a map is $x \mapsto Ux U^\dagger$ where $ B=UAU^\dagger$ for some unitary $U$.
Dec 11, 2019 at 15:39	history	asked	Arnold Neumaier	CC BY-SA 4.0