Mathematical methods of quantum information theory, Lecture 4

In 2017 Reinhard Werner gave a series of lectures on the mathematical methods of quantum information theory at the Leibniz Universität Hannover. These lectures were recorded and I have the pleasure of hosting these videos on my youtube channel. Over the coming weeks I'll be posting these lectures here.

The prerequisites for these lectures are a standard course on quantum mechanics and some familiarity with mathematical analysis, e.g., Hilbert space, operators, etc., although these topics are reviewed in the first lectures.

Lecture notes and exercises will not be distributed.

In this fourth lecture, state space, probabilites, composition, the positive cone, positivity, and the geometry of cones were discussed.

[ArthurParzygnat]

I can't stress enough how excellent these lectures are. Thank you.


Maybe others will come to a similar conclusion, but I came up with an alternative proof of (1) implies the first part of (2) in the Lemma (around 44:04), which I felt would be good to share. I'll use v for my vector instead of phi.


Since <v, Av>≥0 for all v,
<v, Av>=<v, Av>* (the complex conjugate)
=<Av, v> by the definition of a (complex) inner product
=<v, A* v> by definition of the adjoint.
Taking the difference gives
0=<v, (A-A*)v> for all v
which holds if and only if A-A*=0, i.e. A=A*.