Guillaume Aubrun and I wrote a book focused on the interface between mathematical aspects of Quantum Information Theory and local theory of Banach spaces, a field which studies the properties of (very) high-dimensional objects and is also often called Asymptotic Geometric Analysis.
The book appeared in the Mathematical Surveys and Monographs series of the American Mathematical Society in the fall of 2017. If you read Alice and Bob Meet Banach (or even if you did not, but would like to), you may want to consult the related blog.
Here is the book's table of contents, preface, list of tables (we hope some of them will be very useful beyond the present applications), sample chapters: The Mathematics of Quantum Information Theory, Metric Entropy and Concentration of Measure in Classical Spaces, Entanglement of Pure States in High Dimensions, Random Quantum States and, last but not least, Errata.