Resources
Nothing like a good (printed) book to master a subject…
Here is a selection of introductory resources that were useful for me, with a preference for those that mix intuition with mathematics. *= more specialized reference.
Quantum entropies

Trace Inequalities and Quantum Entropy*
by E. Carlen
(PDF)
On the relations between the strong subadditivity, convex and monotone matrix functions, and trace inequalities. 
Quantum Information Processing with Finite Resources*
by M. Tomamichel
(PDF)
Quantum (Renyi) divergences and their properties.
Convex optimisation

Convex Optimisation
by S. Boyd & L. Vanderberghe
(PDF)
Introduction to convex sets, convex problems, algorithms 
Lectures on Convex Optimisation*
by Y. Nesterov
Interiorpoint algorithms, selfconcordant barriers
Invariant theory

Algorithms in Invariant Theory
by B. Sturmfels
On polynomial invariants of groups. A modern treatement of a subject going back to the 19th century. See also this video
Computing
 Computational Complexity by S. Arora & B. Barak
Hardware

Introductory Quantum Optics
by C. Jerry & P. Knight
Gaussian and nonGaussian states, phase vs. Fockspace representation. 
Circuit QED: superconducting qubits coupled to microwave photons
by S. Girvin
(PDF)
I liked this introduction to superconducting circuits. For physicists.
Scientific writing/careers

The Chicago Guide to Communicating Science
by S. Montgommery
Passionate reading, conveys a simple message of what is important in scientific writing. 
Clear and Simple as the Truth*
by FN. Thomas & M. Turner
If you want to know what excellence in scientific writing looks like. NB: Hard to apply! 
A PhD Is Not Enough! A Guide to Survival in Science
by P. Feibelman
Nicely written, relevant for mathphysics related fields