This repository aims to contain definitions and proofs of basic ideas in quantum information theory. Some major goals, in rough order of difficulty, would be:
- Defining most notions of "distance", "entropy", "information", "capacity" that occur in the literature.
- Showing that these reflect the classical notions where applicable
- For instance, that if you embed a clasical probability distribution as a quantum mixed state, then the classical conditional entropy and the quantum conditional entropy are the same number.
- Strong sub-additivity of von Neumann entropy
- Holevo's theorem
- The LSD theorem on quantum capacity
- Non-additivity of quantum capacity
All of this will be done only in the theory finite-dimensional Hilbert spaces. Reasons:
- Most quantum information theory is done in this setting anyway. Not to say that the infinite-dimensional work isn't important, just that this is what more researchers spend their time thinking about.
- Infinite-dimensional quantum theory can be weirdly behaved.
- Dealing with infinite-dimensional quantum theory is just hard. You need e.g. trace-class operators, CTC functions, and people often can't even agree on the definitions. (For instance, does a mixed state necessarily have a finite spectrum? I've seen it both ways.)
There is a tiny bit of infinite-dimensional theory in the QuantumInfo/InfiniteDim folder. But most stuff is in the QuantumInfo/FiniteDim folder.
Docmentation of the main definitions can be found at DOC.md. A majority of the work will be outlining the major definitions and theorems from Mark Wilde's Quantum Information Theory. A correspondence to the definitions and theorems (mostly as todo-list!) are in TODO.md