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 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