The main topic of this thesis is the study of de Finetti methods and their applications in quantum information theory. Those methods include de Finetti representation theorems and De Finetti reductions.

The primary motivation of a de Finetti representation theorem is to represent, or approximate, a mathematical object symmetric under permutation of its components, into a probabilistic ensemble of elementary independent and identically distributed (i.i.d.) constituents.

De Finetti reductions are another class of techniques that are used to take advantage of permutation symmetries. For example, a quantum de Finetti reduction provides an upper bound to a symmetric quantum state in the form of an integral superposition of product states,
weighted by a factor that is polynomial in terms of the number of copies and exponential in terms of the local dimensionality.

Our findings include:

1- The development of general mathematical techniques that can be used to obtain concrete constrained de Finetti representation theorems for the desired application.

2- The application of those methods to the problem of approximate quantum error correction. In particular, we use our framework to develop asymptotically converging SDP hierarchies that can be used to study the average and worst error cases, as given by the quantum
channel fidelity and a channel distance based on the diamond norm, respectively.

3- A new de Finetti reduction in the presence of an additional system carrying side information, that can handle various types of linear constraints.

4- The development of entropic techniques that can be used to generate de Finetti representation theorems from a starting de Finetti reduction. In particular, we use those methods to obtain a new proof for finite quantum de Finetti theorems.