We expose the mathematical model of qubits and quantum circuits.

We will cover the definition and first properties of density matrices in quantum information. We will introduce mixed states, the operations of partial trace, and of purification. If time permits, we will briefly introduce the quantum marginal problem. We loosely follow sections 2.4 and 2.5 of Nielsen, Chuang "Quantum computation and quantum information".

In this seminar I would like to introduce the formalism commonly used to describe quantum computing: from the vectorial definition of multi-qubit states, to the description of their dynamics through the application of quantum gates as unitary matrix multiplication. The main goal is to compare this formalism with a purely tensorial one that is not used by the physics community but which could be useful in the quantum gate decomposition procedure.

In this talk, I will discuss the structure of Shor’s algorithm based on an ingenious use of discrete Fourier transform. 

The degree of entanglement of a pure quantum state can be measured by its distance or angle to the closest product state. This is called Geometric Measure of Entanglement (GME) and can be extended to mixed states. I will try to introduce this notion, by providing examples from the literature.

I will present entanglement as a resource theory where the free operations are local operations and classical communication.

I will not closely follow any particular reference, but all the topics I will present can be found in:

• M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010, sec. 12.5;

• I. Bengtsson, K. Życzkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, 2nd edition, Cambridge University Press, 2017,  chpts. 16-17;

• E. Chitambar, G. Gour, Quantum resource theories,  Rev. Mod. Phys. 91, 025001, 2019.

The ground state of a Hamiltonian is a most suitable state to play with if one is interested in seeing quantum mechanics effects. These kinds of states naturally appear in a special "region" of the tensor space, namely on/very close to Tensor Network varieties. I would like to illustrate these connections and, in dependence on the time, to show one of the few examples where all the construction is exactly possibile, namely the so-called AKLT model.