The first lesson covered the definition and properties of quantum states, on the one hand, and measurement in quantum physics, on the other. The notions of a pure state (represented by a wave function) and a statistical mixture of states (represented by a density operator) were recalled. We then described the simplest system, that of a two-state qubit, introducing the representation of the Bloch vector evolving on the Bloch sphere (pure case) or within it (statistical mixture). This representation identifies qubits with 1⁄2 spins and exploits the angular momentum properties of these spins. The link between the components of the Bloch vector and the mean values of the qubit's Pauli operators has been recalled. The generalization of this description to the case of a d-level system (d > 2; qudit) was briefly discussed. In this case, the Pauli operators are replaced by a basis of irreducible tensor operators on which the density operator of the system can be developed. We then turned to the description of a symmetric N-qubit system by exchanging any two qubits by introducing the basis of Dicke states, eigenstates with maximum eigenvalue J = N/2 of the total angular momentum of the set of N 1⁄2 spins making up the system. Coherent states of angular momentum were then described. The N spins are then aligned in the same direction on a "Bloch hyper sphere" whose given polar angles define the state. The coherent states thus define a continuous basis of states on which any symmetrical state of the N qubits can develop. We recall the analogy of this basis with that of the coherent states of a harmonic oscillator (Glauber coherent states). The properties of coherent angular momentum states (transverse fluctuations, closure relations) were recalled. We concluded this section with a general discussion of the statistical nature of the notion of quantum state, which is impossible to determine from measurements performed on a single realization of the system. Knowledge of the wave function (pure case) or the density matrix (mixture) necessarily requires observations to be made on a large set of systems all prepared in the same state. The corollary of this statistical property is that it is impossible to clone a quantum system perfectly, since such cloning would make it possible to realize an ensemble from a single system, and would introduce an internal contradiction into the theory.
09:30 - 10:30
Lecture
Not recorded
Synthesis and reconstruction of quantum states (1)
Serge Haroche
09:30 - 10:30
