Synthesis and reconstruction of quantum states (1)

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.

The second part of the lesson recalled some fundamental results on the measurement of a quantum system. After a brief statement of the postulates of standard projective measurement (projection onto the eigenstates of the measured observable, probability law of the result, reproducibility of the measurement), we recalled the properties of POVM measurements defined by giving a set of positive Hermitian operators constituting a partition of the unit operator. The projection rule of these POVMs is similar to that of standard measures, but the measurement reproducibility property is no longer satisfied. We then showed how the POVM measurement appears as a standard measurement performed on an environment to which the system is coupled by an appropriate unit operation. To make this analysis more concrete, we then gave a few examples of POVMs. Starting with the measurement of a qubit, we showed that a four-element POVM could be associated with the vectors joining the origin of the Bloch sphere to the vertices of a regular tetrahedron. Statistical measurements performed on this POVM allow us to define the state of the qubit in a way equivalent to the standard definition based on the mean values of the Pauli operators. A symmetrical set of N qubits can be measured using a POVM with an infinite number of elements, built on the basis of the coherent states of the system's collective angular momentum. We have described this POVM and shown how it can be realized, at least in principle, by associating with the Hilbert space of the system the space of positions of a point particle evolving on a sphere image of the Bloch hyper-sphere of the collective angular momentum. The POVM is realized by measuring the position of this particle in the standard way, after coupling it to the angular momentum via a unitary transformation. The final POVM example concerned the measurement of a harmonic oscillator. We defined a two-element POVM by coupling the oscillator to a qubit on which a standard measurement is performed. We then generalized this model to the product of several two-element POVMs, achieved by successively coupling the oscillator to independent qubits on which projective measurements are performed. This model describes the non-destructive quantum measurement of the photon number of a field mode in a cavity, achieved by sending a train of two-level atoms dispersively coupled to the field into the cavity. The results of the binary measurements on the various qubits project the number of photons in the cavity onto a discrete value, the result of the measurement. With this example, we have concluded our first introductory lecture, recalling results established in previous lectures.