Synthesis and reconstruction of quantum states (4)

The fourth lesson continued the study of cloning by describing how to produce M approximate copies of a single qubit in a pure state, and then more generally, M copies from N identical qubits all in the same pure state. Generalizing the results of the previous lesson, we have shown that optimal cloning from 1 to M qubits can be achieved by preparing the M qubits in a state resulting from symmetrization by inter-qubit exchange of the tensor product of the state of the qubit to be cloned with M-1 qubits in a completely depolarized state, described by a density matrix proportional to unity. In the case of cloning one qubit into two, this symmetrization operation is performed by the logic circuit described in the previous lesson. In the more general case of cloning from one to M qubits with M > 2, symmetrization can be achieved by a unitary operation on a system associating a cloning machine with the M qubit system, followed by a trace on the state of this machine. We have shown that the cloning fidelity thus obtained, equal to (2M + 1)/3M, tends towards 2/3 when M tends towards infinity, a result in line with the optimal fidelity of qubit estimation. Indeed, an estimation procedure may consist in first cloning a qbit into an infinite number of copies, before measuring the state of these copies with arbitrarily high precision. As the symmetrization operation is a collective operation on qubits, we found that collective estimation procedures are more efficient than the accumulation of independent measurements on qubits. We then generalized the procedure to the cloning of N to M qubits, again exploiting the symmetrization method. We have shown that the fidelity obtained, F = (MN + M + N)/M(N + 2) generalizes to the case N > 1 the result (2M + 1)/3M obtained for N = 1 and we have discussed this general result in relation to the theory of optimal estimation, illustrating once again the internal consistency of quantum theory.

In the second part of the lesson, we showed that optimal cloning by symmetrization could be naturally achieved not by a complex logic circuit, but simply by exploiting the properties of stimulated emission. Qubit states are then encoded in the horizontal (H) or vertical (V) polarization of photons. The cloning machine can ideally be realized by M-N three-level atoms amplifying H and V photons on two transitions with a common excited level. If N photons of H polarization "pass" over this machine, more than N photons are produced with the same H polarization under the effect of stimulated emission on the corresponding transition, but some are also produced with V polarization under the effect of spontaneous emission on the other transition. A simple counting calculation shows that the fraction produced with the "wrong" polarization returns to the optimal cloning limit. In other words, the approximate nature of cloning is a consequence of the inevitable spontaneous emission that adds its quantum noise to the classical amplification process. The symmetrization of optimal cloning in this model results simply from the bosonic character of the photons, which are automatically in a symmetrical state by exchange. While the theory of this optimal cloning by optical amplification is straightforward, its experimental realization is tricky. We concluded the lesson with a discussion of the problems facing the experimenter. In the case of cloning from N = 1 to M = 2, a single photon must be sent to an excited atom and the two resulting photons detected in coincidence. But the atom emits spontaneously in all vacuum modes, while the incident photon stimulates emission only in its own mode. The probability of cloning is therefore low, the most frequent process being the spontaneous and uncontrollable emission of the atom in a mode different from the photon to be cloned. To solve this mode matching problem, the spontaneous atomic emission of a photon is replaced by a parametric conversion process in a non-linear crystal emitting photons in pairs. Detection of one photon of the pair defines the mode in which the other is emitted. By sending the photon to be cloned into this mode, we can isolate a signal sensitive only to events for which optical amplification is important. We've concluded the lesson with a brief description of the experiment, leaving the details for the next lesson.