Synthesis and reconstruction of quantum states (3)

The third lesson dealt with the question of quantum cloning, closely linked to that of state estimation. We recalled that, unlike the classical situation, the information encoded in the state of a quantum system cannot be cloned exactly. This impossibility is linked to the very nature of the statistical concept of state in quantum physics. However, approximate cloning is possible. In the case of qubits, this can be achieved by a logic circuit combining one- and two-bit gates. In its simplest form, this cloning approximately copies the state of one qubit onto another, giving two qubits sharing quantum information symmetrically, both described by the same density operator reproducing with optimum fidelity the initial state to be cloned. We approached the lesson by first analyzing the reasons for the impossibility of the exact cloning that would lead to the possibility of superluminal communication in an EPR-like experiment. We then described an optimal universal cloning procedure cloning with the same fidelity any state of a qubit. This procedure uses a logic circuit, explicitly described, combining two-qubit CNOT gates and individual qubit rotations. This part of the lesson exploited results on logic circuits established in earlier lectures. We showed that this cloning achieves the fidelity F = 5/6, higher than the limit of the fidelity of the single-copy estimate of a given qubit (S = 2/3). By analyzing this difference, we have shown that a qubit estimation strategy based on cloning remains within the limits allowed by estimation theory.