9 Quantum Operations on Qubitsand Their Characterization

Abstract

Information encoded in quantum system has to obey rules of quantum physics which impose strict bounds on state estimation and on possible manipulations with quantum information. That is, an unknown state of a qubit cannot be precisely determined from a measurement performed on a finite ensemble of identically prepared qubits. As a consequence the perfect, universal cloning map is not allowed. Another map which cannot be performed perfectly on an unknown state of a qubit is the "spin-flip" i.e. the universal-NOT gate. Within this significant framework the contextual experimental realization of the optimal 1 to 2 universal cloning and the optimal universal-NOT (U-NOT) gate by quantum injected optical parametric amplification (QIOPA) through a modified quantum state teleporta-tion protocol is reported. In addition, the realization of a multi-photon "all opti-cal" Schroedinger cat state is investigated by exploiting the information preserving property of the parametric amplification. Finally, as a significant demonstration of the perspectives offered by quantum entanglement to modern measurement theory, we shall present how the total parallelism of a bipartite entangled state can be adopted to extract efficiently the full information that characterizes any unknown "quantum operations". 9.1 Introduction Classical information is represented by bits which can be either 0 or 1. Quantum information is represented by quantum-bits, or qubits, which are two-dimensional quantum systems. A qubit, unlike a classical bit, can exist in a state |Ψ that is a superposition of two orthogonal basis states {| ↑↑, | ↓↓}, i.e. |Ψ = α| ↑↑+ β| ↓↓. The fact that qubits can exist in these superposition states gives to quantum information unusual properties. Specifically, information encoded in a quantum system has to obey rules of quantum physics that impose strict bounds on possible manipulations with quantum information. The common denominator of these bounds is that all quantum-mechanical transformations have to be represented by Completely Positive (CP) maps [1] that in turn impose a constraint on the fidelity of quantum-mechanical measurements. That is, an unknown state of a qubit cannot be precisely determined (or reconstructed) from a measurement performed on a finite ensemble of identically prepared qubits [2-4]. In particular, the mean fidelity of the best possible (optimal) state estimation strategy based on the measurement of N