## Universal optimum quantum cloning and quantum injected optical parametric

Because of the linearity of quantum mechanics an arbitrary quantum state cannot be 'cloned' perfectly, i.e. reproduced with 'fidelity' F=1 into M>1 states identical to the original. A second quantum impossibility process, based on the complete positivity character of any quantum operation, forbids the realization of a universal NOT gate i.e. one that flips exactly any input qubit into an orthogonal one. In the domain of quantum optics the cloning effect is directly associated to a photon amplification process in an Optical Parametric Amplifier (OPA). In the OPA N photons, prepared identically in an arbitrary quantum state |?> of polarization, are injected into the amplifier on the input mode k1.

The amplifier then generates on the same output 'cloning mode' M>N copies, or 'clones' of the input qubit |?>. Correspondingly, the OPA amplifier generates on the output'anticloning (AC) mode, k2 M-N states |?->, thus realizing a quantum NOT gate, which performs the operation to flip a qubit. It can exist simultaneously in the superposition |?>=a|0> + ß|1> of two logical states |0> and |1>, and it is impossible to find a universal transformation which would flip the original state |?> into the perpendicular state |?->=ß|0> - a|1> for all values of complex amplitudes a and ßi.e. for any (unknown) |?>.The first experimental realization of a universal quantum machine performing the best possible approximation of an anti-unitary operation, the Universal NOT (U-NOT) transformation was given in Rome.

The system adopted was a quantum self-injected optical parametric amplifier (QIOPA) of entangled photon states. An important consideration in the field of quantum information theory is which physical transformations to the state of a quantum system are allowed. The investigation of these universal optimal transformations, which are also called universal quantum machines, is important since it reveals bounds on optimal manipulations of information with quantum systems.

Theoretically it can be useful to design new algorithms and protocols, whereas experimental realizations witness an improvement in the manipulation of small ensembles of qubits. Partner 1 has been interested in realizing the quantum analogues of two fundamental processes of classical information: the NOT gate and the cloning (copying) machine. The quantum forms of these two transformations cannot be realized perfectly. However their optimal realizations, with the minimum possible noise, present interesting features.

The quantum U-NOT gate is deeply related to the quantum estimation of an unknown state while the optimal quantum cloning is the best way to redistribute the initial information content into many parts. Quantum cryptography bases its security on the impossibility to clone unknown quantum states. Nevertheless, since experimental imperfections have to be taken into account, there exist some bounds on the noise value that assure a secure communication.

These bounds depend on the information that a spy can obtain interacting with the quantum system. The optimal quantum cloning is the best eavesdropping attack on some quantum cryptography protocols It is the best trade-off between disturbance and acquired knowledge. These two quantum machines, the U-NOT gate and the optimal quantum cloning, were contextually realized by adopting the process of stimulated emission generated by a single photon into an optical parametric amplifier(OPA).

The amplifier then generates on the same output 'cloning mode' M>N copies, or 'clones' of the input qubit |?>. Correspondingly, the OPA amplifier generates on the output'anticloning (AC) mode, k2 M-N states |?->, thus realizing a quantum NOT gate, which performs the operation to flip a qubit. It can exist simultaneously in the superposition |?>=a|0> + ß|1> of two logical states |0> and |1>, and it is impossible to find a universal transformation which would flip the original state |?> into the perpendicular state |?->=ß|0> - a|1> for all values of complex amplitudes a and ßi.e. for any (unknown) |?>.The first experimental realization of a universal quantum machine performing the best possible approximation of an anti-unitary operation, the Universal NOT (U-NOT) transformation was given in Rome.

The system adopted was a quantum self-injected optical parametric amplifier (QIOPA) of entangled photon states. An important consideration in the field of quantum information theory is which physical transformations to the state of a quantum system are allowed. The investigation of these universal optimal transformations, which are also called universal quantum machines, is important since it reveals bounds on optimal manipulations of information with quantum systems.

Theoretically it can be useful to design new algorithms and protocols, whereas experimental realizations witness an improvement in the manipulation of small ensembles of qubits. Partner 1 has been interested in realizing the quantum analogues of two fundamental processes of classical information: the NOT gate and the cloning (copying) machine. The quantum forms of these two transformations cannot be realized perfectly. However their optimal realizations, with the minimum possible noise, present interesting features.

The quantum U-NOT gate is deeply related to the quantum estimation of an unknown state while the optimal quantum cloning is the best way to redistribute the initial information content into many parts. Quantum cryptography bases its security on the impossibility to clone unknown quantum states. Nevertheless, since experimental imperfections have to be taken into account, there exist some bounds on the noise value that assure a secure communication.

These bounds depend on the information that a spy can obtain interacting with the quantum system. The optimal quantum cloning is the best eavesdropping attack on some quantum cryptography protocols It is the best trade-off between disturbance and acquired knowledge. These two quantum machines, the U-NOT gate and the optimal quantum cloning, were contextually realized by adopting the process of stimulated emission generated by a single photon into an optical parametric amplifier(OPA).