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# Research

## Recent and Current Research

My work is mostly in the fields of quantum optics and quantum information, though during the last five years, most of my effort has been in quantum information. This has been done largely in collaboration with Vladimir Buzek and has been centered about quantum copying. It has been known since 1982 that it is impossible to copy, or clone, the state of a quantum system. This theorem, however, only applies if the copies are exact; if the copies do not have to be perfect, then something can be done. We have shown how the approximate copying of quantum states can be accomplished and what the bounds on the performance of such copying processes are. The device that accomplishes this is known as a quantum cloner. It has the property that its behavior is universal in the sense that the quality of the copies it produces does not depend on the state it is copying. The cloner can be represented by a quantum logic circuit, and it has been shown that a parametric down converter acts as a quantum cloner.

In the same spirit we studied the behavior of other quantum machines. We found the optimal behavior of a device that takes a qubit, the basic unit of quantum information, which is a two-level quantum system, and transforms it into the state orthogonal to its original state. We called this device a universal NOT, or U-NOT, gate. It is impossible to construct a perfect U-NOT gate, but we found the transformation that comes closest. It is very closely related to the cloning transformation. We also examined devices that entangle an unknown input state with a known state, and, conversely, a device that takes a state in which a known and an unknown state are entangled, and disentangles it yielding the unknown state.

I have also been interested in quantum cryptography, the use of quantum mechanics to protect information from eavesdropping. With Vladimir Buzek and Andre Berthiaume, I developed a protocol for quantum secret sharing, a procedure in which a message is split into several parts, and can only be reconstructed by someone in possession of all of the parts. I also examined a quantum cryptographic protocol for continuous variables that uses squeezed states.

Not all of the work has been in quantum information, however. With Peter Drummond I am examining the foundations of the quantum theory of nonlinear optics. We developed a microscopic theory of linear optical media, which includes multiple resonances, and a scattering theory to describe how fields propagate into, through and out of it. This provides a theoretical description which is more in accord with what is done in the laboratory, and it allows us to study some basic questions such as the role of dispersion and operator ordering. We are presently working on extending this approach to nonlinear media.