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Recent and Current Research
My research is in the areas of quantum information and quantum optics. The work in quantum information has mostly been concerned with what can be called quantum machines, devices that manipulate quantum information. The work in quantum optics has most recently been in the foundations of the quantum theory of nonlinear optics.
I have collaborated with Vladimir Buzek and his group in Bratislava, and with colleagues at CUNY in the work on quantum information. It was initially centered around quantum copying. 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 copier is universal in the sense that its performance is independent of its input state. We went on to study other ``universal'' gates, such as a NOT gate, which takes a qubit in an arbitrary input state, and transforms it into the state orthogonal to the original one. Both the cloner and the universal-NOT gate have been constructed and demonstrated experimentally [1,2]. We also considered an entangler, and a disentangler. The entangler entangles a qubit in an arbitrary state with another in a known, fixed state, and the disentangler takes a two-qubit state consisting of a qubit in an unknown state and one in a known state, and yields a single qubit in the unknown state. We are presently examining programmable arrays of quantum gates. These quantum circuits have two inputs, a data state and a program state, and the program state controls which operation is performed on the data state.
With a graduate student, Yuqing Sun, Prof.Janos Bergou at Hunter College, and Ulrike Herzog from the Humboldt University in Berlin, I have been examining the problem of discriminating among non-orthogonal quantum states. We studied the problem of unambiguously discriminating among three states and the problem of quantum filtering. In the latter, one is given a quantum system in one of N known quantum states, one of which is distinguished, and the object is to determine whether one has been given the distinguished state or not. We were able to use this procedure to construct a generalization of the Deutsch-Jozsa algorithm. In addition we have proposed one optical device that will distinguish, with a certain probability of failure, between two nonorthogonal quantum states, and a second device that will distinguish three states. We have been collaborating with Aephriam Steinberg's group at the University of Toronto, where one of these devices has been built. With another student, I am presently examining the discrimination between multipartite states using local operations and restricted classical communication. Prof. Bergou and I are also collaborating with Prof. Edgar Feldman at the CUNY Graduate Center on a study of quantum walks.
With Peter Drummond I examined the foundations of the quantum theory of nonlinear optics. We developed a microscopic theory of optical media, which includes multiple resonances, and developed 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 allows us to study some basic questions such as the role of dispersion and operator ordering.
1. A. Lamas-Linares, C. Simon, J. C. Howell, and D. Bouwmeester, Science 296, 712 (2002).
2. F. de Martini, V. Buzek, and C. Slas, Nature 419, 815 (2002).