William K. Wootters

Photo of William K. Wootters

Barclay Jermain Professor of Natural Philosophy, Emeritus

At Williams since 1982

Areas of Expertise

Quantum Information Theory
Information stored in quantum systems behaves very differently from ordinary information. It cannot be copied perfectly, for example, and it is usually degraded by measurement. Despite these restrictions, this ghostly sort of information could be of great value in quantum computation and quantum cryptography. My research aims at learning more about the fundamental properties of quantum information. In past years my students and I have studied the classical capacity of a quantum channel and the amount of information one can extract from a single quantum object, and we have found quantitative rules governing the “entanglement” between two or more quantum objects.

Entanglement is a peculiarly quantum mechanical kind of correlation that has no analogue in classical physics. It is the essential ingredient in such phenomena as superdense coding, in which any of four possible messages can be transmitted via a single binary quantum object, and teleportation, in which a quantum state is transmitted from one location to another without passing through the intervening space. In the last few decades much progress has been made in developing a quantitative theory of entanglement. For example, we now have a well-justified analytic formula for entanglement between simple systems. My students and I have used these developments to identify certain “laws of entanglement.” To give one example: two former students, Valerie Coffman and Joydip Kundu, and I, using a convenient measure of entanglement known as the “concurrence,” showed that for any state of three binary quantum objects (qubits) A, B, and C, there is a simple trade-off between the AB entanglement and the AC entanglement. That is, qubit A has a limited capacity for entanglement, which must be split between the objects with which it is entangled.

Wigner Functions
The density matrix is the most commonly used representation of a general mixed or pure quantum state. However, there is an alternative representation, the Wigner function, which is a real function on phase space. The Wigner function acts in many ways like a probability distribution, but it is not a standard probability distribution in that it can take negative values. In 2004, two students (Kathleen Gibbons and Matthew Hoffman) and I presented a discrete version of the Wigner function applicable to a system of qubits, in which the phase space is a two-dimensional vector space over a finite field. This construction generalized an earlier discrete Wigner function introduced in 1987.  Other students have explored these discrete Wigner functions in more depth, with an eye towards applications in quantum foundations and quantum information.


Scholarship/Creative Work

Selected publications

Other Posts

  • Santa Fe Institute, visiting researcher 1989-90
  • University of Montreal, visiting researcher 1994
  • IBM Watson Research Center, collaborator 1995, 1998
  • Perimeter Institute, visiting professor 2009
  • Kigali Institute of Science and Technology, visiting professor 2010