William Wootters
Professor of Physics
at Williams since 1982
Education/Experience
 Stanford University, B.S. 1973
 University of Texas, Austin, Ph.D. 1980
 Santa Fe Institute, visiting researcher 198990
 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
Contact Information
Courses given 20102011
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. More recently I have been interested in finding
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 recent years much progress has been
made in developing a quantitative theory of entanglement. For example, we
now have a welljustified 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 tradeoff 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 become negative. A few years ago
two students (Kathleen Gibbons and Matthew Hoffman) and I presented a discrete
version of the Wigner function applicable to a system ofqubits, in which
the phase space is a twodimensional vector space over a finitefield. My
current students are exploring this construction in more depth, with an eye
towards applications in foundations of physics and quantum computation.
Selected publications
 W. K. Wootters and C. G. Langton, "Is There a Sharp Phase Transition
for Deterministic Cellular Automata?" Physica D 45, 95104 (1990).
 A. Peres and W. K. Wootters, "Optimal Detection of Quantum Information,"
Phys. Rev. Lett. 66, 11191122 (1991).
 C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W.
K. Wootters, "Teleporting an Unknown Quantum State via Dual Classical and
EinsteinPodolskyRosen Channels," Phys. Rev. Lett. 70, 1895
(1993).
 L. P. Hughston, R. Jozsa, W. K. Wootters, "A complete classification
of quantum ensembles having a given density matrix," Phys. Lett. A
183, 1418 (1993).
 R. Jozsa, D. Robb '93, and W. K. Wootters, "A Lower Bound for Accessible
Information in Quantum Mechanics," Phys. Rev. A 49, 668 (1994).
 W. K. Wootters, "Is Time Asymmetry Logically Prior to Quantum Mechanics?"
in Physical Origins of Time Asymmetry, ed. by J. Halliwell, J. PerezMercader
and W. Zurek (Cambridge Univ. Press, 1994).
 P. Hausladen '93 and W. K. Wootters, "A 'Pretty Good' Measurement
for Distinguishing Quantum States," J. Mod. Optics, 41, 2385
(1994).
 P. Hausladen '93, R. Jozsa, B. Schumacher, M. Westmoreland, and W.
K. Wootters, "Classical information capacity of a quantum channel," Phys.
Rev. A 54, 1869 (1996).
 C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin,
and W. K. Wootters, "Purification of Noisy Entanglement and Faithful Teleportation
via Noisy Channels," Phys. Rev. Lett. 76, 722 (1996).
 S. Hill '97 and W. K. Wootters, "Entanglement of a Pair of Quantum
Bits," Phys. Rev. Lett. 78, 5022 (1997).
 W. K. Wootters, "Entanglement of Formation of an Arbitrary State of
Two Qubits," Phys. Rev. Lett. 80, 2245 (1998).
 V. Coffman, J. Kundu, and W. K. Wootters, "Distributed Entanglement,"
Phys. Rev. A 61, 052306 (2000).
 C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, B. M. Terhal, and W.
K. Wootters, "Remote State
Preparation," Phys. Rev. Lett. 87, 077902 (2000).
 K. M. O'Connor '00 and W. K. Wootters, "Entangled Rings," Phys.
Rev. A 63, 052302 (2001).
 W. K. Wootters, "Entanglement of Formation and Concurrence," Quantum
Information and Computation 1, 27 (2001).
 W. K. Wootters, "Entangled Chains," in
Quantum Computation and Information, S. J. Lomonaco and H. E. Brandt,
editors (American Mathematical Society, Providence, 2002), pp. 299310.
 K. A. Dennison '01 and W. K. Wootters, "Entanglement Sharing among
Particles with More than Two Orthogonal States," Phys. Rev. A
65, 010301 (2002).
 W. K. Wootters, "Parallel Transport in an
Entangled Ring," J. Math. Phys. 43, 4307 (2002).
 N. Linden, S. Popescu, and W. K. Wootters, "Almost Every Pure State of
Three Qubits Is Completely Determined by Its TwoParticle Reduced Density
Matrices," Phys. Rev. Lett. 89, 207901 (2002).
 N. Linden and W. K. Wootters, "The Parts Determine the Whole
in a Generic Pure Quantum State," Phys. Rev. Lett. 89, 277906
(2002).
 S. R. Nichols '03 and W. K. Wootters, "Between Entropy and Subentropy," Quantum Information and Computation 3, 1 (2003).
 W. K. Wootters, "Why Things Fall," Foundations of Physics 33,
1549 (2003).
 W. K. Wootters, "Picturing Qubits in Phase Space,"IBM
Journal of Research and Development 48, no. 1, p. 99 (2004).
 K. S. Gibbons '03, M. J. Hoffman '04, and W. K. Wootters, "Discrete Phase Space Based on
Finite Fields," Phys. Rev. A 70, 062101 (2004).
 W. K. Wootters, "Quantum Measurements and Finite
Geometry," Foundations of Physics 36, 112 (2006).
 W. K. Wootters, "Distinguishing Unentangled States
with an Unentangled Measurement," Int. J. Quant. Inf. 4, 219
(2006).
 W. K. Wootters and D. M. Sussman, "Discrete phase space and minimumuncertainty states," Proceedings of the Eighth International Conference on Quantum Communication, Measurement and Computation, 269
(2007).
 S. Bandyopadhyay, G. Brassard, S. Kimmel, and W. K. Wootters, "Entanglement Cost of Nonlocal Measurements," Phys. Rev. A 80, 012313
(2009).
 C. Chudzicki, O. Oke, and W. K. Wootters, "Entanglement and Composite Bosons," Phys. Rev. Lett. 104, 070402
(2010).
 W. K. Wootters, "Entanglement Sharing in RealVectorSpace Quantum Theory," to appear in Found. Phys.
(2010).
Williams Physics
