## 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**

- W. K. Wootters, “Statistical Distance and Hilbert Space,”
*Phys. Rev. D***23**, 357 (1981). - W. K. Wootters and W. H. Zurek, “A Single Quantum Cannot Be Cloned,”
*Nature***299**, 802 (1982). [The no-cloning theorem was proved independently, in the same year, by Dennis Dieks: D. Dieks, “Communication by EPR Devices,”*Phys. Lett. A***92**, 271 (1982). And it was proved twelve years earlier in a paper by James Park: J. L. Park, “The Concept of Transition in Quantum Mechanics,”*Found. Phys.***1**, 23 (1970).] - D. N. Page and W. K. Wootters, “Evolution without Evolution: Dynamics Described by Stationary Observables,”
*Phys. Rev. D***27**, 2885 (1983). - W. K. Wootters, “Quantum Mechanics without Probability Amplitudes,”
*Found. Phys.***16**, 391 (1986). - W. K. Wootters, “A Wigner-Function Formulation of Finite-State Quantum Mechanics,”
*Annals of Physics***176**, 1 (1987). - W. K. Wootters and B. D. Fields, “Optimal State-Determination by Mutually Unbiased Measurements,”
*Ann. Phys. New York***191**, 363 (1989). - W. K. Wootters, “Local Accessibility of Quantum States,” in
*Complexity, Entropy, and the Physics of Information*, edited by W. H. Zurek (Addison-Wesley, 1990). - W. K. Wootters and C. G. Langton, “Is There a Sharp Phase Transition for Deterministic Cellular Automata?”
*Physica D***45**, 95-104 (1990). - W. K. Wootters, “Random Quantum States,”
*Found. Phys.***20**, 1365 (1990). - A. Peres and W. K. Wootters, “Optimal Detection of Quantum Information,”
*Phys. Rev. Lett.***66**, 1119-1122 (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 Einstein-Podolsky-Rosen 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**, 14-18 (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. Perez-Mercader 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, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed State Entanglement and Quantum Error Correction,”
*Phys. Rev. A***54**, 3824 (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. 299-310. - 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 Two-Particle 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 ’07, “Discrete phase space and minimum-uncertainty states,”
*Proceedings of the Eighth International Conference on Quantum Communication, Measurement and Computation*, 269 (2007). - S. Bandyopadhyay, G. Brassard, S. Kimmel ’08, and W. K. Wootters, “Entanglement Cost of Nonlocal Measurements,”
*Phys. Rev. A***80**, 012313 (2009). - C. Chudzicki ’10, O. Oke ’10, and W. K. Wootters, “Entanglement and Composite Bosons,”
*Phys. Rev. Lett.***104**, 070402 (2010). - S. Bandyopadhyay, R. Rahaman, and W. K. Wootters, “Entanglement cost of two-qubit nonlocal orthogonal measurements,”
*J. Phys. A: Math. Theor.***43**, 455303 (2010). - M. S. Williamson, M. Ericsson. M. Johannson, E. Sjoqvist, A. Sudbery, V. Vedral, and W. K. Wootters, “Geometric local invariants and pure three-qubit states,”
*Phys. Rev. A***83**, 062308 (2011). - W. K. Wootters, “Entanglement Sharing in Real-Vector-Space Quantum Theory,”
*Found. Phys.***42**, 19 (2012). - L. Hardy and W. K. Wootters, “Limited Holism and Real-Vector-Space Quantum Theory,”
*Found. Phys.***42**, 454 (2012). - A. Aleksandrova ’11, V. Borish ’12, and W. K. Wootters, “Real-vector-space quantum theory with a universal quantum bit,”
*Phys. Rev. A***87**, 052106 (2013). - W. K. Wootters, “Communicating through Probabilities: Does Quantum Theory Optimize the Transfer of Information?”
*Entropy***15**, 3220 (2013). - P. Mandayam, S. Bandyopadhyay, M. Grassl, and W. K. Wootters, “Unextendible mutually unbiased bases from Pauli classes,”
*Quantum Information and Computation***14**, 823 (2014). - W. K. Wootters, “The rebit three-tangle and its relation to two-qubit entanglement,”
*J. Phys. A: Math. Theor.***47**, 424037 (2014). - I. Amburg ’14, R. Sharma ’13, D. Sussman ’07, and W. K. Wootters, “States that ‘look the same’ with respect to every basis in a mutually unbiased set,”
*J. Math. Phys.***55**, 122206 (2014). - W. K. Wootters, “Merging contradictory laws: Imagining a constructive derivation of quantum theory,” in
*Information and Interaction: Eddington, Wheeler, and the Limits of Knowledge*, I. T. Durham and D. Rickles, eds. (Springer, 2017). - S. Alterman ’18, J. Choi ’17, R. Durst ’17, S. M. Fleming ’17, and W. K. Wootters, “The Boltzmann distribution and the quantum-classical correspondence,”
*J. Phys. A: Math. Theor.***51**, 345301 (2018). - W. K. Wootters, “A classical interpretation of the Scrooge distribution,”
*Entropy***20**, 619 (2018). - W. F. Braasch, Jr. and W. K. Wootters, “Transition probabilities and transition rates in discrete phase space,” Physical Review A
**102**, 052204 (2020). - W. F. Braasch, Jr. and W. K. Wootters, “A quantum prediction as a collection of epistemically restricted classical predictions,” Quantum
**6**, 659 (2022). - W. F. Braasch, Jr. and W. K. Wootters, “A classical formulation of quantum theory?” Entropy
**24**, 137 (2022).

### 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