Mathematical Physics Research Group

The Mathematical Physics Research Group at Lebanon Valley College conducts collaborative research involving undergraduate students from mathematics, physics, and computer science.

  • Interdisciplinary, student-faculty research. Recent areas of primary focus have been functional programming and quantum information science. Our work is interdisciplinary, involving physics, mathematics, and computer science.
  • Functional Programming. The application of powerful conceptual tools from computer science brings insight and adventure to physics pedagogy and research. Dr. Walck and students use functional programming languages for novel applications in computational physics.
  • Quantum Information Science. The prospect of harnessing quantum mechanical systems to process information promises spectacular applications. These include encrypted communication with fundamentally new levels of security, computing machines that will outperform existing supercomputers, and improved measurement devices that will probe new physical realms. The `quantum advantage' at the heart of these revolutionary technologies arises from exploiting entanglement and other non-local properties that are easy to define, but difficult to analyze. Our research group has achieved a body of results that analyze local symmetry (that is, operations on individual particles of a composite physical system that leave its quantum state unchanged) to quantify and classify entanglement in a way that identifies and characterizes important classes of states. Ongoing work studies the interplay of local symmetry with other types of symmetry and applications.

Faculty and students in the research group have won over one million dollars in competitive grant funding from the National Science Foundation, Arnold Experiential Grant Fund, and Lebanon Valley College. In addition to student-faculty research, these grants have supported student travel to conferences such as the Mathematical Association of America, where several students have presented. They supported research projects ranging from Symmetry, Entanglement, and Application to Structure and Local Equivalence of Stabilizers and States. The group's work has also been published in prestigious science journals like Quantum. See a full list of publications below.

Publications & Presentations

Grants

Title: CQIS: RUI: Quantum State Symmetry and Applications
Award No.: PHY-2011074
Dates: 9/1/2020–8/31/2023
Source: National Science Foundation
Amount: $210,429
PI: D.W. Lyons

Title: Faculty Research Grant
Dates: 2020–2021
Source: Lebanon Valley College
Amount: $2400
Recipient: David W. Lyons
Note: The PI declined this award upon receipt of the above NSF award PHY-2011074

Title: CQIS: RUI: Entanglement and Applications via Local and Permutational Symmetry
Award No.: PHY-1713868
Dates: 9/2017–8/2020
Source: National Science Foundation
Amount: $174,999
PI: D.W. Lyons

Title: Entanglement and Applications via Symmetry
Dates: 5/2017–8/2017
Source: Lebanon Valley College Arnold Student-Faculty Research Grant
Amount: $4500
Recipient: David W. Lyons

Title: Faculty Research Grant
Dates: 6/2017
Source: Lebanon Valley College
Amount: $2500
Recipient: David W. Lyons

Title: Symmetry, Entanglement, and Applications
Dates: 5/2016–8/2016
Source: Lebanon Valley College Arnold Student-Faculty Research Grant
Amount: $7500
Recipient: David W. Lyons

Title: Faculty Research Grant
Dates: 6/2016
Source: Lebanon Valley College
Amount: $2630
Recipient: David W. Lyons

Title: Student presenter travel grant
Dates: October 2014
Source: Lebanon Valley College
Amount: $400
Recipient: Daniel J. Upchurch, Chase D. Yetter

Title: Faculty Research Grant
Dates: 4/2014
Source: Lebanon Valley College
Amount: $1500
Recipient: D.W. Lyons

Title: Student presenter travel grant
Dates: March 2014
Source: Lebanon Valley College
Amount: $400
Recipient: Anthony Hoover, Adam Rosier

Title: Video Lectures for the Calculus Curriculum
Dates: Fall 2013
Source: Lebanon Valley College
Amount: $4,000
PIs: D.W. Lyons 

Title: MAA Student Travel Grant
Dates: January 2013
Source: Mathematical Association of America 
Amount: $900
Recipients: Anthony Hoover, Ian Bond, Kelsey Moore, and Oliver Lyons

Title: Student presenter travel grant
Dates: January 2013
Source: Lebanon Valley College
Amount: $800
Recipients: Anthony Hoover, Ian Bond, Kelsey Moore, and Oliver Lyons

Title: RUI: Structure and Local Equivalence of Stabilizers and States
Award No.: PHY-1211594
Dates: 9/2012–8/2015
Source: National Science Foundation
Amount: $273,975
PIs: D.W. Lyons and S.N. Walck

Title: Structure and Local Equivalence of Stabilizers and States
Dates: Academic Year 2012–2013
Source: Lebanon Valley College Arnold Student-Faculty Research Grant
Amount: $6000
PIs: D.W. Lyons and S.N. Walck
Note: The PIs declined this award upon receipt of the above NSF award PHY-1211594

Title: MAA Student Travel Grant
Dates: January 2012
Source: Mathematical Association of America 
Amount: $300
Recipient: A.M. Skelton

Title: Student presenter travel grant
Dates: January 2012
Source: Lebanon Valley College
Amount: $300
Recipient: A.M. Skelton

Title: MAA Student Travel Grant
Dates: January 2011
Source: Mathematical Association of America 
Amount: $900
Recipient: C.D. Cenci, L.M. Snyder, E.C. Ulicny

Title: Student presenter travel grant
Dates: January 2011
Source: Lebanon Valley College
Amount: $600
Recipient: C.D. Cenci, L.M. Snyder, E.C. Ulicny

Title: Travel Grant Program for the Nordita/Mittag-Leffler International Conference on Quantum Information and Computation
Dates: October 2010
Source: National Science Foundation 
Amount: $3000
Recipient: D.W. Lyons

Title: RUI: Entanglement Classification via Stabilizers and Subsystem States
Award No.: PHY-0903690
Dates: 9/2009–8/2012
Source: National Science Foundation
Amount: $236,198
PIs: D.W. Lyons and S.N. Walck

Title: Student presenter travel grant
Dates: January 2009
Source: Lebanon Valley College
Amount: $450
Recipient: S.A. Blanda

Title: Faculty Research Grant
Dates: 4/2008–3/2009
Source: Lebanon Valley College
Amount: $2000
Recipient: D.W. Lyons

Title: Faculty Research Grant
Dates: 4/2007–3/2008
Source: Lebanon Valley College
Amount: $2500
Recipient: D.W. Lyons

Title: RUI: Space of Multiparticle Entanglement Types 
Award No.: PHY-0555506
Dates: 7/2006–6/2009
Source: National Science Foundation
Amount: $160,000
PIs: D.W. Lyons and S.N. Walck

Title: Topology, geometry, and visualization of the pure three-qubit state space
Dates: 2002–2005
Source: Research Corporation
Amount: $33,218
PI: S.N. Walck

Title: Characterization and Visualization of Quantum Entanglement
Dates: 2002–2003
Source: Lebanon Valley College
Amount: $8,000
PIs: D.W. Lyons and S.N. Walck

People

Dr. Lyons with summer research students

Summer Research in Quantum Information Science

The LVC Mathematical Physics Research Group mentors undergraduate mathematics, computer science, and physics majors in intensive summer research projects in quantum information science (QIS). Note: At the present time, summer research assistant positions are open only to Lebanon Valley College Students.

Additional Resources

Background Notes for Undergraduate Researchers

The minimum background for undergraduate students to begin research in quantum information science is a linear algebra course and some "mathematical maturity". Essential linear algebra background includes the spectral theorem, at least for diagonalization of hermitian and unitary operators on complex vector spaces.

To bridge the gap between an introductory linear algebra course and really working on problems in quantum information, we use Nielsen and Chuang's text together with background notes that we have produced. Our students spend the first two weeks of their summer project working through Chapter 2 of Nielsen and Chuang and our background notes on the essentials of matrix groups, their Lie algebras, and group and algebra actions on vector spaces and tensor products.

We have used versions of our background notes since 2002. We revise and update each year. Please contact us if you would be interested in having a copy.

Software for Stabilizer Calculations

We and our students have written code in Mathematica and GAP for computations involving pure and mixed states, partial trace, local unitary stabilizer, and Pauli stabilizer calculations. Development is ongoing. Please contact us if you would be interested in trying our latest versions.