David Lyons

My current research is in quantum information science—an interdisciplinary area involving mathematics, physics, computer science, and engineering. My thesis work, under the direction of Shrawan Kumar, was in algebraic topology of Lie Groups. My pedagogical scholarship is driven by a long-standing interest in the use of geometry and visualization for teaching mathematical concepts.

Recent Significant Grants

• National Science Foundation grant award PHY-2011074, CQIS: RUI: Quantum State Symmetry and Applications, 9/1/2020 through 8/31/2023, $210,429
• National Science Foundation grant award PHY-1713868, CQIS: RUI: Entanglement and Applications via Local and Permutational Symmetry, 9/1/2017 through 8/31/2020, $174,999
• Lebanon Valley College President’s Innovation Fund, Math Vista video lecture project, 2013, $4000
• National Science Foundation grant award PHY-1211594, RUI: Structure and Local Equivalence of Stabilizers and States (joint with S.N. Walck), 9/2012 through 8/2015, $273,975

Peer-Reviewed Publications

David W. Lyons, Cristina Mullican, Adam Rilatt, and Jack D. Putnam. Werner states from diagrams. Journal of Physics A: Mathematical and Theoretical, 56(22):225301, May 2023. arXiv:2302.05572

Paul Appel, Alexander J. Heilman, Ezekiel W. Wertz, David W. Lyons, Marcus Huber, Matej Pivoluska, and Giuseppe Vitagliano. Finite-Function-Encoding Quantum States. Quantum, 6:708, May 2022. arXiv:2012.00490.

David W. Lyons, Jesse R. Arnold, and Ashley F. Swogger. Local unitary classes of states invariant under permutation subgroups. Phys. Rev. A, 105:032442, 2022. arXiv:2109.06921.

David W. Lyons, Nathaniel P. Gibbons, Mark A. Peters, Daniel J. Upchurch, Scott N. Walck, and Ezekiel W. Wertz. Local Pauli stabilizers of symmetric hypergraph states. Journal of Physics A: Mathematical and Theoretical, 50(24):245303, 2017. arXiv:1609.01306.

David W. Lyons. Undergraduate Research in Quantum Information Science. PRIMUS, 27(4-5):508{516, 2017.
David W. Lyons, Daniel J. Upchurch, Scott N. Walck, and Chase D. Yetter. Local unitary symmetries of hypergraph states. Journal of Physics A: Mathematical and Theoretical, 48(9):095301, February 2015. arXiv:1410.3904.

Curt D. Cenci, David W. Lyons, and Scott N. Walck. Local unitary group stabilizers and entanglement for multiqubit symmetric states. In Dave Bacon, Miguel Martin-Delgado, and Martin Roetteler, editors, Theory of Quantum Computation, Communication, and Cryptography, volume 6745 of Lecture Notes in Computer Science, pages 198{207. Springer, March 2014. arXiv:1011.5229v1.

David W. Lyons and Scott N. Walck. Entanglement verification using local unitary stabilizers. Phys. Rev. A, 87:062321, Jun 2013. arXiv:1303.6497 [quant-ph].

David W. Lyons, Abigail M. Skelton, and Scott N. Walck. Werner state structure and entanglement classification. Advances in Mathematical Physics, 2012:463610, 2012. arXiv:1109.6063v2 [quant-ph].

David W. Lyons and Scott N .Walck. Symmetric mixed states of n qubits: Local unitary stabilizers and entanglement classes. Phys. Rev. A, 84:042340, October 2011. arXiv:1107.1372v1 [quant-ph].

David W. Lyons and Scott N. Walck. Entanglement classes of symmetric Werner states. Phys. Rev. A, 84:042316, October 2011. arXiv:1106.4220v2 [quant-ph].

Curt D. Cenci, David W. Lyons, Laura M. Snyder, and Scott N. Walck. Symmetric states: local unitary equivalence via stabilizers. Quantum Information and Computation, 10:1029{1041, November 2010. arXiv:1007.3920v1 [quant-ph].

Scott N. Walck and David W. Lyons. Only n-qubit Greenberger-Horne-Zeilinger states contain n-partite information. Phys. Rev. A, 79:032326, March 2009. arXiv:0808.0859v1 [quant-ph].

David W. Lyons and Scott N. Walck. Multiparty quantum states stabilized by the diagonal subgroup of the local unitary group. Phys. Rev. A, 78:042314, October 2008. arXiv:0808.2989v2 [quant-ph].

Scott N. Walck and David W. Lyons. Only n-qubit Greenberger-Horne-Zeilinger states are undetermined by their reduced density matrices. Phys. Rev. Lett., 100:050501, 2008. arXiv:0707.4428 [quant-ph].

David W. Lyons, Scott N. Walck, and Stephanie A. Blanda. Classification of nonproduct states with maximum stabilizer dimension. Phys. Rev. A, 77:022309, 2008. arXiv:0709.1105 [quant-ph].

Scott N. Walck and David W. Lyons. Maximum stabilizer dimension for nonproduct states. Phys. Rev. A, 76:022303, 2007. arXiv:0706.1785 [quant-ph].

David W. Lyons and Scott N. Walck. Classi cation of n-qubit states with minimum orbit dimension. J. Phys. A: Math. Gen., 39:2443{2456, 2006. arXiv:quant-ph/0506241.

Richard H. Hammack and David W. Lyons. The alternating series test{visual proof. Oxford Journals: Teaching Mathematics and its Applications, 25(2):58{60, 2006.

David W. Lyons and Scott N. Walck. Minimum orbit dimension for local unitary action on n-qubit pure states. J. Math. Phys., 46:102106, 2005. arXiv:quant-ph/0503052.

Richard H. Hammack and David W. Lyons. Alternating Series Proof Without Words. The College Mathematics Journal, 36(1):72, January 2005.

David W. Lyons. An elementary introduction to the Hopf bration. Mathematics Magazine, 76(2):87{98, 2003. arXiv:2212.01642.

David W. Lyons. Integral Weyl invariants and the Borel homomorphism for Spin(n). Communications in Algebra, 26(7):2221{2239, 1998.

Richard H. Hammack and David W. Lyons. A simple way to teach logarithms. Mathematics Teacher, 85(5):374{375, 1995.

Other Publications

David W. Lyons. Linear and Matrix Algebra for Multivariable Calculus. May 2021 edition, 2021. https://mathvista.org.

David W. Lyons. Introduction to Groups and Geometries. January 2021 edition, 2020. https://mathvista.org.

David W. Lyons. Zeno’s Paradox Part II: Geometric Sequences. Math Vista Video Project, November 2013. http://quantum.lvc.edu/lyons/mathvista/.

David W. Lyons. Zeno’s Paradox Part I: A Puzzle and a Flaw. Math Vista Video Project, November 2013. http://quantum.lvc.edu/lyons/mathvista/.

David W. Lyons. The Cross Product. Math Vista Video Project, November 2013. http://quantum.lvc.edu/lyons/mathvista/.

David W. Lyons. Why Linear Algebra? Math Vista Video Project, November 2013. http://quantum.lvc.edu/lyons/mathvista/.

David W. Lyons. Sequences of Approximations. Math Vista Video Project, November 2013. http://quantum.lvc.edu/lyons/mathvista/.

David W. Lyons. Introduction to Mathematical Proof. 2011.

David W. Lyons. Quantum Information Notes: Introduction to Matrix Groups and Their Lie Algebras. 2010.

David W. Lyons. Survey of Hopf brations and rotation conventions in mathematics and physics. arXiv:0808.3089v2 [math-ph], September 2008.

David W. Lyons. Mathematical Reasoning II: Introduction to Mathematics Beyond Calculus. 2006.

David W. Lyons. Mathematical Reasoning I: Introduction to Mathematics Beyond Calculus. 2005.
David W. Lyons. Mathematics Magazine cover artwork, April 2003.

Presentations

Geometry and Algebra in the Hopf Fibration. AMS Special Session on Quaternions, Joint Mathematics Meetings, Boston, January 2023.

Function-encoding Quantum States. Tetrahedral Geometry and Topology Seminar, Hempfield, PA, February 2020.

Mathematical Symmetries in Quantum Information Science. Eastern Illinois University Perry Lectures, November 2019.

Spooky Action: Scientific and Philosophical Challenges in the Era of Quantum Technology. Eastern Illinois University Perry Lectures, October 2019.

Book Review: “Q is for Quantum” by Terry Rudolf. Cornwall Manor Book Review Series, August 2019.

Function-encoding qudit hypergraph states. Seefeld QI Workshop, Seefeld, Austria, July 2018.

Local Pauli stabilizers of symmetric hypergraph states. Theory of Quantum Computation, Communication and Cryptography (TQC) 2017, Paris, June 2017.

Quantum hypergraph states: local equivalence, nonlocality, and error correction. IQOQI Vienna, Austria, May 2017.

Permutation subgroup invariant states: entanglement and applications. Seefeld QI Workshop, Seefeld, Austria, June 2016.

Harmony and Topology. Sigma Alpha Iota Province Day, Lebanon Valley College, April 2016.

Adventure in the Quantum World. Pi Mu Epsilon (Mathematics Honor Society), Elizabethtown College, PA, April 2015.

Entanglement properties of hypergraph states. British Mathematics Colloquium, University of Cambridge, UK, March 2015.

A Few of My Favorite Things. Sabbatical Talk, Lebanon Valley College, PA, March 2015.

Student-faculty research in mathematics and physics. Faculty Scholarship Celebration, Lebanon Valley College, Pa., February 2015.

Graphs and Hypergraphs, Symmetry and Entanglement. Tetrahedral Geometry and Topology Seminar, Hempfield, PA, September 2014.

Undergraduate research in quantum information science. MathFest, Portland, OR, August 2014.

Local Unitary Symmetries of Hypergraph States. Quantum Information Workshop, Seefeld, Austria, July 2014.

Local Unitary Symmetries of Hypergraph States. University of Cambridge, May 2014.

Local Unitary Symmetries of Hypergraph States. University of Bristol, May 2014.

Local Unitary Symmetries of Hypergraph States. Télécom Paris Tech, May 2014.

Entanglement and local unitary stabilizer subgroups. Quantum Information Seminar, Institute for Quantum Optics and Quantum Information, Innsbruck, Austria, March 2014.

Entanglement verification using local unitary stabilizers. Central European Quantum Information Processing Workshop, Valtice, Czech Republic, June 2013.

Quantum information: An ongoing research program with undergraduate students. The MD-DC-VA Section meeting of the Mathematical Association of America, Virginia Military Institute, Lexington, VA, October 2012.

Writing as a path to understanding and discovery. Elizabethtown College Bowers Writers House, September 2012.

Linear basis for decoherence-free subspace for collective decoherence. Quantum Information Workshop, Seefeld, Austria, July 2012.

Local unitary stabilizers, symmetric states, and Werner states. Télécom Paris Tech, March 2012.

Local Unitary Classes of Symmetric Mixed States and Ongoing Work on Werner States. Quantum Information Processing (QIP) 2012, Montreal, December 2011.

Classifying entanglement for symmetric states of n quantum bits. Tetrahedral Geometry and Topology Seminar, Elizabethtown College, PA, September 2011.

Local unitary group stabilizers and entanglement for multiqubit symmetric states. Theory of Quantum Computation, Communication and Cryptography (TQC) 2011, Madrid, May 2011.

Some questions about local unitary stabilizers. Theory of Quantum Computation, Communication and Cryptography (TQC) 2010, Leeds, UK, April 2010.

Quantum entanglement: When the whole is more than the sum of the parts. Lebanon Valley College Faculty Colloquium, Annville, PA, March 2009.

Classification of Multi-party Quantum Entanglement: Continuing Work. Tetrahedral Geometry and Topology Seminar, Hempfield, PA, October 2008.

An Application of Combinatorics in Quantum Information. Virginia Commonwealth University Discrete Mathematics Seminar, Richmond, VA, October 2008.

Quantum Information and Entanglement. Virginia Commonwealth University Mathematical Expositions Talk, Richmond, VA, October 2008.

Entanglement Classification Using Stabilizer Subalgebras. Susquehanna University Research Experience for Undergraduates in Quantum Information Theory, Selinsgrove, PA, July 2008.

Maximum Stabilizer Dimension for Multiparty States. University of Bristol Quantum Computation and Information Seminar, Bristol, UK, February 2008.

Maximum Stabilizer Dimension for Multiparty States. University of York Quantum Information Seminar, York, UK, January 2008.

Hamilton, Hopf and Bloch. Joint Math Colloquium of Franklin and Marshall College and Millersville University, Millersville, PA, April 2007.

Classification of Multiparticle Entanglement Types with Minimum Orbit Dimension. Joint Meetings of the American Mathematical Society and the Mathematical Association of America, San Antonio, TX, January 2006.

Minimum Orbit for the Local Unitary Group Action on State Space for a System of Qubits. Joint Meetings of the American Mathematical Society and the Mathematical Association of America, Atlanta, GA, January 2005.

Rotations, Quaternions and the Hopf Map. Joint Math Colloquium of Franklin and Marshall College and Millersville University, Lancaster, PA, April 2004.

Problems in Quantum Entanglement. Tetrahedral Geometry and Topology Seminar, Hempfield, PA, October 2003.

Quantum Information. Lebanon Valley College Faculty Colloquium, Annville, PA, April 2003.

Simplified Method for Classi cation of Entanglement Types. Joint Meetings of the American Mathematical Society and the Mathematical Association of America, Baltimore, MD, January 2003.

Convergence, Behold! Mathematical Association of America Southeastern Section Meeting, Memphis, TN, March 1999.

The Inner Life of the 3-sphere. Davidson College, Davidson, NC, October 1998.

Visual Lie Theory. Institute for Advanced Study Park City Mathematics Institute, Park City, UT, July 1998.

Geometry in 3 and 4 Dimensions and Computer Visualization. Wake Forest University Colloquium, Winston-Salem, NC, April 1998.

The Hopf Fibration|Geometric Computer Visualization. Mathematical Association of America Southeastern Section Meeting, Charleston, SC, March 1998.

Weyl Invariants for Compact Lie Groups. Mid-Atlantic Algebra Conference, Virginia Polytechnic Institute, Blacksburg, VA, April 1997.

Symmetry, Geometry and Topology. Ithaca College, Ithaca, NY, April 1997.

Lie Groups and Fiber Bundles. Wake Forest University, Winston-Salem, NC, April 1997.

Towards a Uniform Method for Integral Cohomology of Compact Lie Groups. Mathematical Association of America Southeastern Section Meeting, Atlanta, GA, March 1997.

Integral Weyl Invariants and Cohomology of Compact Lie Groups. Lie Algebra Seminar, NC State University, Raleigh, NC, February 1997.

Groups in Geometry and Topology. Davidson College Math Coffee, Davidson, NC, September 1996.

• Introductory Through Vector Calculus
• Introductory and Intermediate Statistics
• Mathematical Probability and Statistics
• College Algebra
• Trigonometry
• Finite Mathematics
• Liberal Arts Mathematics
• Introduction to Proofs
• Linear Algebra 1 and 2
• Discrete Mathematics
• Differential Equations
• Real Analysis
• Abstract Algebra 1 and 2
• Modern Geometry
• Topology 1 and 2
• Quantum Information and Computation