I work on information-theoretic aspects of nature, specifically about the nature of space, time, and quantum information. I am motivated by questions stemming from physics, and I generally adhere to strict mathematical logic in formulating definitions, making statements, and proving theorems. Category theory provides me with the structure to do this, and it appears heavily in my thinking. Three examples of my most recent works are on time-reversal symmetry, quantum Bayesian inference, retrodiction (making inference about the past), and quantum broadcasting. Preprints can be found online at 2212.08088 [quant-ph], 2210.13531 [quant-ph], and 2310.13049 [quant-ph] (the first two are now published). More of my works can be found by scrolling further down the page or on my Google Scholar.

My research program primarily focuses on extending inference, retrodiction, Bayes’ rule, and entropy to quantum systems and dynamics. There have been many attempts at extending these ideas to quantum systems, and my work presents novel and constructive approaches using ideas from category theory in a way that bypasses many previous no-go results, the latter of which suggested such extensions might not be possible. Currently, I am exploring applications of this program in the quantification of quantum temporal correlations, quantum error-correction, open quantum systems, the entanglement-wedge reconstruction from AdS/CFT, and other subjects. If any of these ideas sound interesting to you, please feel free to contact me as I am happy to discuss and have fruitful and intellectually stimulating collaborations.

Videos on a categorical introduction to (classical) Bayes’ theorem can be watched here: https://www.youtube.com/playlist?list=PLSx1kJDjrLRQksb7H9fqRE8GVMJdkX-4A.

I have also spent some time thinking about K-theory, higher gauge theory, and 2-dimensional algebra. The latter describes a way of associating algebraic data to 2-dimensional surfaces that is compatible with the way that 2-dimensional surfaces can be glued (i.e., “sewn”) together.

During the summer of 2009, while an undergraduate in Queens College (CUNY), I visited Steven G. Johnson at MIT. There, we worked on a project in condensed matter to prove the existence of bound states for certain periodic potentials using tools from functional analysis and measure theory. The summer internship resulted in a publication in Phys. Rev. B.

Below, I include some preprints and publications, organized first by category and then from most recent to older works.

Quantum Information, Bayes, Probability, and Retrodiction

• (with James Fullwood, Francesco Buscemi, and Giulio Chiribella) “Virtual Quantum Broadcasting,” arXiv: 2310.13049 [quant-ph]
• (with James Fullwood) “From time-reversal symmetry to quantum Bayes’ rules,” PRX Quantum 4, 020334 (2023). DOI: 10.1103/PRXQuantum.4.020334, arXiv: 2212.08088 [quant-ph]
• (with Francesco Buscemi) “Axioms for retrodiction: achieving time-reversal symmetry with a prior,” Quantum 7, 1013 (2023).  DOI: 10.22331/q-2023-05-23-1013, arXiv: 2210.13531 [quant-ph]
• (with James Fullwood) “On quantum states over time” Proc. R. Soc. A. 478: 20220104. DOI: 10.1098/rspa.2022.0104, arXiv: 2202.03607 [quant-ph]
• “Conditional distributions for quantum systems,” EPTCS 343, Proceedings 18th International Conference on Quantum Physics and Logic (2021), pp 1–13. DOI: 10.4204/EPTCS.343.1, arXiv: 2102.01529 [quant-ph]
• (with Benjamin P. Russo) “A non-commutative Bayes’ theorem” Linear Algebra Appl. 644 (2022), pp 28–94. DOI: 10.1016/j.laa.2022.02.030, arXiv: 2005.03886 [quant-ph]
• “Inverses, disintegrations, and Bayesian inversion in quantum Markov categories” (91 pages) arXiv: 2001.08375 [quant-ph]
• (with Benjamin P. Russo) “Non-commutative disintegration: existence and uniqueness in finite dimensions” J. Noncommut. Geom. 17, 3, (2023) pp. 899-955. DOI: 10.4171/jncg/493, arXiv: 1907.09689 [quant-ph]

Entropy

• (with Tadashi Takayanagi, Yusuke Taki, and Zixia Wei) “SVD Entanglement Entropy,” arXiv: 2307.06531 [hep-th]
• (with James Fullwood) “On dynamical measures of quantum information,” arXiv: 2306.01831 [quant-ph]
• “A functorial characterization of von Neumann entropy” Cah. Topol. Géom. Différ. Catég. LXIII, 1 (2022), 89–128, arXiv: 2009.07125 [quant-ph]
• (with James Fullwood) “The information loss of a stochastic map,” Entropy 23, no. 8 (2021). DOI: 10.3390/e23081021, arXiv: 2107.01975 [cs.IT]
• “Towards a functorial description of quantum relative entropy,” in: Nielsen F., Barbaresco F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science, vol 12829. Springer, Cham. (2021). DOI: 10.1007/978-3-030-80209-7_60, arXiv: 2105.04059 [quant-ph]

Algebraic quantum theory and operator algebras

• (with Luca Giorgetti, Alessio Ranallo, and Benjamin P. Russo)
  “Bayesian inversion and the Tomita–Takesaki modular group,” Quart. J. Math. 74 , 3, pp 975-1014 (2023). DOI: 10.1093/qmath/haad014, arXiv: 2112.03129 [math.OA]
• “Stinespring’s construction as an adjunction,” Compositionality 1, 2 (2019). DOI: 10.32408/compositionality-1-2, arXiv: 1807.02533 [math.OA]
• “From observables and states to Hilbert space and back: a 2-categorical adjunction,” Appl. Categorical Struct., 26, 6, pp 1123–1157 (2018). DOI: 10.1007/s10485-018-9522-6, arXiv: 1609.08975 [math-ph]

K-theory

• (with Byungdo Park, Corbett Redden, and Augusto Stoffel) “Noncommutative differential K-theory” J. Geom. Phys. 174 (2022), 104446. DOI: 10.1016/j.geomphys.2021.104446, arXiv: 2106.12073 [math.KT]

Analysis

• “Discrete probabilistic and algebraic dynamics: a stochastic Gelfand-Naimark Theorem,”
  https://arxiv.org/abs/1708.00091

Higher gauge theory

• “Two-dimensional algebra in lattice gauge theory,” Journal of Mathematical Physics 60, 043506 (2019); https://doi.org/10.1063/1.5078532. Also available as
  https://arxiv.org/abs/1802.01139
• “Gauge invariant surface holonomy and monopoles,” Theory and Applications of Categories, Vol. 30, No. 42, pp 1319-1428 (2015)
  https://www.tac.mta.ca/tac/volumes/30/42/30-42abs.html

Condensed matter

• “Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps” with Karen K. Y. Lee, Yehuda Avniel, and Steven G. Johnson, Phys. Rev. B 81, 155324 (2010)
  https://journals.aps.org/prb/abstract/10.1103/PhysRevB.81.155324