# Research

## Research interests

Operator algebras

Quantum computing

Quantum information

Quantum cryptography

## Research statement

My current research focus is on operator algebraic characterizations of quantum entanglement. Recent developments have shown that questions about quantum correlations can be reduced to computational problems about non-commutative *-algebras. Understanding the limitations and structure of this algebraic framework for quantum correlations has important applications for cryptographic primitives for quantum devices, quantum complexity theory, and quantum information theory.

## Past projects

### Bounding the quantum value of compiled nonlocal games

Nonlocal games enable several prominent protocols for verifying quantum computations. However, a serious drawback of these protocols is the requisite non-communication assumption between the two quantum players (or provers) as ensuring spatial separations in a single device is not so practical. A cryptographic compiler introduced by Kalai et al. (STOC'23) transforms a nonlocal game into an interactive protocol between a computationally bounded prover and a classical verifier. In the "compiled game" the non-communication assumption between players is replaced by a cryptographic assumption, relieving the need for the spatial separation, as well as removing the need for multiple provers. Although the compilation procedure requires sophisticated cryptographic primitives (such as quantum homomorphic encryption), the black box nature of the compilation procedure is desirable from a theoretical standpoint. In their work, Kalai et al. established both the completeness and soundness of the compiler in the context of classical provers. This means essentially that the compiler preserves the classical value (maximal winning probability) of the nonlocal game. Despite also demonstrating completeness in the context of quantum provers, the soundness (a meaningful upper bound on the compiled quantum value) remained unknown. Several works have made progress and further explored compiled nonlocal games, this has culminated in the the work of Kulpe et al. who established that the (quantum) commuting operator value is an upper bound on the compiled quantum value.

My work on this topic includes:

A Computational Tsirelson's Theorem for the Value of Compiled XOR Games by David Cui, Giulio Malavolta, Arthur Mehta, Anand Natarajan, Connor Paddock, Simon Schmidt, Michael Walter, and Tina Zhang, (Accepted to TQC24), arXiv:2402.17301, 2024.

Self-testing in the compiled setting via tilted-CHSH inequalities by Arthur Mehta, Connor Paddock, and Lewis Wooltorton, arXiv:2406.04986, 2024.

A bound on the quantum value of all compiled nonlocal games by Alexander Kulpe, Giulio Malavolta, Connor Paddock, Simon Schmidt, and Michael Walter, arXiv:2408.06711 2024.

### An operator-algebraic formulation for the self-testing of quantum measurement models

We give a new definition of self-testing for correlations in terms of states on C*-algebras. We show that this definition is equivalent to the standard definition for any nice enough class of finite-dimensional quantum models, provided that the correlation is extremal and has a full-rank model in the class. This last condition automatically holds for the class of POVM quantum models, but does not necessarily hold for the class of projective models by a result of Mancinska and Kaniewski. For extremal binary correlations and for extremal synchronous correlations, we show that any self-test for projective models is a self-test for POVM models. The question of whether there is a self-test for projective models which is not a self-test for POVM models remains open. We propose a C*-algebraic definition of self-testing for commuting operator models, and show that this definition is also equivalent to the standard definition of self-testing in the finite-dimensional case.

https://arxiv.org/abs/2301.11291

### Near-optimal quantum strategies and approximate representations of the nonlocal game algebra

For synchronous, binary constraint systems, and XOR nonlocal games, there is a correspondence between optimal finite-dimensional quantum strategies and matrix representations of the affiliated game algebra with a maximally-entangled state. Based on the work of Slofstra and Vidick we extend the robustness of this correspondence by showing that near-optimal strategies are near or approximate representations of the game algebra with respect to the normalized Hilbert-Schmidt norm.

Paddock, Connor. "Rounding near-optimal quantum strategies for nonlocal games to strategies using maximally entangled states." arXiv preprint, 2022. (arXiv)

### Nonlocal games from graphs

Perfect strategies for linear constraint system nonlocal games manifest in the structure of a group associated with the game. Based on the interesting work of Alex Arkhipov on forbidden graph minors and the existence of perfect quantum strategies of the Magic Square and Magic Pentagram nonlocal games. Vincent Russo, Turner Silverthorne, William Slofstra, and I initiated a study of linear constraint system nonlocal games arising from the incidence matrix of 2-coloured graphs. In this case, we found that certain group properties are characterized by the combinatorial structure of the associated graphs describing the game. This work was part of my M.Math thesis at the University of Waterloo under the supervision of Jon Yard and William Slofstra.

Paddock, Connor. "Algebraic and combinatorial aspects of incidence groups and linear system non-local games arising from graphs." M.Math thesis. University of Waterloo, (2019). (UW space)

Expanding on my master's thesis my co-authors and I have written a more extensive article based on graph minors, linear system nonlocal games, and Arkhipov's theorem.

### Antidegradable qubit channels

Antidegradable quantum channels are an important family of quantum channels due to the interesting information-theoretic properties in a multi-use setting. Using a spectral characterization of the density matrix for bipartite qubit states that admit a symmetric extension and a known connection between antidegradability and symmetric extendibility of the Choi matrix, Jianxin and I derived a characterization for antidegradable qubit channels based on the spectral properties of the Choi matrix. This work was part of an M.Sc. thesis at the Univerisity of Guelph under the supervision of Bei Zeng and Rajesh Pereira.

Paddock, Connor, and Jianxin Chen. "A Characterization of Antidegradable Qubit Channels." arXiv preprint arXiv:1712.03399 (2017). (arXiv link)

### Subspaces of anticoherent spin states

Anticoherence of spin states is a separate but related notion to the degree of entanglement in a symmetric product of fermions. Upon stereographically projecting the Majorana representation, anticoherent spin states appear as collections of points on the 3-sphere with certain symmetries. Rajesh Pereira and I used ideas from polynomial invariant theory to construct subspaces of anticoherent spin states of various degrees. A connection to anticoherent subspaces and the higher-rank numerical range of certain spin observables was also found. This work was partially supported by an NSERC USRA.

Pereira, Rajesh, and Connor Paul-Paddock. "Anticoherent subspaces." Journal of Mathematical Physics 58.6 (2017): 062107. (arXiv link)

For a list of works and links to them checkout my papers page.