Jason Crann
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Jason Crann has a total of 19 co-authors. Here are the top co-authors by number of shared publications:
- David W. Kribs
University of Guelph
6 shared publications - Rupert H. Levene
School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland
5 shared publications - Ivan G. Todorov
School of Mathematical Sciences University of Delaware Newark, DE 19716, USA
4 shared publications - Alex Bearden
University of Texas at Tyler
2 shared publications - Alexandre Conlon
School of Mathematics and Statistics, Carleton University, Ottawa, Canada
2 shared publications - Lyudmila Turowska
Department of Mathematical Sciences Chalmers University of Technology
2 shared publications
Latest Publications
- 11 May 2026
- Published by Springer Nature in Annales Henri Poincare
Operator Algebraic AdS/CFT and Subregion Duality
- 11 May 2026
- Published by Springer Nature in Annales Henri Poincare
Operator Algebraic AdS/CFT and Subregion Duality
- Open Access
An operator system approach to self-testing
We develop a general framework for self-testing, in which bipartite correlations are described by states on the commuting tensor product of a pair of operator systems. We propose a definition of a local isometry between bipartite quantum systems in the commuting operator model, and define self-testing and abstract self-testing in the latter generality. We show that self-tests are in the general case always abstract self-tests and that, in some cases, the converse is also true. We apply our framework in a variety of instances, including to correlations with quantum inputs and outputs, quantum commuting correlations for the CHSH game, synchronous correlations, contextuality scenarios, quantum colourings and Schur quantum channels.
Open AccessAn operator system approach to self-testing
We develop a general framework for self-testing, in which bipartite correlations are described by states on the commuting tensor product of a pair of operator systems. We propose a definition of a local isometry between bipartite quantum systems in the commuting operator model, and define self-testing and abstract self-testing in the latter generality. We show that self-tests are in the general case always abstract self-tests and that, in some cases, the converse is also true. We apply our framework in a variety of instances, including to correlations with quantum inputs and outputs, quantum commuting correlations for the CHSH game, synchronous correlations, contextuality scenarios, quantum colourings and Schur quantum channels.
- Open Access
- 20 March 2025
- Published by Springer Nature in Letters in Mathematical Physics
- Vol. 115 (2) , 1-48
GAUSSIAN QUANTUM INFORMATION OVER GENERAL QUANTUM KINEMATICAL SYSTEMS I: GAUSSIAN STATES
We develop a theory of Gaussian states over general quantum kinematical systems with finitely many degrees of freedom. The underlying phase space is described by a locally compact abelian (LCA) group G with a symplectic structure determined by a 2-cocycle on G. We use the concept of Gaussian distributions on LCA groups in the sense of Bernstein to define Gaussian states and completely characterize Gaussian states over 2-regular LCA groups of the form $$G= F\times \widehat{F}$$ endowed with a canonical normalized 2-cocycle. This covers, in particular, the case of n-bosonic modes, n-qudit systems with odd $$d\ge 3$$ , and p-adic quantum systems. Our characterization reveals a topological obstruction to Gaussian state entanglement when we decompose the quantum kinematical system into the Euclidean part and the remaining part (whose phase space admits a compact open subgroup). We then generalize the discrete Hudson theorem (Gross in J Math Phys 47(12):122107, 2006) to the case of totally disconnected 2-regular LCA groups. We also examine angle-number systems with phase space $$\mathbb {T}^n\times \mathbb {Z}^n$$ and fermionic/hard-core bosonic systems with phase space $$\mathbb {Z}^{2n}_2$$ (which are not 2-regular) and completely characterize their Gaussian states.- 20 March 2025
- Published by Springer Nature in Letters in Mathematical Physics
- Vol. 115 (2) , 1-48
Open AccessGAUSSIAN QUANTUM INFORMATION OVER GENERAL QUANTUM KINEMATICAL SYSTEMS I: GAUSSIAN STATES
We develop a theory of Gaussian states over general quantum kinematical systems with finitely many degrees of freedom. The underlying phase space is described by a locally compact abelian (LCA) group G with a symplectic structure determined by a 2-cocycle on G. We use the concept of Gaussian distributions on LCA groups in the sense of Bernstein to define Gaussian states and completely characterize Gaussian states over 2-regular LCA groups of the form $$G= F\times \widehat{F}$$ endowed with a canonical normalized 2-cocycle. This covers, in particular, the case of n-bosonic modes, n-qudit systems with odd $$d\ge 3$$ , and p-adic quantum systems. Our characterization reveals a topological obstruction to Gaussian state entanglement when we decompose the quantum kinematical system into the Euclidean part and the remaining part (whose phase space admits a compact open subgroup). We then generalize the discrete Hudson theorem (Gross in J Math Phys 47(12):122107, 2006) to the case of totally disconnected 2-regular LCA groups. We also examine angle-number systems with phase space $$\mathbb {T}^n\times \mathbb {Z}^n$$ and fermionic/hard-core bosonic systems with phase space $$\mathbb {Z}^{2n}_2$$ (which are not 2-regular) and completely characterize their Gaussian states. - Open Access
Fejér representations for discrete quantum groups and applications
We prove that a discrete quantum group $\mathbb{G}$ has the approximation property if and only if a Fejér-type representation holds for its $C^*$-algebraic or von Neumann algebraic crossed products. As applications, we extend several results from the literature to the context of discrete quantum groups with the approximation property. Additionally, we provide new characterizations of invariant $L^\infty(\widehat{\mathbb{G}})$-bimodules of $\mathcal{B}(\ell^2(\mathbb{G}))$ and invariant $C(\widehat{\mathbb{G}})$-bimodules of $\mathcal{K}(\ell^2(\mathbb{G}))$, some of which are new in the group setting. Finally, we study Fubini crossed products of discrete quantum group actions.Open AccessFejér representations for discrete quantum groups and applications
We prove that a discrete quantum group $\mathbb{G}$ has the approximation property if and only if a Fejér-type representation holds for its $C^*$-algebraic or von Neumann algebraic crossed products. As applications, we extend several results from the literature to the context of discrete quantum groups with the approximation property. Additionally, we provide new characterizations of invariant $L^\infty(\widehat{\mathbb{G}})$-bimodules of $\mathcal{B}(\ell^2(\mathbb{G}))$ and invariant $C(\widehat{\mathbb{G}})$-bimodules of $\mathcal{K}(\ell^2(\mathbb{G}))$, some of which are new in the group setting. Finally, we study Fubini crossed products of discrete quantum group actions. - Open Access
Algebraic approach to spacetime bulk reconstruction
Motivated by the theory of holographic quantum error correction in the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, together with the kink transform conjecture on the bulk AdS description of boundary cocycle flow, we characterize (approximate) complementary recovery in terms of (approximate) intertwining of bulk and boundary cocycle derivatives. Using the geometric modular structure in vacuum AdS, we establish an operator algebraic subregion-subregion duality of boundary causal diamonds and bulk causal wedges for Klein-Gordon fields in the universal cover of AdS. Our results suggest that, from an algebraic perspective, the kink transform is bulk cocycle flow, which (in the above case) induces the bulk geometry via geometric modular action and the corresponding notion of time. As a by-product, we find that if the von Neumann algebra of a boundary CFT subregion is a type $\mathrm{III}_1$ factor with an ergodic vacuum, then the von Neumann algebra of the corresponding dual bulk subregion, is either $\mathbb{C}1$ (with a one-dimensional Hilbert space) or a type $\mathrm{III}_1$ factor.Open AccessAlgebraic approach to spacetime bulk reconstruction
Motivated by the theory of holographic quantum error correction in the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, together with the kink transform conjecture on the bulk AdS description of boundary cocycle flow, we characterize (approximate) complementary recovery in terms of (approximate) intertwining of bulk and boundary cocycle derivatives. Using the geometric modular structure in vacuum AdS, we establish an operator algebraic subregion-subregion duality of boundary causal diamonds and bulk causal wedges for Klein-Gordon fields in the universal cover of AdS. Our results suggest that, from an algebraic perspective, the kink transform is bulk cocycle flow, which (in the above case) induces the bulk geometry via geometric modular action and the corresponding notion of time. As a by-product, we find that if the von Neumann algebra of a boundary CFT subregion is a type $\mathrm{III}_1$ factor with an ergodic vacuum, then the von Neumann algebra of the corresponding dual bulk subregion, is either $\mathbb{C}1$ (with a one-dimensional Hilbert space) or a type $\mathrm{III}_1$ factor. - Open Access
Values of cooperative quantum games
We develop a resource-theoretical approach that allows us to quantify values of two-player, one-round cooperative games with quantum inputs and outputs, as well as values of quantum probabilistic hypergraphs. We analyse the quantum game values arising from the type hierarchy of quantum no-signalling correlations, establishing tensor norm expressions for each of the correlation types. As a consequence, we provide metric characterisations of state convertibility via LOSR and LOCC.En route, we obtain an alternative description of the maximal tensor products of ternary rings of operators.
Open AccessValues of cooperative quantum games
We develop a resource-theoretical approach that allows us to quantify values of two-player, one-round cooperative games with quantum inputs and outputs, as well as values of quantum probabilistic hypergraphs. We analyse the quantum game values arising from the type hierarchy of quantum no-signalling correlations, establishing tensor norm expressions for each of the correlation types. As a consequence, we provide metric characterisations of state convertibility via LOSR and LOCC.En route, we obtain an alternative description of the maximal tensor products of ternary rings of operators.
- 6 June 2023
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 151 (09) , 3793-3799
Orthogonal unitary bases and a subfactor conjecture
We show that any finite dimensional von Neumann algebra admits an orthonormal unitary basis with respect to its standard trace. We also show that a finite dimensional von Neumann subalgebra of M n ( C ) M_n(\mathbb {C}) admits an orthonormal unitary basis under normalized matrix trace if and only if the normalized matrix trace and standard trace of the von Neumann subalgebra coincide. As an application, we verify a recent conjecture of Bakshi-Gupta [Glasg. Math. J. 64 (2022), pp. 586–602], showing that any finite-index regular inclusion N ⊆ M N\subseteq M of I I 1 II_1 -factors admits an orthonormal unitary Pimsner-Popa basis.
- 6 June 2023
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 151 (09) , 3793-3799
Orthogonal unitary bases and a subfactor conjecture
We show that any finite dimensional von Neumann algebra admits an orthonormal unitary basis with respect to its standard trace. We also show that a finite dimensional von Neumann subalgebra of M n ( C ) M_n(\mathbb {C}) admits an orthonormal unitary basis under normalized matrix trace if and only if the normalized matrix trace and standard trace of the von Neumann subalgebra coincide. As an application, we verify a recent conjecture of Bakshi-Gupta [Glasg. Math. J. 64 (2022), pp. 586–602], showing that any finite-index regular inclusion N ⊆ M N\subseteq M of I I 1 II_1 -factors admits an orthonormal unitary Pimsner-Popa basis.
- 23 December 2022
- Published by Springer Nature in Annales Henri Poincare
- Vol. 24 (5) , 1779-1821
Quantum Teleportation in the Commuting Operator Framework
- 23 December 2022
- Published by Springer Nature in Annales Henri Poincare
- Vol. 24 (5) , 1779-1821
Quantum Teleportation in the Commuting Operator Framework
- Open Access
Quantum teleportation in the commuting operator framework
We introduce a notion of teleportation scheme between subalgebras of semi-finite von Neumann algebras in the commuting operator model of locality. Using techniques from subfactor theory, we present unbiased teleportation schemes for relative commutants $N'\cap M$ of a large class of finite-index inclusions $N\subseteq M$ of tracial von Neumann algebras, where the unbiased condition means that no information about the teleported observables are contained in the classical communication sent between the parties. For a large class of subalgebras $N$ of matrix algebras $M_n(\mathbb{C})$, including those relevant to hybrid classical/quantum codes, we show that any tight teleportation scheme for $N$ necessarily arises from an orthonormal unitary Pimsner-Popa basis of $M_n(\mathbb{C})$ over $N'$, generalising work of Werner. Combining our techniques with those of Brannan-Ganesan-Harris, we compute quantum chromatic numbers for a variety of quantum graphs arising from finite-dimensional inclusions $N\subseteq M$.Open AccessQuantum teleportation in the commuting operator framework
We introduce a notion of teleportation scheme between subalgebras of semi-finite von Neumann algebras in the commuting operator model of locality. Using techniques from subfactor theory, we present unbiased teleportation schemes for relative commutants $N'\cap M$ of a large class of finite-index inclusions $N\subseteq M$ of tracial von Neumann algebras, where the unbiased condition means that no information about the teleported observables are contained in the classical communication sent between the parties. For a large class of subalgebras $N$ of matrix algebras $M_n(\mathbb{C})$, including those relevant to hybrid classical/quantum codes, we show that any tight teleportation scheme for $N$ necessarily arises from an orthonormal unitary Pimsner-Popa basis of $M_n(\mathbb{C})$ over $N'$, generalising work of Werner. Combining our techniques with those of Brannan-Ganesan-Harris, we compute quantum chromatic numbers for a variety of quantum graphs arising from finite-dimensional inclusions $N\subseteq M$. - Open Access
Orthogonal Unitary Bases and a Subfactor Conjecture
We show that any finite dimensional von Neumann algebra admits an orthonormal unitary basis with respect to its standard trace. We also show that a finite dimensional von Neumann subalgebra of $M_n(\mathbb{C})$ admits an orthonormal unitary basis under normalized matrix trace if and only if the normalized matrix trace and standard trace of the von Neumann subalgebra coincide. As an application, we verify a recent conjecture of Bakshi-Gupta, showing that any finite-index regular inclusion $N\subseteq M$ of $II_1$-factors admits an orthonormal unitary Pimsner-Popa basis.Open AccessOrthogonal Unitary Bases and a Subfactor Conjecture
We show that any finite dimensional von Neumann algebra admits an orthonormal unitary basis with respect to its standard trace. We also show that a finite dimensional von Neumann subalgebra of $M_n(\mathbb{C})$ admits an orthonormal unitary basis under normalized matrix trace if and only if the normalized matrix trace and standard trace of the von Neumann subalgebra coincide. As an application, we verify a recent conjecture of Bakshi-Gupta, showing that any finite-index regular inclusion $N\subseteq M$ of $II_1$-factors admits an orthonormal unitary Pimsner-Popa basis.