We prove that the topology on the density space with respect to a unital C*-algebra and a faithful induced by the C*-norm is finer than the Bures metric topology. We also provide an example when this containment is strict. Next, we provide a metric on the density space induced by a quantum metric in the sense of Rieffel and prove that the induced topology is the same as the topology induced by the Bures metric and C*-norm when the C*-algebra is assumed to be finite dimensional. Finally, we provide an example of when the Bures metric and induced quantum metric are not metric equivalent. Thus, we provide a bridge between these aspects of quantum information theory and noncommutative metric geometry.We prove that the topology on the density space with respect to a unital C*-algebra and a faithful induced by the C*-norm is finer than the Bures metric topology. We also provide an example when this containment is strict. Next, we provide a metric on the density space induced by a quantum metric in the sense of Rieffel and prove that the induced topology is the same as the topology induced by the Bures metric and C*-norm when the C*-algebra is assumed to be finite dimensional. Finally, we provide an example of when the Bures metric and induced quantum metric are not metric equivalent. Thus, we provide a bridge between these aspects of quantum information theory and noncommutative metric geometry.

arXiv:2406.07219Jun 11, 20240 citations

Non-local games and quantum symmetries of quantum metric spaces

Kari Eifler

We generalize Banica's construction of the quantum isometry group of a metric space to the class of quantum metric spaces in the sense of Kuperberg and Weaver. We also introduce quantum isometries between two quantum metric spaces, and we show that if a pair of quantum metric spaces are algebraically quantum isometric, then their quantum isometry groups are monoidally equivalent. Motivated by the recent work on the graph isomorphism game, we introduce a new two-player non-local game called the metric isometry game, where players can win classically if and only if the metric spaces are isometric. Winning quantum strategies of this game align with quantum isometries of the metric spaces.We generalize Banica's construction of the quantum isometry group of a metric space to the class of quantum metric spaces in the sense of Kuperberg and Weaver. We also introduce quantum isometries between two quantum metric spaces, and we show that if a pair of quantum metric spaces are algebraically quantum isometric, then their quantum isometry groups are monoidally equivalent. Motivated by the recent work on the graph isomorphism game, we introduce a new two-player non-local game called the metric isometry game, where players can win classically if and only if the metric spaces are isometric. Winning quantum strategies of this game align with quantum isometries of the metric spaces.

arXiv:2011.03867Nov 7, 20201 citation

Quantum maps

H. H. Wills Physics Laboratory

Annals of PhysicsSep 1, 19790 citations

Codes in W\ast-metric Spaces: Theory and Examples

Christopher J. Bumgardner

We introduce a $W^*$ -metric space, which is a particular approach to non-commutative metric spaces where a \textit{quantum metric} is defined on a von Neumann algebra. We generalize the notion of a quantum code and quantum error correction to the setting of finite dimensional $W^*$ -metric spaces, which includes codes and error correction for classical finite metric spaces. We also introduce a class of $W^*$ -metric spaces that come from representations of semi-simple Lie algebras $\mathfrak{g}$ called \textit{ $\mathfrak{g}$ -metric} spaces, and present an outline for code constructions. In turn, we produce specific code constructions for $\mathfrak{su}(2,\mathbb{C})$ -metric spaces that depend upon proving Tverberg's theorem for points on a moment curve constructed from arithmetic sequences. We introduce a \textit{quantum distance distribution}, and we prove an analogue of the MacWilliam's identities for $\mathfrak{su}(2)$ -metric spaces.We introduce a $W^*$-metric space, which is a particular approach to non-commutative metric spaces where a \textit{quantum metric} is defined on a von Neumann algebra. We generalize the notion of a quantum code and quantum error correction to the setting of finite dimensional $W^*$-metric spaces, which includes codes and error correction for classical finite metric spaces. We also introduce a class of $W^*$-metric spaces that come from representations of semi-simple Lie algebras $\mathfrak{g}$ called \textit{$\mathfrak{g}$-metric} spaces, and present an outline for code constructions. In turn, we produce specific code constructions for $\mathfrak{su}(2,\mathbb{C})$-metric spaces that depend upon proving Tverberg's theorem for points on a moment curve constructed from arithmetic sequences. We introduce a \textit{quantum distance distribution}, and we prove an analogue of the MacWilliam's identities for $\mathfrak{su}(2)$-metric spaces.

arXiv:1205.4517May 21, 20120 citations

Quantum maps

Michael Berry, Nandor L. Balazs, M. Tabor +1 more

Annals of PhysicsSep 1, 1979421 citations

We introduce a

W^*

-metric space, which is a particular approach to non-commutative metric spaces where a \textit{quantum metric} is defined on a von Neumann algebra. We generalize the notion of a quantum code and quantum error correction to the setting of finite dimensional

W^*

-metric spaces, which includes codes and error correction for classical finite metric spaces. We also introduce a class of

W^*

-metric spaces that come from representations of semi-simple Lie algebras

\mathfrak{g}

called \textit{

\mathfrak{g}

-metric} spaces, and present an outline for code constructions. In turn, we produce specific code constructions for

\mathfrak{su}(2,\mathbb{C})

-metric spaces that depend upon proving Tverberg's theorem for points on a moment curve constructed from arithmetic sequences. We introduce a \textit{quantum distance distribution}, and we prove an analogue of the MacWilliam's identities for

\mathfrak{su}(2)

-metric spaces.