En 1976, Connes demande si toute algèbre de von Neumann finie se plonge dans un ultraproduit d'algèbres de matrices. En 1980, Tsirelson demande si, dans la formulation mathématique de la mécanique quantique, autoriser des espaces de Hilbert de dimension infinie change fondamentalement le modèle. En 1993, Kirchberg conjecture que le produit tensoriel de deux copies de la C*-algèbre pleine du groupe libre de rang infini dénombrable peut être muni d'une unique norme de C*-algèbre. De manière surprenante et non triviale, ces trois problèmes sont en fait équivalents, c'est maintenant bien compris. Ces problèmes viennent d'être résolus, par la négative, avec des méthodes d'informatique : calculabilité, complexité, et informatique quantique. Je ferai de mon mieux pour raconter les grandes lignes de cette très longue preuve.En 1976, Connes demande si toute algèbre de von Neumann finie se plonge dans un ultraproduit d'algèbres de matrices. En 1980, Tsirelson demande si, dans la formulation mathématique de la mécanique quantique, autoriser des espaces de Hilbert de dimension infinie change fondamentalement le modèle. En 1993, Kirchberg conjecture que le produit tensoriel de deux copies de la C*-algèbre pleine du groupe libre de rang infini dénombrable peut être muni d'une unique norme de C*-algèbre. De manière surprenante et non triviale, ces trois problèmes sont en fait équivalents, c'est maintenant bien compris. Ces problèmes viennent d'être résolus, par la négative, avec des méthodes d'informatique : calculabilité, complexité, et informatique quantique. Je ferai de mon mieux pour raconter les grandes lignes de cette très longue preuve.

Jan 1, 20230 citations

Ensembles paradoxaux et paradoxe de Von Neumann

Laura Reynaerts

Jun 27, 20160 citations

A variation on von Neumann Entropy and a result of Varadarajan

Glenn D. Appleby

Expositiones MathematicaeApr 25, 20241 citation

Quantum Soundness of a Total-Degree Line-versus-Point Test

Honghao Fu, Nengkun Yu, Tianrun Zhao

We prove quantum soundness of the total-degree diagonal line-vs-point test using the individual-degree soundness theorem of Ji, Natarajan, Vidick, Wright, and Yuen. A random change of coordinates yields projective polynomial decoders of total degree at most $md$ . The uniform-line slice of the test bounds the weight of outcomes of degree greater than $d$ , which are removed by a common relabeling. This reduction does not yield a dimension-independent soundness bound: the $\operatorname{poly}(m)$ dependence of the individual-degree theorem persists, as discussed in Section 1.2 of arXiv:2009.12982.We prove quantum soundness of the total-degree diagonal line-vs-point test using the individual-degree soundness theorem of Ji, Natarajan, Vidick, Wright, and Yuen. A random change of coordinates yields projective polynomial decoders of total degree at most $md$. The uniform-line slice of the test bounds the weight of outcomes of degree greater than $d$, which are removed by a common relabeling. This reduction does not yield a dimension-independent soundness bound: the $\operatorname{poly}(m)$ dependence of the individual-degree theorem persists, as discussed in Section 1.2 of arXiv:2009.12982.

quant-pharXiv:2609.27129Sep 24, 20260 citations

Orthogonalization of Positive Operator Valued Measures

Mikael DE LA SALLE

We show that a partition of the unity (or POVM) on a Hilbert space that is almost orthogonal is close to an orthogonal POVM in the same von Neumann algebra. This generalizes to infinite dimension previous results in matrix algebras by Kempe–Vidick and Ji–Natarajan–Vidick–Wright–Yuen. Quantitatively, our result are also finer, as we obtain a linear dependance, which is optimal. We also generalize to infinite dimension a duality result between POVMs and minimal majorants of finite subsets in the predual of a von Neumann algebra.We show that a partition of the unity (or POVM) on a Hilbert space that is almost orthogonal is close to an orthogonal POVM in the same von Neumann algebra. This generalizes to infinite dimension previous results in matrix algebras by Kempe–Vidick and Ji–Natarajan–Vidick–Wright–Yuen. Quantitatively, our result are also finer, as we obtain a linear dependance, which is optimal. We also generalize to infinite dimension a duality result between POVMs and minimal majorants of finite subsets in the predual of a von Neumann algebra.

arXiv:2103.14126Comptes Rendus MathématiqueMay 23, 20223 citations

Nonlocal games, compression theorems, and the arithmetical hierarchy

Hamoon Mousavi, Seyed Sajjad Nezhadi, Henry Yuen

We investigate the connection between the complexity of nonlocal games and the arithmetical hierarchy, a classification of languages according to the complexity of arithmetical formulas defining them. It was recently shown by Ji, Natarajan, Vidick, Wright and Yuen that deciding whether the (finite-dimensional) quantum value of a nonlocal game is 1 or at most 1/2 is complete for the class Σ1 (i.e., ). A result of Slofstra implies that deciding whether the commuting operator value of a nonlocal game is equal to 1 is complete for the class Π1 (i.e., coRE).We investigate the connection between the complexity of nonlocal games and the arithmetical hierarchy, a classification of languages according to the complexity of arithmetical formulas defining them. It was recently shown by Ji, Natarajan, Vidick, Wright and Yuen that deciding whether the (finite-dimensional) quantum value of a nonlocal game is 1 or at most 1/2 is complete for the class Σ1 (i.e., ). A result of Slofstra implies that deciding whether the commuting operator value of a nonlocal game is equal to 1 is complete for the class Π1 (i.e., coRE).

arXiv:2110.04651ACM Symposium on Theory of Computing (STOC)Jun 9, 202211 citations

md

. The uniform-line slice of the test bounds the weight of outcomes of degree greater than

d

, which are removed by a common relabeling. This reduction does not yield a dimension-independent soundness bound: the

\operatorname{poly}(m)

dependence of the individual-degree theorem persists, as discussed in Section 1.2 of arXiv:2009.12982.