Integration against POVMs in von Neumann algebras and Operational Quantum Reference Frames
Abstract
We construct integrals of uniformly bounded ultraweakly measurable functions against positive operator-valued measures (POVMs) in von Neumann algebras, assuming that the integrand algebra has separable predual. The integral lies in the spatial tensor product and is uniquely characterized by its evaluations on product normal states. Integration is unital and completely positive and descends faithfully to the quotient by POVM-null functions. When both preduals are separable, this quotient is a von Neumann algebra and integration is normal, agreeing with the amplification of scalar POVM integration. We establish a dilation factorization, a Leibniz rule and a Fubini theorem, and specify the additional joint-normality hypothesis needed for realization in an ambient algebra. As an application, under the stated separability and joint-normality assumptions, we extend operational quantum reference frame relativization to covariant POVMs on arbitrary measurable spaces carrying a group action. Its domain is the fixed-point von Neumann algebra of system-valued functions modulo null equality. For homogeneous spaces of second countable locally compact Hausdorff groups with closed stabilizers, this domain is identified with the stabilizer-fixed system algebra. For sharp frames, relativization is a normal unital *-homomorphism, faithful when ambient multiplication is injective. For localizable frames on metrizable spaces, restriction recovers continuous integrands at supported configurations. On trivial principal bundles, the domain is identified with system-valued fields on the base. These results provide integration machinery for forthcoming applications to relativistic gauge theories.
Related
- Non-isomorphism of q-Gaussian von Neumann Algebras in Infinite Variables
- Associativity of operator convolutions for group actions on von Neumann algebras
- A uniform commutator bound in finite von Neumann algebras
- Group actions on von Neumann algebras with compact open subgroups
- Quantum channels on duals of von Neumann algebras in the Schrödinger picture
BibTeX
@misc{glowacki2026integration,
title = {{Integration against POVMs in von Neumann algebras and Operational Quantum Reference Frames}},
author = {Jan Głowacki and Yui Kuramochi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.27366},
eprint = {2606.27366},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.27366},
}
RIS
TY - UNPB
TI - Integration against POVMs in von Neumann algebras and Operational Quantum Reference Frames
AU - Jan Głowacki
AU - Yui Kuramochi
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2606.27366
UR - https://arxiv.org/abs/2606.27366
AB - We construct integrals of uniformly bounded ultraweakly measurable functions against positive operator-valued measures (POVMs) in von Neumann algebras, assuming that the integrand algebra has separable predual. The integral lies in the spatial tensor product and is uniquely characterized by its evaluations on product normal states. Integration is unital and completely positive and descends faithfully to the quotient by POVM-null functions. When both preduals are separable, this quotient is a von Neumann algebra and integration is normal, agreeing with the amplification of scalar POVM integration. We establish a dilation factorization, a Leibniz rule and a Fubini theorem, and specify the additional joint-normality hypothesis needed for realization in an ambient algebra. As an application, under the stated separability and joint-normality assumptions, we extend operational quantum reference frame relativization to covariant POVMs on arbitrary measurable spaces carrying a group action. Its domain is the fixed-point von Neumann algebra of system-valued functions modulo null equality. For homogeneous spaces of second countable locally compact Hausdorff groups with closed stabilizers, this domain is identified with the stabilizer-fixed system algebra. For sharp frames, relativization is a normal unital *-homomorphism, faithful when ambient multiplication is injective. For localizable frames on metrizable spaces, restriction recovers continuous integrands at supported configurations. On trivial principal bundles, the domain is identified with system-valued fields on the base. These results provide integration machinery for forthcoming applications to relativistic gauge theories.
ER -