Stability of isometries of quantum states
Abstract
We prove dimension-independent Hyers--Ulam stability for surjective approximate isometries of the positive trace-class cone in both the trace and Bures metrics. Here ε≥0 is a uniform upper bound on the additive error in preserving distances. Such an approximate isometry is uniformly approximated by a Wigner symmetry (a unitary or antiunitary conjugation), with error at most 9ε/2 in the trace metric and 2√2ε in the Bures metric, even if it does not fix zero. For maps fixing zero, the trace bound improves to 3ε even under the hypothesis of δ-surjectivity, independently of δ. In infinite dimensions, we construct spectrum-preserving homeomorphisms of the full state space, including mixed states, whose distortions tend to zero in both metrics but which stay a fixed uniform distance from every Wigner symmetry. For state spaces, stability holds in each fixed finite dimension with a dimension-dependent modulus tending to zero as ε→0, but no such modulus can be chosen independently of dimension.
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BibTeX
@misc{looi2026stability,
title = {{Stability of isometries of quantum states}},
author = {Sam Looi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.37133},
eprint = {2609.37133},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37133},
}
RIS
TY - UNPB
TI - Stability of isometries of quantum states
AU - Sam Looi
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2609.37133
UR - https://arxiv.org/abs/2609.37133
AB - We prove dimension-independent Hyers--Ulam stability for surjective approximate isometries of the positive trace-class cone in both the trace and Bures metrics. Here ε≥0 is a uniform upper bound on the additive error in preserving distances. Such an approximate isometry is uniformly approximated by a Wigner symmetry (a unitary or antiunitary conjugation), with error at most 9ε/2 in the trace metric and 2√2ε in the Bures metric, even if it does not fix zero. For maps fixing zero, the trace bound improves to 3ε even under the hypothesis of δ-surjectivity, independently of δ. In infinite dimensions, we construct spectrum-preserving homeomorphisms of the full state space, including mixed states, whose distortions tend to zero in both metrics but which stay a fixed uniform distance from every Wigner symmetry. For state spaces, stability holds in each fixed finite dimension with a dimension-dependent modulus tending to zero as ε→0, but no such modulus can be chosen independently of dimension.
ER -