Flatness-Preserving Operations
Abstract
A quantum state is called flat if it is proportional to a projector. There has been recent interest in studying antiflatness, the property of diverging from flat states, and establishing a resource theory for it. Identifying the free operations (the Flatness-Preserving Operations (FPOs)) remained an open problem. For this purpose, we first discuss Orthogonality-Preserving Operations (OPOs), of which trivial examples are unitary operations in an isolated system. More generally, we give a simple proof that all OPOs are isometric embeddings consisting of combinations of unitaries/isometries and appending a fixed state. We then show that all FPOs are either constant maps to some fixed flat state or a special case of an OPO, where the appended fixed state must be a flat state. We also show that the only possible flat convex combinations of flat states are those of orthogonal flat states with weights given by their purities.
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BibTeX
@misc{veltheim2026flatnesspreserving,
title = {{Flatness-Preserving Operations}},
author = {Otto Veltheim and Esko Keski-Vakkuri},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30697},
eprint = {2608.30697},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30697},
}
RIS
TY - UNPB
TI - Flatness-Preserving Operations
AU - Otto Veltheim
AU - Esko Keski-Vakkuri
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30697
UR - https://arxiv.org/abs/2608.30697
AB - A quantum state is called flat if it is proportional to a projector. There has been recent interest in studying antiflatness, the property of diverging from flat states, and establishing a resource theory for it. Identifying the free operations (the Flatness-Preserving Operations (FPOs)) remained an open problem. For this purpose, we first discuss Orthogonality-Preserving Operations (OPOs), of which trivial examples are unitary operations in an isolated system. More generally, we give a simple proof that all OPOs are isometric embeddings consisting of combinations of unitaries/isometries and appending a fixed state. We then show that all FPOs are either constant maps to some fixed flat state or a special case of an OPO, where the appended fixed state must be a flat state. We also show that the only possible flat convex combinations of flat states are those of orthogonal flat states with weights given by their purities.
ER -