Deciding the Attainability of the Multiparameter Quantum Fisher Information is NP-Hard
Abstract
The quantum Fisher information (QFI) sets a fundamental bound on the attainable precision when estimating multiple parameters simultaneously. Incompatibility among the individually optimal measurements can, in some cases, imply that the precision limit set by the QFI is not attainable. In certain special cases, including pure states and full-rank states, the exact conditions for when the precision limit set by the QFI can be saturated are known. However, general conditions for the attainability of the QFI with measurements on individual copies of the quantum state have long been sought. Indeed, this was recently stated as one of the five problems in quantum information theory highlighted by [P. Horodecki et al, PRX Quantum 3, 010101 (2022)]. In this work we prove that exact attainability with individual measurements is NP-hard to decide, even for a restricted set of real, constant-rank quantum models. The source problem for our proof is the NP-hard problem of deciding whether a given bipartite density matrix is separable or not. Our result shows that the longstanding difficulty in obtaining general conditions for QFI attainability reflects a fundamental computational obstruction, rather than merely a limitation of existing mathematical techniques: unless P=NP, no efficiently computable necessary-and-sufficient criterion can exist in general.
Related
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BibTeX
@misc{conlon2026deciding,
title = {{Deciding the Attainability of the Multiparameter Quantum Fisher Information is NP-Hard}},
author = {Lorcan O. Conlon and Laura Shou and V Vijendran and Kishor Bharti and Aritra Das and Simon K. Yung and Jacob M. Taylor and Alexey V. Gorshkov and Jun Suzuki and Syed M. Assad},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35975},
eprint = {2609.35975},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35975},
}
RIS
TY - UNPB
TI - Deciding the Attainability of the Multiparameter Quantum Fisher Information is NP-Hard
AU - Lorcan O. Conlon
AU - Laura Shou
AU - V Vijendran
AU - Kishor Bharti
AU - Aritra Das
AU - Simon K. Yung
AU - Jacob M. Taylor
AU - Alexey V. Gorshkov
AU - Jun Suzuki
AU - Syed M. Assad
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35975
UR - https://arxiv.org/abs/2609.35975
AB - The quantum Fisher information (QFI) sets a fundamental bound on the attainable precision when estimating multiple parameters simultaneously. Incompatibility among the individually optimal measurements can, in some cases, imply that the precision limit set by the QFI is not attainable. In certain special cases, including pure states and full-rank states, the exact conditions for when the precision limit set by the QFI can be saturated are known. However, general conditions for the attainability of the QFI with measurements on individual copies of the quantum state have long been sought. Indeed, this was recently stated as one of the five problems in quantum information theory highlighted by [P. Horodecki et al, PRX Quantum 3, 010101 (2022)]. In this work we prove that exact attainability with individual measurements is NP-hard to decide, even for a restricted set of real, constant-rank quantum models. The source problem for our proof is the NP-hard problem of deciding whether a given bipartite density matrix is separable or not. Our result shows that the longstanding difficulty in obtaining general conditions for QFI attainability reflects a fundamental computational obstruction, rather than merely a limitation of existing mathematical techniques: unless P=NP, no efficiently computable necessary-and-sufficient criterion can exist in general.
ER -