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An uncertainty principle for entanglement

  • Mitali Nanda
  • Daochen Wang
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Abstract

Consider two orthonormal bases of a bipartite Hilbert space such that states in one basis are weakly entangled and states in the other are strongly entangled. We establish a quantitative version of the following uncertainty principle: every state that is localized in one basis must be delocalized in the other. More specifically, we lower bound the sum of the Shannon entropies of any state's coefficient distributions in the two bases by the difference between the entanglement Rényi entropies of their basis states. The result follows from new lower and upper bounds on the entanglement of superpositions that may be of independent interest. As an application, our result limits the power of weakly-entangled states in representing ground states of Hamiltonians with strongly-entangled energy eigenstates. We illustrate this application numerically for the Sachdev-Ye-Kitaev model.

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Cite

BibTeXRIS
BibTeX
@misc{nanda2026uncertainty,
  title = {{An uncertainty principle for entanglement}},
  author = {Mitali Nanda and Daochen Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39126},
  eprint = {2609.39126},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39126},
}
RIS
TY  - UNPB
TI  - An uncertainty principle for entanglement
AU  - Mitali Nanda
AU  - Daochen Wang
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39126
UR  - https://arxiv.org/abs/2609.39126
AB  - Consider two orthonormal bases of a bipartite Hilbert space such that states in one basis are weakly entangled and states in the other are strongly entangled. We establish a quantitative version of the following uncertainty principle: every state that is localized in one basis must be delocalized in the other. More specifically, we lower bound the sum of the Shannon entropies of any state's coefficient distributions in the two bases by the difference between the entanglement Rényi entropies of their basis states. The result follows from new lower and upper bounds on the entanglement of superpositions that may be of independent interest. As an application, our result limits the power of weakly-entangled states in representing ground states of Hamiltonians with strongly-entangled energy eigenstates. We illustrate this application numerically for the Sachdev-Ye-Kitaev model.
ER  -