Exact distinguishability between real-valued and complex-valued Haar random quantum states

Tristan Nemoz; Romain Alléaume; Peter Brown

doi:10.22331/q-2026-05-29-2120

Exact distinguishability between real-valued and complex-valued Haar random quantum states

Tristan Nemoz, Romain Alléaume, and Peter Brown

Télécom Paris, LTCI, Institut Polytechnique de Paris, Inria, 19 Place Marguerite Perey, 91 120 Palaiseau, France

Published:	2026-05-29, volume 10, page 2120
Editor:	Tomoyuki Morimae
Eprint:	arXiv:2507.16939v2
Doi:	https://doi.org/10.22331/q-2026-05-29-2120
Citation:	Quantum 10, 2120 (2026).

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Abstract

Haar random states are fundamental objects in quantum information theory and quantum computing. We study the density matrix resulting from sampling $t$ copies of a $d$-dimensional quantum state according to the Haar measure on the orthogonal group. In particular, we analytically compute its spectral decomposition. This allows us to compute exactly the trace distance between $t$-copies of a real Haar random state and $t$-copies of a complex Haar random state. Using this we show a lower-bound on the approximation parameter of real-valued state $t$-designs and improve the lower-bound on the number of copies required for imaginarity testing.

► BibTeX data

@article{Nemoz2026exact,
  doi = {10.22331/q-2026-05-29-2120},
  url = {https://doi.org/10.22331/q-2026-05-29-2120},
  title = {Exact distinguishability between real-valued and complex-valued {H}aar random quantum states},
  author = {Nemoz, Tristan and All{\'{e}}aume, Romain and Brown, Peter},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2120},
  month = may,
  year = {2026}
}

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Cited by

[1] Riccardo Castellano, Dmitry Grinko, Sadra Boreiri, Nicolas Brunner, and Jef Pauwels, "Stronger Welch Bounds and Optimal Approximate $k$-Designs", arXiv:2602.13099, (2026).

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