Exact distinguishability between real-valued and complex-valued Haar random quantum states
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917535026249728 |
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| author | Nemoz, Tristan Alléaume, Romain Brown, Peter |
| author_facet | Nemoz, Tristan Alléaume, Romain Brown, Peter |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16939 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exact distinguishability between real-valued and complex-valued Haar random quantum states Nemoz, Tristan Alléaume, Romain Brown, Peter Quantum Physics 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. |
| title | Exact distinguishability between real-valued and complex-valued Haar random quantum states |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2507.16939 |