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

• URL: http://arxiv.org/abs/2507.16939v1
• Date: Tue, 22 Jul 2025 18:23:46 GMT
• Title: Exact distinguishability between real-valued and complex-valued Haar random quantum states
• Authors: Tristan Nemoz, Romain Alléaume, Peter Brown,
• Abstract summary: We analytically compute the spectral decomposition of a Haar random state.<n>We show a lower-bound on the approximation parameter of real-valued state $t$-designs.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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