{
  "$type": "site.standard.document",
  "bskyPostRef": {
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  "path": "/papers/q-2026-05-29-2120/",
  "publishedAt": "2026-05-29T10:14:38.000Z",
  "site": "https://quantum-journal.org",
  "tags": [
    "Paper",
    "https://doi.org/10.22331/q-2026-05-29-2120"
  ],
  "textContent": "Quantum 10, 2120 (2026).\n\nhttps://doi.org/10.22331/q-2026-05-29-2120\n\nHaar 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"
}