{
  "annotation_id": "8d61f565-1888-4eb8-b0ff-966409e089c6",
  "date_created": "2026-03-02T18:02:30.216000Z",
  "date_modified": "2026-03-02T18:02:30.216000Z",
  "file_hash": "24de5b1bf86a83fcf4783f1b072c338d97109db2339dab0b778ef2f19305b153",
  "private": false,
  "record": {
    "abstract": "Generalising in the sense of Hahn\u0027s spin echo, we completely characterise\nthose unitary propagators of effective multi-qubit interactions that can be\ninverted solely by {\\em local} unitary operations on $n$ qubits\n(spins-$\\tfrac{1}{2}$). The subset of $U\\in \\mathbf{SU}(2^n)$ satisfying\n$U^{-1}=K_1 U K_2$ with pairs of local unitaries $K_1,\nK_2\\in\\mathbf{SU}(2)^{\\otimes n}$ comprises two classes: in type-I, $K_1$ and\n$K_2$ are inverse to one another, while in type-II they are not. {Type-I}\nconsists of one-parameter groups that can jointly be inverted for all times\n$t\\in\\R{}$ because their Hamiltonian generators satisfy $K H K^{-1} = \\Ad K (H)\n= -H$. As all the Hamiltonians generating locally invertible unitaries of\ntype-I are spanned by the eigenspace associated to the eigenvalue -1 of the\n{\\em local} conjugation map $\\Ad K$, this eigenspace can be given in closed\nalgebraic form. The relation to the root space decomposition of\n$\\mathfrak{sl}(N,\\C{})$ is pointed out. Special cases of type-I invertible\nHamiltonians are of $p$-quantum order and are analysed by the transformation\nproperties of spherical tensors of order $p$. Effective multi-qubit interaction\nHamiltonians are characterised via the graphs of their coupling topology.\n  {Type-II} consists of pointwise locally invertible propagators, part of which\ncan be classified according to the symmetries of their matrix representations.\nMoreover, we show gradient flows for numerically solving the decision problem\nwhether a propagator is type-I or type-II invertible or not by driving the\nleast-squares distance $\\norm{K_1 e^{-itH} K_2 - e^{+itH}}^2_2$ to zero.",
    "arxiv_id": "quant-ph/0610061",
    "authors": [
      "T. Schulte-Herbrueggen",
      "A. Spoerl"
    ],
    "categories": [
      "quant-ph"
    ],
    "title": "Which Quantum Evolutions Can Be Reversed by Local Unitary Operations?   Algebraic Classification and Gradient-Flow-Based Numerical Checks",
    "url": "https://arxiv.org/abs/quant-ph/0610061"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "202bcb2f-ff40-4388-9bc4-16b60ba16f08",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}