Doubly Quantum Mechanics

Vittorio D'Esposito; Giuseppe Fabiano; Domenico Frattulillo; Flavio Mercati

doi:10.22331/q-2025-04-24-1721

Doubly Quantum Mechanics

Vittorio D'Esposito^1,2, Giuseppe Fabiano^3,4,5, Domenico Frattulillo², and Flavio Mercati⁶

¹Dipartimento di Fisica Ettore Pancini, Università di Napoli ``Federico II'', Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy
²INFN, Sezione di Napoli, I-80126 Napoli, Italy
³Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA
⁴Department of Physics, University of California, Berkeley, CA 94720, USA
⁵Centro Ricerche Enrico Fermi—Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”, Roma
⁶Departamento de Física, Universidad de Burgos, 09001 Burgos, Spain

Published:	2025-04-24, volume 9, page 1721
Editor:	Maximilian Lock
Eprint:	arXiv:2412.05997v3
Doi:	https://doi.org/10.22331/q-2025-04-24-1721
Citation:	Quantum 9, 1721 (2025).

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Abstract

Motivated by the expectation that relativistic symmetries might acquire quantum features in Quantum Gravity, we take the first steps towards a theory of ''Doubly'' Quantum Mechanics, a modification of Quantum Mechanics in which the geometrical configurations of physical systems, measurement apparata, and reference frame transformations are themselves quantized and described by ''geometry'' states in a Hilbert space. We develop the formalism for spin-$\frac{1}{2}$ measurements by promoting the group of spatial rotations $SU(2)$ to the quantum group $SU_q(2)$ and generalizing the axioms of Quantum Theory in a covariant way. As a consequence of our axioms, the notion of probability becomes a self-adjoint operator acting on the Hilbert space of geometry states, hence acquiring novel non-classical features. After introducing a suitable class of semi-classical geometry states, which describe near-to-classical geometrical configurations of physical systems, we find that probability measurements are affected, in these configurations, by intrinsic uncertainties stemming from the quantum properties of $SU_q(2)$. This feature translates into an unavoidable fuzziness for observers attempting to align their reference frames by exchanging qubits, even when the number of exchanged qubits approaches infinity, contrary to the standard $SU(2)$ case.

Featured image: Visual representation of probability in superposition. In standard Quantum Mechanics, although some properties of a system can be in a quantum superposition, and the outcomes of measurements are uncertain, the probabilities of those outcomes can be determined with arbitrary precision, just by repeating the same measurement a sufficient amount of times. The newly introduced quantumness of the probability manifests itself as the possibility of obtaining, with the same experimental setup, different outcomes for the determination of a probability. In the picture, the probability is determined as a frequency counting of results in a heads-or-tails coin toss (awake cat represents heads, while sleeping cat represents tails), in the ideal limit of infinite coin tosses. The analogue of an experimental setup describing a superposition of two probabilities of obtaining spin up in a spin measurement is represented by a coin with two possible outcomes for the frequency counting of heads results. Two coins prepared in the exact same state (representing two experimental setups prepared in the same geometry state), can lead to two different results for the determination of probability via frequency counting, as long as this state is not a probability eigenstate.

Popular summary

In exploring the interface between Quantum Theory and Gravity, spacetime is expected to acquire quantum features, which have been safely ignored in Quantum Mechanics, but become important in the physical regimes relevant to Quantum Gravity. We investigate an aspect of these quantum features by introducing a novel framework called ''Doubly Quantum Mechanics'', an extension of Quantum Mechanics in which the geometrical configurations of physical systems, measurement apparata and, crucially, reference frame transformations are also quantum. In this work we focus on the process of exchanging information between relatively rotated observers by means of the exchange of spin $\frac{1}{2}$ particles (e.g. electrons). This formalism naturally yields a quantization of the Born rule. This implies that the probabilities of outcomes, which can be determined exactly in standard Quantum Mechanics, are now affected by uncertainties and can fluctuate. This feature prevents two observers, who do not know their relative orientation, from aligning their reference frames sharply in a communication protocol involving spin exchanges, even when the number of exchanged spins approaches infinity, contrary to the standard case.

► BibTeX data

@article{DEsposito2025doublyquantum,
  doi = {10.22331/q-2025-04-24-1721},
  url = {https://doi.org/10.22331/q-2025-04-24-1721},
  title = {Doubly {Q}uantum {M}echanics},
  author = {D'Esposito, Vittorio and Fabiano, Giuseppe and Frattulillo, Domenico and Mercati, Flavio},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1721},
  month = apr,
  year = {2025}
}

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Cited by

[1] Angel Ballesteros, Diego Fernandez-Silvestre, Flaminia Giacomini, and Giulia Gubitosi, "Quantum Galilei group as quantum reference frame transformations", Quantum 9, 1935 (2025).

[2] Gaetano Fiore and Fedele Lizzi, "Mixed states for reference frames transformations", arXiv:2507.05758, (2025).

[3] Michele Arzano, Antonio Del Prete, and Domenico Frattulillo, "Quantum Evolution of Hopf Algebra Hamiltonians", arXiv:2602.07887, (2026).

[4] Angel Ballesteros, Diego Fernandez-Silvestre, and Ivan Gutierrez-Sagredo, "Universal $T$-matrices for quantum Poincaré groups: contractions and quantum reference frames", arXiv:2604.01058, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-09 13:14:57) and SAO/NASA ADS (last updated successfully 2026-08-09 13:14:59). The list may be incomplete as not all publishers provide suitable and complete citation data.

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