Stoica (arXiv:2404.05633, published in Physica Scripta 2025) proved a no-go theorem: if the total Hamiltonian of a closed system-plus-apparatus is bounded from below, as is axiomatic for physical Hamiltonians and assumed for our universe, then no exact ('accurate') measurement of a sharp observable and no exact state preparation can be realized — one can approach perfect accuracy asymptotically but never attain it, without any conservation law being invoked. The theorem is purely qualitative and the paper explicitly states that no quantitative prediction of the achievable accuracy in terms of the size or other properties of the measurement device exists. The open problem is to quantify this impossibility: prove tight bounds on the achievable error of measurement and preparation schemes as an explicit function of the apparatus's physical resources (Hilbert-space dimension, energy spread above the spectral bottom, spectral-gap structure, interaction time), construct explicit semibounded apparatus Hamiltonians attaining the optimal accuracy-resource scaling, and use the resulting scaling law to adjudicate whether the boundedness assumption is compatible with the manifest accuracy and scalability of real information-processing devices, or whether a consistent relaxation of boundedness is required.

Background

Reading and writing quantum data means measuring and preparing quantum states. For a sharp observable — one whose outcomes correspond to mutually orthogonal projection operators Pλ\mathsf{P}_{\lambda}, i.e. a projection-valued measure — an ideal measurement maps an eigenstate ψλ\psi_{\lambda} of the observed observable A\mathsf{A} together with the apparatus' ready pointer state φ∅\varphi_{\varnothing} to a product state in which the pointer observable Z\mathsf{Z} of the apparatus indicates the outcome λ\lambda; an ideal preparation produces, conditioned on the pointer outcome, a system state in the corresponding eigenspace. The study of fundamental obstructions to such ideal operations goes back to Wigner (1952) and Araki and Yanase (1960): the Wigner–Araki–Yanase (WAY) theorem states that if an additive quantity such as total angular momentum is conserved and does not commute with the observable to be measured, the measurement cannot be both accurate and non-disturbing unless the pointer commutes with the conserved quantity (the Yanase condition); Wigner noted that an infinitely large apparatus evades the restriction.

Stoica (2024; published 2025) identified a different obstruction that requires no conservation law. Treat the entire laboratory — observed system SS plus apparatus MM, including everything that switches the interaction on and off — as one closed, autonomous system with a time-independent total Hamiltonian H\mathsf{H}, and make the minimal pointer-stability assumption (Assumption 3 of the source, 'Pointer persistence'): after the interaction ends, the pointer states remain eigenstates of Z\mathsf{Z} throughout some later time interval. A lemma of Hegerfeldt and Ruijsenaars (1980) states that if H\mathsf{H} is bounded from below — its spectrum has a finite lower edge, the standard axiom for physical Hamiltonians since Pauli — and the Schrödinger orbit e−iℏtHΨe^{-\frac{i}{\hbar}t\mathsf{H}}\Psi of a vector stays inside a closed subspace throughout an open time interval, then it stays inside that subspace for all times t∈Rt\in\mathbb{R}. Applied to the pointer eigenspaces, this forces the apparatus to occupy the same pointer eigenspace at all times, contradicting the transition from the ready state φ∅\varphi_{\varnothing} to a distinct outcome state φλ\varphi_{\lambda}. Hence Theorem 1 of the source (for pure states) and Theorem 2 (for mixed states, via the Hilbert–Schmidt space of density operators): if the total Hamiltonian is bounded from below, accurate measurements and accurate preparations are impossible, and so is accurate reading and writing of quantum data. The paper shows this restriction is in a sense stronger than the WAY theorem, since it needs no additive conserved observable.

Boundedness from below is the same assumption behind several other foundational no-go results: the Pauli and Unruh–Wald argument that it forbids a self-adjoint time operator (a clock observable increasing monotonically with time), the Hegerfeldt–Ruijsenaars instantaneous spreading of relativistic wave packets, and the nonvanishing recurrence amplitudes of Stoica (2022). The paper therefore closes on a dilemma: complex, apparently scalable and accurate devices — classical computers, which are themselves quantum systems — seem to contradict the theorem, which would hint that the world's Hamiltonian is not actually bounded from below; yet abandoning boundedness would reintroduce the problematic consequences just listed. But the theorem is deliberately dichotomous, and the paper states the gap explicitly: 'We don't have a quantitative prediction of the possible accuracy in terms of the size or other properties of the measurement device, so it would be rushed to draw some conclusions from this. But we definitely need to explore this and find out.'

Quantitative results exist only in neighboring, mechanism-different settings: conservation-law-constrained gate fidelities and their scaling with apparatus size (Karasawa and Ozawa; Karasawa, Gea-Banacloche and Ozawa), extensions of the WAY theorem to continuous and unbounded conserved observables (Kuramochi and Tajima) and to unsharp observables, i.e. positive-operator-valued measures (Mohammady, Miyadera and Loveridge), thermodynamic results that ideal projective measurements require infinite resource costs (Guryanova, Friis and Huber), and quantum speed limits that bound the pace of evolution using the energy above the ground state (Margolus and Levitin). None of these quantifies the boundedness-only obstruction, which is the subject of this problem.

Problem Statement

Quantify the accuracy limits that a total Hamiltonian bounded from below imposes on quantum state readout and writing. Formally: a closed, autonomous system S+MS+M (observed system plus apparatus, the whole laboratory) evolves under e−iℏtHe^{-\frac{i}{\hbar}t\mathsf{H}} with H\mathsf{H} bounded from below; an interaction scheme takes ψ(0)⊗φ∅\psi(0)\otimes\varphi_{\varnothing} (ready pointer state) during [0,T][0,T] to a state in which the pointer eigenvalue λ\lambda identifies, respectively, the eigenstate of the observed observable that was present before the interaction (measurement) or the eigenspace prepared in the system (preparation) — the source's Definitions 1–2, equations (4)–(5) — together with pointer stability after the interaction (the source's Assumption 3). Since the source's conditions (4)–(5) and its Assumption 3 are strictly dichotomous, the error metric in which these conditions are relaxed — i.e. what exactly ε\varepsilon and δ\delta measure — is itself part of what a quantitative theory must specify. The problem is to determine, for schemes satisfying these conditions up to error ε\varepsilon (and pointer-stability error δ\delta):

(1) Model-independent lower bounds on the achievable error ε\varepsilon as an explicit function of the apparatus's physical resources — e.g. the dimension of the apparatus Hilbert space, the energy spread above the spectral bottom of H\mathsf{H}, the spectral-gap structure, the interaction time TT, and the stability window — establishing how fast accuracy can improve as resources grow, for any Hamiltonian bounded from below.

(2) Attainability: explicit families of system–apparatus Hamiltonians that are bounded from below (semibounded), achieving error ε\varepsilon with a stated resource scaling matching the bounds of (1).

(3) The physical adjudication the source demands from these numbers: whether the resulting accuracy–resource scaling is consistent with the observed reliability and scalability of macroscopic information-processing devices (classical computers, which are quantum systems) in a universe whose Hamiltonian is bounded from below, or whether consistency requires relaxing boundedness — and, in the latter case, whether the relaxation can avoid the known consequences of unbounded spectra (instantaneous wave-packet spreading, absence of monotonically increasing time observables, nonvanishing recurrence amplitudes). This adjudication is settled by comparing the scaling law of (1)–(2) with the component counts, error tolerances and achievable resource budgets of concrete devices.

The common objective of (1)–(3) is a quantitative theory of read/write accuracy under semibounded Hamiltonians: (1) and (2) supply the tight tradeoff, and (3) is its physical interpretation, which is the very conclusion the source states cannot be drawn without such numbers.

References

  1. Ovidiu Cristinel Stoica, 'Can we accurately read or write quantum data?', Physica Scripta 100, 085108 (2025), DOI 10.1088/1402-4896/adf0a5, https://doi.org/10.1088/1402-4896/adf0a5 (preprint arXiv:2404.05633, https://arxiv.org/abs/2404.05633)
  2. Paul Busch (translation), 'Translation of "Die Messung quantenmechanischer Operatoren" by E.P. Wigner' (original: E.P. Wigner, Zeitschrift für Physik 133, 101–108 (1952)), arXiv:1012.4372, https://arxiv.org/abs/1012.4372
  3. Huzihiro Araki and Mutsuo M. Yanase, 'Measurement of Quantum Mechanical Operators', Physical Review 120, 622–626 (1960), DOI 10.1103/PhysRev.120.622, https://doi.org/10.1103/PhysRev.120.622
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  8. Yelena Guryanova, Nicolai Friis and Marcus Huber, 'Ideal Projective Measurements Have Infinite Resource Costs', Quantum 4, 222 (2020), DOI 10.22331/q-2020-01-13-222, https://doi.org/10.22331/q-2020-01-13-222
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  12. Ovidiu Cristinel Stoica, 'The Problem of Irreversible Change in Quantum Mechanics', arXiv:2204.02270 (2022), https://arxiv.org/abs/2204.02270
  13. Ovidiu Cristinel Stoica, 'Exact quantum time compatible with positive energy', arXiv:2607.01296 (2026), https://arxiv.org/abs/2607.01296
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