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arXiv:2610.24056 · Symmetry

Operational Meaning of the Mandelstam-Tamm Time Scale in a Spin-1/2 Clock

Jakub Rog

quant-phSubmitted Oct 16, 2026• Announced Oct 19, 2026• 0 citations
arXivHTMLPDF

Abstract

Georgiev's analysis of a freely precessing spin correctly emphasizes that the Mandelstam-Tamm quantity $ΔT=Δ\hat A/|d\langle\hat A\rangle/d t|$ is not the variance of a time operator. He concludes, however, that its divergence near the extrema of the clock signal is devoid of physical meaning and that $ΔT$ does not disclose how well the system measures the passage of time. For the same projective $\hatσ_x$ readout, we show that $ΔT$ equals the inverse square root of the classical Fisher information in one binary outcome at regular operating points. Its divergence at submaximal-coherence extrema therefore identifies a loss of local sensitivity in the selected readout, not an infinite variance of physical time; a complementary $\hatσ_y$ measurement remains informative. This interpretation also separates Georgiev's exactly known tick--tock period from the statistical precision with which a tick can be located. It preserves his warning against a time-operator interpretation while qualifying his stronger conclusion that the Mandelstam--Tamm scale lacks physical content.

Journal reference
Symmetry 2026, 18(11), 1762
DOI
10.3390/sym18111762
Comments
A Comment on Georgiev, Symmetry (16), 100 (2024)

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Georgiev's analysis of a freely precessing spin correctly emphasizes that the Mandelstam-Tamm quantity ΔT=ΔA^/∣d⟨A^⟩/dt∣ΔT=Δ\hat A/|d\langle\hat A\rangle/d t|ΔT=ΔA^/∣d⟨A^⟩/dt∣ is not the variance of a time operator. He concludes, however, that its divergence near the extrema of the clock signal is devoid of physical meaning and that ΔTΔTΔT does not disclose how well the system measures the passage of time. For the same projective σ^x\hatσ_xσ^x​ readout, we show that ΔTΔTΔT equals the inverse square root of the classical Fisher information in one binary outcome at regular operating points. Its divergence at submaximal-coherence extrema therefore identifies a loss of local sensitivity in the selected readout, not an infinite variance of physical time; a complementary σ^y\hatσ_yσ^y​ measurement remains informative. This interpretation also separates Georgiev's exactly known tick--tock period from the statistical precision with which a tick can be located. It preserves his warning against a time-operator interpretation while qualifying his stronger conclusion that the Mandelstam--Tamm scale lacks physical content.
The time–energy uncertainty relation in nonrelativistic quantum mechanics has been intensely debated with regard to its formal derivation, validity, and physical meaning. Here, we analyze two formal relations proposed by Mandelstam and Tamm and by Margolus and Levitin and evaluate their validity using a minimal quantum toy model composed of a single qubit inside an external magnetic field. We show that the ℓ1 norm of energy coherence C is invariant with respect to the unitary evolution of the quantum state. Thus, the ℓ1 norm of energy coherence C of an initial quantum state is useful for the classification of the ability of quantum observables to change in time or the ability of the quantum state to evolve into an orthogonal state. In the single-qubit toy model, for quantum states with the submaximal ℓ1 norm of energy coherence, C<1, the Mandelstam–Tamm and Margolus–Levitin relations generate instances of infinite “time uncertainty” that is devoid of physical meaning. Only for quantum states with the maximal ℓ1 norm of energy coherence, C=1, the Mandelstam–Tamm and Margolus–Levitin relations avoid infinite “time uncertainty”, but they both reduce to a strict equality that expresses the Einstein–Planck relation between energy and frequency. The presented results elucidate the fact that the time in the Schrödinger equation is a scalar variable that commutes with the quantum Hamiltonian and is not subject to statistical variance.
The article by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen is stimulating and raises some interesting and profound issues regarding the foundations of quantum mechanics and quantum measurements. One point the authors make is that a measurement of spin component √2‾/2 of a spin-1⁄2 particle is unphysical and can be attributed only to errors in weak measurements performed on a collection of N spins. I offer a more mundane interpretation that does not require the introduction of any error or postselection concepts. The physical, textbook picture of spin-1⁄2 is a vector of length √S(S + 1) = √3‾/2 with some distribution of orientations (that is, polar coordinates θ and φ). A conventional single measurement of Sz can yield only one of the eigenvalues ±1⁄2. This may be interpreted in the classical vector model by envisioning that the spin lies on a cone with θ defined by cosθ = √3‾/3. The expectation value of Sx then vanishes due to the uniform distribution of φ. Having some control over φ can yield a finite value of Sx . Thus I would argue that only an expectation value greater than √3‾/2 is unphysical; a value of √2‾/2 is quite physical and can even be interpreted classically in terms of some distribution of φ and θ.I would phrase the important observation of Aharonov and coauthors in a different way. While spin-1⁄2 has a magnitude of √3‾/2, a fundamental limitation of conventional single-point quantum measurements is that they can yield only the values +1⁄2 and −1⁄2 and any expectation value must therefore lie between those two extremes. Classically, therefore, the spin may not be fully aligned along the z-axis with θ = 0. Such alignment is impossible since it would yield well-defined values of the three noncommuting variables Sz = √3‾/2 and Sx = Sy = 0. However, once multiple time measurements are performed, one can define a reduced distribution of some of the measurements that is conditional on the outcome in the other measurements; such a distribution can be interpreted in terms of Sz greater than 1⁄2 but less than or equal to √3‾/2. That interpretation does not violate any of quantum mechanics' fundamental rules, which do not usually consider such quantities.Rather than a new, time-symmetric formulation of quantum mechanics that involves pre- and postselection, one can simply treat the scheme of the authors' figure 1b as a three-point correlation function, whereas figures 2a and 2b show a four-point correlation function, each panel having its own set of conditional probabilities. Error, time reversal, and postselection need not be invoked. Instead, one may think in multiple dimensions and develop the right language for the interpretation of the observables.The combination of pre- and postselection is an attempt to create an artificial ensemble that reproduces the results of multiple-point measurements in a single one-point measurement. As shown by Aharonov and coauthors, that is possible by introducing errors. An alternative physical picture is obtained by retaining the multipoint analysis. Multidimensional thinking is well developed in coherent nonlinear spectroscopy, where a system of spins or optical chromophores are subjected to sequences of short impulsive pulses.11. R. R. Ernst G. Bodenhausen A. Wokaun Principles of Nuclear Magnetic Resonance in One and Two Dimensions Clarendon Press, New York (1978) . , 22. S. Mukamel Annual Review of Physical Chemistry 51 1 691 (2000). https://doi.org/10.1146/annurev.physchem.51.1.691 ; Physical Review A 61 2 21804 (2000). https://doi.org/10.1103/PhysRevA.61.021804 ; Principles of Nonlinear Optical Spectroscopy Oxford U. Press, New York (1995) . Similar ideas may be applied for the interpretation of multiple measurements. One can think of various types of n-point observables obtained by combining n − m perturbations and m measurements. The nonlinear n-point response functions in spectroscopy represent n − 1 impulsive perturbations followed by a single measurement. The objects Aharonov and colleagues considered correspond to n = m. Proper multiple-distribution functions could then be naturally used for the interpretation of such generalized measurements.REFERENCESSection:ChooseTop of pageREFERENCES <<1. R. R. Ernst G. Bodenhausen A. Wokaun Principles of Nuclear Magnetic Resonance in One and Two Dimensions Clarendon Press, New York (1978) . Google Scholar2. S. Mukamel Annual Review of Physical Chemistry 51 1 691 (2000). https://doi.org/10.1146/annurev.physchem.51.1.691 ; Google ScholarCrossref, ISI Physical Review A 61 2 21804 (2000). https://doi.org/10.1103/PhysRevA.61.021804 ; , Google ScholarCrossref, ISI Principles of Nonlinear Optical Spectroscopy Oxford U. Press, New York (1995) . , Google Scholar© 2011 American Institute of Physics.
Quantum state readout on many experimental platforms is performed through a single projective measurement channel rather than full quantum state tomography. We derive a detector-conditioned quantum speed limit that constrains the minimum time for the projective fidelity F(t)=⟨ω∣ρ(t)∣ω⟩F(t) = \langle\omega|\rho(t)|\omega\rangleF(t)=⟨ω∣ρ(t)∣ω⟩ to reach a prescribed discrimination threshold:T  ≥  ∣Θ(0)−Θ∗∣ΔHω,T \;\ge\; \frac{|\Theta(0) - \Theta_*|}{\Delta H_\omega},T≥ΔHω​∣Θ(0)−Θ∗​∣​,where Θ=arccos⁡F\Theta = \arccos\sqrt{F}Θ=arccosF​ is the Bures angle and ΔHω\Delta H_\omegaΔHω​ is the energy standard deviation of the \emph{fixed target state}. This bound, valid for both pure and mixed states under unitary dynamics, expresses the speed constraint in terms of a time-independent quantity determined entirely by the Hamiltonian and the detector projector, rather than the evolving-state variance appearing in standard Mandelstam--Tamm-type bounds. We further show that the resulting bound is structurally distinct from standard Mandelstam--Tamm-type bounds based on evolving-state variance. The bound exhibits commuting-sector invariance, suppressing Hamiltonian components that are energetically large but detection-irrelevant. Exact diagonalization of the transverse-field Ising model for N≤12N \le 12N≤12 spins confirms asymptotic separation between the bounds in the strong-interaction regime. The framework provides a detector-conditioned lower bound for rank-1 projective detection dynamics, applicable to superconducting qubit readout, trapped-ion fluorescence detection, and stabilizer syndrome monitoring. The construction is particularly relevant for architectures in which experimentally accessible information is restricted to a small set of projective readout channels rather than full-state tomography. ## Results & Validation Below is the macro-level data extracted from the 156-qubit **ibm_fez** processor, confirming the asymptotic separation predicted by Theorem 5.2 as the many-body coupling strength scales extensively: [Quantum Speed Limit Validation Benchmark](qsl_validation_plot.png) ## Hardware Compilation Profile The transpiled circuit diagram below illustrates the native pulse-level gate routing on the IBM Quantum Heron architecture. By optimizing the physical register map, the compiler enforces a clean nearest-neighbor coupling loop with zero SWAP gate overhead: [Heron Chip Circuit Native Compilation](circuit-d8uhclsbp3hs73861d7g.png) ## Raw Measurement Diagnostics To evaluate the projective fidelity reduction threshold (F∗=0.15F_* = 0.15F∗​=0.15), raw binary distributions are unpacked and corrected using Matrix-Free Measurement Mitigation (M3). Below is a representative diagnostic histogram of single-outcome state frequencies sampled from the hardware backend: [Error Mitigated Hardware Measurement Histogram](job_d8uhc4ctqbtc73d1qdmg_results_c.png)
This work proposes an operational answer to the question of what time is: time is the process by which a physical system produces mutually distinguishable states, a quantity made precise in quantum mechanics through Quantum Fisher Information and the Mandelstam–Tamm bound. Building on this definition, the paper develops a framework in which inertia and gravitation are read as different derivatives of a single underlying quantity — the quantum phase. The framework points to a natural acceleration scale a⋆ = mc³/ℏ, constructible solely from m, c, and ℏ, and shows that for the electron this coincides algebraically with the acceleration in the Schwinger field. It further explores a phase-coherence factor C(a), its kinematic relation to a finite "phase-proper-time" window (via the Gudermann function), a possible Unruh-related spectral signature, and a scalar/vector decomposition of the four-force under an effective-momentum ansatz. The study positions itself explicitly not as a new theory of physics, but as a consistent rereading of standard physics and a research program that formulates the relevant open problems sharply. Throughout the text, results that are algebraically exact are transparently separated from those that hold open-problem / speculative status. Among the algebraically exact results are the four-force rereading, the recovery of Newtonian gravity from flowing space, the Schwinger-scale coincidence, and the kinematic identity for C(a); among the open problems are the physical mechanism behind the Schwinger coincidence, the status of C(a) as a genuine QFI modulation, and the derivation of the internal tick duration at the level of the QFI or the Wightman function. The framework is compatible with the relational-time and quantum-reference-frame programs (Page–Wootters, Rovelli) and shares conceptual ground with the flowing-space / river-model tradition (Painlevé–Gullstrand, Hamilton–Lisle), while preserving Lorentz invariance and treating the natural acceleration scale as a characteristic scale rather than a maximum-acceleration law.