Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and Theoretical

Harmonic rigidity at fixed spectral gap in one dimension

Arseny Pantsialei

Published Jan 14, 2026• 0 citations
Publisher

Abstract

Abstract We resolve the static isoperimetric problem underlying the Mandelstam–Tamm limit: among one-dimensional confining potentials at fixed gap Δ, the harmonic trap uniquely maximizes Var 0 ( x ) , yielding the exact geometric quantum speed limit g x x ⩽ ( 2 m Δ ) − 1 with an iff criterion. Beyond the extremum we prove quantitative rigidity (Thomas–Reiche–Kuhn-tail and structural L 2 control), extend to magnetic settings (longitudinal iff; transverse guiding-center bounds), and note applications to static polarizability, quantum-metric limits, and trap benchmarking.

DOI
10.1088/1751-8121/ae340a

Related papers

Harmonic rigidity at fixed spectral gap in one dimension

Arseny Pantsialei

We solve the static isoperimetric problem underlying the Mandelstam-Tamm bound. Among one-dimensional confining potentials with a fixed spectral gap, we prove that the harmonic trap is the unique maximizer of the ground-state position variance. As a consequence, we obtain a sharp geometric quantum speed-limit bound on the position-position component of the quantum metric, and we give a necessary-and-sufficient condition for when the bound is saturated. Beyond the exact extremum, we establish quantitative rigidity. We control the Thomas-Reiche-Kuhn spectral tail and provide square-integrable structural stability for potentials that nearly saturate the bound. We further extend the analysis to magnetic settings, deriving a longitudinal necessary-and-sufficient characterization and transverse bounds expressed in terms of guiding-center structure. Finally, we outline applications to bounds on static polarizability, limits on the quantum metric, and benchmarking of trapping potentials.

quant-pharXiv:2512.24790•Jan 1, 2026•0 citations

Center-of-Mass Bounds and Harmonic Extremality

Arseny Pantsialei

We study the center-of-mass observable in one-dimensional many-body systems with translation-invariant interactions and extend the harmonic-rigidity mechanism from the one-body setting to an interacting many-body problem. We prove a sharp upper bound on the ground-state center-of-mass fluctuation in terms of the active spectral gap associated with the center-of-mass probe, and show that this bound does not require any positivity assumption on the ground state. In the positivity class, we characterize the equality case completely. Exact saturation occurs if and only if the external one-body traps are harmonic with a common frequency, while the interaction may remain arbitrary within the translation-invariant class. We also identify a natural rigidity defect measuring deviation from the harmonic extremal situation and prove quantitative near-saturation estimates controlling both the variance deficit and the spectral weight outside the first active shell. In this way, the paper establishes harmonic confinement as the unique static extremizer for the rigid interacting center-of-mass mode at fixed active gap.

quant-pharXiv:2607.08096•Journal of Physics A Mathematical and Theoretical•Jul 10, 2026•0 citations

Unified fidelity-based framework for quantum speed limits in open quantum systems

Tristán M. Osán, Yanet Álvarez, M. Portesi +1 more

In this work, we investigate the role of functionals of generalized fidelity measures in deriving quantum speed limits (QSLs) within a geometric approach. We establish a general theoretical framework and show that, once a specific generalized fidelity is selected, the resulting Margolus-Levitin and Mandelstam-Tamm QSLs for both unitary and nonunitary (Lindblad-type) dynamics depend solely on the chosen fidelity measure. We prove that any monotone, differentiable reparametrization of the chosen fidelity yields exactly the same Margolus-Levitin and Mandelstam-Tamm-type QSL, rendering the QSLs invariant under such transformations and thereby unifying fidelity- and metric-induced geometric formulations. This result highlights the limitations of improving QSLs through functional transformations of fidelity and indicates that genuine improvements must arise from alternative fidelity definitions. We further show that several QSLs reported in the literature are encompassed by our framework and discuss possible extensions based on other generalized fidelity measures beyond those explicitly analyzed. In addition, we consider the damped Jaynes-Cummings model as a concrete physical setting to explore the behavior of the QSLs discussed in this work and analyze their regime-dependent features.

arXiv:2509.04283•Physical Review A•Jan 26, 2026•0 citations

Commutator Obstruction to Quantum Advantage: A Spectral Confinement Theorem for Quantum Heuristics

A. Kim

We establish a necessary condition for quantum heuristics to achieve nontrivial computational advantage relative to a classical problem generator A. We prove that any unitary procedure whose dynamics nearly commute with A is spectrally confined: its ability to transfer amplitude across separated spectral sectors is bounded strictly by the commutator defect divided by the spectral gap. This yields a quantitative operator-theoretic obstruction to global state-space exploration. Consequently, repeated application of such procedures (e.g., in deep variational circuits) produces apparent algorithmic activity without generating meaningful information gain. We classify this regime as a "degenerate quantum heuristic". We apply this framework to Variational Quantum Eigensolvers (VQE) and the Quantum Approximate Optimization Algorithm (QAOA), demonstrating that circuit depth cannot overcome transversality failure. The theoretical results are fully closed within trace-ideal (S₂) operator theory. We extend the spectral confinement bound to the entire resolvent geometry, proving that degenerate heuristics preserve the Birman-Schwinger kernel and regularized Fredholm determinants (det₂), thereby mathematically obstructing any computational phase transition or new eigenvalue emergence. Finally, we demonstrate that genuine quantum speedup requires structural noncommutativity, which can be deterministically enforced via an Ω-Σ dissipative control layer. The manuscript includes complete finite-dimensional numerical validation, exhibiting the exact quantitative rigidity between commutator defect, spectral transport, and resolvent deformation.

•Apr 8, 2026•0 citations

Probing quantum phase transitions via quantum speed limits

Manish Kumar Suman, S. Aravinda, Ranjan Modak

The quantum speed limit (QSL) is the lower bound on the time required for a state to evolve to a desired final state under a given Hamiltonian evolution. Three well-known QSLs exist: the Mandelstam-Tamm (MT), Margolus-Levitin (ML), and dual ML (ML*) bounds. We consider one-dimensional systems that undergo a delocalization-localization transition in the presence of quasiperiodic and linear potential. By performing sudden quenches across the phase boundary, we find that the exact dynamics gets captured very well by QSLs. We show that the MT bound is always tighter in the short-time limit for any arbitrary state, while the optimal bound for the time of orthogonalization (time required to reach the orthogonal state) depends on the choice of the initial state. Further, for extreme quenches, we prove that the MT bound remains tighter for the time of orthogonalization and it can qualitatively describe the nonanalyticity in free energy for the dynamical quantum phase transition. We also demonstrate that the localization-delocalization transition point can be exactly identified from QSLs for two of the models of noninteracting fermions. Further, we also show that even for interacting systems, an ergodic to many-body localization transition can also be efficiently predicted from QSLs, whose computation cost is much less than other diagnostic tools.

arXiv:2311.18579•Physical Review A•Jul 31, 2024•3 citations

Optimal bounds on the speed of subspace evolution*

Sergio A Albeverio, Alexander Konstantinovich Motovilov

Abstract By a quantum speed limit one usually understands an estimate on how fast a quantum system can evolve between two distinguishable states. The most known quantum speed limit is given in the form of the celebrated Mandelstam–Tamm inequality that bounds the speed of the evolution of a state in terms of its energy dispersion. In contrast to the basic Mandelstam–Tamm inequality, we are concerned not with a single state but with a (possibly infinite-dimensional) subspace which is subject to the Schrödinger evolution. By using the concept of maximal angle between subspaces we derive optimal bounds on the speed of such a subspace evolution. These bounds may be viewed as further generalizations of the Mandelstam–Tamm inequality. Our study includes the case of unbounded Hamiltonians.

arXiv:2111.05677•Journal of Physics A Mathematical and Theoretical•Apr 29, 2022•1 citation
Abstract We resolve the static isoperimetric problem underlying the Mandelstam–Tamm limit: among one-dimensional confining potentials at fixed gap Δ, the harmonic trap uniquely maximizes Var 0 ( x ) , yielding the exact geometric quantum speed limit g x x ⩽ ( 2 m Δ ) − 1 with an iff criterion. Beyond the extremum we prove quantitative rigidity (Thomas–Reiche–Kuhn-tail and structural L 2 control), extend to magnetic settings (longitudinal iff; transverse guiding-center bounds), and note applications to static polarizability, quantum-metric limits, and trap benchmarking.
We solve the static isoperimetric problem underlying the Mandelstam-Tamm bound. Among one-dimensional confining potentials with a fixed spectral gap, we prove that the harmonic trap is the unique maximizer of the ground-state position variance. As a consequence, we obtain a sharp geometric quantum speed-limit bound on the position-position component of the quantum metric, and we give a necessary-and-sufficient condition for when the bound is saturated. Beyond the exact extremum, we establish quantitative rigidity. We control the Thomas-Reiche-Kuhn spectral tail and provide square-integrable structural stability for potentials that nearly saturate the bound. We further extend the analysis to magnetic settings, deriving a longitudinal necessary-and-sufficient characterization and transverse bounds expressed in terms of guiding-center structure. Finally, we outline applications to bounds on static polarizability, limits on the quantum metric, and benchmarking of trapping potentials.
We study the center-of-mass observable in one-dimensional many-body systems with translation-invariant interactions and extend the harmonic-rigidity mechanism from the one-body setting to an interacting many-body problem. We prove a sharp upper bound on the ground-state center-of-mass fluctuation in terms of the active spectral gap associated with the center-of-mass probe, and show that this bound does not require any positivity assumption on the ground state. In the positivity class, we characterize the equality case completely. Exact saturation occurs if and only if the external one-body traps are harmonic with a common frequency, while the interaction may remain arbitrary within the translation-invariant class. We also identify a natural rigidity defect measuring deviation from the harmonic extremal situation and prove quantitative near-saturation estimates controlling both the variance deficit and the spectral weight outside the first active shell. In this way, the paper establishes harmonic confinement as the unique static extremizer for the rigid interacting center-of-mass mode at fixed active gap.
In this work, we investigate the role of functionals of generalized fidelity measures in deriving quantum speed limits (QSLs) within a geometric approach. We establish a general theoretical framework and show that, once a specific generalized fidelity is selected, the resulting Margolus-Levitin and Mandelstam-Tamm QSLs for both unitary and nonunitary (Lindblad-type) dynamics depend solely on the chosen fidelity measure. We prove that any monotone, differentiable reparametrization of the chosen fidelity yields exactly the same Margolus-Levitin and Mandelstam-Tamm-type QSL, rendering the QSLs invariant under such transformations and thereby unifying fidelity- and metric-induced geometric formulations. This result highlights the limitations of improving QSLs through functional transformations of fidelity and indicates that genuine improvements must arise from alternative fidelity definitions. We further show that several QSLs reported in the literature are encompassed by our framework and discuss possible extensions based on other generalized fidelity measures beyond those explicitly analyzed. In addition, we consider the damped Jaynes-Cummings model as a concrete physical setting to explore the behavior of the QSLs discussed in this work and analyze their regime-dependent features.
We establish a necessary condition for quantum heuristics to achieve nontrivial computational advantage relative to a classical problem generator A. We prove that any unitary procedure whose dynamics nearly commute with A is spectrally confined: its ability to transfer amplitude across separated spectral sectors is bounded strictly by the commutator defect divided by the spectral gap. This yields a quantitative operator-theoretic obstruction to global state-space exploration. Consequently, repeated application of such procedures (e.g., in deep variational circuits) produces apparent algorithmic activity without generating meaningful information gain. We classify this regime as a "degenerate quantum heuristic". We apply this framework to Variational Quantum Eigensolvers (VQE) and the Quantum Approximate Optimization Algorithm (QAOA), demonstrating that circuit depth cannot overcome transversality failure. The theoretical results are fully closed within trace-ideal (S₂) operator theory. We extend the spectral confinement bound to the entire resolvent geometry, proving that degenerate heuristics preserve the Birman-Schwinger kernel and regularized Fredholm determinants (det₂), thereby mathematically obstructing any computational phase transition or new eigenvalue emergence. Finally, we demonstrate that genuine quantum speedup requires structural noncommutativity, which can be deterministically enforced via an Ω-Σ dissipative control layer. The manuscript includes complete finite-dimensional numerical validation, exhibiting the exact quantitative rigidity between commutator defect, spectral transport, and resolvent deformation.
The quantum speed limit (QSL) is the lower bound on the time required for a state to evolve to a desired final state under a given Hamiltonian evolution. Three well-known QSLs exist: the Mandelstam-Tamm (MT), Margolus-Levitin (ML), and dual ML (ML*) bounds. We consider one-dimensional systems that undergo a delocalization-localization transition in the presence of quasiperiodic and linear potential. By performing sudden quenches across the phase boundary, we find that the exact dynamics gets captured very well by QSLs. We show that the MT bound is always tighter in the short-time limit for any arbitrary state, while the optimal bound for the time of orthogonalization (time required to reach the orthogonal state) depends on the choice of the initial state. Further, for extreme quenches, we prove that the MT bound remains tighter for the time of orthogonalization and it can qualitatively describe the nonanalyticity in free energy for the dynamical quantum phase transition. We also demonstrate that the localization-delocalization transition point can be exactly identified from QSLs for two of the models of noninteracting fermions. Further, we also show that even for interacting systems, an ergodic to many-body localization transition can also be efficiently predicted from QSLs, whose computation cost is much less than other diagnostic tools.
Abstract By a quantum speed limit one usually understands an estimate on how fast a quantum system can evolve between two distinguishable states. The most known quantum speed limit is given in the form of the celebrated Mandelstam–Tamm inequality that bounds the speed of the evolution of a state in terms of its energy dispersion. In contrast to the basic Mandelstam–Tamm inequality, we are concerned not with a single state but with a (possibly infinite-dimensional) subspace which is subject to the Schrödinger evolution. By using the concept of maximal angle between subspaces we derive optimal bounds on the speed of such a subspace evolution. These bounds may be viewed as further generalizations of the Mandelstam–Tamm inequality. Our study includes the case of unbounded Hamiltonians.