Quantum

Spectral measures and sharpness of quantum speed limits

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Briefings

A 2026 arXiv preprint establishes that quantum speed limits are defined by spectral measures composed of two Dirac measures under specific energy constraints. The authors also demonstrate that these limits can be approached arbitrarily closely by measures that are purely absolutely continuous or purely singular continuous with respect to the Lebesgue measure, providing a geometric interpretation via the numerical range of the evolution operator.

Key points

• The unique spectral measure associated with the initial vector that saturates the quantum speed limit is a sum of two Dirac measures.
• Quantum speed limits can be approached arbitrarily closely by spectral measures that are purely absolutely continuous or purely singular continuous with respect to the Lebesgue measure.

Context

This research addresses the theoretical bounds on the time required for a quantum state to evolve into a specific subspace, constrained by energy levels. The findings clarify the mathematical structure of these limits and their relationship to continuous spectral measures.

Limitations

The evidence is an arXiv preprint from November 2026, indicating it is not yet peer-reviewed. The study focuses on specific energy constraints and standard orthogonalization time bounds, limiting the scope to these particular conditions.

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