The quantum speed limit (QSL) is a fundamental concept in quantum mechanics, providing a lower bound on the evolution time. The attainability of the QSL, which greatly depends on the understanding of the QSL, is a long-standing open problem, especially for high-dimensional systems. In this paper, we solve this problem by establishing a QSL suitable and attainable for both open and closed quantum systems based on a new proposed state distance. It is shown that, given any initial state in a certain finite dimension, unitary and nonunitary dynamics can always be found to attain our QSL bound in the corresponding dimension, and for any given Hamiltonian for a unitary evolution a pair of states always exists, attaining the bound. As applications, we demonstrate the QSL time attained by various physical settings. This paper will shed light on the QSL problems.

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• Open quantum systems & decoherence
• Quantum foundations