Quantum speed limits for open system dynamics based on a representation-basis-dependent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ℓ</mml:mi> <mml:mi>w</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> </mml:math> -seminorm

We report a family of quantum speed limits (QSLs) that give evolution time lower bounds between an initial and a final state whose separation is described by a certain representation-basis-dependent norm derived from the weighted ℓ w p -seminorm. These QSLs are applicable to open, closed, time-dependent, or time-independent systems in finite-dimensional Hilbert spaces whose density matrices are piecewise time differentiable. They can be extended to systems over separable Hilbert spaces as well. Crucially, these QSLs are valid for arbitrary operators, not just density matrices, provided that a modest technical condition is fulfilled. When compared to the existing QSLs applied to pure state time-independent Hamiltonian evolution, qubit spontaneous emission, high-fidelity gate implementation, coherent state photon loss, and operator coherence or dephasing, ours consistently show improved sharpness in most cases, along with greater universality and still retaining computational efficiency.