The Margolus-Levitin lower bound on minimal time required for a state to be transformed into an orthogonal state is generalized. It is shown that for some initial states new bound is stronger than the Margolus-Levitin one.Comment: 6 pages, no figures; some comments added; final version accepted for publication in Phys. Rev.

A comodule algebra P over a Hopf algebra H with bijective antipode is called principal if the coaction of H is Galois and P is H -equivariantly projective (faithfully flat) over the coaction-invariant subalgebra P^{\mathrm{co}H} . We prove that principality is a piecewise property: given N comodule-algebra surjections P \rightarrow P_i whose kernels intersect to zero, P is principal if and only if all P_i 's are principal. Furthermore, assuming the principality of P , we show that the lattice these kernels generate is distributive if and only if so is the lattice obtained by intersection with P^{\mathrm{co}H} . Finally, assuming the above distributivity property, we obtain a flabby sheaf of principal comodule algebras over a certain space that is universal for all such N -families of surjections P \rightarrow P_i and such that the comodule algebra of global sections is P .

From N -tensor powers of the Toeplitz algebra, we construct a multi-pullback C*-algebra that is a noncommutative deformation of the complex projective space \mathbb{P}^N\!(\mathbb{C}) . Using Birkhoff’s Representation Theorem, we prove that the lattice of kernels of the canonical projections on components of the multi-pullback C*-algebra is free. This shows that our deformation preserves the freeness of the lattice of subsets generated by the affine covering of the complex projective space.