We study the relation between almost-symmetries and the geometry of Banach spaces. We show that any almost-linear extension of a transformation that preserves transition probabilities up to an additive error admits an approximation by a linear map, and the quality of the approximation depends on the type and cotype constants of the involved spaces.
We study the relation between almost-symmetries and the geometry of Banach spaces. We show that any almost-linear extension of a transformation that preserves transition probabilities up to an additive error admits an approximation by a linear map, and the quality of the approximation depends on the type and cotype constants of the involved spaces.
It d-pends. Wigner’s symmetry theorem implies that transformations that preserve transition probabilities of pure quantum states are linear maps on the level of density operators. We investigate the stability of this implication. On the one hand, we show that any transformation that preserves transition probabilities up to an additive ε in a separable Hilbert space admits a weak linear approximation, i.e., one relative to any fixed observable. This implies the existence of a linear approximation that is 4εd–close in Hilbert-Schmidt norm, with d the Hilbert space dimension. On the other hand, we prove that a linear approximation that is close in norm and independent of d does not exist in general. To this end, we provide a lower bound that depends logarithmically on d.
We prove dimension-independent Hyers--Ulam stability for surjective approximate isometries of the positive trace-class cone in both the trace and Bures metrics. Here ε≥0 is a uniform upper bound on the additive error in preserving distances. Such an approximate isometry is uniformly approximated by a Wigner symmetry (a unitary or antiunitary conjugation), with error at most 9ε/2 in the trace metric and 22ε in the Bures metric, even if it does not fix zero. For maps fixing zero, the trace bound improves to 3ε even under the hypothesis of δ-surjectivity, independently of δ. In infinite dimensions, we construct spectrum-preserving homeomorphisms of the full state space, including mixed states, whose distortions tend to zero in both metrics but which stay a fixed uniform distance from every Wigner symmetry. For state spaces, stability holds in each fixed finite dimension with a dimension-dependent modulus tending to zero as ε→0, but no such modulus can be chosen independently of dimension.