Are almostsymmetries almost linear?
Abstract
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 $\varepsilon$ 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 $4\sqrt{\varepsilon} d$close in HilbertSchmidt 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 $d$.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 August 2019
 DOI:
 10.1063/1.5087539
 arXiv:
 arXiv:1812.10019
 Bibcode:
 2019JMP....60h2101C
 Keywords:

 Mathematical Physics;
 Quantum Physics
 EPrint:
 8 pages, 1 figure