2013/04/30 by Ole Andersson, Andersson, Ole, Hoshang Heydari +1
Computer Science · Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #Quantum optics and atomic interactions #math-ph #math.MP #quant-ph
paper · pdf · doi:10.48550/arxiv.1304.8103
9 pages
arxiv created 2013/04/30 · openalex publication_date 2013/04/30 · arxiv updated 2013/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Geometric effects make evolution time vary for different evolution curves that connect the same two quantum states. Thus, it is important to be able to control along which path a quantum state evolve to achieve maximal speed in quantum calculations. In this paper we establish fundamental relations between Hamiltonian dynamics and Riemannian structures on the phase spaces of unitarily evolving finite-level quantum systems. In particular, we show that the Riemannian distance between two density operators equals the infimum of the energy dispersions of all possible evolution curves connecting the two density operators. This means, essentially, that the evolution time is a controllable quantity. The paper also contains two applied sections. First, we give a geometric derivation of the Mandelstam-Tamm estimate for the evolution time between two distinguishable mixed states. Secondly, we show how to equip the Hamiltonians acting on systems whose states are represented by invertible density operators with control parameters, and we formulate conditions for these that, when met, makes the Hamiltonians transport density operators along geodesics.