Time-optimal quantum dynamics under fixed-norm generators

Abstract

We investigate the dynamics of quantum systems whose evolution is governed by linear generators of fixed unitarily invariant norm. For this broad class of evolutions, we derive a universal time-optimal generator of the dynamics converting one given state into another. It provides a time-optimal non-Hermitian Hamiltonian for pure state conversion. We then use this general result to derive quantum speed limits for mixed-state evolutions governed by Lindbladians and non-Hermitian Hamiltonians. Our findings follow from the geometric properties of the group $GL(n,\mathbb{C})$, in particular the geodesic distance. The derived bounds hold for an arbitrary system size $n$, depend only on the norms of the evolution generator and the initial and final states, and do not require performing any time-averaging.

The paper asks how quickly prescribed quantum states can be connected when the instantaneous generator has a fixed, normalized unitarily invariant norm. It treats unrestricted non-Hermitian evolution of pure-state vectors exactly and derives lower bounds for Lindblad and non-Hermitian mixed-state dynamics. For nonzero, possibly unnormalized vectors, the optimal conversion time is $T_*=\sqrt{\lambda^2+\theta^2}$, where $\lambda=\log(\|\psi_f\|/\|\psi_i\|)$ and $\theta=\arccos[\operatorname{Re}\langle\psi_i|\psi_f\rangle/(\|\psi_i\|\|\psi_f\|)]$. An explicit time-dependent rank-one generator attains this time for every allowed norm. For normalized states with irrelevant final phase, $T_*=\arccos|\langle\psi_i|\psi_f\rangle|$. At fixed Schatten-$p$ norm, this is faster than optimal Hermitian evolution by a factor $2^{1/p}$, including no advantage for the operator norm. The derivation uses a quotient geometry induced from a right-invariant metric on $GL(n,\mathbb C)$. The resulting state-space metric is a product of logarithmic radius and spherical angle. Applying it to vectorized density matrices gives $T_{\mathrm{Lind}}\geq\sqrt{\lambda_\rho^2+\theta_\rho^2}$, with $\lambda_\rho=\log(\|\rho_f\|_{HS}/\|\rho_i\|_{HS})$ and $\theta_\rho=\arccos[\operatorname{Tr}(\rho_i\rho_f)/(\|\rho_i\|_{HS}\|\rho_f\|_{HS})]$. Bounding the Hamiltonian norm instead gives half this lower bound for non-Hermitian mixed-state evolution. The mixed-state bounds generally are not tight because the geometric optimization permits nonphysical maps and states. Simulations show deteriorating Lindblad-bound tightness with increasing dimension; exact Lindblad optima remain open. Non-Hermitian evolution also cannot connect states of different rank.