Bounds in Nonequilibrium Quantum Dynamics

Abstract

We review various bounds concerning out-of-equilibrium dynamics in few-level and manybody quantum systems. We primarily focus on closed quantum systems but will also mention some related results for open quantum systems and classical stochastic systems. We start from the speed limits, the universal bounds on the speeds of (either quantum or classical) dynamical evolutions. We then turn to review the bounds that address how good and how long would a quantum system equilibrate or thermalize. Afterward, we focus on the stringent constraint set by locality in many-body systems, rigorously formalized as the Lieb-Robinson bound. We also review the bounds related to the dynamics of entanglement, a genuine quantum property. Apart from some other miscellaneous topics, several notable error bounds for approximated quantum dynamics are discussed. While far from comprehensive, this topical review covers a considerable amount of recent progress and thus could hopefully serve as a convenient starting point and up-to-date guidance for interested readers.

Speed limits of the trace distance for open quantum system

“…We discuss a system of which Hilbert space is twodimensional. In this case, the state of the system can be written as ρ(t) = 1 2 (1 + r(t) • τ ). Here, τ = (τ x , τ y , τ z ), τ i is the Pauli matrix, and r(t) is the Bloch vector.…”

“…In recent years, studies of time-dependent open systems have been active [1]. These studies relate to quantum pumps [2,3], excess entropy production [4][5][6], the efficiency and power of heat engines [7][8][9][10], shortcuts to adiabaticity [11][12][13], and speed limits [14][15][16][17][18].…”

“…Obtaining a fundamental bound on the speed of state transformation is an important issue relevant to broad research fields including quantum control theory [19] and foundations of nonequilibrium statistical mechanics [20]. Speed limits for time-dependent closed quantum systems have been studied more than half century [1]. Since 1945, the Mandelstam-Tamm relation [21] L/ τ 0 dt ∆E ≤ 1 has been known (In this letter, we set = 1).…”

Quantum adiabaticity in many-body systems and almost-orthogonality in complementary subspace

“…An inequality of quantum speed limit of the Mandelstam-Tamm type [38][39][40][41][42][43][44] sets an upper bound on the Bures angle θ(λ) (5),…”

“…[37] is that the adiabatic fidelity can be estimated by * jhchen@lorentz.leidenuniv.nl exploiting a many-body nature of the problemgeneralized orthogonality catastrophe (GOC), and a fundamental inequality for the evolution of unitary dynamics in Hilbert spacesquantum speed limit (QSL). The GOC refers to the property that the fidelity between instantaneous eigenstates and the initial state decays exponentially as the system size and the value of the evolution parameter increase [36], whereas the QSL sets an intrinsic limit on how fast the time-evolved state can deviate from the initial state [38][39][40][41][42][43][44].…”