Finite Speed of Quantum Scrambling with Long Range Interactions

Abstract

In a locally interacting many-body system, two isolated qubits, separated by a large distance r, become correlated and entangled with each other at a time t ≥ r/v [1]. This finite speed v of quantum information scrambling limits quantum information processing [2], thermalization [3] and even equilibrium correlations [4]. Yet most experimental systems contain long range power law interactions -qubits separated by r have potential energy V (r) ∝ r −α . Examples include the long range Coulomb interactions in plasma (α = 1) and dipolar interactions between spins (α = 3). In one spatial dimension, we prove that the speed of quantum scrambling remains finite for sufficiently large α. This result parametrically improves previous bounds [4][5][6][7], compares favorably with recent numerical simulations [8,9], and can be realized in quantum simulators with dipolar interactions [10,11]. Our new mathematical methods lead to improved algorithms for classically simulating quantum systems [6,12], and improve bounds on environmental decoherence in experimental quantum information processors.Almost five decades ago, Lieb and Robinson proved that spatial locality implies the ballistic propagation of quantum information [1]. Intuitively defining a "scrambling time" t s (r) by the time at which an initially isolated qubit can significantly entangle with another a distance r away, locality implies that t s (r)r. This result has deep implications in theoretical physics: emergent spacetime locality arising from microscopic quantum mechanics without manifest relativistic invariance may play a crucial role in understanding quantum gravity through the holographic correspondence [13]. Moreover, if quantum information can only propagate with a finite speed, a classical computer can efficiently approximate early time quantum dynamics [12], and a quantum information processor with short-range interactions cannot become entangled with an infinite environment arbitrarily quickly [14,15], despite the exponentially large Hilbert space in many-body quantum systems.While the Lieb-Robinson theorem is quite elegant, it is not useful for a typical quantum information processor. A qubit in an experimental device is usually a spin or atomic degree of freedom, or Josephson junction. Such objects generically interact with long range interactions, and until now, whether locality of quantum scrambling necessarily persists in the presence of long range interactions has remained unclear. In 2005, Hastings and Koma used the canonical Lieb-Robinson theorem to prove that when α > d, t s (r) log r [4]; more recently, this bound has been improved for α > 2d to t s (r) r (α−2d)/(α−d) [6]. If such bounds were tight, then insulating a quantum processor from its environment would be absolutely crucial. Yet numerical simulations cast into doubt the tightness of these formal bounds: two groups have recently shown that t s r in one dimensional models with α 1.8 [8] or even α > 1 [9], depending on microscopic details.In this letter, we prove that t s (r) r whenever α > 3...

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