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| Fig. 1: Example of random unitary evolution on a 1D spin chain. At each time step, a random unitary operator is applied to neighbouring spins, which will generically increase their entanglement. (Source: C. Gustin) |
Entanglement is a unique feature of quantum mechanics, with no analogue in classical physics. Initially poorly understood, these quantum correlations that develop between different parts of a composite quantum system are fundamental to the underlying quantum evolution, and are crucial to the majority of the distinctive properties of quantum mechanics.
Entanglement is a critical concept in many different fields of physics - in quantum optics, entanglement between a system (i.e. an atom and the electromagnetic field) leads to decoherence, crucial to the understanding of spontaneous emission. In quantum information theory, it is entanglement which is the key ingredient in quantum computing which allows for exponential advantage in computing speed over classical computing for certain algorithms. In high energy physics the entanglement entropy is important to the study of the black hole information paradox, and in condensed matter physics, entanglement plays a key role in the process of thermalization, which bridges a closed-system quantum mechanical approach to thermal equilibrium with the classical one utilizing thermodynamic ensembles. In both quantum and classical mechanics, a fully isolated system will evolve fully deterministically. However, for many systems, in the thermodynamic limit (large system size, long time elapsed since initial condition), the system is assumed to evolve into a thermal state at a certain temperature in which its observables can be calculated probabilistically from a thermodynamic ensemble, with no memory of the initial conditions of the system. In classical mechanics, this apparent discrepancy can be resolved with chaos theory, assuming the system to have after a large enough time explored its phase space ergodically such that initial condition of the system becomes irrelevant. In the language of quantum mechanics, the information encoded in the initial state is remembered by the system at long times, but becomes spread out in a nonlocal fashion as entanglement is generated across the many degrees of freedom of the system. Thus the initial state determines the properties of the thermalized state (temperature), but memory of the initial observables is hidden as it can no longer be accessed by physical local measurements.
One interested question in condensed matter physics is how the entanglement in a system is dynamically generated under interactions. If a system of composite subsystems (i.e., a 1D array of spins) is initialized in a separable product state and allowed to undergo general unitary evolution in the presence of interactions, it will generally evolve from a state with zero entanglement to an entangled state. A natural question to ask then is what is the rate at which this entanglement will grow. Furthermore, how can one classify systems based on the characteristics of their entanglement growth?
In this paper, we discuss the results of two papers which interrogate and partially answer these questions. [1,2] Both of these papers discuss entanglement growth in a 1D chain of quantum subsystems, starting from an initially unentangled state.
Quantum systems which thermalize do so on the basis of their eigenstates of the system Hamiltonian, and these will typically saturate to a state which has entanglement entropy scaling extensively with the volume of the system. However, there also exist systems which do not thermalize on the basis of disorder, instead forming many-body localized states which retain a memory of their initial conditions in local observables and have entanglement entropies that scale with the area of the system in the steady state. These states are known to exhibit logarithmic growth of entanglement entropy over time.
In quantum systems which do thermalize, those which are also integrable (have extensively many conserved constants of motion), were known prior to the work of Kim and Huse to have ballistic entropy growth. [1] In these systems, this growth of entropy can be understood from a picture of quasiparticles in the system which propagate ballistically. The work Kim and Huse deals with the growth of entanglement in a non-integrable, non-localizing system. [1] The authors exactly diagonalize a spin 1/2 chain and find ballistic entanglement growth when averaged over random initially unentangled states. They use a model which exhibits diffusive energy transfer, and thus they show that in that model entanglement spreads faster than energy (and they conjecture, all local observables).
Nahum et al. study instead a model of a random unitary quantum circuits, in which time is discretized and random unitary evolutions are applied to neighbouring "spins" at each time step. [2] This allows the results to be, in principle, more generic than using any specific Hamiltonian. In this work, the authors again find ballistic entanglement growth, and map this evolution to the Kardar-Parisi-Zhang (KPZ) stochastic differential equation used in three different models in classical mechanics, giving a wide variety of perspectives to view the features of entanglement growth. One of their key findings is that the speed of entanglement growth is generically slower than the speed at which an initially local operator becomes more non-local under Heisenberg evolution, which they summarize as thermalization is slower than operator spreading.
© Chris Gustin. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] H. Kim and D. A. Huse, "Ballistic Spreading of Entanglement in a Diffusive Nonintegrable System," Phys. Rev. Lett. 111, 127205 (2013).
[2] A. Nahum et al., "Quantum Entanglement Growth under Random Unitary Dynamics," Phys. Rev. X 7, 031016 (2017).