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WORLD SCIENTIFIC eBooks

TOWARDS THE FAST SCRAMBLING CONJECTURE

Patrick M. Hayden

Published Oct 1, 2013• 33 citations
Publisher

Abstract

Many proposed quantum mechanical models of black holes include highly nonlocal interactions.The time required for thermalization to occur in such models should reflect the relaxation times associated with classical black holes in general relativity.Moreover, the time required for a particularly strong form of thermalization to occur, sometimes known as scrambling, determines the time scale on which black holes should start to release information.It has been conjectured that black holes scramble in a time logarithmic in their entropy, and that no system in nature can scramble faster.In this article, we address the conjecture from two directions.First, we exhibit two examples of systems that do indeed scramble in logarithmic time: Brownian quantum circuits and the antiferromagnetic Ising model on a sparse random graph.Unfortunately, both fail to be truly ideal fast scramblers for reasons we discuss.Second, we use Lieb-Robinson techniques to prove a logarithmic lower bound on the scrambling time of systems with finite norm terms in their Hamiltonian.The bound holds in spite of any nonlocal structure in the Hamiltonian, which might permit every degree of freedom to interact directly with every other one.

DOI
10.1142/9789814449243_0055

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Many proposed quantum mechanical models of black holes include highly nonlocal interactions.The time required for thermalization to occur in such models should reflect the relaxation times associated with classical black holes in general relativity.Moreover, the time required for a particularly strong form of thermalization to occur, sometimes known as scrambling, determines the time scale on which black holes should start to release information.It has been conjectured that black holes scramble in a time logarithmic in their entropy, and that no system in nature can scramble faster.In this article, we address the conjecture from two directions.First, we exhibit two examples of systems that do indeed scramble in logarithmic time: Brownian quantum circuits and the antiferromagnetic Ising model on a sparse random graph.Unfortunately, both fail to be truly ideal fast scramblers for reasons we discuss.Second, we use Lieb-Robinson techniques to prove a logarithmic lower bound on the scrambling time of systems with finite norm terms in their Hamiltonian.The bound holds in spite of any nonlocal structure in the Hamiltonian, which might permit every degree of freedom to interact directly with every other one.
Given a quantum many-body system with few-body interactions, how rapidly can quantum information be hidden during time evolution? The fast-scrambling conjecture is that the time to thoroughly mix information among N degrees of freedom grows at least logarithmically in N. We derive this inequality for generic quantum systems at infinite temperature, bounding the scrambling time by a finite decay time of local quantum correlations at late times. Using Lieb-Robinson bounds, generalized Sachdev-Ye-Kitaev models, and random unitary circuits, we propose that a logarithmic scrambling time can be achieved in most quantum systems with sparse connectivity. These models also elucidate how quantum chaos is not universally related to scrambling: We construct random few-body circuits with infinite Lyapunov exponent but logarithmic scrambling time. We discuss analogies between quantum models on graphs and quantum black holes and suggest methods to experimentally study scrambling with as many as 100 sparsely connected quantum degrees of freedom.
We prove that the time required for sustained information scrambling in any Hamiltonian quantum system is universally at least logarithmic in the entanglement entropy of scrambled states. This addresses two foundational problems in nonequilibrium quantum dynamics. (1) It sets the earliest possible time for the applicability of equilibrium statistical mechanics in a quantum system coupled to a bath at a finite temperature. (2) It proves a version of the fast scrambling conjecture, originally motivated in models associated with black holes, as a fundamental property of quantum mechanics itself. Our result builds on a refinement of the energy-time uncertainty principle in terms of the infinite temperature spectral form factor in quantum chaos. We generalize this formulation to arbitrary initial states of the bath, including finite temperature states, by mapping Hamiltonian dynamics with any initial state to nonunitary dynamics at infinite temperature. A regularized spectral form factor emerges naturally from this procedure, whose decay is universally constrained by analyticity in complex time. This establishes an exact speed limit on information scrambling by the most general quantum mechanical Hamiltonian without any restrictions on locality or the nature of interactions.
If black holes are not to be dreaded sinks of information but rather fully described by unitary evolution, they must scramble in-falling data and eventually leak it through Hawking radiation. Sekino and Susskind have conjectured that black holes are fast scramblers; they generate entanglement at a remarkably efficient rate, with the characteristic time scaling logarithmically with the entropy. In this work, we focus on Matrix theory---M-theory in the light-cone frame---and directly probe the conjecture. We develop a concrete test bed for quantum gravity using the fermionic variables of Matrix theory and show that the problem becomes that of chains of qubits with an intricate network of interactions. We demonstrate that the black hole system evolves much like a Brownian quantum circuit, with strong indications that it is indeed a fast scrambler. We also analyze the Berenstein-Maldacena-Nastase model and reach the same tentative conclusion.
Random transformations are typically good at "scrambling" information. Specifically, in the quantum setting, scrambling usually refers to the process of mapping most initial pure product states under a unitary transformation to states which are macroscopically entangled, in the sense of being close to completely mixed on most subsystems containing a fraction fn of all n particles for some constant f. While the term scrambling is used in the context of the black hole information paradox, scrambling is related to problems involving decoupling in general, and to the question of how large isolated many-body systems reach local thermal equilibrium under their own unitary dynamics. Here, we study the speed at which various notions of scrambling/decoupling occur in a simplified but natural model of random two-particle interactions: random quantum circuits. For a circuit representing the dynamics generated by a local Hamiltonian, the depth of the circuit corresponds to time. Thus, we consider the depth of these circuits and we are typically interested in what can be done in a depth that is sublinear or even logarithmic in the size of the system. We resolve an outstanding conjecture raised in the context of the black hole information paradox with respect to the depth at which a typical quantum circuit generates an entanglement assisted encoding against the erasure channel. In addition, we prove that typical quantum circuits of poly(log n) depth satisfy a stronger notion of scrambling and can be used to encode alpha n qubits into n qubits so that up to beta n errors can be corrected, for some constants alpha, beta > 0.
We consider the process of diffusion or ``prescrambling'' of information in a quantum system. We define a measure for this spreading or ``prescrambling'' of the wave function in terms of a minimum probability threshold for the states in the system's Hilbert space. We illustrate our findings on the example of a prototype model with enhanced memory capacity. We conjecture: (1) The fastest prescramblers require a time logarithmic in the number of degrees of freedom. (2) The investigated enhanced memory capacity model is a fast prescrambler. (3) (Fast) prescrambling occurs not later than (fast) scrambling. (4) Fast scramblers are fast prescramblers. (5) Black holes are fast prescramblers.
Measurement-induced phase transitions arise due to a competition between the scrambling of quantum information in a many-body system and local measurements. In this work we investigate these transitions in different classes of fast scramblers, systems that scramble quantum information as quickly as is conjectured to be possible---on a timescale proportional to the logarithm of the system size. In particular, we consider sets of deterministic sparse couplings that naturally interpolate between local circuits that slowly scramble information and highly nonlocal circuits that achieve the fast-scrambling limit. We find that circuits featuring sparse nonlocal interactions are able to withstand substantially higher rates of local measurement than circuits with only local interactions, even at comparable gate depths. We also study the quantum error-correcting codes that support the volume-law entangled phase and find that our maximally nonlocal circuits yield codes with nearly extensive contiguous code distance. Use of these sparse, deterministic circuits opens pathways towards the design of noise-resilient quantum circuits and error correcting codes in current and future quantum devices with minimum gate numbers.