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The Second Law of Quantum Complexity

Adam R. Brown, Leonard Susskind

arXiv:1701.01107v2hep-thgr-qcquant-ph

TL;DR

The paper asks how quantum complexity evolves after a system reaches maximum entropy but before it reaches maximum complexity. It models complexity through a classical auxiliary system, finding entropy-like evolution and interpreting uncomplexity as accessible spacetime volume behind a black-hole horizon.

  • Problem

    The paper examines what happens to quantum systems between reaching maximum entropy and reaching the much later maximum complexity.

    Source: §1 · p. 4

  • Method

    The authors represent quantum evolution with a classical auxiliary system and relate average complexity to classical entropy, including Kolmogorov complexity in the correspondence.

    Source: §6.4 · p. 35, §10 · p. 59

  • Results

    Complexity evolution for k-local systems of K qubits closely resembles classical entropy evolution for systems with exp[K] degrees of freedom.

    Source: §10 · p. 59

  • Takeaways & Limitations

    For a boundary state with a black hole, uncomplexity corresponds to the spacetime volume accessible to an observer planning to cross the horizon.

    Source: §10 · p. 60

    The proposed complexity-growth and recurrence pattern is conjectural and cannot currently be proved.

    Source: §1.1 · p. 6

Abstract

from arXiv · show

We give arguments for the existence of a thermodynamics of quantum complexity that includes a "Second Law of Complexity". To guide us, we derive a correspondence between the computational (circuit) complexity of a quantum system of $K$ qubits, and the positional entropy of a related classical system with $2^K$ degrees of freedom. We also argue that the kinetic entropy of the classical system is equivalent to the Kolmogorov complexity of the quantum Hamiltonian. We observe that the expected pattern of growth of the complexity of the quantum system parallels the growth of entropy of the classical system. We argue that the property of having less-than-maximal complexity (uncomplexity) is a resource that can be expended to perform directed quantum computation. Although this paper is not primarily about black holes, we find a surprising interpretation of the uncomplexity-resource as the accessible volume of spacetime behind a black hole horizon.

1 Quantum Complexity and Classical Entropy

The paper asks whether thermodynamics can explain the evolution of quantum complexity and argues that the Second Law of Complexity follows from thermodynamics of an auxiliary classical system. It conjectures complexity growth that resembles classical entropy growth, while proposing uncomplexity as a resource for directed computation.

1 Quantum Complexity and Classical Entropy

  • The paper focuses on quantitative complexity evolution and asks whether physics, especially statistical mechanics and thermodynamics, can help analyze it.The authors distinguish this focus from the mainstream goal of organizing tasks into qualitative complexity classes.Source: §1 · p. 4
  • The paper argues that the Second Law of Complexity for a quantum system follows from the thermodynamic second law for an auxiliary classical system.The motivating question concerns quantum systems after they reach maximum entropy but before they reach maximum complexity.Source: §1 · p. 4

1.1 The Evolution of Complexity

  • Complexity is conjectured to grow linearly at rate K, reach an exponential maximum, fluctuate there, and recur to sub-exponential values on a doubly exponential timescale.The recurrence pattern is explicitly presented as a conjecture that cannot currently be proved.Source: §1.1 · pp. 5–6
  • Quantum complexity for K qubits behaves similarly to the entropy of a classical system with 2^K degrees of freedom, though their growth and recurrence times differ.Classical entropy grows linearly for only a polynomial time, reaches a maximum of order the degrees of freedom, and recurs on an exponential timescale.Source: §1.1 · p. 6
  • Uncomplexity—the gap between a system’s complexity and its maximum—is proposed as a resource that can be expended for directed quantum computation.The authors compare its role to free energy in conventional thermodynamics, under appropriate conditions.Source: §1.1 · p. 6

2 The Quantum System Q

The paper considers generic k-local quantum systems, using ensemble averaging to study their behavior and a classical auxiliary system to represent their evolution. Its focus is high-temperature behavior, where the authors expect standard-qubit and SYK models to behave somewhat similarly.

2.2 k-locality

  • A k-local Hamiltonian contains terms acting on at most k qubits, and generic couplings are fast-scrambling rather than spatially local.Nearest-neighbor lattice Hamiltonians are 2-local, while the generic k-local case couples every qubit to every other through terms involving at most k qubits.Source: §2.2 · pp. 7, 9
  • The comparison between SYK and standard-qubit models is scoped to high temperature; at low temperature, qubit models can differ and may exhibit spin-glass behavior.The authors expect both model types to behave somewhat similarly to uncharged Schwarzschild black holes at high temperature.Source: §2.2 · p. 8

2.3 The Quantum System

  • The quantum system is a K-qubit system with k-local Hamiltonians whose couplings are ensemble-averaged; the coupling variance is normalized using the infinite-temperature TFD state.The variance of the dimensionless energy in that state is tied to its entropy, which is K for each copy.Source: §2.3 · pp. 9–10

2.1 Randomness

  • The authors average over a Gaussian ensemble of time-independent Hamiltonian couplings, while noting that lessons from stochastic averaging may not apply to their focus on time-independent systems.In the ensemble, non-k-local couplings are assumed to vanish.Source: §2.1 · p. 7, §2.3 · p. 9

3 The Classical System A

The auxiliary system represents quantum evolution as classical motion on SU(2^K), then recasts the Hamiltonian couplings as initial velocities. Its geometry, action, and conservation laws provide a classical-mechanics description, while the motion is constrained rather than ergodic across the full group.

3 The Classical System A

  • The auxiliary system represents a quantum time-evolution operator as a classical particle moving on SU(2^K), later using a modified complexity geometry.Each point of the group corresponds to a unitary matrix.Source: §3 · p. 11

3.1 Equations of Motion

  • The Schrödinger equation becomes a second-order geodesic equation on SU(2^K), with the Hamiltonian information encoded in the initial velocity.The first- and second-order formulations are equivalent because H and the initial velocity can be translated into one another.Source: §3.1 · p. 12

3.2 Velocity-Coupling Correspondence

  • The V/J correspondence identifies Hamiltonian couplings J_I with initial velocity components V_I, so the classical mechanics describes the family of k-local Hamiltonians.The couplings are projections of the initial velocity onto tangent-space axes in the Pauli basis.Source: §3.2 · p. 13

3.3 Action

  • The equations of motion can also be derived from an action, which has a simple form when expressed in terms of U.The displayed action expression is accompanied by the statement that it yields the equations of motion.Source: §3.3 · p. 14

3.4 Conservation Laws

  • Auxiliary energy is conserved and equals the Lagrangian, while bi-invariance conserves right charges; the left charges are not functionally independent.The auxiliary Hamiltonian is distinct from the quantum-system Hamiltonian.Source: §3.4 · p. 14

3.5 Ergodicity

  • For a fixed Hamiltonian, evolution is generically ergodic on a 2^K-dimensional torus, but varying k-local Hamiltonians covers only a slightly larger subset of SU(2^K).This is far short of ergodic motion over the full 4^K-dimensional group; time-dependent Brownian circuits instead fill all 4^K dimensions.Source: §3.5 · p. 15

4 Geometry of Complexity

The section develops relative complexity as an operational distance between quantum states and studies the resulting complexity geometry, focusing on penalty-dependent sectional curvature and its implications for the switchback effect and geodesic divergence. It does not compute higher-order behavior through the scrambling time.

4.1 The Distance Between Quantum States

  • Relative complexity defines an operational distance for transitions, interference, and phase measurements, distinguishing states that the Fubini-Study metric places equally far apart.In chaotic systems, initially similar states become exponentially difficult to interfere even after the added qubit thermalizes.Source: §4.1 · pp. 16–18

4.2 Complexity Geometry

  • Complexity geometry uses a non-standard metric on SU(2K) whose shortest geodesic distance tracks relative complexity; its right-invariant metric is homogeneous, but gate-set choices make complexity non-unique.The original Nielsen penalty is too severe to reproduce the switchback effect, motivating a more moderate choice.Source: §4.2 · pp. 19, 21, 22

4.3 A More Moderate Penalty

  • A gradual penalty for operators above weight k, independent of K for weights below K, yields negative typical sectional curvatures of order 1/K when k = 2.The proposed penalty rises smoothly from order 1 to order 4K for weights above k.Source: §4.3 · pp. 22–23

4.4 Sectional Curvature

  • For random 2-local Hamiltonians, locality leaves only about 1/K of terms noncommuting with Δ, giving sectional curvature that is negative and of order 1/K.The curvature calculation assumes k = 2 and random, zero-mean coefficients.Source: §4.4 · pp. 23, 26
  • With Eq. 4.11 governing Iw, I3 is about 4 and curvature is of order 1/K; assuming I3 is of order 4K instead gives curvature of order 4K, incompatible with the switchback effect and scrambling time log K.The latter curvature would cause geodesics to deviate so violently that scrambling would be almost immediate; negative sectional curvature is generic when I3 > 4/3 [9].Source: §4.4 · p. 26
  • For k-local generalizations, sectional curvature is generically ∼k2/K because two randomly chosen terms fail to commute when they share a qubit, with probability roughly k2/K.Source: §4.4 · p. 26
  • Negative curvature makes geodesics diverge exponentially with typical local Lyapunov behavior, suggesting chaotic motion in complexity space; its Lyapunov exponent corresponds to operator growth in out-of-time-order correlations [11].Source: §4.4 · p. 26
  • Following exponential complexity growth through the scrambling time requires computing higher orders in t, which the authors have not yet done.At the origin, sectional curvature depends only on I3; higher-weight penalty factors appear when calculating geodesic deviation at higher orders in t.Source: §4.4 · p. 27

5 Particle on the Complexity Geometry

The section relates k-local geodesics across the standard and complexity metrics, then uses curvature-driven geodesic deviation to connect complexity geometry with chaos; it also postulates an action-based definition of unitary complexity.

5 Particle on the Complexity Geometry

  • For k-local initial velocities, geodesics follow the same curves and have the same lengths under the bi-invariant and complexity metrics.Source: §5 · p. 27

5.1 Geodesic Deviation

  • Typically negative sectional curvature in the complexity geometry produces diverging geodesics, whereas positive curvature in the standard metric produces convergence.Source: §5.1 · p. 28
  • Ergodicity alone is insufficient for classical chaos here; conservation laws confine auxiliary-system chaos to a 2K-dimensional submanifold, and the metric-based appearance of chaos needs further study.Source: §5.1 · p. 28
  • Negative curvature controls the auxiliary model’s Lyapunov exponents, whose largest may be identified with the quantum Lyapunov exponent discussed in [4].Source: §5.1 · p. 28

5.2 Complexity Equals Action

  • Complexity is postulated as the minimum complexity-metric action from identity to U, with the auxiliary particle’s energy fixed to K.Source: §5.2 · p. 29

Action =√

The paper relates quantum complexity to the motion of an auxiliary classical system, proposing that complexity equals action and grows with the system’s Lagrangian. It further proposes that ensemble-averaged complexity tracks the auxiliary system’s entropy.

5.3 The Growth of Complexity

  • Assuming complexity equals action, the paper argues that its growth rate is the auxiliary system’s Lagrangian.The passages state that complexity grows as expected but do not supply the resulting formula.Source: §5.3 · p. 30
  • Ensemble-averaged quantum complexity is proposed to equal the entropy of an auxiliary classical system, up to a factor to be determined.The auxiliary system moves on a sub-manifold of dimension slightly larger than 2K.Source: §5.3 · p. 31
  • For stochastic time dependence in the couplings, the auxiliary system becomes a diffusion problem on the complexity geometry.Source: §5.3 · p. 31

6 Statistical Mechanics of Complexity

The paper proposes a thermodynamics of complexity by relating quantum complexity to positional entropy and Hamiltonian description complexity to kinetic entropy. It supports these conjectures with counting arguments while noting that they are not rigorous.

6.1 Average Complexity Equals Entropy

  • For a factorized classical phase-space distribution, total entropy separates into positional and kinetic terms, without requiring equilibrium.The paper takes the ensemble average over initial velocities or, by the V/J correspondence, over couplings J.Source: §6.1 · pp. 31–32

6.2 Computational Complexity and Positional Entropy

  • The ensemble-averaged computational complexity of Q is conjectured to be proportional to A’s positional entropy, not its total entropy.Complexity concerns position in complexity space; the proportionality allows a multiplicative ambiguity associated with choices such as the gate set.Source: §6.2 · p. 32

6.3 Kinetics

  • The auxiliary system’s kinetic entropy is identified with the entropy of P(J), linking it to the quantum Hamiltonian rather than the quantum state.The couplings correspond to auxiliary velocities, and the Hamiltonian determines those velocities.Source: §6.4 · p. 34, §6.3 · p. 34

6.4 Kolmogorov Complexity and Kinetic Entropy

  • Under mild assumptions, ensemble-average Kolmogorov complexity and entropy agree [26], connecting kinetic entropy to the Hamiltonian’s description complexity.The distribution itself must not be too complex; the paper gives a Gaussian as an example of a simple distribution.Source: §6.4 · pp. 34–35
  • In the time-independent Hamiltonian model, kinetic entropy is time-independent and polynomial in K, while positional entropy can grow to ∼2^K at equilibrium.During early complexity growth the two contributions can compete, but at equilibrium the positional term dominates.Source: §6.4 · pp. 35–36

6.5 A Counting Argument

  • A counting argument supports exponential growth in the number of operators with complexity, but relies on collisions being rare at subexponential times.The authors say this assumption is less secure for unitaries generated by time-independent k-local Hamiltonians than for stochastic circuits.Source: §6.5 · pp. 36, 38

6.6 A State-Complexity Argument

  • A state-complexity counting argument likewise yields that average complexity is auxiliary entropy, but the arguments are far from rigorous.The conclusion follows from the number of states of a given complexity growing exponentially with complexity.Source: §6.6 · p. 39

7 The Second Law

The auxiliary system’s positional entropy can increase even when it does not thermalize. Applying the correspondence between that entropy and quantum complexity, the paper argues that submaximal complexity tends to rise, with decreasing-complexity trajectories unstable to perturbations.

7.1 Obstruction to Thermalization

  • Right-multiplication conservation laws obstruct thermalization, but positional entropy still grows to its maximum within each conserved-quantity leaf.The auxiliary system can therefore fail to reach thermal equilibrium when its initial velocities are not Maxwell-Boltzmann distributed.Source: §7.1 · p. 40

7.2 Second Law of Complexity

  • With overwhelming likelihood, complexity below its maximum increases both into the future and into the past.Source: §7.2 · p. 41
  • The auxiliary system reaches positional equilibrium after a time polynomial in its degrees of freedom; the corresponding quantum system reaches complexity equilibrium after an exponential time in K and recurs after a doubly exponential time.Source: §7.2 · p. 41
  • A small perturbation exponentially spreads through the system and reverses an engineered decrease in complexity.The paper notes that the largest classical Lyapunov exponent of A is the quantum Lyapunov exponent of Q.Source: §7.2 · p. 41

8 Uncomplexity as a Resource

The section examines uncomplexity as a resource, presenting computational uncomplexity for directed quantum computation and Kolmogorov uncomplexity for erasure. It illustrates these uses with a one-clean-qubit computation and compression to reduce erasure costs.

8 Uncomplexity as a Resource

  • The gap between a state’s complexity and its maximum, called uncomplexity, is proposed as a resource for directed computation; at maximal complexity, ordinary operations cannot support useful computation.The computational process is quantum-in to quantum-out, with measurement allowed only at the end; a maximally mixed state is unchanged by any unitary operation.Source: §8 · pp. 43–44

8.1 Combining Systems: A Paradox

  • Combining two K-qubit systems as Q ⊗ Q multiplies auxiliary degrees of freedom rather than adding them, leaving the result far from complexity equilibrium.The resolution is to combine auxiliary systems by adding one qubit, which doubles their degrees of freedom; for independently chosen, maximally complex states, generic interactions yield final complexity equal to the sum of the initial complexities. Identical states instead start with complexity 2K and reach 2 × 2K after interaction, exposing an available resource.Source: §8.1 · pp. 44–46

8.2 One Clean Qubit

  • Adding one pure qubit to a maximally complex system doubles its maximum complexity and replenishes uncomplexity, allowing useful computational work again.In the one-clean-qubit example, applying Hadamard–controlled-G–Hadamard operations prepares a state whose clean-qubit measurement can reveal information about Tr G; measurement occurs only at the end.Source: §8.2 · pp. 47, 48, 51
  • One-clean-qubit computation uses a clean qubit to calculate Tr G while increasing the state’s complexity and expending uncomplexity by approximately the complexity of G.It moves toward a computational goal through a procedure that does not depend on the microscopic starting state; repeated circuit action depletes the resource roughly by the complexity of G each time.Source: §8.2 · p. 52

8.3 Kolmogorov Uncomplexity as a Resource

  • Kolmogorov uncomplexity can be expended to erase another bit string, a distinct use from computational uncomplexity’s role in directed computation.The section identifies erasure as the purpose for which Kolmogorov uncomplexity is useful.Source: §8.3 · p. 53
  • Bennett et al. show that erasure cost is determined by a bit string’s Kolmogorov complexity, so compressible strings save exactly their uncomplexity relative to the naive Landauer cost.The first million digits of π can be reversibly compressed into a shorter program before erasure, unlike a million random digits.Source: §8.3 · p. 53

9 Uncomplexity as Spacetime

The paper interprets uncomplexity as spacetime volume available to an observer entering a black hole. This resource is finite and vanishes when complexity reaches its maximum at the cutoff time.

9 Uncomplexity as Spacetime

  • The behind-horizon WDW action grows linearly with time, consistent with early complexity growth, but its classical description breaks down at tmax = eS.At that cutoff, complexity reaches Cmax and the horizon becomes opaque by developing a firewall.Source: §9 · pp. 54–55
  • The uncomplexity is proportional to a finite blue-region 4-volume that goes to zero as t approaches tmax ∼ eS.The uncomplexity is the difference between the actions of the behind-horizon regions of the maximum-complexity and time-t WDW patches.Source: §9 · pp. 56–57, Figure 10 · p. 57
  • Uncomplexity corresponds to the spacetime volume accessible to an observer who plans to cross the horizon.The blue region represents the union of interior locations Alice can visit by entering after boundary time t.Source: §9 · pp. 54, 57
  • At maximum complexity, the resource is exhausted and Alice encounters an obstruction at the horizon.This parallels attempting computation with a computer that has reached maximal complexity.Source: §9 · pp. 55, 57
  • A thermal quantum restores horizon transparency after a scrambling time, and that transparency lasts for an exponential time.The quantum shifts the shockwave away from the horizon; the authors compare this rejuvenation with restoring computing power by adding one clean qubit.Source: §9 · pp. 57–58, Figure 11 · p. 58

10 Summary

The paper develops a classical auxiliary-system framework for relating quantum complexity to entropy. It argues that this relation supports a second law of complexity and treats uncomplexity as a computational and spacetime resource.

10 Summary

  • The complexity metric better captures differences between states of a chaotic system than the standard inner-product metric.Its calculated sectional curvatures are consistent with the negatively curved toy-model geometry.Source: §10 · p. 59
  • The auxiliary system A is a classical nonrelativistic particle moving on a complexity geometry, with relative complexity given by constrained minimal action.The auxiliary system has a number of classical degrees of freedom exponential in the number of qubits.Source: §10 · p. 59
  • The proposed correspondence identifies ensemble-average quantum complexity with classical entropy, including kinetic entropy and the Hamiltonian's Kolmogorov complexity.The shortest program specifying the Hamiltonian supplies the Kolmogorov-complexity term, which soon becomes negligible compared with gate complexity.Source: §10 · p. 60
  • The paper formulates a second law of complexity as the second law of thermodynamics for auxiliary system A.This explains why complexity growth for K qubits resembles entropy growth for a classical system with exponentially many degrees of freedom.Source: §10 · pp. 59–60
  • Uncomplexity is presented as a resource for computational work, and adding one qubit can double the size of an auxiliary system.The paper illustrates this resource idea with one-clean-qubit computation.Source: §10 · p. 60
  • For a black hole, uncomplexity corresponds to the spacetime volume accessible to an observer planning to cross the horizon.The paper identifies this as the holographic dual of boundary-state uncomplexity.Source: §10 · p. 60

11 Questions

The authors identify unresolved questions about the foundations and scope of complexity thermodynamics. These include its underlying assumptions, complexity-measure choices, chaos, resource interpretation, and possible extensions.

11 Questions

  • The paper leaves open how to choose among the many complexity measures and how their inertia-tensor penalties should scale.Sectional curvatures are generically negative and of order 1/K when penalty factors are not too small.Source: §11 · p. 61
  • The complexity–entropy correspondence assumes that the number of unitaries of complexity at most C grows like e^C, but the supporting arguments are far from a proof.A proof requires counting unitaries on 2^K-dimensional tori in SU(2^K).Source: §11 · p. 61
  • Whether the auxiliary-system motion is chaotic depends on the metric, and the authors guess it is locally chaotic over exponentially long times but not truly chaotic.They note that this distinction may matter little on sufficiently long timescales.Source: §11 · pp. 61–62
  • The authors ask whether thermodynamics can also be defined for classical computational complexity.They suggest that averaging over algorithms may make some difficult average-behavior problems tractable through classical statistical mechanics.Source: §11 · pp. 60–62

Least Action and Least Computation

  • The relationship between the proposed auxiliary-system least-action principle and other action-equals-complexity proposals remains an open question.The paper asks whether the apparently different principles might be the same.Source: §11 · p. 62

Uncomplexity as a Resource

  • The authors ask whether the gap between current and maximal complexity can be precisely characterized as a resource useful for quantum computation.They also raise the unresolved relation between computational and Kolmogorov uncomplexity.Source: §11 · p. 63

First Law of Complexity

  • Identifying a first law of complexity is left for future investigation.The paper argues for a second law but does not identify a corresponding first law.Source: §11 · p. 63

A Some Terminology

The terminology section defines the paper's quantum and auxiliary systems, complexity measures, and notation. It also specifies conventions for time, metrics, and circuit architecture.

A Some Terminology

  • The complexity metric is right-invariant but not left-invariant, unlike the conventional bi-invariant standard metric.The moment of inertia tensor IIJ is a symmetric matrix in the adjoint representation of SU(2^K).Source: A Some Terminology · pp. 64–65
  • Q denotes the K-qubit quantum system, while A denotes the classical auxiliary system; a subscript a marks quantities belonging to A.The auxiliary system is a particle moving on SU(2^K), and its velocity and energy are denoted Va and Ea.Source: A Some Terminology · p. 64
  • Time t is dimensionless, and for an uncharged neutral black hole it is the Rindler boost-angle time.In the Hayden-Preskill circuit convention, each time-step consists of order K gates and sub-exponential complexity growth is ∼K per unit.Source: A Some Terminology · pp. 64–65
  • Circuit complexity C is the gate-count measure, while Cκ denotes Kolmogorov complexity; uncomplexity is the gap ΔC between C and its maximum.The Kolmogorov complexity of a string s is written Cκ(s).Source: A Some Terminology · p. 65

B Some Clarifications

This section clarifies assumptions and conventions behind the complexity framework, including how measurement and computation are treated. It also explains the resource use and equilibration picture for directed computation.

Locality

  • The paper’s Eq. 2.1 uses exact k-locality, containing only weight-k terms, while the standard definition allows terms of weight up to k.The authors say the equations can be generalized to the standard definition.Source: B Some Clarifications · p. 66

Time units

  • Dimensionless Rindler time is used, and the Hamiltonian and J coefficients are dimensionless in those units.The paper relates this time choice to black hole physics and also interprets t as counting steps in a Hayden-Preskill circuit.Source: B Some Clarifications · p. 66

Penalty factors

  • Penalty factors of order 4K would suppress shortcuts so strongly that complexity would grow linearly almost immediately, unlike the discrete-circuit precursor pattern.The authors argue that continuous-Hamiltonian penalties must turn on more smoothly to match the negligible but nonzero growth before scrambling time.Source: B Some Clarifications · p. 66

Measurement

  • Measurement is excluded from the auxiliary description of quantum evolution; global resource tracking would include the observer and, the authors believe, measurement would decrease that resource.They treat measurement globally as entanglement between Q and the rest of the world, requiring complexity tracking for everything, including the observer.Source: B Some Clarifications · pp. 66–67

Computation and the resource

  • Computational work can proceed in small (k+1)-local steps, each using a small amount of uncomplexity.In the 2-gas model, the circuit evolves one component while leaving the initially maximally complex component fixed; long evolution breaks their correlation and can bring the whole system to equilibrium.Source: B Some Clarifications · p. 67

C Action vs. Distance in the Toy Model

This section compares distance and action conventions for complexity geometry in a toy model. It explains how metric normalization reconciles the formulations and relates toy-model curvature to complexity growth and scrambling.

Conventions

  • The distance and action formulations differ by a convention that can be absorbed into a redefinition of the metric scale.The paper notes that [4] used the distance formulation in its analysis, while this paper uses the action formulation.Source: C Action vs. Distance in the Toy Model · pp. 67–68

Toy model

  • In the two-dimensional toy geometry with uniform negative curvature, neighboring geodesics separate exponentially with r, identifying r-dot as the Lyapunov exponent for scrambling.With dimensionless Rindler time, the Lyapunov exponent is 1 and constrains the motion.Source: C Action vs. Distance in the Toy Model · p. 68

Growth conditions

Normalization

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