How to think about Quantum Mechanics—Part 7: Quantum chaos and linear evolution

[Other parts in this series: 1,2,3,4,5,6,7.]

You’re taking a vacation to Granada to enjoy a Spanish ski resort in the Sierra Nevada mountains. But as your plane is coming in for a landing, you look out the window and realize the airport is on a small tropical island. Confused, you ask the flight attendant what’s wrong. “Oh”, she says, looking at your ticket, “you’re trying to get to Granada, but you’re on the plane to Grenada in the Caribbean Sea.” A wave of distress comes over your face, but she reassures you: “Don’t worry, Granada isn’t that far from here. The Hamming distance is only 1!”.

After you’ve recovered from that side-splitting humor, let’s dissect the frog. What’s the basis of the joke? The flight attendant is conflating two different metrics: the geographic distance and the Hamming distance. The distances are completely distinct, as two named locations can be very nearby in one and very far apart in the other.

Now let’s hear another joke from renowned physicist Chris Jarzynski:

The linear Schrödinger equation, however, does not give rise to the sort of nonlinear, chaotic dynamics responsible for ergodicity and mixing in classical many-body systems. This suggests that new concepts are needed to understand thermalization in isolated quantum systems. – C. Jarzynski, “Diverse phenomena, common themes” [PDF]

Ha! Get it? This joke is so good it’s been told by S. Wimberger“Since quantum mechanics is the more fundamental theory we can ask ourselves if there is chaotic motion in quantum systems as well. A key ingredient of the chaotic phenomenology is the sensitive dependence of the time evolution upon the initial conditions. The Schrödinger equation is a linear wave equation, implying also a linear time evolution. The consequence of linearity is that a small distortion of the initial conditions leads only to a small and constant change in the wave function at all times (see Sect. 4.1). Certainly, this is not what we mean when talking about ‘quantum chaos’.” -S. Wimberger, “Nonlinear Dynamics and Quantum Chaos: An Introduction” (2014). a  , H.-J. Stöckmann“The Schrödinger equation is a linear equation leaving no room for chaos.” – H.-J. Stöckmann, “Quantum Chaos: An Introduction” (1999). b  , Casetti et al.“Chaos does not exist in the linear evolution of the quantum state vector, hence different approaches to identify quantum features that correspond to classical chaos have been developed…” – Casetti et al., “Chaos in effective classical and quantum dynamics”, arXiv:hep-th/9707054. c  , Ullmo & Tomsovic“One possible sense of the term “chaos” here could be that two slightly different initial wave functions \phi_1(r, t = 0) and \phi_2(r, t = 0) diverge “exponentially” rapidly from one another with time. It turns out however that one can answer this question under relatively general conditions, and the answer is negative. Indeed, the simple fact that the Schrödinger equation is linear (i.e. that a linear combination of two solutions of Eq. (6) is also a solution of this equation) makes it impossible that chaos, in any sense similar to classical mechanics, develops in quantum mechanics.” – Ullmo & Tomsovic, “Introduction to Quantum Chaos” [PDF]. d  , and of course the great Wikipedia“However, this mechanism of dynamical chaos is absent in Quantum Mechanics, due to the strictly linear time evolution of the Schrödinger equation…This time evolution is manifestly linear, and any notion of dynamical chaos is absent. Thus, it becomes an open question as to whether an isolated quantum mechanical system, prepared in an arbitrary initial state, will approach a state which resembles thermal equilibrium, in which a handful of observables are adequate to make successful predictions about the system.” – Wikipedia: “Eigenstate Thermalization Hypothesis” | “Motivation” (January 2017). Luckily, at least this one I was able to fix myself… e  , who all point to the linearity of the Schrödinger equation to motivate the need for new tools to study quantum chaos. Heck, even people who know better love to tell this knee-slapper“The relation between classical and quantum chaos has been always somewhat unclear and, at times, even strained. The cause of the difficulties can be traced to the fact that the defining characteristic of classical chaos — sensitive dependence on initial conditions — has no quantum counterpart: It is defined through the behavior of neighboring trajectories, a concept which is essentially alien to quantum mechanics. Moreover, when the natural language of quantum mechanics of closed systems is adopted, an analog of the exponential divergence cannot be found.” – Zurek & Paz, “Decoherence, Chaos, and the Second Law”, arXiv:gr-qc/9402006. “In an idealized case, the distance between the two points in phase space grows as e^{\lambda t}, where \lambda is the largest Lyapunov exponent of the system. This does not happen in quantum mechanics. The (oversimplified) reason is that quantum mechanics is linear; thus two ‘‘nearly identical’’ states i.e.,states with a large initial overlap” remain nearly identical—their overlap is constant under unitary evolution—for all time.” – Blume-Kohout & Zurek, ‘Decoherence from a Chaotic Environment: An Upside Down “Oscillator” as a Model’, arXiv:quant-ph/0212153. f  .

The “humor” is grounded in the fact that these jokesters are conflating two completely different metrics:

  • distance in phase space, and
  • distinguishability of states.

Classical chaos is associated with the exponential divergence in phase space of nearby states, but two classical probability distributions remain just as distinguishable under Hamiltonian flow, even when it’s chaotic. Likewise, quantum chaos is associated with the exponential divergence, in phase space, of nearby states when such a distance is well-defined (i.e., when those states are roughly localized in phase space). But two quantum states, pure or mixed, always remain just as distinguishable under unitary evolution.


The two colored regions represent alternative distributions in phase space, either two classical probability distributions or two Wigner functions. Classically, the probability density at any point is preserved under phase-space flow. In quantum mechanics, the analog is only guaranteed for quadratic Hamiltonians. In both cases, distinguishability of the states is always preserved by Hamiltonian evolution and roughly corresponds to the amount of overlap of the two distributions (purple area). This is distinct from the distances between points in phase space, which may diverge exponentially (dotted lines) in chaotic systems.

One way to quantify the (in)distinguishabilityOf course, here by “distinguishable” we mean in the idealized sense of perfectly precise measurements. Chaos is certainly associated with initial distributions that progressively in time become more difficult to distinguish for practical measurements of finite accuracy. But this applies just as well to both classical and quantum mechanics! g   is with quantum relative entropy

(1)   \begin{align*} S (\rho\vert\vert \eta) = \mathrm{Tr} \big[\rho (\ln \rho - \ln\eta)\big] \end{align*}

which reduces to the KL divergence

(2)   \begin{align*} D (P \vert\vert Q) = \sum_i P_i \ln \frac{P_i}{Q_i} \end{align*}

classically, i.e., when \rho = \sum_i P_i \vert i \rangle \langle i \vert and \eta = \sum_i Q_i \vert i \rangle \langle i \vert are diagonal in the same basis. (An individual point in phase space corresponds to the special case of a delta-function probability distribution, and so the KL divergence of two classical states is simply unity when they coincide and vanishes when they don’t.) Both of these distances are manifestly preserved by Hamiltonian evolution. Indeed, the linearity of the Schrödinger equation, \partial_t \psi = -i H \psi, is just the quantum analog of the linearity of the Liouville equation in classical mechanics,

(3)   \begin{align*} \partial_t f = \{H,f\}, \end{align*}

where f(\vec x,\vec p) denotes the classical PDF and \{\cdot,\cdot\} denotes the Poisson bracket.

(The von Neumann equation, \partial_t \rho = - i [H,\rho], which simply generalizes the Schrödinger equation to allow for mixed states, is a more direct quantum analog to the classical Liouville equation. This is another manifestation of the idea that quantum states — both pure and mixed — are much more analogous to classical probability distributions than to classical points in phase space.Another key aspect of this analogy is the size of the spaces: classical probability distributions and quantum states (pure or mixed) live in spaces that are exponentially large in the number of degrees of freedom (and hence take an exponential amount of information to specify), whereas classical points live in phase space itself, which scales linearly. h  )

Now, there are plenty of reasons one might need a different definition of chaos in quantum systems. It is ambiguous how we are to extend the phase-space distance to arbitrary quantum states for the same reason that it’s ambiguous how we would define a phase-space distance between probability distributions that are widely dispersed over classical phase space. The key difference is that no precisely localized states exist in quantum mechanics, and it’s not surprising that a certain phenomenon in a limiting theory (classical mechanics) could be just a special case of a more general and complicated phenomena in the fundamental theory (quantum mechanics); more powerful and abstract tools may be required for the latter.For an example of someone explaining this point well, see M. V. Berry, in New Trends in Nuclear Collective Dynamics, edited by Y. Abe, H. Horiuchi, and K. Matsuyanagi (Springer, Berlin, 1992), p. 183. i   But the linearity of the Schrödinger equation has nothing whatever do to with this.

Indeed, saying “the exponential-sensitivity criterion for identifying chaos doesn’t work in quantum systems because the Schrödinger equation is linear” is just as silly as saying “spatial separation between cities is not useful in China because the Hamming distance is almost always maximum for city names in logographic languages”. On the other hand, it is both analogous and correct to observe that spatial separation between cultures is sometimes poorly defined because cultures don’t need to be well localized in space. But that when they are (e.g., the cultures of Florence, Italy and Hawaii in the 15th century), the distance between them is perfectly sensible.

An aside

So if linearity of evolution in phase space implies no chaos, but this is not the same thing as linearity of evolution in Hilbert space (which always applies), what is an example of the former?: The harmonic oscillator. Abstractly, linearity of an equation means that solutions form a vector space; if v and w are solutions of the equation, then av+bw is too, for a,b \in \mathbb{R} (or \mathbb{C} in the case of a complex vector space). If the classical evolution has two solutions x_1(t) and x_2(t), then the solution (x_1 + x_2)(t) = x_1(t) + x_2(t) is also a solution. This is guaranteed by the equation of motion for the harmonic oscillator \ddot{x} = -k x but does not apply for an anharmonic oscillator like \ddot{x} = -k x - h x^3. The linearity of the classical equation of motion follows from the quadratic form of the Hamiltonian,

(4)   \begin{align*} H = H_0 + H_x x + H_p p + H_{xp} xp + H_{xx} x^2 + H_{pp} p^2, \end{align*}

since this can always be put in the form H = ap^2 + bx^2 through a linear change of variables, and from this we get \ddot{x} \propto \dot{p} \propto \partial H/\partial x \propto x. This caries over immediately to the analogous statement about the Hamiltonian operator of quantum mechanics.

Implications AKA more complaining

One could argue that some of these writers do understand all this but that they just don’t explain it well in their writing. This may be true for some authors. However, my anecdotal experience from speaking in person to physicists is that many of them don’t get it, a testament to the disservice done by authors repeating this misleading idea.

Consider what this says about how physics knowledge is stored in the literature and encoded in humans brains, and the implications for how physicists decide what to work on. This particular step in the motivation for studying quantum chaos is mindlessly repeated in dozens of papers and books on quantum chaos, and it is totally bogus. What makes you think most folks understand the other steps better? I think we have to conclude that many physicists do not quite know why they are doing what they are doing; of those who do know, many apparently cannot explain it even to their peers, much less students.

And yet: there are good reasons to study different quantifiers of quantum chaos! But if many people can’t actually articulate those reasons, what mechanism draws them to the field? It’s some combination of them being interested in the object-level work (as opposed to the big-picture motivation), in the usefulness of the results, in the impressiveness of the other practitioners, and probably many other things I can’t think of. Nonetheless, the fact remains: many people apparently can’t clearly explain why they do what they do. More contentiously, I think we should also conclude that this is an impediment to them carefully reasoning about it as well, and hence that they are less likely to abandon the field when new relevant information comes to light.

There is nothing special about quantum chaos; there are many of these sorts of bad explanations that propagate by being uncritically repeated from one physicist to another. This is just an example where it’s particularly unambiguous.

Footnotes

(↵ returns to text)

  1. “Since quantum mechanics is the more fundamental theory we can ask ourselves if there is chaotic motion in quantum systems as well. A key ingredient of the chaotic phenomenology is the sensitive dependence of the time evolution upon the initial conditions. The Schrödinger equation is a linear wave equation, implying also a linear time evolution. The consequence of linearity is that a small distortion of the initial conditions leads only to a small and constant change in the wave function at all times (see Sect. 4.1). Certainly, this is not what we mean when talking about ‘quantum chaos’.” -S. Wimberger, “Nonlinear Dynamics and Quantum Chaos: An Introduction” (2014).
  2. “The Schrödinger equation is a linear equation leaving no room for chaos.” – H.-J. Stöckmann, “Quantum Chaos: An Introduction” (1999).
  3. “Chaos does not exist in the linear evolution of the quantum state vector, hence different approaches to identify quantum features that correspond to classical chaos have been developed…” – Casetti et al., “Chaos in effective classical and quantum dynamics”, arXiv:hep-th/9707054.
  4. “One possible sense of the term “chaos” here could be that two slightly different initial wave functions \phi_1(r, t = 0) and \phi_2(r, t = 0) diverge “exponentially” rapidly from one another with time. It turns out however that one can answer this question under relatively general conditions, and the answer is negative. Indeed, the simple fact that the Schrödinger equation is linear (i.e. that a linear combination of two solutions of Eq. (6) is also a solution of this equation) makes it impossible that chaos, in any sense similar to classical mechanics, develops in quantum mechanics.” – Ullmo & Tomsovic, “Introduction to Quantum Chaos” [PDF].
  5. “However, this mechanism of dynamical chaos is absent in Quantum Mechanics, due to the strictly linear time evolution of the Schrödinger equation…This time evolution is manifestly linear, and any notion of dynamical chaos is absent. Thus, it becomes an open question as to whether an isolated quantum mechanical system, prepared in an arbitrary initial state, will approach a state which resembles thermal equilibrium, in which a handful of observables are adequate to make successful predictions about the system.” – Wikipedia: “Eigenstate Thermalization Hypothesis” | “Motivation” (January 2017). Luckily, at least this one I was able to fix myself…
  6. “The relation between classical and quantum chaos has been always somewhat unclear and, at times, even strained. The cause of the difficulties can be traced to the fact that the defining characteristic of classical chaos — sensitive dependence on initial conditions — has no quantum counterpart: It is defined through the behavior of neighboring trajectories, a concept which is essentially alien to quantum mechanics. Moreover, when the natural language of quantum mechanics of closed systems is adopted, an analog of the exponential divergence cannot be found.” – Zurek & Paz, “Decoherence, Chaos, and the Second Law”, arXiv:gr-qc/9402006. “In an idealized case, the distance between the two points in phase space grows as e^{\lambda t}, where \lambda is the largest Lyapunov exponent of the system. This does not happen in quantum mechanics. The (oversimplified) reason is that quantum mechanics is linear; thus two ‘‘nearly identical’’ states i.e.,states with a large initial overlap” remain nearly identical—their overlap is constant under unitary evolution—for all time.” – Blume-Kohout & Zurek, ‘Decoherence from a Chaotic Environment: An Upside Down “Oscillator” as a Model’, arXiv:quant-ph/0212153.
  7. Of course, here by “distinguishable” we mean in the idealized sense of perfectly precise measurements. Chaos is certainly associated with initial distributions that progressively in time become more difficult to distinguish for practical measurements of finite accuracy. But this applies just as well to both classical and quantum mechanics!
  8. Another key aspect of this analogy is the size of the spaces: classical probability distributions and quantum states (pure or mixed) live in spaces that are exponentially large in the number of degrees of freedom (and hence take an exponential amount of information to specify), whereas classical points live in phase space itself, which scales linearly.
  9. For an example of someone explaining this point well, see M. V. Berry, in New Trends in Nuclear Collective Dynamics, edited by Y. Abe, H. Horiuchi, and K. Matsuyanagi (Springer, Berlin, 1992), p. 183.
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2 Comments

  1. Jess, nice write up as usual.

    Regarding the classical chaos, is it possible to define classical chaos without referring to phase space trajectories? Asked differently, can one quantify classical chaos from the (time evolution of) density distribution only?

    If the answer to this question is yes, then what would go wrong if we simply replace density distribution by density matrix?

    • Yea, you can use a Kolmogorov-Sinai (KS) entropy to define classical chaos as follows: Fix some coarse-graining of phase space (e.g. a fine grid) and consider the coarse-graining of the probability density as a function of time. For chaotic systems the entropy (~area) of the coarse-grained distribution grows exponentially in time while for integrable systems it does not. The rate of exponential growth reached at long times (i.e., once the entropy of the coarse-grained distributions is much larger than the initial entropy) remains finite in the limit of an infinitely fine grid, and defines the KS entropy.

      Note that taking the limit of an infinitely fine grid still requires a notion of phase-space points, which is problematic in quantum mechanics. And indeed I couldn’t find a definitive definition of a quantum KS entropy after a quick search, but see a discrete definition by Alicki and Fannes and (unnecessarily mathematical?) pessimistic discussion starting from the bottom of page 8 by Wehrl.

      (Amazingly, there’s a PRL where someone seems to have taken the opposite approach of extending phase-space trajectories to arbitrarily small scales in quantum mechanics using Bohmian trajectories and then building up Lyapunov exponents from that.)

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