Over at PhysicsOverflow, Daniel Ranard asked a question that’s near and dear to my heart:

How deterministic are large open quantum systems (e.g. with humans)?

Consider some large system modeled as an open quantum system — say, a person in a room, where the walls of the room interact in a boring way with some environment. Begin with a pure initial state describing some comprehensible configuration. (Maybe the person is sitting down.) Generically, the system will be in a highly mixed state after some time. Both normal human experience and the study of decoherence suggest that this state will be a mixture of orthogonal pure states that describe classical-like configurations. Call these configurations branches.

How much does a pure state of the system branch over human time scales? There will soon be many (many) orthogonal branches with distinct microscopic details. But to what extent will probabilities be spread over macroscopically (and noticeably) different branches?

I answered the question over there as best I could. Below, I’ll reproduce my answer and indulge in slightly more detail and speculation.

This question is central to my research interests, in the sense that completing that research would necessarily let me give a precise, unambiguous answer. So I can only give an imprecise, hand-wavy one. I’ll write down the punchline, then work backwards.

##### Punchline

**The instantaneous rate of branching, as measured in entropy/time (e.g., bits/s), is given by the sum of all positive Lyapunov exponents for all non-thermalized degrees of freedom.**

Most of the vagueness in this claim comes from defining/identifying degree of freedom that have thermalized, and dealing with cases of partial/incomplete thermalization; these problems exists classically.

##### Elaboration

The original question postulates that the macroscopic system starts in a quantum state corresponding to some comprehensible classical configuration, i.e., the system is initially in a quantum state whose Wigner function is localized around some classical point in phase space. The Lyapunov exponents (units: inverse time) are a set of local quantities, each associated with a particular orthogonal direction in phase space. They give the rate at which local trajectories diverge, and they (and their associated directions) vary from point to point in phase space.

Lyapunov exponents are defined by the *linearized* dynamics around a point, and therefore they are constant on scales smaller than the third derivative of the potential. (Perfectly linear dynamics are governed by a quadradic Hamiltonian and hence a vanishing third derivative.) So if the Wigner function for the relevant degree of freedom is confined to a region smaller than this scale, it has a single well-defined set of Lyapunov exponents.^{a }

On the other hand, the Wigner function for degrees of freedom that are completely thermalized is confined only by the submanifold associated with values of conserved quantities like the energy; within this submanifold, the Wigner function is spread over scales larger than the linearization neighborhood and hence is not quasi classical. Degrees of freedom that have fully thermalized do not have branch structure because they are *already* maximally entangled with the external environment.

As mentioned above, I don’t know how to think about degree of freedom which are neither fully thermalized nor confined within the linear neighborhoods.

##### Argument

We want to associate (a) the rate at which nearby classical trajectories diverge with (b) the production of quantum entanglement entropy. The close relationship between these two has been shown in a bunch of toy models. For instance, see the many nice cites in the introduction of

Asplund & Berenstein, “Entanglement entropy converges to classical entropy around periodic orbits”, (2015). arXiv:1503.04857.

especially this paper by my former advisor

Zurek and Paz, “Quantum chaos: a decoherent definition”, Physica D 83, 300 (1995). arXiv:quant-ph/9502029.

The very crude picture is this: an initially pure quantum state with Wigner function localized around a classical point in phase space will spread to much larger phase-space scales at a rate given the Lyapunov exponent. The couplings between systems and environments are smooth functions of the phase space coordinates (i.e., environments monitor/measure some combination of the system’s *x’*s and *p’*s, but not arbitrary superpositions thereof), and the decoherence rate between two values of a coordinate is an increasing function of the difference. Once the Wigner function is spread over a sufficient distance in phase space, it will start to decohere into an incoherent mixture of branches, each of which are localized in phase space^{b }. See, for instance, Fig. 1 in Zurek & Paz:

Hence, the rate of trajectory divergence gives the rate of branching.^{c }

##### Authoritativeness

This answer would certainly *not* be immediately recognized as correct by the typical physicist, but I do think that if we restrict ourselves to the set of researcher that (1) agree that this is a well-posed question and (2) claim to know a non-trivial answer as least as precise as the one I’ve given above, then more than half of these physicists would agree that I’m correct up to the (considerable) inherent vagueness. But I’d guess this set probably has no more than a few dozen people in it.

##### Meta

(This part is more controversial.) The reason this questions was so hard to even formulate is two fold:

- No one has a good definition of what a branch is, nor how to extract predictions for macroscopic observations directly from a
*general*unitarily evolving wavefunction of the universe. (My preferred formulation of this is Kent’s set selection problem in the consistent histories framework.) - Branching is intimately connected to the process of thermalization. Although some recent progress in non-equilbirum thermodynamics has been made for systems
*near*equilibrium (especially the Crooks fluctuation theorem and related work), folks are still very confused about the process of thermalization even classically. See, for instance, the amazingly open question of deriving Fourier’s law^{d }^{e }, a very special case!

### Footnotes

(↵ returns to text)

- The system can then be well approximated as evolving quasi-classically over timescales shorter than the Lyapunov constant, in agreement with Ehrenfest’s theorem.↵
- This is especially transparent when looking at the class of sympectically invariant quantum Brownian motion models that I’ve recently become enamored with, since
*everything*(…Markovian…) looks like QBM if you zoom in far enough.↵ - Daniel also says “If the initial state describes a person in deep sleep, then after a few seconds there will be no high-probability branches in which the person is awake. But if the person has an alarm clock triggered by nuclear decay, there may well be 50/50 branching in which the person is awake or asleep. (Note that you couldn’t get such branching with a regular alarm clock.)”. But note that, although it’s true you couldn’t get such branching with a highly-reliable deterministic alarm clock, you could dispense with the nuclear decay by measuring any macroscopic chaotic degree of freedom on a timescales longer than the associated Lyapunov time constant. In particular, measuring thermal fluctuations of just about anything should be sufficient.↵
- F. Bonetto, J.L. Lebowitz, L. Rey-Bellet. “Fourier’s Law: a Challenge for Theorists” (2000) arXiv:math-ph/0002052.↵
- D. Ruelle, “A Mechanical Model for Fourier’s Law of Heat Conduction”, Communications in Mathematical Physics 311, 755 (2012). arXiv:1102.5488.↵

Hi Jess,

Nice exposition. I have a rather naive question about the postulated relationship between entanglement entropy growth and the (semi-)classical orbits. I understand that it is possible to relate the density of states of a quantum system to its semi-classical orbits (the Gutzwiller trace formula). However, it is hard for me to imagine how one could develop some sort of semi-classical notion of entanglement.

Yuan

Hi, Yuan. The short answer is that the increasing entropy of entanglement of a macroscopic quantum system (with the external environment) is analogous to the increasing phase-space uncertainty of the classical system (induced by it’s coupling to the external environment).

Even when a classical system starts in a maximum knowledge state, so that its probability distribution over phase space is given by a delta function, this distribution will in general be smeared to a distribution with non-zero area as the system evolves in contact with an environment. Likewise, a macroscopic quantum system in a state of maximum knowledge is in a pure quantum state, and — if it has an interpretation as a particular classical state — its Wigner function is concentrated in an area of size ħ/2 around a single phase-space point; as the system evolves, it becomes entangled with the environment, so the mixedness of the system’s state increases, and so that the Wigner function spreads over a larger area. (In the case of Gaussian state, the entropy of entanglement, in bits, is roughly log_2[A/(ħ/2)], where A is the phase-space area. See Blume-Kohout and Zurek in arXiv:quant-ph/0212153.) Quantifying this precisely requires a notion of the differential entropy of continuous variable. In this case, the continuous variable is the phase-space point, and the cut-off scale is ħ.

Schematically:

[Entanglement entropy] ~ [Quantum state mixedness] ~ [Wigner function area] ~ [Differential entropy of classical phase-space PDF]

We could make this rigorous as follows. Start with an exact quantum system. Define the entanglement, and define the differential entropy of the

positivepart of the Wigner function. Bound the difference between the entanglement and differential entropy, and then prove that the difference between two quantities vanishes in the ħ→0 limit. (This will require appropriate assumptions about the negative parts of the Wigner function becoming small/negligible in this limit, which is caused by decoherence and is a necessary part of recovering classical behavior. I have a post on that here.)Thanks, Jess. May I ask one more question? Let’s say we have a system of two coupled harmonic oscillators, and let’s suppose they are in an eigenstate of the two-oscillator Hamiltonian. Can I compute the entanglement between these two oscillators in such a state semi-classically? Where can I find a calculation like this?

I don’t know if that calculation has been done (it wouldn’t surprise me), but I can tell you how to do it. Two harmonic oscillators coupled linearly (i.e., using terms like x_1*x_2 and p_1*x_2) can be solved exactly by diagonalizing the Hamiltonian through a linear change of coordinates: (x_1,p_1,x_2,p_2) –> (x’_1,p’_1,x’_2,p’_2). This decouples the two oscillators. Then all you need to do is take the Wigner function of the energy eigenstate of this system, switch back to the original coordinates, and then trace out one of the systems (i.e., integrate over x_2 and p_2). The Wigner function of the energy eigenstate will be a Gaussian function of the 4 coordinates, so the Wigner function corresponding to the reduced state of system 1 will also be Gaussian. You can then calculate the entropy of this state by using the formula from Blume-Kohout and Zurek (above). You may also find Weedbrook et al. useful, arxiv:1110.3234.

Thanks! It makes sense. I will also check out the paper you linked.