KS entropy generated by entanglement-breaking quantum Brownian motion

A new paper of mine (PRA 93, 012107 (2016), arXiv:1507.04083) just came out. The main theorem of the paper is not deep, but I think it’s a clarifying result within a formalism that is deep: ideal quantum Brownian motion (QBM) in symplectic generality. In this blog post, I’ll refresh you on ideal QBM, quote my abstract, explain the main result, and then — going beyond the paper — show how it’s related to the Kolmogorov-Sinai entropy and the speed at which macroscopic wavefunctions branch.

Ideal QBM

If you Google around for “quantum Brownian motion”, you’ll come across a bunch of definitions that have quirky features, and aren’t obviously related to each other. This is a shame. As I explained in an earlier blog post, ideal QBM is the generalization of the harmonic oscillator to open quantum systems. If you think harmonic oscillator are important, and you think decoherence is important, then you should understand ideal QBM.

Harmonic oscillators are ubiquitous in the world because all smooth potentials look quadratic locally. Exhaustively understanding harmonic oscillators is very valuable because they are exactly solvable in addition to being ubiquitous. In an almost identical way, all quantum Markovian degrees of freedom look locally like ideal QBM, and their completely positive (CP) dynamics can be solved exactly.

To get true generality, both harmonic oscillators and ideal QBM should be expressed in manifestly symplectic covariant form. Just like for Lorentz covariance, a dynamical equation that exhibits manifest symplectic covariance takes the same form under linear symplectic transformations on phase space. At a microscopic level, all physics is symplectic covariant (and Lorentz covariant), so this better hold. In addition to the traditional (symmetric) harmonic oscillator, symplectic generality covers skew oscillators and unstable inverted “oscillators”. The reason a lot of different looking dynamics all get called QBM is because they are all special cases of a single ideal class, or slight modification thereto. Non-Markovian modifications, though very important, are idiosyncratic. Likewise, there are lots of ways to model friction in classical physics, but frictionless dynamics are key conceptual bedrock.

Ideal QBM dynamics of a single continuous degree of freedom is a time-indexed CP map \Phi_t parameterized by three objects:

  • H, a 2×2 matrix describing the harmonic Hamiltonian flow
  • \gamma, a scalar quantifying the dissipation
  • D, a 2×2 matrix describing the diffusion

Importantly, D and \gamma have their normal (macroscopic) classical interpretation. When D = 0 = \gamma, ideal QBM reduces to harmonic motion. One way to parameterize H is

(1)   \begin{align*} H =  \left( \begin{array}{cc}-\mu & -1/m \\ m \omega^2 & \mu\end{array} \right) \end{align*}

where m is the mass, \omega is the oscillator frequency, and \mu is the skewWhen \omega^2 > \mu^2 > 0, the elliptical stable orbits of the (skewed) harmonic oscillator are tilted in phase space. When \mu^2 > \omega^2, the potential forms an upside-down oscillator and exhibits hyperbolic unstable orbits.a  .


Here it is:

Einstein-Smoluchowski diffusion, damped harmonic oscillations, and spatial decoherence are special cases of an elegant class of Markovian quantum Brownian motion models that is invariant under linear symplectic transformations. Here we prove that for each member of this class there is a preferred timescale such that the dynamics, considered stroboscopically, can be rewritten exactly as unitary evolution interrupted periodically by an entanglement-breaking measurement with respect to a fixed overcomplete set of pure Gaussian states. This is relevant to the continuing search for the best way to describe pointer states and pure decoherence in systems with continuous variables, and gives a concrete sense in which the decoherence can be said to arise from a complete measurement of the system by its environment. We also extend some of the results of Diósi and Kiefer to the symplectic covariant formalism and compare them with the preferred timescales and Gaussian states associated with the POVM form.

See the appendix for a concise summary of the QBM formalism, and especially the symplectic index notation.

Explanation of result

For simplicity, assume \gamma = 0, i.e., the dissipation is negligible. Let \alpha = (x,p) be a point in phase space and \vert \alpha \rangle be a coherent state centered at \alpha with covariance matrix V. (We suppress V in the notation.) Then there exists a characteristic length of time T and a characteristic covariance matrix V such that the ideal QBM dynamics can be be written down exactly as

(2)   \begin{align*} %\rho_T = \Phi_T[\rho_0] = \int \! \mathrm{d}\alpha \, \vert U_T(\alpha) \rangle_V \langle U_T(\alpha) \cdot \langle \alpha \vert \rho_0 \vert \alpha \rangle_V  \rho_T = \Phi_T[\rho_0] = \int \! \mathrm{d}\alpha_0 \, \vert \alpha_T \rangle \langle \alpha_T \vert \cdot \langle \alpha_0 \vert \rho_0 \vert \alpha_0 \rangle  \end{align*}

where \alpha_T = U_T(\alpha_0) = e^{TH}\alpha_0 is the (harmonic) Hamiltonian evolutionFor general \gamma, the evolution is \alpha_T = e^{t(H-\gamma I)}\alpha_0 = e^{-t\gamma}(\alpha_T \vert_{\gamma = 0}). This is classical dissipative flow, and therefore not strictly Hamiltonian.b   of the phase space point \alpha_0 in time T. In words: the environment performs a POVM measurement of the system in an overcomplete basis of wavepackets with covariance matrix V, then takes the result \alpha, evolves it forward by a time T using the Hamiltonian flow described by H, and finally prepares the system in the corresponding wavepacket \vert \alpha_T \rangle.

This is an entanglement breaking channel, and gives a very specific and concrete sense in which the environment performs a complete measurement of the state of the system. (If D \to 0, then T \to \infty, which makes sense because unitary evolution isn’t entanglement breaking.)

KS entropy production

So what does this have to do with Kolmogorov-Sinai (KS) entropy?

As I have discussed before, there is good reason to think that the rate at which the wavefunction of macroscopic systems branch is given by the “local” KS entropy, by which we mean the sum of positive local Lyapunov exponentsNote that \Gamma_{\mathrm{KS}} this is an entropy production rate, i.e., in units of bits/time.c  :

(3)   \begin{align*} \Gamma_{\mathrm{KS}} = \sum_{\lambda_i >0} \lambda_i \end{align*}

where the exponents \lambda_i are the eigenvalues of the linearized dynamics H, and give the rate at which local trajectories exponentially diverge in phase space. Recall that linearized Hamiltonian dynamics about any point in phase space has a set of eigenvalues which come in pairs (+\lambda_i,-\lambda_i). Each \lambda_i is either strictly real (unstable, locally hyperbolic trajectories) or strictly imaginary (stable, locally elliptical trajectories). For unstable degrees of freedom, the exponential divergence in one phase-space direction is balanced by an equally rapid convergence in the conjugate direction (in order to satisfy Liouville’s theorem).

We concentrate on the case of one unstable degree of degree of freedom, so the eigenvalues of H are \pm \lambda with

(4)   \begin{align*} \lambda^2 = -\vert H \vert = \Gamma_{\mathrm{KS}}^2 > 0 \end{align*}

where \vert \cdot \vert is the determinant. We now show that when the quantum dynamics of this unstable degree of freedom are almost unitary, then

(5)   \begin{align*} T = \frac{O(1)}{\Gamma_{\mathrm{KS}}} \end{align*}

where O(1) represents a factor of order unity. In other words, when \gamma and D are both small (so that we expect to recover classical Hamiltonian dynamics for macroscopic degrees of freedom), the environment makes an entanglement breaking POVM measurement of the system at a rate consistent with the interpretation that the macroscopic wavefunction of the universe is branching with each measurement. (The small but finite size of the non-unitary dynamics is crucial for understanding how quantum mechanics yields an irreducibly stochastic version of classical mechanics in the anomalous \hbar \to 0 limit.)

To get Eq. (5), we take from the paper that T is identified as the unique positive solution to

(6)   \begin{align*} \vert C_T \vert = \left(\frac{1+e^{-2 \gamma T}}{2}\right)^2 \end{align*}


(7)   \begin{align*} C_t = \int_0^t \! \mathrm{d}\tau\, e^{t(H-\gamma I)} D e^{t(H^\intercal-\gamma I)}. \end{align*}

For sufficiently small \gamma and D, annoying algebra shows that

(8)   \begin{align*} \vert C_t \vert \approx \frac{e^{2 \lambda t}}{4 \lambda^4} \left(\left\vert \frac{H D - D H^\intercal}{2} \right\vert - \vert H \vert \vert D \vert  \right) \end{align*}

where \lambda = \sqrt{\mu^2 - \omega^2} = \Gamma_{\mathrm{KS}} is the rate at which unstable trajectories diverge. Since \gamma \ll \lambda, we have e^{-2 \gamma t} \approx 1 for all relevant times t so solving Eq. (6) gives

(9)   \begin{align*} T = \frac{1}{2 \lambda} \ln \left[ \frac{4 \lambda^4}{ \vert (H D - D H^\intercal)/2 \vert - \vert H \vert \vert D \vert}\right]= \frac{O(1)}{\Gamma_{\mathrm{KS}}}. \end{align*}

The O(1) factor is tied up in my continued confusion about how to think about the branching of continuous degrees of freedom, and I won’t try to understand it here.

The basic idea that weakly open quantum systems generate entanglement entropy at a rate given by the KS entropy is deep and correspondingly well-trodden. (See References [6-21, 34-36] in Asplund and Berenstein (2015) and especially Zurek and Paz (1995).) All that’s new here is the discrete, entanglement-breaking nature of the environmental measurements.


(↵ returns to text)

  1. When \omega^2 > \mu^2 > 0, the elliptical stable orbits of the (skewed) harmonic oscillator are tilted in phase space. When \mu^2 > \omega^2, the potential forms an upside-down oscillator and exhibits hyperbolic unstable orbits.
  2. For general \gamma, the evolution is \alpha_T = e^{t(H-\gamma I)}\alpha_0 = e^{-t\gamma}(\alpha_T \vert_{\gamma = 0}). This is classical dissipative flow, and therefore not strictly Hamiltonian.
  3. Note that \Gamma_{\mathrm{KS}} this is an entropy production rate, i.e., in units of bits/time.
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