# What is quantum Brownian motion?

It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is this one by László Erdös, but the closest the quantum Brownian motion comes to appearing is in this conjecture (p. 30):

[Quantum Brownian Motion Conjecture]: For small [disorder] $\lambda$ and [dimension] $d \ge 3$, the location of the electron is governed by a heat equation in a vague sense: $$\partial_t \big|\psi_t(x)\big|^2 \sim \Delta_x \big|\psi_t(x)\big|^2 \quad \Rightarrow \quad \langle \, x^2 \, \rangle_t \sim t, \quad t \gg 1.$$ The precise formulation of the ﬁrst statement requires a scaling limit. The second statement about the diﬀusive mean square displacement is mathematically precise, but what really stands behind it is a diﬀusive equation that on large scales mimics the Schrödinger evolution. Moreover, the dynamics of the quantum particle converges to the Brownian motion as a process as well; this means that the joint distribution of the quantum densities $\big|\psi_t(x)\big|^2$ at diﬀerent times $t_1 < t_2 < \dots < t_n$ converges to the corresponding ﬁnite dimensional marginals of the Wiener process.

This is the Anderson model in $\mathbb R^d$ with disordered Hamiltonian $H = -\Delta + \lambda V$. The potential $V$ is disordered, and is generated by i.i.d. random fields; the parameter $\lambda$ controls the scale of the disorder.

Classical Brownian motion admits many characterizations and generalizations. For example, Wiener measure leads to the construction of an abstract Wiener space, which is the appropriate setting for the powerful Mallivin calculus. The structure theorem of Gaussian measures says that all Gaussian measures are abstract Wiener measures in this way. I would love to know what all this theory looks like in the language of non-commutative probability theory.

The QBM Conjecture states roughly that a quantum particle in a weakly disordered environment should behave like a quantum Brownian motion. This is an important open problem, but it doesn't quite capture what a QBM is, nor what different types of QBM may exist. Thus my question:

What kind of precise mathematical object is a quantum Brownian motion?

• I think the review by Erdos has nothing to do with non-commutative probability. The goal is to show that in some limit the quantum particle in a disordered medium behaves according to classical Brownian motion, with emphasis on the word "classical". Feb 21, 2013 at 13:31
• @Abdelmalek Abdesselam: to clarify, while the review has nothing do with a non-commutative probabilistic description of quantum Brownian motion, I think that non-comm. prob. theory might be one framework in which to precisely describe QBM. I make no claims that this approach is necessary but it may be useful. Feb 21, 2013 at 16:18

(In words explained below:) Quantum Brownian motion (QBM) is a class of possible dynamics for an open, quantum, continuous degree of freedom in which the reduced dynamics are specified by a quadratic Hamiltonian and linear Lindblad operators in the phase-space variables $x$ and $p$.

Consider the arbitrary time-evolution of a system's density matrix when it is in contact with an environment: \begin{align} \rho = \rho_{\mathcal{S}}(t) = \mathrm{Tr}_{\mathcal{E}} [U_t \sigma^0_{\mathcal{SE}} U_t^\dagger], \end{align} where $\sigma^0_{\mathcal{SE}}$ is the initial global state (both $\mathcal{S}$ and $\mathcal{E}$) and $U_t$ is the unitary governing the global evolution. Then the system is said to evolve according to a special case of quantum Brownian motion — a QBM quantum dynamical semigroup — when the evolution of its density matrix obeys a Lindblad master equation \begin{align} \partial_t \rho = -i [\hat{H},\rho] + \sum_i \left(V_i \rho V_i^\dagger - \frac{1}{2} \{V_i^\dagger V_i, \rho\} \right), \end{align} generated by a time-independent Hamiltonian that is a quadratic polynomial in $x$ and $p$ \begin{align} \hat{H} = \frac{1}{2m}\hat{p}^2 + \frac{\mu}{2} \{\hat{x},\hat{p}\} + \frac{m\omega^2}{2} \hat{x}^2, \end{align} with $\mu$, $m$, and $\omega^2$ real, and by time-independent Lindblad operators that are linear polynomials in the same \begin{align} V_i = a_i \hat{p} + b_i \hat{x}, \qquad (i=1,2) \end{align} with $a_i$ and $b_i$ complex. The master equation can be re-written as \begin{align} \partial_t \rho = -i &[\hat{H},\rho] + i (\lambda/2) [\hat{p},\{\hat{x},\rho\}] - i (\lambda/2) [\hat{x},\{\hat{p},\rho\}] \\ &- D_{pp}[\hat{x},[\hat{x},\rho]] - D_{xx}[\hat{p},[\hat{p},\rho]] + D_{xp}[\hat{p},[\hat{x},\rho]] + D_{px}[\hat{x},[\hat{p},\rho]] \end{align} with coefficients \begin{align} D_{xx} &= \frac{\vert a_1 \vert^2 + \vert a_2 \vert^2}{2} \quad , \quad & D_{pp} &= \frac{\vert b_1 \vert^2 + \vert b_2 \vert^2}{2},\\ D_{xp} &= D_{px} = -\mathrm{Re} \frac{a_1^* b_1 + a_2^* b_2}{2} \quad , \quad & \lambda &= \mathrm{Im} (a_1^* b_1 + a_2^* b_2), \end{align}

More generally, we say a system undergoes quantum Brownian motion when it evolves according to the above master equation, regardless of whether it forms a quantum dynamical semigroup. If it obeys the master equation with time-independent coefficients then the QBM is time-homogeneous (in the sense of a Markov process); otherwise it is time-inhomogeneous. The class of all possible instantaneous QBM dynamics is parameterized by $\mu$, $m$, $\omega^2$, $a_i$, and $b_i$.

The resulting dynamics take a particularly beautiful form in the Wigner representation. The above master equation for $\rho$ is equivalent to the following dynamical equation for the Wigner function $W(x,p)$: \begin{align} \partial_t W = -\frac{p}{m}\partial_x W + m\omega^2 & x \partial_p W + (\lambda - \mu)\partial_x (x W) + (\lambda + \mu)\partial_p (p W)\\ &+D_{pp} \partial^2_x W + D_{xx} \partial^2_p + (D_{xp}+D_{px}) \partial_x \partial_p W. \end{align} More compactly: \begin{align} \partial_t W (\alpha) &= \left[ F_{ab} \partial_a \alpha_b + D_{ab} \partial_a \partial_b \right] W(\alpha) \end{align} where \begin{align} F_{ab} = \left( \begin{array}{cc} \lambda - \mu & -1/m \\ m \omega^2 & \lambda+\mu \end{array} \right) \quad, \quad D_{ab} = \left( \begin{array}{cc} D_{xx} & D_{xp} \\ D_{px} & D_{pp} \end{array} \right) \end{align} are matrices with real elements. Above, the phase-space indices $a,b$ take the values $x,p$, with Einstein summation assumed, so that $\alpha_a$ is a vector in phase space. (The directional derivative $\partial_a$ is just shorthand for $\partial_{\alpha_a}$.)

This is identical in form to a Klein-Kramers equation (more generally a Fokker-Planck-type equation) for the phase-space probability distribution of a classical point particle undergoing Brownian motion.

This is remarkable because such equations were originally derived for true probability distribution, but they also apply to the Wigner function. As a bonus, this gives us an immediate and simple physical interpretation for each of the terms in the QBM master equation

The best comprehensive modern statement of the above definition is probably

which includes comparisions to important special cases discussed by other authors. Here are some more references I found useful in compiling the above:

A. To the extent that you think of Brownian motion as a random walk, the natural quantum extension is the quantum random walk. For a physics perspective, see Quantum random walks - an introductory overview, but you might prefer the more math-oriented exposition of Martin boundary theory of some quantum random walks and On algebraic and quantum random walks.

We give a concise prescription of the concept of a quantum random walk (QRW), using the example of QRW on integers as paradigm. It briefly explains the notion of quantum coin system and the coin tossing map, and summarizes two emblematic properties of that walk, namely the quadratic enhancement of its diffusion rate due to quantum entanglement between the walker and the entropy increase without majorization effect of its probability distributions. We conclude with a group theoretical scheme of classification of various known QRW's.

B. Concerning the relation between Wiener processes and quantum Brownian motion: A quantum version of the wavelet expansion of a Wiener process has been developed in A Levy-Cielsielski expansion for quantum Brownian motion and the construction of quantum Brownian bridges.

Classical Brownian motion has a delightful wavelet expansion obtained by combining the Schauder system with a sequence of i.i.d. standard normals. Our main technical result is to obtain a quantum version of this expansion and so construct quantum Brownian motion in Fock space. Consequently, only the discrete skeleton provided by a "quantum random walk" is required to generate the continuous time process. Our result seems easier to establish than the classical one of Lévy-Cielsielski as we don’t require logarithmic growth estimates on the squares of i.i.d. Gaussians, thanks to the nice action of annihilation operators on exponential vectors.

C. Concerning a mathematical description of the physical phenomenon of Brownian motion: We are then concerned with the effect of an environment having a large (infinite) number of degrees of freedom on the dynamics of a particle with a few degrees of freedom. So we are seeking a quantum theory of friction, diffusion, and thermalization. The seminal paper here is the path integral theory of Caldeira and Leggett. The literature is very extensive, an older but still relevant review is Quantum Brownian Motion: The Functional Integral Approach.

The quantum mechanical dynamics of a particle coupled to a heat bath is treated by functional integral methods and a generalization of the Feynman-Vernon influence functional is derived. The extended theory describes the time evolution of nonfactorizing initial states and of equilibrium correlation functions. The theory is illuminated through exactly solvable models.

• Thanks Carlo. Althoug I certainly want to understand the connection to random walks, I'm more interested in actual Brownian motion (continuous time, continuous space, and dynamic that rigorously arise from microscopic collisions). Sep 20, 2014 at 15:02
• added some more viewpoints beyond random walks. Sep 20, 2014 at 21:45
• I've just posted an answer that I think is the precise definition that Tom LaGatta is looking for, and in any case is what I wanted when I made the bounty. Oct 11, 2014 at 23:37

Is quantum Brownian motion related to Quantum Noise or the quantum Wiener process? I think these notions have a well-established mathematical theory, e.g. there are quantum stochastic integrals defined for them.

For a more physical approach see

Gardiner, Zoller, Quantum Noise, Springer, 2004,

for more mathematical literature see, e.g.,

K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Springer, 1992,

P.A. Meyer, Quantum Probability for Probabilists, Lect. Notes in Math. 1538, Springer, 1995.

The quantum Wiener process has applications to quantum filtering, see, e.g.

L. Bouten, R. van Handel, M. James, An introduction to quantum filtering, http://arxiv.org/abs/math/0601741.

• Good point, @Uwe Franz. I just found another MathOverflow question on the quantum Wiener process: mathoverflow.net/questions/15973/… Feb 21, 2013 at 16:06