PARTICLE TRAJECTORIES FOR QUANTUM MAPS YONAH BORNS-WEIL AND IZAK OLTMAN Abstract. We study the trajectories of a semiclassical quantum particle under re-

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PARTICLE TRAJECTORIES FOR QUANTUM MAPS

YONAH BORNS-WEIL AND IZAK OLTMAN

Abstract. We study the trajectories of a semiclassical quantum particle under re-

peated indirect measurement by Kraus operators, in the setting of the quantized

torus. In between measurements, the system evolves via either Hamiltonian prop-

agators or metaplectic operators. We show in both cases the convergence in total

variation of the quantum trajectory to its corresponding classical trajectory, as de-

ﬁned by the propagation of a semiclassical defect measure. This convergence holds

up to the Ehrenfest time of the classical system, which is larger when the system is

“less chaotic”. In addition, we present numerical simulations of these eﬀects.

In proving this result, we provide a characterization of a type of semi-classical

defect measure we call uniform defect measures. We also prove derivative estimates

of a function composed with a ﬂow on the torus.

Figure 1. A numerical simulation demonstrating this paper’s main result.

Each grey curve represents the trajectory of a quantum particle (under the

evolution procedure discussed in this paper) on a torus with identical initial

conditions and a ﬁxed Hamiltonian. A single trajectory is plotted in red and

the corresponding classical trajectory is plotted as a blue dotted line. Up until

time 0.4, most of the positions of the quantum particles are within 0.1 of the

classical trajectory, but shortly after become decoherent. More numerics are

provided in §4. This simulation is discussed at the end of §1.1 and §4.

arXiv:2210.03224v3 [math-ph] 30 Oct 2023

PARTICLE TRAJECTORIES FOR QUANTUM MAPS 2

1. Introduction

The framework of “quantum trajectories” has gotten increasing attention in recent

years. Instead of treating measurement as a one-time event, researchers consider many

measurements obtained over time and study their eﬀects on the particle being mea-

sured, described mathematically via quantum instruments. The sequence of measured

values is called the quantum trajectory, and is a random variable worthy of study.

In this paper, we take quantum trajectories into the setting of the semiclassical

quantized torus, as studied in Bouzouina–De Bi´evre [BD96], Schenck [Sch09], Dyat-

lov–Jezequel [DJ21], and others. The spaces involved are all ﬁnite-dimensional, which

makes numerical simulations easier than in PDE-based models. Our result is inspired

by the recent work of Benoist, Fraas, and Fr¨ohlich [BFF22], who prove in the PDE

setting that the trajectory of a repeatedly observed quantum particle in the semi-

classical limit approaches the natural notion of a classical trajectory. See Figure 1

for a simulation of this quantum-classical correspondence of particle trajectories on

the quantized torus and §4for more numerics. We strengthen the result in [BFF22]

in our setting by allowing the time to depend on the semiclassical parameter, and

we relate the rate of convergence to the “chaotic behavior” of the underlying classi-

cal system. This matches the intuition that a less chaotic transformation should be

well-approximated for longer by its classical counterpart. Additionally, we illustrate

the results numerically to demonstrate in the chaotic case that our time restriction is

essentially optimal.

1.1. Model. In our setting, a quantum particle is modeled by a density operator (see

Deﬁnition 2.13)ρNon the quantized 2d−dimensional torus HN(see (2.8)) where N∈N

is proportional to the reciprocal of the semi-classical parameter h. This particle is

“indirectly measured” by conjugating by Kraus operators (see Deﬁnition 2.14) OpN(fq)

where {fq}q∈Ωare ﬁxed elements of C∞(T2d) with uniformly bounded derivatives (such

that OpN(fq) provide a resolution of the identity), Ω is a compact metric space with

ﬁnite Borel measure ν, and OpNis the quantization of functions on the torus as

described in §2.4. The measurement procedure is further described in §2.5.

After observing the particle, we evolve it for one unit of time by conjugating ρNby

a unitary operator U, which is either a quantization of a ﬁxed symplectic matrix M

(deﬁned in §2.2) or Schr¨odinger evolution e−i

hPfor a ﬁxed Hamiltonian P. If we repeat

this process ntimes, we get a resulting measure on the space of trajectories. Suppose

F⊂Ωn+1, then the probability of obtaining the set of trajectories Fis given by:

trHNΦ∗

N,qn◦ ··· ◦ Φ∗

N,q0(ρN)dν(q0)···dν(qn)

PARTICLE TRAJECTORIES FOR QUANTUM MAPS 3

where

Φ∗

N,qi(ρ) := UOpN(fqi)ρNOpN(fqi)∗U∗.

Therefore, we have a “quantum probability measure” given by

P(n)

N,ρ := trHNΦ∗

N,qn◦ ··· ◦ Φ∗

N,q0ρdν(q0)···dν(qn) (1.1)

We now describe the trajectory of a corresponding classical particle. Let µbe the

defect measure (deﬁned in Deﬁnition 2.5) of ρN. We interpret µas the probability

distribution of the initial position and momentum of a classical particle. We apply

an “approximate measurement,” which computes an observable quantity qto be in

E0⊆Ω with probability RE0|fq0(ζ)|2dν(q0).The particle is then allowed to classically

evolve for one unit of time by a ﬂow ϕton T2dwhich is either multiplication by

the symplectic matrix Mor else the nonlinear Hamiltonian evolution exp(Hp). We

again repeat the process of alternating measurement and evolution, though we must

remark that unlike in the quantum case, the measurement does not aﬀect the location

of the particle. Conditioned on initially having position/momentum ζ, we have the

probability of classically measuring (q0,...qn) in a measurable set F⊂Ωn+1 to be

ZF|fqn(ϕn(ζ))|2···|fq0(ζ)|2dν(q0)···dν(qn).

One can easily check that (qn)nis a process of independent random variables (although

not identically distributed). As the initial value of ζwas taken to be randomly chosen

from the distribution µ, we take the classical probability of measuring (q0, . . . , qn)∈F

to be ZT2dZF|fqn(ϕn(ζ))|2···|fq0(ζ)|2dν(q0)···dν(qn) dµ(ζ).

We therefore have a “classical probability measure” given by

P(n)

µ:= ZT2d|fqn(ϕn(ζ))|2···|fq0(ζ)|2dµ(ζ) dν(q0)···dν(qn).(1.2)

Note that the functions fqare allowed to overlap so that P(n)

µis a probability of

trajectories in Ωn+1 while ϕtis the unique trajectory on the torus.

The goal of this paper is to describe in what sense, and at what quantitative rate,

the measure P(n)

N,ρ approaches P(n)as Ngrows large.

We can state our main result.

Theorem (Main result).Suppose ρ=ρN:HN→HNis a density operator with

semiclassical defect measure µ. Fix n∈N, and let (qN,0, qN,1, . . . , qN,n)be random

variables with law P(n)

N,ρ given by (1.1)and let (q0, q1, . . . , qn)be random variables with

law P(n)

µgiven by (1.2). Then as N→ ∞,(qN,0, qN,1, . . . , qN,n)converges weakly to

(q0, q1, . . . , qn).

PARTICLE TRAJECTORIES FOR QUANTUM MAPS 4

In proving this, we show uniform convergence of P(n)

N,ρ to P(n)

µand hence

P(n)

N,ρ

N→∞

−−−→ P(n)

in total variation. For a stronger version of Theorem 1.1, see Theorem 1, in which we

provide a quantitative rate of convergence in the case where the number of steps n

approaches the Ehrenfest time (deﬁned in 2.4).

Figure 1numerically samples 60 quantum trajectories from such an evolution pro-

cedure (see the end of §4for more details) for 100 time steps of 0.01 units of time.

Observe that up until t= 0.4, most trajectories are within 0.1 of the classical trajec-

tory1, while shortly after this time, they disagree. This paper’s main result estimates

how long they agree, and to what distribution they agree with.

1.2. Previous Work. The understanding that a measurement aﬀects a quantum state

goes back to the early days of quantum mechanics. In his paper on the uncertainty

principle, Heisenberg [Hei27] (see [Hei83] for an English translation) discussed the so-

called “collapse” of the wave function, which was later rephrased in mathematical terms

by von Neumann [Von13]. When a system is only partially or indirectly measured, the

necessary framework is that of positive operator-valued measures, which were ﬁrst

studied by Naimark [Neu43]. The analog of wave-function collapse was introduced by

Davies and Lewis [DL70] with the development of quantum instruments.

Further research focused on multiple measurements as a probabilistic process, which

could be modeled in either discrete or continuous time. The ﬁrst analysis of continuous-

time measurements was due to Davies [Dav69], and required extensive machinery from

stochastic calculus (See [BG09] or [Hol03] for a brief introduction). Meanwhile, the

discrete-time model consisting of repeated applications of a quantum instrument, while

requiring substantially fewer technicalities, has nonetheless demonstrated a rich vari-

ety of behavior and is a subject of current study. K¨ummerer and Maassen [KM03]

demonstrated ergodicity of a repeated measurement process and showed that the states

approach a pure state provided the measurement operators do not have a “dark” sub-

space [KM06]. Ballesteros et al. [Bal+18] show that when the dynamic is trivial, the

state of the system localizes in space.

If the state is allowed to evolve in-between measurements, the situation becomes

more complex. The mathematical analysis of such a quantum trajectory, which ap-

proximates a classical one, was done by Ballesteros et al. [Bal+21] following extensive

physical evidence of the phenomenon, see the exposition by Figari [FT13]. That pa-

per studied particles, initially in the set of normal states, evolving under a quadratic

1Our result shows that before the Ehrenfest time, at each ﬁxed time, the distribution of quantum

positions is a Gaussian centered at the classical trajectory position with variance given by the precision

of the indirect measurement.

PARTICLE TRAJECTORIES FOR QUANTUM MAPS 5

Hamiltonian. The motivation for our work came from the study of the semiclassical

case under a more general Hamiltonian by Benoist, Fraas, and Fr¨ohlich in [BFF22].

We adapt their framework to the setting of quantum maps. This has two advan-

tages: we can improve on the number of measurements exhibiting classical/quantum

correspondence and we can provide (more easily than in the PDE setting) numerical

simulations. In particular, the numerics indicate the accuracy of the dynamical bound

on the number of measurements.

2. Background

In this section, we provide a background on the measurement procedure as well as

the semiclassical analysis tools required to prove our main result. Throughout this

paper, we consider h= (2πN)−1>0 as the semiclassical parameter where N∈N.

2.1. Semiclassical Analysis on the Real Line. Physically, we are often concerned

with how our quantum mechanical world approximately produces classical mechanics.

This limit is achieved by taking a semiclassical parameter h∈(0,1) to be very small,

and the math involved is called semiclassical analysis.

Deﬁnition 2.1 (symbol class).For each δ∈[0,1/2), the symbol class Sδis deﬁned

Sδ:= {a∈C∞(T∗Rd) : ∀α, β ∈Nd,|∂α

x∂β

ξa(x, ξ)|⩽Cα,βh−δ|α+β|}.

Of most importance is the case where δ= 0, though technical steps of the proof

of Theorem 1will require us to consider general δ. Nevertheless, if δis omitted it

will be taken to be 0. Elements of Sδ(called symbols) may depend on h, though

we require that the constants Cα,β in the deﬁnition be uniform in h. Real-valued

symbols correspond to classical observables, which can be “quantized” to get quantum

observables as follows.

Deﬁnition 2.2 (Weyl quantization).For δ∈[0,1/2), the Weyl quantization of a

symbol a∈Sδis an operator Opha:S(Rd)→ S′(Rd)(sometimes written a(x, hD))

given by

(Oph(a)u)(x) := 1

(2πh)dZZ ei

h⟨x−y,ξ⟩ax+y

2, ξu(y) dydξ,

for u∈ S(Rd).

A theorem of Calder´on and Vaillancourt states that if δ∈[0,1/2) and a∈Sδ, then

Oph(a) is a bounded operator on L2(Rd), with bound

∥Oph(a)∥L2(Rd)→L2(Rd)⩽C∥a∥L∞(Rd)+CX

|α|+|β|⩽K

h|α|+|β|

2∥∂α

x∂β

ξa∥L∞(Rd)(2.1)

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PARTICLETRAJECTORIESFORQUANTUMMAPSYONAHBORNS-WEILANDIZAKOLTMANAbstract.Westudythetrajectoriesofasemiclassicalquantumparticleunderre-peatedindirectmeasurementbyKrausoperators,inthesettingofthequantizedtorus.Inbetweenmeasurements,thesystemevolvesviaeitherHamiltonianprop-agatorsormetaplecticoperators.W...

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PARTICLE TRAJECTORIES FOR QUANTUM MAPS YONAH BORNS-WEIL AND IZAK OLTMAN Abstract. We study the trajectories of a semiclassical quantum particle under re-

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