# Measurement, information, and disturbance
in Hamiltonian mechanics

David Theurel<EMAIL_ADDRESS>Department of Physics, Massachusetts Institute
of Technology, Cambridge, MA 02139

###### Abstract

Measurement in classical physics is examined here as a process involving the
joint evolution of object-system and measuring apparatus. For this, a model of
measurement is proposed which lends itself to theoretical analysis using
Hamiltonian mechanics and Bayesian probability. At odds with a widely-held
intuition, it is found that the ideal measurement capable of extracting finite
information without disturbing the system is ruled out (by the third law of
thermodynamics). And in its place a Heisenberg-like precision-disturbance
relation is found, with the role of $\hbar/2$ played by $k_{B}T/\Omega$; where
$T$ and $\Omega$ are a certain temperature and frequency characterizing the
ready-state of the apparatus. The proposed model is argued to be maximally
efficient, in that it saturates this Heisenberg-like inequality, while various
modifications of the model fail to saturate it. The process of continuous
measurement is then examined; yielding a novel pair of Liouville-like master
equations—according to whether the measurement record is read or
discarded—describing the dynamics of (a rational agent’s knowledge of) a
system under continuous measurement. The master equation corresponding to
discarded record doubles as a description of an open thermodynamic system. The
fine-grained Shannon entropy is found to be a Lyapunov function (i.e.
$\dot{S}\geq 0$) of the dynamics when the record is discarded, providing a
novel H-theorem suitable for studying the second law and non-equilibrium
statistical physics. These findings may also be of interest to those working
on the foundations of quantum mechanics, in particular along the lines of
attempting to identify and unmix a possible epistemic component of quantum
theory from its ontic content. More practically, these results may find
applications in the fields of precision measurement, nanoengineering and
molecular machines.

##  I Introduction

It is commonly held among the wider physics community that the topic of
classical measurement is essentially trivial. I don’t mean the modeling in
physical detail of any one laboratory setup, which of course can get very
complicated, but just the examination of “measurement” as a bare-bones
physical process, idealized away from as many complications as possible; a
theoretical physicist’s model of measurement. One way of stating the wide-held
intuition is that there is in principle no obstruction in classical physics to
measuring any observable of a system with arbitrary precision while disturbing
the system arbitrarily little. This intuition is in sharp contrast to the
situation in quantum physics, where the Heisenberg uncertainty principle
(specifially the Ozawa inequality [1]) asserts just such a limit. Surely
influenced by this attitude, there is a correspondingly sharp contrast between
the little attention ever payed to the measurement process in classical
physics, and the large attention payed over the decades (deservedly) to that
same process in quantum physics. To the best of my knowledge, only a handful
of examples can be associated with the first category: Heisenberg’s own
thought experiments in the late 1920’s [2] (particularly Heisenberg’s
microscope); although they served as the motivation for his quantum
uncertainty principle, they were essentially classical arguments, augmented
only by Einstein’s theory of the photon. In 1996 Lamb and Fearn [3] set up the
problem of a classical point particle (the system) in interaction with a
second point particle (the “apparatus”) subject to noise. They stopped short
of a thorough analysis; their primary interest being the quantum case.
Recently Morgan [4] and Katagiri [5] made use of KvN formalism in separate
attempts to use quantum measurement theory to examine measurement in classical
mechanics.

The only long-lasting foray into classical measurement seems to be the body of
work surrounding Maxwell’s demon and the foundations of thermodynamics. The
demon was first conceptualized by Maxwell in 1867 [6] as a “very observant and
neat-fingered being” capable of monitoring the molecules of a gas, and, by
opening and closing a small door without exerting any work, of sorting the
high-energy molecules from the low, thus creating a temperature gradient. This
amplifier of fluctuations, if it existed, could then be used to run a
perpetual motion machine of the second kind, violating the second law. Writing
in 1929 Szilárd [7] realized that, to save the second law, somewhere in the
measurement process entropy had to be produced. Soon afterwards von Neumann
[8], in his reading of Szilárd, pointed to information acquisition as the key
step incurring entropy cost—a claim that would later be developed by Brillouin
[9, 10]. Building on work by Landauer [11], Bennett in 1982 [12, 13] argued
against Brillouin, pointing instead to erasure of the measurement record as
the key step incurring entropy cost. This 150-year-long inquiry seems to have
finally near a close in recent years, with the realization that the entropy
cost of measurement can in fact be traded between the acquisition and erasure
steps, as reviewed in [14]. Crucially, in the absence of a mature theory of
classical measurement, the latter rigorous analyses had to rely on quantum
measurement theory to settle the problem. Earlier attempts had it worse: with
neither a mature classical nor quantum theory of measurement available, most
of them proceeded by contradiction; requiring that measurement (whatever its
mechanism may be) not be incompatible with the laws of thermodynamics.

The above illustrates three points which I would like to contend: (i) despite
the wide-held intuition, measurement in classical physics is far from trivial;
(ii) it is a woefully underdeveloped subject; and (iii) unacknowledged, it is
a subject whose absence has long held back progress in physics. To address the
issue, a reasonable aim would be a theory of measurement in the context of
Hamiltonian mechanics, which is the mathematical framework at the foundation
of classical physics. The research program I’m suggesting can be summarized
as: to bring Bayesian probability to bear on an ontology governed by
Hamiltonian mechanics, with the full strength, and no more, that is permitted
by the geometro-algebraic structure of the ontology.

The present paper aims to kickstart this program. We begin by noting that the
assumption of perfect information regarding the initial state of the measuring
apparatus is unrealistic. In fact it is ruled out as a matter of principle by
the third law of thermodynamics; initial uncertainty must be present if for
nothing other than for finite-temperature thermal noise. Next we posit a model
of the measurement as a physical process. While some assumptions are made
concerning the systems that can be used as measuring apparatuses, no
restrictions are placed on the system under measurement. This model enjoys
substantial generality while at the same time lending itself to Bayesian
analysis. We then show that, in the process of measurement, the uncertainty in
the state of the apparatus propagates into two uncertainties regarding the
object-system: one is the precision of the measurement; and the other an
uncertainty in the magnitude of the disturbance caused upon the system—that
is, an observer effect. And we find that these two are bound by a Heisenberg-
like precision-disturbance relation. In particular, while we find no obstacle
in principle to making a measurement arbitrarily precise, we do find an
obstruction to realizing such a measurement without disturbance. Thus our
findings are at odds with the wide-held intuition. We argue that our
measurement model is maximally efficient, in the sense that it saturates the
precision-disturbance relation, while deviations from it decrease efficiency.
Next, we derive a novel pair of Liouville-like master equations describing the
dynamics of (a rational agent’s knowledge of) a system under continuous
measurement, according to whether the measurement record is read or discarded.
And we show that the fine-grained Shannon entropy is a Lyapunov function (i.e.
$\dot{S}\geq 0$) of the dynamics when the record is discarded. Finally, we
suggest that the master equation for the case of discarded record doubles as a
description of an open thermodynamic system. In this case the above result
constitutes a novel H-theorem detailing entropy increase in non-equilibrium
thermodynamics. I hope our topic will be of interest to several fields of
physics, particularly to (non-equilibrium) statistical physics and to the
foundations of quantum mechanics.

The rest of the paper is organized as follows. We begin in Section II by
reminding the reader of the basic concepts and equations of Hamiltonian
mechanics. Section III does the conceptual heavy lifting; there we construct
our measurement model and obtain the basic results on which the rest of the
paper is based. In Section IV we arrive at the precision-disturbance relation.
Section V considers the problem of continuous weak measurement over time. The
method of analysis there is drawn directly from the field of continuous
quantum measurement. In Section VI we discuss a number of relevant topics in
the new light of our results: the similarities, and likely coexistence in the
real world, of the classical and quantum uncertainty relations; the approach
to thermal equilibrium in statistical physics; the subtle interplay between
ontology and epistemology in a theory of measurement; and the epistemic
limitations inherent to classical Hamiltonian ontology. The paper ends by
contemplating some of the many possibilities ahead; both the concrete and the
speculative.

##  II Brief recap of Hamiltonian mechanics

Hamiltonian mechanics is a confluence of differential, algebraic and
symplectic geometry, Lie algebra and Lie groups. A wonderful resource for the
topic is [15].

We consider a continuous-time dynamical system over a $2n$-dimensional
symplectic manifold, called _phase space_. The _observables_ of the system
(e.g. position, momentum, angular momentum, etc) are the smooth, single-
valued, real-valued functions defined globally over phase space. By convention
we take observables to not depend explicitly on time. (With this convention,
any explicit time-dependence is regarded as specifying a different observable
at each moment in time.) The points in phase space can be expressed in local
_canonical coordinates_ $(q,p)=(q_{1},\dots,q_{n},p_{1},\dots,p_{n})$
(Darboux’s theorem). The state of the system evolves over time according to
_Hamilton’s equations_ ,

$\dot{q}(t)=\frac{\partial H}{\partial p}(q(t),p(t);t),\ \
\dot{p}(t)=-\frac{\partial H}{\partial q}(q(t),p(t);t),$ (1)

where at each moment the system’s _Hamiltonian_ , $H$, is an observable.
Notice that “Hamiltonian” and “$H$” are indexical terms; they don’t specify
any concrete function over phase space, but refer to whichever observable
happens to serve as the generator of time-evolution (as in (1)) for a given
system at a given time. At each moment Hamilton’s equations describe a _flow_
$\Phi^{H}_{\tau}$ on phase space. Along the integral curves of this flow the
value of any observable $A(q,p)$ changes as

$\dot{A}=\\{A,H\\},$ (2)

where $\\{A,H\\}$ denotes the _Poisson bracket_ ,

$\\{A,H\\}\triangleq\sum_{j=1}^{n}\left(\frac{\partial A}{\partial
q_{j}}\frac{\partial H}{\partial p_{j}}-\frac{\partial A}{\partial
p_{j}}\frac{\partial H}{\partial q_{j}}\right).$ (3)

(Note that (2) follows from (1) after application of the chain rule to
$\frac{d}{dt}A(q,p)$; but also contains (1) as special cases when $A$ equals
one of the canonical coordinates.) Two observables $A,B$ for which $\\{A,B\\}$
is identically zero are said to be in _involution_ with each other. In this
case, by (2), the value of $A$ remains constant along the integral curves of
the flow $\Phi^{B}_{t}$ (and vice versa). It follows that any observable in
involution with the Hamiltonian is a _constant of the motion_. In particular,
if $H$ is not explicitly time-dependent then it is itself a constant of the
motion (conservation of energy). Including itself, a given observable can be
in involution with as few as one and as many as $2n-1$ independent
observables, but only as many as $n$ can be all in involution with one
another. On the other hand, if $\\{A,B\\}=1$ identically then $A,B$ are said
to be _conjugate_ to each other. In this case $B$ is also said to be “the”
_generator of translations_ in $A$ (and vice versa); because, by (2), the
value of $A$ changes monotonically at unit rate along the integral curves of
the flow $\Phi^{B}_{t}$. A given observable, $A$, may fail to have a conjugate
observable. In this case it is still possible to speak of a locally-defined
conjugate “quantity”, $B$, which satisfies $\\{A,B\\}=1$ but fails to satisfy
the stringent definition of a bona fide observable. This is illustrated on the
2D phase space by the observable $I=\frac{1}{2}(q^{2}+p^{2})$ (the Hamiltonian
for the simple harmonic oscillator); whose conjugate quantity
$\phi=\arg(q+ip)$ (the phase of oscillation for the sho) either fails to be
globally continuous, or else fails to be single-valued (depending on one’s
choice of definition).

Notice that the components of $(q,p)$ satisfy the _canonical relations_

$\\{q_{i},q_{j}\\}=\\{p_{i},p_{j}\\}=0,\quad\\{q_{i},p_{j}\\}=\delta_{ij},$
(4)

so each canonical coordinate is in involution with all other coordinates but
one, to which it is conjugate. A diffeomorphism of phase space,
$(q,p)\mapsto(q^{\prime},p^{\prime})$, such that $(q^{\prime},p^{\prime})$
again satisfy these canonical relations is said to be a _canonical
transformation_. Canonical transformations have Jacobian determinant equal to
1, so they preserve the _Liouville measure_ of phase space volume,
$d^{n}qd^{n}p=d^{n}q^{\prime}d^{n}p^{\prime}$. For any flow parameter, $\tau$,
the Hamiltonian flow $\Phi^{H}_{\tau}$ is an example of an (active) canonical
transformation; in particular, Hamiltonian flow preserves the Liouville
measure (Liouville’s theorem). Changes of coordinates implemented by (passive)
canonical transformations are particularly convenient since they preserve the
simple form of the Liouville measure, the equations of motion (1, 2), and the
Poisson bracket (3).

##  III A model of measurement in a Hamiltonian world

Suppose we wished to measure an observable $A(q,p)$ of the system (1) at time
$t_{0}$. In the world of Hamiltonian mechanics this can only be done by
coupling the system to a measuring apparatus, where the joint system ($=$
object-system $+$ apparatus) is itself a Hamiltonian system, with

$\displaystyle H_{\text{joint}}(q,p,x,y;t)$
$\displaystyle=H(q,p;t)+H_{\text{app}}(x,y;t)$
$\displaystyle\quad+H_{\text{int}}(q,p,x,y;t).$ (5)

Here $(x,y)$ are canonical coordinates on the $2m$-dimensional phase space of
the apparatus; $H_{\text{app}}$ is the apparatus’ Hamiltonian; and
$H_{\text{int}}$ is the interaction between system and apparatus, which we
will assume to be switched on only briefly around $t=t_{0}$. We now stipulate
a model for the measurement.

###  III.1 System-apparatus coupling

Consider the “gauge”, or “pointer display”, of the apparatus; by which I mean
the observable of the apparatus which, after interaction with the system, we
want to reflect the sought-after value of $A$ at time $t_{0}$. Denote this
observable of the apparatus by $Q(x,y)$. Suppose $Q$ has a conjugate
observable, $P(x,y)$. For the interaction to imprint the value of $A$ on $Q$,
the interaction Hamiltonian must involve $A$ and the conjugate quantity to
$Q$, namely $P$; because this is the generator of translations in $Q$ [16].
The simplest interaction of this form is the product

$H_{\text{int}}(q,p,x,y;t)=\sqrt{k}\delta(t-t_{0})A(q,p)P(x,y),$ (6)

where the constant $k>0$ represents the _strength_ of the measurement (see
Section III.3), and $\delta(t-t_{0})$ is the Dirac delta function indicating
that the measurement is idealized as taking place instantaneously at $t_{0}$.
Without loss of generality we will assume that $P^{2}$ has physical dimensions
of energy. (Otherwise this can be achieved by scaling $(Q,P)\mapsto(aQ,P/a)$
for some suitable choice of constant $a$.) The physical dimensions of $k$ are
then $[k]=\text{energy}\cdot\text{time}^{2}/[A]^{2}$.

###  III.2 Readying the apparatus

Let us take a step back to consider how to initialize the apparatus into its
“ready state” prior to interaction at $t_{0}$. Being, as we are, in the
process of defining what we mean by “measurement”, on pain of circularity we
shouldn’t appeal to measurement to assess the state of the apparatus, as might
be needed to actively manipulate it into a state ready for measurement of the
system. This difficulty can be circumvented by letting low-temperature
thermalization take care of confining the state of the apparatus to a narrow
region of its phase space. The region in question can be specified
experimentally by setting up a deep energetic well there—a “trap”. This trap
could be due to a confining gravitational or electrostatic potential; a
combination of near-field electric and magnetic fields; a light field; atomic
chemical bonds; etc. We write

$H_{\text{app}}(x,y;t)=H_{\text{app}}^{\text{own}}(x,y)+\Pi(t)H_{\text{trap}}(x,y),$
(7)

where $H_{\text{app}}^{\text{own}}$ is the apparatus’ own, or internal,
Hamiltonian, which we take to be time-independent; and $\Pi(t)$ is a
rectangular step-function taking only the values $1/0$, describing the on/off
switch of the trap. The trap will be switched off for all $t>t_{0}$; it is
only switched on in the time leading up to $t_{0}$, to help bring the
apparatus into its ready state, as we will now describe. The trap consists of
a deep energetic well which, when switched on ($\Pi(t)=1$), sets the ground
state of the apparatus at some point $(x^{*},y^{*})$. Without loss of
generality we may set our coordinates such that $(x^{*},y^{*})=(0,0)$, and we
may assume that the corresponding energy is
$H_{\text{app}}(x^{*},y^{*})|_{\text{trap on}}=0$. (Otherwise these conditions
can be met by shifted redefinitions of $x,y,H_{\text{app}}$.) We Taylor-expand
$H_{\text{app}}(x,y)|_{\text{trap on}}$ around the ground state, obtaining a
positive-definite quadratic form:

$\displaystyle H_{\text{app}}(x,y)\Big{|}_{\text{trap on}}$
$\displaystyle=H_{\text{app}}^{\text{own}}(x,y)+H_{\text{trap}}(x,y)$
$\displaystyle=\frac{1}{2}\begin{pmatrix}x&y\end{pmatrix}\hat{M}\begin{pmatrix}x\\\
y\end{pmatrix}+\mathcal{O}(3),$ (8)

where $\hat{M}$ is a symmetric positive-definite $2m$-by-$2m$ matrix of
coefficients, and $\mathcal{O}(3)$ denotes all higher-degree terms in the
series. As shown by Whittaker [17] (see also theorem by Williamson [18],
explained in [15, appendix 6]), there exists a local linear canonical
transformation $(x,y)\mapsto(z,w)$ which reduces (8) to the normal form

$\displaystyle H_{\text{app}}(z,w)\Big{|}_{\text{trap on}}$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{m}(b_{i}^{2}z_{i}^{2}+w_{i}^{2})+\mathcal{O}(3).$
(9)

Here $b_{1}\geq b_{2}\geq\dots\geq b_{m}>0$ are constants with physical
dimensions of angular frequency; they are the natural frequencies of
oscillation of the apparatus around its trapped ground state.

Now to ready the apparatus: while the trap is on, the apparatus is brought
into contact with a thermal bath at some temperature $T=1/\beta k_{B}$,
allowed to equilibrate, and then isolated again [19]. After this our knowledge
about the state of the apparatus is given by the Boltzmann probability
distribution

$\rho(z,w)\,d^{m}zd^{m}w\propto e^{-\beta H_{\text{app}}(z,w)|_{\text{trap
on}}}\,d^{m}zd^{m}w.$ (10)

Note that in the time between isolation from the bath and measurement at
$t_{0}$ the evolution of the apparatus will preserve this distribution, as
opposed to spoiling the preparation, since $H_{\text{app}}|_{\text{trap on}}$
is constant under the phase-space flow generated by itself and such flow
preserves the Liouville measure $d^{m}zd^{m}w$.

At this point we make three requirements that constrain the apparatuses,
traps, and temperatures allowed by our model. (i) We require that the trap be
harmonic enough, or the temperature be low enough, that in the Boltzmann
distribution (10) the higher-degree terms in (9) can be neglected. (ii) We
require that at least one of the coordinates $z_{i}$ be in involution with
$H_{\text{app}}^{\text{own}}$. Let $i=i^{*}$ be the index of this special
coordinate. (If given a choice, we want the associated frequency $b_{i^{*}}$
to be as large as possible, for a reason to be seen in Section IV.) The
condition means that $z_{i^{*}}$ will be a constant of the motion of the
apparatus when the trap is switched off—a desirable property for the pointer
$Q$ (introduced in Section III.1); so that the measurement record is stable
after the interaction has past. We thus identify the pointer $Q\triangleq
z_{i^{*}}$ and its conjugate $P\triangleq w_{i^{*}}$. We denote the
corresponding frequency by $\Omega\triangleq b_{i^{*}}$. Note the physical
interpretation of $\Omega$ as the natural frequency of oscillation of the
pointer around its trapped state. Since $Q,P$ are observables, in making these
identifications we’re implicitly assuming that (iii) the pair of conjugate
local quantities $(z_{i^{*}},w_{i^{*}})$ are globally extendable to smooth
single-valued functions on phase space. From now on $Q,P$ are the only
observables of the apparatus with which we will be concerned. With the above
requirements met, we can easily marginalize over all other variables in (10)
to find the probability distribution over the pointer and its conjugate:

$\rho(Q,P)\,dQ\,dP=\frac{\beta\Omega}{2\pi}e^{-\frac{\beta\Omega^{2}}{2}Q^{2}-\frac{\beta}{2}P^{2}}dQ\,dP.$
(11)

This is the apparatus ready state. It describes a preparation in which the
pointer and its conjugate have been set independently to zero, but there
remains some uncertainty on their exact values.

###  III.3 Integrating Hamilton’s equations

Integrating Hamilton’s equations for the joint system, the effect of the
interaction (6) is to instantaneously change the state of both object-system
and apparatus as [20]

$\displaystyle\begin{pmatrix}q\\\ p\end{pmatrix}_{t_{0}^{+}}$
$\displaystyle=\Phi^{A}_{\sqrt{k}P}\begin{pmatrix}q\\\
p\end{pmatrix}_{t_{0}^{-}}$ (12a) $\displaystyle\begin{pmatrix}Q\\\
P\end{pmatrix}_{t_{0}^{+}}$ $\displaystyle=\begin{pmatrix}Q+\sqrt{k}A(q,p)\\\
P\end{pmatrix}_{t_{0}^{-}}$ (12b)

where $\Phi^{A}_{\tau}$ is the transformation on the system’s phase space that
implements flowing for a “time” $\tau$ under the Hamiltonian flow generated by
$A$. Having initialized the apparatus to its ready state (11) prior to the
interaction, then, in view of (12b), after the interaction our state of
knowledge of the apparatus, conditional on a given state of the system at the
time of measurement, is

$\displaystyle\rho(Q,P|q,p)dQdP$
$\displaystyle=\frac{\beta\Omega}{2\pi}e^{-\frac{\beta\Omega^{2}}{2}\left(Q-\sqrt{k}A(q,p)\right)^{2}-\frac{\beta}{2}P^{2}}dQdP.$
(13)

Note that the dependence on $(q,p)$ is only through $A(q,p)$.

The trap on the apparatus is released at the moment of measurement ($\Pi(t)=0$
for $t>t_{0}$), so that the apparatus Hamiltonian returns to its internal
setting $H_{\text{app}}^{\text{own}}$. By construction the pointer $Q$ is in
involution with this Hamiltonian, so it constitutes a stable record of the
measurement. At this time (i.e. any time after $t_{0}$) we read the pointer on
the apparatus, yielding some definite value $Q^{*}$, or equivalently

$A^{*}\triangleq\frac{Q^{*}}{\sqrt{k}}.$ (14)

($A^{*}$ is just the reading on the pointer with the scale set appropriately.)
Note that this does not mean that the value of $A$ at the time of measurement
is $A^{*}$! Rather, given this datum, the likelihood function for the value of
$A$ at the time of measurement is, from (13),

$\displaystyle\rho(A^{*}|A)dA^{*}=\sqrt{\frac{\beta
k\Omega^{2}}{2\pi}}e^{-\frac{\beta
k\Omega^{2}}{2}\left(A^{*}-A\right)^{2}}dA^{*}.$ (15)

This completes our model of measurement. The _measurement record_ $A^{*}$, or
equivalently the likelihood function (15) (with $A^{*}$ specified),
constitutes the outcome of the measurement.

###  III.4 Consuming the measurement

There are two operations that one, as a recipient, should perform to consume
the information of the measurement. The first is triggered by the information
that the observable $A$ of the system was measured at time $t_{0}$ by the
stipulated procedure, with specified settings $(\beta,k,\Omega)$. As seen in
(12a), the interaction involved in this measurement affects the state of the
system by causing it to move along the flow generated by $A$ for some unknown
“time” $\sqrt{k}P$. If one knew the value of $P$ then one should change their
probability distribution about the state of the system at time $t_{0}$
according to

$\rho(q,p;t_{0}^{-})\mapsto\rho(q,p;t_{0}^{+})=\left[\left(\Phi^{A}_{\sqrt{k}P}\right)_{*}\rho\right](q,p;t_{0}^{-}),$

where $(\Phi^{A}_{\tau})_{*}$ denotes the push-forward of the transformation
$\Phi^{A}_{\tau}$, defined as
$[(\Phi^{A}_{\tau})_{*}\rho](q,p)\triangleq\rho(\Phi^{A}_{-\tau}(q,p))$; and
the $+/-$ superscripts on $t_{0}$ are meant as a reminder that this update
reflects a physical transition of the system that took place in a short time
interval around $t_{0}$. But one does not know the value of $P$; all that is
know about it is expressed by the probability distribution (11). One folds
this in by marginalizing over $P$:

$\rho(q,p;t_{0}^{+})=\sqrt{\frac{\beta}{2\pi}}\int
dP\,e^{-\frac{\beta}{2}P^{2}}\left[\left(\Phi^{A}_{\sqrt{k}P}\right)_{*}\rho\right](q,p;t_{0}^{-}).$
(16)

The second operation is triggered by the information of the measurement
outcome (15). One assimilates this by performing the Bayesian update
$\rho_{\text{prior}}(q,p;t_{0})\mapsto\rho_{\text{posterior}}(q,p;t_{0})$,
with

$\displaystyle\rho$
${}_{\text{posterior}}(q,p;t_{0})\propto\rho_{\text{prior}}(q,p;t_{0})\rho(A^{*}|A(q,p))$
$\displaystyle\qquad\qquad\quad\propto\rho_{\text{prior}}(q,p;t_{0})e^{-\frac{\beta
k\Omega^{2}}{2}\left(A^{*}-A(q,p)\right)^{2}},$ (17)

where the omitted factor of proportionality is just the normalization,
obtained by integrating the expression shown over the system’s phase space
($\int d^{n}qd^{n}p$). Since multiplication by a function of $A$ commutes with
the push-forward $\left(\Phi^{A}_{\tau}\right)_{*}$, operations (16, 17) can
be performed in either order to the same effect. If (17) is performed first,
it corresponds to updating one’s knowledge about the state the system was in
before the measurement was made (i.e. at $t_{0}^{-})$; if second, about the
state the system was left in by the measurement. Notice that if only the fact
of the measurement is revealed but not the outcome (in this case we say the
outcome was _discarded_), then one should only perform operation (16), not
(17).

Finally, if a single number is desired as an objective quantification of the
measured observable (i.e. not biased by anyone’s prior), the maximum-
likelihood estimate can be given, from (15):

$\displaystyle A\Big{|}_{t_{0}}=A^{*}\pm\frac{1}{\sqrt{\beta k\Omega^{2}}}$
(18)

(mean $\pm$ standard deviation) [21]. We will refer to

$\epsilon_{A}\triangleq\frac{1}{\sqrt{\beta k\Omega^{2}}}$ (19)

as the _precision_ of the measurement, or of the measurement’ outcome. (But
notice that to translate this to an uncertainty in a given agent’s knowledge
of $A$ we must first combine the likelihood function with the agent’s prior,
as in (17).) At this point it is clear why $k$ represents the strength of the
measurement: the larger $k$ the higher the measurement’s precision [22].

##  IV A Heisenberg-like precision-disturbance relation in Hamiltonian
mechanics

The measurement model we’ve just stipulated is characterized by the triad
$(\beta,k,\Omega)$; respectively the (inverse) temperature of the thermal bath
($=$ temperature of apparatus), the measurement strength, and the frequency of
oscillation of the apparatus’ pointer around its trapped ground state. Notice,
from (19), that for a given bath and trap (i.e. fixed $\beta,\Omega$), we can
make our measurement of $A$ more precise by cranking up the strength, $k$,
which we might think of as a knob on our experimental setup. However, notice
that the more precisely $A$ is measured (i.e. the larger $k$), the more the
system is affected by the measurement (i.e. the longer the “flow time”
$\sqrt{k}P$ in (12a)), and more importantly, the more uncertain we are about
the magnitude of said observer effect (since the uncertainty in $P$,
$1/\sqrt{\beta}$ from (11), translates into uncertainty $\sqrt{k/\beta}$ in
the “flow time”). It’s worth emphasizing that this disturbance of the system
is not arbitrary, but has the form of time-evolution along the Hamiltonian
flow generated by the measured observable, $A$; the only thing uncertain is
how much “time” the system flowed. We can see that this disturbance will
affect some observables of the system more than others: in particular, any
observable $B$ in involution with $A$ will emerge undisturbed in the immediate
aftermath of the measurement; although subsequent time-evolution under the
system’s own dynamics will cause the initial disturbance to “leak into” such a
$B$, unless $B$ is also in involution with $H$.

Concretely, we find that the precision of a measurement (19), and the
_disturbance_ caused by the measurement upon the system,

$\eta_{A}\triangleq\sqrt{k/\beta},$ (20)

obey the inverse relation

$\epsilon_{A}\eta_{A}=\frac{1}{\beta\Omega},$ (21)

which is fixed for a given bath and trap; independent of the identity of the
system measured, of that of the system used as measuring apparatus, of the
measurement strength and of the choice of observable measured. The product on
the left-hand side can easily be made larger (see discussion in Section VI.1)
but not smaller, as far as I can tell. This Heisenberg-like precision-
disturbance relation (or “uncertainty relation” for short) suggests an
obstruction to how close we can come in a world governed by Hamiltonian
mechanics to the idealization of measurement without disturbance. To the
extent that our model of measurement has a claim to generality, relation (21)
will be a general principle. Note that this relation is softer than the
Heisenberg uncertainty principle of quantum mechanics: for any given bath and
trap one will have a finite obstruction on the right-hand side of (21), but
one can always endeavor to make the obstruction smaller by cooling the
apparatus further or tightening the trap. Instead this obstruction is of a
kind with the third law, to which it is clearly related: it suggests that it
is impossible by any procedure, no matter how idealized, to reduce the
observer effect of measurement to zero in a finite number of operations.

##  V Continuous measurement over time: new master equations and H-theorem

Extracting information about the system by measurement increases our knowledge
about some aspect of it. However, we’ve seen that any such measurement
according to our model will disturb the system to an extent that we cannot
monitor; and this decreases our knowledge about some other aspect of the
system. For a single measurement this tradeoff is expressed by the precision-
disturbance relation (21), or in more detail by the updates (16, 17). In this
section we explore the compound effect of such tradeoff due to multiple
measurements; specifically, a continuous succession of vanishingly-weak
measurements. The method of analysis we follow is drawn from the field of
continuous quantum measurement, which addresses the corresponding problem in
that setting. (See for example [23].)

Subdivide a finite interval of time $[0,T]$ into $N$ equal subintervals
demarcated by $t_{0}=0<t_{1}<t_{2}<\dots<t_{N}=T$, with $t_{j}=j\Delta t$. For
each $j\in\\{1,\dots,N\\}$, select an observable $A_{j}=A_{j}(q,p)$ of the
system, and prepare for it a measurement $(\beta_{j},k_{j}\Delta
t,\Omega_{j})$ to be carried out at time $t_{j}$. Notice that we’ve scaled the
strength according to the size of the subintervals; smaller $\Delta t$ means
each individual measurement is weaker, but a greater number of them fit into
$[0,T]$. We will see that this is the right scaling for the effects to
converge when we take the limit of smaller and smaller $\Delta t$. (Note that
this changes the physical dimensions of $k_{j}$; they are now
$[k_{j}]=\text{energy}\cdot\text{time}/[A_{j}]^{2}$.) The resulting tuple of
pointer readings
$\mathbf{A}^{*}\triangleq(A^{*}_{1},A^{*}_{2},\dots,A^{*}_{N})$ constitutes
the measurement record for the entire succession of measurements. To
assimilate the $j$-th measurement we perform the two operations (16, 17),
resulting in the update

$\displaystyle\rho$ $\displaystyle(q,p;t_{j+1})\propto
e^{-\frac{\beta_{j}(k_{j}\Delta
t)\Omega_{j}^{2}}{2}\left(A_{j}^{*}-A_{j}(q,p)\right)^{2}}$
$\displaystyle\quad\cdot\sqrt{\frac{\beta_{j}}{2\pi}}\int
dP_{j}\,e^{-\frac{\beta_{j}}{2}P_{j}^{2}}\left[\left(\Phi^{A_{j}}_{\sqrt{k_{j}\Delta
t}P_{j}}\right)_{*}\rho\right](q,p;t_{j}).$ (22)

For small $\tau$, the push-forward

$\left[\left(\Phi^{A}_{\tau}\right)_{*}\rho\right](q_{0},p_{0})=\rho\left(\Phi^{A}_{-\tau}(q_{0},p_{0})\right)=\rho(q(-\tau),p(-\tau))$
(23)

can be calculated by Taylor-expanding the function
$\tau\mapsto\rho(q(-\tau),p(-\tau))$ around $\tau=0$; using the chain rule to
pass all time-derivatives onto $q,p$, and calculating the latter from
Hamilton’s equations with Hamiltonian $A$. The result is

$\left(\Phi^{A}_{\tau}\right)_{*}\rho=\rho+\tau\\{A,\rho\\}+\frac{\tau^{2}}{2}\\{A,\\{A,\rho\\}\\}+\mathcal{O}(\tau^{3}).$
(24)

Putting this into (22), the integral over $P_{j}$ can then be done order-by-
order. The odd-order terms all vanish by symmetry, leaving us with

$\displaystyle\rho$ $\displaystyle(q,p;t_{j+1})\propto
e^{-\frac{\beta_{j}k_{j}\Omega_{j}^{2}\Delta
t}{2}\left(A_{j}^{*}-A_{j}\right)^{2}}$
$\displaystyle\qquad\cdot\left(\rho+\frac{k_{j}\Delta
t}{2\beta_{j}}\\{A_{j},\\{A_{j},\rho\\}\\}+\mathcal{O}(\Delta
t^{2})\right)\Bigg{|}_{(q,p;t_{j})}.$ (25)

###  V.1 Case of discarded measurement record

Figure 1: Master equation dynamics in various measurement regimes. Evolution
of the state of knowledge $\rho(q,p;t)$ of a rational agent under master
equation (38) is illustrated in a simple example: the system under measurement
is a 1D simple harmonic oscillator (sho); the measurement is characterized by
constant $\beta,k,\Omega$ and fixed $A$; the measured observable is the energy
$A=H\triangleq\frac{1}{2}(\omega^{2}q^{2}+p^{2})$; and the initial
distribution over phase space is unimodal. Although not proven here, three
timescales are involved: that of internal dynamics, $\tau_{\text{dyn}}\sim
1/\omega$; that of diffusion due to observer effect,
$\tau_{\text{dif}}\sim\beta/k\omega^{2}$; and that of collapse due to Bayesian
update on the measurement record, $\tau_{\text{col}}\sim 1/\beta
k\Omega^{2}\Delta E^{2}$, where $\Delta E$ is the target certainty on $H$
(i.e. $\tau_{\text{col}}$ is the characteristic timescale for the variance of
$\rho(H;t)$ to fall below $\Delta E^{2}$). (a) Phase portrait showing level
sets of the sho Hamiltonian. (b–e) From left to right, snapshots of
$\rho(q,p;t)$ at successive times, indicated at top in units of the sho
period, for four different measurement regimes (rows). For ease of
visualization the color scheme (left) is normalized anew for each plot. (b)
Regime $\tau_{\text{dyn}}\ll\tau_{\text{dif}},\tau_{\text{col}}$; describes an
isolated system; (38) reduces to the Liouville equation $\partial\rho/\partial
t=\\{H,\rho\\}$. (c) Regime
$\tau_{\text{dyn}}\sim\tau_{\text{dif}}\ll\tau_{\text{col}}$; describes case
of discarded measurement record; (38) reduces to (26). Notice entropy
increase, in accordance with (26), due to diffusion along the flow generated
by $A$. (d) Regime
$\tau_{\text{dyn}}\sim\tau_{\text{col}}\ll\tau_{\text{dif}}$; describes an
approximation to ideal classical measurement with minimal disturbance. Notice
the trend of decreasing entropy, in accordance with (41), due to collapse
towards the measurement outcome. (e) Regime
$\tau_{\text{dyn}}\sim\tau_{\text{col}}\sim\tau_{\text{dif}}$; describes the
three processes (dynamics, diffusion and collapse) happening together. Notice
the tradeoff between information about $A$ and information about the conjugate
quantity (sho phase). (f) Evolution of the first four cumulants of $\rho(A;t)$
in regime d (equivalently regime e). For ease of visualization each cumulant
is rescaled to 1 at $t=0$. Note qualitative agreement with (43).

Let’s pause to consider the case in which the measurement record
$\mathbf{A}^{*}$ is discarded. In this case we should skip update (17), which
amounts to dropping the exponential factor and the omitted proportionality
factor in (25). Taking then the limit $\Delta t\to dt$ describing a continuous
succession of vanishingly-weak measurements, we arrive in this case (discarded
measurement record) at

$\displaystyle\frac{\partial\rho}{\partial t}$
$\displaystyle=\underbrace{\\{H,\rho\\}}_{\begin{subarray}{c}\text{internal
dynamics}\\\ \text{Hamiltonian flow}\\\ \text{info
preserved}\end{subarray}}+\underbrace{\frac{k}{2\beta}\\{A,\\{A,\rho\\}\\}}_{\begin{subarray}{c}\text{observer
effect}\\\ \text{diffusion along flow $\Phi^{A}_{t}$}\\\ \text{info about $A$
preserved}\\\ \text{other info lost}\end{subarray}}$ (26)

where we’ve introduced the well-known Liouville term $\\{H,\rho\\}$ accounting
for the internal dynamics of the system under $H$ [24], which we had been
ignoring until now; and all quantities shown may be explicit functions of
time. This is a Liouville-like master equation, with an additional second-
order term due to the observer effect of measurement. We can get some sense
for the effect of this new term as follows. Let $B(q,p;t)$ denote any function
over phase space, possibly explicitly time-dependent. Here and throughout
let’s use $\langle\,\cdot\,\rangle$ to denote the phase-space average:

$\langle B\rangle\triangleq\int d^{n}qd^{n}p\,\rho(q,p;t)B(q,p;t).$ (27)

In Appendix A we prove that under master equation (26) any such phase-space
average evolves as

$\displaystyle\frac{d}{dt}\langle B\rangle$
$\displaystyle=\langle\\{B,H\\}\rangle+\left\langle\frac{\partial B}{\partial
t}\right\rangle-\frac{k}{2\beta}\left\langle\\{A,\log\rho\\}\\{A,B\\}\right\rangle.$
(28)

The first term on the right-hand side of this equation is due to the Liouville
term in (26); the second term is due to any explicit time-dependence of $B$;
and the third term is due to the second-order term in (26). As a special
application of this equation consider $B=-\log\rho$, in which case the phase-
space average is the Shannon entropy:

$S(t)\triangleq\langle-\log\rho\rangle.$ (29)

It is not hard to show that the first two terms on the right-hand side of (28)
vanish in this case [25]. Thus we find that under dynamics (26),

$\displaystyle\dot{S}$
$\displaystyle=\frac{k}{2\beta}\left\langle\\{A,\log\rho\\}^{2}\right\rangle\geq
0.$ (30)

It is a well-known result that $S(t)$ remains constant ($\dot{S}=0$) under the
Liouville equation $\partial\rho/\partial t=\\{H,\rho\\}$ (see example in
Figure 1b). In breaking with that, we have just found that entropy generally
increases over time under (26) on account of the new term. Thus the Liouville
term preserves information, while the second-order term causes information
loss. Indeed, in accordance with our discussion in Section IV concerning the
nature of the observer effect, this term describes diffusion along the flow
lines generated by the instantaneous observable $A(q,p;t)$ (see example in
Figure 1c). This diffusion preserves instant-to-instant information pertaining
to that instant’s observable $A(q,p;t)$, while it erases information
pertaining to observables not in involution with it.

We note here, in passing, that (30) may alternatively be construed as an
H-theorem detailing entropy increase in an open thermodynamic system in
contact with a thermal bath (described here as the collection of apparatuses).
In this light (26, 28, 30) may help further our understanding of non-
equilibrium statistical physics and the approach to thermal equilibrium. We
discuss this topic further in Section VI.2.

###  V.2 Case of simulated measurement record

Returning now to (25), suppose instead that the measurement record is not
discarded but that we have only yet read up to the $(j-1)$-th entry; i.e.
$A^{*}_{1}$ through $A^{*}_{j-1}$ are known while $A^{*}_{j}$ onward are not.
We would like to simulate ahead of time (say, on a computer) how our state of
knowledge will evolve as we continue to read more of the record. However,
without the benefit of hindsight the upcoming record entries appear to us as
random variables. The language for this kind of simulation is stochastic
calculus. (See tutorial on stochastic calculus in [23].) Let us first ask:
what should be our probability distribution for the $j$-th outcome,
$A_{j}^{*}$? Making use of the likelihood function (15), this question can be
answered in terms of our current knowledge of the value of $A_{j}$:

$\displaystyle\rho(A_{j}^{*};t_{j})$ $\displaystyle=\int
dA_{j}\,\rho(A_{j};t_{j})\rho(A_{j}^{*}|A_{j})$ $\displaystyle\propto\int
dA_{j}\,\rho(A_{j};t_{j})e^{-\frac{\beta_{j}(k_{j}\Delta
t)\Omega_{j}^{2}}{2}\left(A_{j}^{*}-A_{j}\right)^{2}}.$ (31)

As $\Delta t\to dt$, the Gaussian in this expression becomes very wide and
spread out as a function of $A_{j}$. The distribution $\rho(A_{j};t_{j})$
becomes very narrow by comparison, and can be replaced by a Dirac delta, which
must be centered at $\langle A_{j}\rangle$ for the means to match. Using the
delta to do the integral over $A_{j}$ we have, up to a normalization factor,

$\displaystyle\rho(A_{j}^{*};t_{j})$ $\displaystyle\underset{\Delta t\to
dt}{\longrightarrow}e^{-\frac{\beta_{j}k_{j}\Omega_{j}^{2}\Delta
t}{2}\left(A_{j}^{*}-\langle A_{j}\rangle\right)^{2}}.$ (32)

By a simple change of variables we introduce $\Delta W_{j}$, our probability
distribution of which is a zero-mean Gaussian with variance $\Delta t$, and in
terms of which

$A_{j}^{*}=\langle
A_{j}\rangle+\frac{1}{\sqrt{\beta_{j}k_{j}\Omega_{j}^{2}}}\frac{\Delta
W_{j}}{\Delta t}.$ (33)

The value of expressing (32) this way is two-fold. From a simulation
standpoint, we can use a random number generator to sample $\Delta W_{j}$ from
its Gaussian distribution, and (33) then tells us how to convert this into a
sample of $A_{j}^{*}$. And from an analysis standpoint, this expression
enables a very convenient form of calculation: in the limit $\Delta t\to dt$
we write

$A^{*}=\langle A\rangle+\frac{1}{\sqrt{\beta k\Omega^{2}}}\frac{dW}{dt},$ (34)

where $W(t)\triangleq\int_{0}^{t}dW$ is a standard Wiener process, with $dW$
obeying the basic rule of Itô calculus $dW^{2}=dt$. Notice also that $\Delta
W_{j}$ is statistically-independent from $A_{j}$. Using
$\llangle\,\cdot\,\rrangle$ to denote averaging over the Wiener process we
thus have, for any function $f(A)$:

$\llangle f(A)dW\rrangle=f(A)\llangle dW\rrangle=0.$ (35)

Taking stock: given $\rho(q,p;t_{k})$ for a given time $t_{k}$ we can use it
to calculate $\langle A_{k}\rangle$, and combine this with the output of a
random number generator as in (33) to simulate the upcoming entry of the
measurement record $A_{k}^{*}$. We can then use (25) to calculate what our
updated state of knowledge $\rho(q,p;t_{k+1})$ would be upon reading that
entry, and iterate the process. Analytically we proceed as follows. Substitute
(33) into (25); expand the square in the exponent, discarding the overall
factor $\exp{-\Delta W_{j}^{2}/2\Delta t}$ which is independent of $(q,p)$;
and Taylor-expand the exponential, keeping in mind that powers of $\Delta
W_{j}$ count for “half an order”, to obtain

$\displaystyle\rho(q,p$ $\displaystyle;t_{j+1})\propto$
$\displaystyle\mspace{-55.0mu}\Bigg{(}1-\frac{\beta_{j}k_{j}\Omega_{j}^{2}}{2}(A_{j}-\langle
A_{j}\rangle)^{2}\Delta t+\sqrt{\beta_{j}k_{j}\Omega_{j}^{2}}(A_{j}-\langle
A_{j}\rangle)\Delta W_{j}$
$\displaystyle+\frac{\beta_{j}k_{j}\Omega_{j}^{2}}{2}(A_{j}-\langle
A_{j}\rangle)^{2}\Delta W_{j}^{2}+\mathcal{O}(\Delta t\,\Delta W_{j})\Bigg{)}$
$\displaystyle\cdot\Bigg{(}\rho+\frac{k_{j}\Delta
t}{2\beta_{j}}\\{A_{j},\\{A_{j},\rho\\}\\}+\mathcal{O}(\Delta
t^{2})\Bigg{)}\Bigg{|}_{(q,p;t_{j})}.$ (36)

In the limit of continuous measurement $\Delta t\to dt,\Delta W_{j}\to
dW,\Delta W_{j}^{2}\to dt$ this reduces to

$\displaystyle\rho(q,p;t+dt)$
$\displaystyle\propto\rho+\frac{k}{2\beta}\\{A,\\{A,\rho\\}\\}dt$
$\displaystyle\quad+\sqrt{\beta k\Omega^{2}}(A-\langle
A\rangle)\rho\,dW\Bigg{|}_{(q,p;t)},$ (37)

where again all quantities shown may be explicit functions of time. One can
check that the right-hand side is already normalized, so the omitted factor of
proportionality is $1$. We arrive in this case (simulated measurement record)
at

$\displaystyle\frac{\partial\rho}{\partial t}$
$\displaystyle=\underbrace{\\{H,\rho\\}}_{\begin{subarray}{c}\text{internal
dynamics}\\\ \text{Hamiltonian flow}\\\ \text{info
preserved}\end{subarray}}+\underbrace{\frac{k}{2\beta}\\{A,\\{A,\rho\\}\\}}_{\begin{subarray}{c}\text{observer
effect}\\\ \text{diffusion along flow $\Phi^{A}_{t}$}\\\ \text{info about $A$
preserved}\\\ \text{other info lost}\end{subarray}}$
$\displaystyle\quad+\underbrace{\sqrt{\beta k\Omega^{2}}(A-\langle
A\rangle)\rho\,\frac{dW}{dt}}_{\begin{subarray}{c}\text{Bayesian update}\\\
\text{collapse towards measurement outcome}\\\ \text{non-linear \& non-
local}\\\ {\mathbb{\llangle}\Delta\text{info}\rrangle\geq 0}\end{subarray}},$
(38)

where again we’ve re-introduced the Liouville term $\\{H,\rho\\}$ accounting
for the internal dynamics of the system. Compared to (26) we now have a new
stochastic term appearing, which is due to assimilation of the simulated
measurement record via Bayesian update. To get some sense for the effect of
this new term, in Appendix B we prove that under master equation (38) the
Shannon entropy (29) evolves as

$\displaystyle\dot{S}$
$\displaystyle=\underbrace{\frac{k}{2\beta}\left\langle\\{A,\log\rho\\}^{2}\right\rangle}_{\begin{subarray}{c}\text{observer
effect}\\\ \Delta\text{entropy}\geq 0\end{subarray}}$
$\displaystyle\quad\underbrace{-\frac{\beta
k\Omega^{2}\sigma_{A}^{2}}{2}-\sqrt{\beta k\Omega^{2}}\left\langle(A-\langle
A\rangle)\log\rho\right\rangle\frac{dW}{dt}}_{\begin{subarray}{c}\text{Bayesian
update}\\\ \text{can be positive or negative}\end{subarray}},$ (39)

where

$\sigma_{A}^{2}\triangleq\langle(A-\langle A\rangle)^{2}\rangle$ (40)

is the variance in our knowledge of $A$. The first term on the r.h.s. of (39)
is familiar from (30); it describes increasing entropy due to the observer
effect of measurement. The remaining two terms are due to the stochastic term
in (38); these two together may be positive for particular measurement
outcomes, but they are non-positive on average, as can be seen by invoking
(35):

$\llangle\dot{S}\rrangle=\underbrace{\frac{k}{2\beta}\left\langle\\{A,\log\rho\\}^{2}\right\rangle}_{\begin{subarray}{c}\text{observer
effect}\\\ \Delta\text{entropy}\geq 0\end{subarray}}\underbrace{-\ \
\frac{\beta k\Omega^{2}\sigma_{A}^{2}}{2}}_{\begin{subarray}{c}\text{Bayesian
update}\\\ \llangle\Delta\text{entropy}\rrangle\leq 0\end{subarray}}.$ (41)

Thus the stochastic term in (38) leads, on average, to increasing information
(see example in Figure 1d,e). It is also interesting to note that this term is
both non-linear and non-local in $\rho$, since $\langle A\rangle$ depends on
the value of $\rho$ everywhere on phase space. To gain further insight into
the effects of this term, suppose the measured observable is fixed $A=A(q,p)$,
and consider our PDF over this observable, $\rho(A;t)$, which is just the
marginal

$\rho(A^{\prime};t)\triangleq\int
d^{n}qd^{n}p\,\delta(A(q,p)-A^{\prime})\rho(q,p;t).$ (42)

Let $\kappa_{i}$ denote the $i$-th cumulant of this distribution. In Appendix
C we prove the following hierarchy of equations describing the contribution of
the stochastic term in (38) to the evolution of these cumulants:

$\displaystyle d\kappa_{1}$ $\displaystyle=\sqrt{\beta
k\Omega^{2}}\,\kappa_{2}\,dW,$ (43a) $\displaystyle d\kappa_{2}$
$\displaystyle=\sqrt{\beta k\Omega^{2}}\,\kappa_{3}\,dW-\frac{\beta
k\Omega^{2}}{2}(2\kappa_{2}^{2})dt,$ (43b) $\displaystyle d\kappa_{3}$
$\displaystyle=\sqrt{\beta k\Omega^{2}}\,\kappa_{4}\,dW-\frac{\beta
k\Omega^{2}}{2}(6\kappa_{2}\kappa_{3})dt,$ (43c) $\displaystyle d\kappa_{4}$
$\displaystyle=\sqrt{\beta k\Omega^{2}}\,\kappa_{5}\,dW-\frac{\beta
k\Omega^{2}}{2}(8\kappa_{2}\kappa_{4}+6\kappa_{3}^{2})dt,$ (43d)
$\displaystyle\dots$

Notice in particular the trends $\llangle\dot{\kappa}_{1}\rrangle=0$,
$\llangle\dot{\kappa}_{2}\rrangle\sim-\llangle\kappa_{2}\rrangle^{2}$,
$\llangle\dot{\kappa}_{3}\rrangle\sim-\llangle\kappa_{3}\rrangle$,
$\llangle\dot{\kappa}_{4}\rrangle\sim-\llangle\kappa_{4}\rrangle$, …. These
trends tell us that (supposing $A$ is not explicitly time-dependent and the
Liouville term does not intervene too strongly) the stochastic term in (38)
causes all cumulants of $\rho(A;t)$ higher than second to vanish exponentially
fast, leaving $\rho(A;t)$ a Gaussian; it then causes the variance to vanish
like $\sim 1/t$, while the mean jiggles around in a random walk of zero drift
and volatility decaying with the variance. In the limit in which the
measurement process is complete, $\rho(A;t)$ converges to a delta distribution
centered at the simulation’s putative true value of $A$. (See example in
Figure 1d–f.)

###  V.3 Simultaneous measurements

Simultaneous weak measurement of multiple observables
$A_{1}(q,p),\dots,A_{s}(q,p)$, whether these are in involution or not, can be
handled by letting $A(q,p;t)$ switch between these observables on a fast time
scale. By averaging (38) over this fast time scale this equation then becomes

$\displaystyle\frac{\partial\rho}{\partial t}$
$\displaystyle=\\{H,\rho\\}+\sum_{j=1}^{s}\frac{k_{j}}{2\beta_{j}}\\{A_{j},\\{A_{j},\rho\\}\\}$
$\displaystyle\qquad+\sum_{j=1}^{s}\sqrt{\beta_{j}k_{j}\Omega_{j}^{2}}(A_{j}-\langle
A_{j}\rangle)\rho\,\frac{dW_{j}}{dt},$ (44)

where $(\beta_{j},k_{j},\Omega_{j})$ describes the measurement setup for the
$j$-th observable, and the $W_{j}(t)\triangleq\int_{0}^{t}dW_{j}$ are
independent Wiener processes for $j\neq j^{\prime}$. The analogues of (39) and
(41) for this equation are

$\displaystyle\dot{S}=\sum_{j=1}^{s}\frac{k_{j}}{2\beta_{j}}\left\langle\\{A_{j},\log\rho\\}^{2}\right\rangle$
$\displaystyle-\sum_{j=1}^{s}\left(\frac{\beta_{j}k_{j}\Omega_{j}^{2}\sigma_{A_{j}}^{2}}{2}+\sqrt{\beta_{j}k_{j}\Omega_{j}^{2}}\left\langle(A_{j}-\langle
A_{j}\rangle)\log\rho\right\rangle\frac{dW_{j}}{dt}\right),$ (45)

and

$\llangle\dot{S}\rrangle=\sum_{j=1}^{s}\frac{k_{j}}{2\beta_{j}}\left\langle\\{A_{j},\log\rho\\}^{2}\right\rangle-\sum_{j=1}^{s}\frac{\beta_{j}k_{j}\Omega_{j}^{2}\sigma_{A_{j}}^{2}}{2}.$
(46)

If some of the measurement outcomes are discarded the corresponding terms
should be dropped from the right-most sums in (44–46). If all the outcomes are
discarded we are left with

$\displaystyle\frac{\partial\rho}{\partial t}$
$\displaystyle=\\{H,\rho\\}+\sum_{j=1}^{s}\frac{k_{j}}{2\beta_{j}}\\{A_{j},\\{A_{j},\rho\\}\\},$
(47)

which is linear, local, and deterministic; and

$\displaystyle\dot{S}$
$\displaystyle=\sum_{j=1}^{s}\frac{k_{j}}{2\beta_{j}}\left\langle\\{A_{j},\log\rho\\}^{2}\right\rangle\geq
0.$ (48)

##  VI Discussion

###  VI.1 Comparing the quantum and classical uncertainty relations

How does our classical precision-disturbance relation (21) compare to the
Heisenberg uncertainty principle of quantum mechanics? The latter can be
stated in a few different forms. We will consider the Kennard-Weyl-Robertson
form in section VI.4, where we discuss the epistemology of Hamiltonian
ontology. Here we consider the “joint measurement form” [26, 27, 28, 29],
pertaining to simultaneous measurement of two observables, $A$ and $B$. When
$A,B$ are conjugate to each other this reads:

$\epsilon_{A}\epsilon_{B}\geq\frac{\hbar}{2},$ (49)

where $\epsilon_{A}$ and $\epsilon_{B}$ denote the precisions in the
measurement of $A$ and $B$, respectively [30]; and $\hbar$ is the reduced
Planck constant.

One superficial difference between (21) and (49) is that one is an equality
while the other an _in_ equality. However, this difference is illusory. The
product on the left-hand side of (21) can easily be made larger than the
right-hand side, so that for a more general class of measurement models I
indeed expect us to have

$\epsilon_{A}\eta_{A}\geq\frac{1}{\beta\Omega}.$ (50)

Let us call a measurement that saturates this bound _maximally efficient_ ,
while one failing to saturate it, _inefficient_ (or not as efficient). In
these terms any measurement following our model is maximally efficient, as
we’ve seen. The most obvious modification to our model that leads to
inefficiency is if the measurement outcome is discarded
($\epsilon_{A}\to\infty;\eta_{A}$ unchanged). Another way to reduce efficiency
is if the apparatus’ pointer fails to be in involution with
$H_{\text{app}}^{\text{own}}$, so that some amount of “deterioration” of the
measurement record can happen between the time of the system-apparatus
interaction and whenever the record is read. In the opposite direction, one
might ask: could not a higher efficiency than (21) be reached, say, by using a
pointer and coupling, $(Q,P)$, that are correlated in the apparatus ready
state? To achieve the latter, one would need $(Q,P)$ to _not_ diagonalize the
quadratic form (9); namely, instead of choosing $(Q,P)=(z_{i^{*}},w_{i^{*}})$,
one would choose $(Q,P)$ related to $(z_{i^{*}},w_{i^{*}})$ by some linear
canonical transformation. In fact, although not proven here, I find that this
approach leads to the same precision-disturbance relation (21); the only
difference (aside from the Bayesian analysis becoming more involved) is that a
systematic component is added to the observer effect. This component can be
corrected given the measurement outcome $A^{*}$; so it doesn’t count towards
the disturbance $\eta_{A}$ [31]. In summary, these remarks suggest that, while
it is easy to do worse than (21), it may not be possible to do better; i.e.
they suggest inequality (50) to be the general principle.

A second difference, which remains between (49) and (50), is that one involves
the product of two precisions, while the other the product of a precision with
a disturbance. This difference can be bridged as well. Recall that the
disturbance in question amounts to flowing along $\Phi^{A}_{\tau}$ for an
unknown “time” $\tau$ whose uncertainty is $\eta_{A}$. Under this flow the
“rate” of change of any observable $B$ is as given by (2):
$\frac{d}{d\tau}B=\\{B,A\\}$. In particular, if $B$ is the conjugate to $A$,
so that $\\{B,A\\}=1$, then $B$ increases monotonically at the steady rate of
$1$; and the net effect of the flow on $B$ is simply to displace its value by
$\tau$. (This final step can fail if $B$ has a discontinuity somewhere; so it
is important that $B$ be a bona fide observable, not just a local quantity
such as the phase $\phi$ of an oscillator.) So the uncertainty in the “flow
time”, $\eta_{A}$, translates directly into a disturbance in the value of the
conjugate observable, $B$. This places a bound on the precision,
$\epsilon_{B}$, with which any subsequent measurement can hope to determine
the original value of $B$: $\epsilon_{B}\geq\eta_{A}$, with equality holding
only if the measurement of $B$ is done at full strength. Thus we have

$\epsilon_{A}\epsilon_{B}\geq\frac{1}{\beta\Omega},$ (51)

and the parallel with (49) becomes apparent. Historically it seems that
Heisenberg’s own interpretation of the uncertainty principle was as a
precision-disturbance relation [2], not very different in spirit from (50).
And in recent years work in quantum mechanics has payed considerable attention
to precision-disturbance relations, yielding formulas similar to (50) but with
$\hbar/2$ on the right-hand side [1, 32, 33, 34, 35].

Figure 2: Coexistence of quantum and classical uncertainty relations in the
real world. One may expect the quantum and classical uncertainty relations to
coexist in reality based on the observation that a Hamiltonian world
effectively emerges from quantum mechanics at macroscopic scales. The quantum
relation dominates at low apparatus temperature and/or tightly-trapped pointer
in the apparatus ready state; when $1/\beta\Omega<\hbar/2$ (below the dashed
diagonal). The classical relation dominates in the other direction. Here
$f=\Omega/2\pi$ and $T=1/k_{B}\beta$. Notice that for the range of $(f,T)$
shown, the obstruction is never larger than $\sim 10^{-17}$J$\cdot$s.

The real world is no doubt quantum mechanical, and so the Heisenberg
uncertainty principle is fundamental. But as we know, as one “zooms out” to
larger scales somehow an approximately Hamiltonian world effectively emerges
(Bohr’s correspondence principle and the quantum-to-classical transition).
Hand in hand with the emergence of this effective Hamiltonian world I expect
our classical uncertainty relation to gain traction. Figure 2 illustrates how
the classical and quantum relations then must coexist. For a tight enough trap
and/or cold enough bath (below the dashed diagonal), the obstruction in (51)
is brought below $\hbar/2$ and becomes unreachable; the quantum obstruction
acts like rock bottom. For less tight traps and/or less cold baths the
obstruction in (51) rises above $\hbar/2$ and begins to dominate. Taken
together, one may expect to have in the real world an obstruction that
interpolates between these two; something along the lines of

$\displaystyle\epsilon_{A}\epsilon_{B}$ $\displaystyle\geq\
\frac{\hbar}{2}+\frac{1}{\beta\Omega}\ \ \text{or perhaps}\ \
\frac{\hbar/2}{1-e^{-\frac{1}{2}\beta\hbar\Omega}};$ (52)

it will take a detailed quantum calculation to work out the precise formula
(see Section VI.5 for a germ of how this might be done). To gain some
perspective for the scales involved, note from Figure 2 that, at room
temperature, trap frequencies any lower than about $12$THz (corresponding to
light wave-lengths $\gtrsim 25\mu$m) are already enough to put us in the
classical regime. At the same time, even for the highest temperatures and
lowest frequencies shown in the top-left of Figure 2, the classical
obstruction hardly becomes larger than $\sim 10^{-17}$J$\cdot$s; an extremely
small quantity by macroscopic standards. And yet, even in more moderate
regimes towards the center of Figure 2, the classical obstruction may be
relevant in the contexts of precision measurement, nanoengineering and
molecular machines.

###  VI.2 On the foundations of statistical physics

As obtained here, master equation (26) describes a rational agent’s knowledge
of a Hamiltonian system under continuous measurement when the measurement
record is discarded. As briefly mentioned in Section V.1, this equation can be
repurposed to describe (a rational agent’s knowledge of) a Hamiltonian system
$(q,p)$ in interaction with a thermal bath at (inverse) temperature $\beta$.
Seen in this light, corollary (30) constitutes a novel H-theorem detailing how
the Shannon entropy increases over time for an open thermodynamic system. Note
that this H-theorem deals directly with the fine-grained entropy, without
coarse-graining phase space as in Gibbs’ H-theorem [36]. We should summarize
the inputs that go into justifying this application of (26): (i) equilibrium
thermodynamics to give us prior knowledge about the state of the bath, in
(10); (ii) Hamiltonian mechanics to describe the system-bath dynamics, in
(12); (iii) Bayesian probability to translate ontic into epistemic dynamics,
in (16) [37]. A fourth and final ingredient perhaps deserves the most scrutiny
in future work: (iv) modeling of the bath as a collection of temporally-
uncorrelated systems, each initially in thermal equilibrium, each momentarily
“minimally coupled” to the object-system, as in (6) [38].

Figure 3: Thermalization under master equation (26) according to the ergodic
program. Cartoon of the $2n$-dimensional phase space of system $(q,p)$,
showing a level set of $A(q,p)$ ($(2n-1)$-dimensional), and a single integral
curve of the flow $\Phi^{A}_{t}$ on this level set ($1$-dimensional). Note
that the apparent self-intersections of the curve are an artifact of our low-
dimensional cartoon. For $A$ ergodic, almost all its level sets are densely
filled by almost all integral curves on them. Assuming $H(q,p)\triangleq 0$,
dynamics (26) just describes diffusion along these integral curves. (a–c) The
value of $\rho(q,p;t)$ along one integral curve is illustrated at three
different times. For ease of visualization the color scheme (below) is
normalized anew for each plot. The depicted process happens in parallel along
each integral curve (only one shown). (a) An initial state of knowledge is
localized in a small neighborhood on the level set. (b) As time passes
probability diffuses out along the integral curve. Because the curve is
ergodic, this can lead to rapid spread of probability on the level set. Taking
all integral curves into account amplifies the speed of spreading on the level
set. (c) At late times the distribution converges to a steady state which is
uniform all along the curve. Taking all integral curves into account, the
distribution is now uniform on the level set.

There is ongoing debate among different programs of statistical mechanics for
understanding the approach to thermal equilibrium [39]. Equation (26), and its
corollaries (28, 30), seem ideally poised for informing this debate. For
instance, consider the ergodic program [39], which posits that thermal
equilibrium corresponds to a steady state of $\rho(q,p;t)$. A criticism levied
against this program is that, even if the ergodic hypothesis is
granted—according to which the integral curves of the Hamiltonian flow
$\Phi^{H}_{t}$ will typically fill the level sets of $H$ densely—it still
remains the case that under the Liouville equation it is impossible for a non-
steady state to evolve into a steady state (because this equation implies
$\dot{S}=0$); which seems to lead to the absurd conclusion that thermal
equilibrium could never follow from non-equilibrium. I think this objection
can be entirely resolved by the use of master equation (26) in place of
Liouville’s equation. To see how this comes about consider the special case in
which $A=A(q,p)$ is not explicitly time-dependent, and the system Hamiltonian
is trivial, $H(q,p)\triangleq 0$. In this case (26) simply describes diffusion
along the integral curves of the flow $\Phi^{A}_{t}$. Clearly, from any given
initial condition a steady state will be reached in which probability is
distributed uniformly along each integral curve. If we suppose further, as in
the ergodic hypothesis, that such integral curves typically fill the level
sets of $A$, then we see that an initially-localized distribution (Figure 3a)
will converge as time passes to a distribution uniform on the level sets of
$A$ (Figure 3c). In the general case, the features of the steady state
(including whether or not there is one) will depend on the interplay between
$H$ and $A(t)$ (alternatively, between $H,A_{1},\dots,A_{s}$ in (47)). I
expect that settings can be found corresponding to the various ensembles of
statistical mechanics. For example, Figure 1c illustrates a situation in which
the coupling to the bath does not disturb the energy of the system. As shown,
the steady state is uniformly distributed on the energy level sets. If the
value of the energy is precisely known (Figure 1e, late time), this is just
the micro-canonical ensemble.

We should noted that, despite the time-irreversibility present in (26), it
would be mistaken to suggest this equation as an answer to Loschmidt’s
reversibility objection [39] and the issue of the arrow of time. Rather, I
think the arrow of time has been baked into (26) by our model of the thermal
bath; according to which in each infinitesimal interval of time the system
comes into contact with a fresh uncorrelated thermal system and leaves it a
little bit correlated. This is not unlike the effect that Boltzmann’s
molecular chaos hypothesis has for his H-theorem [36, 24]. Personally I’m not
too troubled by this, since I consider the “past hypothesis” [39] a better
avenue for pursuing this issue.

Finally, I would emphasize that the equations we have derived are valid for
systems of any size, not just for the large-size limit; they are well suited
for studying fluctuations from the mean in finite-size thermodynamic systems,
both in and out of equilibrium.

###  VI.3 Reading the measurement record; is it turtles all the way down?

We return to a complication that was tactically overlooked in Section III. In
our measurement model, after the apparatus had interacted with the
system—let’s call that step “pre-measurement”—, we stipulated that the pointer
on the apparatus should be read, which would yield some definite value
$Q^{*}$. But what could it mean to “read $Q$” if not to measure this
observable of the apparatus? This seems to lead us down an infinite regression
in which the system is pre-measured by an apparatus, which must then be pre-
measured by an apparatus, which must then be…the passage from “systems
interacting” to “agent being informed” never quite taking place. In a sense
this predicament is similar to the quantum measurement problem, particularly
as articulated by the Wigner’s friend thought experiment [40]. In both
settings, the paradoxical step is the passage from what seems to be best
regarded as an ontic-level description (Hamiltonian or unitary dynamics) to
what seems to be best regarded as an epistemic-level operation (Bayesian
updating or collapse of the wave function). At the epistemic level we speak
freely of agents, observers, measurements, observations, reading the
measurement record, information, probability, Bayesian updating, collapse of
the wave function. But at the ontic level all of these are complicated
phenomena, resisting precise characterization. Until such characterizations
are available [41] we are stuck with the shifty split—to use a term coined by
Bell [42].

In our model, the shifty split was introduced at one degree of separation from
the system under study: we described the system-apparatus interaction at the
ontic level; then we described the apparatus-agent interaction at the
epistemfic level. Such a once-removed approach can be very useful, as
demonstrated by the example of the theory of general quantum measurement [43].
However, there is a pair of consistency tests that should be checked of such a
theory. Notice that the once-removed theory contains the twice-removed and
higher theories. To see this, simply let what we have been calling the object-
system instead be used as an apparatus of some kind to (pre-)measure another
system. (Now we have an object-system which is pre-measured by an apparatus,
which is pre-measured by a second apparatus, whose pointer is “read” by an
agent.) This maneuver uses only the rules of the ontology, yet succeeds in
shifting away the split by one degree. In light of this feature of the theory,
for a first test (T1) we ask: is there any such way to shift away the split
that increases the efficiency of our measurement (i.e. decreases the
obstruction in (50))? If the answer is yes then test (T1) is failed; it is a
sign that the way we are bridging the shifty split does not fully exploit the
possibilities allowed by the ontology, and we should strengthen it. (This
strengthening is necessary. So long as test (T1) is not passed, bounds such as
(50) derived from once-removed measurement schemes cannot be taken seriously,
since they can be circumvented by better use of the operations allowed by the
ontology.) On the other hand, for a second test (T2) we ask: is it the case
that when we shift away the split, no matter how we do it, we find that it
always decreases the efficiency of our measurement (i.e. increases the
obstruction in (50))? If the answer is yes then test (T2) is failed; it is a
sign that the way we are bridging the shifty split requires an operation that
is not allowed by the ontology, and must be revised.

How does our model fair on these tests? Consider test (T2) first. We have
bridged the shifty split by stipulating: (i) “read” the pointer on the
apparatus, yielding some definite value $Q^{*}$. (This is after the apparatus
has pre-measured the object-system.) Here’s a way to shift away the split
without reducing efficiency. Instead of (i) do: (ii) use a second apparatus,
operated according to our same model with parameters
$(\beta^{(2)},k^{(2)},\Omega^{(2)})$, to pre-measure at full strength
$(k^{(2)}\to\infty)$ the value of $Q$, recording it on its own pointer
$Q^{(2)}$ which we then “read”, yielding some definite value $Q^{(2)*}$. To
see that procedure (ii) is just as efficient as (i) notice, first, that
neither of the two involve further disturbance of the object-system, so
$\eta_{A}$ is the same for both. Second, since the final measurement in
procedure (ii) is at full strength, we have $\epsilon^{(2)}_{Q}\propto
1/\sqrt{k^{(2)}}\to 0$, so this measurement reveals the exact value of $Q$;
$Q^{(2)*}=Q^{*}$. Thus procedure (ii) leads to the same likelihood function
(15) as (i), and hence to the same $\epsilon_{A}$. This gives us proof-by-
example that it is possible to shift away the split without reducing our
model’s efficiency, so test (T2) is passed. It is harder to prove that test
(T1) is passed since this requires proving that efficiency remains unimproved
_for all_ ways of shifting away the split. I don’t know how to do that, but I
conjecture that our model passes this test too. In support of this conjecture,
note one way in which efficiency might have been increased by shifting away
the split, but isn’t. Turn again to procedure (ii) just discussed. Imagine
that, after having measured the pointer of the first apparatus ($Q$) as
described, it were possible to do another measurement on this apparatus to
determine $P$. If we could gain even a little bit of information about the
value that $P$ had at the time of interaction with the object-system
($t_{0}$), we could combine it with what we know of $P(t_{0})$ from the
thermal distribution (11) to reduce our uncertainty $\sigma_{P}$, and hence
reduce $\eta_{A}=\sqrt{k}\sigma_{P}$. If this were possible our model would
fail test (T2). That it is impossible follows from the fact that the
measurement of $Q$ in procedure (ii) had to be done at full strength
$(k^{(2)}\to\infty)$, which leads to an infinite disturbance of the first
apparatus ($\eta^{(2)}_{Q}\propto 1/\epsilon^{(2)}_{Q}\to\infty$); and this
infinitely disturbs the value of $P$ (since $\\{Q,P\\}=1$). So we see that in
the course of carrying out procedure (ii) all information about $P(t_{0})$ is
lost beyond recovery.

###  VI.4 On the epistemology of Hamiltonian ontology

Consider the Kennard-Weyl-Robertson (KWR) form of the Heisenberg uncertainty
principle of quantum mechanics [44]. For a pair of conjugate observables, $A$
and $B$, it reads:

$\sigma_{A}\sigma_{B}\geq\frac{\hbar}{2},$ (53)

where $\sigma_{A}$ and $\sigma_{B}$ denote the standard deviations at a given
time in our knowledge of $A$ and $B$, respectively [30]. This form of the
uncertainty principle speaks directly to the limits of what can be known about
the state of a quantum system; that is, to the epistemology of quantum
ontology. In this section we ask whether the present developments allow us to
establish an analogous result about the epistemology of Hamiltonian ontology.

Notice, first of all, the sense in which we must understand such a question.
Unlike in the quantum formalism, there is nothing in our classical formalism
that rules out the possibility of _starting_ with perfect information about
conjugate observables:

$\rho(A^{\prime},B^{\prime};t)=\delta(A(q(t),p(t))-A^{\prime})\delta(B(q(t),p(t))-B^{\prime}),$
(54)

where

$\displaystyle\rho(A^{\prime},B^{\prime};t)$ $\displaystyle\triangleq\int
d^{n}qd^{n}p\,\delta(A(q,p)-A^{\prime})$
$\displaystyle\qquad\qquad\cdot\delta(B(q,p)-B^{\prime})\rho(q,p;t).$ (55)

Rather, the question is whether it is at all possible to _arrive_ at such a
state of perfect information from a state of less information. In particular:
suppose we were handed a Hamiltonian system of which we knew nothing at all,
so that $\rho$ were initially uniform on phase space. Does there exist a
sequence of measurements on the system that would take $\rho$ into the
perfect-information state (54)?

Consider the direct approach of performing simultaneous measurement of $A$ and
$B$, with respective measurement settings $(\beta_{A},k_{A},\Omega_{A})$ and
$(\beta_{B},k_{B},\Omega_{B})$. The evolution of $\rho$ is as given by master
equation (44). Suppose that the measurements are strong enough that they come
to completion on a much faster timescale than that of the system’s dynamics,
so that the Liouville term in (44) can be neglected. It is a bit tricky,
because one must be mindful of the rules of Itô calculus, but one can check
that, starting from an uncorrelated Gaussian distribution in $A$ and $B$ (of
which the uniform distribution is the special case of infinite variances), the
general solution to (44) is

$\rho(A,B;t)=\frac{1}{2\pi\sigma_{A}\sigma_{B}}e^{-\frac{(A-\mu_{A})^{2}}{2\sigma_{A}^{2}}-\frac{(B-\mu_{B})^{2}}{2\sigma_{B}^{2}}},$
(56)

where the means $\mu_{A},\mu_{B}$ are stochastic functions of time evolving as

$\displaystyle d\mu_{A}$
$\displaystyle=\sqrt{\beta_{A}k_{A}\Omega_{A}^{2}}\sigma_{A}(t)^{2}dW_{A},$
(57a) $\displaystyle d\mu_{B}$
$\displaystyle=\sqrt{\beta_{B}k_{B}\Omega_{B}^{2}}\sigma_{B}(t)^{2}dW_{B};$
(57b)

while the variances $\sigma_{A}^{2},\sigma_{B}^{2}$ are the deterministic
functions of time

$\displaystyle\sigma_{A}(t)^{2}$
$\displaystyle=\frac{\left[\coth{\sqrt{(\beta_{A}/\beta_{B})k_{A}k_{B}\Omega_{A}^{2}}(t-t_{0})}\right]^{l_{A}}}{\sqrt{\beta_{A}\beta_{B}(k_{A}/k_{B})\Omega_{A}^{2}}},$
(58a) $\displaystyle\sigma_{B}(t)^{2}$
$\displaystyle=\frac{\left[\coth{\sqrt{(\beta_{B}/\beta_{A})k_{A}k_{B}\Omega_{B}^{2}}(t-t_{0})}\right]^{l_{B}}}{\sqrt{\beta_{A}\beta_{B}(k_{B}/k_{A})\Omega_{B}^{2}}},$
(58b)

where $t_{0}<t$ is a constant of integration, and $l_{A},l_{B}\in\\{+1,-1\\}$.
We see that, under simultaneous measurement of conjugate observables, an
initially-Gaussian PDF remains Gaussian for all time. Also, much like we saw
in (43), the mean of the distribution executes a random walk (this time in two
dimensions) of volatilities proportional to the variances. However, unlike in
(43), now the variances converge to non-zero values as the measurement runs to
completion:

$\displaystyle\sigma_{A}^{2}$
$\displaystyle\to\frac{1}{\sqrt{\beta_{A}\beta_{B}(k_{A}/k_{B})\Omega_{A}^{2}}},$
(59a) $\displaystyle\sigma_{B}^{2}$
$\displaystyle\to\frac{1}{\sqrt{\beta_{A}\beta_{B}(k_{B}/k_{A})\Omega_{B}^{2}}}.$
(59b)

This comes about because the measurement of $A$ causes collapse “along the
$A$-direction” (along the integral curves of $\Phi^{B}_{\tau}$) and diffusion
“perpendicular to the $A$-direction” (along the integral curves of
$\Phi^{A}_{\tau}$); while the simultaneous measurement of $B$ causes the
converse; and at completion of the measurement the effects precisely cancel
out. Notice that (59) gives

$\displaystyle\sigma_{A}\sigma_{B}$
$\displaystyle\to\frac{1}{\sqrt{\beta_{A}\Omega_{A}}\sqrt{\beta_{B}\Omega_{B}}},$
(60)

which begins to resemble (53). Is it the case that the product
$\sigma_{A}(t)\sigma_{B}(t)$ remains above this limit at all times? That
depends on the exponents $l_{A},l_{B}$. The case $l_{A}=+1$ gives
$\sigma_{A}(t_{0}^{+})\to\infty$; it describes complete ignorance about $A$ at
some past time $t_{0}$. On the other hand, the case $l_{A}=-1$ gives
$\sigma_{A}(t_{0}^{+})=0$; it describes perfect information about $A$ at the
past time $t_{0}$. Likewise for $l_{B}$. Since we are interested in beginning
from a state of ignorance, the relevant solution for us has $l_{A}=l_{B}=+1$.
It then follows from (58) that the inequality

$\displaystyle\sigma_{A}\sigma_{B}\geq\frac{1}{\sqrt{\beta_{A}\Omega_{A}}\sqrt{\beta_{B}\Omega_{B}}}$
(61)

holds for all times. If both measurements are characterized by the same
(inverse) temperature $\beta$ and trap frequency $\Omega$, this further
reduces to

$\displaystyle\sigma_{A}\sigma_{B}\geq\frac{1}{\beta\Omega}.$ (62)

We have derived this uncertainty-uncertainty relation by considering
simultaneous measurement of the pair of conjugate observables $A,B$. Could
this be a general epistemic obstruction, or is there some different sequence
of measurements that fares better? We leave the question open for future
investigation.

###  VI.5 Future directions

In closing we look to some of the many questions and possibilities ahead. We
have argued that our model of measurement is maximally efficient; i.e. that it
is impossible to do better than (50) in the way of measuring without
disturbing. It would be of fundamental interest to have a proof of this claim
at the level of rigor of mathematical physics. Complementing this, it would be
valuable to have experimental tests of (50, 51, 62) and of the bigger picture
outlined in Figure 2. Moving forward it will be worth honing our intuition
about the range of possible dynamics of a Hamiltonian system under measurement
(or more accurately, of the epistemic state of a rational agent about a
Hamiltonian system under measurement). For this it would be good to see
numerical studies of equations (26, 38) in more interesting scenarios than the
one-dimensional simple harmonic oscillator explored in Figure 1. For this task
it might be useful if somebody reproduced in the classical setting the steps,
discovered already in the quantum setting, to transform a non-linear
stochastic master equation like (38) into an equivalent linear equation [45,
23]. Another calculation that I think should be attempted is the derivation of
the precise version of (52), which I suspect can be obtained by reproducing
our measurement model from Section III in the quantum setting.

As discussed in Section VI.2, equations (26, 28, 30) seem like a promising
basis for a better understanding of equilibrium and non-equilibrium
statistical physics. I think this research program deserves much attention. To
name just two topics of inquiry in this direction: (i) Can the various
ensembles of equilibrium statistical mechanics indeed be obtained as the
equilibrium solutions to (26) under suitable choices of the coupling to the
bath, $A$? And if so, what does this teach us about the approach to thermal
equilibrium in each case? (ii) How do the present results relate to the body
of work on Maxwell’s demon reviewed in Section I? Can the old debate now be
resolved within classical physics, without recourse to quantum physics?

In connection with the quantum measurement problem and the interpretation of
quantum mechanics, there is a program dating back to Einstein [46, 47] of
attempting to identify and unmix a possible epistemic component of quantum
theory from its ontic content. In recent times this program has made promising
progress at the hands of Caves, Fuchs, and others [48, 49, 47]. In particular
Spekkens [50, 51], and Bartlett, Rudolph, and Spekkens [52], have illustrated
how an uncircumventable epistemic limitation in an otherwise classical world,
much like what is suggested by our discussion in Section VI.4, can lead to
several of the phenomena usually regarded as characteristic of quantum
mechanics. It will be interesting to see what these two programs can
contribute to each other.

Finally I would like to venture the following speculative suggestions. (i) As
we know from general relativity, gravity couples directly to energy. Perhaps a
system subject to a strong external gravitational field can, in certain cases,
be reasonably modeled by (26) with $A=H$. If so, could this tell us something
interesting about the entropy of a system falling onto the event horizon of a
black hole? Could this be a useful tool for studying black hole
thermodynamics? (A quantum version of this master equation (c.f. earlier
comments in connection to (52)) might be an even better tool.) (ii) To the
best of my knowledge, theoretical computer science grounds its notions of
computability and complexity in concrete (if highly abstracted and idealized)
physical models. Does the existence of an obstruction to ideal measurement
without disturbance in Hamiltonian mechanics have some bearing on those
notions of computer science grounded in the world of classical physics [53]?
(iii) Hamilton’s equations and their underlying geometro-algebraic structure
are not unique to physics; they emerge wherever the equations of a theory can
be gotten out of a variational principle [54]. Indeed, in classical physics
they emerge in just this way from Hamilton’s principle of stationary action.
In particular, optimal control theory uses essentially the same equations
under the name of Pontryagin’s minimum principle [55]. Could the present
results have consequences for aspects of optimal control under partial
information and, by extension, for artificial intelligence? At the least,
these musings illustrate the breadth of potential implications of our subject.

###### Acknowledgements.

It is a pleasure to thank Omar Eulogio López, Kurt Jacobs, and Pavel Chvykov
for reading versions of the typescript and providing many useful suggestions.
I’m particularly grateful to Matthew A. Wilson, for believing in me when I
most needed it and for his continued mentorship and patience. While conducting
this research I was supported by the Picower Neurological Disorder Research
Fund.

##  Appendix A Derivation of equation (28)

Our objective is to derive equation (28). For brevity of notation we will omit
the integration measure $d^{n}qd^{n}p$ in integrals over phase space. We will
make use of the identity

$\int A\\{B,C\\}=\int B\\{C,A\\}=\int C\\{A,B\\},$ (63)

which is valid for any smooth functions $A(q,p;t),B(q,p;t),C(q,p;t)$ as long
as their product decays to zero as $\|(q,p)\|\to\infty$, so that boundary
terms from integration by parts can be discarded. This identity is readily
verified:

$\displaystyle\int A\\{B,C\\}$ $\displaystyle=\int
A\sum_{i=1}^{n}\left(\frac{\partial B}{\partial q_{i}}\frac{\partial
C}{\partial p_{i}}-\frac{\partial B}{\partial p_{i}}\frac{\partial C}{\partial
q_{i}}\right)$ $\displaystyle=\int C\sum_{i}\left(-\frac{\partial}{\partial
p_{i}}\left(A\frac{\partial B}{\partial q_{i}}\right)+\frac{\partial}{\partial
q_{i}}\left(A\frac{\partial B}{\partial p_{i}}\right)\right)$
$\displaystyle=\int C\sum_{i}\left(-\frac{\partial A}{\partial
p_{i}}\frac{\partial B}{\partial q_{i}}+\frac{\partial A}{\partial
q_{i}}\frac{\partial B}{\partial p_{i}}\right)$ $\displaystyle=\int
C\\{A,B\\}.$ (64)

In our applications of the identity one of the factors will always be
homogeneous in $\rho$, which it is safe to assume decays fast enough for the
identity to hold (e.g. for each $t$, $\rho(q,p;t)$ can be assumed to have
compact support over phase space without any loss of physical generality.)

Now, the phase-space average of $B$ is $\langle B\rangle=\int\rho B$, and the
time derivative of this is

$\displaystyle\frac{d}{dt}\langle
B\rangle=\int\left(B\frac{\partial\rho}{\partial t}+\rho\frac{\partial
B}{\partial t}\right)=\int B\frac{\partial\rho}{\partial
t}+\left\langle\frac{\partial B}{\partial t}\right\rangle.$ (65)

Working with the first term on the r.h.s. here, we substitute into it from
(26):

$\displaystyle\int B\frac{\partial\rho}{\partial t}=\int
B\left(\\{H,\rho\\}+\frac{k}{2\beta}\\{A,\\{A,\rho\\}\\}\right).$ (66)

Using identity (63), the first term on the r.h.s. here can be written as
$\int\rho\\{B,H\\}=\langle\\{B,H\\}\rangle$. Turning to the remaining term on
the r.h.s. of (66), we let $C\triangleq\\{A,\rho\\}$ and again use identity
(63), so that the integral in this term can be written as

$\displaystyle\int B\\{A,C\\}$ $\displaystyle=\int
C\\{B,A\\}=-\int\\{A,\rho\\}\\{A,B\\}$
$\displaystyle=-\int\rho\\{A,\log\rho\\}\\{A,B\\}$
$\displaystyle=-\left\langle\\{A,\log\rho\\}\\{A,B\\}\right\rangle.$ (67)

All together we have

$\displaystyle\frac{d}{dt}\langle B\rangle$
$\displaystyle=\langle\\{B,H\\}\rangle+\left\langle\frac{\partial B}{\partial
t}\right\rangle-\frac{k}{2\beta}\left\langle\\{A,\log\rho\\}\\{A,B\\}\right\rangle,$
(68)

which is (28), as desired.

##  Appendix B Derivation of equation (39)

Our objective is to derive equation (39). For brevity of notation we will omit
the integration measure $d^{n}qd^{n}p$ in integrals over phase space.
Expanding the differential of $S$ (from (29)) to second order in $d\rho$:

$\displaystyle dS$ $\displaystyle=-\int d(\rho\log\rho)$
$\displaystyle=-\int\big{(}(\rho+d\rho)\log(\rho+d\rho)-\rho\log\rho\big{)}$
$\displaystyle=-\int\Bigg{(}(\rho+d\rho)\bigg{(}\log\rho+\frac{d\rho}{\rho}-\frac{1}{2}\frac{d\rho^{2}}{\rho^{2}}\bigg{)}-\rho\log\rho\Bigg{)}$
$\displaystyle=-\int\bigg{(}(\log\rho+1)d\rho+\frac{1}{2}\frac{d\rho^{2}}{\rho}\bigg{)}$
$\displaystyle=-\int\bigg{(}\log\rho\,d\rho+\frac{1}{2}\frac{d\rho^{2}}{\rho}\bigg{)}.$
(69)

(In the last step we used the fact that $\int d\rho=d\int\rho=d1=0$.) We will
now substitute into here for $d\rho$ from (38); however, notice that the non-
stochastic terms from that equation will only contribute linearly (since terms
of order $dt\,dW$ and $dt^{2}$ are negligible), so their final contribution to
$dS$ will be same as already deduced in connection to master equation (26)
(c.f. (30)). We therefore need only consider here the contribution to $dS$ of
the stochastic term in (38), that is $d\rho=\sqrt{\beta k\Omega^{2}}(A-\langle
A\rangle)\rho\,dW$. Substituting this into (69), and in the following step
using the rule of Itô calculus $dW^{2}=dt$:

$\displaystyle dS$ $\displaystyle=-\int\bigg{(}\log\rho\left(\sqrt{\beta
k\Omega^{2}}(A-\langle A\rangle)\rho\,dW\right)$
$\displaystyle\qquad\qquad+\frac{1}{2}\frac{\left(\sqrt{\beta
k\Omega^{2}}(A-\langle A\rangle)\rho\,dW\right)^{2}}{\rho}\bigg{)}$
$\displaystyle=-\sqrt{\beta k\Omega^{2}}\,dW\int(A-\langle
A\rangle)\rho\log\rho$ $\displaystyle\qquad\qquad-\frac{\beta
k\Omega^{2}}{2}dt\int(A-\langle A\rangle)^{2}\rho$ $\displaystyle=-\sqrt{\beta
k\Omega^{2}}\,dW\left\langle(A-\langle A\rangle)\log\rho\right\rangle$
$\displaystyle\qquad\qquad-\frac{\beta k\Omega^{2}}{2}dt\langle(A-\langle
A\rangle)^{2}\rangle.$ (70)

This, together with the contribution (30) due to the non-stochastic terms from
(38), gives us (39), as desired.

##  Appendix C Derivation of the hierarchy of equations (43)

Our objective is to derive the hierarchy of equations (43), which describes
the contribution of the stochastic term in (38) to the evolution of the
cumulants of $\rho(A;t)$ when $A=A(q,p)$ is not explicitly time-dependent. For
brevity of notation we will omit the integration measure $d^{n}qd^{n}p$ in
integrals over phase space. Consider the cumulant-generating function for
$\rho(A;t)$:

$f(z;t)\triangleq\log\left\langle
e^{zA}\right\rangle\triangleq\kappa_{1}(t)\frac{z}{1!}+\kappa_{2}(t)\frac{z^{2}}{2!}+\kappa_{3}(t)\frac{z^{3}}{3!}+\dots.$
(71)

Let $df$ denote the differential of this function with respect to time, and
$f^{\prime}$ denote its derivative with respect to the dummy variable $z$.
Expanding the differential of $f$ to second order in $d\rho$:

$\displaystyle df$
$\displaystyle=d\left(\log\int\rho\,e^{zA}\right)=\log\int(\rho+d\rho)e^{zA}-\log\int\rho
e^{zA}$ $\displaystyle=\left(\frac{\int
d\rho\,e^{zA}}{\int\rho\,e^{zA}}\right)-\frac{1}{2}\left(\frac{\int
d\rho\,e^{zA}}{\int\rho\,e^{zA}}\right)^{2}.$ (72)

Substituting into here the stochastic term from (38) (that is
$d\rho=\sqrt{\beta k\Omega^{2}}(A-\langle A\rangle)\rho\,dW$), and using the
rule of Itô calculus $dW^{2}=dt$:

$\displaystyle df$ $\displaystyle=\sqrt{\beta
k\Omega^{2}}\,dW\left(\frac{\int(A-\langle
A\rangle)\rho\,e^{zA}}{\int\rho\,e^{zA}}\right)$
$\displaystyle\qquad\qquad-\frac{\beta
k\Omega^{2}dt}{2}\left(\frac{\int(A-\langle
A\rangle)\rho\,e^{zA}}{\int\rho\,e^{zA}}\right)^{2}$
$\displaystyle=\sqrt{\beta k\Omega^{2}}\,dW\left(f^{\prime}-\langle
A\rangle\right)-\frac{\beta k\Omega^{2}dt}{2}\left(f^{\prime}-\langle
A\rangle\right)^{2}.$ (73)

Writing $f$ in terms of its cumulant expansion (71), and noting that $\langle
A\rangle=\kappa_{1}$:

$\displaystyle
d\kappa_{1}\frac{z}{1!}+d\kappa_{2}\frac{z^{2}}{2!}+d\kappa_{3}\frac{z^{3}}{3!}+\dots$
$\displaystyle=\sqrt{\beta
k\Omega^{2}}\,dW\left(\kappa_{2}\frac{z}{1!}+\kappa_{3}\frac{z^{2}}{2!}+\kappa_{4}\frac{z^{3}}{3!}+\dots\right)-\frac{\beta
k\Omega^{2}dt}{2}\left(\kappa_{2}\frac{z}{1!}+\kappa_{3}\frac{z^{2}}{2!}+\kappa_{4}\frac{z^{3}}{3!}+\dots\right)^{2}.$
(74)

Expanding the square on the r.h.s. and equating coefficients of corresponding
powers of $z$ yields the hierarchy of equations (43), as desired.

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