# Ticking clocks in quantum theory

Ralph Silva  Institute for Theoretical Physics, ETH Zürich, Wolfgang-Pauli-
Strasse 27, 8093 Zürich, Switzerland  Nuriya Nurgalieva  Institute for
Theoretical Physics, ETH Zürich, Wolfgang-Pauli-Strasse 27, 8093 Zürich,
Switzerland  Henrik Wilming  Leibniz Universität Hannover, Appelstraße 2,
30167 Hannover, Germany

(August 31, 2024)

###### Abstract

We present a derivation of the structure and dynamics of a ticking clock by
showing that for finite systems a single natural principle serves to
distinguish what we understand as ticking clocks from time-keeping systems in
general. As a result we recover the bipartite structure of such a clock: that
the information about ticks is a classical degree of freedom. We describe the
most general form of the dynamics of such a clock, and discuss the additional
simplifications to go from a general ticking clock to models encountered in
literature. The resultant framework encompasses various recent research
results despite their apparent differences. Finally, we introduce the
information theory of ticking clocks, distinguishing their abstract
information content and the actually accessible information.

> Is time the wheel that turns, or the track it leaves behind?
>
> Robin Hobb, Fool’s Errand

## Content

###### Contents

  1. I Introduction
  2. II General ticking clocks
    1. II.1 The notion of a clock and the principle of independence
    2. II.2 The structure of a ticking clock: emergence of a classical register
    3. II.3 The dynamics of a ticking clock: a minimum uncertain time-scale between measurements
    4. II.4 Discrete ticking clocks vs quantum metrology
  3. III From general to elementary ticking clocks
    1. III.1 Self-timing
    2. III.2 Clockwork independence
    3. III.3 Serial registers
    4. III.4 Irreversible ticks
  4. IV The information content of a ticking clock
    1. IV.1 Asymptotic stationarity and full counting statistics.
    2. IV.2 Irreversible ticks: time of arrival and waiting time.
    3. IV.3 Reset clocks: i.i.d. ticks.
    4. IV.4 Precision of ticks.
    5. IV.5 Discretisation of dynamics: maps-based picture of ticking clocks.
    6. IV.6 Abstract vs operational information.
  5. V Multiple clocks: tick sequences
    1. V.1 Abstract vs operational information revisited
  6. VI Previous literature
    1. VI.1 Independence vs a classical register
    2. VI.2 Bounds on the performance of clocks
  7. VII Discussion & Outlook
    1. VII.1 General vs elementary clocks
    2. VII.2 Thermodynamic cost of clocks
    3. VII.3 The emergent continuity of classical clocks
  8. VIII Acknowledgements.
  9. A Koashi-Imoto decomposition
  10. B Elementary reset clocks: connecting asymptotic rates to moments of the waiting time
  11. C Asymptotic Allan variance for self-timed clocks
  12. D Continuity of the set of measurement outcomes for a non-independent clock

##  I Introduction

Figure 1: Structure of a general ticking clock. The principle of independence
implies a Koashi-Imoto decomposition for the clock structure, featuring a
classical degree of freedom that stores the time information; to maintain this
structure the clock’s unitary dynamics have to be coarse grained over,
implying a minimal spacing and uncertainty for measurements upon the clock.

The passage of time is measured by us via clocks. While the word ‘clock’
usually brings to mind the specialised devices humans have constructed, in a
broader sense one can include any system in nature that changes appreciably in
time. In fact, animals and plants — including humans — employ simpler clocks
to regulate their behaviour, such as the circadian rhythm and the solar and
lunar cycles.

Starting from the premise that nature is well described by quantum mechanics,
we would describe a clock as a system whose state is undergoing some dynamical
evolution with respect to background time, and which is occasionally measured
by an observer, human or otherwise. However, we cannot usually label as
‘clock’ only the evolving system — the measurements can disturb the system,
and so to understand the behaviour of the clock we must include in the
description both the evolving system and the measurements.

This motivates the central theme of this work: how does one get ‘independent’
clocks, i.e. those clocks defined by the property of not being disturbed by
the measurements made upon them? This is not only an abstract query: everyday
clocks appear to work independently of us observing them. To explain the
emergence of such systems from first principles is to explain the theoretical
foundation of time-keeping.

In this paper we realize the above in an axiomatic manner. Using existing
results from quantum theory, we first derive the general structure of an
independent clock, featuring a classical degree of information on which the
time information is stored, and an internal inaccessible degree of freedom
that may be quantum mechanical. Furthermore, it is argued that these clocks
can only arise as a semi-classical limit of a quantum mechanical clock; their
evolution is a coarse-grained version of detailed unitary evolution.

With the general form of an independent clock in place we then discuss four
natural properties that a clock may possess, each simplifying the description
of the clock. Taken all together, they pinpoint what we term an ‘elementary
clock’: these include the everyday clocks we are used to, such as wall clocks
and wristwatches.

As a consequence, this paper also explains the foundations of ‘ticking
clocks’. This is a recent manner of modelling clocks with the following
features: that the clock measures time in discrete steps (ticks) and that
these ticks are stored in a classical degree of freedom. Indeed while the
original motivation of this work was to establish the theoretical foundations
of ticking clocks, our conclusion is that independent clocks are ticking
clocks. Specifically: the defining traits of ticking clocks follow naturally
by assuming an independent clock.

Finally, we glimpse at the information theory of such clocks: how to
characterise the clocks behaviour over time and how this is reflected in
measurements upon it. A key point here is that observers in quantum mechanical
theory cannot directly measure background time, thus the only way to follow a
clocks behaviour is to compare it to other clocks. This leads to the picture
of ‘tick sequences’ from multiple clocks — these are also seen to arise as a
specific case of independent clocks.

All in all, this paper serves the following purposes:

  * •

to determine the properties and explain the emergence of independent — i.e.
ticking — clocks,

  * •

to classify the various types of ticking clocks, zeroing in upon the
properties of the elementary one,

  * •

to discuss the foundations for an information theory of these clocks, and
finally,

  * •

to place recent models of these clocks and results therein into context.

The paper is organised as follows. Sec. II derives the structure and a
constraint on the dynamics of a general ticking clock from the independence
principle. In Sec. III, we discuss four further properties that a ticking
clock may possess, ending with the model of the elementary ticking clock
typically encountered in the literature. Sec. IV discusses the many ways of
quantifying the abstract information content of a ticking clock, followed by
Sec. V that discusses the operational scenario of measuring clocks w.r.t. one
another. Finally, we wrap up with a discussion and outlook in Sec. VII.

##  II General ticking clocks

###  II.1 The notion of a clock and the principle of independence

We take a clock to be any object that can be observed in order to make an
inference about time, which is assumed to be the continuous background
parameter $t$ in the Schrödinger picture of quantum mechanics. Namely,

  * •

a clock is a system $S$, whose state depends on $t$;

  * •

there exists at least one measurement corresponding to the extraction of time
information.

The above description includes almost all objects in nature, as it should —
most objects in nature do reflect the passage of time in some fashion and so
are in the most basic sense a clock. We now state our central thesis to pick
out ticking clocks: the principle of independence.

######  Principle 1 (Independent clocks).

A ticking/independent clock is one for which the measurement of time
information does not disturb the clock or affect the result of future
measurements.111If our use of the word ”ticking” falls in any way afoul of the
reader’s preferred semantics vis-a-vis clocks, one may replace it with
”independent” in the sense implied by the principle, and consider this entire
paper to pertain to “independent clocks”.

While this principle could be applied to a broad range of physical theories,
in this paper we focus on clocks that have a quantum mechanical description.
The state of the clock is then described by a time-dependent density matrix
${\rho}_{t}\in L({\mathcal{H}})$, where ${\mathcal{H}}$ is the Hilbert space
of the clock with (bounded) linear operators $L(\mathcal{H})$, and the
measurement on the clock by a set of Kraus operators $\\{{A}_{k}\\}_{k}$, with
${A}_{k}\in L({\mathcal{H}})$ and
$\sum_{k}{A}_{k}^{\dagger}{A}_{k}={\mathbbm{1}}$ 222The analysis can be
generalized to a set of measurements, however a single instance suffices to
demonstrate the results..

Furthermore, the dynamics of the clock between measurements is described by
the Schrödinger equation; this need not be upon the clock alone but upon a
larger system of which the clock is a part of. Labelling the rest of the
system as the environment $E$, the state of the clock evolves as

$\displaystyle\rho_{t}$
$\displaystyle=\operatorname{Tr}_{E}\left[e^{-iH_{SE}t}\rho_{0}^{SE}e^{+iH_{SE}t}\right],$
(1)

where $\rho_{SE}^{(0)}$ is the joint state of the clock and its environment at
some time labelled as $t=0$ for simplicity, and $H_{SE}$ is the total
Hamiltonian operator.

In the following two subsections we discuss the two consequences of the
independence principle: the Koashi-Imoto decomposition of the clock structure
and the uncertain timescale of measurements. The logical steps are visualised
in Fig. 1.

Before we proceed, note that the principle is only consequential in the
quantum regime, as the disturbance of quantum systems is fundamentally
unavoidable while measuring. In a regime where classical physics suffices to
describe both the clock and the measurement, there is no apparent difference
between clocks in general and ticking clocks. We comment further upon this in
the discussion.

###  II.2 The structure of a ticking clock: emergence of a classical register

The non-disturbance part of the independence principle is expressed
technically as

$\displaystyle\sum_{k}{A}_{k}{\rho}_{t}{A}_{k}^{\dagger}$
$\displaystyle={\rho}_{t}\;\forall t.$ (2)

A direct consequence of the above is that the clock and the measurement must
admit a decomposition given by the Koashi-Imoto theorem [1, 2, 3, 4]. We
provide a brief formal introduction to the theorem in Appendix A, and state
its consequences for clocks below.

The Hilbert space of an independent clock splits into three distinct degrees
of freedom, one of which is not accessed by the measurement, the second
storing no information vis-à-vis $t$ and the the third being classical and
accessed in a read-only manner. The first degree of freedom corresponds to a
set of ‘clockwork’ spaces $\\{\mathcal{H}_{C_{n}}\\}_{n}$, and the second one
to a set of ‘display’ spaces $\\{\mathcal{H}_{F_{n}}\\}_{n}$. We also identify
the index $n$ as the third degree of freedom, called the ‘register’ and
denoted by $T$. More precisely:

######  Theorem 1 (Koashi-Imoto decomposition of a ticking clock).

The Hilbert space of a ticking clock is of the following form:

$\displaystyle\mathcal{H}$
$\displaystyle=\bigoplus_{n}\mathcal{H}_{C_{n}}\otimes\mathcal{H}_{F_{n}},$
(3)

and the states of the clock $\rho_{t}$ are block-diagonal w.r.t. the register
index $n$:

$\displaystyle{\rho}_{t}$
$\displaystyle=\bigoplus_{n}p_{n|t}\;\rho_{n|t}\otimes\omega_{n},$ (4)

where

  * •

$\omega_{n}$ are independent of $t$,

  * •

$p_{n|t}$ are probability distributions: $p_{n|t}\geq 0$ and
$\sum_{n}p_{n|t}=1\;\forall t$,

  * •

$\rho_{n|t}$ are normalised states: $\rho_{n|t}\geq 0$ and
$\operatorname{Tr}[\rho_{n|t}]=1\;\forall n$.

Furthermore, the measurement channel is subject to the same decomposition:
each Kraus operator is of the form

$\displaystyle{A}_{k}$
$\displaystyle=\bigoplus_{n}\mathds{1}_{\mathcal{C}_{n}}\otimes A_{k,n},$ (5)
$\displaystyle\omega_{n}$
$\displaystyle=\sum_{k}A_{k,n}\omega_{n}A_{k,n}^{\dagger}\quad\forall n.$ (6)

We see that the clockwork degree of freedom is not accessed by the
measurement, while the display degree of freedom holds no temporal information
— the states $\omega_{n}$ therein are independent of $t$. The measurement only
accesses the register index $n$: the probability of obtaining the outcome $k$
from the measurement is

$\displaystyle P_{t}(k)$
$\displaystyle=\sum_{n}p_{n|t}\operatorname{Tr}\left[A_{k,n}^{\dagger}A_{k,n}\omega_{n}\right]$
(7) $\displaystyle=\sum_{n}p_{n|t}a_{k,n},$ (8)

where for each $n$ the $\\{a_{k,n}\\}_{k}>0$ are a probability distribution
that is independent of time. Any measurement of this form is a coarse-graining
of the direct measurement of $p_{n|t}$; this is the measurement that simply
checks which of the spaces w.r.t. the index $n$ the clock is in, i.e the one
with projective Kraus operators:

$\displaystyle{\Pi}_{k}$
$\displaystyle=\bigoplus_{n}\delta_{kn}\mathds{1}_{\mathcal{C}_{n}}\otimes\mathds{1}_{\mathcal{F}_{n}},$
(9)

for which the probability of the outcome $k$ is simply $P_{t}(k)=p_{k|t}$.

Figure 2: Two clocks in the same register state. Even though the display
systems are different, one mechanical and the other digital, the time
information is equivalent — 10:10.

To illustrate the roles of the three parts of the clock, take the two clocks
as depicted in Fig. 2, one mechanical and the other digital, both measuring
time down to a minute. Within each there is an internal clockwork that is
unobserved by us, a mechanical setup in the former and an electronic circuit
in the latter. There is also a set of display states for each clock: the
different possible configurations of the hands in one case, and the possible
states of the digital display in the second case.

Something that is intuitively clear to us from observation and which the KI
decomposition captures elegantly is that the display acts only as an indicator
for the time measurement, while not partaking in the actual dynamical time-
keeping of the clock. In this example, the time indicated by both clocks is
10:10 — this corresponds to the index $n$ of the register. If a person wearing
one of these clocks is asked what the time is, the answer is 10:10, and not
“10:10 indicated by angles between the hands” or “10:10 indicated by
electronic ink”.

Thus if one is interested (as we are) in modelling the time-keeping properties
of a clock and not the manner in which the information is presented to the
environment, then the display part of the clock can be ignored. Mathematically
this corresponds to reducing all of the $\mathcal{F}_{n}$ to trivial spaces,
resulting in the following ‘sufficient’ structure for a ticking clock and the
measurement upon it:

######  Corollary 1.1 (Minimal ticking clock).

For every ticking clock with Hilbert-space $\tilde{\mathcal{H}}$, quantum
states $\tilde{\rho}_{t}$ and measurement operators $\tilde{\Pi}_{k}$ there
exists a _minimal clock_ $(\mathcal{H},\rho_{t},\Pi_{k})$ with associated
quantum states $\rho_{t}$ and measurement operators $\Pi_{k}$ on a Hilbert-
space $\mathcal{H}$ together with completely positive trace-preserving maps
$S$ and $R$ fulfilling

$\displaystyle S(\tilde{\Pi}_{k}\tilde{\rho}_{t})=\Pi_{k}\rho_{t},\
R(\Pi_{k}\rho_{t})=\tilde{\Pi}_{k}\tilde{\rho}_{t}\quad\forall t.$ (10)

In particular $S(\tilde{\rho}_{t})=\rho_{t}$ and
$R(\rho_{t})=\tilde{\rho}_{t}$ for all $t$. Moreover, the Hilbert-space,
quantum states and measurement operators of the minimal clock decompose as

$\displaystyle\mathcal{H}$ $\displaystyle=\bigoplus_{n}\mathcal{H}_{C_{n}},$
(11a) $\displaystyle\rho_{t}$
$\displaystyle=\bigoplus_{n}p_{n|t}\;\rho_{n|t},$ (11b) $\displaystyle\Pi_{k}$
$\displaystyle=\bigoplus_{n}\delta_{kn}\;\mathds{1}_{n}.$ (11c)

One may wonder how unique the minimal clock. To explore this, let us define
two ticking clocks $\rho_{t}$ and $\sigma_{t}$ with measurement operators
$\Pi_{k}$ and $\Pi^{\prime}_{k}$ to be _dynamically equivalent_ (shortened to
d-equivalent) if there exist completely positive trace-preserving maps $S$ and
$R$ such that

$\displaystyle S(\Pi_{k}\rho_{t})=\Pi^{\prime}_{k}\sigma_{t}\
R(\Pi^{\prime}_{k}\sigma_{t})=\Pi_{k}\rho_{t}\quad\forall t.$ (12)

In particular, a ticking clocks is d-equivalent to its associated minimal
clock and two d-equivalent ticking clocks must also have d-equivalent minimal
clocks. In fact, a result about sufficiency in quantum statistical experiments
implies that the associated minimal clocks are _unitarily_ equivalent [3, 4,
5]:

######  Corollary 1.2 (Dynamical equivalence of clocks).

Two ticking clocks $\tilde{\rho}_{t}$ and $\tilde{\sigma}_{t}$ with
measurement operators $\tilde{\Pi}_{k}$ and $\tilde{\Pi}^{\prime}_{k}$ are
dynamically equivalent if and only if the associated minimal clocks
$(\mathcal{H},\rho_{t},\Pi_{k})$ and
$(\mathcal{H}^{\prime},\sigma_{t},\Pi^{\prime}_{k})$ are unitarily equivalent:
There exists a unitary operator $U:\mathcal{H}\rightarrow\mathcal{H}^{\prime}$
independent of $t$ and such that $U\Pi_{k}=\Pi^{\prime}_{k}U$ and

$\displaystyle U\rho_{t}U^{\dagger}=\sigma_{t}\quad\forall t.$ (13)

Thus two $d$-equivalent clocks essentially correspond to a choice of basis on
the Hilbert-spaces for their minimal clocks.

A direct consequence of the d-equivalence of clocks is that $p_{k|t}$ is the
same function for all equivalent clocks. In particular, if a clock is
equivalent to itself when shifted by some period $\tau$, we find that
$p_{k|t}=p_{k|t+\tau}$, which should indeed be the case for any periodic
clock.

###  II.3 The dynamics of a ticking clock: a minimum uncertain time-scale
between measurements

The second part of the independence principle is that the measurement of time
information does not affect the result of future statistics. This requires a
discussion of the dynamics of the clock in-between measurements. Our starting
point is that of standard quantum mechanics: the evolution is described by the
Schrödinger equation, either upon the clock itself or a larger system that the
clock is part of. In the latter case we label the rest of the system as the
environment $E$, and represent the evolution of the clock alone as the channel
(1).

For the clock to be non-trivial, $p_{n|t}$ should not be a constant. Consider
one such time $t$ at which the probability is changing, so that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}p_{n|t}$
$\displaystyle=-i\operatorname{Tr}\left[\left({\Pi}_{n}\otimes\mathds{1}_{E}\right)[H,\rho]\right]$
(14)
$\displaystyle=-i\operatorname{Tr}\left[[\rho,{\Pi}_{n}\otimes\mathds{1}_{E}]H\right]\neq
0.$ (15)

Note that although the state of the clock alone is block-diagonal w.r.t. the
register index $n$, this need not be true of the joint state $\rho_{SE}$,
which is why the above expression can be non-zero. However, if a time
measurement is performed at $t$, then the final state will indeed be decohered
w.r.t. $n$ on the full state of clock and environment. But then the expression
above becomes zero, and the measurement has disturbed future statistics.

This is not a novel insight, rather it is arises from the same origin as that
of the Zeno effect [6]. Just as in the Zeno case, here too we have that the
evolution of a quantum system from one state to an orthogonal one — different
$n$ spaces — is disturbed by measurements in that basis; the evolution can be
slowed down by measurements, and even frozen in the limit that the
measurements are infinitely dense in time.

As a consequence an independent clock in the sense of our principle cannot
arise if measurements of time information are unconstrained. The question is
therefore, what constraint is necessary to allow for a non-trivial independent
clock?

Clearly one necessary condition is that the measurements are spaced out enough
to allow for the clock to evolve. But this is not sufficient on its own, one
also requires an uncertainty in when the measurements happen. Consider the
case that the measurements are spaced out but perfectly timed. Then the only
manner in which the future statistics of the clock are not disturbed is if the
time period of the measurement is fine-tuned to that of the joint Hamiltonian
$H_{SE}$, so that at the moment of measurement there is always zero coherence
w.r.t. the register index. This violates the independence principle in spirit
as the clock and measurement are constructed to match each other rather than
the clock existing as an separate entity; also the measurement itself would
require a perfect clock, and one that renders the clock of interest redundant.

We thus arrive at a necessary condition for an independent clock:

######  Consequence 1 (Minimal and uncertain measurement timescale).

If one has a clock whose evolution is independent of the extraction of time
information, then it must evince a spacing between measurements that is both
lower bounded as well as uncertain.

Having an uncertainty in the measurement times allows for a coarse-graining of
the dynamics: within the unitary evolution, there are oscillations of various
frequencies, corresponding to (differences of) energy eigenvalues in the full
Hamiltonian $H_{SE}$. Those oscillations whose time period is much smaller
than that of the uncertainty in the measurement timescale (i.e. of high enough
frequency) can be averaged out. The resultant description of the dynamics is
not one for infinitesimal $dt$ but rather for $\delta t$ that is of the order
of uncertainty in the measurement spacing.

This allows for the emergence of classical degrees of freedom within the clock
which is required for the register degree of freedom to exist. Even though the
transition between states of the register is effected by underlying unitary
dynamics, these could be fast enough w.r.t. the measurement uncertainty so
that the coherences between register states average out, leaving it apparently
classical. This is a standard manner of describing the effective dynamics of
open quantum systems [7], and typically requires the environment to be ideal
in several senses: infinite, with short correlation timescale, etc.

The conclusion is that ticking clocks are an emergent phenomenon, in much the
same way that classical physics at the macroscopic scale is understood to be
emergent from microscopic dynamics that is quantum mechanical. In particular,
the frequency with which the clock can be measured or interacted with by an
observer is constrained. In a similar way, we think of the Koashi-Imoto
decomposition arising from the principle of independence as an idealized
emergent property that only holds to very high precision in practice.

###  II.4 Discrete ticking clocks vs quantum metrology

A key feature of ticking clocks is that they appear to measure time in terms
of discrete events, hence the name. This arises naturally from the
independence principle: the clock structure is such that the only observable
change in the clock is when it changes the index of the register. One can thus
associate the ‘tick’ to the event of jumping from one state of the register to
another.

A direct consequence is that the observed information about time is in terms
of a fundamental unit, encoded in the index $n$ of the register. This
distinguishes ticking clocks from the general prepare-and-measure scenarios
studied extensively in quantum metrology. There too a measurement can be made
upon a state $\rho_{t}$ to estimate the parameter $t$, but without the
constraint that the measurement is non-disturbing. As a result, the set of
estimated values of $t$ corresponding to all of the possible measurement
outcomes need not be discrete, even though each individual measurement
typically has a finite — and thus discrete — set of outcomes. In appendix D we
illustrate this distinction for the simple quantum clock of Salecker, Wigner
and Peres [8, 9].

Thus results from the field of quantum metrology are not automatically
applicable to ticking clocks, even though both are heavily concerned with the
accuracy of time-keeping. Whether one can still draw a fundamental connection
between them remains an interesting open question.

##  III From general to elementary ticking clocks

Figure 3: The assumptions that simplify a general ticking clock to an
elementary one. Listed clockwise from top-left they are: self-timing, the
requirement that the clock dynamics are self-contained and memory-less —
resulting in Lindbladian-generated dynamics
$\frac{\mathrm{d}}{\mathrm{d}t}{\rho}_{t}={\mathcal{L}}[{\rho}_{t}]$;
clockwork independence, the assumption that the dynamics of the clockwork is
independent of the register — resulting in an explicit register system; serial
registers, the simplification that register states form a chain that the clock
only shifts one step upon — resulting in registers that are tick counters;
tick irreversibility, the property that the clock only shifts the register
forward — resulting in the concept of a time of arrival for the ticks of a the
clock.

We have derived the structure of a general ticking clock together with a
restriction on its dynamical description. This model differs from the ones
typically encountered in the literature [10, 11, 12, 13], which have a simpler
structure and regular dynamics. In this section, we describe four additional
properties that a clock can possess that lead to a simpler reduced model. We
call the clock satisfying all four properties an elementary ticking clock.

For each property, we provide the motivation thereof, which types of clocks it
excludes, and how the reduction affects the information-theoretic aspects of
the clock.

To preview the simplifying properties we state the resulting definition of an
elementary ticking clock. The assumptions leading to it are depicted in Fig.
3.

######  Definition 1 (Elementary ticking clock).

An elementary ticking clock is a (minimal) ticking clock having the following
Hilbert space, state and representative measurement operators:

$\displaystyle{\mathcal{H}}$
$\displaystyle=\mathcal{H}_{C}\otimes\mathcal{H}_{T},$ (16a)
$\displaystyle{\rho}_{t}$
$\displaystyle=\sum_{n}p_{n|t}\;\rho_{n|t}\otimes\ket{n}\\!\bra{n}_{T},$ (16b)
$\displaystyle{A}_{k}$
$\displaystyle=\mathds{1}_{C}\otimes\ket{n}\\!\bra{n}_{T},$ (16c)

and whose dynamics is governed by a Lindbladian master equation of the form

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}{\rho}_{t}={\mathcal{L}}[\rho_{t}]$
$\displaystyle=-i\left[H_{C}\otimes\mathds{1}_{T},\rho_{t}\right]+\sum_{k}\mathcal{D}_{L_{k}\otimes\mathds{1}_{T}}[\rho_{t}]$
$\displaystyle\quad\quad\quad\quad\quad\quad\;\;+\sum_{j}\mathcal{D}_{J_{j}\otimes\Gamma_{T}}\left[\rho_{t}\right].$
(17)

Here, $\mathcal{D}_{X}[\rho]$ is the standard dissipator
$\mathcal{D}_{X}[\rho]=X\rho
X^{\dagger}-\frac{1}{2}\left\\{X^{\dagger}X,\rho\right\\}$ and $\Gamma$ is the
shift operator on the register $\Gamma=\sum_{n}\ket{n+1}\\!\bra{n}_{T}$.

###  III.1 Self-timing

A general dynamical description of a clock requires one to include and keep
track of the degrees of freedom within the environment that interact with the
clock. However, in investigating the fundamental nature of ticking clocks one
is typically interested in how the temporal behaviour of the register is
determined by the properties of the clock only – which is impossible to
conclude if the said behaviour derives from some external timing, whether it
is realized via significant correlations with environment, or via the time-
dependence of the dynamics of the clock. If we assume that the dynamics of the
clock are Markovian (memoryless), and do not require keeping track of the
environment, we arrive to a simpler description of the clock’s dynamics. We
call this pair of assumptions self-timing in the same manner as Ref. [13].

######  Property 1 (Self-timing).

A clock is self-timed if

  1. 1.

it does not require to keep track of the environment: if the state at some
time $t=0$ is ${\rho}_{0}$, the state at any other time $t>0$ depends only
upon the clock itself via

$\displaystyle{\rho}_{t}$ $\displaystyle={\Lambda}_{t}[{\rho}_{0}],$ (18)

where $\\{\Lambda_{t}\\}$ is a family of completely positive trace preserving
maps;

  2. 2.

its dynamics are Markovian: the maps $\Lambda_{t}$ are CP-divisible, i.e. for
all $s,t$ such that $0\leq s\leq t$,

$\displaystyle{\Lambda}_{t}$
$\displaystyle={\Lambda}_{s}\circ{\Lambda}_{t-s}.$ (19)

Together these imply that the set of maps $\\{\Lambda_{t}\\}_{t}$ form a
dynamical semi-group [14] whose generator is labelled a Lindbladian operator
${\mathcal{L}}$:

$\displaystyle{\Lambda}_{t}$ $\displaystyle=e^{{\mathcal{L}}t},$ (20)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}{\rho}_{t}$
$\displaystyle={\mathcal{L}}[{\rho}_{t}].$ (21)

However, not every Lindbladian generator is suitable to describe a ticking
clock: the only allowed dynamics are those which preserve the algebra of the
KI decomposition of the clock’s state (11). Employing a recent result [15],
which provides the description of generators that have an invariant atomic
algebra, we arrive to the following characterization of their dynamics.

######  Theorem 2 (Dynamics of a self-timed ticking clock).

For a minimal ticking clock whose dynamics is described by a semi-group, the
generator is of the form

$\displaystyle{\mathcal{L}}\left[{\rho}_{t}\right]$
$\displaystyle=-i\left[{H},{\rho}_{t}\right]+\sum_{j}\gamma_{j}\mathcal{D}_{{L}_{j}}\left[{\rho}_{t}\right],$
(22a) where the Hamiltonian is block-diagonal w.r.t. the register index $n$:
$\displaystyle{H}$ $\displaystyle=\bigoplus_{n}H^{(n)}_{C_{n}},$ (22b) and the
‘jump operators’ ${L}_{j}$ may only connect a single clockwork space to a
single other one (possibly the same), i.e. for every ${L}_{j}$ there exists
only one pair of register indices $\\{m,n\\}$ (possibly equal) such that
$\displaystyle{L}_{j}$ $\displaystyle=\Pi_{m}{L}_{j}\Pi_{n},$ (22c)

where $\Pi_{m}=\bigoplus_{k}\delta_{km}\mathds{1}_{\mathcal{C}_{k}}$ is the
projector upon the $m^{th}$ subspace of the clockwork.

We see that the dynamics of the clock decomposes into two parts: one
corresponding to general Lindbladian dynamics within each clockwork space —
encoded in the block-diagonal Hamiltonian ${H}$ and the jump operators
${L}_{j}$ that keep the clock in the same clockwork space — and the other
corresponding to jumps between clockwork spaces. Both sets of operators leave
the register classical, i.e. they introduce no coherence between clockwork
subspaces.

Self-timing is a strong condition and therefore self-timed ticking clocks
exclude some examples of ticking clocks, for instance: clocks driven by
externally timed signals; clocks that amplify the accuracy of an incoming
temporal signal; and clocks for which one wishes to describe only a sub-part
of the entirety of the clock. For an example of all three see [16].

In a very strict sense, self-timing is not even fulfilled by the best clocks
that we use today, namely atomic clocks. These rely on synchronizing the
frequency of an oscillator (a laser or maser) to atomic transition
frequencies. While the bare atomic transition frequencies are in principle
fixed and an _ideal_ atomic clock may be considered self-timed, in reality the
frequency of the oscillator drifts in time due to environmental influences.

Therefore, one should view self-timing as an idealisation that is fulfilled up
to some precision, in the same way as the KI decomposition.

###  III.2 Clockwork independence

In the general model (11), we distinguish between distinct clockwork spaces
for each $n$, corresponding to distinct dynamics for each state of the
register. This is useful for descriptions of multi-stage processes that can be
interpreted to be a ticking clock by associating the label of the stage to the
register index. The dynamics of each stage is in general distinct, and their
modelling would require distinct clockwork spaces. A decay chain of
radioactive isotopes is one example of such a process, as is protein synthesis
within organisms.

It is no coincidence that the above examples feature only a small number of
stages — a clock that requires a separate clockwork for each value of the
register index grows in size as the register size gets larger. A useful
property for a ticking clock is thus that of clockwork independence: that the
dynamics of the clockwork is independent of the state of the register. This
property allows for the same clockwork to be used throughout, which in turn
allows for larger registers, possibly even of an unbounded size.

######  Property 2 (Clockwork independence).

The clockworks of a ticking clock are independent of the state of the register
if its description can be simplified to a separate Hilbert space and state for
the register, taking us from the general minimal clock (11) to that of the
simple model (16):

$\displaystyle\mathcal{H}$ $\displaystyle=\bigoplus_{n}\mathcal{H}_{C_{n}}$
$\displaystyle\longrightarrow\mathcal{H}_{C}\otimes\mathcal{H}_{T},$ (23a)
$\displaystyle\rho_{t}$ $\displaystyle=\bigoplus_{n}p_{n|t}\;\rho_{n|t}$
$\displaystyle\longrightarrow\sum_{n}p_{n|t}\;\rho_{n|t}\otimes\ket{n}\\!\bra{n}_{T},$
(23b) $\displaystyle A_{k}$
$\displaystyle=\bigoplus_{n}\delta_{kn}\;\mathds{1}_{n}$
$\displaystyle\longrightarrow\mathds{1}_{C}\otimes\ket{n}\\!\bra{n}_{T},$
(23c)

with the dynamics simplifying analogously, the Hamiltonian in (22a) now being
the tensor product

$\displaystyle{H}$ $\displaystyle=H_{C}\otimes\mathds{1}_{T},$ (24)

and the Lindblad jump operators splitting similarly:

$\displaystyle{L}_{k,\Delta}$
$\displaystyle=L_{k|\Delta}\otimes\Gamma^{\Delta},$ (25)

where each $\Gamma^{\Delta}$ is a translation by $\Delta\in\mathbb{Z}$ steps
of the register

$\displaystyle\Gamma^{\Delta}$
$\displaystyle=\sum_{n}\ket{n+\Delta}\\!\bra{n}.$ (26)

With the assumptions of self-timing and clockwork independence we gain a
significant feature of the model: the stabilisation of the clock dynamics over
large times (more precisely, the property of asymptotic stationarity). This is
discussed further in Sec. IV.

Clockwork independence also appears as an axiom in [13], where it is termed
the ‘time invariance symmetry condition’.

###  III.3 Serial registers

In our general model, the dynamics of the clock can involve jumps between any
pair of states of the register. An example of such a process in nature is a
multi-stage chemical reaction: one may skip a stage via an enzymatic shortcut.
Here too by identifying the stage with the register index we get a ticking
clock with jumps that are not just on sequential states of the register. We
will also see that non-serial registers are important for the theory of
multiple clocks and tick sequences, Sec. V.

The clocks that we are used to do typically have serial registers, i.e. the
states of the register are arranged in a totally ordered chain with jumps only
possible between nearest neighbours.

######  Property 3 (Serial registers).

A ticking clock has serial registers if its Lindblad jump operators in (22c),
(25) are non-zero for at most three values $m-n=\Delta\in\\{-1,0,+1\\}$.

This assumption also naturally strengthens the interpretation of the register
index as the ‘number of ticks’ of the clock.

This property is closely related to the ‘leading order condition’ axiom in
[13].

| Self-timing | Clockwork independence | Serial registers | Irrev. ticks
---|---|---|---|---
Elementary ticking clock (Def. 1) | ✓ | ✓ | ✓ | ✓
Clocks driven by externally timed signals (e.g. [16]) | ✗ | ✓ | ✓ | ✓
Atomic clocks | ✗ | ✓ | ✓ | ✓
Brownian clock of [11] | ✓ | ✓ | ✓ | ✗
Circadian rhythm (see e.g. [17]) | ✗ | ✗ | ✗ | ✓
Table 1: Different kinds of clocks and their properties.

###  III.4 Irreversible ticks

The final simplification to consider is that of the irreversibility of the
register dynamics: that the clock dynamics can only change the register index
in a particular direction. For serial registers, as the index $n$ is
associated to the number of ticks, this property is equivalent to the
statement that once a tick is recorded it can never be reversed. One may also
have irreversibility for non-serial registers as along as a partial order is
in place.

######  Property 4.

A ticking clock is irreversible if there exists a partial order $\leq$ on the
set of register indices $\mathcal{I}$ such that if the register index is
measured to be $n$ at any time $t$, then for all $t^{\prime}>t$ the only
possible outcome of a time measurement is $m\geq n$, where
$m,n\in\mathcal{I}$.

This property appears to matches our experience: every clock that we are used
to employing from the pendulum clock down to atomic clocks have ticks that
once recorded cannot be ’unrecorded’ — the tick is recorded into the
environment in a manner that does not allow for reversal. This has the same
flavour as the irreversibility of measurements in quantum mechanics — from the
viewpoint of an observer, the measurement process is irreversible and the
outcome cannot be erased 333Celebrated thought experiments regarding the
internal inconsistency of quantum mechanics always involve toying with the
reversibility (or lack thereof) of the measurement..

The observational irreversibility goes hand-in-hand with thermodynamic
irreversibility. Clocks that tick “loudly” — i.e. whose ticks are recorded on
a macroscopic scale — dissipate entropy in their environment as a result. This
is because the process of recording a tick on a macroscopic scale involves the
production of a macroscopic amount of entropy [18]. The reversal of these
processes would involve a large decrease in entropy that is in practice not
observed. In other words the reversal of the tick is possible but improbable,
and absent in all practical cases.

This outlook also fits with our understanding of the thermodynamic theory of
fluctuations and detailed balance [19]. Given any dynamical process, there
exists its reverse process, and the rates of the two are related to each other
by the difference in entropy production in either direction. Thus to eliminate
the possibility of the reverse of a tick one must have the tick-process create
a large amount of entropy in the environment.

All of this suggests the assumption of irreversibility is judicious, with one
exception: in the case one would like to understand the fundamental entropic
cost of a ticking clock. While not currently a question relevant to practical
clocks it is still of interest from the point of view of fundamental physics,
as it concerns the “arrow of time”: the observation that entropy increases as
we observe time passing. For clocks this turns into a quantifiable question:
Does the precision of a clock come at a fundamental entropic cost [20]?

One thus has a choice: if entropic considerations are not central then
assuming irreversible ticks is appropriate; this takes us to the model of the
elementary ticking clock as described in Def. 1. This is also the model
employed by [10, 12, 13, 20, 21]. On the other hand, if entropy is the
fundamental resource (either directly or in the form of heat or other thermal
processes that keep the clock out of equilibrium) then it makes sense to keep
the ticks reversible, as is done in [22, 11].

It is also important to understand a finer distinction between thermodynamic
reversibility of the ticks, and thermodynamic reversibility within the
clockwork: it is quite possible to keep reversibility for processes within the
clockwork but not for the ticks. For instance, one may demand that detailed
balance is obeyed by the jump operators that do not change the register state,
while restricting those that change it to only go forward. This middle ground
is the form used in [20, 21]. It is justified if the entropic dissipation
within the clockwork is high enough as to render the ignored entropic cost of
the backward tick insignificant; the conclusions drawn in such models are
hence applicable in the high dissipation regime.

The major advantage in having irreversible ticks is that there is now a single
(but random) time that the register arrives to an index $n$. This time of
arrival of a tick is an alternative description of the register dynamics to
that of the number of ticks. We discuss this in greater detail in the next
section.

All of the properties discussed in the section can be considered independently
of each other; in Table 1 we list examples of different clocks and indicate
which of the four properties they satisfy.

##  IV The information content of a ticking clock

In this section, we lay out the basic principles of an information theory of
ticking clocks; namely, we identify the objects encoding their information
content, and connect various measures of the accuracy of ticking clocks. The
full treatment of the information theory is beyond the scope of this paper,
and will be addressed in future work [23, 24].

How do we extract the information from a ticking clock? The first thing to
note is that the only way we access the clock’s information content is via
reading off the state of its register. Thus, the information theory of a clock
is, strictly speaking, the information theory of the clock’s register [10,
12], and the question of how the properties of the clockwork affect the
behaviour of the ticks [12, 20, 25] could only be answered by tracking the
behaviour of the register.

For each time $t$, a measurement of the register will return the answer $n$
with probability $p_{n|t}$. If one performs many such measurements at
different times, there is an associated joint probability distribution
$p_{n_{1},n_{2},...n_{m}|t_{1},t_{2},...,t_{m}}$. This joint distribution
encodes the information content of a single clock.

For ticking clocks which satisfy the properties of self-timing (Property 1)
and clockwork independence (Property 2), the theory simplifies considerably.
The dynamics are given by a time-independent Lindbladian generator that is
translation invariant w.r.t. the register. One can split the Lindblad operator
(22a) w.r.t. how much each term shifts the register:

$\displaystyle{\mathcal{L}}\left[{\rho}_{t}\right]$
$\displaystyle=\sum_{\Delta}\left(\mathcal{L}_{\Delta}\otimes
X_{\Delta}\right)\left[{\rho}_{t}\right],$ (27a) $\displaystyle\quad
X_{\Delta}\big{[}\ket{n}\\!\bra{n}\big{]}$
$\displaystyle=\ket{n+\Delta}\\!\bra{n+\Delta}\;\forall n.$ (27b) For each
$\Delta\neq 0$ the $\mathcal{L}_{\Delta}$ is an unnormalised channel
$\displaystyle\mathcal{L}_{\Delta}[\rho]$
$\displaystyle=\sum_{k}\gamma_{k,\Delta}L_{k|\Delta}\rho
L_{k|\Delta}^{\dagger},$ (27c) while the $\Delta=0$ operator collects all
terms in the generator that leave the register unchanged ($X_{0}=\mathds{1}$),
$\displaystyle\mathcal{L}_{0}\left[\rho\right]$
$\displaystyle=-i\left[H,\rho\right]-\frac{1}{2}\sum_{k,\Delta}\gamma_{k,\Delta}\left\\{L_{k|\Delta}^{\dagger}L_{k|\Delta},\rho\right\\}.$
(27d)

It is customary to refer to $\mathcal{L}_{0}$ as a non-tick generator, and
$\mathcal{L}_{\Delta}$ as ticking generators [12], especially in the case of
an elementary clock, for which there is only
$\mathcal{L}_{0},\mathcal{L}_{1}$. One could correspondingly view the process
of each tick of the clock as comprised of two processes working in tandem: the
internal evolution of the clockwork generated by $\mathcal{L}_{0}$ (labelled
temporal probability concentration [21]) and the tick process generated by
$\mathcal{L}_{1}$.

The set $\\{\mathcal{L}_{\Delta}\\}_{\Delta}$ allow for the description of the
clock without the need of an explicit register, by following the evolution of
the set of clockwork states:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(p_{n|t}\rho_{n|t}\right)$
$\displaystyle=\sum_{\Delta}\mathcal{L}_{\Delta}\left[p_{n-\Delta|t}\rho_{n-\Delta|t}\right].$
(28)

###  IV.1 Asymptotic stationarity and full counting statistics.

In general, in order to capture the full information content of a ticking
clock which runs indefinitely, one has to consider the joint probability
$p_{n_{1},n_{2},...n_{m}|t_{1},t_{2},...,t_{m}}$ for an infinite range of
times $t_{1}$ to $t_{m}$ and an infinite density of measurements in between.
However, this is rendered unnecessary for self-timed clocks with an
independent clockwork: as the dynamics are generated by a time-independent
Lindbladian the behaviour of the clock in the long-time limit is typically
independent of the initial state of clock and register, deriving only from the
properties of the generator444A sufficient condition is that the reduced
Linbladian upon the clockwork has a unique steady state. This ergodic behavior
is a well-known concept in the theory of Markov processes. Self-timed clocks
in particular fall within the theory of full counting statistics [26] 555To be
precise, the consideration of [26] also assumes the clock having serial
registers, as it splits the relevant Lindblad clock dynamics into
$\mathcal{L}_{+}$ and $\mathcal{L}_{-}$, corresponding to increasing and
decreasing the tick count of the clock’s register. One can however also carry
out a similar analysis without demanding that property, with
$e^{i*j*{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}$
being the appropriate phase field for $\mathcal{L}_{j}$ terms in (27).. We
summarize the main points of the theory below.

Given the probabilities $p_{n|t}$ one can define the moments
$\braket{n^{k}}_{t}=\sum_{n}n^{k}p_{n|t}$. For large $t$, these approach
polynomials in $t$, the degree of which is the degree of the moment.

$\displaystyle\braket{n^{k}}_{t}$ $\displaystyle=\sum_{j=0}^{k}a_{k,j}t^{j}.$
(29)

Every coefficient $a_{k,j}$ with the exception of the constant term $a_{k,0}$
is independent of the initial state of the clockwork and register, and
determined solely by the set of Lindbladian operators
$\\{\mathcal{L}_{\Delta}\\}_{\Delta}$. The asymptotic behaviour of the clock
is thus encoded in these moments.

In particular, the mean and variance of the number of ticks scales linearly
for large values of $t$

$\displaystyle\braket{n}_{t}$ $\displaystyle\rightarrow\nu\;t,$ (30a)
$\displaystyle\braket{n^{2}}_{t}-\braket{n}^{2}_{t}$
$\displaystyle\rightarrow\Sigma\;t,$ (30b)

where $\nu,\Sigma$ are constants determined by $\\{\mathcal{L}_{\Delta}\\}$.
The mean and variance are in fact the first and second _cumulants_ of
$p_{n|t}$, every one of these are asymptotically linear, and the set of them
contains the same information as the moments. [26]

###  IV.2 Irreversible ticks: time of arrival and waiting time.

If the clock also satisfies the properties of having serial registers and
irreversible ticks (Properties 3 and 4), in other words, it is an elementary
clock by Def. 1, one can also characterize it via a time of arrival for the
$n^{th}$ tick of the clock, denoted by $T_{n}$. This denotes the moment of
time at which the state of the register changes from $\ket{n-1}_{T}$ to
$\ket{n}_{T}$. Unless the clock is perfect, the time of arrival does not have
a definite value, but is distributed according to the probability density
$P(T_{n}=t)$. The time-of-arrival and number-of-ticks pictures are connected
by the following transformation

$\displaystyle P(T_{n}=t)$
$\displaystyle=-\frac{\mathrm{d}}{\mathrm{d}t}\sum_{m<n}p_{m|t}.$ (31)

Just as in the case of the number of ticks, the full information of the clock
is not encoded in a single time of arrival, but rather in a joint probability
density $P(T_{1}=t_{1},T_{2}=t_{2},...,T_{n}=t_{n})$. In the literature, the
time of arrival $T_{n}$ is referred to as a waiting time for the $n^{th}$
tick, and its distribution as a waiting-time distribution, also sometimes
referred to as a delay function [20, 12].

###  IV.3 Reset clocks: i.i.d. ticks.

If the clock has the further simplifying property that it goes to the same
fixed state immediately after every tick (we call such clocks reset clocks),
then the behaviour of each tick becomes identical and independent of the ticks
before. The waiting-time distribution of a single tick is hence sufficient to
describe everything about the clock.

If we denote this distribution as $\omega_{t}$, then the joint waiting-time
distribution of any number of consecutive ticks is simply the product,

$\displaystyle P(T_{1}=t_{1},...,T_{n}=t_{n})$
$\displaystyle=\omega_{t_{1}}\cdot\omega_{t_{2}-t_{1}}\cdot...\cdot\omega_{t_{n}-t_{n-1}},$
(32)

and the distribution for the waiting-time of solely the $n^{th}$ tick is the
n-fold convolution of $\omega_{t}$.

###  IV.4 Precision of ticks.

There are several ways to quantify how good a clock is; the simplest one is
based on comparing the mean value of some clock quantity to the fluctuations
of the same [11, 20, 12, 25, 16, 27]. Both the number of ticks $n_{t}$ and the
time of arrival $T_{n}$ can be chosen the quantity.

The number-of-ticks picture is more general, as it does not require the clock
to have serial registers or irreversible ticks. As the asymptotic rates of the
mean and the variance approach constant values (30), the ratio between said
rates is a valid measure of precision:

$\displaystyle R_{1}$
$\displaystyle=\lim_{t\to\infty}\frac{\braket{n}_{t}}{\braket{n^{2}}_{t}-\braket{n}^{2}_{t}}=\frac{\nu}{\Sigma}.$
(33)

This is the inverse of the ‘Fano factor’ and is the measure employed in [22,
11], and widely used in the literature on thermodynamic uncertainty relations
[28].

An elegant property of the above is its time-scale invariance — if we
transform $t\to t/a$, or equivalently multiply the dynamical generator
$\mathcal{L}$ by $a$, the precision remains unchanged. As such it measures the
fluctuations of the ticks of the clock w.r.t. the clock’s own timescale.

On the other hand, consider that we have an elementary reset clock. Here too
there is an analogous time-scale invariant measure of precision:

$\displaystyle R_{2}$ $\displaystyle=\frac{\mu^{2}}{\sigma^{2}},$ (34)

where $\mu,\sigma^{2}$ are the mean and variance of the waiting-time
distribution $\omega_{t}$. This is the measure of precision referred to in
[20, 12, 16, 25, 21, 27], which corresponds to the number of ticks seen from
the clock before the uncertainty in the interval between successive ticks
becomes as large as the mean time between them.

As expected, these two precision measures are equal $R_{1}=R_{2}$ (see
Appendix B for a derivation), in particular:

$\displaystyle\nu$
$\displaystyle=\frac{1}{\mu},\quad\Sigma=\frac{\sigma^{2}}{\mu^{3}}.$ (35)

Another standard of precision is the Allan variance, a measure of the
stability of the frequency of the clock. When applied to the case of self-
timed clocks operating in the stationary (asymptotic) limit the Allan variance
takes a simple form:

$\displaystyle A_{\tau}$
$\displaystyle=\frac{\Sigma}{\tau}=\frac{\nu}{R_{1}\tau},$ (36)

where $\tau$ is the averaging time (for a derivation, see Appendix C). Thus
for self-timed clocks, the Allan variance does not contain more information
than the usual precision $R_{1}$ and decreases with $\tau$. It becomes an
important measure in its own right in the case of non-self-timed clocks, in
particular those that exhibit a frequency drift. In this case, the Allan
variance for larger values of $\tau$ reflects the frequency drift.

###  IV.5 Discretisation of dynamics: maps-based picture of ticking clocks.

Another way to view the dynamics of the clock is to discretise it into many
applications of a family of dynamical maps. This approach represented one of
the earlier efforts to model clocks from a quantum information perspective
[10], and proved to be useful in deriving fundamental bounds on the precision
of clocks [25]. Here we describe it from the perspective of our work.

By taking the evolution of an elementary clock for a small but finite amount
of time $\delta$ one can construct a quantum operation as follows. Consider
the state of the clock at some time $t$ conditioned on the register being in
the state $n$:

$\displaystyle\rho_{t}$ $\displaystyle=\rho_{n|t}\otimes\ket{n}\\!\bra{n}$
(37)

After a time $\delta$, the above transforms to

$\displaystyle{\rho}_{t+\delta}$
$\displaystyle=\sigma^{(0)}_{t+\delta}\otimes\ket{n}\\!\bra{n}+\sum_{m=1}^{\infty}\sigma^{(m)}_{t+\delta}\otimes\ket{n+m}\\!\bra{n+m},$
(38)

where the $\sigma_{t+\delta}^{(m)}$ are sub-normalized density matrices. The
state of the clockwork alone is

$\displaystyle\rho_{t+\delta}$
$\displaystyle=\sigma^{(0)}_{t+\delta}+\sum_{m=1}^{\infty}\sigma^{(m)}_{t+\delta}$
(39)
$\displaystyle=\mathcal{M}^{(0)}_{\delta}[\rho_{t}]+\mathcal{M}^{(1)}_{\delta}[\rho_{t}],$
(40)

where we have collected all of the states corresponding to one or more ticks
having being generated in the second term. The separation above defines two
operators, by expanding the evolution operator $e^{{\mathcal{L}}\delta}$ for
small $\delta$ these are seen to be of the form:

$\displaystyle\mathcal{M}^{(0)}_{\delta}$
$\displaystyle=\mathds{1}+\delta\mathcal{L}_{0}+O\left(\delta^{2}\right),$
(41) $\displaystyle\mathcal{M}^{(1)}_{\delta}$
$\displaystyle=\delta\mathcal{L}_{+}+O\left(\delta^{2}\right),$ (42)

where $\mathcal{L}_{0},\mathcal{L}_{1}$ are the clock operators corresponding
to not shifting and shifting the register respectively, (27).

This abstract division between the two parts of the clock state can be made
into a full quantum operation by including a bit register that indicates
whether the clock ticked or not in the time interval $\delta$:

$\displaystyle\mathcal{M}_{C\rightarrow CR}[\rho]$
$\displaystyle=\mathcal{M}^{(0)}_{\delta}[\rho]\otimes\ket{0}\\!\bra{0}_{R}+\mathcal{M}^{(1)}_{\delta}[\rho]\otimes\ket{1}\\!\bra{1}_{R}.$
(43)

One may now repeat the above map, each time with a new register $R_{i}$,
resulting in a sequence of bits $\\{R_{i}\\}_{i}$ that encode the clock’s
ticking probability in small time intervals [10, 12].

As has been noted before [13], this picture is not a constructive description:
in order to apply each map one needs to disconnect register $R_{i}$ after an
interval of precisely $\delta$ and connect $R_{i+1}$, requiring the presence
of a precise clock of high frequency in the background, in addition to a high
number of bit registers and connecting machinery. In the limit
$\delta\rightarrow 0$ — which is required for the state of the sequence of
$R$’s to exactly match the waiting-time distribution — both the precision of
the background clock and the amount of memory become infinite.

Furthermore, recall that a ticking clock can only emerge in a regime where the
measurements on the register are both spaced out and uncertain enough in time.
Thus, while $p_{n|t}$ and all associated objects are correct predictors of the
observable probabilities, they rely on the assumption that one is not
measuring frequently enough to invalidate the coarse-grained dynamics. With a
high enough density of measurements on the clock one would invalidate the
principle of independence, invalidating most of the analysis so far.

However, as an abstract representation of the information content of the clock
the model works well. The probability distribution over the sequence of
registers is a discretisation (to step size $\delta$) of the distribution of
waiting-times. In fact, it can be shown [23] that for $\delta$ small enough
(but still non-zero) the discrete statistics are in on-to-one correspondence
to those in the continuous picture. It is also worth noting that by starting
from discrete maps one can derive that the clock must have Lindbladian
dynamics as described in [12].

###  IV.6 Abstract vs operational information.

All of the quantities of the clock we discuss so far in this section were
defined w.r.t. the background time $t$:
$p_{n|t},P(T_{n}=t),\braket{n}_{t},\omega_{t}$, and so on. However, in
realistic scenarios we don’t have access to perfect background time, but
rather infer it from measurements on clocks. In other words, in a strictly
operational sense the above quantities are not in fact measurable, as we can
only measure quantities related to comparing the ticks of different clocks to
each other [10].

The latter is the focus of the next section. For now we distinguish between
what can be called the abstract information content of a single clock,
encompassing any such quantity defined w.r.t. background time $t$ and the
operational information content, which is accessible to us when we have
multiple clocks to compare ticks from.

##  V Multiple clocks: tick sequences

When we have multiple clocks, we have access not only to the individual ticks,
but also to their relative ordering; as a result, the main object considered
is the tick sequence capturing that ordering. This is however only applicable
to irreversible clocks, as it requires the notion of ‘recording a tick’ to be
well-defined.

To illustrate this, let us take two elementary clocks $A$ and $B$ and compose
them 666The model presented in this section can be analogously extended to an
arbitrary number of considered clocks.. Their joint state belongs to a Hilbert
space consisting of four subsystems, namely, two clockworks and two
corresponding registers. The dynamics can then be separated into two parts

  * •

the internal dynamics: the dynamics for each clockwork that do not shift the
register, encoded by the Hamiltonian
$H_{C_{A}}\otimes\mathds{1}_{\text{rest}}$ and set of jump operators
$\\{L_{k|0,C_{A}}\otimes\mathds{1}_{T_{A}}\otimes\mathds{1}_{\text{rest}}\\}_{k}$
(analogously for $B$),

  * •

the ticking dynamics of each clock, encoded by the set of
operators$\\{L_{j|1,C_{A}}\otimes\Gamma_{T_{A}}\otimes\mathds{1}_{\text{rest}}\\}_{j}$
(analogously for $B$).

We denote the first part (which does not concern the register) by
$\mathcal{L}_{0,A}[\rho_{t}]$ (similarly for $B$), splitting the total
Lindbladian as follows:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\rho_{t}$
$\displaystyle=-i\left[H_{C_{A}}\otimes\mathds{1}_{T_{A}C_{B}T_{B}},\rho_{t}\right]+\sum_{k}\mathcal{D}_{L_{k|0,C_{A}}\otimes\mathds{1}_{T_{A}C_{B}T_{B}}}[\rho_{t}]+\sum_{j}\mathcal{D}_{L_{j|1,C_{A}}\otimes\Gamma_{T_{A}}\otimes\mathds{1}_{C_{B}T_{B}}}[\rho_{t}]+\text{similar
for $B$.}$ (44)
$\displaystyle=\mathcal{L}_{0,A}[\rho_{t}]+\mathcal{L}_{0,B}[\rho_{t}]+\sum_{j}\mathcal{D}_{L_{j|1,C_{A}}\otimes\Gamma_{T_{A}}\otimes\mathds{1}_{C_{B}T_{B}}}[\rho_{t}]+\sum_{j}\mathcal{D}_{\mathds{1}_{C_{A}T_{A}}\otimes{L}_{j|1,C_{B}}\otimes\Gamma_{T_{B}}}[\rho_{t}].$
(45)

To incorporate the ordering between ticks, we replace the separate registers
for $A$ and $B$ by a single register for both that will store the tick
sequence. Its Hilbert space can be by a composition of qutrits, three basis
states of which $\\{0,A,B\\}$ correspond to an unwritten register, the tick of
$A$ and the tick of $B$ respectively,

$\displaystyle\mathcal{H}_{T}=\mathcal{H}_{T_{1}}\otimes\mathcal{H}_{T_{2}}\otimes...\
\text{with}\ \mathcal{H}_{T_{n}}=\mathbb{C}^{3}.$ (46)

We assume that at $t=0$ no ticks have been recorded, and hence the register
starts out with all qutrits initialized in the 0 state,
$\bigotimes_{n}\ket{0}_{T_{n}}$. This modification does not affect the physics
of the clocks, only the manner in which ticks are recorded.

The dynamics are modified accordingly. All operators previously containing
shifts in individual registers of each clocks are replaced by operators where
these shifts write the label of the ticked clock upon the first unwritten
register:
$\Gamma_{T_{A}}\otimes\mathds{1}_{T_{B}}\longrightarrow\Theta^{(A)}_{T}$,

$\displaystyle\Theta^{(A)}_{T}$
$\displaystyle=\sum_{n=1}^{\infty}\left(\prod_{m=1}^{n-1}\left(\mathds{1}-\ket{0}\\!\bra{0}\right)_{T_{m}}\right)\otimes\ket{A}\\!\bra{0}_{T_{n}}\otimes\mathds{1}_{\text{rest}},$
(47)

and analogously for $B$. This form of the operator allows to ignore the part
of the register already written upon, write the label of the clock upon the
first unwritten slot, and leave the rest unchanged.

Figure 4: Multiple clocks writing a tick sequence upon a shared register. All
possible tick sequences observed for two clocks $A$ and $B$ can be represented
as a tree, where each new level indicates the number of observed ticks
increasing by one.

The result is illustrated in Fig. 4. The register state space is best
visualised as a tree, with the initial node being the unwritten sequence,
which then branches out, each branch corresponding to the next tick being $A$
or $B$. At any time $t>0$, the register reflects the order of all the previous
ticks from $A$ and $B$.

By observing the tick sequence we can easily reconstruct the statistics
$p(n_{A}|t),p(n_{B}|t)$ of the individual clocks by counting the total number
of ticks of each clock within the sequence. However, the information contained
in the tick sequence picture is strictly greater as it also includes the
ordering between ticks.

An elegant feature of the KI decomposition for clocks is that it includes this
structure of multiple clocks and a tick sequence. Namely, the structure of
Hilbert space and dynamics above satisfies the KI structure, with the only
difference from an elementary clock being that the states of the register that
the clocks move between are not arranged in a serial chain, but rather as a
tree, see Fig. 4. Thus playing ticking clocks vs one another to form a tick
sequence — which is necessary to form an operational theory of ticking clocks
— is seen to be not a separate model but rather another instance of a ticking
clock, with a non-serial register.

###  V.1 Abstract vs operational information revisited

A key observation in the section on the information content of a single clock
was that the objects were abstract: as they are all defined w.r.t. background
time $t$ they are not directly observable. An operational information theory
can only be based on comparing clocks to each other which is precisely the
tick sequence we have just discussed. One can build operational analogues to
abstract objects — a simple example is $P(T_{n}=t)$, the probability density
that the $n^{th}$ tick of the clock arrives at $t$. The operational analog of
this would be to have a second clock $B$, and when the $n^{th}$ tick of $A$
occurs one observes the number of ticks so far from $B$ rather than background
time, corresponding to a probability distribution $P(N_{B}^{(n)}=m)$. This
does not need to be measured immediately on ticking, the advantage of the tick
sequence is that the information once recorded can be observed at any later
time.

Developing an information theory of tick sequences is worth a research work on
its own and is the main focus of upcoming work [24]. One immediate question
given the abstract vs operational distinction is whether one must lose
information going from the former picture to the latter, and if so, how much.
Clearly if we know the behaviour of a set of clocks $\\{C_{j}\\}_{j}$ w.r.t.
background time $t$ — from their times of arrival
$P_{j}(T_{1}=t_{1},T_{2}=t_{2},...)$ for instance — then we can recover all of
the tick sequences that they would generate together. The converse is not
obviously true and the gap between the two is an important question. A
positive result in that direction is [23], where it is shown that, using
arguably the simplest reference clock (a Poisson process), one can recover the
abstract waiting-time distribution of an unknown reset clock from the tick
sequence it generates with the reference.

##  VI Previous literature

###  VI.1 Independence vs a classical register

The bipartite structure of a ticking clock featuring a classical register is
not a novel concept; for example, it is an integral part of the models
presented in [10, 29], where ticking clocks are studied from the viewpoint of
quantum information theory. There, the authors argue that for a clock that
marks the passage of time in an operational manner – that is, the time read
off the clock is interpreted as its measurable output – the time information
must be written upon a classical register. The bipartite structure of the
clock is also implicit in the models of [12, 20] and is postulated in an
explicit manner in [13], where the author also discusses an axiomatic
framework for ticking clocks.

If one starts out by assuming the bipartite structure, then the independence
of the clock — i.e. the fact that the clock is not disturbed by time
measurements — follows immediately. This is the opposite direction to the path
we have taken in this paper, where we postulate the independence principle as
fundamental, and the structure is then implied by it.

At first glance this may appear to be a redundant chicken-and-egg conundrum:
why should one favour either of the above as a starting point? Here we argue
why independence is indeed a more appropriate fundamental starting point.

Firstly, there are a number of implicit assumptions in the model of an
elementary clock that are exposed in detail here. The KI decomposition of the
clock does not feature an explicit system that can be called a register,
rather only an index corresponding to different states of a display. The
explicit ‘register’ system in an elementary clock is an an abstraction of this
degree of freedom. This insight is neither deep nor novel, but it is missed
when one begins already with an explicit register. In the same light, every
real clock has a ‘display’ system whose dynamical behaviour does not
participate in the time-keeping properties of the clock. Again, this is
entirely redundant while discussing the information theory of clocks, but it
is elegant to have the true form of the register and display appear in a
construction from first principles.

Moreover, by beginning from the independence principle we have that ticking
clocks must be ‘emergent’: i.e. that we can only have dynamics that maintains
the classical nature of the register as a coarse-grained approximation. Under
fully unitary dynamics, no such dynamics could exist. Thus every ticking clock
has a limit to the frequency with which time-information can be gleaned from
it, and cannot be continuously observed, contrary to the statement of [13].

This frequency limit comes as no surprise, as similar coarse-graining
assumptions, such as Born, Markov and rotating wave approximations, are
required for the derivation of master equations describing dynamics of open
quantum systems. If any of these assumptions is dropped, the reduced
description of the system’s dynamics will be rendered erroneous. It is thus
important for an axiomatic approach to justify or derive these assumptions
from first principles.

Our approach also allows us to first arrive to a description of general
ticking clocks that is broader than the usual model, the elementary clock. As
we have pointed out, this is necessary in order to include some time-keeping
systems that we would like to take into the consideration. In turn, each of
the four additional properties needed to reduce our description to the
elementary clock can be seen as a physical restriction that one can choose
rather than a technical necessity for a ticking clock. As a result, we account
for how the theory changes with each simplification.

For example, our construction does not apriori assume the irreversibility of
ticks, which is a key assumption in all previous quantum-informational models
of clocks [10, 12, 20, 16, 25, 13, 21, 27], as opposed to the thermodynamic
models [22, 11] that have clocks with reversible ticks. This distinction is
linked to the entropy production associated with macroscopic irreversiblity.
We note that in both cases, the asymptotic behaviour of the clock is accounted
for by full counting statistics, weakening the distinction between them.

###  VI.2 Bounds on the performance of clocks

A clock is, ultimately, a tool that can be used in any operation requiring to
keep track of time, and one of its defining properties is its precision, which
characterizes its performance. Here we list some of the existing results and
open questions regarding the bounds on precision of clocks:

  * •

Fundamental thermodynamic / entropic cost. Refs. [11, 20, 30] suggest a trade-
off between a clocks precision and its entropy dissipation, albeit for a
restricted class of clocks. Proving a general trade-off would represent
understanding the arrow of time from the perspective of the devices that
actually measure said arrow. Promising directions in this regard include the
promising fields of thermodynamic and kinetic uncertainty relations, as also
information-theoretic approaches such as [25];

  * •

Dimension of clockwork space. [12] proves a bound on the precision of
clockworks with incoherent dynamics as a function of the Hilbert space
dimension of the clockwork, and that there exist quantum clockworks that
outperform them. Subsequently, [25] proved a bound for quantum clockworks as
well; the quantum clock has a quadratic advantage over a classical one w.r.t.
the dimension of the clockwork Hilbert space. While the optimal construction
of the classical (incoherent) clockworks is known, the optimal quantum
clockworks remain an interesting open question;

  * •

Precision vs. frequency trade-off. In [27] the authors take the tick dynamics
of the clock to be fundamental (represented by the rates $\gamma_{j}$ attached
to the Lindblad jump operators that move the register), and demonstrate that
in this case the precision of the clock can only be increased by sacrificing
its frequency of ticking. In [21], the trade-offs are shown for a clock the
transitions of which are driven by a thermal engine. The quantum clock
construction in [12] satisfies the scaling of said trade-off, and it remains
open as to whether there exists a quantum over classical advantage here too.

##  VII Discussion & Outlook

###  VII.1 General vs elementary clocks

In this paper, we derive a structure for ticking clocks that is more general
than the usual model in the literature, namely elementary clocks. It is not
clear whether the additional restrictions in the elementary model have a
significant influence on the achievable performance of the clocks. For
completeness, here we recap all open questions related to this loss of
generality mentioned in the main text.

  * •

Self-timing. Even the best clocks — atomic clocks — are not strictly self-
timed, rather their dynamics exhibit a frequency drift. Modelling the long
term behaviour of such systems, i.e. with drifting dynamics is necessary to
connect the research on ticking clocks to practical implementations.

  * •

Clockwork independence. It remains an open question whether one can obtain
greater precision by using different dynamics for different ticks. The results
of [12] suggest that this is impossible in the case that the clockwork
dynamics is incoherent; the argument is that the precision over many ticks of
such a clock is bound by the sum of the precision of each tick in turn. It is
unclear whether quantum clockworks allow one to circumvent this bound.

  * •

Serial registers. The same arguments from [12] suggest that given a network of
register states, the precision of moving from one state to another is
maximised when there is only one possible path rather than a convex mixture of
the same. This picks out the optimal register as a chain of states with only
nearest neighbour jumps. Proving this in generality would be useful; if it is
indeed not the case, then there would exist more precise clock constructions
that employ the ability to jump many steps on the register.

  * •

Reversible ticks. There are indications that reversible ticks cannot increase
the precision of clocks. An example is Ref. [11] where they posit a bound on
the precision of such clocks that is highest when the clock only ticks in one
direction. Proving this also remains an interesting open question.

###  VII.2 Thermodynamic cost of clocks

Our starting point for this paper was the fact that clocks — specifically
those apparently undisturbed by measurement — exist, and our goal was to model
them.

A property that holds for all ticking clocks is that they are out-of-
equilibrium systems, and are maintained so. This implies that thermal
resources must be expended in the working of any clock, as noted in the
previous section.

In quantifying this connection one needs to precisely define the notion of an
autonomous clock, i.e. a model where the resources involved in keeping the
clock out of equilibrium appear explicitly and can be accounted for. While it
is typically straightforward to argue whether a specific model is or is not
autonomous, a general criterion does not exist in the case of ticking clocks
with a quantum clockwork777For a clockwork with only incoherent dynamics, it
is arguably autonomous iff the Lindbladian generator satisfies local detailed
balance.. An important task is thus to derive the general construction of an
autonomous quantum ticking clock, both in and beyond the weak-coupling limit.

###  VII.3 The emergent continuity of classical clocks

Consider a classical system such as a pendulum. This can be interpreted as a
clock: by measuring the angle of the pendulum from the vertical one obtains
information about time (modulo the time period).

This system appears to contradict a central conclusion of our work: on the one
hand it appears to be an independent clock undisturbed by us looking at it, on
the other hand the time information — the angle $\theta$ — is continuous
rather than than taken from a discrete set; thus not a ticking clock.

This apparent continuity is however only an approximation. Observing the angle
of the pendulum involves interacting with photons of a particular wavelength.
The wavelength sets the imprecision $\delta\theta$ of the measurement. This
implies that the time information is effectively discrete: one can divide the
range of $\theta$ into intervals of $\delta\theta$. For true continuity one
would require vanishing wavelength, which in turn implies photons of unbounded
energy that would disturb the motion of the pendulum.

In a similar manner, every classical clock is thus also a discrete ticking
clock, with the property that the measurement precision is high enough to make
the time information appear continuous while low enough as to not disturb the
clock.

##  VIII Acknowledgements.

The authors acknowledge insightful discussions with Paul Erker, Marcus Huber,
Maximilian Lock, Florian Meier, Renato Renner, Emanuel Schwarzhans and Mischa
Woods. We also thank Florian Meier for pointing out the connection to full
counting statistics. R.S. acknowledges funding from the Swiss National Science
Foundation via an Ambizione grant PZ00P2_185986. N.N. acknowledges support
from the Swiss National Science Foundation through SNSF project No.
200021_188541 and through the the National Centre of Competence in Research
Quantum Science and Technology (QSIT). H.W. acknowledges support by the DFG
through SFB 1227 (DQ-mat), Quantum Valley Lower Saxony, and funding by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy EXC-2123 QuantumFrontiers 390837967.

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## Appendix

##  Appendix A Koashi-Imoto decomposition

Below we discuss the theorem postulating Koashi-Imoto decomposition. The
description (adapted to our settings) is borrowed from [2], the results of
which are identical to those of Koashi-Imoto [1]. Refs. [3, 4] show that the
results also transfer to infinite-dimensional Hilbert-spaces.

Suppose that we have a set of density matrices $\\{\rho_{k}\\}_{k}\in
L(\mathcal{H})$, and want to find the structure of the set of quantum channels
(trace-preserving and completely positive linear maps)
$\mathcal{E}:L(\mathcal{H})\to L(\mathcal{H})$ which leave these states
invariant,

$\displaystyle\mathcal{E}(\rho_{k})=\rho_{k}\ \forall k.$ (48)

According to the Stinespring dilation theorem, each such channel admits a
representation of a partial trace over a unitary evolution of the system $S$
together with the environment,

$\displaystyle\mathcal{E}(\rho_{S})=\operatorname{Tr}\left(U_{SE}(\rho_{S}\otimes\sigma_{E})U_{SE}^{\dagger}\right),$
(49)

as well as a decomposition in terms of Kraus operators,

$\displaystyle\mathcal{E}(\rho_{S})=\sum_{j}A_{j}\rho_{S}A_{j}^{\dagger},$
(50)

with $\sum_{j}A_{j}^{\dagger}A_{j}=\mathds{1}$. The Koashi-Imoto decomposition
then implies that the Hilbert space of the system can be written as

$\displaystyle\mathcal{H}=\bigoplus_{n}\mathcal{H}_{C_{n}}\otimes\mathcal{H}_{F_{n}},$
(51)

and the states w.r.t. this decomposition look like

$\displaystyle\rho_{k}$
$\displaystyle=\bigoplus_{n}p_{n|k}\;\rho^{(n)}_{k}\otimes\omega_{n}.$ (52)

Moreover, for every quantum channel $\mathcal{E}$ which leaves the states
$\rho_{k}$ invariant, the associated unitary has the form

$\displaystyle U_{SE}=\bigoplus_{n}\mathds{1}_{C_{n}}\otimes U_{F_{n}E}$ (53)

with unitaries $U_{F_{n}E}$ acting on
$\mathcal{H}_{F_{n}}\otimes\mathcal{H}_{E}$ that don’t change the state
$\omega_{n}$,

$\displaystyle\operatorname{Tr}\left(U_{F_{n}E}(\omega_{n}\otimes\sigma_{E})U_{F_{n}E}^{\dagger}\right)=\omega_{n}.$
(54)

The above defines a quantum channel on states of subspaces
$\mathcal{H}_{F_{n}}$, and can be described via Kraus decomposition,

$\displaystyle\sum_{j}A_{j,n}\omega_{n}A_{j,n}^{\dagger}=\omega_{n}.$ (55)

This allows us to write down the explicit form of Kraus operators $A_{j}$ in
50 as

$\displaystyle A_{j}$
$\displaystyle=\bigoplus_{n}\mathds{1}_{\mathcal{C}_{n}}\otimes A_{j,n}.$ (56)

##  Appendix B Elementary reset clocks: connecting asymptotic rates to moments
of the waiting time

The probability of the number of ticks at time $t$ being equal to $n$ is

$\displaystyle p_{n|t}=F_{n|t}-F_{n+1|t},$ (57)

where $F_{n|t}$ is the cumulative distribution function (CDF) of the waiting
time of the $n^{\mathrm{th}}$ tick. For elementary reset clocks this is the
$n$-fold convolution of the waiting time distribution $\omega_{t}$ for a
single tick,

$\displaystyle
F_{n|t}=\int_{0}^{t}\underbrace{\omega*\dots*\omega_{t^{\prime}}}_{n}dt^{\prime}.$
(58)

From this expression, we can derive the expectation value of $n^{k}$ ($k$th
moment in the number of ticks), $k\geq 1$,

$\displaystyle\braket{n^{k}}_{t}$
$\displaystyle=\sum_{n=0}^{\infty}n^{k}p_{n|t}$ (59)
$\displaystyle=\sum_{n=0}^{\infty}n^{k}\left(F_{n|t}-F_{n+1|t}\right)$ (60)
$\displaystyle=\sum_{n=1}^{\infty}\left(n^{k}-(n-1)^{k}\right)F_{n|t}$ (61)
$\displaystyle=\sum_{n=1}^{\infty}\sum_{l=0}^{k-1}\binom{k}{l}(-1)^{k-l+1}n^{l}F_{n|t}.$
(62)

Now we apply the Laplace transform to both sides of the equation, which maps
all convolutions to the powers of Laplace transforms of single ticks waiting-
time distributions (we denote the Laplace transform of a function by adding a
superscript $\cdot^{*}$ and the change of variable $t\mapsto s$),

$\displaystyle F_{n|t}\mapsto\frac{\left(\omega^{*}_{s}\right)^{n}}{s}.$ (63)

Thus, we arrive to

$\displaystyle\langle
N^{k}\rangle^{*}_{s}=\sum_{n=1}^{\infty}\sum_{l=0}^{k-1}\binom{k}{l}(-1)^{k-l+1}n^{l}\frac{\left(\omega^{*}_{s}\right)^{n}}{s}.$
(64)

Now we use the formula for calculating the sum
$\sum_{n=1}^{\infty}n^{k}x^{n}$,

$\displaystyle\sum_{n=1}^{\infty}n^{k}x^{n}=\frac{x\cdot
A_{k}(x)}{(1-x)^{k+1}},$ (65)

where $A_{k}(x)$ are Eulerian polynomials defined recursively,

$\displaystyle A_{0}(x)$ $\displaystyle=0;$ (66) $\displaystyle A_{k}(x)$
$\displaystyle=x(1-x)A_{k-1}^{\prime}(x)+A_{k-1}(x)\left(1+(k-1)x\right).$
(67)

This allows us to rewrite

$\displaystyle\langle N^{k}\rangle^{*}_{s}$
$\displaystyle=\frac{1}{s}\sum_{l=0}^{k-1}\binom{k}{l}(-1)^{k-l+1}\sum_{n=1}^{\infty}n^{l}\left(\omega^{*}_{s}\right)^{n}$
(68)
$\displaystyle=\frac{1}{s}\sum_{l=0}^{k-1}\binom{k}{l}(-1)^{k-l+1}\frac{\omega^{*}_{s}\cdot
A_{l}\left(\omega^{*}_{s}\right)}{(1-\omega^{*}_{s})^{l+1}}.$ (69)

To find the asymptotic behaviour, we expand the Laplace transform of the
single tick distribution in terms of its moments,

$\displaystyle\omega^{*}_{s}=1-\mu_{1}s+\frac{\mu_{2}s^{2}}{2}-\dots,$ (70)

and apply the inverse Laplace transform, which maps

$\displaystyle\frac{1}{s^{j}}\mapsto\frac{t^{j-1}}{(j-1)!},\ j\in\mathbb{N}.$
(71)

Asymptotics of the first two moments. For the expected number of ticks at time
$t$ and its Laplace transform approximation, we obtain

$\displaystyle\langle n\rangle_{t}$
$\displaystyle=\sum_{n=1}^{\infty}F_{n|t};$ (72) $\displaystyle\langle
n\rangle^{*}_{s}$
$\displaystyle=\frac{1}{s}\sum_{n=1}^{\infty}\left(\omega^{*}_{s}\right)^{n}$
(73) $\displaystyle=\frac{\omega^{*}_{s}}{s(1-\omega^{*}_{s})}$ (74)
$\displaystyle\approx\frac{1}{\mu_{1}s^{2}}\left(1-\mu_{1}s+\frac{\mu_{2}s}{2\mu_{1}}+O(s^{2})\right).$
(75)

Applying the inverse transform, we get the desired asymptotic rate (current of
ticks)

$\displaystyle\braket{n}_{t}$
$\displaystyle\to\frac{t}{\mu_{1}}+\left(\frac{\mu_{2}}{2\mu_{1}^{2}}-1\right),$
(76) $\displaystyle\lim_{t\to\infty}\frac{\braket{n}_{t}}{t}$
$\displaystyle=\frac{1}{\mu_{1}}.$ (77)

We perform a similar analysis for the second moment, ending with the
asymptotic behaviour:

$\displaystyle\langle n^{2}\rangle_{t}$
$\displaystyle\to\frac{t^{2}}{\mu_{1}^{2}}+\left(\frac{2(\mu_{2}-\mu_{1}^{2})}{\mu_{1}^{3}}-\frac{1}{\mu_{1}}\right)t$
$\displaystyle\quad\quad+\left(\frac{3\mu_{2}^{2}}{2\mu_{1}^{4}}-\frac{2\mu_{3}}{3\mu_{1}^{3}}-\frac{\mu_{2}}{\mu_{1}^{2}}\right),$
(78)

so that the variance (second cumulant) is linear in $t$ for large times,

$\displaystyle\langle n^{2}\rangle_{t}-\langle n\rangle^{2}_{t}$
$\displaystyle\to\left(\frac{\mu_{2}}{\mu_{1}^{3}}-\frac{1}{\mu_{1}}\right)t+\text{const}$
(79) $\displaystyle=\frac{\sigma^{2}}{\mu_{1}^{3}}t+\text{const}.$ (80)

where $\sigma^{2}$ is the variance of the waiting-time $\omega_{t}$.

The ratio of asymptotic rates thus satisfies the condition stated in the main
text:

$\displaystyle R_{1}$
$\displaystyle=\lim_{t\to\infty}\frac{d_{t}\braket{n}_{t}}{d_{t}\left(\braket{n^{2}}-\braket{n}^{2}\right)}=\frac{\mu^{2}}{\sigma^{2}}=R_{2}.$
(81)

##  Appendix C Asymptotic Allan variance for self-timed clocks

Allan variance is used as a measure of frequency stability in clocks and
similar devices, which require signal regularity. To calculate it in the
simplest setting without a dead-time between measurements, one partitions the
total observation time $T$ into $M$ bins of length $\tau$ (the averaging
time). One then then determines for each bin ("sample") the average frequency
and takes the average over the collection of $2$-sample variances. For
simplicity, let us assume that we start the sampling process for frequencies
$\nu$ at $t=0$. The first 2-sample variance is defined by

$\displaystyle\sigma^{2}_{\nu,1}$
$\displaystyle=\left(\nu_{1}-\frac{\nu_{1}+\nu_{2}}{2}\right)^{2}+\left(\nu_{1}-\frac{\nu_{1}+\nu_{2}}{2}\right)^{2}$
(82) $\displaystyle=\frac{(\nu_{1}-\nu_{2})^{2}}{2},$ (83)

where $\nu_{1}$ and $\nu_{2}$ are the observed average frequencies in the time
intervals $[0,\tau]$ and $[\tau,2\tau]$ respectively. In terms of the numbers
of observed ticks at times $0,\tau$ and $2\tau$, they can be written as

$\displaystyle\nu_{1}$ $\displaystyle=\frac{n_{\tau}-n_{0}}{\tau},$ (84)
$\displaystyle\nu_{2}$ $\displaystyle=\frac{n_{2\tau}-n_{\tau}}{\tau}.$ (85)

Substituting into the 2-sample variance above, we obtain

$\displaystyle\sigma^{2}_{\nu,1}$
$\displaystyle=\frac{1}{2\tau^{2}}\left(n_{2\tau}-2n_{\tau}+n_{0}\right)^{2}.$
(86)

We repeat this procedure for each consecutive pairs of intervals
$\left[k\tau,(k+1)\tau\right]$ and $\left[(k+1)\tau,(k+2)\tau\right]$ for
$k\in\\{0,\dots,M-1\\}$. The average over the expectation value of all
2-sample variances gives us the Allan variance $A$ of the process,

$\displaystyle A$
$\displaystyle=\frac{1}{M}\sum_{k=0}^{M-1}\langle\sigma^{2}_{\nu,k}\rangle$
(87)
$\displaystyle=\frac{1}{M}\sum_{k=0}^{M-1}\frac{1}{2\tau^{2}}\langle\left(n_{(k+2)\tau}-2n_{(k+1)\tau}+n_{k\tau}\right)^{2}\rangle.$
(88)

We can calculate the above if we assume that the clock is in its unique steady
state, i.e. in the asymptotic limit $t\rightarrow\infty$. In this limit, the
two-time correlations appearing in 88 can be calculated using the quantum
regression theorem [26],

$\displaystyle\frac{d}{d\tau}\langle n_{t}n_{t+\tau}\rangle$
$\displaystyle=\frac{1}{\mu}\langle
n_{t}1\\!\\!1_{t+\tau}\rangle=\frac{t}{\mu^{2}}\quad(t\gg 1).$ (89)

The boundary condition for $\tau=0$ reads $\langle
n_{t}^{2}\rangle=\frac{t^{2}}{\mu^{2}}+\Sigma t$, which follows from the
general asymptotic properties of Markovian processes (see Sec. IV.1) and
results in the solution

$\displaystyle\langle n_{t}n_{t+\tau}\rangle$ $\displaystyle=\langle
n_{t}^{2}\rangle+\frac{t\tau}{\mu^{2}}=\frac{t(t+\tau)}{\mu^{2}}+\Sigma t.$
(90)

Calculating the Allan variance, we obtain

$\displaystyle A$
$\displaystyle=\frac{1}{M}\sum_{k=0}^{M-1}\frac{\Sigma}{\tau}=\frac{\Sigma}{\tau}.$
(91)

##  Appendix D Continuity of the set of measurement outcomes for a non-
independent clock

A simple clock construction due to Salecker and Wigner [8], later expounded by
Peres [9], is the following Hamiltonian: an equally spaced, non-degenerate and
finite spectrum,

$\displaystyle H$ $\displaystyle=\sum_{n=0}^{d-1}n\omega\ket{n}\\!\bra{n}.$
(92)

Its likeness to a clock is seen through the evolution of the following basis
of states:

$\displaystyle\ket{\theta_{k}}$
$\displaystyle=\frac{1}{\sqrt{d}}\sum_{n=0}^{d-1}e^{-i2\pi nk/d}\ket{n},$ (93)

where $k\in\\{0,1,2,...,d-1\\}$ (in the following $k\equiv k\mod d$). One
finds that every ‘angle’ state $\ket{\theta_{k}}$ evolves to
$\ket{\theta_{k+1}}$ in the time period $2\pi/(\omega d)$,

$\displaystyle e^{-iH\frac{2\pi}{\omega d}}\ket{\theta_{k}}$
$\displaystyle=\ket{\theta_{k+1}},$ (94)

Hence the label ‘angle states’, they may be arranged on a circular clock face,
each $\ket{\theta_{k}}$ at an angle of $2\pi k/d$.

A simple manner of employing this system as a clock would be to initialise it
in the state $\ket{\theta_{0}}$ at $t=0$, and measure in the $\theta$ basis at
a later time, corresponding the set of projectors and corresponding outcomes:

$\displaystyle\Pi_{k}$
$\displaystyle=\ket{\theta_{k}}\\!\bra{\theta_{k}},\quad t_{k}=\frac{2\pi
k}{\omega d}.$ (95)

At this point it appears that the clock has a discrete structure in that it
measures time in terms of a fundamental unit. However we are free to pick
another set of angle states: take $\alpha\in(0,1)$ and define a new set via

$\displaystyle\ket{\lambda_{k}}$
$\displaystyle=\frac{1}{\sqrt{d}}\sum_{n=0}^{d-1}e^{-i2\pi
n(k+\alpha)/d}\ket{n}$ (96)

For every $\alpha$ the above is an orthonormal basis, and one can check that
$\ket{\theta_{k}}$ evolves into $\ket{\lambda_{k}}$ in a time step of
$\frac{2\pi\alpha}{\omega d}$. Thus the new set is shifted w.r.t. the original
one by the angle $\frac{2\pi\alpha}{\omega d}$.

We can now construct a measurement that incorporates the new basis as well,
for instance the following set of $2d$ POVM elements:

$\displaystyle\Pi_{k}$
$\displaystyle=\frac{1}{2}\ket{\theta_{k}}\\!\bra{\theta_{k}},\quad
t_{k}=\frac{2\pi k}{\omega d},$ (97) $\displaystyle\Pi^{\prime}_{l}$
$\displaystyle=\frac{1}{2}\ket{\lambda_{l}}\\!\bra{\lambda_{l}},\quad
t_{l}=\frac{2\pi(l+\alpha)}{\omega d}.$ (98)

Indeed one can generalise to an arbitrary number of values of
$\alpha\in[0,1)$. Thus the time measured by this clock does not have a
fundamental unit, even though each individual measurement might have a finite
number of outcomes.