# Tomography of clock signals using the simplest possible reference

Nuriya Nurgalieva<EMAIL_ADDRESS>Institute for Theoretical Physics, ETH
Zürich, 8093 Zürich, Switzerland  Department of Physics, University of Zurich,
8057 Zurich, Switzerland  Ralph Silva<EMAIL_ADDRESS>Institute for
Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland  Renato Renner
Institute for Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland

###### Abstract

We show that finite physical clocks always have well-behaved signals, namely
that every waiting-time distribution generated by a physical process on a
system of finite size is guaranteed to be bounded by a decay envelope.
Following this consideration, we show that one can reconstruct the
distribution using only operationally available information, namely, that of
the ordering of the ticks of one clock with the respect to those of another
clock (which we call the reference), and that the simplest possible reference
clock — a Poisson process — suffices.

> Time and I have quarrelled. All hours are midnight now. I had a clock and a
> watch, but I destroyed them both. I could not bear the way they mocked me.
>
> Jonathan Strange & Mr Norrell
> Suzanna Clark

###### Contents

  1. I Introduction
  2. II Finite clocks always have well behaved signals
  3. III A Poisson process: the simplest ticking clock
  4. IV Reconstruction of the characteristic function of an unknown clock
  5. V Finite sampling: frequency as a resource
  6. VI Reconstructing a non i.i.d. clock
  7. VII Discussion & Outlook
  8. A Proof of existence of moment-generating function for finite clockworks
    1. A.1 First waiting-time of an elementary ticking clock
    2. A.2 Compression of tick dynamics
    3. A.3 Eigenmatrix decomposition w.r.t. reduced Lindbladian
    4. A.4 Summary
  9. B Moments of Poisson process
  10. C Finite sampling errors
  11. D Tomography of non-reset clocks
  12. E Sub-Poissonian processes as a reference

##  I Introduction

#### Motivation.

Clocks come in various shapes and sizes, but the most recognisable are those
that ‘tick’, that is to say, they occasionally emit a clearly observable
signal to indicate that a particular unit of time has passed, be it a second,
an hour or any other time interval. By doing so, the time of an observer is
split into segments, each marked by a different number of ticks counted by the
clock.

An ideal clock would mark these intervals with perfect regularity, each tick
arriving after exactly the same change in the background time. In reality, no
clock is able to do so, whether for reasons of finite size [1, 2, 3], finite
resources [4], or just experimental imperfections. For a non-ideal clock, the
timing of the ticks is not a fixed stable constant, but rather a fluctuating
quantity. Mathematically we quantify this by speaking of the probability
density function of the time of arrival of the ticks, a function
$\Pr(T_{1}=t_{1},T_{2}=t_{2},...)$ that measures the likelihood that the first
tick $T_{1}$ happened at $t_{1}$, the second at $t_{2}$ and so on. For a clock
whose ticks form an i.i.d. sequence, this is determined by the waiting-time
distribution of a single tick, $\Pr(T_{n}-T_{n-1}=t)$. More generally, as long
as the ticks are identically distributed, the asymptotic behaviour of the
clock is encapsulated in its full counting statistics, namely the moments of
the number of ticks of the clock for large enough time $t$.

When we observe a clock, however, we cannot observe these distributions
directly as we have no access to $t$: the background time that our systems are
dancing to the tune of. The only manner in which we can access $t$ is via a
clock itself — thus the only way for us to observe the ticks of one clock is
to compare them to those of another. Thus arises a natural question: is it
possible to use the operationally available information — that of the ordering
of the ticks of one clock with respect to those of another reference clock —
to reconstruct the full abstract information of the first clock, i.e. its
probability density function?

#### Setting.

In this paper, we show that for a clock whose ticks form an identically
distributed sequence, we can indeed reconstruct the distribution by using
arguably the simplest possible reference clock: an i.i.d. Poisson process. If
the ticks of the clock of interest are also independent — so that its
probability density function for many ticks is determined by that of a single
tick — then we recover the waiting-time distribution of a single tick.111Also
referred as the delay function in some literature. More generally, we can
always recover the full counting statistics of the clock.

The method of reconstruction works by determining the moments of the waiting-
time distribution, and thus relies on them being well-behaved in a certain
sense. In that light, we also prove here that every waiting-time distribution
generated by a physical process on a system of finite size (as measured by its
Hilbert space) is guaranteed to fulfill this requirement, and provide a
sufficient criterion for the infinite-dimensional case by considering the
Lindbladian dynamics of the system.

#### Contribution.

Our result is a key conceptual step in the operational characterisation of
clocks, a ‘tomography of time-signals’ so to speak. It shows that even the
simplest clocks are good references given enough measurements; that describing
a clock signal w.r.t. background time does make sense since we can
operationally access it; and finally — unsurprisingly — that the frequency of
the reference together with the memory of multiple measurements are resources
in this process.

#### Structure.

The paper is structured as follows. First, in Sec. II we clarify what we mean
by by a clock and show that all clocks whose dynamics are generated by a
finite state space have well-behaved delay functions. In Sec. III we provide
both mathematical and physical reasons for regarding the Poisson process as
the simplest and most imprecise reference clock. This is followed in Sec. IV
by the method of reconstruction of the moments and characteristic function,
and Sec. V discusses the rate of convergence in the finite sampling case,
based on the rate of the Poisson reference. Sec. VI considers the
reconstruction of non-iid clocks, before we conclude in Sec. VII.

##  II Finite clocks always have well behaved signals

#### Modelling the dynamics of a clock.

To understand the ticks generated by an imperfect clock, we have to model both
the ticks and the physical system that generates them. This is customarily
done via a bipartite structure: one has an internal clockwork ($C$) that is
coupled to a classical integer-labelled register ($T$), the latter of which
indicates how many ticks the clock has recorded. Any measurement of the
observer is only upon $T$, the internal clockwork is left undisturbed.
Crucially, while the register is strictly classical, the clockwork may be
quantum-mechanical in nature. For a deeper discussion of this model and how it
arises among time-keeping models in general, see [5]. We summarise the central
points here.

At any given time, the information about the state of the clockwork and
register is encoded in the following statistical ensemble or density matrix
upon their joint Hilbert space $\mathcal{H}_{C}\otimes\mathcal{H}_{T}$:

$\displaystyle\tilde{\rho}_{CT}(t)$
$\displaystyle=\sum_{n}\rho^{(n)}_{C}(t)\otimes\ket{n}\\!\bra{n}_{T},$ (1)

where the $\rho_{C}^{(n)}(t)$ are semi-positive matrices whose sum is
normalised. The information solely related to ticks is encoded only upon the
register, this is obtained by tracing out (ignoring) the state of the
clockwork:

$\displaystyle\eta_{T}(t)$
$\displaystyle=\sum_{n}\operatorname{Tr}_{C}\left[\rho^{(n)}_{C}(t)\right]\ket{n}\\!\bra{n}_{T}\equiv\left\\{\Pr(n|t)\right\\}_{n}.$
(2)

The distribution is generally not deterministic, reflecting the uncertainty in
the number of recorded ticks of an imperfect clock.

Furthermore, it is assumed that the register evolves _serially_ and
_irreversibly_ , i.e. that the register state can only move one step at a time
from $n$ to $n+1$. Both of these simplifications suggest themselves directly
from the behaviour of real clocks that we observe; we employ them in this
work. Note that both can be relaxed for more general models, in particular the
latter assumption of irreversibility when working with entropic clocks.

One can describe the behaviour of the ticks of the clock by their _time of
arrival_ , i.e. the moment in time $T_{n}$ that the register state changes
from $n-1$ to $n$. For imperfect clocks, these are not determined times;
rather, one has a probability distribution for them, denoted by
$\Pr(T_{n}=t)$. One can shift between the time-of-arrival picture and of the
number-of-ticks picture above (2) via the relation:

$\displaystyle\Pr(T_{n}=t)$
$\displaystyle=-\frac{d}{dt}\sum_{m=0}^{n-1}\Pr(m|t).$ (3)

#### Self-contained clocks and the assumption of self-timing.

It is customary when describing the dynamics of a ticking clock to take it to
be _self-timed_ , i.e., that its dynamics is time-independent and dependent
only upon its own state. This ensures that all of the resources and time-
keeping of the clock are contained within its description rather than arising
from an external hidden clock. Under this assumption, the dynamics can be
simply expressed as arising from a Lindbladian generator $\tilde{L}_{CT}$,
albeit one that maintains the bipartite structure from (1), in particular the
classical nature of the tick register.

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

We take as a starting point the case of _reset_ clocks (equivalently _i.i.d._
clocks). These have the property that the moment the clock ticks the clockwork
returns to a particular fixed state regardless of the dynamics leading up to
the tick. This ensures that the process of each tick is independent of the
previous one as well as identical. As a result, the distribution of the time
of arrival of every tick is entirely determined by that of a single tick, that
of the clock when it begins in the fixed post-tick state (referred to as the
reset state), via the $n$-fold convolution:

$\displaystyle\Pr(T_{n}=t)$ $\displaystyle=\omega(t)^{\circ n},$ (4)

where $\omega(t)$ is the _waiting-time distribution_ of a single tick
(alternatively, as the _delay function_).

#### Moments and the moment generating function.

One manner of recovering the waiting-time distribution $\omega(t)$ of an
unknown clock is by finding its moments $M_{j}$:

$\displaystyle M_{j}$ $\displaystyle=\int_{0}^{\infty}t^{j}\omega(t)dt,$ (5)

where $j\in\mathbb{Z}^{+}$. If the distribution is well-behaved in a certain
sense, then it is uniquely determined by the full set of moments
$\\{M_{j}\\}$. In particular, the relation is one-to-one in the case that the
moment generating function $\mathcal{M}$ is well-defined in a finite region.
More precisely, given

$\displaystyle\mathcal{M}(x)$
$\displaystyle=\int_{0}^{\infty}e^{tx}\omega(t)dt,$ (6)

the distribution $\omega$ is uniquely determined by its moments $M_{j}$ if
there exists some $a>0$ such that $\mathcal{M}(x)$ is finite for $x<a$.

#### The moment generating function of finite clocks.

Our first result is that for clocks whose dynamics are generated by a finite
state space — so that $\mathcal{H}_{C}$ can be taken to be of finite size —
and which are guaranteed to tick eventually, the moment generating function is
well-defined in a finite neighbourhood. We outline the argument here, the full
proof is detailed in Appendix A. We prove the statement by showing that
$\omega(t)$ falls under an exponential envelope $e^{-at}$ for some $a>0$, so
that $\mathcal{M}(x)$ exists for $x<a$.

Proof outline. Under Lindbladian evolution the state of the clock and register
at any time is the exponential $e^{\tilde{\mathcal{L}}t}[\tilde{\rho}_{0}]$,
where $\tilde{\rho}_{0}$ is an arbitrary initial state. The generator is
infinite-dimensional — thanks to the register — however, as we are only
interested in a single tick of the clock, we can construct a reduced scenario
wherein the dynamics after the first tick is ignored, giving us a finite state
space of size $d+1$, where $d$ is the dimension of the clock Hilbert space and
the extra space is to denote the clock having ticked, a state we label as
$\ket{\mathrm{tick}}$. The reduced Lindbladian $\mathcal{L}^{\prime}$ is also
finite and has the same dynamics for the first tick as the original.

The eigenvalues of the generator are either zero (corresponding to steady
states) or negative (corresponding to decaying states). If the clock is
guaranteed to tick eventually, then the only steady state is
$\ket{\mathrm{tick}}$. One can now express the difference between the initial
state of the clock and the steady state in terms of the eigenmatrices of
$\mathcal{L}^{\prime}$:222If the generator is not diagonalisable then one has
generalised eigenmatrices. The proof in Appendix A deals with this general
case.

$\displaystyle\rho_{0}^{\prime}-\ket{\mathrm{tick}}\\!\bra{\mathrm{tick}}$
$\displaystyle=\sum_{i}c_{i}\sigma_{i}.$ (7)

We argue that every eigenmatrix appearing in the above sum must correspond to
an eigenvalue with a strictly negative real part; this follows by applying the
evolution operator and taking $t\to\infty$. The left side goes to zero as
$\rho_{0}^{\prime}\to\ket{\mathrm{tick}}\\!\bra{\mathrm{tick}}$ while
$\ket{\mathrm{tick}}\\!\bra{\mathrm{tick}}$ is preserved. The right side gains
exponentials $e^{\lambda_{i}t}$ where $\lambda_{i}$ are the eigenvalues of
$\mathcal{L}^{\prime}$. For this to go to zero, every $\lambda_{i}$ must have
a negative real part.

By picking the eigenvalue with real part closest to zero, we obtain a lower
bound on the decay rate of the above expression. This, in turn, allows us to
obtain the same lower bound on the waiting-time distribution. From this, we
can show that the moment generating function exists for $x<a$ where
$a=-\max\Re(\lambda_{i})>0$ (the zero eigenvalue is excluded in the
maximisation).

##  III A Poisson process: the simplest ticking clock

Next, we turn to the reference, which we pick to be arguably the simplest and
most widely available resource: a Poisson process. Viewed as a clock, it is
specified by the Poisson distribution of rate $\gamma$: in a time interval of
$t$ the probability distribution of the number of ticks seen from this clock
is

$\displaystyle\Pr(N_{R}=n|t)$ $\displaystyle=\frac{(\gamma t)^{n}e^{-\gamma
t}}{n!}.$ (8)

An equivalent description of a Poisson process is via its waiting-time
distribution: the probability density function of the time between one tick
and the next is the exponential decay:

$\displaystyle\Pr(T_{n}-T_{n-1}=t)$ $\displaystyle=\gamma e^{-\gamma t}.$ (9)

The most distinctive example of a Poisson process is the decay of a
radioactive atom.

The Poisson process has the property of being memoryless in the sense that in
an infinitesimal interval of time $dt$, the probability of a tick being seen
is $\gamma dt$ and is entirely independent of the history of ticks and thus
there are no correlations across time. This is in contrast to a perfectly
regular clock whose probability density of ticking is highly correlated with
the history of ticks.

The simplicity of the Poisson process is also seen with respect to the
precision measure $P=\mu^{2}/\sigma^{2}$, where $\mu$ is the first moment of
the waiting-time and $\sigma^{2}$ is the variance of the same. This measure
derives from the question: how many ticks of an i.i.d. clock (which the
Poisson process is) must we see before the variance of the ticks is as large
as the expected interval between them? The better the clock the higher $P$ is
expected to be. For the Poisson process, we have $P=1$ only.

We can also argue that the Poisson process is trivial from a complexity
viewpoint: one can describe its dynamics without the need of a clockwork,
i.e., its time evolution depends only upon the entry in the register $T$:

$\displaystyle\eta_{T}$
$\displaystyle=\sum_{n=0}^{\infty}\Pr(n|t)\ket{n}\\!\bra{n}_{T},$ (10)
$\displaystyle\frac{d}{dt}\eta=\mathcal{L}\left[\eta\right]$
$\displaystyle=\gamma\left(\Gamma\eta\Gamma^{\dagger}-\frac{1}{2}\left\\{\Gamma^{\dagger}\Gamma,\eta\right\\}\right),$
(11) $\displaystyle\text{where}\quad\Gamma$
$\displaystyle=\sum_{0}^{\infty}\ket{n+1}\\!\bra{n}.$ (12)

Thus, from the property of being memoryless, having minimal precision, and the
emergence in systems of minimal complexity, the Poisson process suggests
itself as the simplest possible clock. And yet, as we proceed to demonstrate,
it makes a perfectly good reference.

The simplicity of the Poisson process allows for a useful property: increasing
the effective rate of the process by taking many together. If one has ticks
produced by two processes at rates $\gamma_{1}$ and $\gamma_{2}$, then by
combining the two — i.e. counting a tick from either one as a single tick — we
obtain a new Poisson process of rate $\gamma_{1}+\gamma_{2}$. This is handy as
we will show that the frequency of our reference process is a resource, the
higher the better.

##  IV Reconstruction of the characteristic function of an unknown clock

Figure 1: Measuring one clock against the other. We characterize the clock $A$
by means of measuring it relative to a reference clock $R$, which we model as
a Poisson clock. The observed statistics is then the number of ticks of $R$
between successive ticks of $A$.

Our main result concerns the following scenario: we have an imperfect clock
$A$ whose ticks form an i.i.d. sequence so that the probability density
function of multiple ticks is determined by that of a single tick:

$\displaystyle\Pr(T_{1}=t_{1},$ $\displaystyle T_{2}=t_{2},T_{3}=t_{3},...)$
$\displaystyle=\omega(t_{1})\cdot\omega(t_{2}-t_{1})\cdot\omega(t_{3}-t_{2})...$
(13)

$\omega(t)$ is called the waiting-time distribution, or waiting-time in short,
of the clock. We characterize it by means of measuring it against a reference
clock that we take to be a Poisson clock $R$ of rate $\gamma$. Our observed
quantity is the number of ticks of $R$ between successive ticks of $A$; see
Fig. 1. This itself is an i.i.d. variable: the probability of seeing $n$ ticks
of $R$ between the $j^{th}$ and $j+1^{th}$ tick of $A$ is simply the product
probability of the waiting-time being $t$ and the Poisson process probability
of $n$ given parameter $\gamma t$:

$\displaystyle\Pr(N_{R}=n,T_{j+1}-T_{j}=t)$
$\displaystyle=\Pr(N_{R}=n|t)\cdot\omega(t).$ (14)

As the number of measurements increases, we can estimate the probability of
observing $n$ ticks of $R$ given the clock $A$:

$\displaystyle p_{n}$
$\displaystyle=\int_{0}^{+\infty}Pr(N_{R}=n|T=t)\;\omega(t)\;dt$ (15)
$\displaystyle=\int_{0}^{+\infty}\frac{(\gamma t)^{n}e^{-\gamma
t}}{n!}\omega(t)\;dt$ (16)

We can write down the $k^{th}$ moments of the distribution $\\{p_{n}\\}_{n}$
(which can be interpreted as the moments of the reference $R$ relative to the
clock $A$),

$\displaystyle m_{k}(\omega,\gamma)$
$\displaystyle=\sum_{n=0}^{+\infty}n^{k}p_{n}=\sum_{n=1}^{+\infty}n^{k}\int_{0}^{+\infty}\frac{(\gamma
t)^{n}e^{-\gamma t}}{n!}\omega(t)dt.$ (17)

To simplify this expression, we make use of the moments of the Poisson
distribution $\Pr(N_{R}=n|T=t)$, which can be written compactly as 333For
completeness, we include the derivation in Appendix B.

$\displaystyle M_{k}(t)$
$\displaystyle=\sum_{n=1}^{+\infty}n^{k}\Pr(N_{R}=n|T=t)$ (18)
$\displaystyle=\sum_{j=1}^{k}(\gamma t)^{j}\genfrac{\\{}{\\}}{0.0pt}{}{k}{j},$
(19)

where the braces $\genfrac{\\{}{\\}}{0.0pt}{}{k}{j}$ denote the Stirling
numbers of the second kind, which count the number of ways to partition a set
of $k$ objects into $j$ non-empty subsets. Substituting into (17), we obtain

$\displaystyle m_{k}(\omega,\gamma)$
$\displaystyle=\int_{0}^{+\infty}\sum_{j=1}^{k}\genfrac{\\{}{\\}}{0.0pt}{}{k}{j}(\gamma
t)^{j}\omega(t)dt$ (20)
$\displaystyle=\sum_{j=1}^{k}\genfrac{\\{}{\\}}{0.0pt}{}{k}{j}\gamma^{j}M_{j}(\omega),$
(21)

where $M_{j}(\omega)$ is the $j$th moment of the waiting-time of the measured
clock $A$.

Hence, having access to all relative moments $m_{k}(\omega,\gamma)$, we are
able to reconstruct all the moments of the measured clock. The explicit
expression is given by the inverse Stirling transform of the sequence
$\\{m_{k}(\omega,\gamma)\\}$,

$\displaystyle M_{k}(\omega)$
$\displaystyle=\frac{1}{\gamma^{k}}\sum_{j=1}^{k}(-1)^{k-j}\genfrac{[}{]}{0.0pt}{}{k}{j}m_{j}(\omega,\gamma),$
(22)

where the square brackets $\genfrac{[}{]}{0.0pt}{}{k}{j}$ denote the unsigned
Stirling numbers of the first kind, counting the number of permutations of $k$
elements with $j$ cycles. For example, the first and the second clock moments
have the form

$\displaystyle\gamma M_{1}(\omega)$ $\displaystyle=m_{1}(\omega,\gamma);$ (23)
$\displaystyle\gamma^{2}M_{2}(\omega)$
$\displaystyle=m_{2}(\omega,\gamma)-m_{1}(\omega,\gamma),$ (24)

with the following moments formed in an analogous manner. The moment
generating and characteristic functions can then be reconstructed as

$\displaystyle\mathcal{M}(x)$
$\displaystyle=\sum_{j=0}^{\infty}\frac{x^{j}}{j!}M_{j}(\omega);$ (25)
$\displaystyle\mathcal{C}(x)$
$\displaystyle=\sum_{j=0}^{\infty}\frac{(ix)^{j}}{j!}M_{j}(\omega).$ (26)

The above analysis implicity assumes that the rate $\gamma$ of the Poisson
process is known. However, even if $\gamma$ is unknown, one may still access
the moments of the clock relative to the timescale of the reference, i.e.
$\gamma^{j}M_{j}$. One may then reconstruct the moment-generating and
characteristic functions $\mathcal{M}(\gamma x)$ and $\mathcal{C}(\gamma x)$
analogously to the above. This is consistent with a fundamental fact about
time-keeping: all measured time intervals are in relation to the timescale of
some physical process; our current standard is the ground-state hyperfine
transition frequency of the Cs-133 atom. Thus any characterisation of a clock
must necessarily be expressed w.r.t. the timescale of another reference clock,
as is the case in this work.

##  V Finite sampling: frequency as a resource

The results of the previous section show that one could recover the waiting-
time distribution of an unknown clock using any Poisson process as a
reference, albeit in the asymptotic limit of infinite ticks. In this section
we derive error bounds on the estimates of the clock moments in the case of a
finite number of ticks, proving the robustness of the results. Of specific
interest is how the error depends upon the finite frequency $\gamma$ of the
reference clock. An important conclusion we draw is that the error has
independent contributions from both the innate fluctuations of the clock of
interest as well as the finite nature of $\gamma$.

Consider that one has a finite sample size of $N$ with elements
$X_{1},\dots,X_{N}$. Note that each sample corresponds to the interval between
one tick of the unknown clock and the next, with $X_{i}$ the number of ticks
of the reference clock in between. We can then use (22) and (17) to construct
an estimator $\tilde{M}_{k}$ for the $k^{th}$ moment $M_{k}(\omega)$ of the
waiting time of the clock,

$\displaystyle\tilde{M}_{k}$
$\displaystyle=\frac{1}{\gamma^{k}}\sum_{j=1}^{k}(-1)^{k-j}\genfrac{[}{]}{0.0pt}{}{k}{j}\sum_{m=1}^{N}\frac{n_{m}^{j}}{N}$
(27)

We analyse this estimator in Appendix C. It is unbiased, so that
$\braket{\tilde{M}_{k}}=M_{k}(\omega)$. By calculating its variance, we can
apply Chebyshev’s inequality for the probability that the estimate is off by a
fraction $\theta$

$\displaystyle
Pr\left(\left|\frac{\tilde{M}_{k}}{M_{k}}-1\right|\geq\theta\right)$
$\displaystyle\leq\frac{1}{N\theta^{2}}\Big{(}\frac{1}{P^{(k)}}+\frac{1}{\gamma}\frac{M_{2k-1}(\omega)}{M_{k}^{2}(\omega)}$
$\displaystyle\quad\quad\quad\quad+O\left(\frac{1}{\gamma^{2}}\right)\Big{)},$
(28)

where we have defined the “precision of the $k^{th}$ clock moment” as the
ratio

$\displaystyle P^{(k)}$
$\displaystyle=\frac{M_{k}^{2}(\omega)}{M_{2k}(\omega)-M_{k}^{2}(\omega)}$
(29)

In the case of $k=1$, this corresponds to a standard precision/accuracy
measure, see [3, 4, 2, 5, 6], and corresponds to the inverse of the Fano
Factor for stochastic processes [7, 5].

We can understand (V) as the confluence of multiple factors. Firstly, the
product $(N\theta^{2})^{-1}$ implies that the same size scales quadratically
with our confidence interval $\theta$. Within the error there are — to first
order in $\gamma^{-1}$ — two independent contributions. Firstly, the precision
$P^{(k)}$: it is scale invariant, i.e. it does not change under speeding up
the clock by a factor of $a$ via the transformation $\omega(t)\to
a\;\omega(at)$.444If $\omega(t)\to a\;\omega(at)$, then $M_{k}(\omega)\to
a^{-k}M_{k}(\omega)$ Thus $P^{(k)}$ is a measure of the fluctuations of the
clock moment w.r.t. its own timescale, and contributes to the error
independently of the reference. Indeed one could repeat this analysis with a
perfect reference instead of a Poissonian, in this case the error reduces to
only the term involving $P^{(k)}$. The second term may be understood as a
comparison between the scale of the clock moment and that of the reference, as
the ratio $M_{2k-1}(\omega)/M_{k}^{2}(\omega)$ scales by $a$ under the
speeding up of the clock as described above; here is where the frequency of
the reference clock is important. As an illustrative example, the expression
for the first moment is

$\displaystyle\Pr\left(\left|\frac{\tilde{M}_{1}}{m_{1}}-1\right|\geq\theta\right)$
$\displaystyle=\frac{1}{N\theta^{2}}\left(\frac{1}{P^{(1)}}+\frac{\nu}{\gamma}\right),$
(30)

where $P^{(1)}$ is the aforementioned precision and $\nu=1/M_{1}(\omega)$ is
the average frequency of the measured clock (there are no contributes of
$O(\gamma^{-2})$ in this case). There are thus two independent contributions
to the error: the clocks own uncertainty given by $1/P^{(1)}$ and the relative
frequency $\nu/\gamma$ between the measured clock and the reference Poisson
process. An important consequence is that increasing $\gamma$ is useful only
up to a point: once $\nu/\gamma\ll 1/P^{(1)}$ the clock’s own fluctuations
will dominate the error rendering a further increase in the Poisson rate
superfluous, a similar statement holds for the higher moments (bounds derived
in Appendix C).

##  VI Reconstructing a non i.i.d. clock

The clock of interest has so far been taken to be _irreversible_ and reset,
the first implying that a tick cannot be reversed, and the second resulting in
every tick being identical to the rest. While these are idealised properties,
real clocks may not fulfill them perfectly. This is, for instance, the case
for clocks constructed out of thermodynamic processes [8, 7, 4]. Here, every
stochastic jump present in the dynamics — such as the one that makes the
register of the clock increase by $1$ — implies the presence of the reverse
process, which cannot be set to zero unless one assumes an infinite entropy
production.

For a clock whose register can move both ways, there is no notion of a time-
of-arrival of a tick, but the number of ticks recorded by the register is
still a well-defined quantity. Assuming that the clock is still self-timed, so
that a time-independent Lindbladian operator exists to describe the behaviour
of the clock, the long-time behaviour of the moments of the number of ticks is
given by polynomials in $t$:

$\displaystyle\braket{n^{j}}$
$\displaystyle=\sum_{i=0}^{j}\alpha_{i}^{(j)}t^{i}.$ (31)

All of the coefficients, with the exception of $\alpha_{0}^{(j)}$, are
uniquely determined by the Lindbladian; the latter is dependent on the choice
of the initial state of the clock. Thus, the long-time behaviour is entirely
encapsulated by the set of coefficients $\\{\alpha_{i}^{j}\\}_{i,j}$ ($i\geq
1$), the so-called full counting statistics of the process.

These may also be recovered using a Poisson reference clock. In this case, the
tick of the target clock is not well-defined, so we rather use the ticks of
the reference. One may wait for $n$ ticks of the reference clock and then
measure the number of ticks $k$ of the target. By repeating this procedure,
one may calculate the moments $\braket{k^{j}}$ (which we label by
$m_{j}(A,\gamma,n)$) from which the coefficients $\smash{\alpha_{i}^{(j)}}$
are determined. The full details are found in Appendix D.

For instance, the current of ticks and the rate of variance of the current
(i.e., the first two cumulants of the full counting statistics) of the target
clock are estimated from

$\displaystyle J_{\infty}$ $\displaystyle=\gamma\frac{m_{1}(A,\gamma,n)}{n},$
(32) $\displaystyle\Sigma_{\infty}$
$\displaystyle=\gamma\left(m_{2}(A,\gamma,n)-m_{1}(A,\gamma,n)^{2}\left(1+\frac{1}{n}\right)\right),$
(33)

from which the precision $P=J_{\infty}/\Sigma_{\infty}$ may be calculated —
this is the same as the precision introduced earlier, reducing to the earlier
expression in the case of reset clocks.

##  VII Discussion & Outlook

#### Justifying abstract information measures.

Most recent literature on ticking clocks quantifies their accuracy via the
waiting-time distribution, especially the mean and variance thereof [3, 4, 9,
6, 2]. It is usually easier to deal with than operational measures such as in
[1]. However, this is only justified if the waiting-time is recoverable from
an operational scenario, and without too much difficulty. The contribution of
our work is a strong positive result in that direction, placing the abstract
information-theoretical characterisation of clocks performed in the literature
so far on a sound operational footing.

On its own, it also represents a type of operational clock tomography: the
process of recovering the properties of a clock from measurements made w.r.t.
to other clocks. Note that our method of reconstruction may be considered to
be incoherent and in the time domain: incoherent, because it works via a tick
register that both clocks interact with incoherently; in the time domain,
because the relevant object is the sequence of ticks. In contrast, the current
method of feedback in atomic clocks is coherent and in the frequency domain.
There, the (laser) clock interacts coherently with the reference (Cs atoms),
and the output is a measure of how well the frequencies of the target and the
reference match.

Our work thus opens up a new type of tomography for clocks. A relevant
question is whether one can also perform feedback on clocks in a similar
manner, i.e. by using the tick sequence rather than an interferometric method
as is the current standard.

#### The assumption of i.i.d.

Throughout this work, we have restricted ourselves to i.i.d. clocks for the
reference and self-timed clocks as targets. While for random processes in
general this is typically a highly restrictive assumption, it is not entirely
so for clocks. Firstly, there are fundamental processes that are i.i.d., such
as the decay of unstable atomic and nuclear states; indeed, our current
definition of the second is in terms of such a transition. Thus, the notion of
an i.i.d. reference clock is not without merit. As for the target being self-
timed, consider that this is not the case. Then one has two regimes, loosely
speaking: either the drifts in the dynamics of the clock are slow w.r.t. its
own timescale, or they are fast.

In the former case, which includes all of the clocks that we currently
construct, one may still perform tomography as described in this paper. On
short timescales, this gives us a snapshot of the clock as it is; repeating it
on longer timescales provides a measurement of the drift in the dynamics. This
is the point of measures of frequency stability, such as the Allan variance,
meant to isolate both the intrinsic “shot noise” and the drift.

In the latter case of fast-drifting dynamics, applying the method described
here will lead to results of repeated measurements that do not converge at
all, showing that the clock is, in fact, unstable.

Thus, the results described in this work are still applicable to non-i.i.d.
targets if used carefully.

#### Operational time.

While our work has employed the notion of background Schrödinger time to
ensconce the results, it would be interesting to translate them into a setting
where time is treated more _operationally_. That is, where there is no notion
of a ‘continuous background time’; rather, the sequence of ticks from various
clocks is the fundamental object from which a time reference is defined.

There are natural analogues within this picture to the background case. For
instance, consider a Poisson process, which represents a clock with no memory
of $t$. In the absence of background time there is no such object, however
there is a relational equivalent: a Bernoulli sequence [10]. This is a
collection of two or more clocks with the property that given the sequence of
ticks so far, the probability that the next tick is from any one of the clocks
is a fixed probability independent of the history of ticks. Thus the analogue
of having a Poissonian reference $R$ would be to have a pair of clocks $Q,R$
that form a Bernoulli sequence of ticks. Either of them may then take over the
role of an operational background.

Similarly, the simplest case of an unknown clock — a reset clock — has a
relational analogue as well. We may look at the sequence of ticks formed by
the clock of interest $A$ with $R$, and separately with $Q$, with
probabilities $p^{(m)}_{n}$ and $q^{(m)}_{n}$ of there being $n$ ticks of $R$
(or $Q$) between the $m^{th}$ and $(m+1)^{(th)}$ tick of $A$. If these are
independent of $m$, then $A$ can be said to be a reset clock w.r.t. both $Q$
and $R$.

In such a scenario we may repeat the analysis in Sec. IV to recover the
moments of the clock of interest, either w.r.t. $Q$ or to $R$. However, in the
absence of background time, the analysis is complicated by additional
correlations. That is, unlike in the case of continuous $t$, here the moments
w.r.t. $Q$ may not be recoverable from the moments w.r.t. $R$ due to the
possible presence of correlations in the tick sequence of all three clocks
that are not visible in any of the marginal tick sequences of a pair of them.

An interesting direction of future research is to explore this operational
scenario in more detail.

#### General reference clocks.

The Poisson process suggests itself as a reference clock due to its
simplicity, it is in a sense the most freely available one, in addition to
being very analytically tractable as our results demonstrate. It is
nonetheless interesting to consider how other references might be employed to
characterise an unknown clock. For instance, in Appendix E we show that
estimates and bounds can be constructed if the reference is sub-Poissonian —
i.e. with a precision $P^{(1)}>1$. Deriving sharper bounds for relevant
reference clocks would be an interesting future question.

#### Relativistic generalizations.

Finally, an important future direction is to consider scenarios where the
clocks are embedded in some spacetime structure: here the goal will be to
reconstruct the the clock properties w.r.t. the spacetime background $x,t$.

## Acknowledgements

The authors thank Henrik Wilming for interesting discussions. We acknowledge
support from the Swiss National Science Foundation (SNSF) through project No.
200021_188541 and through the National Centre of Competence in Research
SwissMAP. R.S. further acknowledges funding from the Swiss National Science
Foundation via an Ambizione grant PZ00P2_185986.

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## Appendix

##  Appendix A Proof of existence of moment-generating function for finite
clockworks

###  A.1 First waiting-time of an elementary ticking clock

To start off, we take the standard model of a ticking clock with a register:
the Hilbert space is a direct product of clockwork and register spaces, the
latter being a classical degree of freedom that only contains information
about the number of ticks:

$\displaystyle\tilde{\mathcal{H}}$
$\displaystyle=\mathcal{H}_{C}\otimes\mathcal{H}_{T},$ (34)
$\displaystyle\tilde{\rho}_{t}$
$\displaystyle=\sum_{n=0}^{\infty}\rho^{(n)}_{t}\otimes\ket{n}\\!\bra{n}_{T},$
(35)

with dynamics described by a Lindbladian generated semigroup that preserves
the classicality of the register, in addition to the other properties
mentioned in the main text namely, that the register only moves forward and by
one step at a time:

$\displaystyle\tilde{\mathcal{L}}\left[\tilde{\rho}_{t}\right]=-i\left[H\otimes\mathds{1}_{T},\tilde{\rho}_{t}\right]$
$\displaystyle+\sum_{k}\mathcal{D}_{L_{k}\otimes\mathds{1}_{T}}\left[\tilde{\rho}_{t}\right]$
$\displaystyle+\sum_{j}\mathcal{D}_{J_{j}\otimes\Gamma_{T}}\left[\tilde{\rho}_{t}\right],$
(36)

where $\mathcal{D}_{A}$ is the standard dissispator:

$\displaystyle\mathcal{D}_{A}\left[X\right]$
$\displaystyle=AXA^{\dagger}-\frac{1}{2}\left\\{A^{\dagger}A,X\right\\},$ (37)

and $\Gamma_{T}$ is the raising operator on the register,

$\displaystyle\Gamma_{T}$
$\displaystyle=\sum_{n=0}^{\infty}\ket{n+1}\\!\bra{n}_{T}.$ (38)

For clarity of notation, we use $\tilde{\mathcal{L}},\tilde{\rho}$ to denote
the Lindbladian and state on the joint Hilbert space of clockwork and
register, with the $\mathcal{L},\rho$ denoting the same upon the clockwork
space alone.

It is more useful to express the dynamics w.r.t. the clockwork states
$\rho_{t}^{(n)}$, each corresponding to a definite number $n$ of ticks
recorded by the register. To do so we define the following operators on the
clockwork alone:

$\displaystyle\mathcal{L}_{0}\left[\rho\right]$
$\displaystyle=-i\left[H,\rho\right]+\sum_{k}\mathcal{D}_{L_{k}}\left[\rho\right],$
(39a) $\displaystyle V$ $\displaystyle=\sum_{j}J_{j}^{\dagger}J_{j},$ (39b)
$\displaystyle\mathcal{L}_{+}\left[\rho\right]$
$\displaystyle=\sum_{j}J_{j}\rho J_{j}^{\dagger}.$ (39c)

From these we find the equation of motion for the clockwork states to be:

$\displaystyle d_{t}\rho_{t}^{(n)}$
$\displaystyle=\mathcal{L}_{0}\left[\rho_{t}^{(n)}\right]-\frac{1}{2}\left\\{V,\rho_{t}^{(n)}\right\\}+\mathcal{L}_{+}\left[\rho_{t}^{(n-1)}\right],$
(40)

where the latter term does not appear for $n=0$,

$\displaystyle d_{t}\rho_{t}^{(0)}$
$\displaystyle=\mathcal{L}_{0}\left[\rho_{t}^{(0)}\right]-\frac{1}{2}\left\\{V,\rho_{t}^{(0)}\right\\}.$
(41)

Importantly, the equation of motion for $n=0$ does not depend on any other
clockwork states. This is expected as the clock is fundamentally irreversible
— once it ticks it never goes back. Thus the dynamics of each clockwork state
only depends on itself and the previous ones in the sequence of states.

The initial state of the register is taken to correspond to no tick having
been seen,

$\displaystyle\tilde{\rho}_{0}$
$\displaystyle=\rho_{0}\otimes\ket{0}\\!\bra{0}_{T}.$ (42)

To find the waiting-time distribution of the first tick we may track the rate
of change of the survival probability, which is the same as the probability of
being in the $n=0$ state:

$\displaystyle\omega_{t}$
$\displaystyle=-d_{t}Q_{t}=-d_{t}\operatorname{Tr}\left[\rho_{t}^{(0)}\right]$
(43)
$\displaystyle=\operatorname{Tr}\left[\mathcal{L}_{+}\left[\rho_{t}^{(0)}\right]\right]$
(44) $\displaystyle=\operatorname{Tr}\left[V\rho_{t}^{(0)}\right].$ (45)

###  A.2 Compression of tick dynamics

As it stands the clock Lindbladian is infinite-dimensional and hence
cumbersome to work with. For the purpose of studying the first waiting-time
distribution we do not require all of the dynamics, in particular the dynamics
after the first tick. In view of this we construct a reduced scenario where we
keep the dynamics of the first tick the same and compress the entire space of
the clock after ticking into a single one-dimensional Hilbert space.

In technical terms, this corresponds to the following transformation of the
Hilbert space. The original Hilbert space may be expressed as a direct sum:

$\displaystyle\mathcal{H}_{C}\otimes\mathcal{H}_{T}=\bigoplus_{n}\mathcal{H}_{C}^{(n)},$
(46)

where $n$ is the index of the register. We keep the $n=0$ space and compress
the rest into a one-dimensional space $\mathbb{C}$, so that the new space is
$\mathcal{H}_{C}\oplus\mathbb{C}$, a $(d+1)$-dimensional Hilbert space.

For ease of notation we will use the vector $\ket{T}$ and operator
$\ket{T}\\!\bra{T}$ to refer to two things depending on the expression they
fall in:

  1. 1.

as a one-dimensional state/operator from the new space in expressions of the
form $\ket{\psi}\oplus\ket{T}\in\mathcal{H}_{C}\oplus\mathbb{C}$,

  2. 2.

as a normalised $d+1$-dimensional state or operator thereupon with support
only upon the additional space $\mathbb{C}$, in expressions of the form
$\ket{T}\in\mathcal{H}\otimes\mathbb{C}$.

Our goal is to construct dynamics upon this new space in such a way that the
state at any time $t$ would be given by

$\displaystyle\rho^{(0)}_{t}\oplus\left(1-Q_{t}\right)\ket{T}\\!\bra{T},$ (47)

so that the ‘tick’ space corresponds to the part of the dynamics where the
first tick has already happened.

Rather than derive the correct reduced form of the Lindbladian, we state it
directly from intuition and prove that it corresponds to the correct dynamics.
Starting from the three operators in (39), we begin by keeping the dynamics
that are only upon the clockwork, simply appending the null operator on the
tick space,

$\displaystyle\mathcal{L}_{0}^{\prime}=\mathcal{L}_{0}\oplus
0\ket{T}\\!\bra{T}.$ (48)

For the tick dynamics we note that the post-tick state is no longer relevant,
thus all of the required information is in the positive operator $V$ from
(39). Given the spectral decomposition of the original $V$ as
$\sum_{k}v_{k}\ket{v_{k}}\\!\bra{v_{k}}$, we first modify the eigenvectors by
adding a null state in the tick space

$\displaystyle\ket{v_{k}^{\prime}}=\ket{v_{k}}\oplus 0\ket{T},$ (49)

and from these we get a set of operators $\\{A_{k}\\}_{k}$ that are sufficient
for the tick dynamics, replacing the original set $\\{J_{j}\\}_{j}$:

$\displaystyle A_{k}$ $\displaystyle=\ket{T}\\!\bra{v_{k}^{\prime}}.$ (50)

By constructing a dissipator from this set we find:

$\displaystyle\sum_{k}\mathcal{D}_{A_{k}}\left[X\right]$
$\displaystyle=\operatorname{Tr}\left[V^{\prime}X\right]\ket{T}\\!\bra{T}-\frac{1}{2}\left\\{V^{\prime},X\right\\},$
(51)

where $V^{\prime}$ is the new form of $V$,

$\displaystyle V^{\prime}$ $\displaystyle=\sum_{k}A_{k}^{\dagger}A_{k}=V\oplus
0\ket{T}\\!\bra{T}.$ (52)

Thus the new reduced Lindbladian in full is

$\displaystyle\mathcal{L}^{\prime}\left[\rho_{t}^{\prime}\right]$
$\displaystyle=\mathcal{L}_{0}^{\prime}\left[\rho_{t}^{\prime}\right]-\frac{1}{2}\left\\{V^{\prime},\rho_{t}^{\prime}\right\\}+\operatorname{Tr}\left[V^{\prime}\rho_{t}^{\prime}\right]\ket{T}\\!\bra{T},$
(53)

where each term is the direct reduction of the three operators in (39) in the
original description.

To see that the dynamics are the correct one, we note first that by
construction the dynamics do not create any coherence between the tick space
and the rest. By picking the initial state to be incoherent w.r.t. the direct
sum,

$\displaystyle\rho_{0}^{\prime}=\eta_{0}\oplus P_{0}\ket{T}\\!\bra{T},$ (54)

we can express the state at a later time $t$ in the same decomposition:

$\displaystyle\rho_{t}^{\prime}=\eta_{t}\oplus P_{t}\ket{T}\\!\bra{T}.$ (55)

We can now straightforwardly derive the equations of motion for
$\eta_{t},P_{t}$ by writing the equation of motion (53) w.r.t. the
decomposition:

$\displaystyle d_{t}\left(\eta_{t}\oplus P_{t}\ket{T}\\!\bra{T}\right)$
$\displaystyle=\left(\mathcal{L}_{0}\oplus 0\right)\left[\eta_{t}\oplus
P_{t}\ket{T}\\!\bra{T}\right]-\frac{1}{2}\left\\{V\oplus 0,\eta_{t}\oplus
P_{t}\ket{T}\\!\bra{T}\right\\}+\operatorname{Tr}\left[\left(V\oplus
0\right)\left(\eta_{t}\oplus P_{t}\right)\right]\ket{T}\\!\bra{T},$ (56)

which is seen to split neatly w.r.t. the direct sum into two separate
equations:

$\displaystyle d_{t}\eta_{t}$
$\displaystyle=\mathcal{L}_{0}\left[\eta_{t}\right]-\frac{1}{2}\left\\{V,\eta_{t}\right\\},$
(57a) $\displaystyle d_{t}P_{t}$
$\displaystyle=\operatorname{Tr}\left[V\eta_{t}\right].$ (57b)

The first of these is identical to (41) for the evolution of the clockwork
state corresponding to no ticks (41), while the second is the equation of
motion for the cumulative probability of ticking — compare to (43). Thus if we
pick $\eta_{0}=\rho_{0},P_{0}=0$ it will be that $\eta_{t}=\rho_{t}^{(0)}$ for
all $t$, and that $P_{t}$ will be the probability of having ticked,
$P_{t}=1-Q_{t}$.

###  A.3 Eigenmatrix decomposition w.r.t. reduced Lindbladian

We proceed to study the new reduced Lindbladian, using the tools from [11] to
describe the eigenmatrices of the Lindbladian. Since the Lindbladian is not
diagonalisable in general, it will have both proper and generalised
eigenvectors. We employ the following notation. Consider every instance of an
eigenvalue with a proper eigenmatrix to be denoted by $\lambda_{i}$ and its
proper eigenmatrix by $\sigma_{i,0}$, so that

$\displaystyle\left(\mathcal{L}-\lambda_{i}\mathds{1}\right)\left[\sigma_{i,0}\right]$
$\displaystyle=0.$ (58)

To each of these eigenvalues there may also be generalised eigenmatrices that
we can take to be a Jordan chain $\\{\sigma_{i,j}\\}$, so that

$\displaystyle\left(\mathcal{L}-\lambda_{i}\mathds{1}\right]\left[\sigma_{i,j}\right]$
$\displaystyle=\sigma_{i,j-1},$ (59)

where $j$ runs from $1$ to the dimension of the corresponding Jordan block of
the Lindbladian when written in Jordan form. The entire set of eigenmatrices
over $i,j$ form a linearly independent basis for the space of matrices, there
are $(d+1)^{2}$ of them. Importantly, each Jordan chain is of finite size.

The evolution of each eigenmatrix under the Lindbladian operator is simple
given its Jordan chain:

$\displaystyle e^{\mathcal{L}t}\left[\sigma_{i,j}\right]$
$\displaystyle=e^{\lambda_{i}t}\sum_{k=0}^{j}\frac{t^{j-k}}{(j-k)!}\sigma_{j,k}=e^{\lambda_{i}t}\theta(t)_{i,j},$
(60)

where we have denoted by $\theta_{i,k}$ the fixed polynomial of finite degree
in $t$ formed by the eigenmatrices.

By construction, the tick state is a stationary state, i.e. a proper
eigenmatrix of eigenvalue $0$,

$\displaystyle\mathcal{L}^{\prime}\left[\ket{T}\\!\bra{T}\right]$
$\displaystyle=0.$ (61)

Under the assumption that the clock eventually ticks, it is the steady state
corresponding to the chosen initial state of the clock, i.e.

$\displaystyle\lim_{t\to\infty}e^{\mathcal{L}^{\prime}t}\left[\rho_{0}^{\prime}\right]$
$\displaystyle=\ket{T}\\!\bra{T}.$ (62)

Consider therefore the difference between the initial state and the stationary
state in terms of the eigenmatrices of $\mathcal{L}$,

$\displaystyle\rho_{0}^{\prime}-\ket{T}\\!\bra{T}$
$\displaystyle=\sum_{i,j}c_{i,j}\sigma_{i,j}.$ (63)

Applying the evolution operator $e^{\mathcal{L}^{\prime}t}$ for large $t$ we
see that the LHS goes to $0$.

We now argue that the eigenmatrices $\sigma_{i}$ that appear in the RHS of the
expression must all correspond to eigenvalues with negative real part (i.e.
decaying eigenmatrices). Consider that this is not the case. Then applying the
evolution operator $e^{\mathcal{L}^{\prime}t}$ on both sides and taking $t$
large, we would be left with only the non-decaying (zero real part)
eigenvalues on the RHS:

$\displaystyle 0$
$\displaystyle=\sum_{i:\Re(\lambda_{i})=0}\sum_{j}c_{i,j}e^{\lambda_{i}t}\theta(t)_{i,j}.$
(64)

But this is a contradiction, and thus it must be that $c_{i,j}=0$ for all
$\lambda_{i}$ with zero real part.

Returning to (63), we consider the evolution for arbitrary $t$ to get:

$\displaystyle
e^{\mathcal{L}^{\prime}t}\left[\rho_{0}^{\prime}\right]-\ket{T}\\!\bra{T}$
$\displaystyle=\sum_{i}e^{\lambda_{i}t}\sum_{j}c_{i,j}\theta(t)_{i,j}.$ (65)

We can obtain the waiting-time distribution by applying $V$ and tracing,
resulting in:

$\displaystyle\omega_{t}$
$\displaystyle=\operatorname{Tr}\left[Ve^{\mathcal{L}t}\left[\rho_{0}\right]\right]$
(66)
$\displaystyle=\operatorname{Tr}\left[V^{\prime}e^{\mathcal{L}^{\prime}t}\left[\rho_{0}^{\prime}\right]\right]$
(67)
$\displaystyle=\operatorname{Tr}\left[V^{\prime}\left(e^{\mathcal{L}^{\prime}t}\left[\rho_{0}^{\prime}\right]-\ket{T}\\!\bra{T}\right)\right]$
(68)
$\displaystyle=\operatorname{Tr}\left[V^{\prime}\sum_{i}e^{\lambda_{i}t}\sum_{j}c_{i,j}\theta(t)_{i,j}\right].$
(69)

Since all of the $\lambda_{i}$ in the above expression have negative real
part, the waiting-time distribution must decay exponentially, albeit
multiplied by a finite polynomial in $t$. To complete the proof that the
moment generating function is well-defined, we consider its definition:

$\displaystyle M_{x}$
$\displaystyle=\lim_{a\to\infty}\int_{0}^{a}\omega_{t}e^{xt}dt$ (70)
$\displaystyle=\lim_{a\to\infty}\int_{0}^{a}\operatorname{Tr}\left[V^{\prime}\sum_{i}e^{(x+\lambda_{i})t}\sum_{j}c_{i,j}\theta(t)_{i,j}\right]dt.$
(71)

If we denote by $-\Delta$ the real part of the eigenvalue that is closest to
zero (least negative), then for all $x\in(-\Delta,+\Delta)$ the above limit
exists. The integrand is a finite sum of terms, the largest of which (in the
large $t$ limit) is at worst of the form $Ae^{(x-\Delta)t}t^{N}$, where $N$ is
the largest degree of $t$ appearing in any of the polynomials $\theta_{i,j}$,
and the constants $A$ are finite given the finite size of $V^{\prime}$ and the
eigenmatrices $\sigma_{i,j}$.

###  A.4 Summary

We have demonstrated that for any combination of clock initial state and
Lindbladian such that the clock definitely ticks eventually, the moment
generating function $M_{x}$ of the waiting time distribution $\omega_{t}$ has
a neighbourhood around $0$ where it exists, thus the moments of $\omega_{t}$
uniquely determine it.

We did this by proving that the state of the clock corresponding to not having
ticked yet must itself decay exponentially with a rate that is at least as
large as the minimal negative real part of those eigenvalues of the
Lindbladian that have negative real part.

In the absence of the clockwork being finite, it need not be necessary that
there is a minimal negative real part for the eigenvalues of the Lindbladian —
one may have an infinite sequence approaching $0$. However if there was such a
minimal decay rate for an infinite clockwork, it may be the case that the
moment generating function is well-defined, though in this case one also has
to bound the now infinite sum in (71) to complete the same proof.

##  Appendix B Moments of Poisson process

For completeness, here we derive of the moments of Poisson process in terms of
the Stirling numbers. To do so, we need several ingredients.

A $k$th falling power of $x$ is defined as

$\displaystyle
x^{\underline{k}}=x\cdot(x-1)\cdot\dots\cdot(x-k+1)=\frac{x!}{(x-k)!}.$ (72)

The Stirling numbers of the second kind $\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}$
can then be used to convert usual powers to falling powers via

$\displaystyle
x^{n}=\sum_{k=0}^{n}x^{\underline{k}}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}.$ (73)

The factorial moment of the Poisson distribution (the expectation value of
$X^{\underline{k}}$) with mean $\lambda$ is equal to $\lambda^{k}$,

$\displaystyle\mathbb{E}[X^{\underline{k}}]=\sum_{x=k}^{\infty}\frac{x!}{(x-k)!}\frac{\lambda^{x}e^{-\lambda}}{x!}=e^{-\lambda}\lambda^{k}\underbrace{\sum_{x=0}^{\infty}\frac{\lambda^{x}}{x!}}_{e^{\lambda}}=\lambda^{k}.$
(74)

Hence, the expectation value of $X^{n}$ is given by

$\displaystyle\mathbb{E}[X^{n}]=\sum_{k=0}^{n}\mathbb{E}[X^{\underline{k}}]\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}=\sum_{k=0}^{n}\lambda^{k}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}.$
(75)

##  Appendix C Finite sampling errors

Consider that one is estimating the $k^{th}$ moment of the unknown clock from
a finite sample of size $N$, i.e. we have observed the numbers
$\\{n_{1},...,n_{N}\\}$ of ticks of the reference between ticks of the clock
of interest. From (17) in the main text, we form an estimator first for the
relative moments of the reference,

$\displaystyle\tilde{m}_{k}$
$\displaystyle=\sum_{m=1}^{N}\frac{n_{m}^{k}}{N},$ (76)

and using (22), we obtain the estimate for the clock moment,

$\displaystyle\tilde{M}_{k}$
$\displaystyle=\frac{1}{\gamma^{k}}\sum_{j=1}^{k}(-1)^{k-j}\genfrac{[}{]}{0.0pt}{}{k}{j}\sum_{m=1}^{N}\frac{n_{m}^{j}}{N}.$
(77)

To apply Chebyeshev’s inequality for this moment, we require the average and
variance of the above estimator, for which we need its first two moments. The
first moment is $M_{k}(\omega)$ itself:

$\displaystyle\braket{\tilde{M}_{k}}$
$\displaystyle=\frac{1}{\gamma^{k}}\sum_{j=1}^{k}(-1)^{k-j}\genfrac{[}{]}{0.0pt}{}{k}{j}\sum_{m=1}^{N}m_{k}=M_{k}(\omega),$
(78)

where we have used (22) again; this confirms the estimator as unbiased. It’s
second moment is then

$\displaystyle\braket{\tilde{M}_{k}^{2}}$
$\displaystyle=\frac{1}{\gamma^{2k}}\sum_{j=1}^{k}\sum_{r=1}^{k}(-1)^{2k-j-r}\genfrac{[}{]}{0.0pt}{}{k}{j}\genfrac{[}{]}{0.0pt}{}{k}{r}\sum_{m=1}^{N}\sum_{l=1}^{N}\frac{\braket{n_{m}^{j}n_{l}^{r}}}{N^{2}}$
(79)
$\displaystyle=\frac{1}{\gamma^{2k}}\sum_{j=1}^{k}\sum_{r=1}^{k}(-1)^{2k-j-r}\genfrac{[}{]}{0.0pt}{}{k}{j}\genfrac{[}{]}{0.0pt}{}{k}{r}\frac{1}{N^{2}}\left(N(N-1)m_{j}(\omega,\gamma)m_{r}(\omega,\gamma)+Nm_{j+r}(\omega,\gamma)\right)$
(80)
$\displaystyle=M_{k}^{2}(\omega)\left(1-\frac{1}{N}\right)+\frac{1}{\gamma^{2k}N}\sum_{j=1}^{k}\sum_{r=1}^{k}(-1)^{2k-j-r}\genfrac{[}{]}{0.0pt}{}{k}{j}\genfrac{[}{]}{0.0pt}{}{k}{r}\sum_{s=1}^{j+r}\genfrac{\\{}{\\}}{0.0pt}{}{j+r}{s}\gamma^{s}M_{s}(\omega),$
(81)

where we have used (22) again to turn the sum involving
$m_{j}(\omega,\gamma)m_{r}(\omega,\gamma)$ into $M_{k}^{2}(\omega)$, and (21)
to express $m_{j+r}(\omega,\gamma)$ in terms of the clock moments.

We are primarily interested in how the errors scale w.r.t. the frequency
$\gamma$ of our reference, so we keep the zeroth and first order terms, which
compute to

$\displaystyle\braket{\tilde{M}_{k}^{2}}$
$\displaystyle=M_{k}^{2}(\omega)\left(1-\frac{1}{N}\right)+\frac{1}{\gamma^{2k}N}\left(\gamma^{2k}M_{2k}(\omega)+\gamma^{2k-1}M_{2k-1}(\omega)+O\left(\gamma^{2k-2}\right)\right)$
(82)
$\displaystyle=M_{k}^{2}(\omega)+\frac{1}{N}\left(M_{2k}(\omega)-M_{k}^{2}(\omega)\right)+\frac{1}{N\gamma}M_{2k-1}(\omega)+\frac{1}{N}O\left(\frac{1}{\gamma^{2}}\right),$
(83)

from which the variance of the estimator is

$\displaystyle\braket{\tilde{M}_{k}^{2}}-\braket{\tilde{M}_{k}}^{2}$
$\displaystyle=\frac{1}{N}\left(M_{2k}(\omega)-M_{k}^{2}(\omega)\right)+\frac{1}{N\gamma}M_{2k-1}(\omega)+\frac{1}{N}O\left(\frac{1}{\gamma^{2}}\right).$
(84)

Chebyshev’s inequality for the estimate of the $k^{th}$ clock moment thus
reads:

$\displaystyle Pr\left(\left|\tilde{M}_{k}-M_{k}\right|\geq\epsilon\right)$
$\displaystyle\leq\frac{1}{N\epsilon^{2}}\left(M_{2k}(\omega)-M_{k}^{2}(\omega)+\frac{1}{\gamma}M_{2k-1}(\omega)+O\left(\frac{1}{\gamma^{2}}\right)\right).$
(85)

We convert it to a fractional error by expressing $\epsilon=\theta M_{k}$,

$\displaystyle
Pr\left(\left|\frac{\tilde{M}_{k}}{M_{k}}-1\right|\geq\theta\right)$
$\displaystyle\leq\frac{1}{N\theta^{2}M_{k}^{2}(\omega)}\left(M_{2k}(\omega)-M_{k}^{2}(\omega)+\frac{1}{\gamma}M_{2k-1}(\omega)+O\left(\frac{1}{\gamma^{2}}\right)\right)$
(86)
$\displaystyle=\frac{1}{N\theta^{2}}\left(P^{(k)}+\frac{1}{\gamma}\frac{M_{2k-1}(\omega)}{M_{k}^{2}(\omega)}+O\left(\frac{1}{\gamma^{2}}\right)\right),$
(87)

where we have denoted the “precision” of the $k^{th}$ moment by

$\displaystyle P^{(k)}$
$\displaystyle=\frac{M_{2k}(\omega)-M_{k}^{2}(\omega)}{M_{k}^{2}(\omega)}.$
(88)

The precision is a scale-invariant quantity, i.e. if one changes the waiting-
time of the clock of interest from $\omega(t)$ to $a\omega(a*t)$ — essentially
speeding up the clock by a factor of $a$ — the precision is left unaffected.
On the other hand, the ratio of moments $M_{2k-1}(\omega)/M_{k}^{2}(\omega)$
scales as $a$ under such a transformation.

Note that one could perform the same analysis for estimators of
$m_{k}(\omega,\gamma)$ rather than $M_{k}(\omega)$. Since the underlying
distribution of the $\\{X_{m}\\}$ has moments $\\{m_{k}(\omega,\gamma)\\}$,
according to Chebyshev’s inequality

$\displaystyle
Pr\left(\left|\bar{X^{k}}-m_{k}(\omega,\gamma)\right|\geq\epsilon\right)\leq\frac{m_{2k}(\omega,\gamma)-m_{k}^{2}(\omega,\gamma)}{N\epsilon^{2}}.$
(89)

Dividing by the rate $\gamma^{k}$, and substituting explicit expressions for
the moments, we obtain

$\displaystyle
Pr\left(\left|\frac{\bar{X^{k}}}{\gamma^{k}}-\frac{m_{k}(\omega,\gamma)}{\gamma^{k}}\right|\geq\epsilon\right)$
$\displaystyle\leq\frac{\sum_{j=1}^{2k}\genfrac{\\{}{\\}}{0.0pt}{}{2k}{j}\gamma^{j}M_{j}(\omega)-\left(\sum_{j=1}^{k}\genfrac{\\{}{\\}}{0.0pt}{}{k}{j}\gamma^{j}M_{j}(\omega)\right)^{2}}{N\epsilon^{2}\gamma^{2k}}$
(90)
$\displaystyle=\frac{M_{2k}(\omega)-M_{k}^{2}(\omega)}{N\epsilon^{2}}+\frac{k}{\gamma}\cdot\frac{(2k-1)M_{2k-1}(\omega)-2(k-1)M_{k-1}M_{k}}{N\epsilon^{2}}+\frac{1}{N\epsilon^{2}}\cdot\mathcal{O}\left(\frac{1}{\gamma^{2}}\right)$
(91)
$\displaystyle=\frac{M_{2k}(\omega)-M_{k}^{2}(\omega)}{N\epsilon^{2}}+\frac{1}{N\epsilon^{2}}\cdot\mathcal{O}\left(\frac{1}{\gamma}\right),$
(92)

which is similar in form to that obtained for the estimators of
$M_{k}(\omega)$.

##  Appendix D Tomography of non-reset clocks

Let us generalize the procedure for finding out the parameters of non-reset
clocks. In this case, we can no longer count just the number of ticks of the
reference clock $R$ between the ticks of the target clock $A$, as each tick of
$A$ is not an independent event. However, we can still reconstruct the waiting
time distribution of $A$ by counting the total number of ticks of the target
clock $A$ against the total number of ticks of the reference clock $R$, which
we still model as a Poisson clock.

The probability of having observed $n$ ticks of the reference clock and $k$
ticks of the target clock at any background time $t$ is given by

$\displaystyle P(n,k|t)=P_{R}(n|t)\cdot P_{A}(k|t).$ (93)

Since the ticks of $R$ are i.i.d. and their waiting time is given by the
exponential decay distribution $\gamma e^{-\gamma t}$, the probability of
having counted $n$ ticks at time $t$ is described by the Erlang distribution

$\displaystyle P_{R}(n|t)=\frac{e^{-\gamma t}\gamma^{n}t^{n-1}}{(n-1)!}.$ (94)

Then the conditional probability of observing $k$ ticks of $A$ given $n$ ticks
of $R$ can be written as

$\displaystyle P(k|n)$
$\displaystyle=\int_{0}^{+\infty}P_{R}(n|t)P_{A}(k|t)dt$ (95)
$\displaystyle=\int_{0}^{+\infty}\frac{e^{-\gamma
t}\gamma^{n}t^{n-1}}{(n-1)!}P_{A}(k|t)dt.$ (96)

Similarly to the consideration for reset clocks, we can write the relative
moments of $A$ with respect to $R$

$\displaystyle m_{j}(A,\gamma,n)$
$\displaystyle=\sum_{k=0}^{\infty}k^{j}P(k|n)$ (97)
$\displaystyle=\int_{0}^{+\infty}P_{R}(n|t)P_{A}(k|t)dt$ (98)
$\displaystyle=\int_{0}^{+\infty}\frac{e^{-\gamma
t}\gamma^{n}t^{n-1}}{(n-1)!}\sum_{k=0}^{\infty}k^{j}P_{A}(k|t)dt.$ (99)

We assume that we are already operating in the regime where the target clock
$A$ is in its steady state; this can be done, for example, by waiting for a
sufficiently long time555this would be $t>>\lambda_{min}^{-1}$, the inverse of
the eignenvalue of the Lindbladian with the smallest real part. before
starting the statistics count. In this asymptotic case, we can write $A$’s
counting statistics as polynomials in $t$ [12]

$\displaystyle\langle
X^{j}\rangle=\sum_{k=0}^{\infty}k^{j}P_{A}(k|t)=\sum_{i=0}^{j}\alpha_{i}^{(j)}t^{i}.$
(100)

For example, the first two cumulants of the counting statistics - the
asymptotic current and rate of variance of ticks - are given by

$\displaystyle J_{\infty}$ $\displaystyle=\alpha_{1}^{(1)},$ (101)
$\displaystyle\Sigma_{\infty}$
$\displaystyle=\alpha_{1}^{(2)}-2\alpha_{0}^{(1)}\alpha_{1}^{(1)},$ (102)

from which the precision of $A$ can be calculated using

$\displaystyle R=\frac{J_{\infty}}{\Sigma_{\infty}}.$ (103)

Substituting the polynomial into the relative moments expression, we obtain

$\displaystyle m_{j}(A,\gamma,n)$
$\displaystyle=\int_{0}^{+\infty}\frac{e^{-\gamma
t}\gamma^{n}t^{n-1}}{(n-1)!}\sum_{i=0}^{j}\alpha_{i}^{(j)}t^{i}dt$ (104)
$\displaystyle=\sum_{i=0}^{j}\alpha_{i}^{(j)}\int_{0}^{+\infty}\frac{e^{-\gamma
t}\gamma^{n}t^{i+n-1}}{(n-1)!}dt$ (105)
$\displaystyle=\sum_{i=0}^{j}\alpha_{i}^{(j)}\frac{i!}{\gamma^{i}}\binom{n+i-1}{i}$
(106)

Taking the limit $n\to\infty$

$\displaystyle\lim_{n\to\infty}\frac{m_{1}(A,\gamma,n)}{n}=\frac{J_{\infty}}{\gamma}.$
(107)

Hence, collecting the statistics for the relative moments, we can eventually
identify the coefficients $\alpha_{i}^{(j)}$, which are related to individual
moments of the waiting time distribution of $A$.

It is important to note that this method works not only for target clocks $A$
which have a well-defined waiting time distribution, but also for those which
don’t – for example, reversible clock, which have a probability of ticking
backward [5].

#### Error estimation.

Using Chebyshev’s inequality, if we repeat the sampling $M$ times, then for
the first moment:

$\displaystyle\Pr\left(\left|\frac{\bar{X}}{n}-\frac{J_{\infty}}{\gamma}\right|\geq\epsilon\right)$
$\displaystyle\leq\frac{\langle k^{2}\rangle-\langle
k\rangle^{2}}{Mn^{2}\epsilon^{2}}$ (108)
$\displaystyle=\frac{J_{\infty}^{2}\frac{n(n+1)}{\gamma^{2}}+\Sigma_{\infty}\frac{n}{\gamma}-J_{\infty}^{2}\frac{n^{2}}{\gamma^{2}}+\alpha_{0}^{(2)}-(\alpha_{0}^{(1)})^{2}-2\alpha_{0}^{(1)}J_{\infty}\frac{n}{\gamma}}{Mn^{2}\epsilon^{2}}$
(109)
$\displaystyle=\frac{J_{\infty}^{2}\frac{n}{\gamma^{2}}+\Sigma_{\infty}\frac{n}{\gamma}+\alpha_{0}^{(2)}-(\alpha_{0}^{(1)})^{2}}{Mn^{2}\epsilon^{2}}$
(110)

To compare this to the error analysis in the original case (see Sec. V), we
label by $N$ the average number of ticks of the unknown clock that pass in the
whole duration of the experiment, this is given by

$\displaystyle N$ $\displaystyle=M*n*\frac{J_{\infty}}{\gamma},$ (111)

and we label by $\theta$ the fractional error in the relative frequency of the
target to the reference,

$\displaystyle\epsilon$ $\displaystyle=\theta*\frac{J_{\infty}}{\gamma}.$
(112)

After some algebra, we find that the error bound is

$\displaystyle\Pr\left(\left|\frac{\bar{X}/n}{J_{\infty}/\gamma}-1\right|\geq\theta\right)$
$\displaystyle\leq\frac{1}{N\theta^{2}}\left(\frac{1}{P}+\frac{\nu}{\gamma}+\frac{\alpha_{0}^{(2)}-\left(\alpha_{0}^{(1)}\right)^{2}}{nJ/\gamma}\right).$
(113)

This is analogous to the error in the reset case (30) with the additional term
due to the influence of the initial state of a non-reset clock upon the
statistics. If we do have a reset clock and begin the experiment at the moment
of a tick of the target clock, then $\alpha_{0}^{j}=0$ for all $j$, rendering
the two equations identical.

For the above expression, we note that the error is smallest for $n$
maximised. Substituting the maximum value of $n$, corresponding to having only
one single sample $M=1$, we get

$\displaystyle\Pr\left(\left|\frac{\bar{X}/n}{J_{\infty}/\gamma}-1\right|\geq\theta\right)$
$\displaystyle\leq\frac{1}{N\theta^{2}}\left(\frac{1}{P}+\frac{\nu}{\gamma}+\frac{\alpha_{0}^{(2)}-\left(\alpha_{0}^{(1)}\right)^{2}}{N}\right).$
(114)

Note that multiple samples are required to estimate the higher moments, so we
need $M>1$ unless we are only interested in the first moment (the frequency).

##  Appendix E Sub-Poissonian processes as a reference

The bound similar to the one discussed for the case when the reference clock
is described by a Poisson distribution, applies to another class of processes
called sub-Poissonian [13]. These are described by a distribution with the
mean $\mu$ and the moment-generating function bounded by

$\displaystyle\mathcal{M}(x)\leq exp\left(\mu(e^{s}-1)\right)\ \forall s>0.$
(115)

For example, if all moments of some distribution either coincide or are
bounded by corresponding moments of the Poisson distribution, this
distribution is sub-Poissonian.

The moments of sub-Poissonian random variables can be upper bounded by [13]

$\displaystyle
M_{k}(X)\leq\mu^{k}\left(\frac{k/\mu}{\log\left(1+\frac{k}{\mu}\right)}\right)^{k},$
(116)

where $M_{k}(X)=\mathbb{E}(X^{k})$ is the $k$th moment of $X$. Moreover, one
can also establish a lower bound for the moments using Jensen’s inequality

$\displaystyle\left(\mathbb{E}(X)\right)^{k}\leq\mathbb{E}(X^{k})\
\Rightarrow\ M_{k}(X)\geq\mu^{k}.$ (117)

In our consideration, the mean is given via the average rate of the process
$\gamma$, that is, $\mu=\gamma t$, for the relative moments of the measured
clock with delay function $\omega(t)$ relative to the sub-Poissonian clock the
bounds read

$\displaystyle\int_{0}^{+\infty}(\gamma t)^{k}\omega(t)dt\leq
m_{k}(\omega,\text{sub-Poiss.})\leq\int_{0}^{+\infty}(\gamma
t)^{k}\left(\frac{k/\gamma t}{\log\left(1+\frac{k}{\gamma
t}\right)}\right)^{k}\omega(t)dt.$ (118)

The lower bound is proportional to the $k$th moment of the measured clock; to
simplify the upper bound, we use that for all $y>0$,
$\log(1+y)\geq\frac{y}{1+y}$,

$\displaystyle\frac{k}{\log\left(1+\frac{k}{\gamma t}\right)}\leq k+\gamma t,$
(119)

and the final bound on the relative moments reduced to

$\displaystyle\gamma^{k}M_{k}(\omega)\leq m_{k}(\omega,\text{sub-
Poiss.})\leq\int_{0}^{+\infty}(k+\gamma
t)^{k}\omega(t)dt=\sum_{j=0}^{k}\binom{k}{j}\gamma^{j}k^{k-j}M_{j}(\omega).$
(120)

This allows us to derive the bound on the variance of $k$th relative moment
for the sample of size $N$ with the sample average $\bar{X}$,

$\displaystyle
P\big{(}\left|\frac{\bar{X^{k}}}{\gamma^{k}}-\frac{m_{k}(\omega,\text{sub-
Poiss.})}{\gamma^{k}}\right|\geq\epsilon\big{)}$
$\displaystyle\leq\frac{m_{2k}(\omega,\text{sub-
Poiss.})-m_{k}^{2}(\omega,\text{sub-Poiss.})}{N\gamma^{2k}\epsilon^{2}}$ (121)
$\displaystyle\leq\frac{1}{N\gamma^{2k}\epsilon^{2}}\left(\sum_{j=0}^{2k}\binom{2k}{j}\gamma^{j}(2k)^{2k-j}M_{j}(\omega)-\gamma^{2k}M_{k}^{2}(\omega)\right)$
(122)
$\displaystyle=\frac{M_{2k}(\omega)-M_{k}^{2}(\omega)}{N\epsilon^{2}}+\frac{1}{N}\cdot\mathcal{O}\left(\frac{1}{\gamma}\right),$
(123)

where, similarly to the Poisson-distributed process, in the limit of
$\gamma\to+\infty$ the relative measurement of the moments is as good as the
direct one (as given by Chebyshev’s inequality).