# Measuring time with stationary quantum clocks

Sergii Strelchuk  DAMTP, University of Cambridge, UK  Mischa P. Woods
Institute for Theoretical Physics, ETH Zurich, Zurich, Switzerland

###### Abstract

Time plays a fundamental role in our ability to make sense of the physical
laws in the world around us. The nature of time has puzzled people — from the
ancient Greeks to the present day — resulting in a long running debate between
philosophers and physicists alike to whether time needs change to exist (the
so-called relatival theory), or whether time flows regardless of change (the
so-called substantival theory). One way to decide between the two is to
attempt to measure the flow of time with a stationary clock, since if time
were substantival, the flow of time would manifest itself in the experiment.
Alas, conventional wisdom suggests that in order for a clock to function, it
cannot be a static object, thus rendering this experiment seemingly
impossible. Here we show, with the aid of counterfactual measurements, the
surprising result that a quantum clock can measure the passage of time even
while being switched off, thus lending constructive support for the
substantival theory of time.

##  I Introduction

Time is an essential ingredient in the world we inhabit. Not so surprisingly,
it has played a central role in all of our physical theories. Yet this role
was rather mundane until the advent of modern physics. In particular, out of
the three pillars of modern physics — quantum mechanics, special and general
relativity — it was only the latter two which forced us to change our
preconceptions about the nature of time. Quantum mechanics, on the other hand,
while very strange and mysterious in many ways, did not bring any novel
insights as far as time is concerned: time is just a parameter which increases
in line with any mundane classical clock — just like in Newton’s laws. Even
more recently, it has been proven, in the context of quantum mechanics, that
the time-analogue of Bell’s inequality always has a perfectly sound classical
explanation Purves and Short (2021); Wechs  _et al._ (2021).

Conversely, in the philosophy of physics domain, the notion of time occupied a
prominent role in debates dating back millennia which are still very active
today. One of the ongoing debates is whether time necessitates change to
exist. The substantival theory of time says that time exists and provides an
invisible container in which matter lives, regardless of whether the matter is
moving. On the other hand, the relational theory of time says that time is a
set of relationships among the events of physical material in space — that is
to say, time ceases to exist if matter ceases to change.

Views have continuously shifted through the ages: Many believe that Greek
atomists such as Democritus thought time was substantival. The first written
account dates back to Aristotle, who advocated for a relational theory: “ _But
neither does time exist without change; …_ ” Aristotle (2006). Newton and
Leibniz had opposed views, with Newton on the substantive camp Rynasiewicz
(2014), and Leibniz on the relational camp McDonough (2019). Ernst Mach
attacked Newton’s arguments in favour of a more relative theory Pojman (2020).
Einstein credited Mach’s views as being highly influential in his guiding
principles when developing his theory of general relativity; although later
shifted his stance to a more substantival interpretation of his theory Hoefer
(1994).

Arguments for and against either theories are still ongoing Maudlin (2007);
Rovelli (2018); Smolin (2013); Barbour (2001). One of the biggest problems in
this debate is the apparent inability to distinguish experimentally between
the two scenarios — indeed, it is widely accepted that any clock (or physical
matter used as a rudimentary clock), would need to change its state in order
to measure the passing of time, hence rendering a clock useless for detecting
the substantival’s hypothesised passage of time without change.

Here we show that nonrelativistic quantum mechanics renders this widely held
belief wrong. This is to say, we demonstrate by designing special clocks and
adhering to well established interpretations of quantum theory, that it is
possible to measure the passage of time between two events even when said
clock has been always off, in other words, never evolving. Our result thus
allows one to precisely detect the presence of the flow of time without
evolution, which we argue provides strong and experimentally testable evidence
for the substantival theory of time.

##  II Results

Overview: We will start by considering the simplest systems naturally
occurring in nature which can serve as elementary clocks when dynamically
evolving. We then show how they can be used to tell the time while being
switched off via a more advanced protocol and measurement scheme. This is
followed by an analysis of the interpretations used and a demonstration of how
this strange phenomena cannot be explained if one assumes an underlying “real”
description of nature, using classical variables. Finally, in section III, we
explore the implications of our results in relation to the long standing
historical debates on the nature of time.

Elementary timekeeping systems: While we tend to think of timekeeping devices
as engineered systems, we can use the dynamics of naturally occurring
processes as elementary clocks. The simplest of which can be thought of as an
arbitrary state $\ket{\psi}$ at time $t=0$ whose evolution takes it through a
sequence of mutually orthogonal states
$\ket{\tau_{0}}\\!,\,\ket{\tau_{1}}\\!,\ldots,\,\ket{\tau_{N_{T}}}$ at times
$\tau_{0},\tau_{1},\ldots,\tau_{N_{T}}$ consecutively. If we know that the
system is initially in the state $\ket{\psi}$, and we let it evolve for an
unknown time $t$ given the promise that $t$ belongs to the set
$\\{\tau_{0},\tau_{1},\tau_{2},\ldots,\tau_{N_{T}}\\}$, then we can determine
precisely what the elapsed time is by measuring in the basis
$\ket{\tau_{0}}\\!,\,\ket{\tau_{1}}\\!,\,\ket{\tau_{2}}\\!,\ldots,\,\ket{\tau_{N_{T}}}$,
since at said times, the measurement will deterministically allow one to
distinguish between said states. To use this timekeeping system in practice,
consider two external events, which we will call _1st event_ and _2nd event_ ,
with some unknown elapsed time
$t\in\\{\tau_{0},\tau_{1},\tau_{2},\ldots,\tau_{N_{T}}\\}$ between them. One
initialises the system in the state $\ket{\psi}$ when the 1st event occurs and
then waits until the 2nd event occurs, at which point the measurement would be
performed; revealing the elapsed time. When a clock is revealed to have been
dynamically evolving upon measurement as in this example, we say it is
operating in _standard fashion_.

This is arguably one of the most elementary types of timekeeping devices one
can conceive of. It has no inherently quantum features and should be quite
ubiquitous in nature since the states of systems tend to become completely
distinguishable if one waits long enough. As a simple illustration of how it
could be used, imagine being on a boat at sea heading in to port. There is a
lighthouse whose pulsing flashes of light reach you every 3 seconds, but thick
fog rolls in obscuring the flashes for some time before clearing. By setting
the times $\\{\tau_{n}\\}$ to coincide with the flashes and choosing the 1st
and 2nd events to be flashes before and after the presence of the fog, the
clock can deduce the duration of the foggy period.

Suppose that, in addition, there is a state denoted $\ket{E}$ which is
invariant under time evolution, in other words, it remains in the state
$\ket{E}$ at all times. Furthermore, suppose it is orthogonal to $\ket{\psi}$
and its time evolved state at all times. We refer to this state as an _off_
state, in contrast to any state which evolves in time, which we refer to as an
_on_ state. The state $\ket{E}$ could be, for example, an energy eigenstate of
the system. If one starts in the state $\ket{E}$ and then measures the state
later, clearly there is no possible measurement one can perform which would
reveal any information about the elapsed time — it would be analogous to
taking the batteries out of a wall clock, and then trying to use it to tell
the time.

Telling the time when the clock is off: We will now show, with the aid of some
additional stationary ancilla qubits, how quantum mechanics allows one to
determine precisely which time $\tau_{n}$ it is, even though the timekeeping
device was never on (in the runs of the experiment in which we determined
$\tau_{n}$). For the purpose of illustration, consider the most basic example
of ${N_{T}}\\!=\\!1$, in which the system in question merely has two times
$\tau_{0},\tau_{1}$. We relegate the ${N_{T}}\geq 2$ version to appendix A. In
this case we will only need one ancillary state, which we denote $\ket{A}$. It
could be, for example, chosen to be another energy eigenstate which is
orthogonal to the other states, or if such a state is not available, we can
use an ancilla qubit with a trivial Hamiltonian so as to ensure its states do
not evolve. In the latter case, denoting a basis for the ancilla by
$\ket{\uparrow}\\!,\,\ket{\downarrow}$, the states would then be associated
with $\ket{\psi(t)}\equiv\ket{\psi(t)}\ket{\uparrow}$,
$\ket{E}\equiv\ket{E}\ket{\uparrow}$, and the new ancilla state with
$\ket{A}=\ket{E}\ket{\downarrow}$ (Here $\ket{\psi(t)}$ denotes $\ket{\psi}$
after evolving for a time $t$).

We will use the same retrodictive arguments as in other famous experiments,
such as the Elitzur-Vaidman bomb tester or Hardy’s paradox. However, let us
first describe the protocol, before discussing the interpretation: Initially
the system is set to the off state $\ket{E}$. Then, when the 1st event occurs,
we apply a unitary $U_{0}$ such that the system is now in a superposition of
on and off, namely, of $\ket{E}$ and $\ket{\psi}$. We use branching notation
to indicate the orthogonal branches of the superposition associated with
static and dynamical terms (Upper branch is static, lower branch is dynamic).
The state reads:

$\ket{E}$$c\ket{E}$$s\ket{\psi}$

where $c=\cos(\theta),$ $s=\sin(\theta)$. One then waits until the 2nd event
occurs, at either time $\tau_{0}$ or $\tau_{1}$, at which point one applies a
judiciously chosen unitary $U_{\textup{m}}$ followed by immediately measuring
in the $\ket{E}$, $\ket{\tau_{0}}$, $\ket{\tau_{1}}$, $\ket{A}$ basis, which
we will call the measurement basis for short. This completes the protocol —
modulo specification of the required constraints on $U_{0}$ and
$U_{\textup{m}}$. Diagrammatically, up to the point of measurement, this
protocol and unitary $U_{\textup{m}}$ have the form:

(1)

at time $\tau_{0}$ and

(2)

at time $\tau_{1}$, where $|cA_{1}^{0}|>0$, $|cA_{2}^{0}|>0$ and other
amplitudes are arbitrary. Squiggly arrows represent the passing of an amount
of time $\tau_{0}$ or $\tau_{1}$, while straight arrows the application of
$U_{\textup{m}}$.

Suppose the $E$ outcome is obtained.111We use the convention in which “outcome
$x$” means that the post-measurement state is $\ket{x}$. At time $\tau_{0}$, a
non zero amplitude associated with $\ket{E}$ exists only for the off branch,
while for time $\tau_{1}$, this amplitude from the off branch cancels with an
on branch amplitude due to destructive interference. As such, when $E$ is
obtained, we can deduce that the clock has collapsed to a branch of the wave
function that was always off _and_ that the unknown time $t$ must be
$\tau_{0}$. Similarly, when outcome $A$ is obtained, we deduce that the clock
was collapsed to a branch of the wave function that was always off _and_ that
the time $t$ must be $\tau_{1}$. However, if outcome $\tau_{0}$ or $\tau_{1}$
is obtained, we cannot conclude anything useful: the clock may have been on
and we cannot deduce whether the time is $\tau_{0}$ or $\tau_{1}$.

Hence whenever outcome $E$ or $A$ is obtained, we can deduce what the time is,
even though the system was always off, that is to say, always in a stationary
state (since $\ket{E}$ and $\ket{A}$ are orthogonal to all dynamical branches
whenever $E$, $A$ have a non zero probability of being obtained).

Interpretation: One may object that the system has certainly been in a
superposition of on and off, and in this sense, has actually been evolving in
time. However, the outcomes $E$ and $A$ can only arise via an always off
branch of the wave function so if it is seen then we have been confined to a
part of the total quantum state in which the system is always stationary.
Furthermore, this interpretation is independent of the basis used to represent
the state during the different stages of the protocol — we represented it in
the measurement basis purely for convenience.

Our explanation of the results has relied on a certain kind of retrodictive
interpretation of quantum mechanics. This interpretation can be most readily
captured by Schrödinger’s proverbial cat experiment: one starts with an alive
cat, denoted $\ket{\text{alive}}$, which when put in the closed box, takes on
the form
$\frac{1}{\sqrt{2}}\left(\ket{\text{alive}}+\ket{\text{dead}}\right)$. Suppose
that upon opening the box, we make a measurement of alive versus dead, and
obtain the outcome “alive”. Then in the conventional collapse of the wave
function formalism, the state of the cat changes discontinuously into
$\ket{\text{alive}}$ and the $\ket{\text{dead}}$ component ceases to have any
further physical existence or further consequence — see fig. 1. This is a form
of retrodiction, since it says that upon observing an alive cat we collapse to
a state in which the cat was _always_ alive, since the dead and alive branches
were orthogonal at all times. In our setup, this retrodiction takes on the
form of a so-called _counterfactual measurement_ ,222Also known as
“interaction-free measurement” or “quantum interrogation”. since when outcome
$E$ or $A$ is obtained, it allows one to deduce properties of a dynamical
clock by reasoning counterfactually, while collapsing the wave function to a
state where those eventualities never actually took place. Counterfactual
measurements were first discovered by Elitzur and Vaidman in their famous bomb
tester experiment Elitzur and Vaidman (1993); Vaidman (2003); Penrose (1994),
and later applied to other scenarios including counterfactual computation
Jozsa (1998); Mitchison and Jozsa (2001); Hasenöhrl and Wolf (2020); Kwiat
_et al._ (1995), and experimentally verified Weinfurter  _et al._ (1995);
Kwiat  _et al._ (1999); White  _et al._ (1998). Prior examples have played out
in space, while ours is the first protocol to do so in time. Another example
of retrodiction and counterfactual measurement in quantum mechanics is the
celebrated Hardy paradox Hardy (1992). Weakly measuring the distinct branches
can provide theoretical and experimental evidence for the validity of such
retrodictive interpretations Aharonov  _et al._ (2002); Lundeen and Steinberg
(2009); Yokota  _et al._ (2009).

We will call our clock when operated in this retrodictive fashion a
_counterfactual clock_. Likewise, the measurement outcomes of interest in a
counterfactual measurement ($E$ and $A$), are referred to as counterfactual
outcomes. If the above protocol is implemented and a counterfactual outcome is
obtained, we say that the clock was _always off_. In fig. 2 we compare one
instance of the Elitzur and Vaidman bomb test with one instance of the
counterfactual clock.

Our analysis thus far has been presented _forward in time_ , namely starting
from the initial state $\ket{E}$ and analysing the different steps of the
protocol until post-selecting on a counterfactual outcome. We can also examine
the _backwards in time_ process where we start with one of the post-selected
states, and examine how it evolves backwards in time to the initial state
$\ket{E}$. Whenever analysing paradoxes involving post-selection, it is
prudent to check that the paradox still exits when the problem is analysed
backwards in time, since this can reveal any “hidden” assumptions
inadvertently made due to post-selecting. It is important to do so in the case
of counterfactual experiments, as shown in Vaidman (2007). We prove in
supplementary section IV.1 that when the counterfactual outcomes are analysed
backwards in time, the pre-selected wave function is collapsed onto an always
off branch. This guarantees that our counterfactual measurements have been
properly implemented.

Of course, one can apply alternative interpretations of quantum mechanics. In
the many-worlds interpretation of Schrödinger’s cat, upon measurement, reality
splits into two parallel worlds — one in which the component
$\ket{\text{alive}}$ prevails and the cat is alive, the other, in which the
component $\ket{\text{dead}}$ prevails and the cat is dead. Conversely, the
explanation of the counterfactual clock also changes if one uses this latter
interpretation. Specifically, in the language of many-worlds, whenever the
outcome $E$ or $A$ is obtained, we will be living in a world in which the
system never dynamically evolved (i.e. never in an on state) yet in this world
we learn information about the dynamical states (i.e. the on branch), namely
what the time is. The fact that the system was evolving in another world is of
no consequence to _us_.

Figure 1: Schrödinger’s cat. A1) Schrödiger’s cat starts out alive. At the 1st
dotted line, the box is closed and the cat is in a superposition of dead and
alive until just before the box is opened (2nd dotted line). A2) The box has
now been opened, causing the cat to be measured. Here we show the case in
which we see an alive cat. The history of the cat from before the box was
closed to present is now set in stone, with the dead cat branch of the wave
function ceasing to have ever existed.

A Elizur-Vaidman bomb test

B The counterfactual clock

Figure 2: Comparison between the Elizur-Vaidman bomb test and the
counterfactual clock
A1) Mach-Zehnder interferometer with 50/50 beam splitters and a photosensitive
bomb placed in the bottom arm. Photon trajectories are black lines. A photon
enters at a) and splits into a superposition travelling down the upper and
lower arms upon interacting with the 1st beam splitter. The photon in the
bottom arm is coherently absorbed by the bomb, hence exploding. The photon in
the upper arm encounters the 2nd beam splitter and coherently splits into
trajectories b), c) without bomb exploding. A2) Continuing on from A1), a
measurement is now performed via the photodetectors D1 and D2. There are 3
possible outcomes: If photon took path c), D1 clicks. If photon took path b),
D2 clicks. If bomb exploded, photon is lost and neither photodetector clicks.
Fig A2 depicts the scenario that D2 clicks (the counterfactual outcome). Upon
clicking, the wave function collapses into an eventuality where the bomb never
exploded. B1) The counterfactual clock starts out off. At the 1st dotted line,
the 1st event occurs and the clock is put into a superposition of on and off
until just before the 2nd event occurs (2nd dotted line). B2) The 2nd event
now occurs, causing the clock to be measured. Here we show the case in which
we see a counterfactual outcome. The history of the clock from before the 1st
event occurred to present is not set in stone, with the on clock branch of the
wave function ceasing to have ever existed. Comparison between A and B: Both
use a counterfactual measurement to avoid the eventualities of branch 1 of the
wave function occurring when obtaining certain outcomes.

The role of the ancilla is more subtle than may appear at first sight: indeed,
one might envisage a situation with no ancilla and where we interpret outcomes
$E$ and $\tau_{1}$ as counterfactual outcomes by choosing the unitary
$U_{\textup{m}}$ differently to as presented. Notwithstanding, we prove in
appendix A.5, that there does not exist a counterfactual clock for which there
is no ancilla and the probabilities of the two counterfactual outcomes are
both non zero. This implies a more fundamental role for the ancilla state: by
including it, we can reach a larger class of unitary transformations which are
necessary for the counterfactual clock to function. This is important for
understanding what is the minimal model for making the appropriate comparison
with classical descriptions, which we will do shortly.

In this set-up, the optimal protocol is the one which maximises the
probabilities of obtaining either measurement outcomes $E$ or $A$ over
unitaries $U_{\textup{m}}$ and branching amplitudes $c,s$. In the case in
which both outcomes are equally likely, $|cA_{1}^{0}|^{2}=|cA_{2}^{0}|^{2}$;
we prove in appendix A.5 that there exists a protocol which assigns
probability $1/6$ to obtaining the outcome $E$ at time $\tau_{0}$ and
probability $1/6$ to outcome $A$ at time $\tau_{1}$. For later reference, we
will refer to the sum of the probabilities corresponding to the counterfactual
outcomes, the _total counterfactual probability_. In this case, it is $1/3$.

One may wonder whether a way out of this conundrum is via the existence of an
underlying description respecting realism, that is, a theory where the
relevant physical properties have well defined values at all times prior to
the measurement in our protocol. For example, such a theory could have hidden
variables which evolve dynamically when the clock is supposedly off. We prove
that there cannot exist a non-contextual ontic model for the relevant degrees
of freedom for the above counterfactual clock with probability $1/6$ for each
counterfactual outcome. This rules out all classical models under natural
assumptions; see supplementary IV.2 for details.

Engineered clocks: in this section we will discuss a more advanced type of
engineered clock. This is more of an aside, and one can skip this section
heading straight to the discussion in section III, if only interested in
interpretation and consequences for the nature of time.

While the protocol thus far discussed allows one to use the most elementary of
systems as a timekeeping device, it is limited in that it can only distinguish
between times $\tau_{0},$ $\tau_{1},\ldots,\tau_{N_{T}}$ given the guarantee
that the elapsed time between the two events coincides with one of these
times. This limitation is due to an elementary clock design — indeed, this
feature was present even when the clock was run in standard fashion. This
limitation means that while it can be used for some applications (recall,
e.g., the lighthouse example), it cannot be used for others, such as
determining whether the winner of a race set a new record, because there in no
reason to believe that the interval between starting the race and finishing
it, will belong to any prior chosen set $\\{\tau_{0},$
$\tau_{1},\ldots,\tau_{N_{T}}\\}$. To overcome these restrictions, we
demonstrate the existence of a counterfactual clock which has counterfactual
outcomes with a protocol identical to the one described prior, modulo a few
distinctions: 1) The elapsed time between the two events can be arbitrary. 2)
The unitaries $U_{0}$ and $U_{\textup{m}}$ are chosen differently. 3) The
projective measurement is in a different basis.

The physically significant difference in the protocol is clearly 1).
Furthermore, it enjoys the analogous physical interpretation when the
counterfactual outcomes are obtained to that described in the previous case
above. Moreover, the Hamiltonian we use in the construction is time
independent. This is important since otherwise the protocol would likely
require an external timekeeping device for its implementation — rendering the
entire experiment pointless. Another physically relevant feature of the
engineered clock, is that when the clock is on, it cannot change
instantaneously between distinct states representing the distinct ticks of the
clock. This introduces a small error — regardless of whether it is run
counterfactually or in standard fashion. The error is analogous to the
situation we face with a wall clock: the second hand (i.e. the hand the
records the passing of seconds) cannot instantaneously transition between one
clock face marking and the next; therefore, if you happen to read the clock
around the milliseconds interval in which the second hand is transitioning,
you will likely be off by a second.

In particular, when the clock is run counterfactually, and one of the
counterfactual outcomes is obtained, this error means that there is a small
probability of a false positive: the quantum destructive interference is not
perfect, leading to the possibility that the clock was actually on after all.
Luckily, this error can be made arbitrarily small — although at the expense of
a decrease in the probability of obtaining a counterfactual outcome.
Notwithstanding, reasonable probabilities can be obtained. For example, in the
case of one tick, and a total counterfactual probability of $1/12$, the
probability that the clock was on when one of the counterfactual outcomes is
obtained is of order $10^{-14}$. See Supplementary IV.3 for details.

Physically speaking, this tiny error is due to imperfect destructive
interference between on and off breaches. Such events are key to all quantum
experiments using counterfactual measurements and due to engineering
imperfections, will always be present. For example, in the quantum
counterfactual experiments involving photons Weinfurter  _et al._ (1995);
White  _et al._ (1998); Hosten  _et al._ (2006); Kwiat  _et al._ (1999); Jang
(1999), small deviations in the reflectivity of the beam splitters lead to
ports with photodetectors which were not completely dark.

##  III Discussion

With the aid of counterfactual measurements, we have shown that quantum
mechanics allows one to predict the time passed between two events, even
though no dynamics has taken place between the two events. The most elementary
form of this clock presented in the main text should be obtainable from
ubiquitous systems in nature (a four dimensional system with two stationary
and two non-stationary states), while the more advanced clocks providing a
larger number of distinguishable times (large ${N_{T}}$) require more ancillas
and should be realisable with state of the art current technologies such as
those developed for quantum computation.

While it is readily clear that a classical clock cannot tell the time without
dynamical evolution, it could have been that our quantum clock had hidden
variables which were changing when the clock was supposedly off — we ruled out
such a possibility under reasonable assumptions.

We will now discuss the implications for the relational and substantival
theories of time which were presented in the introduction. As explained
previously, one of the reasons why the debate has been ongoing for more that
two thousand years, is due to the lack of the possibility to experimentally
distinguish between the two theories due to an apparent inability to measure
the flow of time without evolution. Our results can remedy this dilemma by the
following experiment:

Alice has a random number generator which outputs heads or tails. If she finds
heads (tails), she sends two light pulses to Bob separated by an elapsed time
$\tau_{0}$ ($\tau_{1}$). Bob possesses a counterfactual clock in the energy
eigenstate $\ket{E}$, which can distinguish between two elapsed times
$\tau_{0}$ and $\tau_{1}$, but does not know the outcome of the random number
generator. Alice tunes her two light pulses so that when they come into
contact with Bob’s clock, the 1st pulse implements $U_{0}$ while the 2nd
implements $U_{\textup{m}}$. Bob measures his clock in the basis $\ket{E}$,
$\ket{\tau_{0}}$, $\ket{\tau_{1}}$, $\ket{A}$ upon seeing the second light
pulse. Alice repeats this experiment until Bob reports to her that he found a
counterfactual outcome. There is a second system near Bob which remains in an
energy eigenstate during the entire experiment and does not interact with the
light pulses; bob will use it for reference purposes only. Consider the last
run of the experiment: the clock did not change between the two light pulses
(relative to the second system) but was nevertheless responsible for the
detection and quantification of an elapsed time which is identical to what a
clock which was dynamically evolving would have predicted (had one been
present). We argue that in a relative theory of time, from the perspective of
the clock relative to the second system, any value for the elapsed time would
have to be arbitrary, since time would not exist from it’s perspective.
Contrarily, the experiment is perfectly consistent with a substantival theory
of time, since in this case, from the clock’s perspective, there is a well-
defined elapsed time between the events, as time passes independently of the
clock’s state of motion.

We can formalise this observation via the following assumptions and theorem:

  * (A)

Counterfactual measurements exist.

  * (B)

If time can be measured via a clock which is always off, then time is
substantival.

######  Theorem 1.

Time is substantival if assumptions (A) and (B) hold.

We prove the theorem in supplementary IV.4. If (A) is false, then the other
phenomena which uses counterfactual measurements, such as counterfactual
computing or the bomb test, would require serious re-evaluation. Similarly, if
(A) holds yet (B) is false, then arguably, the very meaning of a substantival
theory of time requires substantial review.

Finally, we conclude with a discussion how one might circumvent or modify our
conclusions. Our results relied on the validity of the counterfactual
measurement interpretation in quantum mechanics, and we discussed how in the
alternative interpretation of many-worlds, when one of the counterfactual
outcomes is obtained, while _we_ would be in a world where the clock was
always off, there would be another world where it would be on. Regarding the
theory of time debate, this interpretation would still lead to difficulties
for a relatival theory of time, since in our thought experiment the second
system remains in Bob’s world, so when he obtains the counterfactual outcome,
there would have been no relative change between the clock and the second
system in Bob’s world, yet Bob would know how much time had passed in his
world from the measurement outcome. Another well-known alternative
interpretation of quantum theory is Bohmian mechanics. Here there is always a
real state associated with the wave function even when not observed. It is a
contextual interpretation of quantum mechanics, and thus is not ruled out via
our no-go results regarding a classical interpretation.

As alluded to in the introduction, our beliefs about the nature of time have
been highly influential in the development of our physical theories. Going
forward, perhaps one of the key ingredients to finding a correct theory of
quantum gravity is contingent of asserting an appropriate belief about the
nature of time. We hope that our work will bring much needed attention to this
debate.

###### Acknowledgements.

We are grateful to Caslav Brukner, Rob Spekkens and Victor Gitton for
discussions. M.W. acknowledges funding from the Swiss National Science
Foundation (AMBIZIONE Fellowship, No. PZ00P2 179914) in addition to supported
from the Swiss National Science Foundation via the National Centre for
Competence in Research “QSIT”. S.S. acknowledges support from the QuantERA
ERA-NET Cofund in Quantum Technologies implemented within the European Union’s
Horizon 2020 Programme (QuantAlgo project), and administered through EPSRC
Grant No. EP/R043957/1, and support from the Royal Society University Research
Fellowship scheme. M.W. and S.S. both acknowledge funding from the Royal
Society International Exchanges program (grant No. IES\R1\1801060).  AUTHOR
CONTRIBUTIONS M.W. derived the protocols and wrote the paper. S.S. implemented
the numerical proof for the non existence of a noncontextual ontic model. Both
authors contributed to ideas and discussions.

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##  IV Supplementary

Here we provide more details concerning the claims made in the main text.
There are three subsections. The first, section IV.1, describes the two state
formalism analysis in which the states evolve both forwards and backwards in
time. The second, section IV.2, describes the setup and how we performed our
calculations which led us to conclude that our counterfactual clock has no
noncontextual ontic description. The third, section IV.3, provides more
details about the engineered clocks. The fourth, section IV.4, contains a
proof of theorem 1.

###  IV.1 Backwards in time analysis

Here we analyse our protocol backwards in time for the post-selected outcomes
of interest. The objective is to show that the clock was off when viewed from
this perspective. For simplicity, we analyse the case ${N_{T}}=1$ presented in
the main text where the probability of obtaining either counterfactual outcome
is $1/6$. The general case is relegated to appendix A.4. The physical
interpretation is the same in all cases.

The amplitudes associated with obtaining the counterfactual outcomes are given
by

$\displaystyle\braket{E}{U_{\textup{m}}U(\tau_{0})U_{0}}{E}\qquad\text{ and
}\qquad\braket{A}{U_{\textup{m}}U(\tau_{1})U_{0}}{E},$ (3)

for times $t=\tau_{0}$ and $t=\tau_{1}$ respectively, where $U(\tau_{j})$ is
the unitary dynamics of the clock (when it is on) for a time $\tau_{j}$. In
the main text we interpreted these processes going forwards in time, namely
starting in state $\ket{E}$ and analysing the state after applying $U_{0}$,
$U(\tau_{j})$, $U_{\textup{m}}$ and finally post-selecting on measurement
outcomes $\ket{E}$ and $\ket{A}$. Here we will analyse the reverse process:
starting with the post-selected state $\bra{E}$ or $\bra{A}$, evolving
backwards in time by analysing the affect of the sequence of unitaries
$U_{\textup{m}}$, $U(\tau_{j})$, $U_{0}$, followed by pre-selecting onto the
initial state $\ket{E}$. This process is known as the two-state vector
formalism. This formalism, and many other variants on this idea, permit time-
symmetric interpretations of quantum mechanics (see review Aharonov and
Vaidman (2008) for more details).

For the example under consideration, $U_{\textup{m}}$ is provided at the end
of appendix A.3. Using it we find that the backwards evolved post-selected
counterfactual outcome state just prior to dynamics take on the form:

$\displaystyle\bra{E}U_{\textup{m}}=\frac{1}{\sqrt{3}}\big{(}\bra{E}-\bra{\tau_{1}}-\bra{A}\big{)}\qquad\text{
and
}\qquad\bra{A}U_{\textup{m}}=\frac{1}{\sqrt{3}}\big{(}\bra{E}-\bra{\tau_{0}}-\bra{A}\big{)}.$
(4)

We now have to evolve backwards in time for an amount $\tau_{0}$. Recall that
$U(\tau_{0})$ induces the following transitions: $U(\tau_{0})\ket{E}=\ket{E}$,
$U(\tau_{0})\ket{A}=\ket{A}$, $U(\tau_{0})\ket{\psi}=\ket{\tau_{0}}$. Consider
the ansatz
$\bra{\tau_{1}}U(\tau_{0})=c_{0}\bra{\psi_{\perp}}+c_{2}\bra{E}+c_{3}\bra{\psi}$,
where $\ket{\psi_{\perp}}$ is some state which is perpendicular to $\ket{E}$
and $\ket{\psi}$. We have
$0=\braket{\tau_{1}}{E}=\braket{\tau_{1}}{U(\tau_{0})U(\tau_{0})^{\dagger}}{E}=c_{2}$.
Similarly we find that $c_{3}=0$ and thus that
$\bra{\tau_{1}}U(\tau_{0})=\bra{\psi_{\perp}}$. Furthermore, since
$U(\tau_{1})\ket{E}=\ket{E}$, $U(\tau_{1})\ket{A}=\ket{A}$,
$U(\tau_{1})\ket{\psi}=\ket{\tau_{1}}$, it follows in an analogous fashion
that $\bra{\tau_{0}}U(\tau_{1})=\bra{\psi_{\perp}^{\prime}}$, where
$\bra{\psi_{\perp}^{\prime}}$ is some state perpendicular to $\ket{E}$ and
$\ket{\psi}$. Therefore, from eq. 4, the backwards in time dynamics yields the
states

$\displaystyle\bra{E}U_{\textup{m}}U(\tau_{0})=\frac{1}{\sqrt{3}}\big{(}\bra{E}-\bra{\psi_{\perp}}-\bra{A}\big{)}\qquad\text{
and
}\qquad\bra{A}U_{\textup{m}}U(\tau_{1})=\frac{1}{\sqrt{3}}\big{(}\bra{E}-\bra{\psi_{\perp}^{\prime}}-\bra{A}\big{)}.$
(5)

The unitary $U_{0}$ only acts non trivially in the subspace spanned by
$\ket{E}$ and $\ket{\psi}$: $U_{0}\ket{E}=c\ket{E}+s\ket{\psi}$,
$U_{0}\ket{\psi}=-s\ket{E}+c\ket{\psi}$, and thus it acts trivially on the
dynamical branches $\bra{\psi_{\perp}}$ and $\bra{\psi_{\perp}^{\prime}}$. We
thus have

$\displaystyle\bra{E}U_{\textup{m}}U(\tau_{0})U_{0}=\frac{1}{\sqrt{3}}\big{(}c\bra{E}-s\bra{\psi}-\bra{\psi_{\perp}}-\bra{A}\big{)}\quad\text{
and
}\quad\bra{A}U_{\textup{m}}U(\tau_{1})U_{0}=\frac{1}{\sqrt{3}}\big{(}c\bra{E}-s\bra{\psi}-\bra{\psi_{\perp}^{\prime}}-\bra{A}\big{)}.$
(6)

Finally, by post-selecting on $\ket{E}$, and recalling that $c=1/\sqrt{2}$, we
find that the amplitudes associated with the counterfactual outcomes is
$1/\sqrt{6}$ in both cases, hence leading to a probability of $1/6$ for each
counterfactual outcome as stated in the main text. Importantly, during the
entire backwards evolution, we see that the dynamical branches of the wave
function has not contributed to the pre-selected state $\ket{E}$. Another way
to see this is to note that had we measured whether the clock was on or off
during the evolution in either of the above pre and post selected cases, we
would have found that the probability of finding the clock on was zero.
Namely, we can define the binary outcome projective measurement
$P_{\textup{on}}:=|\psi\rangle\\!\langle\psi|+|\tau_{0}\rangle\\!\langle\tau_{0}|+|\tau_{1}\rangle\\!\langle\tau_{1}|$
and $P_{{\scriptscriptstyle\backslash}\textup{on}}:=|E\rangle\\!\langle
E|+|A\rangle\\!\langle A|$, where $P_{\textup{on}}$ corresponds to a
projection onto the on subspace. From Aharonov and Vaidman (2008), it follows
that the probability of the clock being on at any time during the dynamical
phase, that is any $t\in(0,\tau_{0})$, for the run of the experiment
corresponding to post-selecting on $\ket{E}$ is

$\displaystyle\textup{Prob}\big{(}\textup{on},E\big{)}$
$\displaystyle:=\frac{\big{|}\braket{E}{U_{\textup{m}}U(\tau_{0}-t)P_{\textup{on}}U(t)U_{0}}{E}\big{|}^{2}}{\big{|}\braket{E}{U_{\textup{m}}U(\tau_{0}-t)P_{\textup{on}}U(t)U_{0}}{E}\big{|}^{2}+\big{|}\braket{E}{U_{\textup{m}}U(\tau_{0}-t)P_{{\scriptscriptstyle\backslash}\textup{on}}U(t)U_{0}}{E}\big{|}^{2}}.$
(7)

Note that since the ancilla and energy eigenstate are stationary, we have that
$U(t)=U_{\perp E,A}(t)+{\mathbbm{1}}_{E,A}$, where ${\mathbbm{1}}_{E,A}$ is
the identity operator on the subspace spanned by $\ket{E}$ and $\ket{A}$, and
that $U_{\perp,A}(t)$ is orthogonal to said subspace. Therefore, if we assume
that the dynamics is Markovian, namely $U(t_{1})U(t_{2})=U(t_{1}+t_{2})$ for
all $t_{1},t_{2}\in{\mathbbm{R}}$, it follows that
$U(\tau_{0}-t)P_{\textup{on}}U(t)=U_{\perp E,A}(\tau_{0})$ and hence that

$\displaystyle\textup{Prob}\big{(}\textup{on},E\big{)}=0$ (8)

for all $t\in(0,\tau_{0})$. Similarity, it follows that the probability of the
clock being on at any time during the dynamical phase, that is any
$t\in(0,\tau_{1})$, for the run of the experiment corresponding to post-
selecting on $\ket{A}$ is

$\displaystyle\textup{Prob}\big{(}\textup{on},A\big{)}$
$\displaystyle:=\frac{\big{|}\braket{A}{U_{\textup{m}}U(\tau_{1}-t)P_{\textup{on}}U(t)U_{0}}{E}\big{|}^{2}}{\big{|}\braket{A}{U_{\textup{m}}U(\tau_{1}-t)P_{\textup{on}}U(t)U_{0}}{E}\big{|}^{2}+\big{|}\braket{A}{U_{\textup{m}}U(\tau_{1}-t)P_{{\scriptscriptstyle\backslash}\textup{on}}U(t)U_{0}}{E}\big{|}^{2}}=0.$
(9)

###  IV.2 No classical analogue

The interpretation of measuring time when the clock was off using
counterfactual reasoning, relied on a basic concept in quantum mechanics,
namely the superposition principle, in which if a system is in a superposition
of two states, it cannot be regarded as being in either until measured. As we
discussed, Schrödinger famously popularised this point with his thought
experiment concerning a cat. However, what if underlying our counterfactual
clock protocol, there was a “real” description; that is to say if one could
associate the states in our clock protocols with probability distributions
$\lambda$ over some underlying states — a so-called _ontic state apace_
$\Lambda$ — and the measurements with update rules for said distributions, in
a meaningful way? While such a result would not invalidate the current
interpretation provided, it would, at least in-principle, provide for an
alternative interpretation in which the clock might have been dynamically
evolving even when a measurement collapsing the clock to the off branch is
obtained. We will now consider such a possibility for the simplest elementary
timekeeping systems.

To start with, lets consider the case where we only run the clock when it is
on; that is, the clock is never in a superposition of on and off. Then the
question is, can we find an ontic state space on which there exists
appropriate probability distributions that represent our states
$\\{|E\rangle\\!\langle
E|,|\tau_{0}\rangle\\!\langle\tau_{0}|,|\tau_{1}\rangle\\!\langle\tau_{1}|\\}$
and measurements (with PVM elements $\\{|E\rangle\\!\langle E|$,
$|\tau_{0}\rangle\\!\langle\tau_{0}|$, $|\tau_{1}\rangle\\!\langle\tau_{1}|$,
$|A\rangle\\!\langle A|\\}$). Moreover, if an underlying ontic state space
does exist, one should expect to be able to take probabilistic mixtures of the
states and measurements in it: if your lab assistant walked in to your lab at
some unknown stage of the protocol’s implementation, the assistant would
attribute such a description. Thus denoting the convex hull by conv, we
required that when a state
$\rho\in\mathtt{s}_{\textup{on}}=\textup{conv}\big{(}\\{|E\rangle\\!\langle
E|,|\tau_{0}\rangle\\!\langle\tau_{0}|,|\tau_{1}\rangle\\!\langle\tau_{1}|\\}\big{)}$
is prepared in our protocol, the probability that the ontic state is in state
$\lambda$, is $P\big{[}\lambda\big{|}\rho\big{]}$. Likewise, when we make the
canonical measurement with corresponding POVM element
$\mathtt{E}\in\mathtt{e}_{\textup{on}}=\textup{conv}\big{(}\\{|E\rangle\\!\langle
E|,|\tau_{0}\rangle\\!\langle\tau_{0}|,|\tau_{1}\rangle\\!\langle\tau_{1}|,|A\rangle\\!\langle
A|\\}\big{)}$, then the probability that the outcome associated with
$\mathtt{E}$ occurred, given that the ontic state is in state $\lambda$, is
$P\big{[}\mathtt{E}\big{|}\lambda\big{]}$. Furthermore, we require the usual
convexity relationships of mixtures of quantum states and measurements hold at
the ontic level: for all $\lambda\in\Lambda$, for all $p\in[0,1]$, for all
$\rho_{1},\rho_{2}\in\mathtt{s}_{\textup{on}}$:
$P\big{[}\lambda\big{|}p\rho_{1}+(1-p)\rho_{2}\big{]}=pP\big{[}\lambda\big{|}\rho_{1}\big{]}+(1-p)P\big{[}\lambda\big{|}\rho_{2}\big{]}$
and similarly, that for all $\lambda\in\Lambda$, for all $p\in[0,1]$, for all
$\mathtt{E}_{1},\mathtt{E}_{2}\in\mathtt{e}_{\textup{on}}$:
$P\big{[}p\mathtt{E}_{1}+(1-p)\mathtt{E}_{2}\big{|}\lambda\big{]}=pP\big{[}\mathtt{E}_{1}\big{|}\lambda\big{]}+(1-p)P\big{[}\mathtt{E}_{2}\big{|}\lambda\big{]}$.
Finally, we require that the ontic states can reproduce the measurement
statistics of our protocol: for all $\rho\in\mathtt{s}_{\textup{on}}$, for all
$\mathtt{E}\in\mathtt{e}_{\textup{on}}$,

$\displaystyle\textup{tr}\big{[}\rho\,\mathtt{E}\big{]}=\int_{\Lambda}\\!\\!d\lambda\,P\big{[}\lambda\big{|}\rho\big{]}P\big{[}\mathtt{E}\big{|}\lambda\big{]},$
(10)

where $\textup{tr}[\cdot]$ denotes the trace, and the r.h.s. is the
probability of obtaining outcome associated with $\mathtt{E}$ for quantum
state $\rho$, according to quantum mechanics.

Notice how we have assumed that the probability distributions
$P\big{[}\lambda\big{|}\rho\big{]}$ do not depend on how the quantum state
$\rho$ was prepared nor do $P\big{[}\mathtt{E}\big{|}\lambda\big{]}$ depend on
how the measurements were implemented. Different preparation procedures which
lead to the same quantum state and different measurement procedures which lead
to the same POVM elements, are called different _contexts_. We are thus
looking for a non-contextual ontic variable model. This notion of
contextuality was pioneered by Spekkens Spekkens (2005). Prior notions of
classicality based on local realism Bell (1964) or its generalisation to non-
contextual realism Kochen and Specker (1967), cannot be tested in our case
since we do not dispose of a local structure, nor relevant communing
measurements, in our clock protocols. However, in keeping with this philosophy
of preparation and measurement non-contextuality, our current formulation of
an ontic variable model requires some refinement: since the set
$\mathtt{e}_{\textup{on}}$ is not tomographically complete, nor the set
$\mathtt{s}_{\textup{on}}$ span all quantum states in the Hilbert space,
neither all the d.o.f. of the density matrices nor those in the POVM elements,
contribute to the measurement statistics. Indeed,
$\textup{tr}\big{[}\rho\mathtt{E}\big{]}=\textup{tr}\big{[}\mathcal{P}_{\mathcal{R}}(\rho)\mathcal{P}_{\mathcal{R}}(\mathtt{E})\big{]}$
for all $\rho\in\mathtt{s}_{\textup{on}}$,
$\mathtt{E}\in\mathtt{e}_{\textup{on}}$, where $\mathcal{P}_{\mathcal{R}}$ is
a projection onto the vector space $\mathcal{R}$, generated by projecting the
span of $\mathtt{s}_{\textup{on}}$ onto the span of
$\mathtt{e}_{\textup{on}}$. We should thus make the replacements
$P\big{[}\lambda\big{|}\cdot\big{]}\mapsto
P\big{[}\lambda\big{|}\mathcal{P}_{\mathcal{R}}(\cdot)\big{]}$ and
$P\big{[}\cdot\big{|}\lambda\big{]}\mapsto
P\big{[}\mathcal{P}_{\mathcal{R}}(\cdot)\big{|}\lambda\big{]}$ in the above
theory. This completes the description of our would-be non-contextual ontic
variable model. Finally, if a classical description exists, one would also
need a stochastic map describing the dynamics of the ontic variables
throughout the protocols. This imposes an additional constraint on our would-
be ontic description, which we will not need to consider.

In this case, it can readily be seen that such a model _does_ exist; for
example, one may simply choose the vectors
$\\{\ket{E}\\!,\ket{\tau_{0}}\\!,\ket{\tau_{1}}\\!,\ket{A}\\}$ as a basis for
the ontic state space $\Lambda$ with each step and measurement allocated to
deterministic distributions $\lambda$ on it.

We now turn to the case is which the clock can also be used to tell the time
when off via counterfactual measurements. We need to supplement the sets
$\mathtt{s}_{\textup{on}},\mathtt{e}_{\textup{on}}$ with the other relevant
elements which are now required, namely, for states,

$\displaystyle\mathtt{s}_{\textup{cf}}=\textup{conv}\big{(}\mathtt{s}_{\textup{on}}\cup\\{|\textup{cf}_{0}\rangle\\!\langle\textup{cf}_{0}|,|\textup{cf}_{1}\rangle\\!\langle\textup{cf}_{1}|,U_{\textup{m}}|\textup{cf}_{0}\rangle\\!\langle\textup{cf}_{0}|U_{\textup{m}}^{\dagger},U_{\textup{m}}|\textup{cf}_{1}\rangle\\!\langle\textup{cf}_{1}|U_{\textup{m}}^{\dagger}\\}\big{)},$
(11)

where $\ket{\textup{cf}_{0}}:=c\ket{E}+s\ket{\tau_{0}}$,
$\ket{\textup{cf}_{1}}:=c\ket{E}+s\ket{\tau_{1}}$, since these are the states
which appear in our protocol. When it comes to the measurements, in addition
to those required in the final measurement, namely $\mathtt{e}_{\textup{on}}$,
we want to include measurements corresponding to ontic degrees of freedom
which are able to describe the paradoxical aspects of our protocol. Before
applying $U_{\textup{m}}$, at times $\tau_{0}$, $\tau_{1}$, when we measure in
the measurement basis, the measurement determined which branch (off or on) we
collapsed to, but we can only deduce the time when we happen to collapse onto
an on branch. This situation is analogous to a statistical mixture over on and
off. What is surprising, is that after $U_{\textup{m}}$ is applied, we can
deduce not only which branch we were on, but also the time when we collapse to
an off branch. So our would-be ontic model should have a variable which
determines whether we are in the off branch, or the on branch at the times
$\tau_{0}$, $\tau_{1}$. In other words, a variable which, after the
application of $U_{\textup{m}}$, plays the same role that the measurement
basis played before the application of $U_{\textup{m}}$. Including this in the
set of things which are measurable in our would-be non-contextual theory,
gives us

$\displaystyle\mathtt{e}_{\textup{cf}}=\textup{conv}\big{(}\mathtt{e}_{\textup{on}}\cup\\{U_{\textup{m}}^{\dagger}|E\rangle\\!\langle
E|U_{\textup{m}},U_{\textup{m}}^{\dagger}|\tau_{0}\rangle\\!\langle\tau_{0}|U_{\textup{m}},U_{\textup{m}}^{\dagger}|\tau_{1}\rangle\\!\langle\tau_{1}|U_{\textup{m}},U_{\textup{m}}^{\dagger}|A\rangle\\!\langle
A|U_{\textup{m}}\\}\big{)}.$ (12)

An alternative motivation for the inclusion of the additional ontic degrees of
freedom, is that, quantum mechanically, the unitary channel generated by
$U_{\textup{m}}$ could have been applied to rotate the measurement basis,
rather than being applied to the state. This alternative protocol, is
equivalent to the one we study here, from the perspective of quantum
mechanics. We can thus think of these two alternative implementations of our
protocol as different contexts which are indistinguishable according to the
laws of quantum mechanics, in an analogous way to how would-be ontic variable
model is preparation and measurement non-contextual by design.

In Gitton and Woods (2020) an algorithm was developed which, given a set
$\mathtt{s}$ of states and set $\mathtt{e}$ of POVM elements, can determine
whether an ontic variable model as described here exits. When said sets have
finitely many extremal points, as in the case for $\mathtt{s}_{\textup{cf}}$,
$\mathtt{e}_{\textup{cf}}$, it provably runs in finite time. For the simple
case where the probability of each counterfactual outcome is $1/6$ as
described in the main text, the unitary $U_{\textup{m}}$ is given by eq. 23a
and $c,s$ by eq. 23b. For this case, we used the algorithm to provide a
computer assisted proof that the sets
$\mathtt{s}_{\textup{cf}},\mathtt{e}_{\textup{cf}}$ — and hence our
counterfactual clock — do _not_ admit a non-contextual ontic variable model as
per the above description; see appendix B for details.

A crucial aspect of our protocol for using the clock to tell the time when
off, via counterfactual measurements, was the existence of negative amplitudes
allowing for destructive interference between the on and off branches. Since
probabilities are nonnegative, one might have thought that this aspect of our
protocol automatically rules out any non contextual realistic theory. However,
this is definitely not the case since other experiments using counterfactual
measurements, such as the Elizur-Vaidman bomb test Elitzur and Vaidman (1993),
have been shown to admit a non contextual ontic model description analogous to
the one ruled out here for our setup; see Spekkens  _et al._ .

###  IV.3 Engineered clocks

Here we give a more detailed account of the engineered clocks outline in the
main text. As before, we will discuss the simplest case here of just one tick
while relegating the full details of the construction and multiple tick
scenario, which is qualitatively the same, to the appendix (appendix C). As
with the elementary clock, we have a stationary state,
$\ket{\psi_{\textup{off}}}=\mathrm{e}^{-\mathrm{i}t\hat{H}}\ket{\psi_{\textup{off}}}$,
and a dynamical one,
$\ket{\psi_{\textup{on}}(t)}=\mathrm{e}^{-\mathrm{i}t\hat{H}}\ket{\psi_{\textup{on}}}$,
which are mutually orthogonal at all times:
$\braket{\psi_{\textup{off}}}{\psi_{\textup{on}}(t)}=0$. We can run the clock
in standard fashion by applying a unitary to $\ket{\psi_{\textup{off}}}$ which
maps it to $\ket{\psi_{\textup{on}}}$ when the 1st event occurs, followed by
measurement via an appropriate projection-valued measure when the 2nd event
occurs. We use the convention that when the clock is on, it ticks at time
$t_{1}>0$, and that the elapsed time between the 1st and 2nd events is at most
$2t_{1}$. Parameter $t_{1}$ can be chosen to be any value in our construction.
Unlike with the elementary clock, we now have that the state
$\ket{\psi_{\textup{on}}(t)}$ before the tick takes place at time $t_{1}$,
will not be exactly orthogonal to the state after the tick occurs. Therefore,
in order to unambiguously predict whether the clock has ticked, one must
perform an unambiguous quantum discrimination measurement Barnett and Croke
(2009). This is a projective measurement with three possible outcomes: clock
has not ticked yet, it has ticked, or I do not know. The last outcome can be
thought of as an error, since when it is obtained we cannot say what time it
is. This setting allows for more flexibility than in the prior clock setups.
In the counterfactual case, this aspect will not be detrimental to its
functioning, since, as with previous cases, the counterfactual outcomes will
only occur with some probability.

We will now explain how to run this clock in a counterfactual manner. The
protocol proceeds similarly to in prior cases: we start the clock in
$\ket{\psi_{\textup{off}}}$ and apply a unitary which maps it to a suitable
superposition of off and on when the 1st event occurs, namely to
$c\ket{\psi_{\textup{off}}}+s\ket{\psi_{\textup{on}}}$. When the 2nd event
occurs at some unknown time $t\in[0,2t_{1})$, we apply a unitary
$U_{\textup{m}}$ and measure using the projection-valued measure
$|\psi_{\textup{off}}\rangle\\!\langle\psi_{\textup{off}}|$,
$|A\rangle\\!\langle A|$,
$\text{\large${\mathbbm{1}}$}-|\psi_{\textup{off}}\rangle\\!\langle\psi_{\textup{off}}|-|A\rangle\\!\langle
A|$, where $\ket{A}$ is stationary under the Hamiltonian evolution. After
applying $U_{\textup{m}}$ to the on and off clock states at time $t$, the
states take on the form

$\displaystyle U_{\textup{m}}\ket{\psi_{\textup{off}}}=$
$\displaystyle\frac{A_{1}}{N}\ket{\psi_{\textup{off}}}+\frac{A_{1}}{N}\ket{A}+A_{2}\ket{A_{\textup{off}}},$
(13a) $\displaystyle U_{\textup{m}}\ket{\psi_{\textup{on}}(t)}=$
$\displaystyle-\frac{c}{s}A_{1}\ket{\bar{\psi}_{\textup{off}}(t)}-\frac{c}{s}A_{1}\ket{\bar{A}(t)}+A_{3}\ket{A_{\textup{on}}(t)},$
(13b)

where $A_{2}=\sqrt{1-2(A_{1}/N)^{2}}$, $A_{3}=\sqrt{1-2(c/s)^{2}A_{1}^{2}}$
are normalisation parameters and the other parameters will be discussed
shortly. All kets in the superpositions are orthonormal except for the
overlaps $\braket{\psi_{\textup{off}}}{\bar{\psi}_{\textup{off}}(t)}$,
$\braket{A}{\bar{A}(t)}$ and $\braket{A_{\textup{off}}}{A_{\textup{on}}(t)}$.
The kets $\ket{A_{\textup{off}}}$, $\ket{A_{\textup{on}}(t)}$ play the role of
ancilla states that are chosen to guarantee
$\braket{\psi_{\textup{off}}}{\psi_{\textup{on}}(t)}=0$ holds at all times.
The overlaps $\braket{\psi_{\textup{off}}}{\bar{\psi}_{\textup{off}}(t)}$,
$\braket{A}{\bar{A}(t)}$ are chosen such that the counterfactual clock
functions properly: consider the state of the clock just before the
measurement

$\displaystyle
U_{\textup{m}}\mathrm{e}^{-\mathrm{i}t\hat{H}}\left(c\ket{\psi_{\textup{off}}}+s\ket{\psi_{\textup{on}}}\right)=c\,U_{\textup{m}}\ket{\psi_{\textup{off}}}+s\,U_{\textup{m}}\ket{\psi_{\textup{on}}(t)},$
(14)

there the latter quantities are provided by eqs. 13a and 13b. If the 2nd event
occurs in interval $t\in[0,t_{1})$, we require

$\displaystyle\braket{\psi_{\textup{off}}}{\bar{\psi}_{\textup{off}}(t)}=\begin{cases}0&\mbox{
for }t\in[0,t_{1})\\\ 1/N&\mbox{ for }t\in[t_{1},2t_{1})\,,\end{cases}$ (15a)
in order to be sure that the clock was off and $t\in[0,t_{1})$, when the
measurement outcome $\psi_{\textup{off}}$ is obtained (This can readily be
seen from eqs. 13b, 13a, 14 and 15a and the same counterfactual measurement
reasoning presented in the analysis of the elementary timekeeping systems).
Similarly, if the 2nd event occurs in time interval $t\in[t_{1},2t_{1})$, we
require $\displaystyle\braket{A}{\bar{A}(t)}=\begin{cases}1/N&\mbox{ for
}t\in[0,t_{1})\\\ 0&\mbox{ for }t\in[t_{1},2t_{1})\,,\end{cases}$ (15b)

in order to be sure that the clock was off and $t\in[t_{1},2t_{1})$, when the
outcome $A$ is obtained, according to a counterfactual measurement. The role
of $N$ now becomes apparent: it quantifies the overlap between the projectors
associated with the counterfactual outcomes and the dynamical kets
$\ket{\bar{\psi}_{\textup{off}}(t)}$ and $\ket{\bar{A}(t)}$.

Figure 3: Plot of the overlap $\braket{\tilde{x}_{0}}{\bar{x}_{0}(t)}$ in eq.
16 for different values of $\sigma$ (note that
$\braket{\tilde{x}_{1}}{\bar{x}_{1}(t)}$ is identical upto a shift to the left
by an amount $t_{1}$). Observe how the difference between the ideal case
(black) and that of $\sigma=0.01$ (red) are practically indistinguishable. The
small deviation between the black and coloured plots induces a small error.
Note how the error is centred around $t_{1}$, which is the time at which the
clock would tick if it were on, and $2t_{1}$, which is the time interval
corresponding to the 1st tick would ends if the clock were on.

It readily follows that the probability of knowing whether the 2nd event
occurred in time interval $[0,t_{1})$ or $[t_{1},T_{0})$ when the clock was
off, that is to say, the probability of obtaining measurement outcomes
$\psi_{\textup{off}}$ or $A$ in our protocol, is
$P_{\textup{cf}}=2c^{2}A_{1}^{2}/N^{2}$. One would want to choose $A_{1},s$
and $N$ such as to maximise this probability while satisfying all the
constraints. It happens that one can find states $\ket{\psi_{\textup{off}}}$,
$\ket{\psi_{\textup{on}}}$, and a Hamiltonian $\hat{H}$, such that all the
constraints can be satisfied other than eqs. 15a and 15b, which appears to
lead to unnormalisable states $\ket{\psi_{\textup{off}}}$, $\ket{A}$. From a
physical perspective, the problem appears to be related to requiring the
overlaps to transition from zero to a finite constant instantaneously while
having a time evolution governed by a time independent Hamiltonian. However, a
minor modification resolves the conflict: One can replace the r.h.s. of eqs.
15a and 15b with approximate version using the _error function_
$\operatorname{erf}$. Specifically,
$\braket{\psi_{\textup{off}}}{\bar{\psi}_{\textup{off}}(t)}=f_{0}(t)$,
$\braket{A}{\bar{A}(t)}=f_{1}(t)$, with

$\displaystyle
f_{l}(t)=\frac{1}{2N}\\!\left[\operatorname{erf}\\!\left(\frac{t/t_{1}\\!+l\\!-\\!1}{\sqrt{2}\sigma}\right)\\!-\operatorname{erf}\\!\left(\frac{t/t_{1}\\!+l\\!-\\!2}{\sqrt{2}\sigma}\right)\\!\right],$
(16)

and where $\sigma>0$ controls the approximation. In the limit $\sigma\to 0$
the above overlaps are equal to eqs. 15a and 15b. We can now find solutions
for all $\sigma>0$. We plot eq. 16 in fig. 3 for varying approximation values
$\sigma>0$. We can quantify the error by the difference in probabilities
associated with the counterfactual outcomes, between the actual clock state
which is measured, and an “idealised” clock state:
$\textup{Dif}_{p}(\sigma,t):=\Big{|}\,\big{|}\bra{x_{p}}\big{(}c\ket{\psi_{\textup{off}}}+s\ket{\psi_{\textup{on}}(t)}\\!\big{)}\big{|}^{2}-\big{|}\bra{x_{p}}\big{(}c\ket{\psi_{\textup{off}}}+s\ket{\psi_{\textup{on}}I(t)}\\!\big{)}\big{|}^{2}\,\Big{|}$,
where we have used the short-hand ${x_{0}}={\psi_{\textup{off}}}$ for $p=0$,
${x_{1}}={A}$ for $p=1$. Here the “idealised on state”, namely
$\ket{\psi_{\textup{on}}I(t)}$, is identical to $\ket{\psi_{\textup{on}}(t)}$
given by eq. 13b, except that the kets
$\ket{\bar{\psi}_{\textup{off}}(t)},\ket{\bar{A}(t)}$, satisfy eqs. 15a and
15b rather than eq. 16. Importantly, the quantification of the error is
meaningful, since if one were to use this idealised state in our protocol, it
would result in zero false predictions: it would always predict the correct
time and the clock would have always been off when a counterfactual outcome
had been obtained. Additionally, since the time $t$ at which one measures the
clock could be any time in $[0,2t_{1})$, we time-average
$\textup{Dif}_{p}(\sigma,t)$ over the time interval $[pt_{1},(p+1)t_{1})$ in
which the outcome $x_{p}$ should have occurred, resulting in what we call the
type-1 error (This error is thus due to the clock predicting the correct time,
but being on); and we time-average over the remaining time, in which the
outcome $x_{p}$ should not have occurred, resulting in what we call the type-2
error (This error is thus due to the clock predicting the incorrect time). In
the case considered here, in which the clock ticks once when turned on, one
only needs to consider the type-1 and type-2 errors for the $p=0$ case, since
the error types are identical for the $p=1$ case. Hence we will forgo the
label $p$ in the following discussion.

We show in the appendix C.3 that the difference in fidelities can be made
arbitrarily small for both error types, while always having a non zero
probability of obtaining the counterfactual outcome. In particular, if we
choose $\sigma=0.019$, we can obtain a total counterfactual probability of
half what it was in the previous optimal case, namely to $1/6$, and only incur
type one and two errors of $1.0\times 10^{-3}$ and $3.7\times 10^{-4}$
respectively. If we further reduce $\sigma$ to $\sigma=0.0012$, the total
counterfactual probability only drops by half again, to $1/12$, and the type
one and two error probabilities are now merely $7.3\times 10^{-14}$ and
$2.8\times 10^{-4}$ respectively.

###  IV.4 Proof of Theorem 1

Here we prove the theorem from the main text:

See 1

Note that while we will only need to consider the counterfactual clock
explained in the main text (i.e. the ${N_{T}}=1$ case), the proof is analogous
in the case one uses the clocks generalised for ${N_{T}}\geq 2$ found in
appendix A. In the latter case, the definitions of a _counterfactual
measurement_ and of an _always off_ clock which are found in the main text and
used in assumptions (A) and (B), would be substituted for the more general
definitions found in the appendix; i.e. definitions 1 and 2.

###### Proof.

Assumption (A) together with the example of a counterfactual clock for
${N_{T}}=1$ provided in the main text (the values for $U_{\textup{m}}$ and
$c,s$ which fully determine the protocol are provided at the end of A.3 in the
appendix), prove that for any two elapsed times $\tau_{0}$, $\tau_{1}$ between
two events, a counterfactual clock exists and can determine said elapsed time
with the clock always off. Therefore, the theorem follows by invoking
assumption (B). ∎

Appendices

##  Appendix A Elementary time keeping devices

In subsections A.1 and A.2, we detail the protocol one implements when using
the clock to tell the time counterfactually. In subsection A.3, we
characterise the measurement basis used at the end of the protocol from
subsections A.1 and A.2. In subsection A.4 we prove some consequences of the
protocols and definitions made in the previous subsections. Finally in
subsection A.5, we provide two structure theorems which prove that ancillas
are necessary and that counterfactual clocks exist for arbitrary
${N_{T}}\in{\mathbbm{N}_{>0}}$.

###  A.1 Setup and protocol

In this subsection, we explain the full protocol which allows one to determine
the time $t\in\\{\tau_{0},\tau_{1},\tau_{2},\ldots,\tau_{N_{T}}\\}$ using
counterfactual measurements. For the purpose of performing general projective
measurements, we append an $m$-dimensional ancilla space
$\mathcal{H_{\textup{A}}}$ to the Hilbert $\mathcal{H}_{\textup{S}}$ of the
system, leading to a total Hilbert space
$\mathcal{H}_{\textup{S}}\oplus\mathcal{H}_{\textup{A}}$. We will require the
number of ancillas to be grater or equal to the number of distinguishable time
intervals, namely $m\geq{N_{T}}$. The ancilla states are always stationary,
namely they do not change in time. When the clock is tuned on, it is
initialised to another state denoted $\ket{\psi}$. Thus in total, we have the
orthonormal basis states
$\big{\\{}\ket{\psi},\ket{E},\ket{\tau_{0}},\ket{\tau_{1}},\ldots,\ket{\tau_{N_{T}}},\ket{A_{1}},\ket{A_{2}},\ldots,\ket{A_{m}}\big{\\}}$,
which we call the measurement basis (Since the clock never has support on the
basis state $\ket{psi}$ when measured, it will often be omitted for
simplicity). Observe that the $m$ ancillas could be other energy eigenstates
of the system which are orthogonal to the states
$\ket{\tau_{0}},\ket{\tau_{1}},\ldots,\ket{\tau_{N_{T}}},\ket{E}$, or they
could be produced via a separate $m+1$ dimensional system via the
identification
$\ket{\tau_{0}}\equiv\ket{\tau_{0}}\otimes\ket{d_{0}},\ket{\tau_{1}}\equiv\ket{\tau_{1}}\otimes\ket{d_{0}},\ldots,\ket{\tau_{N_{T}}}\equiv\ket{\tau_{N_{T}}}\otimes\ket{d_{0}}$,
and $\ket{E}\equiv\ket{E}\otimes\ket{d_{0}}$, for original states of the
system and
$\ket{A_{1}}\equiv\ket{E}\otimes\ket{d_{1}},\ket{A_{2}}\equiv\ket{E}\otimes\ket{d_{2}},\ldots,\ket{A_{m}}\equiv\ket{E}\otimes\ket{d_{m}}$,
for the ancillary states, where
$\\{\ket{d_{0}},\ket{d_{1}},\ldots,\ket{d_{m}}\\}$, are orthonormal states of
the separate $m+1$ dimensional system.

Initially we set the system to be in the energy eigenstate, $\ket{E}$. Then,
when the 1st event occurs, a unitary is applied to take the state to a
superposition of $\ket{E}$ and $\ket{\psi}$. Using the branching notation from
the main text to distinguish between the two orthogonal states, this has the
form:

(17)

where $c:=\cos\theta$ and $s:=\sin{\theta}$. We then wait until the 2nd event
occurs at unknown time $\tau_{l}$. The system now is of the form

(18)

where $\tau_{l}\in\\{\tau_{0},\tau_{1},\ldots,\tau_{N_{T}}\\}$. We now perform
a unitary $U_{\textup{m}}$ over the system. The quantum states have support on
the ancillas after the application of the unitary. The system now takes on the
form:

(19)

where the amplitudes $A_{q}^{0}$, $A_{q}^{1}(\tau_{l})$ are to be
characterised in appendix A.3. To finalise the protocol, we immediately
measure in the measurement basis after $U_{\textup{m}}$ is applied. Here we
will use the convention that the counterfactual outcomes are associated with
the post-measurement states
$\big{\\{}\ket{E},\ket{A_{1}},\ldots,\ket{A_{{N_{T}}}}\big{\\}}$.

###  A.2 Counterfactual measurement (Formal definition)

Given the presentation of the general protocol in appendix A.1, we can now
formally define the counterfactual measurement which is appropriate for our
setting. These form a generalisation of the main text to the ${N_{T}}>1$ case.
In the following, we label the energy $E$ by $A_{0}:=E$ for simplicity of
notation (since the symbol $A_{0}$ has not been used previously, this should
not lead to confusion).

######  Definition 1 (Counterfactual measurement for clocks for
${N_{T}}\in{\mathbbm{N}_{>0}}$).

We call a projective measurement a _counterfactual measurement_ if it has
outcomes $A_{0}$, $A_{1}$, $A_{2}$, …, $A_{N_{T}}$ (e.g. post-measurement
states $\ket{A_{0}}$, $\ket{A_{1}}$, $\ket{A_{2}}$, …, $\ket{A_{N_{T}}}$
respectively) of which at least one of these outcomes is a _counterfactual
outcome_. Furthermore, we say that the outcome $A_{k}$ is a counterfactual
outcome if the following two equations are both satisfied.

$\displaystyle A_{k}^{1}(\tau_{k})=0,$ (20)

and

$\displaystyle A_{k}^{1}(\tau_{l})=-\frac{c}{s}A^{0}_{k}$ (21)

for all $l=0,1,2,\ldots,k-1,k+1,\ldots,{N_{T}}$ and where
$k\in\\{0,1,2,\ldots,{N_{T}}\\}$.

######  Definition 2 (Clock always off for ${N_{T}}\in{\mathbbm{N}_{>0}}$).

When a counterfactual outcome $A_{k}$ is obtained, we say that the elapsed
time between the two events was $\tau_{k}$, and that the clock was _always
off_.

The reasoning behind these definitions is based on the usual arguments of
counterfactual measurements: eq. 20 allows us to conclude that if one obtains
measurement outcome $A_{k}$ _and_ $t=\tau_{k}$, the clock must have been
always off, since the amplitude in the on branch associate with the ket
$\ket{A_{k}}$ is zero during said time interval (see eq. 19). Meanwhile, eq.
21 guarantees that if $\tau_{k}$ is obtained, then we _must_ have
$t=\tau_{k}$, since the amplitude $A_{k}^{1}(\tau_{l})$ cancels out with the
amplitude $A_{k}^{0}$ for all times $t\neq\tau_{k}$, (see eq. 19). In appendix
A.5 we show that these definitions are not vacuous, specifically, that clocks
exist for any ${N_{T}}\in{\mathbbm{N}_{>0}}$ with the probabilities of
obtaining their counterfactual outcomes all non-zero. In appendix A.4 we will
see that these definitions have other interpretational implications.

###  A.3 Characterisation of unitary $U_{\textup{m}}$.

We consider the case in which we associate all of the measurement outcomes
$E$, $\tau_{1}$, $\tau_{2}$, …, $\tau_{N_{T}}$ with the elapsed time being
$\tau_{0}$, $\tau_{1}$, $\tau_{2}$, …, $\tau_{N_{T}}$ respectively, and the
clock being off (this is to say, we associate $E$, $\tau_{1}$, $\tau_{2}$, …,
$\tau_{N_{T}}$ with counterfactual outcomes). The constraints imposed by eqs.
20 and 21 constrain the matrix coefficients of the unitary matrix
$U_{\textup{m}}$. Specifically, by writing them out explicitly, we have that
$U_{\textup{m}}$ is of the form
$U_{\textup{m}}=U_{\textup{ex}}U_{\textup{m}}^{\prime}$ where
$U_{\textup{ex}}$ implements a simple change of basis, defined via the
relations: $U_{\textup{ex}}\ket{E}=\ket{E}$,
$U_{\textup{ex}}\ket{\tau_{0}}=\ket{A_{1}}$,
$U_{\textup{ex}}\ket{\tau_{1}}=\ket{A_{2}}$, $\ldots$,
$U_{\textup{ex}}\ket{\tau_{{N_{T}}-1}}=\ket{A_{N_{T}}}$,
$U_{\textup{ex}}\ket{A_{1}}=\ket{\tau_{0}}$,
$U_{\textup{ex}}\ket{A_{2}}=\ket{\tau_{1}}$, $\ldots$,
$U_{\textup{ex}}\ket{A_{N_{T}}}=\ket{\tau_{{N_{T}}-1}}$, and
$U_{\textup{ex}}\ket{\tau_{N_{T}}}=\ket{\tau_{N_{T}}}$,
$U_{\textup{ex}}\ket{A_{{N_{T}}+1}}=\ket{A_{{N_{T}}+1}}$,
$U_{\textup{ex}}\ket{A_{{N_{T}}+2}}=\ket{A_{{N_{T}}+2}}$, …,
$U_{\textup{ex}}\ket{A_{m}}=\ket{A_{m}}$. The other matrix
$U_{\textup{m}}^{\prime}$ has the following matrix representation:

where we have denoted $r:=c/s=\cos(\theta)/\sin(\theta)$, and all matrix
entries are arbitrary complex numbers such that
$U_{\textup{m}}^{\prime}\,{U_{\textup{m}}^{\prime}}^{\\!{\dagger}}={U_{\textup{m}}^{\prime}}^{\\!{\dagger}}\,U_{\textup{m}}^{\prime}={\mathbbm{1}}$.
In writing matrix section A.3, we have used the convention that kets are row
vectors and bras are column vectors. We will used this matrix representation
convention throughout. The horizontal solid inner line and the vertical dotted
inner line are visual aids only.

The probability of finding the register off and the time to be $\tau_{l}$ upon
measurement is $P_{\textup{cf}}^{(l)}=|cA^{0}_{l}|^{2}$,
$l\in{\mathbbm{N}_{\geq 0}}$.

The example in the main text where there are just two distinguishable times
(i.e. ${N_{T}}=1$) and requires one ancilla state corresponds to the unitary
matrix $U_{\textup{m}}=U_{\textup{ex}}U_{\textup{m}}^{\prime}$, with
$U_{\textup{ex}}$ determined by $U_{\textup{ex}}\ket{E}=\ket{E}$,
$U_{\textup{ex}}\ket{\tau_{0}}=\ket{A}$,
$U_{\textup{ex}}\ket{A}=\ket{\tau_{0}}$,
$U_{\textup{ex}}\ket{\tau_{1}}=\ket{\tau_{1}}$. The unitary
$U_{\textup{m}}^{\prime}$ is given by

$\begin{matrix}\hskip
18.77892pt\vline&\ket{E}&\quad\ket{\tau_{0}}&\quad\ket{\tau_{1}}&\quad\ket{A}\\\
\hline\cr\ket{E}\hskip
4.26773pt\vline&\sqrt{1/3}&\sqrt{1/3}&\sqrt{1/3}&\large{\brokenvert}\hskip
12.80365pt0\\\ \ket{\tau_{0}}\,\,\vline&0&-\sqrt{1/3}&\sqrt{1/3}&\hskip
6.82881pt\large{\brokenvert}\hskip 8.2511pt-\sqrt{1/3}\\\
\ket{\tau_{1}}\,\,\vline&-\sqrt{1/3}&0&\sqrt{1/3}&\hskip
15.6491pt\large{\brokenvert}\hskip 11.09654pt\sqrt{1/3}\\\
\cline{2-4}\cr\ket{A\hskip 0.28436pt}\hskip
4.41014pt\vline&\sqrt{1/3}&-\sqrt{1/3}&0&\hskip 8.5359pt\sqrt{1/3}\\\
\end{matrix}$ (23a) and coefficients $c$ and $s$ by $\displaystyle
c=s=1/\sqrt{2}.$ (23b)

###  A.4 Backwards in time analysis in the ${N_{T}}\geq 2$ scenario

Here we extend the analysis presented in section IV.1 to the general case of
multiple distinguishable times (${N_{T}}\in{\mathbbm{N}_{>0}}$). Since the
results are quantitatively the same as in section IV.1, we will not discuss at
length the interpretation to avoid repetition. The amplitude corresponding to
the $k$th counterfactual outcome is

$\displaystyle\braket{A_{k}}{U_{\textup{m}}U(\tau_{k})U_{0}}{A_{0}},$ (24)

for $k=0,1,2,\ldots,{N_{T}}$ and where we are again using the convention
$A_{0}:=E$. Using $U_{\textup{m}}=U_{\textup{ex}}U_{\textup{m}}^{\prime}$ and
section A.3, we have

$\displaystyle\begin{split}\bra{A_{k}}U_{\textup{m}}=&A_{k}^{0}\bra{A_{0}}-\\!A_{k}^{0}r\bra{\tau_{0}}-\\!A_{k}^{0}r\bra{\tau_{1}}-\ldots-\\!A_{k}^{0}r\bra{\tau_{k-1}}-\\!A_{k}^{0}r\bra{\tau_{k+1}}-\\!A_{k}^{0}r\bra{\tau_{k+2}}-\\!A_{k}^{0}r\bra{\tau_{k+3}}-\ldots-\\!A_{k}^{0}r\bra{\tau_{N_{T}}}\\\
&+B_{{N_{T}}+2,k}\bra{A_{1}}+B_{{N_{T}}+2,k}\bra{A_{2}}+B_{{N_{T}}+3,k}\bra{A_{3}}+\ldots+B_{{N_{T}}+m,k}\bra{A_{m}}\end{split}$
(25)

where $\bra{\tau_{-1}}:=0$. Recall that $U(\tau_{k})\ket{A_{0}}=\ket{A_{0}}$,
$U(\tau_{k})\ket{A_{1}}=\ket{A_{1}}$, $U(\tau_{k})\ket{A_{2}}=\ket{A_{2}}$, …,
$U(\tau_{k})\ket{A_{m}}=\ket{A_{m}}$, and
$U(\tau_{k})\ket{\psi}=\ket{\tau_{k}}$. Therefore, analogously to section
IV.1, it follows that $\bra{\tau_{l}}U(\tau_{k})=\bra{\psi_{\perp}^{(l,k)}}$
for all $k,l=0,1,\ldots,{N_{T}}$ with $k\neq l$ , where
$\\{\bra{\psi_{\perp}^{(l,k)}}\\}_{l,k}$ are orthogonal to $\ket{A_{0}}$ and
$\ket{\psi}$. This follows from writing the ansatz
$\bra{\tau_{l}}U(\tau_{k})=c_{0}^{(l,k)}\bra{\psi_{\perp}^{(l,k)}}+c_{2}^{(l,k)}\bra{A_{0}}+c_{3}^{(l,k)}\bra{\psi}$,
followed by observing the two equalities
$\delta_{k,l}=\braket{\tau_{l}}{\tau_{k}}=\braket{\tau_{l}}{U(\tau_{k})U(\tau_{k})^{\dagger}}{\tau_{k}}=\big{(}c_{0}^{(l,k)}\bra{\psi_{\perp}^{(l,k)}}+c_{2}^{(l,k)}\bra{A_{0}}+c_{3}^{(l,k)}\bra{\psi}\big{)}\ket{\psi}=c_{3}^{(l,k)}$,
where $\delta_{k,l}$ is the Kronecker delta, and
$0=\braket{\tau_{l}}{A_{0}}=\braket{\tau_{l}}{U(\tau_{k})U(\tau_{k})^{\dagger}}{A_{0}}=\big{(}c_{0}^{(l,k)}\bra{\psi_{\perp}^{(l,k)}}+c_{2}^{(l,k)}\bra{A_{0}}+c_{3}^{(l,k)}\bra{\psi}\big{)}\ket{A_{0}}=c_{2}^{(l,k)}$.
Thus from eq. 25 we arrive at

$\displaystyle\begin{split}\bra{A_{k}}U_{\textup{m}}U(\tau_{k})=&A_{k}^{0}\bra{A_{0}}-\\!A_{k}^{0}r\bra{\psi_{\perp}^{(0,k)}}-\\!A_{k}^{0}r\bra{\psi_{\perp}^{(1,k)}}-\ldots-\\!A_{k}^{0}r\bra{\psi_{\perp}^{(k\\!-\\!1,k)}}-\\!A_{k}^{0}r\bra{\psi_{\perp}^{(k+1,k)}}-\\!A_{k}^{0}r\bra{\psi_{\perp}^{(k+2,k)}}\\\
&-\ldots-\\!A_{k}^{0}r\bra{\psi_{\perp}^{({N_{T}},k)}}+B_{{N_{T}}+2,k}\bra{A_{1}}+B_{{N_{T}}+2,k}\bra{A_{2}}+B_{{N_{T}}+3,k}\bra{A_{3}}+\ldots+B_{{N_{T}}+m,k}\bra{A_{m}}\\\
&=A_{k}^{0}\bra{A_{0}}+\bra{\psi_{\perp}^{\prime\prime(k)}},\end{split}$ (26)

where $\bra{\psi_{\perp}^{\prime\prime(k)}}$ is another (unnormalised) state
orthogonal to $\ket{A_{0}}$ and $\ket{\psi}$. Thus recalling that $U_{0}$ only
acts nontrivially on the subspace spanned by $\ket{A_{0}}$ and $\ket{\psi}$,
we find that

$\displaystyle\begin{split}\bra{A_{k}}U_{\textup{m}}U(\tau_{k})=&A_{k}^{0}\big{(}c\bra{A_{0}}-s\bra{\psi}\big{)}+\bra{\psi_{\perp}^{\prime\prime(k)}}.\end{split}$
(27)

Thus upon pre-selecting onto the initial energy eigenstate $\ket{A_{0}}$, we
observe that the only terms contributing to
$\braket{A_{k}}{U_{\textup{m}}U(\tau_{k})}{A_{0}}$ come from non dynamical
branches of the wave function.

As with the case in which ${N_{T}}=1$ presented in supplementary IV.1, we can
analyse the probability that a measurement of whether the clock was
dynamically evolving during the dynamical stage of our pre and post selected
setup. For the case of post-selecting on counterfactual outcome $\ket{A_{k}}$,
following Aharonov and Vaidman (2008), we have the probability of the clock
being on at time $t\in(0,\tau_{k})$ is given by

$\displaystyle\textup{Prob}\big{(}\textup{on},A_{k}\big{)}$
$\displaystyle:=\frac{\big{|}\braket{A_{k}}{U_{\textup{m}}U(\tau_{k}-t)P_{\textup{on}}U(t)U_{0}}{A_{0}}\big{|}^{2}}{\big{|}\braket{A_{k}}{U_{\textup{m}}U(\tau_{k}-t)P_{\textup{on}}U(t)U_{0}}{A_{0}}\big{|}^{2}+\big{|}\braket{A_{k}}{U_{\textup{m}}U(\tau_{k}-t)P_{{\scriptscriptstyle\backslash}\textup{on}}U(t)U_{0}}{A_{0}}\big{|}^{2}},$
(28)

where
$P_{\textup{on}}:=|\psi\rangle\\!\langle\psi|+|\tau_{0}\rangle\\!\langle\tau_{0}|+|\tau_{1}\rangle\\!\langle\tau_{1}|+\ldots+|\tau_{N_{T}}\rangle\\!\langle\tau_{N_{T}}|$
is the projection onto the on subspace and
$P_{{\scriptscriptstyle\backslash}\textup{on}}:=|A_{0}\rangle\\!\langle
A_{0}|+|A_{1}\rangle\\!\langle A_{1}|+\ldots+|A_{m}\rangle\\!\langle A_{m}|$
projects onto the complementary space (the subspace where no dynamics occurs).
Note that since the ancillas and energy eigenstate are stationary, we have
that $U(t)=U_{\perp
A_{0},\ldots,A_{m}}(t)+{\mathbbm{1}}_{A_{0},\ldots,A_{m}}$, where
${\mathbbm{1}}_{A_{0},\ldots,A_{m}}$ is the identity operator on the subspace
spanned by $\ket{A_{0}}$, $\ket{A_{1}}$, …, $\ket{A_{m}}$, and that $U_{\perp
A_{0},\ldots,A_{m}}(t)$ is orthogonal to said subspace. Therefore, if we
assume that the dynamics is Markovian, namely
$U(t_{1})U(t_{2})=U(t_{1}+t_{2})$ for all $t_{1},t_{2}\in{\mathbbm{R}}$, it
follows that $U(\tau_{k}-t)P_{\textup{on}}U(t)=U_{\perp
A_{0},\ldots,A_{m}}(\tau_{k})$ and hence that

$\displaystyle\textup{Prob}\big{(}\textup{on},A_{k}\big{)}=0$ (29)

for all $t\in(0,\tau_{k})$ and for all $k=0,1,2,\ldots,{N_{T}}$.

###  A.5 Structure and achievability proofs

In this subsection we present two propositions. The first one shows that
ancillary states are necessary in order for the counterfactual clock to
function, while the second shows that counterfactual clocks exist which can
distinguish between an arbitrary number of times (i.e. for all
${N_{T}}\in{\mathbbm{N}_{>0}}$).

To start with, we need to motivate why it could have been that the ancillary
states were not strictly necessary. The most straightforward way to see this
is by recalling eq. 24 from the previous section, and noting that the unitary
$U_{\textup{m}}$ has the form
$U_{\textup{m}}=U_{\textup{ex}}U_{\textup{m}}^{\prime}$. It follows that

$\displaystyle\braket{A_{k}}{U_{\textup{m}}U(\tau_{k})U_{0}}{E}=\begin{cases}\braket{E}{U_{\textup{m}}^{\prime}U(\tau_{0})U_{0}}{E}&\text{if
}k=0\vspace{0.1cm}\\\
\braket{\tau_{k-1}}{U_{\textup{m}}^{\prime}U(\tau_{k})U_{0}}{E}&\text{if
}k=1,2,3,\ldots,{N_{T}}.\end{cases}$ (30)

The physical interpretation of this is that rather than post-selecting on
$\ket{E}=\ket{A_{0}}$ and ancillas $\ket{A_{1}}$, …, $\ket{A_{N_{T}}}$, we
could have implemented the unitary $U_{\textup{m}}^{\prime}$, in stead of
$U_{\textup{m}}$, and post-selected on $\ket{E}$ and $\ket{\tau_{0}}$, …,
$\ket{\tau_{{N_{T}}\\!-\\!1}}$. In this latter method, since the states
$\ket{\tau_{k}}$ evolve in time, upon implementing the counterfactual
measurement, the clock would start evolving unless it was immediately turned
off (note that this does not happen when post-selecting onto the ancillas,
since these are by definition stationary states). However, in the latter
method, we see from the r.h.s. of eq. 30 that one could have used unitaries in
which the number of ancillas is zero, i.e. $m=0$, since the ancillas only
appear implicitly (no pre or post selection onto them is required). The
following proposition proves that in the case of ${N_{T}}=1$, the unitary
$U_{\textup{m}}^{\prime}$ needs at least one ancilla. This result implies that
the ancillas are necessary.

######  Proposition 1 (Ancillas are necessary).

Consider the counterfactual clock protocol described by the two state
formalism on the r.h.s. of eq. 30 for the case where the clock can distinguish
between two times (${N_{T}}=1$) and the number of ancilla states is zero
($m=0$), namely

$\displaystyle\braket{E}{U_{\textup{m}}^{\prime}U(\tau_{0})U_{0}}{E}\qquad\text{and
}\qquad\braket{\tau_{0}}{U_{\textup{m}}^{\prime}U(\tau_{1})U_{0}}{E}.$ (31)

There is no solution for which the probability of the elapsed time being
$\tau_{0}$, namely
$P_{\textup{cf}}^{(0)}:=|\braket{E}{U_{\textup{m}}^{\prime}U(\tau_{0})U_{0}}{E}|^{2}$,
and the probability of the elapsed time being $\tau_{1}$, namely
$P_{\textup{cf}}^{(1)}=|\braket{\tau_{0}}{U_{\textup{m}}^{\prime}U(\tau_{1})U_{0}}{E}|^{2}$,
are both non-zero.

###### Proof.

For the case ${N_{T}}=1$, $m=0$, the matrix representation of
$U_{\textup{m}}^{\prime}$ in section A.3 reduces to

$\begin{matrix}\hskip
19.0633pt\vline&\ket{E}&\quad\ket{\tau_{0}}&\quad\ket{\tau_{1}}\\\
\hline\cr\ket{E}\hskip 4.55254pt\vline&A_{0}^{0}&A_{1}^{0}&\gamma_{0}\\\
\ket{\tau_{0}}\hskip 3.69899pt\vline&0&-A_{1}^{0}r&\gamma_{1}\\\
\ket{\tau_{1}}\hskip 3.69899pt\vline&-A_{0}^{0}\,r&0&\gamma_{2}\\\
\end{matrix}$ (32)

Unitary matrices require their row and column vectors to be orthogonal.
Applying this constraint to the 1st two columns, one finds
$A_{0}^{0}\,(A_{1}^{0})^{\dagger}=0$, which implies $|A_{0}^{0}||A_{1}^{0}|=0$
and hence $A_{0}^{0}=0$ and/or $A_{1}^{0}=0$. Therefore, since
$P_{\textup{cf}}^{(0)}=|cA_{0}^{0}|^{2}$, and
$P_{\textup{cf}}^{(1)}=|cA_{1}^{0}|^{2}$, there is no solution for which
$P_{\textup{cf}}^{(0)}>0$ and $P_{\textup{cf}}^{(1)}>0$. ∎

We now prove that for all ${N_{T}}\in{\mathbbm{N}}^{+}$, there exits a finite
dimensional ancilla system and unitary $U_{\textup{m}}$ such that there is a
non zero probability of finding the clock off at all measurement times.

######  Proposition 2 (Counterfactual clocks with arbitrarily many
distinguishable times exist).

Let ${N_{T}}\in{\mathbbm{N}_{>0}}$. Then, for any
$\\{\tilde{A}^{0}_{0},\tilde{A}^{0}_{1},\tilde{A}^{0}_{2},\ldots,\tilde{A}^{0}_{N_{T}},\tilde{\gamma}_{0},\tilde{\gamma}_{1},\tilde{\gamma}_{2},\ldots,\tilde{\gamma}_{{N_{T}}+1}\\}\in{\mathbbm{R}}^{2{N_{T}}+3}$,
and $r\neq 0$, there exists a $\gamma>0$ for which a unitary matrix of the
form section A.3 exists, with the amplitudes corresponding to counterfactual
outcomes given by $A^{0}_{0}=\gamma\tilde{A}^{0}_{0}$,
$A^{0}_{1}=\gamma\tilde{A}^{0}_{1}$, $A^{0}_{2}=\gamma\tilde{A}^{0}_{2}$, …,
$A^{0}_{N_{T}}=\gamma\tilde{A}^{0}_{N_{T}}$, and gamma coefficients given by
$\gamma_{0}=\gamma\,\tilde{\gamma}_{0}$,
$\gamma_{1}=\gamma\,\tilde{\gamma}_{1}$,
$\gamma_{2}=\gamma\,\tilde{\gamma}_{2}$, …,
$\gamma_{{N_{T}}+1}=\gamma\,\tilde{\gamma}_{{N_{T}}+1}$, and $m=2({N_{T}}+2)$
ancillary states.

###### Proof.

It is by construction, and hence can be used to work out particular values of
$\gamma$ for any instance of the problem. Denote by
$\vec{F}_{j}=\gamma\vec{f}_{j}\,$ ($j=1,2,3,\ldots,m$) the $j$th column vector
of section A.3. As is well known, since section A.3 is a square matrix, it
follows that it is a unitary matrix if the vectors
$\\{\vec{F}_{j}\\}_{j=1}^{m}$ form an orthonormal family. We therefore need to
prove that the vectors $\vec{F}_{j}$ can be made orthonormal.

To start with, we will decompose the vector
$\\{\vec{f}_{j}\\}_{j=1}^{{N_{T}}+2}$ into a direct sum of three other
vectors, namely for $j=1,2,3,\ldots,{N_{T}}+2$, let
$\vec{f}_{j}=\vec{e}_{j}\oplus\vec{X}_{j}\oplus\vec{Y}_{j}$, where
$\vec{e}_{j},\vec{X}_{j},\vec{Y}_{j}\in{\mathbbm{R}}^{{N_{T}}+2}$. Since the
matrix in section A.3 is a square matrix, this choice fixes the number of
ancillas to be $m=2({N_{T}}+2)$. Note that the vectors
$\\{\vec{e}_{j}\\}_{j=1}^{{N_{T}}+2}$ are completely fixed, up to the constant
$\gamma$, by the given parameters in the proposition statement. Meanwhile the
vectors $\vec{X}_{j},\vec{Y}_{j}$ are complete undetermined at this stage. We
first fix the $\vec{Y}_{j}$ vectors: for $j,k=1,2,3,\ldots,{N_{T}}+2$, let
$[\vec{Y}_{j}]_{k}=\delta_{j,k}c_{j}$, where $\delta_{j,k}$ is the Kronecker
delta, and the coefficients
$\\{c_{j}\\}_{j=1}^{{N_{T}}+2}\in{\mathbbm{R}}^{{N_{T}}+2}$ are to be
determined. We will now fix the $\vec{X}_{j}$ vectors. We will use them for
so-called _dimensional lifting_ of the vectors $\vec{e}_{j}$ to an orthogonal
set. The algorithm in Havlicek and Svozil (2018) shows how to choose vectors
$\\{\vec{x}_{j}\\}_{j=1}^{{N_{T}}+2}$ so that the vectors
$\\{\vec{e}_{j}\oplus\vec{x}_{j}\\}_{j=1}^{{N_{T}}+2}$ form an orthogonal
family for any given set $\\{\vec{e}_{j}\\}_{j=1}^{{N_{T}}+2}$. It thus
follows due to the form of the vectors $\vec{Y}_{j}$, that the vectors
$\\{\vec{F}_{j}=\gamma\,\vec{e}_{j}\oplus\vec{X}_{j}\oplus\vec{Y}_{j}\\}_{j=1}^{{N_{T}}+2}$
form an orthogonal family for all
$\\{c_{j}\\}_{j=1}^{{N_{T}}+2}\in{\mathbbm{R}}^{{N_{T}}+2}$ and for all
$\gamma>0$. Imposing normalisation on the vectors
$\\{\vec{F}_{j}\\}_{j=1}^{{N_{T}}+2}$, we find for $j=1,2,3\ldots,{N_{T}}+2$:

$\displaystyle\frac{1}{\gamma^{2}}-|c_{j}|^{2}=(\vec{X}_{j})^{\dagger}\,\vec{X}_{j}+(\vec{Y}_{j})^{\dagger}\,\vec{Y}_{j}.$
(33)

Now denote by
$D:=\max_{j\in\\{1,2,\ldots,{N_{T}}+2\\}}\left\\{(\vec{X}_{j})^{\dagger}\,\vec{X}_{j}+(\vec{Y}_{j})^{\dagger}\,\vec{Y}_{j}\right\\}$,
and the value of $j$ which solves the maximization by $j^{*}$. Furthermore,
set $c_{j^{*}}=0$ so that it follows from eq. 33, that $\gamma=1/\sqrt{D}$. We
can solve eq. 33 for all $j\neq j^{*}$, with a solution for the coefficients
$c_{j}$ satisfying $0\leq|c_{j}|^{2}\leq 1/\gamma^{2}=D$.

We now have an orthonormal set of vectors
$\\{\vec{F}_{j}\\}_{j=1}^{{N_{T}}+2}$, and all that remains is to find the
remaining vectors $\\{\vec{F}_{j}\\}_{j={N_{T}}+3}^{{N_{T}}+2+m}$, with
$m=2({N_{T}}+2)$. Since this latter set of vectors contains elements without
any constraint on them, other than those which allow eq. 33 to be a unitary
matrix, we can simply apply the Gram-Schmidt orthonormalization procedure on
the input sequence
$\big{(}\vec{F}_{j}\big{)}_{j=1}^{{N_{T}}+2}\,{}^{\frown}\big{(}\vec{z}_{j}\big{)}_{j={N_{T}}+3}^{{N_{T}}+2+m}$,
where $\\{\vec{z}_{j}\\}_{j={N_{T}}+3}^{{N_{T}}+2+m}$ is an arbitrary set of
vectors linearly independent of $\\{\vec{F}_{j}\\}_{j=1}^{{N_{T}}+2}$, and ⌢
denotes sequence concatenation. The output of the Gram-Schmidt
orthogonalisation procedure is then the orthonormal family
$\big{(}\vec{F}_{j}\big{)}_{j=1}^{3({N_{T}}+2)}$ where the 1st ${N_{T}}+2$
elements are identical to the first ${N_{T}}+2$ orthonormal vectors of the
input sequence to the Gram-Schmidt orthogonalisation procedure. This completes
the proof. ∎

##  Appendix B Implementation of the proof of no classical model

Here we describe the numerical implementation of the proof that there does not
exist a non contextual ontic model for the counterfactual clock described in
supplementary IV.2. The entire problem is fully determined by the set
$\mathtt{s}_{\textup{cf}}$ of states (eq. 11) and the set
$\mathtt{e}_{\textup{cf}}$ of effects (eq. 12), with $U_{\textup{m}}$ and
$c,s$ given by eqs. 23a and 23b.

We will follow the outline of the algorithm Gitton and Woods (2020) and
perform the following steps. The inner product is taken to be the Hilbert-
Schmidt inner product:

  1. 1.

Project the set of extreme points of the set $\mathtt{s}_{\textup{cf}}$ onto
$\textup{span}(\mathtt{e}_{\textup{cf}})$, were span denotes the linear span.
Let $P_{\mathtt{e}}(\mathtt{s})$ denote the resulting set vectors. We call
$\textup{span}\big{(}P_{\mathtt{e}}(\mathtt{s})\big{)}$ the _Reduced space_
and denote it by $\mathcal{R}$. It is the effective vector space of our clock.

  2. 2.

Using the Gram-Schmidt orthogonalization procedure, construct an orthonormal
basis for $\mathcal{R}$.

  3. 3.

Project the extreme points of the sets $\mathtt{s}_{\textup{cf}}$ and
$\mathtt{e}_{\textup{cf}}$ onto $\cal R$. Denote the corresponding new sets as
$P_{\cal R}(\mathtt{e})$ and $P_{\cal R}(\mathtt{s})$. The elements of the
latter sets represent rays (also known as half-lines) emanating from the
origin (which is the point $\mathbf{0}:=(0,\ldots,0)$ with dimension
$\text{dim}(\mathcal{R})$ entries in the orthonormal basis for
$\mathcal{R}$.).

  4. 4.

Run Vertex Enumeration algorithm Avis (2000) twice. Once for rays $P_{\cal
R}(\mathtt{e})$ and again for the rays $P_{\cal R}(\mathtt{s})$. The input to
the algorithm is given using the V-representation which has the format (list
of vertices, list of rays) and in our case it simplifies to ($\mathbf{0}$,
list of rays). Note that it differs from the H-representation which is set of
linear inequalities corresponding to the intersection of halfspaces. We used
Matlab wrapper GeoCalcLib (http://worc4021.github.io/) and performed all the
computations using Matlab R2020a. The two runs of the algorithm on inputs
($\mathbf{0}$, $P_{\cal R}(\mathtt{s})$) and ($\mathbf{0}$, $P_{\cal
R}(\mathtt{e})$) produce the following sets of extreme rays: $\text{Ray}_{S}$,
$\text{Ray}_{E}$ respectively.

  5. 5.

Form a new set $\text{Ray}_{final}=\\{a\otimes
b\,\,|\,\,\forall\,a\in\text{Ray}_{S}\text{ and
}\forall\,b\in\text{Ray}_{E}\\}$.

  6. 6.

Run Vertex Enumeration algorithm on ($\mathbf{0}\otimes\mathbf{0}$,
$\text{Ray}_{final}$) using the V-representation to obtain a set of extreme
rays $W$. The set $W$ contains the list of potential non-classicality
witnesses.

  7. 7.

To verify that the pair of sets $\mathtt{s}_{\textup{cf}}$,
$\mathtt{e}_{\textup{cf}}$ does not admit a non contextual ontic model, it
suffices to check whether there exists $w\in W$ such that

$\Bigg{\langle}\sum_{j=1}^{\textup{dim}(\mathcal{R})}\mathcal{R}_{j}\otimes\mathcal{R}_{j}\,\,,\,\,w\Bigg{\rangle}<0,$
(34)

where $\big{(}\mathcal{R}_{j}\big{)}_{j=1}^{\textup{dim}(\mathcal{R})}$ is the
sequence of orthonormal basis elements for $\mathcal{R}$ calculated in step 2,
and $\langle\cdot,\cdot\rangle$ represent the Hilbert-Schmidt inner product.

Note that computation times for the last instance of vertex enumeration (with
input ($\mathbf{0}\otimes\mathbf{0}$,$\text{Ray}_{final}$) runs for a long
time, so we employed a slight optimization by preprocessing the sets
$\text{Ray}_{S}$, $\text{Ray}_{E}$ by applying the GeoCalcLib vertex Reduction
routine which removed non-extremal (redundant) rays of the polyhedrons in
V-representation. This allowed us to run the Vertex Enumeration algorithm with
input ($\mathbf{0}$,T), where $T\subset\text{Ray}_{final}$.

We were able to successfully identify an element of $w\in W$ which provides a
violation which is two orders of magnitude larger than any error due to
rounding and approximations that stem from using floating-point arithmetic.

##  Appendix C Engineered counterfactual clock

Here we will explain in detail how the counterfactual engineered clock works.
The material is divided into four subsections to aid comprehension.

###  C.1 Preliminaries

Here we will describe the required dynamics of the on and off states of the
clock, and show that such dynamics is indeed achievable. In particular, we
will need to consider two orthonormal states $\ket{\psi_{\textup{off}}}$ and
$\ket{\psi_{\textup{on}}}$, on an infinite dimensional Hilbert space, whose
dynamics is generated via a Hamiltonian $\hat{H}$ of the form

$\displaystyle\hat{H}=\hat{H}_{\textup{on}}-\hat{H}_{\textup{on}}|\psi_{\textup{off}}\rangle\\!\langle\psi_{\textup{off}}|\hat{H}_{\textup{on}}/r_{0},$
(35)

where $\hat{H}_{\textup{on}}$ and $\ket{\psi_{\textup{off}}}$ are arbitrary so
long as
$r_{0}:=\braket{\psi_{\textup{off}}}{\hat{H}_{\textup{on}}}{\psi_{\textup{off}}}\neq
0$. We require that the dynamics of $\ket{\psi_{\textup{on}}}$ under
$\hat{H}_{\textup{on}}$ is orthogonal to $\ket{\psi_{\textup{off}}}$ at all
times relevant to the experiment:

$\displaystyle\braket{\psi_{\textup{off}}}{\psi_{\textup{on}}(t)}=0,\quad\forall\,t\in[0,x_{0}T_{0}]$
(36)

where

$\displaystyle\ket{\psi_{\textup{on}}(t)}:=\mathrm{e}^{-\mathrm{i}t\hat{H}_{\textup{on}}}\ket{\psi_{\textup{on}}},$
(37)

and $[0,x_{0}T_{0}]$ is the time during which the clock will function and will
be detailed later at the beginning of section C.2). We now demonstrate a
simple proposition which shows how $\ket{\psi_{\textup{off}}}$ and
$\ket{\psi_{\textup{on}}(t)}$ evolve under the total Hamiltonian $\hat{H}$:

######  Proposition 3.

For all $t\in[0,x_{0}T_{0}]$,

$\displaystyle\mathrm{e}^{-\mathrm{i}t\hat{H}}\ket{\psi_{\textup{off}}}$
$\displaystyle=\ket{\psi_{\textup{off}}}$ (38)
$\displaystyle\mathrm{e}^{-\mathrm{i}t\hat{H}}\ket{\psi_{\textup{on}}}$
$\displaystyle=\ket{\psi_{\textup{on}}(t)},$ (39)

where $\ket{\psi_{\textup{on}}(t)}$ is given by eq. 36.

###### Proof.

From eq. 37 it follows

$\displaystyle
0=\frac{d}{dt}\braket{\psi_{\textup{off}}}{\psi_{\textup{on}}(t)}=-\mathrm{i}\braket{\psi_{\textup{off}}}{\hat{H}_{\textup{on}}}{\psi_{\textup{on}}(t)}.$
(40)

We thus conclude
$\braket{\psi_{\textup{off}}}{\hat{H}_{\textup{on}}}{\psi_{\textup{on}}(t)}=0$
for all $t\in[0,x_{0}T_{0}]$. Therefore, using eq. 35 we find

$\displaystyle\hat{H}\ket{\psi_{\textup{on}}(t)}$
$\displaystyle=\hat{H}_{\textup{on}}\ket{\psi_{\textup{on}}(t)},\quad
t\in[0,x_{0}T_{0}]$ (41) $\displaystyle\hat{H}\ket{\psi_{\textup{off}}}$
$\displaystyle=\bf{0},$ (42)

where $\bf{0}$ is the zero vector. We can use eq. 41 to obtain eq. 38:

$\displaystyle\mathrm{e}^{-\mathrm{i}t\hat{H}}\ket{\psi_{\textup{on}}}$
$\displaystyle=\lim_{N\to\infty}\left({\mathbbm{1}}-\mathrm{i}t\hat{H}/N\right)^{N}\ket{\psi_{\textup{on}}}=\lim_{N\to\infty}\left({\mathbbm{1}}-\mathrm{i}t\hat{H}/N\right)^{N-1}\Big{(}\ket{\psi_{\textup{on}}(t/N)}+\mathcal{O}(t/N)^{2}\Big{)}$
(43)
$\displaystyle=\lim_{N\to\infty}\left({\mathbbm{1}}-\mathrm{i}t\hat{H}/N\right)^{N-2}\Big{(}\ket{\psi_{\textup{on}}(2t/N)}+\mathcal{O}(t/N)^{2}\Big{)}=\lim_{N\to\infty}\Big{(}\ket{\psi_{\textup{on}}(t)}+N\mathcal{O}(t/N)^{2}\Big{)}$
(44) $\displaystyle=\ket{\psi_{\textup{on}}(t)}.$ (45)

Likewise, eq. 39 follows from eq. 42.∎

###  C.2 Protocol and model derivation

We 1st state the general dynamical properties of the clock at a qualitative
level when turned on (i.e. operated in standard fashion). This will allow for
a mental picture which will aid comprehension of the quantitative study which
is to follow. When the 1st event occurs, the clock is turned to the on state,
$\ket{\psi_{\textup{on}}(0)}$, and it starts ticking at elapsed times $t_{1}$,
$2t_{1}$, $3t_{1}$, …, ${N_{T}}t_{1}$. Then, at time $T_{0}=({N_{T}}+1)t_{1}$,
the dynamics of the clock repeats itself. The periodic behaviour is repeated
$x_{0}\in{\mathbbm{N}_{>0}}$ times. Evidently, this clock can only tell the
time modulo the period $T_{0}$. Importantly, the time at which the 2nd event
occurs can be _any_ time in the interval $[0,x_{0}T_{0})$, with the answer
being an estimate on the number of ticks which have occurred between the 1st
and 2nd events.

One starts with the clock in the off state $\ket{\psi_{\textup{off}}}$ and
applies a unitary to turn it to $\ket{\psi_{\textup{on}}(t)}$ when the 1st
event occurs. The clock will then evolve unitarily until the 2nd event occurs
at some time $t$, at which point we measure the state
$\ket{\psi_{\textup{on}}(t)}$ using an appropriately chosen measurement. The
engineered clock’s on state, $\ket{\psi_{\textup{on}}(t)}$, will not be
orthogonal to itself after ticking:
$\braket{\psi_{\textup{on}}(t)}{\psi_{\textup{on}}(t^{\prime})}\neq 0$ for
$t\in\big{[}lt_{1},(l\\!+\\!1)t_{1}\big{)}$,
$t^{\prime}\in\big{[}l^{\prime}t_{1},(l^{\prime}\\!+\\!1)t_{1}\big{)}$, with
$l\neq l^{\prime}$, and $l,l^{\prime}\in\\{0,1,2,\ldots,{N_{T}}\\}$. Since
quantum measurements cannot perfectly distinguish non-orthogonal states, this
implies that the clock will not be able to tell the time perfectly when used.
However, the overlaps will decrease with increasing $|l-l^{\prime}|$, and
hence a well chosen measurement can still provide a good estimate of the
number of ticks occurred. This point will not be of much importance when using
the clock counterfactually, since in this modus operandi, the clock can only
achieve the counterfactual outcomes with a probability less than one anyway
(analogously to the elementary clocks from appendix A) and we will effectively
be performing a form of unambitious quantum state discrimination.

While the quantum system used as a counterfactual clock is much more complex
in the engineered quantum clock case, the actual protocol is very similar to
the one we have seen already in the elementary clock case, namely, starting
with the clock in the off state, $\ket{\psi_{\textup{off}}}$, a unitary
$U_{0}$ is applied when the 1st event occurs transforming the clock to
$c\ket{\psi_{\textup{off}}}+s\ket{\psi_{\textup{on}}(0)}$, followed by
applying a unitary $U_{\textup{m}}$ when the 2nd event occurs and measuring in
the measurement basis. The main physically important new feature is that the
elapsed time between the two events can be _any_ time in the interval
$[0,x_{0}T_{0}]$. It is useful to introduce a set of orthonormal ancillary
states, $\\{\ket{\tilde{A}_{l}}\\}_{l=1}^{N_{T}}$, which are orthogonal to the
Hamiltonian, namely $\hat{H}\ket{\tilde{A}_{l}}=0$. These form part of the
basis in which we measure. In this section, the measurement basis, is the set
of orthogonal projectors $\\{$
$|\Psi_{\textup{off}}\rangle\\!\langle\Psi_{\textup{off}}|$,
$|\tilde{A}_{1}\rangle\\!\langle\tilde{A}_{1}|$,
$|\tilde{A}_{2}\rangle\\!\langle\tilde{A}_{2}|$, …,
$|\tilde{A}_{N_{T}}\rangle\\!\langle\tilde{A}_{N_{T}}|$,
${\mathbbm{1}}-|\Psi_{\textup{off}}\rangle\\!\langle\Psi_{\textup{off}}|-|\tilde{A}_{1}\rangle\\!\langle\tilde{A}_{1}|-|\tilde{A}_{2}\rangle\\!\langle\tilde{A}_{2}|-\ldots-|\tilde{A}_{N_{T}}\rangle\\!\langle\tilde{A}_{N_{T}}|$
$\\}$, where ${\mathbbm{1}}$ denotes the identity operator.333Strictly
speaking, we are not projecting onto a basis for the entire Hilbert space, but
only onto the relevant space for us. (In the case considered in main text, for
which ${N_{T}}=1$, we denoted $\ket{\tilde{A}_{1}}$ by $\ket{A}$ for
simplicity). Diagrammatically, just before the measurement, the protocol is as
follows:

(46)

where $N>0$, $A_{1}>0$, are to be determined and we have defined the
amplitudes

$\displaystyle
A_{2}=\sqrt{1-({N_{T}}\\!+\\!1)\left(\frac{A_{1}}{N}\right)^{2}},\quad
A_{3}=\sqrt{1-({N_{T}}\\!+\\!1)\left(\frac{c}{s}\right)^{2}\\!A_{1}^{2}},$
(47)

and where all kets are normalised, and we will show that
$\braket{\tilde{A}_{r}}{\psi_{\textup{off}}}=\braket{A_{\textup{off}}}{\psi_{\textup{off}}}=\braket{\bar{A}_{r}(t)}{\psi_{\textup{off}}}=\braket{A_{\textup{on}}(t)}{\psi_{\textup{off}}}=\braket{\tilde{A}_{r}}{\tilde{A}_{l}}=\braket{A_{\textup{off}}}{\tilde{A}_{l}}=\braket{\bar{\psi}_{\textup{off}}}{\tilde{A}_{l}}=\braket{\bar{A}_{r}(t)}{\tilde{A}_{l}}=\braket{A_{\textup{on}}(t)}{\tilde{A}_{l}}=\braket{\bar{\psi}_{\textup{off}}(t)}{A_{\textup{off}}}=\braket{\bar{A}_{r}(t)}{A_{\textup{off}}}=\braket{\bar{A}_{r}(t)}{\bar{\psi}_{\textup{off}}(t)}=\braket{A_{\textup{on}}(t)}{\bar{\psi}_{\textup{off}}(t)}=\braket{\bar{A}_{r}(t)}{\bar{A}_{l}(t)}=\braket{A_{\textup{on}}(t)}{\bar{A}_{l}(t)}=0$
for all $t\geq 0$, and $l,r\in 0,1,2,\ldots,{N_{T}}-1$ such that $l\neq r$.

The remaining overlaps, namely
$\braket{\bar{\psi}_{\textup{off}}(t)}{\psi_{\textup{off}}}$,
$\braket{\bar{A}_{l}(t)}{A_{l}}$,
$\braket{A_{\textup{on}}(t)}{A_{\textup{off}}}$, are to be determined.

As we will soon see, we will associate the counterfactual outcomes with
measurement outcomes $\psi_{\textup{off}}$, $\tilde{A}_{1}$, $\tilde{A}_{2}$,
…, $\tilde{A}_{N_{T}}$, resultant from a measurement in the measurement basis
performed immediately after $U_{\textup{m}}$ is applied. To make the analysis
a bit easier, we can consider the mathematically equivalent scenario in which,
rather than applying the unitary $U_{\textup{m}}$ to the state before
measuring in the measurement basis, we can apply $U_{\textup{m}}$ to the
projectors onto the measurement basis instead, rotating the basis in which me
measure to:
$|\tilde{x}_{0}\rangle\\!\langle\tilde{x}_{0}|:=U_{\textup{m}}^{\dagger}|\psi_{\textup{off}}\rangle\\!\langle\psi_{\textup{off}}|U_{\textup{m}}$,
$|\tilde{x}_{1}\rangle\\!\langle\tilde{x}_{1}|:=U_{\textup{m}}^{\dagger}|\tilde{A}_{1}\rangle\\!\langle\tilde{A}_{1}|U_{\textup{m}}$,
$|\tilde{x}_{2}\rangle\\!\langle\tilde{x}_{2}|:=U_{\textup{m}}^{\dagger}|\tilde{A}_{2}\rangle\\!\langle\tilde{A}_{2}|U_{\textup{m}}$,
…,
$|\tilde{x}_{N_{T}}\rangle\\!\langle\tilde{x}_{N_{T}}|:=U_{\textup{m}}^{\dagger}|\tilde{A}_{{N_{T}}}\rangle\\!\langle\tilde{A}_{{N_{T}}}|U_{\textup{m}}$,
${\mathbbm{1}}-\sum_{l=0}^{{N_{T}}}|\tilde{x}_{l}\rangle\\!\langle\tilde{x}_{l}|$.

We additionally chose $U_{\textup{m}}$ to act trivially on
$\ket{A_{\textup{off}}}$ and $\ket{A_{\textup{on}}(t)}$, for all
$t\in[0,x_{0}T_{0})$. Writing the states $\ket{\psi_{\textup{off}}}$ and
$\ket{\psi_{\textup{on}}(t)}$ in the new basis, we find

$\displaystyle\ket{\psi_{\textup{off}}}$
$\displaystyle=\frac{A_{1}}{N}\ket{\tilde{x}_{0}}+\frac{A_{1}}{N}\ket{\tilde{x}_{1}}+\ldots+\frac{A_{1}}{N}\ket{\tilde{x}_{N_{T}}}+\sqrt{1-({N_{T}}\\!+\\!1)\left(\frac{A_{1}}{N}\right)^{2}}\ket{\textup{A}_{\textup{off}}}$
(48) $\displaystyle\ket{\psi_{\textup{on}}(t)}$
$\displaystyle=\mathrm{e}^{-\mathrm{i}t\hat{H}_{\textup{on}}}\\!\left(-\left(\frac{c}{s}\right)\\!A_{1}\ket{\bar{x}_{0}(0)}-\left(\frac{c}{s}\right)\\!A_{1}\ket{\bar{x}_{1}(0)}-\ldots-\left(\frac{c}{s}\right)\\!A_{1}\ket{\bar{x}_{N_{T}}(0)}+\sqrt{1-({N_{T}}\\!+\\!1)\left(\frac{c}{s}\right)^{2}\\!A_{1}^{2}}\ket{\textup{A}_{\textup{on}}(0)}\\!\right)$
(49)
$\displaystyle=-\left(\frac{c}{s}\right)\\!A_{1}\ket{\bar{x}_{0}(t)}-\left(\frac{c}{s}\right)\\!A_{1}\ket{\bar{x}_{1}(t)}-\ldots-\left(\frac{c}{s}\right)\\!A_{1}\ket{\bar{x}_{N_{T}}(t)}+\sqrt{1-({N_{T}}\\!+\\!1)\left(\frac{c}{s}\right)^{2}\\!A_{1}^{2}}\ket{\textup{A}_{\textup{on}}(t)}$
(50)

where
$\ket{\bar{x}_{0}(t)}:=U_{\textup{m}}^{\dagger}\ket{\psi_{\textup{off}}(t)}$,
$\ket{\bar{x}_{1}(t)}:=U_{\textup{m}}^{\dagger}\ket{\bar{A}_{1}(t)}$,
$\ket{\bar{x}_{2}(t)}:=U_{\textup{m}}^{\dagger}\ket{\bar{A}_{2}(t)}$,
$\ldots$,
$\ket{\bar{x}_{{N_{T}}}(t)}:=U_{\textup{m}}^{\dagger}\ket{\bar{A}_{N_{T}}(t)}$.

Hence
$\braket{\bar{\psi}_{\textup{off}}(t)}{\psi_{\textup{off}}}=\braket{\tilde{x}_{0}}{\bar{x}_{0}(t)}$
and
$\braket{\bar{A}_{l}(t)}{\tilde{A}_{l}}=\braket{\tilde{x}_{l}}{\bar{x}_{l}(t)}$.
We impose the constraint that the overlaps must satisfy

$\displaystyle\braket{\tilde{x}_{l}}{\bar{x}_{l}(t)}=\frac{1}{N}\,G_{\sigma}\\!\left(\frac{t-(\theta
l+1)t_{1}}{{N_{T}}}-\frac{t_{1}}{2}\right),\quad l=0,1,\ldots,{N_{T}}$ (51)

where $\theta=-1$ if ${N_{T}}=1$, and $\theta=1$ otherwise; and where

$\displaystyle
G_{\sigma}(x):=\sum_{q=-(\theta+1)/2}^{x_{0}-1}G_{0,\sigma}\left(x-q\frac{T_{0}}{{N_{T}}}\right)$
(52)

with $T_{0}=({N_{T}}+1)t_{1}$ the clock period, $x_{0}\in{\mathbbm{N}_{>0}}$
the number of cycles through the period the clock performs and $G_{0,\sigma}$
is an approximation to the top hat function; namely

$\displaystyle
G_{0,\sigma}(t):=\frac{1}{2}\left[\operatorname{erf}\left(\frac{t/t_{1}+1/2}{\sqrt{2}\sigma}\right)-\operatorname{erf}\left(\frac{t/t_{1}-1/2}{\sqrt{2}\sigma}\right)\right],$
(53)

where $\operatorname{erf}$ is the _error function_ and $\sigma>0$ controls the
quality of the approximation. Approximations to the top hat function are know
as “bells”; see Boyd (2006) and references in the introduction for other
possibilities. The overlap $\braket{\tilde{x}_{l}}{\bar{x}_{l}(t)}$ is plotted
in fig. 4 in the limit of small positive $\sigma$. By considering the
conditions for a counterfactual measurement, we see that this is precisely the
necessary form of the function $G_{\sigma}$. Namely, it guarantees that for
$t\notin\big{[}lt_{1},(l+1)t_{1}\big{)}$, the probability of obtaining outcome
$\tilde{x}_{l}$ is zero, while for $t\in\big{[}lt_{1},(l+1)t_{1}\big{)}$, the
probability of obtaining outcome $\tilde{x}_{l}$ is non zero but the overlap
with the on branch is zero, namely
$\braket{\tilde{x}_{l}}{\psi_{\textup{on}}(t)}=\braket{\tilde{x}_{l}}{\bar{x}_{l}(t)}=0$.

Figure 4: Plot of $\braket{\tilde{x}_{l}}{\bar{x}_{l}(t)}$ over one period:
the function is zero in the time interval $[lt_{1},(l+1)t_{1})$ and $1/N$ at
all other times.

Expanding the kets $\ket{\tilde{x}_{l}},\ket{\bar{x}_{l}}$ in the energy
basis, $\ket{\tilde{x}_{l}}=\int\textup{d}E\,\tilde{x}_{l}(E)\ket{E}$,
$\ket{\bar{x}_{l}}=\int\textup{d}E\,\bar{x}_{l}(E)\ket{E}$, the overlap
$\braket{\tilde{x}_{l}}{\bar{x}_{l}(t)}$ becomes

$\displaystyle\braket{\tilde{x}_{l}}{\bar{x}_{l}(t)}=\int\textup{d}E\,\tilde{x}^{*}_{l}(E)\bar{x}_{l}(E)\,\mathrm{e}^{-\mathrm{i}2\pi
Et},$ (54)

which we identify as the Fourier Transform of
$\tilde{x}^{*}_{l}(E)\bar{x}_{l}(E)$. Denoting the Fourier Transform and its
inverse by
$\mathcal{F}[f(t)](E):=\int\textup{d}Ef(E)\mathrm{e}^{-\mathrm{i}2\pi Et}$ and
$\mathcal{F}^{-1}[f(E)](t):=\int\textup{dt}f(t)\mathrm{e}^{\mathrm{i}2\pi Et}$
respectively, and using the shift and rescaling properties of the Fourier
Transform, we find

$\displaystyle\tilde{x}^{*}_{l}(E)\bar{x}_{l}(E)$
$\displaystyle=\mathcal{F}^{-1}\big{[}\braket{\tilde{x}_{l}}{\bar{x}_{l}(t)}\big{]}(E)=\frac{1}{N}\,\mathcal{F}^{-1}\left[G_{\sigma}\\!\left(\frac{t-(\theta
l+1)t_{1}}{{N_{T}}}-\frac{t_{1}}{2}\right)\right](E)$ (55)
$\displaystyle=\frac{{N_{T}}}{N}\,\mathrm{e}^{\mathrm{i}\pi E\big{(}2(\theta
l+1)+{N_{T}}\big{)}t_{1}}\mathcal{F}^{-1}\big{[}G_{\sigma}(t)\big{]}\big{(}E{N_{T}}\big{)},$
(56)

Using Lemma 1, we find the inverse Fourier transform of $G_{\sigma}(t)$:

$\displaystyle\mathcal{F}^{-1}\big{[}G_{\sigma}(t)\big{]}\big{(}E{N_{T}}\big{)}$
$\displaystyle=\mathcal{F}^{-1}\big{[}G_{0,\sigma}(t)\big{]}\big{(}E{N_{T}}\big{)}\,\frac{\sin\Big{(}\pi\big{(}x_{0}+(\theta+1)/2\big{)}ET_{0}\Big{)}}{\sin\big{(}\pi
ET_{0}\big{)}}\,\mathrm{e}^{-\mathrm{i}\pi ET_{0}(\theta+1)(x_{0}+2)/2}$ (57)
$\displaystyle=\mathcal{F}^{-1}\big{[}G_{0,\sigma}(t)\big{]}\big{(}E{N_{T}}\big{)}\,\frac{\sin\Big{(}\pi({N_{T}}+1)\big{(}x_{0}+(\theta+1)/2\big{)}Et_{1}\Big{)}}{\sin\big{(}\pi({N_{T}}+1)Et_{1}\big{)}}\,\mathrm{e}^{-\mathrm{i}\pi
Et_{1}({N_{T}}+1)(\theta+1)(x_{0}+2)/2},$ (58)

where by direct calculation one finds

$\displaystyle\mathcal{F}^{-1}\big{[}G_{0,\sigma}(t)\big{]}(x)=t_{1}\mathrm{e}^{-2\pi^{2}\sigma^{2}(xt_{1})^{2}}\,\operatorname{sinc}(\pi
xt_{1}),$ (59)

where $\operatorname{sinc}(x)=\sin(x)/x$ is the _sinc_ function. We now make
the trial solutions

$\displaystyle\begin{split}\tilde{x}^{*}_{l}(E)&=\sqrt{\frac{{N_{T}}}{N}\,\mathcal{F}^{-1}\big{[}G_{\sigma}(t)\big{]}\big{(}E{N_{T}}\big{)}\,\frac{\sin\Big{(}\pi\big{(}x_{0}+(\theta+1)/2\big{)}ET_{0}\Big{)}}{\sin\big{(}\pi
ET_{0}\big{)}}\,}\,\mathrm{e}^{\mathrm{i}l\lambda Et_{1}}\\\
\bar{x}_{l}(E)&=\sqrt{\frac{{N_{T}}}{N}\,\mathcal{F}^{-1}\big{[}G_{\sigma}(t)\big{]}\big{(}E{N_{T}}\big{)}\,\frac{\sin\Big{(}\pi\big{(}x_{0}+(\theta+1)/2\big{)}ET_{0}\Big{)}}{\sin\big{(}\pi
ET_{0}\big{)}}\,}\,\mathrm{e}^{-\mathrm{i}l\lambda
Et_{1}}\mathrm{e}^{\mathrm{i}\pi E\big{(}2(\theta
l+1)+{N_{T}}\big{)}t_{1}}\,\mathrm{e}^{-\mathrm{i}\pi
ET_{0}(\theta+1)(x_{0}+2)/2}\end{split}$ (60)

for $l=0,1,2,\ldots,{N_{T}}$; and where $\sqrt{\cdot}$ denotes the principle
square root and $\lambda\in{\mathbbm{R}}$ is to be determined. We can now
calculate the normalisation constant $N$:

$\displaystyle
1=\braket{\tilde{x}_{l}}{\tilde{x}_{l}}=\int\textup{d}E\,\tilde{x}^{*}_{l}(E)\tilde{x}_{l}(E)=\frac{{N_{T}}}{N}\int\textup{d}E\,\left|\sqrt{\mathcal{F}^{-1}\big{[}G_{\sigma}(t)\big{]}\big{(}E{N_{T}}\big{)}\,\frac{\sin\Big{(}\pi\big{(}x_{0}+(\theta+1)/2\big{)}ET_{0}\Big{)}}{\sin\big{(}\pi
ET_{0}\big{)}}\,}\,\right|^{2},$ (61)

hence using the identity $|\sqrt{z}|^{2}=|z|$ for all $z\in{\mathbbm{C}}$,

$\displaystyle N$
$\displaystyle={N_{T}}\int\textup{d}E\,\left|\mathcal{F}^{-1}\big{[}G_{\sigma}(t)\big{]}\big{(}E{N_{T}}\big{)}\,\frac{\sin\Big{(}\pi\big{(}x_{0}+(\theta+1)/2\big{)}ET_{0}\Big{)}}{\sin\big{(}\pi
ET_{0}\big{)}}\right|$ (62)
$\displaystyle={N_{T}}\int\textup{d}E\,t_{1}\mathrm{e}^{-2\pi^{2}\sigma^{2}(E{N_{T}}t_{1})^{2}}\,\left|\operatorname{sinc}(\pi
E{N_{T}}t_{1})\,\frac{\sin\Big{(}\pi\big{(}x_{0}+(\theta+1)/2\big{)}ET_{0}\Big{)}}{\sin\big{(}\pi
ET_{0}\big{)}}\,\right|$ (63)
$\displaystyle={N_{T}}\int\textup{d}y\,\mathrm{e}^{-2(\pi{N_{T}}\sigma)^{2}y^{2}}\,\left|\operatorname{sinc}(\pi{N_{T}}y)\,\frac{\sin\Big{(}\pi({N_{T}}+1)\big{(}x_{0}+(\theta+1)/2\big{)}y\Big{)}}{\sin\big{(}\pi({N_{T}}+1)y\big{)}}\,\right|.$
(64)

Observe that this choice for $N$ also implies
$\braket{\bar{x}_{l}(t)}{\bar{x}_{l}(t)}=1$ for all $t\in{\mathbbm{R}}$. We
now examine the implications of the requirement that the states
$\ket{\psi_{\textup{off}}}$ and $\ket{\psi_{\textup{on}}(t)}$ have to be
orthogonal at all times:

$\displaystyle
0=\braket{\psi_{\textup{off}}}{\psi_{\textup{on}}(t)}=-\left(\frac{c}{s}\right)\frac{A_{1}^{2}}{N}\left(\sum_{l=0}^{N_{T}}\braket{\tilde{x}_{l}}{\bar{x}_{l}(t)}\right)+\sqrt{\left(1-({N_{T}}\\!+\\!1)\left(\frac{A_{1}}{N}\right)^{2}\right)\left(1-({N_{T}}\\!+\\!1)\left(\frac{c}{s}\right)^{2}A_{1}^{2}\right)}\braket{A_{\textup{off}}}{A_{\textup{on}}(t)}.$
(65)

Therefore, $A_{1}$ must satisfy the equation

$\displaystyle
c_{0}=\frac{\left(\frac{c}{s}\right)\frac{A_{1}^{2}}{N}}{\sqrt{\left(1-({N_{T}}\\!+\\!1)\left(\frac{A_{1}}{N}\right)^{2}\right)\left(1-({N_{T}}\\!+\\!1)\left(\frac{c}{s}\right)^{2}A_{1}^{2}\right)}}\in{\mathbbm{R}},$
(66)

where $c_{0}$ satisfies

$\displaystyle\braket{A_{\textup{off}}}{A_{\textup{on}}(t)}=c_{0}\sum_{l=0}^{N_{T}}\braket{\tilde{x}_{l}}{\bar{x}_{l}(t)}.$
(67)

We will now find a value of $c_{0}$ which satisfies eq. 67 and come back to
eq. 66 later. Expanding $\ket{A_{\textup{off}}}$ and
$\ket{A_{\textup{on}}(t)}$ in the energy basis,
$\ket{A_{\textup{off}}}=\int\textup{d}EA_{\textup{off}}(E)\ket{E}$,
$\ket{A_{\textup{on}}(0)}=\int\textup{d}EA_{\textup{on}}(E)\ket{E}$, we have
that
$\braket{A_{\textup{off}}}{A_{\textup{on}}(t)}=\int\textup{d}EA_{\textup{off}}^{*}(E)A_{\textup{on}}(E)\mathrm{e}^{-\mathrm{i}2\pi
Et}$ and hence

$\displaystyle
A_{\textup{off}}^{*}(E)A_{\textup{on}}(E)=c_{0}\,\mathcal{F}^{-1}\left[\sum_{l=0}^{N_{T}}\braket{\tilde{x}_{l}}{\bar{x}_{l}(t)}\right](E).$
(68)

We calculate the summation before proceeding with taking the inverse Fourier
transform. Taking into account eqs. 51 and 52 one finds

$\displaystyle\sum_{l=0}^{N_{T}}\braket{\tilde{x}_{l}}{\bar{x}_{l}(t)}$
$\displaystyle=\sum_{l=0}^{N_{T}}\frac{1}{N}\,G_{\sigma}\\!\left(\frac{t-(\theta
l+1)t_{1}}{{N_{T}}}-\frac{t_{1}}{2}\right)=\frac{1}{N}\sum_{l=0}^{N_{T}}\sum_{q=-(\theta+1)/2}^{x_{0}-1}G_{0,\sigma}\\!\left(\frac{t-(\theta
l+1)t_{1}}{{N_{T}}}-\frac{t_{1}}{2}-q\frac{T_{0}}{{N_{T}}}\right)$ (69)
$\displaystyle=\frac{1}{N}\sum_{l=0}^{N_{T}}\sum_{q=-(\theta+1)/2}^{x_{0}-1}G_{0,\sigma}\\!\left(\frac{t}{{N_{T}}}-\frac{t_{1}}{2}-\big{(}(\theta
l+1)+q({N_{T}}+1)\big{)}\frac{t_{1}}{{N_{T}}}\right)$ (70)
$\displaystyle=\frac{1}{N}\sum_{q=-{N_{T}}-(\theta-1)/2}^{{N_{T}}+(x_{0}-1)({N_{T}}+1)-(\theta+1)/2}G_{0,\sigma}\\!\left(\frac{t}{{N_{T}}}-\frac{t_{1}}{2}-q\frac{t_{1}}{{N_{T}}}\right)=\frac{1}{N}G_{\sigma}^{(2)}\\!\left(\frac{t}{{N_{T}}}-\frac{t_{1}}{2}\right),$
(71)

where in going from line 70 to line 71, we have use the identity
$\sum_{n=0}^{d-1}\sum_{m=a}^{b}f(n+md)=\sum_{n=ad}^{d-1+bd}f(n)$, which holds
for arbitrary function $f$ and $d\in{\mathbbm{N}_{>0}}$, $a\leq b,$
$a,b\in{\mathbbm{Z}}$ and we have fined

$\displaystyle
G_{\sigma}^{(2)}(x):=\sum_{q=-{N_{T}}-(\theta-1)/2}^{{N_{T}}+(x_{0}-1)({N_{T}}+1)-(\theta+1)/2}G_{0,\sigma}\\!\left(x-q\frac{t_{1}}{{N_{T}}}\right).$
(72)

We now return to the task of taking the inverse Fourier transform in eq. 68:

$\displaystyle A_{\textup{off}}^{*}(E)A_{\textup{on}}(E)$
$\displaystyle=c_{0}\,\mathcal{F}^{-1}\left[\sum_{l=0}^{{N_{T}}}\braket{\tilde{x}_{l}}{\bar{x}_{l}(t)}\right](E)=\frac{c_{0}}{N}\mathcal{F}^{-1}\left[G_{\sigma}^{(2)}\\!\left(\frac{t}{{N_{T}}}-\frac{t_{1}}{2}\right)\right](E)$
(73) $\displaystyle=c_{0}\frac{{N_{T}}}{N}\mathrm{e}^{\mathrm{i}\pi
Et_{1}{N_{T}}}\mathcal{F}^{-1}\left[G_{\sigma}^{(2)}(t)\right](E{N_{T}}),$
(74)

where by direct calculation using lemma 1, one has

$\displaystyle\mathcal{F}^{-1}\left[G_{\sigma}^{(2)}(t)\right](E{N_{T}})$
$\displaystyle=\mathcal{F}^{-1}\big{[}G_{0,\sigma}(t)\big{]}(E{N_{T}})\,\frac{\sin\big{(}\pi(x_{0}+1)ET_{0}\big{)}}{\sin(\pi
Et_{1})}\,\mathrm{e}^{-\mathrm{i}\pi
Et_{1}\big{(}({N_{T}}+1)(x_{0}+1)-1\big{)}\big{(}{N_{T}}+(\theta-1)/2\big{)}},$
(75)
$\displaystyle=\mathcal{F}^{-1}\big{[}G_{0,\sigma}(t)\big{]}(E{N_{T}})\,\frac{\sin\big{(}\pi({N_{T}}+1)(x_{0}+1)Et_{1}\big{)}}{\sin(\pi
Et_{1})}\,\mathrm{e}^{-\mathrm{i}\pi
Et_{1}\big{(}({N_{T}}+1)(x_{0}+1)-1\big{)}\big{(}{N_{T}}+(\theta-1)/2\big{)}},$
(76)

where recall that $\mathcal{F}^{-1}\big{[}G_{0,\sigma}(t)\big{]}(E{N_{T}})$ is
given by eq. 59. We thus make the following trial solutions for
$A_{\textup{off}}^{*}(E)$ and $A_{\textup{on}}(E)$

$\displaystyle\begin{split}A_{\textup{off}}^{*}(E)&=\sqrt{c_{0}\frac{{N_{T}}}{N}\,\mathcal{F}^{-1}\big{[}G_{0,\sigma}(t)\big{]}(E{N_{T}})\,\frac{\sin\big{(}\pi({N_{T}}+1)(x_{0}+1)Et_{1}\big{)}}{\sin(\pi
Et_{1})}\,}\,\mathrm{e}^{\mathrm{i}\lambda Et_{1}/2}\\\
A_{\textup{on}}(E)&=\sqrt{c_{0}\frac{{N_{T}}}{N}\,\mathcal{F}^{-1}\big{[}G_{0,\sigma}(t)\big{]}(E{N_{T}})\,\frac{\sin\big{(}\pi({N_{T}}+1)(x_{0}+1)Et_{1}\big{)}}{\sin(\pi
Et_{1})}\,}\,\mathrm{e}^{-\mathrm{i}\lambda
Et_{1}/2}\,\mathrm{e}^{-\mathrm{i}\pi
Et_{1}\big{(}({N_{T}}+1)(x_{0}+1)-1\big{)}\big{(}{N_{T}}+(\theta-1)/2\big{)}},\end{split}$
(77)

where, as before, we use the principle square root. Normalisation of
$\ket{A_{\textup{off}}}$ and $\ket{A_{\textup{on}}(t)}$ imply

$\displaystyle
1=\frac{{N_{T}}}{N}\int\textup{d}E\,\left|\sqrt{c_{0}\,\mathcal{F}^{-1}\big{[}G_{0,\sigma}(t)\big{]}(E{N_{T}})\,\frac{\sin\big{(}\pi({N_{T}}+1)(x_{0}+1)Et_{1}\big{)}}{\sin(\pi
Et_{1})}\,}\,\right|^{2},$ (78)

from which, using the identity $|\sqrt{z}\,|^{2}=|z|$ for all
$z\in{\mathbbm{C}}$, we determine the value of $1/|c_{0}|$ to be

$\displaystyle 1/|c_{0}|$
$\displaystyle=\frac{{N_{T}}}{N}\int\textup{d}E\left|\mathcal{F}^{-1}\big{[}G_{0,\sigma}(t)\big{]}(E{N_{T}})\,\frac{\sin\big{(}\pi({N_{T}}+1)(x_{0}+1)Et_{1}\big{)}}{\sin(\pi
Et_{1})}\,\right|$ (79)
$\displaystyle=\frac{{N_{T}}}{N}\int\textup{d}y\,\mathrm{e}^{-2(\pi{N_{T}}\sigma)^{2}y^{2}}\,\left|\,\operatorname{sinc}(\pi{N_{T}}y)\,\frac{\sin\big{(}\pi({N_{T}}+1)(x_{0}+1)y\big{)}}{\sin(\pi
y)}\,\right|,$ (80)

and where the sign of $c_{0}\in{\mathbbm{R}}$ is given by the sign of $c/s$;
this follows from eq. 66. Now that we have a value for $c_{0}$, we can
determine $A_{1}^{2}$ using eq. 66, which is a quadratic equation in
$A_{1}^{2}$. We thus find

$\displaystyle\frac{A_{1}^{2}}{N^{2}}=\frac{({N_{T}}+1)\Big{(}\left(\frac{c}{s}\right)^{2}N^{2}+1\Big{)}+\Gamma_{\pm}\sqrt{({N_{T}}+1)^{2}\Big{(}\left(\frac{c}{s}\right)^{2}N^{2}+1\Big{)}^{2}-4\left(\frac{c}{s}\right)^{2}\Big{(}({N_{T}}+1)^{2}-1/c_{0}^{2}\Big{)}}}{2\left(\frac{c}{s}\right)^{2}N^{2}\Big{(}({N_{T}}+1)^{2}-1/c_{0}^{2}\Big{)}},$
(81)

where $\Gamma_{\pm}$ is -1 or 1 depending on which solution to the equation we
choose. Since we have assumed the amplitudes associated with the kets
$\ket{A_{\textup{off}}}$ and $\ket{A_{\textup{on}}}$ to be both real in eqs.
48 and 50 respectively, we need to take the $\Gamma_{\pm}=-1$ solution. Since
for small $\sigma$, the probability of measuring a tick counterfactually is
$P_{cf}=c^{2}A_{1}^{2}/N^{2}$ for any one of the ${N_{T}}$ ticks, we have that

$\displaystyle
P_{cf}=s^{2}\frac{({N_{T}}+1)\Big{(}\left(\frac{c}{s}\right)^{2}N^{2}+1\Big{)}-\sqrt{({N_{T}}+1)^{2}\Big{(}\left(\frac{c}{s}\right)^{2}N^{2}+1\Big{)}^{2}-4\left(\frac{c}{s}\right)^{2}\Big{(}({N_{T}}+1)^{2}-1/c_{0}^{2}\Big{)}}}{2N^{2}\Big{(}({N_{T}}+1)^{2}-1/c_{0}^{2}\Big{)}},$
(82)

where recall that $c=\cos(\theta),s=\sin(\theta)$, and $N=N(\sigma)$,
$c_{0}=c_{0}(\sigma)$ are given by eqs. 64 and 80 respectively. The numerical
values of $P_{cf}$ for a given $\sigma$ presented in the main text and are
calculated by setting ${N_{T}}=x_{0}=1$ and evaluating $N$ and $c_{0}$
numerically for this $\sigma$, followed by maximising $P_{cf}$ numerically
over $\theta\in[0,2\pi]$.

Finally, in order for the trial solutions for the wave functions
${\tilde{x}_{l}}(E)$, ${\bar{x}_{l}}(E)$, ${A_{\textup{off}}}(E)$ and
${A_{\textup{on}}}(E)$ to be valid, we must verify their orthogonality
relations; recall that we have so far assumed
$\braket{\tilde{x}_{l}}{\tilde{x}_{r}}=\braket{\tilde{x}_{l}}{\bar{x}_{r}(t)}=\braket{\tilde{x}_{l}}{A_{\textup{on}}(t)}=\braket{\bar{x}_{l}(t)}{A_{\textup{on}}(t)}=\braket{\tilde{x}_{l}}{A_{\textup{off}}}=\braket{\bar{x}_{l}(t)}{A_{\textup{off}}}=0$
for all $t\geq 0$, and $l,r\in 0,1,2,\ldots,{N_{T}}$ such that $l\neq r$. This
is where the phase factors $\lambda$ come in to play. From eqs. 60 and 77, we
see that these overlaps are proportional in absolute value, to integrals of
the two following different forms:

  * Case 1:

$\displaystyle F_{1}(\lambda):=\left|\,\int\textup{d}x\,\mathrm{e}^{-2(\pi
A\sigma)^{2}x^{2}}\,\operatorname{sinc}(\pi B_{1}x)\,\frac{\sin\big{(}\pi
C_{1}x\big{)}}{\sin(\pi
D_{1}x)}\,\mathrm{e}^{\mathrm{i}H_{1}x}\mathrm{e}^{\mathrm{i}H_{2}\lambda
x}\,\right|,$ (83)

where $A,B_{1},C_{1},D_{1},C_{1}/D_{1}\in{\mathbbm{N}_{>0}}$,
$H_{1}\in{\mathbbm{Z}}$, while $H_{2}$ is either a non zero integer or half
integer.

  * Case 2:

$\displaystyle F_{2}(\lambda):=\left|\,\int\textup{d}x\,\mathrm{e}^{-2(\pi
A\sigma)^{2}x^{2}}\,\sqrt{\operatorname{sinc}(\pi
B_{1}x)\,\frac{\sin\big{(}\pi C_{1}x\big{)}}{\sin(\pi
D_{1}x)}}{\left(\\!\sqrt{\operatorname{sinc}(\pi B_{2}x)\,\frac{\sin\big{(}\pi
C_{2}x\big{)}}{\sin(\pi
D_{2}x)}}\,\right)\\!\\!}^{*}\,\mathrm{e}^{\mathrm{i}H_{1}x}\mathrm{e}^{\mathrm{i}H_{2}\lambda
x}\,\right|,$ (84)

where $C_{1},D_{1},H_{1},H_{2}$ satisfy the same constraints as in Case 1,
while $B_{2},C_{2},D_{2},C_{2}/D_{2}\in{\mathbbm{N}_{>0}}$.

We will now verify that we can make $F_{1}(\lambda),F_{2}(\lambda)$
arbitrarily small by choosing $\lambda>0$ large enough.

For case 1, since the integrand of $F_{1}(\lambda)$ is smooth and converges
absolutely, it follows from the principle of stationary phase, that
$F_{1}(\lambda)$ tends to zero as $\lambda$ tends to infinity.

For case 2, the integrand is not absolutely continuous, due to the changes in
sign of the functions under the square roots. As such, the principle of
stationary phase does not directly apply. However, a variant of the usual
stationary phase type argument still applies. For this we will make use of
lemma 2. We start by choosing a consistent expression for $c,f,G,$ and $g$:

$\displaystyle c(x)$
$\displaystyle=\sqrt{\operatorname{sign}\left(\operatorname{sinc}(\pi
B_{1}x)\frac{\sin\big{(}\pi C_{1}x\big{)}}{\sin(\pi
D_{1}x)}\right)}\left(\sqrt{\operatorname{sign}\left(\operatorname{sinc}(\pi
B_{2}x)\,\frac{\sin\big{(}\pi C_{2}x\big{)}}{\sin(\pi
D_{2}x)}\right)}\,\,\right)^{\\!\\!\\!*},$ (85) $\displaystyle f(x)$
$\displaystyle=\sqrt{\left|\,\operatorname{sinc}(\pi
B_{1}x)\frac{\sin\big{(}\pi C_{1}x\big{)}}{\sin(\pi
D_{1}x)}\right|}{\left(\\!\sqrt{\left|\operatorname{sinc}(\pi
B_{2}x)\,\frac{\sin\big{(}\pi C_{2}x\big{)}}{\sin(\pi
D_{2}x)}\right|}\,\right)\\!\\!}$ (86) $\displaystyle=\sqrt{\left|\,\sin(\pi
B_{1}x)\sin(\pi B_{2}x)\frac{\sin\big{(}\pi C_{1}x\big{)}\sin\big{(}\pi
C_{2}x\big{)}}{\sin(\pi D_{1}x)\sin(\pi D_{2}x)}\right|}$ (87) $\displaystyle
G(x)$ $\displaystyle=\mathrm{e}^{-2(\pi A\sigma)^{2}x^{2}},$ (88)
$\displaystyle g(x)$
$\displaystyle=\mathrm{e}^{\mathrm{i}H_{1}x}\mathrm{e}^{\mathrm{i}H_{2}\lambda
x},$ (89)

where $\operatorname{sign}(\cdot)$ is the _sign_ function. Lemma 2 allows us
to upper bound $F_{2}(\lambda)$. Since
$\sum_{n\in{\mathbbm{Z}}}G(nT)=\sum_{n\in{\mathbbm{Z}}}\mathrm{e}^{-2(\pi
A\sigma)^{2}T^{2}n^{2}}<\infty$, the right hand side of eq. 107 is finite in
this case. Moreover, the only $\lambda$ dependency enters in the $v_{0}$ term.
Since
$v(x)=\int\textup{d}x\,\mathrm{e}^{\mathrm{i}H_{1}x}\mathrm{e}^{\mathrm{i}H_{2}\lambda
x}=\mathrm{e}^{\mathrm{i}H_{1}x+\mathrm{i}H_{2}\lambda
x}/(\mathrm{i}H_{1}+\mathrm{i}H_{2}\lambda)$, we can choose
$v_{0}=1/|H_{1}+H_{2}\lambda|$. Therefore, $F_{2}(\lambda)$ tends to zero as
$\lambda$ tends to infinity.

###  C.3 Precision quantification

We now show how to calculate the fidelity between the prescribed protocol in
appendix C and that of a hypothetical “idealised” version of the clock which
satisfies both of the following:

  * 1)

The probability that the clock had been on when one of the outcomes associated
with the projectors
$\Big{\\{}|\tilde{x}_{l}\rangle\\!\langle\tilde{x}_{l}|\Big{\\}}_{l=0}^{N_{T}}$
is obtained, is exactly zero.

  * 2)

The probability that the clock would have ticked $n$ times, had it been on,
when the outcome associated with projector
$|\tilde{x}_{n}\rangle\\!\langle\tilde{x}_{n}|$ is obtained, is exactly one.

The only difference between the idealised version of the clock and that of
appendix C, will be the dynamics of the $\ket{\psi_{\textup{on}}(t)}$ state.
Specifically, we replace eq. 50, namely,

$\displaystyle\ket{\psi_{\textup{on}}(t)}$
$\displaystyle=-\left(\frac{c}{s}\right)\\!A_{1}\ket{\bar{x}_{0}(t)}-\left(\frac{c}{s}\right)\\!A_{1}\ket{\bar{x}_{1}(t)}-\ldots-\left(\frac{c}{s}\right)\\!A_{1}\ket{\bar{x}_{N_{T}}(t)}+\sqrt{1-({N_{T}}\\!+\\!1)\left(\frac{c}{s}\right)^{2}\\!A_{1}^{2}}\ket{\textup{A}_{\textup{on}}(t)},$
(90)

with

$\displaystyle\ket{\psi_{\textup{on}}I(t)}$
$\displaystyle:=-\left(\frac{c}{s}\right)\\!A_{1}\ket{\bar{x}_{0}I(t)}-\left(\frac{c}{s}\right)\\!A_{1}\ket{\bar{x}_{1}I(t)}-\ldots-\left(\frac{c}{s}\right)\\!A_{1}\ket{\bar{x}_{N_{T}}I(t)}+\sqrt{1-({N_{T}}\\!+\\!1)\left(\frac{c}{s}\right)^{2}\\!A_{1}^{2}}\ket{\textup{A}_{\textup{on}}I(t)},$
(91)

where _I_ stands for idealised, and the kets $\ket{\bar{x}_{m}I(t)}$ satisfy
$\braket{\tilde{x}_{l}}{\bar{x}_{m}I(t)}=\frac{1}{N}\delta_{l,m}\delta_{l}(t)$,
where $\delta_{l,m}$ is the Kronecker delta, and where $\delta_{l}(t)=0$ if
$t\in\cup_{q=0}^{x_{0}-1}\big{[}l\,t_{1}+qT_{0},\,(l+1)t_{1}+qT_{0}\big{)}$
and $\delta_{l}(t)=1$ otherwise. The ket $\ket{\textup{A}_{\textup{on}}I(t)}$
is orthogonal to the kets
$\big{\\{}\ket{\tilde{x}_{l}}\big{\\}}_{l=0}^{N_{T}}$ and obeys
$\braket{\textup{A}_{\textup{off}}}{\textup{A}_{\textup{on}}I(t)}=c_{0}\sum_{l=0}^{{N_{T}}}\braket{\tilde{x}_{l}}{\bar{x}_{l}I(t)}$
with $c_{0}$ given by eq. 66, in analogy with eq. 67 satisfied by
$\ket{\textup{A}_{\textup{on}}(t)}$.

Note that while if one performs the protocol associated with the
counterfactual clock laid out in section C.2, one finds that the clock works
“perfectly”, that is to say, satisfies 1) and 2) above, it is unclear whether
such dynamics are achievable with a time independent Hamiltonian. This is the
justification for calling it idealised. Its purpose is to show that our
protocol which is realised via a time independent Hamiltonian, approximates
the idealised clock in fidelity up to an arbitrary precision, by choosing
$\delta>0$ sufficiently small. In the special case in which the clock ticks
just once (${N_{T}}=1$), this description of an idealised clock is the same as
eqs. 15a and 15b in the main text. In particular, we are only concerned with
the differences in fidelity when obtaining one of the outcomes
$\big{\\{}\ket{\tilde{x}_{l}}\big{\\}}_{l=0}^{N_{T}}$, that is to say, the
outcomes which are relevant for the counterfactual operation of the clock. We
start by evaluating the difference in fidelities at time $t$ between what is