# Universal quantum modifications to general relativistic time dilation in
delocalised clocks

Shishir Khandelwal  Institute for Theoretical Physics, ETH Zürich, Switzerland
Group of Applied Physics, University of Geneva, Switzerland  Maximilian P. E.
Lock  Institute for Quantum Optics and Quantum Information (IQOQI), Vienna,
Austria  Mischa P. Woods  Institute for Theoretical Physics, ETH Zürich,
Switzerland  Department of Computer Science, University College London, United
Kingdom

###### Abstract

The theory of relativity associates a proper time with each moving object via
its world line. In quantum theory however, such well-defined trajectories are
forbidden. After introducing a general characterisation of quantum clocks, we
demonstrate that, in the weak-field, low-velocity limit, all “good” quantum
clocks experience time dilation as dictated by general relativity when their
state of motion is classical (i.e. Gaussian). For nonclassical states of
motion, on the other hand, we find that quantum interference effects may give
rise to a significant discrepancy between the proper time and the time
measured by the clock. The universality of this discrepancy implies that it is
not simply a systematic error, but rather a quantum modification to the proper
time itself. We also show how the clock’s delocalisation leads to a larger
uncertainty in the time it measures — a consequence of the unavoidable
entanglement between the clock time and its center-of-mass degrees of freedom.
We demonstrate how this lost precision can be recovered by performing a
measurement of the clock’s state of motion alongside its time reading.

##  1 Introduction

One of the most important programs in theoretical physics is the pursuit of a
successful theory unifying quantum mechanics and general relativity. Arguably,
many of the difficulties arising in this pursuit stem from a lack of
understanding of the nature of time [1, 2, 3], particularly the conflict
between how it is conceived in the two theories [4]. For example, in general
relativity a given object’s proper time is defined geometrically according to
that object’s world line, but according to quantum mechanics such well-defined
trajectories are impossible. How do we then assign a proper time to the
delocalised objects described by quantum theory?

Confusion over the nature of time is not limited to the context of quantum
gravity. Indeed, even in non-relativistic quantum mechanics, there is much to
clarify. In the latter, time is not treated on the same footing as other
observable quantities in the theory — it is a parameter, rather than being
represented by a self-adjoint operator. This aspect of the theory of quantum
mechanics troubled its founders; Wolfgang Pauli, for example, famously noted
the impossibility of a self-adjoint operator corresponding to time [5].
Specifically, a quantum observable with outcome $t\in{\mathbbm{R}}$ equal to
the time parameterising the system’s evolution, can only be achieved in the
limiting case of infinite energy, and is therefore unphysical, as we discuss
in Sec. 2. It is however possible to construct self-adjoint operators whose
outcome is approximately $t$, with an error that can in principle be made
arbitrarily small, in systems of finite energy (see e.g. [6]).

If one takes an operationalist viewpoint, one can draw conclusions about time
by discussing the behaviour of clocks. This underlies Einstein’s development
of special relativity [7], the prototypical example of operationalism [8]. In
a quantum setting, the clocks must of course be quantised, leading naturally
to a time operator. This operational approach has revealed fundamental
limitations to time-keeping in non-relativistic quantum systems [9, 6, 10,
11], and novel effects in a variety of relativistic settings [12, 13, 14, 15,
16].

The present work concerns slowly-moving generic quantum clocks embedded in a
weakly-curved spacetime. We are interested in the phenomenon of time dilation,
and how quantizing the clock may result in predictions distinct from those of
general relativity alone. We take the operational viewpoint noted above,
associating time with the outcomes of measurements performed on generic
quantum clocks. In this setting, in contrast with the study of clocks in non-
relativistic quantum mechanics, we must consider the position and momentum of
the clock. Quantum theory dictates that it be subject to some degree of
spatial delocalisation in addition to its “temporal delocalisation” (i.e. the
indeterminacy of its time-reading), as depicted in Fig. 1.

Figure 1: Depiction of a spatially-delocalised quantum clock exibiting
temporal indeterminacy. In the theory of relativity, the time measured by a
clock is defined geometrically by integrating along a well-defined spacetime
path, and the clock may be arbitrarily precise and accurate. This contrasts
with time in quantum mechanics, where a fully quantum treatment requires
clocks to be delocalised in space and experience an indeterminacy in their
time. They cannot be meaningfully assigned a single trajectory. Relativity
dynamically couples the temporal and kinematic quantum degrees of freedom,
resulting in time measurements which are influenced by the delocalised spatial
trajectory.

Relativistic effects manifest in this quantum setting via a coupling between
the clock’s kinematic and internal (in our case, time-measuring) degrees of
freedom. The form of this coupling has been derived in a number of different
ways: from the relativistic dispersion relation and mass-energy equivalence
[17], from the Klein-Gordon equation in curved spacetime [18] (or its
corresponding Lagrangian density [16]), and as a consequence of classical time
dilation between the clock and the laboratory frame of reference [19]. It has
been used to predict a relativistically-induced decoherence effect [20], and a
corresponding reduction in the visibility of quantum interference experiments
[19]. This approach treats spacetime classically, making no reference to a
quantum theory of gravity. It is applicable only in the low-energy, weak-
gravity limit, so that we may consider relativistic effects as perturbative
corrections to non-relativistic quantum mechanics.

We begin by introducing a way of characterising the accuracy of a generic
quantum clock, before describing how to incorporate the phenomenon of time
dilation into quantum dynamics. This is followed by a description of the
average time dilation experienced by a quantum clock for two different states
of motion. We then discuss how the incorporation of relativity increases the
uncertainty associated with the measurement of time, and how this can be
mitigated by gaining knowledge of the clock’s location/motion, illustrating
this with a numerical example. We finish with a discussion of our results and
the questions that arise from them.

##  2 Non-Relativistic Quantum Clocks

Before we discuss relativistic effects, it is convenient to introduce a
characterisation of the extent to which a given quantum clock measures time
accurately. We associate the relevant degree of freedom with a Hilbert space
$\mathcal{H}_{\textup{c}}$, and consider an initial clock state
${\rho}_{\textup{c}}(0)$ which evolves according to the Schrödinger equation,
with Hamiltonian $\hat{H}_{\textup{c}}$. We use the term _laboratory time_ to
refer to the parameter appearing in the Schrödinger equation. At laboratory
time $t$, a measurement is performed, which corresponds to a Positive-Operator
Valued Measure (POVM) $\\{\hat{F}(s)\\}_{s\in\,\mathcal{S}}$ with
$\mathcal{S}\subseteq{\mathbbm{R}}$. We refer to the outcome $s$ of this
measurement as the _clock time_. A quantum clock is then defined by the tuple
$\big{\\{}\\{\hat{F}(s)\\}_{s\in\,\mathcal{S}},\hat{H}_{\textup{c}},{\rho}_{\textup{c}}(0)\big{\\}}$.
In this article however, it suffices to consider
$\big{\\{}\hat{T}_{\textup{c}},\hat{H}_{\textup{c}},{\rho}_{\textup{c}}(0)\big{\\}}$,
where ${\hat{T}_{\textup{c}}:=\int_{\mathcal{S}}s\,\hat{F}(s)ds}$ is the
first-moment operator of the POVM, which determines the expected value of the
measurement. In the case where the POVM is a projection-valued measure (in
other words, where the clock times are perfectly distinguishable) the
definition of $\hat{T}_{\textup{c}}$ above is simply the spectral
decomposition of the self-adjoint operator corresponding to the measurement.
For simplicity, we choose the laboratory time $t$ to be such that
$\langle\hat{T}_{\textup{c}}\rangle(0)=0$.

If a quantum clock satisfies the Heisenberg form of the canonical commutation
relation,
$\big{[}\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}\big{]}=i\,\mathbbm{1}_{\textup{c}}$,111Here
and throughout this article, we use units such that $\hbar=1$. The symbol
$\mathbbm{1}_{\textup{c}}$ denotes the identity operator on
$\mathcal{H}_{\textup{c}}$. the mean clock time will exactly follow the
laboratory time, i.e. $\langle\hat{T}_{\textup{c}}\rangle(t)=t\text{ mod
}T_{0}$ $\forall t$, for some (possibly infinite) clock period $T_{0}>0$.
Furthermore, the standard deviation of such clocks is constant and determined
by the initial state which can be arbitrarily small. We refer to such a clock
as _idealised_. If one demands only this commutation relation, one can
construct an idealised clock on a dense subset of the Hilbert space, with
finite period $T_{0}$ and with $\hat{H}_{\textup{c}}$ bounded below (e.g. the
phase operators in [21]). If we further impose canonical commutation relations
of the Weyl form, and consider measurements corresponding to projection-valued
measures, we necessarily have that $\hat{H}_{\textup{c}}$ is unbounded below
(Pauli’s theorem [5, 22]), which is clearly unphysical, and ${T_{0}=\infty}$.
These are both examples of _covariant_ POVMs [23, 9], i.e. those satisfying
the covariance property
$\hat{F}(s)=\mathrm{e}^{-\mathrm{i}s\hat{H}_{\textup{c}}}\hat{F}(0)\mathrm{e}^{\mathrm{i}s\hat{H}_{\textup{c}}}$,
$s\in{\mathbbm{R}}$; where in the case that $\mathcal{S}$ is bounded,
$\hat{F}(s)$ has been periodically extended to ${\mathbbm{R}}$. Furthermore,
they satisfy the canonical commutator relation of the Weyl form when
$\mathcal{S}$ is unbounded, and of the Heisenberg form if
${\rho}_{\textup{c}}(0)$ and $\hat{F}(0)$ have disjoint support (see Appendix
C for proof).

To derive results that apply universally to all quantum clocks, regardless of
whether they exhibit the idealised behaviour discussed above, we first
quantify the extent to which a given clock deviates from an idealised one. To
this end, we introduce a clock’s _error operator_ $\hat{E}(t)$; for any clock
characterised by
$\big{\\{}\hat{T}_{\textup{c}},\hat{H}_{\textup{c}},{\rho}_{\textup{c}}(0)\big{\\}}$,
we define $\hat{E}(t)$ via

$-i\left[\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}\right]{\rho}_{\textup{c}}(t)={\rho}_{\textup{c}}(t)+\hat{E}(t)$
(1)

where
${\rho}_{\textup{c}}(t)=e^{-it\hat{H}_{\textup{c}}}{\rho}_{\textup{c}}(0)\,e^{it\hat{H}_{\textup{c}}}$.
While there are a variety of ways one might characterise the deviations of any
clock, it will turn out that for our purposes, the definition of $\hat{E}(t)$
is particularly convenient. As initial motivation, observe that the error
operator $\hat{E}(t)$ allows us to quantify the discrepancy between the
average clock time and the laboratory time $t$ via

$\langle\hat{T}_{\textup{c}}\rangle_{\textup{\tiny{NR}}}(t)=t+\int_{0}^{t}dt^{\prime}\mathrm{tr}\left[\hat{E}(t^{\prime})\right]$
(2)

where the subscript NR denotes that the clock does not account for
relativistic effects (i.e. it evolves under the action of
$\hat{H}_{\textup{c}}$ alone), and we have assumed for convenience that
$\langle\hat{T}_{\textup{c}}\rangle_{\textup{\tiny{NR}}}(0)=0$. For an
idealised clock, we have
$-i\big{[}\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}\big{]}=\mathbbm{1}_{\textup{c}}$,
and therefore $\hat{E}(t)=0$ $\forall t$. We therefore call a clock _good_
when $\hat{E}(t)$ is small in norm relative to the other quantities involved.

An interesting case is when the clock is a $d$-dimensional spin system, with
evenly-spaced energy eigenvalues (corresponding to the spin-projection
states). Constructing $\hat{T}_{\textup{c}}$ such that its eigenvectors form
an appropriate mutually-unbiased basis with respect to the spin-projection
states, and choosing ${\rho}_{\textup{c}}(0)$ to be one of the eigenvectors of
$\hat{T}_{\textup{c}}$, one arrives at the well-known Salecker-Wigner-Peres
clock [12, 24] — a quantum version of the common dial clock (see Fig. 1). In
this case, $\hat{E}(t)$ is not always small; one has
$\mathrm{tr}\big{[}\hat{E}(t)\big{]}=-1$ at regular intervals of laboratory
time, regardless of the dimension $d$ or energy. This arises due to the clock
states’ lack of coherence in the eigenbasis of $\hat{T}_{\textup{c}}$;
specifically $[\hat{T}_{\textup{c}},{\rho}_{\textup{c}}(t)]$ is zero at
regular intervals of laboratory time. This issue can be removed by choosing
${\rho}_{c}(0)$ to have some quantum coherence in this basis, as in the
_Quasi-Ideal_ clock [6]. In the latter case, $\|\hat{E}(t)\|_{1}$ is
exponentially small in both the dimensionality $d$ and the mean energy of the
clock $\forall t$. This is a consequence of the Quasi-Ideal clock’s
approximately dispersionless dynamics in the eigenbasis of
$\hat{T}_{\textup{c}}$. In addition to the Quasi-Ideal clock, the Salecker-
Wigner-Peres and qubit-phase clocks, which do not posses this property, are
discussed in Appendix D.

##  3 Relativistic Time Dilation

Classically, in the post-Newtonian approximation of general relativity, the
proper time $\tau$ experienced by an observer is determined by [25]

$d\tau=d\tilde{t}\left(1-\frac{v^{2}}{2c^{2}}+\frac{\Phi(r)}{c^{2}}\right),$
(3)

to first order in $v^{2}/c^{2}$ and $\Phi/c^{2}$, where $v$ is the observer’s
velocity, $r$ is the distance from a gravitating body of mass $M$,
$\Phi(r)=-GM/r$ is the Newtonian gravitational potential, and $\tilde{t}$ is
the proper time of a fictional observer at rest at $r=\infty$. We take the
common approximation of the Newtonian potential around a point $r_{0}$, i.e.
$\Phi(r)\approx\Phi(r_{0})+gx$, where $g$ is the gravitational acceleration at
the point $r=r_{0}$ (e.g. $g=9.81$ m s-2 at the surface of the Earth), and
$x:=r-r_{0}$. The reference frame corresponding to the laboratory is then
given by $(t,x)$, where the laboratory time $t$ is the proper time of an
observer at rest at $r=r_{0}$. Then, for a classical clock with initial
position $x_{0}$ and velocity $v_{0}$, Eq. (3) allows us to relate the clock’s
proper time $\tau$ to the laboratory time:

$\tau=\left[1-\frac{v_{0}^{2}}{2c^{2}}+\frac{gx_{0}}{c^{2}}+\frac{v_{0}gt}{c^{2}}-\frac{1}{3}\left(\frac{gt}{c}\right)^{2}\right]t.$
(4)

If we now consider the clock to be subject to the laws of quantum mechanics,
we must describe how its temporal degree of freedom interacts with its
kinematic degrees of freedom. To that end, we consider a total Hilbert space
$\mathcal{H}=\mathcal{H}_{\textup{c}}\otimes\mathcal{H}_{\textup{\scriptsize{k}}}$,
where the space $\mathcal{H}_{\textup{\scriptsize{k}}}$ corresponds to the
kinematic variables in one dimension, namely position $\hat{x}$ and momentum
$\hat{p}$, with $[\hat{x},\hat{p}]=i\,\mathbbm{1}_{\textup{\scriptsize{k}}}$.
The coupling between the temporal and kinematic spaces (which can be
understood as a consequence of mass-energy equivalence [26]) is then given by
the interaction Hamiltonian [18, 20, 17]

$\hat{H}_{\textup{c}\textup{\scriptsize{k}}}=\hat{H}_{\textup{c}}\otimes\left(-\frac{\hat{p}^{2}}{2m^{2}c^{2}}+\frac{g\hat{x}}{c^{2}}\right),$
(5)

where $m$ is the mass of the clock, and where we continue the approximation in
Eq. (3), neglecting terms proportional to $1/c^{4}$ (see Appendix A). The two
terms in Eq. (5) represent the lowest-order contributions to the time dilation
due to motion (c.f. $v^{2}/2c^{2}$ in Eq. (3)) and gravity respectively. The
clock is then subject to the total Hamiltonian
$\hat{H}=\hat{H}_{\textup{c}}+\hat{H}_{\textup{\scriptsize{k}}}+\hat{H}_{\textup{c}\textup{\scriptsize{k}}}$,
with
${\hat{H}_{\textup{\scriptsize{k}}}=mc^{2}+mg\hat{x}+\hat{p}^{2}/2m-\hat{p}^{4}/8m^{3}c^{2}}$,
where the final term is the usual lowest-order relativistic correction to the
kinetic energy.

##  4 Time Dilation in Quantum Clocks

We assume the initial state of the clock to be uncorrelated across
$\mathcal{H}_{\textup{c}}$ and $\mathcal{H}_{\textup{\scriptsize{k}}}$, i.e.
that
$\rho_{\textup{c}\textup{\scriptsize{k}}}(0)=\rho_{\textup{c}}(0)\otimes\rho_{\textup{\scriptsize{k}}}(0)$
for some $\rho_{\textup{c}}(0)$ and $\rho_{\textup{\scriptsize{k}}}(0)$. After
evolving for an amount of laboratory time $t$, the average time dilation
experienced by the clock (i.e. the average clock time) is

$\langle\hat{T}_{\textup{c}}\rangle(t)=\langle\hat{T}_{\textup{c}}\rangle_{\textup{\tiny{NR}}}(t)+tR(t)\left\\{1+\mathrm{tr}\left[\hat{E}(t)\right]\right\\},$
(6)

where
$R(t):=\mathrm{tr}\left[\left(-\frac{\hat{p}^{2}}{2m^{2}c^{2}}+\frac{g\hat{x}}{c^{2}}+\frac{\hat{p}gt}{mc^{2}}-\frac{g^{2}t^{2}}{3c^{2}}\right)\rho_{\textup{\scriptsize{k}}}(0)\right]$
is determined by the initial kinematic state. Recall that $\hat{E}(t)$ was
defined according to the non-relativistic evolution of the clock (i.e. under
$\hat{H}_{\textup{c}}$ alone). We now consider two possibilities for
$\rho_{\textup{\scriptsize{k}}}(0)$ — a classical (i.e. Gaussian) and a
nonclassical state.

###  4.1 A Gaussian state of motion

We take the initial kinematic state $\rho_{\textup{\scriptsize{k}}}(0)$ to be
a pure Gaussian wavepacket, denoting the mean momentum by $\bar{p}_{0}$, the
standard deviation of the momentum by $\sigma_{p}$, and the mean position by
$\bar{x}_{0}$. This is the most classical choice in the sense that such states
are the only ones with a non-negative Wigner function [27], and saturate the
position-momentum uncertainty relation. We then have

$R(t)=-\frac{\bar{p}_{0}^{2}+\sigma_{p}^{2}}{2m^{2}c^{2}}+\frac{g\bar{x}_{0}}{c^{2}}+\frac{\bar{p}_{0}gt}{mc^{2}}-\frac{1}{3}\left(\frac{gt}{c}\right)^{2}.$
(7)

In this case, setting $\hat{E}(t)=0$ and using Eqs. (2) and (6), one finds
that the average time measured by an idealised quantum clock is identical to
the expectation value of the classical proper time (Eq. (4)) for an observer
with a momentum following the same (i.e. Gaussian) probability distribution. A
non-idealised clock will exhibit a non-relativistic quantum correction (the
second term in Eq. (2)) as well as a contribution arising from both relativity
and quantum mechanics (the final term in Eq. (6)). However, since both these
effects are proportional to $\mathrm{tr}\big{[}\hat{E}(t)\big{]}$, they can be
made arbitrarily small, for example by using the Quasi-Ideal clock discussed
in Sec. 2 with an appropriately high dimensionality and mean energy [6].
Interestingly, for the Salecker-Wigner-Peres clock discussed Sec. 2,
$\mathrm{tr}\big{[}\hat{E}(t)\big{]}=-1$ whenever the clock is in an
eigenstate of $\hat{T}_{\textup{c}}$, exactly cancelling the usual classical
relativistic effect in Eq. (6) (see Appendix D). This property of the Saleker-
Wigner-Peres clock states has also been identified as the cause for suboptimal
performance in other areas, related to quantum error correction with clocks
[28].

###  4.2 A superposition of heights

A consideration of a nonclassical initial kinematic state can result in a
significant modification of the classical time-dilation effect found in Sec.
4.1. We consider an initial kinematic state constructed by superposing two
Gaussian wavepackets with different mean initial heights, namely

$\ket{\psi}_{\textup{\scriptsize{k}}}=\frac{1}{\sqrt{N}}\left(\sqrt{\alpha}\ket{\psi_{1}}_{\textup{\scriptsize{k}}}+\sqrt{1-\alpha}\ket{\psi_{2}}_{\textup{\scriptsize{k}}}\right)$
(8)

for some $0<\alpha<1$, where $\ket{\psi_{1}}_{\textup{\scriptsize{k}}}$ and
$\ket{\psi_{2}}_{\textup{\scriptsize{k}}}$ are Gaussian states differing only
in the value of $\langle\hat{x}\rangle$. Specifically,
$\ket{\psi_{1}}_{\textup{\scriptsize{k}}}$ and
$\ket{\psi_{2}}_{\textup{\scriptsize{k}}}$ have mean positions $\bar{x}_{0}$
and $\bar{x}_{0}+\Delta x_{0}$ respectively, standard deviation in position
$\sigma_{x}$, standard deviation in momentum $\sigma_{p}$, and for simplicity
we take both wavepackets to have the same initial mean momentum. This is
illustrated in Fig. (2). The normalisation factor $N$ is then given by

$N=1+2\sqrt{\alpha(1-\alpha)}e^{-\frac{1}{2}\left(\frac{\Delta
x_{0}}{2\sigma_{x}}\right)^{2}}.$ (9)

Note that the exponential factor in Eq. (9) is the overlap of the constituent
states, $\braket{\psi_{1}}{\psi_{2}}_{\textup{\scriptsize{k}}}$ (c.f. the grey
region in Fig. (2)).

We denote the average clock time in this case by
$\langle\hat{T}_{\textup{c}}\rangle_{\mathrm{sup}}(t)$. To extract the part of
the effect arising from quantum coherence, we contrast this with the case of a
classical mixture of two such states according to probabilities $\alpha$ and
$1-\alpha$. The average clock time in the latter case, which we denote
$\langle\hat{T}_{\textup{c}}\rangle_{\mathrm{mix}}(t)$, can easily be
calculated via Eq. (6) by taking the corresponding weighted sum, i.e.

$\langle\hat{T}_{\textup{c}}\rangle_{\mathrm{mix}}(t)=\alpha\langle\hat{T}_{\textup{c}}\rangle_{\psi_{1}}(t)+(1-\alpha)\langle\hat{T}_{\textup{c}}\rangle_{\psi_{2}}(t),$
(10)

where $\langle\hat{T}_{\textup{c}}\rangle_{\psi_{i}}(t)$ is the expectation
value for state $\ket{\psi_{i}}_{\textup{\scriptsize{k}}}(t)$. For simplicity
of expression, we consider good clocks in the sense discussed in Sec. 2, so
that the contribution arising from a nonzero $\hat{E}(t)$ is neglible. One
then has

$\langle\hat{T}_{\textup{c}}\rangle_{\mathrm{sup}}(t)=\langle\hat{T}_{\textup{c}}\rangle_{\mathrm{mix}}(t)+T_{\textup{coh}}(t),$
(11)

where $T_{\textup{coh}}(t)$, the contribution due to coherence between the two
constituent Gaussian states, is given by

$T_{\textup{coh}}(t):=\frac{N-1}{N}\left[\left(\frac{\Delta
x_{0}}{2\sigma_{x}}\right)^{2}\frac{\sigma_{v}^{2}}{c^{2}}-\frac{g\Delta
x_{0}}{c^{2}}(1-2\alpha)\right]\frac{t}{2}$ (12)

where ${\sigma_{v}:=\sigma_{p}/m}$ is their standard deviation in their
initial “velocity”. We recall that, since the wavepackets saturate the
uncertainty relation, we have that $\sigma_{p}=1/(2\,\sigma_{x})$.

Figure 2: A cartoon of the (magnitude squared of) the wavefunctions used to
construct delocalised initial (kinematic) states of the clock.

In Eq. (12) we see two contributions to $T_{\textup{coh}}(t)$: one
proportional to $\sigma_{v}^{2}$ and one due to the average difference in
gravitational potential experienced by the wavepackets ($g\Delta x_{0}$). In
other words, we see separately how motional and gravitational time dilation
act on the wavefunction to generate quantum coherence effects. Noting that we
can write $g\Delta x_{0}=v_{g}^{2}/2$, where $v_{g}$ is the velocity accrued
by a classsical object falling a distance of $\Delta x_{0}$ from rest, we see
that for a given $\Delta x_{0}/\sigma_{x}$ and $\alpha$, the relative strength
of the motional and gravitational parts is determined by the comparative
magnitudes of $\sigma_{v}$ and $v_{g}$.

Furthermore, considering Eq. (9), one can see that $T_{\textup{coh}}(t)$
decreases exponentially with increasing $\Delta x_{0}$, a consequence of the
decreasing overlap between the two wavepackets. From this we infer that
$T_{\textup{coh}}(t)$ arises due to interference between
$\ket{\psi_{1}}_{\textup{\scriptsize{k}}}$ and
$\ket{\psi_{2}}_{\textup{\scriptsize{k}}}$. On the other hand, we evidently
have $T_{\textup{coh}}(t)=0$ when $\Delta x_{0}=0$, as in that case
$\ket{\psi_{1}}_{\textup{\scriptsize{k}}}=\ket{\psi_{2}}_{\textup{\scriptsize{k}}}$,
and we return to the classical-motion scenario described in Sec. 4.1.
Consequently there exists, for a given $\sigma_{v}$ and $\alpha$, an
intermediary value of $\Delta x_{0}/\sigma_{x}$ which maximises
$T_{\textup{coh}}(t)$ by finding a balance between the wavefunction overlap
and the “nonclassicality” of the kinematic state.

Given that modern-day atomic clocks are accurate and precise enough to observe
classical relativistic time dilation in tabletop experiments [29], it is
natural to ask whether the quantum contribution to the time dilation might
also be observable. For the purposes of illustration, let us consider an
aluminium atom (whose mass is approximately $27$ u). This is inspired by a
state-of the-art “quantum logic clock” based on optical transitions in a
single aluminium ion [30]. For clarity, we use the (Van der Waals) radius of
aluminium, specifically $r_{\mathrm{Al}}=184$ pm [31], as a unit of length.
Then from Eq. (12), we see that for a balanced superposition (i.e.
$\alpha=1/2$), taking $\sigma_{x}=2$ $r_{\mathrm{Al}}$ and $\Delta x_{0}=4$
$r_{\mathrm{Al}}$, we have $T_{\textup{coh}}(t)\sim 10^{-16}$ s in a
laboratory time of $t=1$ s,222This is well within the coherence times
achievable in both trapped-ion and neutral-atom optical atomic clocks [32].
Maintaining a coherent superposition of heights for $\sim 1$ s is a separate
challenge, achieved in e.g. [33]. which is well within the measurement
capability of state-of-the-art clocks. We have of course, assumed good clocks
so that the contribution due to the error term $\hat{E}(t)$, is neglectable.
While we have demonstrated that this is a justified approximation for clocks
with approximately non-dispersive dynamics, such as the Quasi-Ideal clock,
this approximation might not be so well suited to the qubit-phase and Seleker-
Wigner-Peres clocks previously discussed. This appears very promising, though
we stress that this example is illustrative. A careful analysis concerning
potential experimental platforms is required before conclusions can be drawn
about the prospect of observing the effect.

##  5 Clock Precision and the Coupling of Temporal and Kinematic Degrees of
Freedom

In this section we show how the clock’s precision is modified by the
relativistic coupling between its kinematic and temporal degrees of freedom.
In particular we show how the entanglement generated by the interaction
Hamiltonian in Eq. (5) increases the uncertainty associated with measurements
of the temporal variable.

We quantify the clock’s precision via the standard deviation of the clock
time, which we denote $\scalebox{1.3}{$\sigma$}_{\tiny{T}}(t)$. For
simplicity, we ignore the contribution due to gravity by setting $g=0$, and
consider only temporal measurements which correspond to a projection-valued
measure. In order to calculate the relativistic contribution to
$\scalebox{1.3}{$\sigma$}_{\tiny{T}}(t)$ to leading order, we now include
Hamiltonian terms up to order $\hat{p}^{4}/(mc)^{4}$ (see Appendix E). Let
$\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NR}}}(t)$ denote the
standard deviation of the clock time in the absence of the special
relativistic coupling. Then, assuming the clock’s kinematic and temporal
degrees of freedom to be uncorrelated at $t=0$, one finds that
$\scalebox{1.3}{$\sigma$}_{\tiny{T}}(t)$ separates into

$\scalebox{1.3}{$\sigma$}_{\tiny{T}}(t)=\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NR}}}(t)+\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{I}}}(t)+\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NI}}}(t),$
(13)

where the term $\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{I}}}(t)$ is a
contribution that remains finite in the case of an idealised clock, given by

$\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{I}}}(t):=\frac{t^{2}}{8\,\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NR}}}(t)}\frac{\langle\hat{p}^{4}\rangle+\sigma_{p^{2}}^{2}}{m^{4}c^{4}},$
(14)

where $\sigma_{p^{2}}$ is the standard deviation of the observable
$\hat{p}^{2}$, and the term
$\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NI}}}(t)$ is the
relativistic contribution arising due the clock’s non-idealised nature. Its
full form is given in Appendix E. Note that
$\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{I}}}(t)>0$, and the effect
of the relativistic coupling on good clocks (in the sense discussed in Sec. 2)
is therefore to increase the clock’s temporal uncertainty. In the case of an
idealised clock, $\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NR}}}(t)$
is constant in laboratory time (see Appendix C) and can be made arbitrarily
small by an appropriate choice of ${\rho}_{\textup{c}}(0)$. This feature can
also be well approximated with the Quasi-Ideal clock. Consequently, according
to Eq. (14), the standard deviation of the clock time increases quadratically
with the laboratory time in these cases.

The decrease in precision is a consequence of the clock’s temporal state
losing information via its entanglement with the kinematic degrees of freedom.
We now show how this effect can be reduced by recovering some of the lost
information via a measurement of the clock’s momentum. We consider course-
grained momentum measurements, and show how the uncertainty in the clock time
decreases as the measurement is made more precise. We again choose a classical
(i.e. Gaussian) state of motion, as in Sec. 4.1, and we consider an idealised
clock for simplicity, as this can be approximated arbitrarily well (see the
discussion in Sec. 2). We define a set of projection operators
$\\{\hat{\Pi}_{n,\delta p}\\}_{n}$ acting on
$\mathcal{H}_{\textup{\scriptsize{k}}}$, with $n\in\mathds{Z}$ and $\delta
p>0$, which correspond to a partition of the range of momentum values into
bins of width $\delta p$, with bin $n$ centred on momentum value $n\,\delta
p$, i.e.

$\hat{\Pi}_{n,\delta p}:=\int_{(n-1/2)\delta p}^{(n+1/2)\delta
p}dp\ket{p}\\!\\!\bra{p}_{\textup{\scriptsize{k}}}.$ (15)

The bin width $\delta p$ therefore characterises the amount of knowledge that
we gain from the measurement, and the degree of localisation in momentum space
of the post-measurement clock state. As $\delta p$ approaches zero,
$\\{\hat{\Pi}_{n,\delta p}\\}_{n}$ approaches the set of projectors onto
momentum eigenstates. As $\delta p\to\infty$, on the other hand, the
projectors tend to the identity operator, i.e. the case where no measurement
is performed. Fig. 3 shows examples of
$\scalebox{1.3}{$\sigma$}_{\tiny{T}}(t)$ conditioned on a given outcome of the
measurement, at different laboratory times $t$, and for different values of
$\delta p$. Quantifying the coarseness of the measurement by $q:=\delta
p/\sigma_{p}$, we find that for $q\to 0$, we recover all of the information
leaked into the kinematic state, i.e. after the measurement,
$\scalebox{1.3}{$\sigma$}_{\tiny{T}}(t)=\scalebox{1.3}{$\sigma$}_{\tiny{T}}(0)=\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NR}}}(t)$.
For $q\to\infty$, on the other hand, no information is recovered, as no
measurement had been performed.

Figure 3: The standard deviation of the clock time, conditioned on discovering
that the clock’s momentum lies within the central (most likely) bin of
variously coarse-grained measurements. The momentum follows a Gaussian
distribution centred around $p=0$, and the (idealised) clock is initialised
such that its time also follows a Gaussian distribution. For illustration, we
have taken $\scalebox{1.3}{$\sigma$}_{\tiny{T}}(0)=1$ ns, and chosen the other
initial parameters to correspond to an electron with $\sigma_{x}=1$ nm. The
coarseness of the measurement is determined by $q:=\delta p/\sigma_{p}$, i.e.
the ratio of the bin size to the standard deviation in momentum.

##  6 Discussion

Throughout this work, we have made reference to the laboratory time $t$, which
can be interpreted as the time marked by a classical clock in the laboratory
frame. One may however wish to consider a fully quantum set-up, without
reference to an extrinsic classical time, as in the Page and Wootters
formalism [34], which can be employed in relativistic contexts [35, 36]. In
our framework, one may simply treat $t$ as a bookkeeping coordinate, to be
used to calculate the average clock times of multiple quantum clocks, and thus
their average clock times relative to each other. In doing so, all our derived
results can be written in terms of the average readings of quantum clocks
without any reference to a classical clock or extrinsic time.

The effect that we describe in Sec. 4.2 is distinct from the manifestation of
relativistic time dilation between paths in quantum interferometry
experiments. Examples of the latter include phase shifts observed in a neutron
matter-wave interferometer [37], the prediction that redshift-induced which-
path information leads to a reduction in interferometric visibility [19], and
the proposal to observe gravitational phase shifts in an atom interferometer
[38]. We note that the latter proposal is distinct from the reinterpretation
of measurements of atom-interferometry-based accelerometers as a gravitational
redshift effect (see [39, 40] and subsequent criticisms [41, 42, 43]). These
works concern the phase accrued between distinct paths in space as a
consequence of relativistic time dilation, which is a distinct phenomenon from
the one that we consider in this article. Here we ask what time is measured by
a clock delocalised across distinct spacetime trajectories; a phase between
two spatial components of that delocalisation affects the answer to this
question, but it is not itself the quantity of interest.

The present work has been concerned with the effect of relativity in a
stopwatch scenario, that is, we are interested in the time elapsed between two
events. This contrasts with the ticking clocks discussed in [10, 44, 45, 11,
46, 47]. The incorporation of relativity into the latter is an open problem.

The reduction in clock precision that we predict is both a quantum and
relativistic effect, as the general theory of relativity allows for clocks
which are always perfectly precise, and clocks in non-relativistic quantum
theory are generally uncoupled from their kinematic degrees of freedom. This
leads to the interesting statement that relativity requires us to ask how a
clock is moving as well as the time it measures, or we must pay a penalty in
its precision. Though our explicit calculation of the decreased clock
precision was carried out neglecting the effect of gravity, the same principle
will hold for a nonzero gravitational field. A general initial clock state
which is uncorrelated between temporal and kinematic degrees of freedom will
only typically remain in the set of separable states if either the eigenstates
of the total Hamiltonian are separable (which they are not), or if the initial
state is particularly mixed [48]. One can therefore only escape the
relativistic decrease in precision by increasing the mixedness of the initial
state (and thus decreasing its precision anyway). The process of recovering
the precision via measurements of the kinematic state will likely be modified
by gravity, however. In particular, the gravitational coupling between the
temporal degree of freedom and the clock’s spatial position will mean that a
perfectly accurate measurement of the momentum degree of freedom will no
longer totally restore the clock’s precision.

The decreased clock precision is related to the effective decoherence of
quantum systems discussed in [20], though, contrary to our work, it is the
decoherence of the kinematic degrees of freedom which is studied in that case.
Other work has predicted a limit of the ability of multiple clocks to jointly
measure time due to a gravity-mediated interaction between them [49], or
focused on special relativistic effects [50]. Furthermore, our work shows that
the qubit phase shift seen in [16] is in fact a specific instance of a
universal time-dilation effect. We model the effect of mass-energy equivalence
in the same manner as these authors, and in particular the resulting
relativistic coupling between a system’s internal and centre-of-mass degrees
of freedom described by the interaction Hamiltonian in Eq. (5) in the limit of
low velocities and a weak gravitational field. Outside of this regime, a fully
relativistic theory is necessary. An analysis and rebuttal of some of the
criticisms of the model can be found in [51], including a discussion of the
validity of describing relativistic effects via the Schrödinger equation.
Since a full theory of quantum gravity is yet to be experimentally
established, all results at the interface between quantum theory and gravity
are naturally controversial. Ultimately, future experiments will determine the
full scope of validity for these results.

##  7 Conclusion

In the non-relativistic limit, we characterised a generic quantum clock by an
initial state ${\rho}_{\textup{c}}(0)$, a Hamiltonian $\hat{H}_{\textup{c}}$
and a POVM $\\{\hat{F}(s)\\}_{s\in\mathcal{S}}$ corresponding to the
measurement of the clock time. Not all such systems can serve as clocks; for
example, a clock where ${\rho}_{\textup{c}}(0)$ is an energy eigenstate will
not evolve in time. An unavoidable consequence of quantum theory is that any
finite-energy clock with perfectly distinguishable temporal states cannot be
perfectly accurate (a constraint which is distinct from the uncertainty
principle). We quantified the inaccuracy of a generic clock, with arbitrary
temporal states, by defining an error operator. We call a clock good, when its
error operator is small in norm.

Setting this framework into a relativistic context, we considered the average
time dilation according to a generic quantum clock, finding that for Gaussian
states of motion, any quantum clock will experience the usual classical
relativistic time dilation with some quantum error (given by the trace of the
error operator) which can be made arbitrarily small. Taking a nonclassical
state of motion, namely a superposition of Gaussian states, we found that
interference results in an average clock time which differs from the
prediction of classical relativity, even when the error operator of a clock is
zero (an idealised clock). Considering the example of an aluminium ion in such
a state of motion, we found that this quantum relativistic time dilation
discrepancy should be of an observable magnitude.

We then discussed how relativity leads to entanglement between the clock’s
time-measuring and motional degrees of freedom. We calculated the
corresponding reduction in clock precision for classical states of motion,
showing how it increases over time. We demonstrated how the information lost
from the clock’s time-measuring degrees of freedom can be recovered by
performing a measurement of the clock’s motion in addition to the clock time.

It is our hope that these results will fuel interest from the experimental
community, so that experiments might be devised to observe combined quantum
relativistic effects, perhaps shedding light on the obscure relationship
between the two theories.

## Acknowledgments

The authors would like to thank Alexander Smith for useful discussions and
Yanglin Hu for correcting typos. S.K. acknowledges financial support from ETH
Zürich during his Master’s degree (Birkigt Scholarship). M.P.E.L. acknowledges
support from the ESQ Discovery Grant _Emergent time - operationalism, quantum
clocks and thermodynamics_ of the Austrian Academy of Sciences (ÖAW), as well
as from the Austrian Science Fund (FWF) through the START project Y879-N27.
M.P.W. acknowledges funding from the Swiss National Science Foundation
(AMBIZIONE Fellowship PZ00P2_179914), the NCCR QSIT, and FQXi grant _Finite
dimensional Quantum Observers_ (No. FQXi-RFP-1623) for the programme _Physics
of the Observer_.

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## Appendices

##  Appendix A Relativistic coupling of the clock’s temporal and kinematic
degrees of freedom

For completeness, we give a derivation of the coupling between the clock’s
kinematic and time-measuring degrees of freedom, based on the one in [17]. We
use the post-Newtonian metric to approximate a weak gravitational field. In
coordinates $\tilde{x}^{\mu}=\left(\tilde{t},\vec{r}\right)$ corresponding to
an observer at rest at $r=\infty$, this metric is given by

$\displaystyle\tilde{g}_{00}=-\left[1+\frac{2\Phi(r)}{c^{2}}+\frac{2\Phi(r)^{2}}{c^{4}}\right]+\mathcal{O}\left(\left(\frac{\Phi(r)}{c^{2}}\right)^{3}\right)\quad\text{
and
}\quad\tilde{g}_{ij}=\delta_{ij}\left[1-\frac{2\Phi(r)}{c^{2}}\right]+\mathcal{O}\left(\left(\frac{\Phi(r)}{c^{2}}\right)^{3}\right),$
(16)

where $r=|\vec{r}|$, $i$ and $j$ label spatial coordinates, $\Phi(r)=-GM/r$ is
the gravitational potential of the Earth and $M$ is its mass. Let us now
consider the laboratory frame $x^{\mu}=(t,\vec{r})$, where $t$ is the proper
time of an observer at rest at $\vec{r}=\vec{r}_{0}$. Transforming the metric
in Eq. (16) to the laboratory frame, we have,

$g_{00}=\left[1-\frac{\Phi_{0}}{c^{2}}\right]^{2}\tilde{g}_{00}+\mathcal{O}\left(\left(\frac{\Phi(r)}{c^{2}}\right)^{3}\right)\qquad\text{
and }\qquad g_{ij}=\tilde{g}_{ij},$ (17)

where $\Phi_{0}:=\Phi(r_{0})$. In the laboratory frame the clock has energy
$E_{\mathrm{lab}}$ and spatial momentum $\vec{p}$, with $p=|\vec{p}|$. Now
consider the rest frame of the clock, in which it has energy
$E_{\mathrm{rest}}$ and zero spatial momentum. The norm of the clock’s four-
momentum is a scalar quantity, and therefore the same in both frames. Equating
this norm in both frames and rearranging, one finds

$E_{\mathrm{lab}}^{2}=\frac{\bar{g}^{00}E_{\mathrm{rest}}^{2}-p_{j}p^{j}c^{2}}{g^{00}}=-g_{00}\left(E_{\mathrm{rest}}^{2}+p_{j}p^{j}c^{2}\right),$
(18)

where $\bar{g}^{00}=-1$ is the $00$-component of the metric in the clock’s
rest frame, $p_{j}p^{j}=g^{ij}p_{i}p_{j}$ (using the Einstein summation
convention), and we have used the fact that $g_{00}=1/g^{00}$. Now, in order
to mark the passage of time the clock must have some evolving internal
structure, which is associated with some internal energy. Denoting the clock’s
rest mass by $m$ and its internal energy by $E_{\textup{c}}$, then by mass-
energy equivalence,

$E_{\mathrm{rest}}=mc^{2}+E_{\textup{c}}.$ (19)

Let us now restrict to one spatial dimension, namely the radial one, define
the coordinate $x:=r-r_{0}$ and take the common approximation of the Newtonian
potential around the point $x=0$, i.e. $\Phi(x)\approx\Phi_{0}+gx$, where $g$
is the gravitational acceleration at $x=0$. Then, inserting Eqs. (16), (17)
and (19) into Eq. (18), one finds that

$E_{\mathrm{lab}}=mc^{2}+mgx+\frac{p^{2}}{2m}-\frac{p^{4}}{8m^{3}c^{2}}+E_{\textup{c}}\left(1-\frac{p^{2}}{2m^{2}c^{2}}+\frac{gx}{c^{2}}\right),$
(20)

where we have neglected terms of order $(E_{\textup{c}}/mc^{2})^{2}$,
$(p/mc)^{4}$, $(gx)^{2}/c^{4}$ and $gxp^{2}/m^{2}c^{4}$ and higher. In the
following, we refer to such terms with the notation $\mathcal{O}(1/c^{4})$.
Quantising Eq. (20), one finds that the clock evolves subject to the
Hamiltonian

$\hat{H}=\hat{H}_{\textup{c}}+\hat{H}_{\textup{\scriptsize{k}}}+\hat{H}_{\textup{c}\textup{\scriptsize{k}}},$
(21)

where upon quantisation $E_{\mathrm{lab}}\to\hat{H}$ (which acts upon the
space $\mathcal{H}_{\textup{c}}\otimes\mathcal{H}_{\textup{\scriptsize{k}}}$)
and $E_{\textup{c}}\to\hat{H}_{\textup{c}}$ (which acts upon the space
$\mathcal{H}_{\textup{c}}$),

$\hat{H}_{\textup{\scriptsize{k}}}=mc^{2}+mg\hat{x}+\frac{\hat{p}^{2}}{2m}-\frac{\hat{p}^{4}}{8m^{3}c^{2}}$
(22)

represents the rest-mass-energy, gravitational potential energy and kinetic
energy (with a lowest-order relativistic correction) of the clock, and

$\hat{H}_{\textup{c}\textup{\scriptsize{k}}}=\hat{H}_{\textup{c}}\otimes\left(-\frac{\hat{p}^{2}}{2m^{2}c^{2}}+\frac{g\hat{x}}{c^{2}}\right),$
(23)

is a relativistic coupling of the temporal and kinematic degrees of freedom.
We stress that $\hat{H}$ generates evolution with respect to the proper time
of a classical observer at $x=0$, i.e. the laboratory time $t$.

##  Appendix B Time dilation in generic quantum clocks

We now calculate the average time dilation experienced by a generic quantum
clock evolving under the Hamiltonian given in Eq. (21). We begin by employing
perturbation theory to calculate the evolution of the clock state in the low-
velocity, weak-gravity limit described in Appendix A (i.e. neglecting
$\mathcal{O}(1/c^{4})$ terms). As in the main text, we choose units such that
$\hbar=1$.

Defining a _free_ Hamiltonian by
$\hat{H}_{0}:=\hat{H}_{\textup{c}}+\hat{H}_{\textup{\scriptsize{k}}}$, so the
total Hamiltonian can be written
$\hat{H}=\hat{H}_{0}+\hat{H}_{{\textup{c}}\textup{\scriptsize{k}}}$, where
$\hat{H}_{{\textup{c}}\textup{\scriptsize{k}}}$ is given in Eq. (23), one can
find the evolution of the clock’s state via the interaction picture. Denoting
the evolution of the state in the interaction picture by $\rho^{I}(t)$, and
following standard perturbation theory (see, e.g. [52]), we have that it is
related to the Schrödinger picture by

$\rho^{I}(t)=U_{0}^{\dagger}(t)\rho(t)U_{0}(t),\quad\rho(t)=U(t)\rho(0)U(t)^{\dagger},$
(24)

where $U_{0}(t)=e^{-it\hat{H}_{0}}$ and $U(t)=e^{-it\hat{H}}$. Then
$\rho^{I}(t)$ satisfies,

$\frac{\partial}{\partial
t}\rho^{I}(t)=\left[\hat{H}^{I}_{{\textup{c}}\textup{\scriptsize{k}}}(t),\rho^{I}(t)\right],$
(25)

where
$\hat{H}_{{\textup{c}}\textup{\scriptsize{k}}}^{I}(t):=e^{i\hat{H}_{0}t}\hat{H}_{{\textup{c}}\textup{\scriptsize{k}}}e^{-i\hat{H}_{0}t}$
is the interaction Hamiltonian in the interaction picture. A solution to this
differential equation can be obtained using the Dyson series. Neglecting terms
of order $\mathcal{O}(1/c^{4})$, as in Appendix A, it yields

$\rho^{I}(t)=\rho(0)-i\,\int_{0}^{t}dt^{\prime}[\hat{H}_{{\textup{c}}\textup{\scriptsize{k}}}^{I}(t^{\prime}),\rho(0)].$
(26)

Thus going back into the Schrödinger picture,

$\rho(t)=e^{-i\hat{H}_{0}t}\rho(0)e^{i\hat{H}_{0}t}-i\,e^{-i\hat{H}_{0}t}\left\\{\int_{0}^{t}dt^{\prime}[\hat{H}_{{\textup{c}}\textup{\scriptsize{k}}}^{I}(t^{\prime}),\rho(0)]\right\\}e^{i\hat{H}_{0}t}.$
(27)

Tracing over the kinematic space $\mathcal{H}_{\textup{\scriptsize{k}}}$, we
find the reduced state corresponding to the temporal degree of freedom
$\rho_{\textup{c}}(t):=\mathrm{tr}_{\textup{\scriptsize{k}}}[\rho(t)]$, which
we write as

$\displaystyle\rho_{\textup{c}}(t)=\rho_{\textup{c},\textup{\tiny{NR}}}(t)+\rho_{\textup{c}}^{(1)}(t),$
(28)

where
$\rho_{\textup{c},\textup{\tiny{NR}}}(t):=e^{-i\hat{H}_{\textup{c}}t}\rho_{\textup{c}}(0)\,e^{i\hat{H}_{\textup{c}}t}$
is the non-relativistic evolution of the reduced state, and the lowest-order
relativistic correction to it is given by

$\displaystyle\rho_{\textup{c}}^{(1)}(t)=-i\int_{0}^{t}dt^{\prime}\mathrm{tr}_{\textup{\scriptsize{k}}}\big{[}e^{-i\hat{H}_{0}t}[\hat{H}_{{\textup{c}}\textup{\scriptsize{k}}}^{I}(t^{\prime}),\rho(0)]e^{i\hat{H}_{0}t}\big{]},$
(29)

Now, we consider a clock whose temporal and kinematic degrees of freedom are
initially uncorrelated, writing

$\displaystyle\rho(0)=\rho_{\textup{c}}(0)\otimes\rho_{\textup{\scriptsize{k}}}(0),$
(30)

for some initial reduced states $\rho_{\textup{c}}(0)$ and
$\rho_{\textup{\scriptsize{k}}}(0)$. Then, writing
$\hat{H}_{{\textup{c}}\textup{\scriptsize{k}}}=\hat{H}_{\textup{c}}\otimes\hat{V}_{\textup{\scriptsize{k}}}$,
i.e.
$\hat{V}_{\textup{\scriptsize{k}}}:=-\frac{\hat{p}^{2}}{2m^{2}c^{2}}+\frac{\Phi(\hat{x})}{c^{2}}$,
we calculate the integrand of Eq. (29) to find

$\mathrm{tr}_{\textup{\scriptsize{k}}}\big{[}e^{-i\hat{H}_{0}t}[\hat{H}_{{\textup{c}}\textup{\scriptsize{k}}}^{I}(t^{\prime}),\rho(0)]e^{i\hat{H}_{0}t}\big{]}=\left[\hat{H}_{\textup{c}},\rho_{\textup{c},\textup{\tiny{NR}}}(t)\right]\,\mathrm{tr}\left[e^{i\hat{H}_{\textup{\scriptsize{k}}}t}\hat{V}_{\textup{\scriptsize{k}}}e^{-i\hat{H}_{\textup{\scriptsize{k}}}t}\rho_{\textup{\scriptsize{k}}}(0)\right].$
(31)

Using the relation

$e^{\hat{X}}Ye^{-\hat{X}}=\hat{Y}+\left[\hat{X},\hat{Y}\right]+{\frac{1}{2!}}[\hat{X},[\hat{X},\hat{Y}]]+{\frac{1}{3!}}[\hat{X},[\hat{X},[\hat{X},\hat{Y}]]]+\cdots$
(32)

for some $\hat{X}$ and $\hat{Y}$ (see Prop. 3.35 in [53]), one finds, by
neglecting terms of order $1/c^{4}$ and higher, that

$e^{i\hat{H}_{\textup{\scriptsize{k}}}t}\hat{V}_{\textup{\scriptsize{k}}}e^{-i\hat{H}_{\textup{\scriptsize{k}}}t}=\frac{1}{c^{2}}\left(-\frac{\hat{p}^{2}}{2m^{2}}+g\hat{x}+\frac{2gt}{m}\hat{p}-g^{2}t^{2}\right).$
(33)

Performing the integration in Eq. (29), we find

$\rho_{\textup{c}}^{(1)}(t)=-i\left[\hat{H}_{\textup{c}},\rho_{\textup{c},\textup{\tiny{NR}}}(t)\right]\,tR(t),$
(34)

where
$R(t):=\mathrm{tr}\left[\left(-\frac{\hat{p}^{2}}{2m^{2}c^{2}}+\frac{g\hat{x}}{c^{2}}+\frac{\hat{p}gt}{mc^{2}}-\frac{g^{2}t^{2}}{3c^{2}}\right)\rho_{\textup{\scriptsize{k}}}(0)\right]$.
Having obtained the evolution of the clock’s temporal state, one can calculate
the average clock time, giving

$\langle\hat{T}_{\textup{c}}\rangle(t)=\langle\hat{T}_{\textup{c}}\rangle_{\textup{\tiny{NR}}}(t)+tR(t)\left\\{1+\mathrm{tr}\big{[}\hat{E}(t)\big{]}\right\\}$
(35)

where
$\langle\hat{T}_{\textup{c}}\rangle_{\textup{\tiny{NR}}}:=\mathrm{tr}\big{[}\hat{T}_{\textup{c}}\,\rho_{\textup{c},\textup{\tiny{NR}}}(t)\big{]}$
is the clock time in the absence of relativistic effects, and where the error
operator
$\hat{E}(t):=-i[\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}]\rho_{\textup{c},\textup{\tiny{NR}}}(t)-\rho_{\textup{c},\textup{\tiny{NR}}}(t)$
was introduced in Sec. 2 of the main text.

##  Appendix C Idealised clocks, covariant POVMs and higher moments (in the
absence of relativistic effects)

In the main text, we defined an idealised clock as one whose first-moment
operator satisfies the canonical commutation relations
$\big{[}\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}\big{]}=i\,\mathbbm{1}_{\textup{c}}$
(in the Heisenberg form), and used this to motivate the definition of the
error operator $\hat{E}(t)$ in Eq. (1). We now find covariant POVMs
$\\{\hat{F}(s)\\}_{s\in\mathcal{S}}$ which are absolutely continuous, that
either form an idealised clock, or whose deviation from this condition is
covered by a particular form of the error operator $\hat{E}(t)$. In doing so,
we show that our choice of the definition of the error operator is well
motivated. We will look at both cases in which $\mathcal{S}$ is bounded or
unbounded.

We start with the case that $\mathcal{S}$ is bounded, and denote the
boundaries by $\mathcal{S}=[s_{0},s_{1}]$. We expand the domain of $\hat{F}$
to ${\mathbbm{R}}$ by periodic extension:
$\hat{F}(s+n(s_{1}-s_{0})):=\hat{F}(s)$, $s\in\mathcal{S}$,
$n\in{\mathbbm{Z}}$.

To start with, we will only need the covariance property of the POVMs and the
assumption that $\mathcal{S}$ is bounded. Writing the evolution of
$\hat{T}_{\textup{c}}$ in the Heisenberg picture, and using the covariant
property $\hat{F}(s)=U_{\textup{c}}(s)\hat{F}(0)U^{\dagger}_{\textup{c}}(s)$;
$U_{\textup{c}}(s):=e^{-i\hat{H}_{\textup{c}}s}$, we find for all
$t\in{\mathbbm{R}}$:

$\displaystyle\hat{T}_{\textup{c}}(t)$
$\displaystyle=U_{\textup{c}}^{\dagger}(t)\hat{T}_{\textup{c}}U_{\textup{c}}(t)=\int_{\mathcal{S}}ds\,s\,\hat{F}(s-t)=\int_{s_{0}}^{s_{1}}ds\,f(s)\hat{F}(s-t)$
(36)
$\displaystyle=\int_{s_{0}-t}^{s_{1}-t}ds\,f(t+s)\hat{F}(s)=\int_{s_{0}}^{s_{1}}ds\,f(t+s)\hat{F}(s),$

where $f(s)$ is the sawtooth function: $f(s+n(s_{1}-s_{0})):=s$,
$s\in\mathcal{S}$, $n\in{\mathbbm{Z}}$. By taking limits
$s_{0}\rightarrow-\infty$ and/or $s_{1}\rightarrow+\infty$ in Eq. (36) and
noting that POVMs form a resolution of the identity,
$\int_{s\in\mathcal{S}}\hat{F}(s)=\mathbbm{1}_{\textup{c}}$; we find

$\displaystyle\hat{T}_{\textup{c}}(t)=t\,\mathbbm{1}_{\textup{c}}+\hat{T}_{\textup{c}},$
(37)

for the three unbounded cases:

  * 1)

$\mathcal{S}=[s_{0},\infty)$ and $t\geq 0$. 2) $\mathcal{S}=(-\infty,s_{1}]$
and $t\leq 0$. 3) $\mathcal{S}=(-\infty,\infty)$ and $t\in{\mathbbm{R}}$.

On the other hand, the Heisenberg-picture equation of motion is
$\frac{d}{dt}\hat{T}_{\textup{c}}(t)=-i\,U_{\textup{c}}^{\dagger}(t)\big{[}\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}\big{]}U_{\textup{c}}(t).$
Thus, equating this with the time derivative of
$\hat{T}_{\textup{c}}(t)=t\,\mathbbm{1}_{\textup{c}}+\hat{T}_{\textup{c}},$ we
arrive at

$\displaystyle\left[\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}\right]=i\,\mathbbm{1}_{\textup{c}}$
(38)

for cases 1), 2) and 3) above. Eq. (38) represents the canonical commutation
relations of a specialised Heisenberg form called the Weyl form.

We now investigate the case in which $\mathcal{S}$ is bounded. From Eq. (36)
we find

$\displaystyle\hat{T}_{\textup{c}}(t)$
$\displaystyle=\int_{s_{0}}^{s_{1}-s_{0}-t}ds\,(t+s)\hat{F}(s)+\int_{s_{1}-s_{2}-t}^{s_{1}}ds\,\left(t+s-(s_{1}-s_{0})\right)\hat{F}(s)$
(39)
$\displaystyle=t\,\mathbbm{1}_{\textup{c}}+\hat{T}_{\textup{c}}+(s_{1}-s_{0})\int_{s_{1}-s_{0}-t}^{s_{1}}ds\hat{F}(s).$

Differentiating under the integral sign, we find
$\frac{d}{dt}\hat{T}_{\textup{c}}(t)=\mathbbm{1}_{\textup{c}}+(s_{0}-s_{1})\hat{F}(s_{1}-s_{0}-t)=\mathbbm{1}_{\textup{c}}+(s_{0}-s_{1})U_{\textup{c}}^{\dagger}(t)\hat{F}(0)U_{\textup{c}}(t).$
Hence taking into account Heisenberg-picture equation of motion as before, we
obtain

$\displaystyle\left[\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}\right]=i\,\mathbbm{1}_{\textup{c}}+(s_{0}-s_{1})\hat{F}(0).$
(40)

The family of quantum states for which the commutator
$\big{[}\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}\big{]}$ is well defined, is
the set
$\mathcal{S}_{\textup{Comm}}:=\big{\\{}\rho\in\mathcal{D}(\mathcal{H_{\textup{c}}})\,\big{|}\,\,\hat{T}_{\textup{c}}\rho\in\textup{Dom}(\hat{H}_{\textup{c}})\,\,\&\,\,\hat{H}_{\textup{c}}\rho\in\textup{Dom}(\hat{T}_{\textup{c}})\big{\\}}$,
where $\mathcal{D}(\mathcal{H_{\textup{c}}})$ is the set of density operators
on $\mathcal{H}_{\textup{c}}$ and Dom is the domain of the self-adjoint
operator in question. Thus Eq. (40) satisfies the canonical commutation
relations on the subset of states
$\mathcal{C}:=\\{\rho\in\mathcal{S}_{\textup{comm}}\,|\,\hat{F}(0)\rho=0\\}\subseteq\mathcal{S}_{\textup{comm}}$.
If $\mathcal{C}$ is a dense subset of $\mathcal{S}_{\textup{comm}}$, one says
that the pair $\hat{T}_{\textup{c}}$ and $\hat{H}_{\textup{c}}$ satisfy the
Heisenberg form of the canonical commutation relation. The special case of
$\mathcal{S}$ bounded and POVMs which are projective values measures, was
studied in [21], where it was shown that $\mathcal{C}$ is indeed a dense
subset of $\mathcal{S}_{\textup{comm}}$. Another example of a Heisenberg form
of the canonical commutation relations is a “particle in a box” which is also
reviewed in [21]. These (and other) aspects of covariant POVMs are discussed
in detail in [54]. Moreover, our definition of an error operator $\hat{E}(t)$
in Eq. (1) readily accommodates the situation in Eq. (40). In particular, we
have $\hat{E}(t)=-i(s_{0}-s_{1})\hat{F}(0)\rho(t)$.

We now move on to examine higher moments of covariant POVMs. Assuming all
moments up to and including the $n$th moment are well defined, we have

$\displaystyle\langle\hat{T}_{\textup{c}}^{(n)}\rangle(t)$
$\displaystyle:=\mathrm{tr}\left[\hat{T}_{\textup{c}}^{(n)}\rho_{\textup{c}}(t)\right]=\mathrm{tr}\left[U_{\textup{c}}^{\dagger}(t)\hat{T}_{\textup{c}}^{(n)}U_{\textup{c}}(t)\rho_{\textup{c}}(0)\right]$
(41)
$\displaystyle=\mathrm{tr}\left[\int_{\mathcal{S}}ds\,f(s)^{n}\,U_{\textup{c}}(s-t)\hat{F}(0)U_{\textup{c}}^{\dagger}(s-t)\rho_{\textup{c}}(0)\right]$
$\displaystyle=\mathrm{tr}\left[\int_{\mathcal{S}}ds^{\prime}\,f(s^{\prime}+t)^{n}\,U_{\textup{c}}(s^{\prime})\hat{F}(0)U_{\textup{c}}^{\dagger}(s^{\prime})\rho_{\textup{c}}(0)\right]=\sum_{k=0}^{n}\binom{n}{k}t^{n-k}\langle\hat{T}_{\textup{c}}^{(k)}\rangle(0),$

where $\langle\hat{T}_{\textup{c}}^{(0)}\rangle(0)=1$ for all initial states.
From this Eq., it follows, for example, that the standard deviation of
$\hat{T}_{\textup{c}}$ is time independent and only depends in the initial
state $\rho_{\textup{c}}$.

For an idealised clock, a projection-valued measure is an example of a
covariant POVM. To see this, observe that one can write in this case
$\hat{F}(s)=\ket{s}\\!\\!\bra{s}_{\textup{c}}$, with
$\hat{T}_{\textup{c}}^{(1)}\ket{s}=s\ket{s}$. From there, one can use the fact
that an idealised clock satisfies
$U_{\textup{c}}^{\dagger}(s)\hat{T}_{\textup{c}}^{(1)}U_{\textup{c}}(s)=T_{\textup{c}}^{(1)}+s$
to show that $U_{\textup{c}}(s^{\prime})\ket{s}=\ket{s+s^{\prime}}$, and
therefore $U_{\textup{c}}(s)\hat{F}(0)U_{\textup{c}}^{\dagger}(s)=\hat{F}(s)$,
i.e. the covariance property. If, in addition, the possible clock times span
the entire real line (as in the case considered by Pauli [5]), then the higher
moments satisfy Eq. (41), and again the variance is therefore time-
independent, regardless of the initial clock state.

##  Appendix D Examples of relativistic time dilation in quantum clocks

###  D.1 An idealised quantum clock

Recall that an idealised clock is one satisfying the canonical commutation
relation, $[\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}]=i$, and consequently
$\hat{E}(t)=0$. Let
$\hat{T}_{\textup{c}}=\hat{x}_{\textup{c}}/c+a\mathbbm{1}_{\textup{c}}$ and
$\hat{H}_{\textup{c}}=c\hat{p}_{\textup{c}}$, where $\hat{x}_{\textup{c}}$ and
$\hat{p}_{\textup{c}}$ are position and momentum operators and
$a\in{\mathbbm{R}}$ is chosen such that
$\langle\hat{T}_{\textup{c}}\rangle(0)=0$. We first consider the case of
“classical” motion, i.e. where the initial kinematic state of the clock is a
pure Gaussian wavepacket with mean momentum $\bar{p}_{0}$, standard deviation
of the momentum $\sigma_{p}$ and mean position $\bar{x}_{0}$, i.e.
$\rho_{\textup{\scriptsize{k}}}(0)=\ket{\psi}\\!\\!\bra{\psi}_{\textup{\scriptsize{k}}}$
with

$\displaystyle\ket{\psi}_{\textup{\scriptsize{k}}}=\int
dp\;\psi(p)\ket{p}_{\textup{\scriptsize{k}}};\qquad\psi(p):=\frac{1}{(2\pi\sigma_{p}^{2})^{1/4}}e^{-\left(\frac{p-\bar{p}_{0}}{2\sigma_{p}}\right)^{2}}e^{-i\bar{x}_{0}(p-\bar{p}_{0})}.$
(42)

We then have that
${R}(t)=-\frac{\bar{p}_{0}^{2}+\sigma_{p}^{2}}{2m^{2}c^{2}}+\frac{g\bar{x}_{0}}{c^{2}}+\frac{\bar{p}_{0}gt}{mc^{2}}-\frac{1}{3}\left(\frac{gt}{c}\right)^{2}$
and therefore

$\langle\hat{T}_{\textup{c}}\rangle(t)=\left[1-\frac{\bar{p}_{0}^{2}+\sigma_{p}^{2}}{2m^{2}c^{2}}+\frac{g\bar{x}_{0}}{c^{2}}+\frac{\bar{p}_{0}gt}{mc^{2}}-\frac{1}{3}\left(\frac{gt}{c}\right)^{2}\right]t.$
(43)

This expression is the average proper time of a classical clock (see Eq. (4)
in the main text) with position $\bar{x}_{0}$ and a random momentum following
a Gaussian distribution with mean $\bar{p}_{0}$ and standard deviation
$\sigma_{p}$. To summarise, an idealised clock in a classical (i.e. Gaussian)
state of motion experiences the usual classical relativistic time dilation.

Let us now consider a nonclassical initial kinematic state, specifically a
superposition of two Gaussian states, sometimes referred to as a _cat_ state.
We will contrast this with the case of comparable classical probabilistic
mixture of the two states. Let

$\displaystyle\ket{\psi_{j}}_{\textup{\scriptsize{k}}}:=\int
dp\;\psi_{j}(p)\ket{p}_{\textup{\scriptsize{k}}};\qquad\psi_{j}(p):=\frac{1}{(2\pi\sigma_{p}^{2})^{1/4}}e^{-\left(\frac{p-\bar{p}}{2\sigma_{p}}\right)^{2}}e^{-i\bar{x}_{j}(p-\bar{p})};\qquad
j\in\\{1,2\\}.$ (44)

We now take the initial kinematic state to be in the superposition
$\ket{\psi}_{\textup{\scriptsize{k}}}=\frac{1}{\sqrt{N}}\big{(}\sqrt{\alpha}\ket{\psi_{1}}_{\textup{\scriptsize{k}}}+e^{i\theta}\sqrt{1-\alpha}\ket{\psi_{2}}_{\textup{\scriptsize{k}}}\big{)}$
for some $\alpha\in(0,1)$ and $\theta\in{\mathbbm{R}}$, and find the average
clock time, which we denote
$\langle\hat{T}_{\textup{c}}\rangle_{\textup{sup}}$, using Eq. (35). We then
repeat this procedure for the initial kinematic state in the probabilistic
mixture,
$\rho_{\textup{\scriptsize{k}}}(0)=\alpha\ket{\psi_{1}}\\!\\!\bra{\psi_{1}}+(1-\alpha)\ket{\psi_{2}}\\!\\!\bra{\psi_{2}}$,
and denote the average clock time in that case by
$\langle\hat{T}_{\textup{c}}\rangle_{\textup{mix}}(t)$. From the linearity of
the trace, we see immediately that

$\displaystyle\langle\hat{T}_{\textup{c}}\rangle_{\textup{mix}}(t)=\alpha\langle\hat{T}_{\textup{c}}\rangle_{\psi_{1}}(t)+(1-\alpha)\langle\hat{T}_{\textup{c}}\rangle_{\psi_{2}}(t),$
(45)

where $\langle\hat{T}_{\textup{c}}\rangle_{\psi_{j}}(t)$ is the average clock
time of a clock with initial kinematic state
$\ket{\psi_{j}}_{\textup{\scriptsize{k}}}$. One then finds that

$\displaystyle\langle\hat{T}_{\textup{c}}\rangle_{\textup{sup}}(t)=\langle\hat{T}_{\textup{c}}\rangle_{\textup{mix}}(t)+T_{\textup{coh}}(t),$
(46)

where

$T_{\textup{coh}}(t):=\frac{N-1}{N}\left[\left(\frac{\Delta
x_{0}}{2\sigma_{x}}\right)^{2}\frac{\sigma_{v}^{2}}{c^{2}}-\frac{g\Delta
x_{0}}{c^{2}}(1-2\alpha)-2\frac{\sigma_{v}^{2}}{c^{2}}\Delta
x_{0}(\overline{p}-mgt)\tan\theta\right]\frac{t}{2}$ (47)

is a contribution to the clock time arising due to interference between the
two Gaussian states constituting the coherent superposition (see main text),
and where ${\sigma_{v}=\sigma_{p}/m}$ and $\sigma_{x}=1/2\sigma_{p}$ are
respectively the standard deviation of the velocity and position of the
Gaussian wavepackets used to construct the initial state, and $\Delta
x_{0}=\bar{x}_{2}-\bar{x}_{1}$. We recall that $\hbar=1$, and note that the
normalisation factor $N$ is given by

$N=1+2\sqrt{\alpha(1-\alpha)}e^{-\frac{1}{2}\left(\frac{\Delta
x_{0}}{2\sigma_{x}}\right)^{2}}\cos\theta.$ (48)

Setting $\theta=0$ in Eq. (47), one arrives at Eq. (12) in the main text.

###  D.2 Salecker-Wigner-Peres clock

Consider a $d$-dimensional Hilbert space and a clock Hamiltonian with equally-
spaced energy levels, i.e.
${\hat{H}_{\textup{c}}=\sum_{j=0}^{d-1}j\omega\ket{e_{j}}\\!\\!\bra{e_{j}}}$,
for some $\omega>0$. This corresponds, for example, to the case where the
clock is a spin-$j$ system, and then $d=2j+1$. One can obtain a so-called
_time basis_ $\\{\ket{\theta_{m}}\\}$ by taking the discrete Fourier transform
of the energy eigenstates,

$\ket{\theta_{m}}=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}e^{-i2\pi
jm/d}\ket{e_{j}},\quad m=0,1,\ldots,d-1$ (49)

which are used to construct a clock-time operator
$\hat{T}_{\textup{c}}=\sum_{m=0}^{d-1}m\frac{T_{0}}{d}\ket{\theta_{m}}\\!\\!\bra{\theta_{m}}$,
with $T_{0}:=\frac{2\pi}{\omega}$. When the clock is in the state
$\ket{\theta_{m}}$, we have $\langle\hat{T}_{\textup{c}}\rangle=mT_{0}/d$.
Choosing an initial state
$\rho_{\textup{c}}(0)=\ket{\theta_{0}}\\!\\!\bra{\theta_{0}}$, then the clock
state will at regular time intervals $t_{m}:=mT_{0}/d$, $m\in{\mathbbm{Z}}$
“focus” into one of the time eigenstates, i.e.
$\langle\hat{T}_{\textup{c}}\rangle_{\textup{\tiny{NR}}}\left(t_{m}\right)=mT_{0}/d\,(m\text{
mod }d)$. At intermediate times $t\neq t_{m}$, on the other hand, the clock
will be in a superposition of its time eigenstates (see e.g. [6], Appendix B).
This setup is known as the Salecker-Wigner-Peres clock (or alternatively, the
Larmor clock) [12, 24].

A straightforward calculation gives
$\mathrm{tr}[[\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}]\ket{\theta_{m}}\\!\\!\bra{\theta_{m}}]=0$
for all $m=0,1,\ldots,d-1$, and therefore from the definition of the error
operator $\hat{E}(t)$, we have $\mathrm{tr}[\hat{E}(t_{m})]=-1$ for all
$m\in{\mathbbm{Z}}$ and for all clock dimensions $d\in{\mathbbm{N}}^{+}$.
Consequently, when the Salecker-Wigner-Peres clock is in a time eigenstate,
Eq. (35) tells us that
$\langle\hat{T}_{\textup{c}}\rangle(t_{m})=\langle\hat{T}_{\textup{c}}\rangle_{\textup{\tiny{NR}}}(t_{m})$
for _any_ state of the clock’s kinematic degrees of freedom. To reiterate,
this clock is a special case where the average relativistic time-dilation is
exactly cancelled by the quantum error of the clock at regular time intervals
regardless of how large $d$ is and regardless of the motional state. In this
regard, we note that the Salecker-Wigner-Peres clock was used in [50], where
it was concluded that “quantum clocks do not witness classical time dilation”.

###  D.3 The Quasi-Ideal clock

Consider the same setting as for the Salecker-Wigner-Peres clock described
above, namely the clock Hilbert space, Hamiltonian $\hat{H}_{\textup{c}}$ and
clock-time operator $\hat{T}_{\textup{c}}$, but now with the initial temporal
state $\rho_{\textup{c}}(0)=\ket{\Psi(m_{0})}\\!\\!\bra{\Psi(m_{0})}$, with

$\ket{\Psi(m_{0})}=\sum_{m=0}^{d-1}g_{m_{0}}(m)\ket{\theta_{m}},$ (50)

where $g_{m_{0}}(m)$ is a complex Gaussian distribution centred on $m_{0}$ and
with standard deviation $\bar{\sigma}_{\textup{c}}\in(0,d)$. The choice
$\bar{\sigma}_{\textup{c}}=\sqrt{d}$ corresponds to the case in which the
standard deviation in both the time basis $\\{\ket{\theta_{m}}\\}$ and energy
basis time basis $\\{\ket{e_{j}}\\}$ is approximately the same. For other
choices, one obtains a clock whose behaviour tends to that of the idealised
clock when $\bar{\sigma}_{\textup{c}}\to\infty$ and
$d/\bar{\sigma}_{\textup{c}}\to\infty$ as $d\to\infty$. This is then the
_Quasi-Ideal_ clock, introduced in [6]. We now show that the contribution of
the quantum error to the relativistic time dilation, i.e.
$\mathrm{tr}[\hat{E}(t)]$ (see Eq. (35)), tends exponentially to zero with
increasing clock dimensionality. In other words, we show that the relativistic
time dilation experienced by the idealised clock can be approximated
arbitrarily well by a Quasi-Ideal clock of high enough dimension. Physically
speaking, this is because the Quasi-Ideal clock can move at constant
“velocity” in the time basis with minimal dispersion.

From the definition of the error operator $\hat{E}(t)$, we have

$\mathrm{tr}[\hat{E}(t)]=-i\mathrm{tr}\big{[}[\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}]\rho_{\textup{c},\textup{\tiny{NR}}}(t)\big{]}-1$
(51)

Choosing the initial clock state $\ket{\Psi(0)}\\!\\!\bra{\Psi(0)}$, the non-
relativistic evolution of the state $\rho_{\textup{c},\textup{\tiny{NR}}}(t)$
is given by,

$\displaystyle\rho_{\textup{c},\textup{\tiny{NR}}}(t)=e^{-i\hat{H}_{\textup{c}}t}\ket{\Psi(0)}\\!\\!\bra{\Psi(0)}e^{i\hat{H}_{\textup{c}}t},$
(52)

Now, Theorem 8.1 in [6] states that

$\displaystyle
e^{-i\hat{H}_{\textup{c}}t}\ket{\Psi(0)}=\ket{\Psi(t/T_{0})}+\ket{\varepsilon},$
(53)

where the specific form of $\ket{\varepsilon}$ is irrelevant for the present
purpose, only that as $d\to\infty$ and $d/\bar{\sigma}_{\textup{c}}\to\infty$,

$\displaystyle\sqrt{\braket{\varepsilon}{\varepsilon}}=\mathcal{O}\Big{(}\text{poly}(d)\,\big{(}e^{-\frac{\pi}{4}\frac{d^{2}}{{\bar{\sigma}_{\textup{c}}}^{2}}}+e^{-\frac{\pi}{4}{\bar{\sigma}_{\textup{c}}}^{2}}\big{)}\Big{)},$
(54)

and furthermore Theorem 11.1 in [6] states that

$\displaystyle[\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}]\ket{\Psi(m)}=i\ket{\Psi(m)}+\ket{\varepsilon_{comm}}\quad\forall\,m\in{\mathbbm{R}}$
(55)

where, again, the specific form of $\ket{\varepsilon_{comm}}$ is irrelevant,
only that as $d\to\infty$,

$\displaystyle\sqrt{\braket{\varepsilon_{comm}}{\varepsilon_{comm}}}=\mathcal{O}\Big{(}\text{poly}(d)\,\big{(}e^{-\frac{\pi}{4}\frac{d^{2}}{{\bar{\sigma}_{\textup{c}}}^{2}}}+e^{-\frac{\pi}{4}{\bar{\sigma}_{\textup{c}}}^{2}}\big{)}\Big{)}.$
(56)

Using these two theorems, one finds that

$\displaystyle\left|\mathrm{tr}[\hat{E}(t)]\right|$
$\displaystyle=\left|-i\mathrm{tr}\bigg{[}\ket{\varepsilon_{comm}}\left\langle{\Psi(t/T_{0})}\right|+[\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}]\Big{(}\ket{\varepsilon}\\!\\!\bra{\Psi(t/T_{0})}+\ket{\Psi(t/T_{0})}\\!\\!\bra{\varepsilon}+\ket{\varepsilon}\\!\\!\bra{\varepsilon}\Big{)}\bigg{]}\,\right|$
(57)
$\displaystyle\leq\left|\mathrm{tr}\Big{[}\ket{\varepsilon_{comm}}\left\langle{\Psi(t/T_{0})}\right|\Big{]}\,\right|+\left|\mathrm{tr}\Big{[}[\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}]\Big{(}\ket{\varepsilon}\\!\\!\bra{\Psi(t/T_{0})}+\ket{\Psi(t/T_{0})}\\!\\!\bra{\varepsilon}+\ket{\varepsilon}\\!\\!\bra{\varepsilon}\Big{)}\Big{]}\,\right|$
(58)

We can now bound the terms on the r.h.s. of Eq. (58). To start with

$\displaystyle\big{|}\mathrm{tr}[\ket{\varepsilon_{comm}}\\!\\!\bra{\Psi(t/T_{0})}]\,\big{|}$
$\displaystyle=\big{|}\braket{\varepsilon_{comm}}{\Psi(t/T_{0})}\big{|}\leq\sqrt{\braket{\varepsilon_{comm}}{\varepsilon_{comm}}}$
(59)
$\displaystyle=\mathcal{O}\Big{(}\text{poly}(d)\,\big{(}e^{-\frac{\pi}{4}\frac{d^{2}}{{\bar{\sigma}_{\textup{c}}}^{2}}}+e^{-\frac{\pi}{4}{\bar{\sigma}_{\textup{c}}}^{2}}\big{)}\Big{)}$

Using the Cauchy-Schwarz inequality,
$\left|\mathrm{tr}[\hat{A}\hat{B}]\right|^{2}\leq\mathrm{tr}[\hat{A}^{\dagger}\hat{A}]\mathrm{tr}[B^{\dagger}B]$
for bounded linear operators $\hat{A}$, $\hat{B}$; one finds that for two kets
$\ket{a}$, $\ket{b}$

$\displaystyle\Big{|}\mathrm{tr}\left[[\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}]\ket{a}\\!\\!\bra{b}\right]\Big{|}^{2}$
$\displaystyle\leq-\mathrm{tr}\left[[\hat{T}_{\textup{c}},\hat{H}_{\textup{c}}]^{2}\right]\braket{a}{a}\braket{b}{b}$
(60) $\displaystyle\leq
2\Big{(}\big{|}\mathrm{tr}\left[(\hat{T}_{\textup{c}}\hat{H}_{\textup{c}})^{2}\right]\big{|}+\big{|}\mathrm{tr}\left[\hat{T}_{\textup{c}}^{2}\hat{H}_{\textup{c}}^{2}\right]\big{|}\Big{)}\braket{a}{a}\braket{b}{b}$
(61) $\displaystyle\leq
2\Big{(}\left\|(\hat{T}_{\textup{c}}\hat{H}_{\textup{c}})^{2}\right\|+\left\|\hat{T}_{\textup{c}}^{2}\hat{H}_{\textup{c}}^{2}\right\|\Big{)}\braket{a}{a}\braket{b}{b}$
(62) $\displaystyle\leq
4\left\|\hat{T}_{\textup{c}}\right\|^{2}\left\|\hat{H}_{\textup{c}}\right\|^{2}\braket{a}{a}\braket{b}{b}\leq
4(T_{0}d)^{2}(d^{2}\omega)^{2}\braket{a}{a}\braket{b}{b}$ (63)
$\displaystyle\leq 16\pi^{2}d^{6}\braket{a}{a}\braket{b}{b}.$ (64)

Therefore, using Eqs. (59), (64), and (54); one finds from Eq. (58)

$\left|\mathrm{tr}[\hat{E}(t)]\right|=\mathcal{O}\Big{(}\text{poly}(d)\,\big{(}e^{-\frac{\pi}{4}\frac{d^{2}}{{\bar{\sigma}_{\textup{c}}}^{2}}}+e^{-\frac{\pi}{4}{\bar{\sigma}_{\textup{c}}}^{2}}\big{)}\Big{)}$
(65)

for all $t\in{\mathbbm{R}}$. We therefore find that for any initial clock
width of the form $\bar{\sigma}_{\textup{c}}=d_{\textup{c}}^{\eta}$ for some
fixed $\eta>0$, $\mathrm{tr}[\hat{E}(t)]$ is upper-bounded by a quantity which
decreases exponentially with dimension. Furthermore, note that for
$0<\eta<1/2$, the standard deviation of $\hat{T}_{\textup{c}}$ for the initial
Quasi-Ideal clock state approaches zero as $d\to\infty$. Thus in the limit of
large dimension, the relativistic time dilation (Eq. (35)) is that of the
idealised clock, up to errors that decay exponentially in the dimension.

###  D.4 The qubit-phase clock

Another clock model sometimes found in the literature (e.g. [19, 49, 16]) uses
the phase difference between the two states of a qubit system to estimate
elapsed time. This can be considered a simplified model of the Ramsey method
sometimes used in atomic clocks (see e.g. [32] Sec. IVA for a discussion of
clock interrogation schemes), where it is assumed that the laser is perfectly
tuned to the reference frequency. The qubit is put into the initial state
$\frac{1}{\sqrt{2}}\left(\ket{0}+\ket{1}\right)$, whose non-relativistic
evolution according to the Hamiltonian
$\hat{H}_{\textup{c}}=-\frac{\omega}{2}\ket{0}\\!\\!\bra{0}+\frac{\omega}{2}\ket{1}\\!\\!\bra{1}$
results in the state

$\ket{\theta}:=\frac{1}{\sqrt{2}}\left(\ket{0}+e^{-i\theta}\ket{1}\right)$
(66)

where $\theta=\omega t$. The covariant (see Sec. 2 and Appendix C) POVM
corresponding to the estimation of this phase is given by
$\\{\hat{F}(\theta)\\}_{\theta\in[0,2\pi)}$ with
$F(\theta)=\frac{1}{\pi}\ket{\theta}\\!\\!\bra{\theta}$. This measurement
corresponds to a maximum-likelihood estimation of $\theta$, and therefore the
laboratory time (see e.g. [55], Sec. 4.4). One can then directly calculate the
error operator:

$\hat{E}(t)=-(\mathbbm{1}_{\textup{c}}+\sigma_{x})\rho_{\textup{c}}(t).$ (67)

Since $\mathrm{tr}[\hat{E}(t)]=1+\cos(\omega t)$, we see that this is only a
good clock in the sense defined in Sec. 2 around $\theta=\pi$, which
corresponds to the time at which the clock state $\ket{\theta}$ is orthogonal
to the initial state. This limitation can be overcome by performing the phase
estimation on many copies of the qubit system (see e.g. [56]). Now, carrying
this analysis into a relativistic context, Eq. (6) gives the average time
dilation experienced by the qubit-phase clock as

$\langle\hat{T}_{\textup{c}}\rangle(t)=\langle\hat{T}_{\textup{c}}\rangle_{\textup{\tiny{NR}}}(t)+tR(t)\left[2+\cos(\omega
t)\right],$ (68)

which is of course equal to the average time dilation experienced by the
idealised clock when $\theta=\pi$.

##  Appendix E The effect of the coupling on a clock’s precision

We now show how a clock’s precision, as quantified by the standard deviation
$\sigma_{T}:=\sqrt{\langle\hat{T}_{\textup{c}}^{2}\rangle-\langle\hat{T}_{\textup{c}}\rangle^{2}}$,
changes over time as a result of the relativistic coupling between temporal
and kinematic degrees of freedom. For simplicity, we now consider flat
(Minkowski) spacetime. Here we work to a precision one order higher than
before, i.e. neglecting only $\mathcal{O}\left((\hat{p}/mc)^{6}\right)$ terms.
This is because, as we shall see, relativistic effects on the precision of the
idealised clock vanish below $\mathcal{O}(1/c^{4})$. Repeating the procedure
outlined in Appendix A, one finds that the relativistic coupling between
temporal and kinematic degrees of freedom is now

$\hat{H}_{\textup{c}\textup{\scriptsize{k}}}=\hat{H}_{\textup{c}}\otimes\hat{W}_{\textup{\scriptsize{k}}}+\mathcal{O}\left(\left(\frac{\hat{p}}{mc}\right)^{6}\right)$
(69)

with
$\hat{W}_{\textup{\scriptsize{k}}}:=-\hat{p}^{2}/2m^{2}c^{2}+3\hat{p}^{4}/8m^{4}c^{4}$.

Now, since in this case
$[\hat{H}_{\textup{c}\textup{\scriptsize{k}}},\hat{H}_{\textup{\scriptsize{k}}}]=[\hat{H}_{\textup{c}\textup{\scriptsize{k}}},\hat{H}_{\textup{c}}]=0$,
we can simply calculate the relevant observables without going into the
interaction picture. Using Eq. (32), we have

$\langle\hat{T}_{\textup{c}}^{n}\rangle(t)=\mathrm{tr}\left[\,\sum_{q=0}^{\infty}\frac{1}{q!}[i(\mathbbm{1}_{\textup{\scriptsize{k}}}+\hat{W}_{\textup{\scriptsize{k}}})t]^{q}\,[\hat{H}_{\textup{c}},\hat{T}_{\textup{c}}^{n}]_{q}\right]$
(70)

where $[\cdotp,\cdot]_{q}$ denotes the $q^{\mathrm{th}}$-order nested
commutator, i.e. $[\hat{A},\hat{B}]_{0}:=\hat{B}$ and
$[\hat{A},\hat{B}]_{q}:=[\hat{A},[\hat{A},\hat{B}]_{q-1}]$ for $q>0$. Applying
the Binomial theorem to
$(\mathbbm{1}_{\textup{\scriptsize{k}}}+\hat{W}_{\textup{\scriptsize{k}}})^{q}$
and truncating at the appropriate order, one finds, after some algebra

$\langle\hat{T}_{\textup{c}}^{n}\rangle(t)=\langle\hat{T}_{\textup{c}}^{n}\rangle_{\textup{\tiny{NR}}}(t)+\mathrm{tr}\Big{[}\left\\{it[\hat{H}_{\textup{c}},\hat{T}_{\textup{c}}^{n}]\otimes\hat{W}_{\textup{\scriptsize{k}}}-t^{2}[\hat{H}_{\textup{c}},[\hat{H}_{\textup{c}},\hat{T}_{\textup{c}}^{n}]]\otimes\hat{W}_{\textup{\scriptsize{k}}}^{2}\right\\}\rho_{\textup{\tiny{NR}}}(t)\Big{]}$
(71)

where
$\rho_{\textup{\tiny{NR}}}(t):=e^{-i(\hat{H}_{\textup{c}}+\hat{H}_{\mathrm{k}})t}\rho(0)e^{i(\hat{H}_{\textup{c}}+\hat{H}_{\mathrm{k}})t}$
is the non-relativistic evolution of the clock’s total state and
$\langle\hat{T}_{\textup{c}}^{n}\rangle_{\textup{\tiny{NR}}}(t):=\mathrm{tr}[\hat{T}_{\textup{c}}^{n}\rho_{\textup{c},\textup{\tiny{NR}}}(t)]$,
and where it is understood that $\hat{W}_{\textup{\scriptsize{k}}}^{2}$
contains an $\mathcal{O}\left((\hat{p}/mc)^{6}\right)$ term to be neglected.
Assuming an uncorrelated initial state as in Appendix B, i.e.
$\rho(0)=\rho_{\textup{c}}(0)\otimes\rho_{\textup{\scriptsize{k}}}(0)$, and
further defining
$\langle\hat{W}_{\textup{\scriptsize{k}}}^{n}\rangle_{0}:=\mathrm{tr}[\hat{W}_{\textup{\scriptsize{k}}}^{n}\rho_{\textup{\scriptsize{k}}}(0)]$,
we have

$\langle\hat{T}_{\textup{c}}^{n}\rangle(t)=\langle\hat{T}_{\textup{c}}^{n}\rangle_{\textup{\tiny{NR}}}(t)+it\langle\hat{W}_{\textup{\scriptsize{k}}}\rangle_{0}\,\mathrm{tr}\big{[}[\hat{H}_{\textup{c}},\hat{T}_{\textup{c}}^{n}]\rho_{\textup{c},\textup{\tiny{NR}}}(t)\big{]}-t^{2}\langle\hat{W}_{\textup{\scriptsize{k}}}^{2}\rangle_{0}\,\mathrm{tr}\big{[}[\hat{H}_{\textup{c}},[\hat{H}_{\textup{c}},\hat{T}_{\textup{c}}^{n}]]\rho_{\textup{c},\textup{\tiny{NR}}}(t)\big{]}.$
(72)

Then by a straightforward, if lengthy, calculation, one finally finds that
$\scalebox{1.3}{$\sigma$}_{\tiny{T}}(t)$, separates into

$\scalebox{1.3}{$\sigma$}_{\tiny{T}}(t)=\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NR}}}(t)+\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{I}}}(t)+\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NI}}}(t),$
(73)

where $\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NR}}}(t)$ is the
clock’s standard deviation in the absence of relativistic effects,
$\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{I}}}(t)$ is a contribution
which remains in the case of an idealised clock, given by

$\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{I}}}(t):=\frac{t^{2}}{8\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NR}}}(t)}\frac{\langle\hat{p}^{4}\rangle+\sigma_{p^{2}}^{2}}{m^{4}c^{4}}$
(74)

where $\sigma_{p^{2}}$ is the standard deviation of $\hat{p}^{2}$, and
$\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NI}}}(t)$ is a contribution
arising due to the non-idealised nature of the clock, given by

$\displaystyle\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NI}}}(t):=$
$\displaystyle\frac{\langle\hat{W}_{\textup{\scriptsize{k}}}\rangle_{0}\,t}{2\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NR}}}(t)}\left\\{\mathrm{tr}[(\hat{E}(t)+\hat{E}^{{\dagger}}(t))\hat{T}_{\textup{c}}]-2\langle\hat{T}_{\textup{c}}\rangle_{\textup{\tiny{NR}}}(t)\mathrm{tr}[\hat{E}(t)]\right\\}$
(75)
$\displaystyle-\frac{\left(\langle\hat{W}_{\textup{\scriptsize{k}}}\rangle_{0}\,t\right)^{2}}{8\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NR}}}(t)^{3}}\left\\{\mathrm{tr}[(\hat{E}(t)+\hat{E}^{{\dagger}}(t))\hat{T}_{\textup{c}}]-2\langle\hat{T}_{\textup{c}}\rangle_{\textup{\tiny{NR}}}(t)\mathrm{tr}[\hat{E}(t)]\right\\}^{2}$
$\displaystyle-\frac{\left(\langle\hat{W}_{\textup{\scriptsize{k}}}\rangle_{0}\,t\right)^{2}}{2\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NR}}}(t)}\left\\{2\mathrm{tr}[\hat{E}(t)]+\mathrm{tr}[\hat{E}(t)]^{2}\right\\}$
$\displaystyle-\frac{\langle\hat{W}_{\textup{\scriptsize{k}}}^{2}\rangle_{0}\,t^{2}}{2\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{NR}}}(t)}\Big{\\{}2\mathrm{tr}[\hat{E}(t)]+i\mathrm{tr}[(\hat{H}_{\textup{c}}\hat{e}\,\hat{T}_{\textup{c}}-\hat{T}_{\textup{c}}\hat{e}\,\hat{H}_{\textup{c}})\rho_{\textup{c},\textup{\tiny{NR}}}(t)+\hat{H}_{\textup{c}}\hat{T}_{\textup{c}}\hat{E}(t)-\hat{E}^{{\dagger}}(t)\hat{T}_{\textup{c}}\hat{H}_{\textup{c}}]$
$\displaystyle\qquad\qquad\qquad+2i\langle\hat{T}_{\textup{c}}\rangle_{\textup{\tiny{NR}}}(t)\mathrm{tr}[\hat{H}_{\textup{c}}(\hat{E}(t)-\hat{E}^{{\dagger}}(t))]\Big{\\}}$

where $\hat{E}(t)$ was defined above, and
$\hat{e}:=i[\hat{H}_{\textup{c}},\hat{T}_{\textup{c}}]-\mathbbm{1}_{\textup{c}}$.
Note that
$\scalebox{1.3}{$\sigma$}_{\tiny{T},\textup{\tiny{I}}}(t)=\mathcal{O}\left((\hat{p}/mc)^{4}\right)$,
which necessitated that we work to one order higher of precision than when we
calculated the clock time (Appendix B).