Ladina Hausmann

Alexander Schmidhuber

Esteban Castro-Ruiz

Measurement events relative to temporal quantum reference frames

The Page-Wootters formalism is a proposal for reconciling the backgrounddependent, quantum-mechanical notion of time with the background independence of general relativity. However, there has been much discussion regarding the physical meaning of the framework. In this work, we compare two consistent approaches to the Page-Wootters formalism to clarify the operational meaning of evolution and measurements with respect to a quantum temporal reference frame.

The so-called "twirled observable" approach implements measurements as operators that are invariant with respect to the Hamiltonian constraint. The "purified measurement" approach instead models measurements dynamically by modifying the constraint itself. While both approaches agree in the limit of ideal clocks, a natural generalization of the purified measurement approach to the case of non-ideal, finite-resource clocks yields a radically different picture. We discuss the physical origin of this discrepancy and argue that they describe operationally distinct situations. Moreover, we show that, for non-ideal clocks, the purified measurement approach yields time non-local, non-unitary evolution and implies a fundamental limitation to the operational definition of the temporal order of events. Nevertheless, unitarity and definite temporal order can be restored if we assume that time is discrete.

I. INTRODUCTION

In standard quantum mechanics, systems evolve via the Schrödinger equation relative to an external background time. In general relativity, on the other hand, the metric determines the reading of rods and clocks, irrespective of any external background. The contrast between these two notions of time calls for a conceptual framework in which quantum systems evolve in the absence of an external time parameter.

The Page-Wootters formalism proposes a solution to this problem. Inspired by the Wheeler-De-Witt equation [START_REF] Archibald | The "past" and the "delayed-choice" double-slit experiment[END_REF],

Ĥ |Ψ⟩ = 0, (1)
the Page-Wotters formalism [START_REF] Page | Evolution without evolution: Dynamics described by stationary observables[END_REF][START_REF] William | time" replaced by quantum correlations[END_REF] poses that time evolution emerges from the correlations between the system of interest and a temporal reference frame -a clock. The total system, arXiv:2308.10967v1 [quant-ph] 21 Aug 2023 composed by the system of interest and the frame, satisfies the so-called Hamiltonian constraint, eq. ( 1). Because |Ψ⟩ does not depend on any external background time, the approach based on eq. ( 1) is referred to as "timeless" quantum mechanics.

Although the Page-Wootters formalism reproduces time evolution in a timeless setting, the formalism has been criticized [START_REF] Kuchař | Time and Interpretations of Quantum Gravity[END_REF][START_REF] Unruh | Time and the interpretation of canonical quantum gravity[END_REF]. In particular, Kuchař noted that the postmeasurement states of the formalism do not satisfy eq. ( 1) and predict inconsistent probabilities for two time measurements [START_REF] Kuchař | Time and Interpretations of Quantum Gravity[END_REF]. A more detailed review of Kuchař's criticisms is given in appendix B.

In the current literature, there are two different ways to resolve Kuchař's criticisms.

The first route, which we call the twirled observable (TO) approach, has been taken by [START_REF] Philipp | Trinity of relational quantum dynamics[END_REF] (see also [START_REF] Carl E Dolby | The conditional probability interpretation of the hamiltonian constraint[END_REF]). It is based on time-translation invariant operators, which are inspired by Dirac quantization [START_REF] Dirac | Lectures on quantum mechanics[END_REF][START_REF] Henneaux | Quantization of Gauge Systems[END_REF] and time-reparametrization invariant observables in quantum gravity [START_REF] Bojowald | Effective approach to the problem of time: General features and examples[END_REF][START_REF] Bojowald | An effective approach to the problem of time[END_REF][START_REF] Gambini | A relational solution to the problem of time in quantum mechanics and quantum gravity: a fundamental mechanism for quantum decoherence[END_REF][START_REF] Gambini | Conditional probabilities with dirac observables and the problem of time in quantum gravity[END_REF][START_REF] Rovelli | Quantum mechanics without time: A model[END_REF][START_REF] Rovelli | Covariant loop quantum gravity[END_REF][START_REF] Rovelli | forget time[END_REF]. The second approach, which we call the purified measurement (PM) approach, was pursued by [START_REF] Giovannetti | Quantum time[END_REF]. In this approach, the idea is to modify the constraint to describe the measurement dynamically (the same idea was also pursued in [START_REF] Hellmann | Multiple-event probability in general-relativistic quantum mechanics[END_REF][START_REF] Mondragon | Multiple-event probability in general-relativistic quantum mechanics. II. a discrete model[END_REF]).

More specifically, the modified constraint couples an ancillary system to the clock and the system to be measured, and a measurement on the ancilla reveals the outcome of the measurement on the system.

Both approaches produce a theory that addresses Kuchař 's criticisms. This leads to the question: are these two proposals equivalent? The goal of this paper is to shed light on the operational interpretation of measurement events in timeless quantum mechanics by comparing both approaches.

Our analysis is motivated by recent literature on quantum reference frames transformations [START_REF] Castro-Ruiz | Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems[END_REF][START_REF] Giacomini | Quantum mechanics and the covariance of physical laws in quantum reference frames[END_REF][START_REF] Philipp | Effective relational dynamics of a nonintegrable cosmological model[END_REF][START_REF] Philipp | How to switch between relational quantum clocks[END_REF][START_REF] Philipp | Equivalence of approaches to relational quantum dynamics in relativistic settings[END_REF][START_REF] Giacomini | Spacetime quantum reference frames and superpositions of proper times[END_REF][START_REF] Loveridge | Relative quantum time[END_REF][START_REF] Höhn | Switching internal times and a new perspective on the 'wave function of the universe[END_REF]. In these works, an important goal is to understand, intuitively speaking, how physics "looks like" as described from the perspective of a quantum reference frame.

We explore this question in the case of temporal reference frames, and in particular, in the case where the clocks have finite resources, that is, when clocks are not ideal.

After presenting the clock models used in this work, we review the twirled observable and purified measurement approaches. We show that in physical scenarios where the clocks can be treated as ideal, both models give the same predictions, and are equivalent to ordinary quantum mechanics with an external time parameter. Therefore, for experiments where the ideal-clock approximation is valid, any difference between the TO and PM models is interpretational. In appendix D, we study in detail how to translate between the two frameworks.

However, there are fundamental limitations to the ideal-clock approximation, most notably in view of gravity. Specifically, ideal clocks require an infinite energy spread, creating a large back reaction on space time. This leads not only to the well-known, fun-damental operational limitations on space time measurements at the Plank scale [START_REF] Bronstein | Republication of: Quantum theory of weak gravitational fields[END_REF], but also to limitations stemming from gravitationally-induced entanglement between clocks [START_REF] Castro Ruiz | Entanglement of quantum clocks through gravity[END_REF].

Inspired by these considerations, we construct a natural generalization of measurements in the PM approach for non-ideal clocks. We obtain a time-non-local evolution equation for the system with respect to the clock, as reported in [START_REF] Alexander | Quantizing time: Interacting clocks and systems[END_REF]. We compare our model with its TO counterpart [START_REF] Philipp | Trinity of relational quantum dynamics[END_REF][START_REF] De La Hamette | Perspective-neutral approach to quantum frame covariance for general symmetry groups[END_REF], and show that the difference between the TO and PM approaches is no longer interpretational but leads to different physical predictions.

We argue that this difference arises because each approach has a different notion of measuring a system at a certain time 1 , and therefore describes a different experimental situation.

In the TO approach, measurements and state update proceed basically as in ordinary quantum mechanics with external time evolution. Importantly, time evolution with respect to the non-ideal clock is unitary, and the temporal order between events can be defined sharply. The extension of the PM model to non-ideal clocks leads to a radically different picture. Here, the clock can no longer resolve single time instances, leading to important measurable consequences. In particular, the time evolution of the system "as seen" by the clock is no longer unitary, and the probability rule is modified. Moreover, we show that measurement interactions can occur in an indefinite temporal order, pointing to a fundamental limitation to the notion of temporal order of events, with potential implications to the notion of indefinite causal structure [START_REF] Oreshkov | Quantum correlations with no causal order[END_REF][START_REF] Baumann | Noncausal page-wootters circuits[END_REF][START_REF] Oreshkov | Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics[END_REF] Finally, we construct a discrete-time version of the PM model, and show that unitarity is recovered for general measurement interactions. If unitary evolution is to be held at all costs, our analysis can be seen as an argument in favor of the idea that time is fundamentally discrete.

II. IDEAL AND NONIDEAL CLOCKS

Before analyzing the TO and PM approaches, we summarize the clock models we use in this paper. Our summary is based on Ref. [START_REF] Philipp | Trinity of relational quantum dynamics[END_REF].

In this work, a clock is a quantum system C with Hilbert space H C and a Hamiltonian

ĤC = σ de e |e⟩⟨e| C , (2)
where σ = [-E, E] is the spectrum of the clock, for some energy E. The measurement of time corresponds to a covariant positive operator-valued measure (POVM). In the case of time, a covariant POVM consists of a set of positive operators φt , for t ∈ R, satisfying e -it ĤC φt ′ e it ĤC = φt+t ′ and forming a resolution of the identity. In this work, we take φt = |ϕ t ⟩⟨ϕ t | C , with

|ϕ t ⟩ C = 1 √ 2E E -E e -iet |e⟩ C . ( 3
)
The states |ϕ t ⟩ C transform as e -it ′ ĤC |ϕ t ⟩ C = |ϕ t ′ +t ⟩ C , and resolve the identity as

N C dt |ϕ t ⟩⟨ϕ t | C = 1 C , ( 4
)
for a normalization constant

N C = E/π.
The overlap between two clock states at different times ⟨ϕ

t |ϕ t ′ ⟩ C is N C ⟨ϕ t |ϕ t ′ ⟩ C = sin(E(t -t ′ )) π(t -t ′ ) . ( 5
)
It is easy to see from eq. ( 5) that the distinguishability of clock states for two different times, |ϕ t ⟩ C and |ϕ t ′ ⟩ C , increases as the energy E increases. In this sense, energy is a resource for temporal quantum reference frames. A clock with finite E is then an instance of a quantum reference frame with finite resources [START_REF] Ahmadi | The wigner-araki-yanase theorem and the quantum resource theory of asymmetry[END_REF]. In the limit E → ∞, the spectrum of the clock becomes σ C = R. In this special case, we denote the clock states by |t⟩ C . In contrast to the case of finite E, the states |t⟩ C can be perfectly distinguished, and we can write their inner product as

⟨t|t ′ ⟩ C = δ(t -t ′ ). (6)
In this work, we refer to a clock defined by the states |t⟩ as perfect or ideal. where the approximation of eq. ( 6) is physically meaningful. On the other hand, we refer to the (more realistic) clocks corresponding to states |ϕ t ⟩ C as imperfect or non-ideal.

III. REVIEW OF THE TWIRLED OBSERVABLE AND PURIFIED MEASUREMENT

APPROACHES

We briefly summarize the TO and PM approaches for the case of ideal clocks, as defined in section II. We show that, in this idealized setting, both approaches lead to the same physical predictions.

A. Twirled observable approach

Let us start with the TO approach, introduced in [START_REF] Höhn | Switching internal times and a new perspective on the 'wave function of the universe[END_REF]. For a more detailed account, see appendix C.

The key point of this approach is to substitute the standard Schrödinger equation by a constraint ( ĤC + ĤS ) |Ψ⟩ = 0.

The dynamics of the system S is described by its correlations with a clock C, with Hamiltonians ĤS and ĤC , respectively. The Hamiltonian of the clock, ĤC , is ideal in the sense of section II.

The constraint recovers the Schrödinger equation by contracting eq. ( 7) with ⟨t|. Solving the resulting equation for the initial condition |ψ 0 ⟩ S at t = 0, we find the solution to eq. ( 7):

|Ψ⟩ = R dt |t⟩ C ⊗ e -it ĤS |ψ 0 ⟩ S . ( 8
)
Measurable physical quantities correspond to operators that commute with the constraint ĤS + ĤC , as they should leave the state in the space of states respecting the constraint. For any operator FS on S and any time τ ,

FSC (τ ) = R dt e -it ĤC |τ ⟩⟨τ | C e it ĤC ⊗ e -it ĤS FS e it ĤS . ( 9
)
The operator FSC (τ ) has the physical interpretation of "the operator FS on the system at time τ ". It relies on the well-known G-twirl procedure (see, e.g. [START_REF] Bartlett | Reference frames, superselection rules, and quantum information[END_REF]) for constructing invariant observables -hence the name "twirled observable (TO)" approach. By construction, FCS (τ ) commutes with the constraint, so the state FSC (τ ) |Ψ⟩ satisfies eq. ( 7) as well.

This allows us to define probabilities for consecutive measurements in an analogous way as in ordinary quantum mechanics. For example, let ΠK S and ΠQ S be projectors corresponding to measurement outcomes k and q, respectively, of two observables KS and QS .

Construct the operators ΠK

CS (τ 1 ) and ΠQ CS (τ 2 ), labeled by times τ 1 and τ 2 , via eq. ( 9). The joint probability of measuring p followed by q, at times τ 1 and τ 2 respectively, is defined by (see appendix C)

P (k, q|τ 1 τ 2 ) = ⟨t = 0| C ⟨ψ 0 | S Πq CS (τ 2 ) Πk CS (τ 1 ) |Ψ⟩ 2 . ( 10
)
A straightforward calculation yields

P (k, q|τ 1 τ 2 ) = Πq S exp(-iτ 2 ĤS ) Πk S exp(-iτ 1 ĤS ) |ψ 0 ⟩ 2 . ( 11
)
Therefore, if τ 2 > τ 1 , eq. ( 10) gives the standard quantum mechanical probabilities for measuring k at time τ 1 and then q at time τ 2 .

B. Purified measurement approach

Instead of constructing covariant observables, the PM approach describes measurements dynamically, at the level of the constraint. In this sense, we say that the measurement is "purified" -hence the name "purified measurement (PM)" approach. Specifically, the measurement is modelled by adding to the constraint an interaction term between the clock, the system and an ancilla:

Ĥ |Ψ⟩ CSA = ĤC + ĤS + dt |t⟩⟨t| C ⊗ KSA (t) |Ψ⟩ CSA = 0. ( 12
)
The ancillary system A can be thought of as the pointer of a measurement device that records the measurement outcome. ĤC is the Hamiltonian of a perfect clock.

A standard choice for the measurement interaction is KSA (t) = δ(t -τ ) ĤSA , which models a measurement interaction happening sharply at time t = τ . If we choose ĤSA such that e -i ĤSA |ψ(τ )⟩ S ⊗ |r⟩ A → a Ka S |ψ(τ )⟩ S ⊗ |a⟩ A , then this interaction is a purification of measuring the POVM {K a S } a . We assume this interaction for the rest of this section. In appendix C 1, we show that the states satisfying the constraint Ĥ |Ψ⟩ = 0 and boundary conditions such that ⟨t| C |Ψ⟩ CSA = |ψ(t)⟩ S ⊗ |r⟩ A , for t ≤ τ , have the following form

|Ψ⟩ CSA = τ -∞ dt |t⟩ C ⊗ |ψ(t)⟩ S ⊗ |r⟩ A + ∞ τ dt |t⟩ C ⊗ ÛS (t -τ )e -i ĤSA |ψ(τ )⟩ S ⊗ |r⟩ A = τ -∞ dt |t⟩ C ⊗ |ψ(t)⟩ S ⊗ |r⟩ A + ∞ τ dt |t⟩ C ⊗ ÛS (t -τ ) a Ka S |ψ(τ )⟩ S ⊗ |a⟩ A . ( 13
)
The system first evolves freely according to ĤS . The ancilla is assumed to start in a "ready" state |r⟩ A and evolve trivially until the measurement interaction, which occurs when the clock shows time τ , resulting in the application of the unitary exp(-i ĤSA ). Afterwards, the system evolves freely again. By adding more measurement interactions with additional memories to the Hamiltonian we can model multi-time measurements.

The probability that the measurement outcome a has been written into the memory of the measurement apparatus at time t is given by

P (a|τ ) t = ⟨Ψ| |t⟩⟨t| C ⊗ 1 S ⊗ |a⟩⟨a| A |Ψ⟩ ⟨Ψ| |t⟩⟨t| C ⊗ 1 S ⊗ 1 A |Ψ⟩ . ( 14
)
For ideal clocks, the denominator of eq. ( 14) is independent of t. As we will see in Section IV, this is no longer the case when the perfect-clock approximation is not valid. (For alternative probability rules applied to Wigner's Friend scenarios within the Page and Wootters formalism, see [START_REF] Baumann | Generalized probability rules from a timeless formulation of wigner's friend scenarios[END_REF].) The rule of eq. ( 14) can be extended to multiple-time measurements in a natural way by adding one ancillary system for each measurement [START_REF] Giovannetti | Quantum time[END_REF]. If we construct a purification for the observables K S and Q S , set t > τ 2 > τ 1 , it is easy to see that eq. ( 14) leads to eq. ( 11), showing that both frameworks give the same predictions for ideal clocks. A constructive proof of this result, showing how the two approaches are formally related, can be found in appendix D.

Note that the time t is not the time of the modeled measurement interaction, but the time at which the register is measured. This only needs to happen once, after all of the measurement interactions have occurred.

From the joint probability distribution for all outcomes, we can construct the con-

ditional probabilities P ((a|τ )|(b|τ ′ )) t ,

IV. NON-IDEAL CLOCKS

In this section we study the case where the idealization of a perfect clock is no longer valid. How do we describe the dynamics of a system with respect to a finite-resource temporal reference frame? To answer this question, we focus on the clock model defined by eq. ( 2) in the case where the energy E is finite. We show that the answer depends on whether we use the TO or the PM approach, as in this finite case the two approaches give different physical predictions. The non-ideal TO approach was studied in [START_REF] Philipp | Trinity of relational quantum dynamics[END_REF]; the non-ideal PM approach is the main technical contribution of this work.

A. Twirled observable approach for non-ideal clocks

The central aspects of the TO approach do not depend on whether the clocks are ideal or not. In particular, the constraint is still given by eq. ( 7), but with the bounded-spectrum

Hamiltonian ĤC replacing the ideal one. Likewise, covariant extensions of operators are defined by eq. ( 9), and the probability rule remains the one of eq. ( 10) replacing the ideal clock by a non-ideal one.

Remarkably, the clock does not need to be ideal for the TO approach to reproduce the predictions of standard quantum mechanics for consecutive measurements. Indeed, if clock spectrum is "large enough" with respect to the system spectrum, in a precise sense specified in [START_REF] Philipp | Trinity of relational quantum dynamics[END_REF], all predictions yield the same answers as in the ideal-clock case.

B. Purified measurement approach for non-ideal clocks

In contrast to the TO approach, the PM approach is fundamentally different when clocks are non-ideal. Even with a natural extension of eq. ( 12), as the one presented below, the evolution equation changes substantially, and the evolution becomes non-unitary. The probability rule remains formally the same as eq. ( 14) (changing |t⟩ C → |ϕ t ⟩ C ), with the important difference that the denominator is now not always equal to 1. In appendix F we show how to compute PM probabilities in terms of TO probabilities. The resulting invariant observables in the TO approach are time-non-local.

The Hamiltonian

A natural generalization of eq. ( 12), for the case of non-ideal clocks reads

Ĥ |Φ⟩ CSA = ĤC + ĤS + N C dt |ϕ t ⟩⟨ϕ t | C ⊗ KSA (t) |Φ⟩ CSA = 0, (15)
where N C |ϕ t ⟩⟨ϕ t | C is an element of the clock's covariant POVM described in section II, which corresponds to the measurement of time. We can then interpret the last term of eq. ( 15) as a simplified description that an external observer would give of an internal observer looking at her clock and triggering a measurement interaction KSA (t), which depends on the time t that she observes.

Evolution equation

As in the case of perfect clocks, we can also consider the state

i d dt |ψ(t)⟩ SA = ĤS |ψ(t)⟩ SA + N C dt ′ ⟨ϕ t |ϕ t ′ ⟩ C KSA (t ′ ) |ψ(t ′ )⟩ SA . ( 16
)
If the clock is indeed imperfect, then eq. ( 16) is non-local in time. That is, d dt |ψ(t)⟩ SA does not depend only on on quantities evaluated at time t, but also quantities evaluated at different times. For simplicity, we will in the following consider the case where H S = 0, as this case already captures the conceptual issues relevant for this work. For the general case, we can simply substitute KSA (t) with KSA (t) + ĤS . We solve eq. ( 16) perturbatively in appendix E, using similar methods to those of [START_REF] Alexander | Quantizing time: Interacting clocks and systems[END_REF].

We find

|ψ(t)⟩ = ∞ N =0 (-i) N R • • • R dt N . . . dt 1 F (t -t N ) . . . F (t 2 -t 1 ) KSA (t N ) . . . KSA (t 1 ) |ψ 0 ⟩ (17)
where

F (t) = N C t -∞ dt ′ ⟨ϕ t ′ |ϕ 0 ⟩. 2 F (t)
is a measure of the time-non-locality of eq. ( 16) and its solution, eq. ( 17). In fig. 1, we plot F (t) for different values of E. As E increases, the spectrum broadens and the clock becomes more resourceful. As a consequence, timenon-local effects are suppressed. Note, however, that E cannot increase indefinitely, because gravitational effects eventually become relevant [START_REF] Bronstein | Republication of: Quantum theory of weak gravitational fields[END_REF][START_REF] Castro Ruiz | Entanglement of quantum clocks through gravity[END_REF]. 2 In general this evolution is non-unitary. Consider the case KSA (t) = KSA δ(t -τ ). Then, have |ψ(t)⟩ =

|ψ 0 ⟩ + 2F (t -τ ) ∞ N =1 (-i) N 1 2 N K N SA |ψ 0 ⟩ = (1 + 2F (t -τ ) i KSA 1-i 2 KSA
) |ψ 0 ⟩, which shows that |ψ 0 ⟩ undergoes a nonunitary evolution.

0 0 1 t dt 1 k(t 1 )F (t -t 1 ) E = 1 E = 4 E = 10 ideal 1

FIG. 2:

Magnitude of the order term of the time evolution

dt 1 KSA (t 1 )F (t -t 1 ) for KSA (t) = k(t) K, for clocks with spectrum σ C = [-E, E], with k(t) = χ [0,1] (t).
As expected, we see that for a broader spectrum of the clock, the non-ideal solutions approximate the ideal case better.

Therefore, the limit E -→ ∞ is only valid as an approximation. Roughly, this approximation is suitable when the minimum time δt such that ⟨ϕ 0 |ϕ δt ⟩ ≈ 0 is much smaller that the minimum time ∆t such that ⟨ψ(t)|ψ(t + ∆t)⟩ SA ≈ 0, if we assume that |ψ(t)⟩ SA evolves with the time-dependent Hamiltonian KAS (t) + H S .

Taking E -→ ∞ in eq. ( 17) enforces F (t) = Θ(t), and yields unitary evolution:

|ψ(t)⟩ = ← T e -i t 0 dsK SA (s) |ψ 0 ⟩ , ( 18
)
where ← T denotes the time-ordering operator. After taking the limit E -→ ∞, it is valid to take the limit of KSA (t) being an instantaneous interaction, i.e. KSA (t) = δ(t -τ ) ĤSA .

In this limit we obtain the solution of section III B. Importantly, the limit E -→ ∞ and KSA (t) → δ(t -τ ) ĤSA should be taken precisely in that order, as a clock with finite resources cannot control an operation at a sharp time τ .

Indefinite temporal order of events

The bounds on the energy of our clock have important consequences on the time ordering of events, understood as the measurement interactions triggered by the clock. To see this, consider two measurement interactions, modeled by

KSA (t) = k(t -τ 1 ) K1 + k(t -τ 2 ) K2
in eq. [START_REF] Rovelli | Covariant loop quantum gravity[END_REF]. For the following discussion we assume that τ 1 < τ 2 . We now look at the state of the system "as seen" by the clock in this scenario. To second order in the expansion of

term k(τ 1 -t 1 ) K1 k(τ 2 -t 2 ) K2 of eq. (19), with k(t) = χ [0,1] (t)
, is not zero. Here, the two times τ 1 and τ 2 are far apart enough to ensure that the support of k(t -τ 1 ) is disjoint from the support of k(t -τ 2 ). As these supports do not overlap in time, a nonzero contribution of k(τ 1 -t 1 ) K1 k(τ 2 -t 2 ) K2 means an indefinite temporal order of events. The integration over t 1 , t 2 of

F (t -t 1 )F (t 1 -t 2 )k(τ 1 -t 1 )k(τ 2 -t 2 ) K1 K2
gives the contribution of this term. In the ideal clock case, we can see that, because there is no overlap in time of the two measurements, this term does not contribute.

In the non-ideal case the situation is different. Because the clock states are not orthogonal, even if there is no overlap of the two measurements in time, the contribution k(τ 1 -t 1 ) K1 k(τ 2 -t 2 ) K2 to eq. ( 19) is not zero.

eq. ( 17), we have

|ψ(t)⟩ = |ψ 0 ⟩ + dt 1 F (t -t 1 ) k(t 1 -τ 1 ) K1 + k(t 1 -τ 2 ) K2 + dt 1 dt 2 F (t -t 1 )F (t 1 -t 2 ) k(t 1 -τ 1 ) K1 + k(t 1 -τ 2 K2 k(t 2 -τ 1 ) K1 + k(t 2 -τ 2 ) K2 |ψ 0 ⟩ (19)
In fig. 2 we plot the magnitude of the first order term for one of the measurements in the case where k(t) = χ [0,1] (t). To second order, this expansion contains terms where K1 is applied before K2 , that is, terms of the form

dt 1 dt 2 F (t -t 1 )F (t 1 -t 2 )k(t 1 -τ 1 )k(t 2 -τ 2 ) K1 K2 . ( 20
)
This on its own is not yet a feature stemming only from the imperfection of the clocks, as even for perfect clocks these terms appear if the support of k(t -τ 1 ) and k(t -τ 2 ) overlap.

However, if we choose k(t -τ 1 ) and k(t -τ 2 ) such that their supports do not overlap, then in the ideal case this term should not contribute. An illustration of this case is shown in fig. 3 for perfect and imperfect clocks. As F (t 1 -t 2 ) approximates, but is not equal, to Heaviside's theta function Θ(t 1 -t 2 ), the term in eq. ( 20) is not zero even if t 1 -t 2 < 0.

We conclude that, if the clocks are no longer perfect, and events are defined according to the PM approach, then there is a limitation to the notion of time ordering of events, stemming from time-non-locality. As discussed above, the PM definition of event corresponds to the description that an external observer would give of an internal experimenter looking at a clock and, conditioned on measuring certain time, performing a measurement on a quantum system. From this perspective, these considerations might be relevant to the study of indefinite causal structure in the sense of process matrices and related operational frameworks [START_REF] Oreshkov | Quantum correlations with no causal order[END_REF][START_REF] Baumann | Noncausal page-wootters circuits[END_REF][START_REF] Oreshkov | Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics[END_REF].

Comparison with the non-ideal twirled observable approach

It is clear from the analysis above that, away from the limit of ideal clocks, the TO and PM approaches are physically different. In the TO approach, we can interpret events as time-translation invariant operations performed by an external experimenter on a timetranslation invariant system. The state of the system is given by eq. ( 8) and the operations are given by eq. ( 9) (suitably modifying each of them to the case of non-ideal clocks). Interestingly, for an imperfect yet sufficiently "good" clock (see [START_REF] Philipp | Trinity of relational quantum dynamics[END_REF] for the exact conditions), this description is mathematically equivalent, by means of conditioning on the clock to be on a state |ϕ t ⟩ C , to the standard quantum mechanical description of a system evolving in an external background time. Specifically,

⟨ϕ t | C FCS (τ 2 ) ĜCS (τ 1 ) |Ψ⟩ CS = e -i ĤS (t-τ 2 ) FS e -i ĤS (τ 2 -τ 1 ) ĜS e -i ĤS τ 1 |ψ 0 ⟩ S . (21)
Importantly, however, the order of events in this approach is given externally, and is a priori unrelated to the parameter τ in eq. ( 9) (or τ 1 and τ 2 in eq. ( 21)). In particular, the resources of the clock do not play any role for deciding the order in which the operations FCS (τ 2 ) and ĜCS (τ 1 ) (or equivalently FS and ĜS ) are applied. It is as if the experimenter had access to a continuous set of operators labeled by a parameter τ , and she had the power to choose unambiguously specific operators from the set and apply them in any order she wants. In this sense, she has infinite resources, even if these are not provided by a temporal reference frame.

On the other hand, the PM approach can be interpreted as the external description of an internal observer who looks at a clock and triggers a measurement interaction on the system when they see the clock showing a certain time. This action constitutes the event. By contracting the solution to eq. ( 15) with ⟨ϕ t |, we obtain eq. ( 16), which we can interpret as the description of events "as seen" from the perspective of the quantum clock C (although in a "purified" sense). In this case, the time order between events is determined dynamically, by the interaction between the clock and the systems.

In the PM approach, the resources of the clock become all important. For clocks with finite resources, the measurement interaction induces time-non-local effects, which imply that the evolution of the system "as seen" by the clock is not unitary, and that there is a fundamental limitation to the operational time order of events, as seen in subsection IV B 3. (Non-unitarity can also arise in the σ = R case, see [START_REF] Alexander | Quantizing time: Interacting clocks and systems[END_REF][START_REF] Paiva | Non-inertial quantum clock frames lead to non-hermitian dynamics[END_REF].) Moreover, the denominator of the probability rule, eq. ( 14) becomes non-trivial, which is a departure from the Born rule of ordinary quantum mechanics.

However, this is not the only possible interpretation. In appendix F we show how to compute PM probabilities in terms of TO conditional probabilities, where evolution is unitary. Not surprisingly, the resulting invariant observables in the TO approach are time-non-local. Thus, the PM approach can be interpreted in terms of states which are solutions to the constraint given by the TO approach, but with a restriction on the available class of operators. This restriction, which also implies a form of time-non-locality, is a consequence of the limited resources of the clock.

To illustrate the difference between the two approaches in an extreme case, imagine that the clock has no dynamics, that is, set ĤC = 0. In the TO approach, it is still meaningful to talk about measurements. Even if the state of the system does not evolve relative to the clock, the operator FCS (τ ) will modify the state in a non trivial way. In contrast, setting ĤC = 0 in the PM approach makes measurements meaningless, as a working clock is necessary to trigger the interaction by which the ancilla A gathers information about the measurement outcome.

V. NON-UNITARITY AND DISCRETENESS OF TIME

In the above discussion, we tacitly assumed that time -and as a consequence, time evolution -is continuous. This assumption led us to conclude that time evolution in the presence of measurements, relative to a non-ideal clock, is time-non-local and non-unitary and does not depend on a single instance in time. However, we can drop the assumption that time evolution is continuous.

In the following, we construct a model for the PM approach to measurements in the case where time is discrete. A similar model was previously constructed in [START_REF] Mondragon | Multiple-event probability in general-relativistic quantum mechanics. II. a discrete model[END_REF]. If we consider only a discrete set of times, we are only interested in the system being invariant under discrete time evolution

Û = k |t k+1 ⟩⟨t k | ⊗ Û (k) SA , (22)
with {|t k ⟩ |t k = kπ E k ∈ Z} being an orthonormal basis of clock states. We demand that the global state

|Ψ⟩ CSA satisfies Û |Ψ⟩ = |Ψ⟩ . ( 23
)
As we show in appendix G, the solution to eq. ( 23) reads

k |t k ⟩ C ⊗ |ψ(t k )⟩ SA with |ψ(t k+1 )⟩ SA = k j=0 Û (j) SA |ψ(t 0 )⟩ SA . ( 24
)
Note that, in eq. ( 24), the evolution of the system with respect to the clock is completely unitary. Based on this observation, we arrive at the following statement: if, fundamentally, the best clocks have a finite energy spectrum, and measurements of a system at a certain time couple the clock to the system, and we demand that the system evolves unitarily with respect to the clock, then the underlying time has to be discrete.

VI. DISCUSSION

We have analyzed two recent approaches to measurements in the Page-Wootters formalism -the TO approach and the PM approach. Each approach solves Kuchař's criticisms in a different way: the TO approach defines measurements in terms of time-translation invariant operators, and the PM approach includes explicitly a dynamical description of the measurement process. We have shown that, whereas the two approaches are physically equivalent in the limit of ideal clocks, they differ in the non-ideal case. We have compared the two solutions with the purpose of reaching a better operational understanding of measurement events in the "timeless" approach to quantum mechanics.

The TO approach, closely related to Dirac quantization, is based on time translation invariant operators applied to static state. The operators are applied in an order which is chosen externally, without using the resources of the clock to distinguish between them.

The resulting picture can be "internalized" and reinterpreted as ordinary quantum mechanics with external background time. In this reconstruction of events and time evolution, we have infinite resources to distinguish between different operations and determine their order, even if the clock is not ideal.

The PM approach describes measurements dynamically, and the resources of the clock are important to determine the order in which measurement interactions happen. It can be interpreted as the external description of an internal observer who measures a system when she "sees" her clock reading a certain time. When the clocks are far from the ideal case, time evolution looks very different to the unitary evolution of ordinary quantum mechanics. Specifically, it leads to a time-non-local, non-unitary evolution. In this approach, the fact that the clock has finite resources becomes very important. Most strikingly, the lack of resources to distinguish perfectly between different clock states leads to fundamental limitations to the notion of time ordering.

Therefore, the reconstruction of measurement events in the TO approach coincides with the operational notion of "looking" at a clock and measuring the system only in the limit where clocks can be treated as ideal. In the non-ideal case, the two approaches should be interpreted as describing different physical situations.

Our analysis opens the question of how to properly describe the purification of measurements in timeless approaches to quantum theory. Measurement interactions carry energy, and this has non-negligible consequences at the level of the constraint. The question of purification is important given the role of purification as a foundational principle for quantum mechanics [START_REF] Chiribella | Informational derivation of quantum theory[END_REF].

What are the implications of the PM model, in the non-ideal case, for space-time? If time is defined operationally, in terms of what a clock measures, then a fundamental limit on our clock's resources would also imply a fundamental limit on our notion of time as we understand it classically. Considering the gravitational back reaction of a quantum clock on space time strengthens the operational point of view. In such scenarios, there are physical situations that fundamentally prohibit the measurement of time with more than a certain accuracy, fundamentally limiting the resources of our clocks.

This operational stance resonates with the analysis of Peres in [START_REF] Peres | Measurement of time by quantum clocks[END_REF], who noted that:

It thus seems that the Schrödinger wave function ψ(t), with its continuous time evolution given by iℏ ψ = Hψ, is an idealization rooted in classical theory.

It is operationally ill defined (except in the limiting case of stationary states)

and should probably give way to a more complicated dynamical formalism, perhaps one non-local in time. Thus, in retrospect, the Hamiltonian approach to quantum physics carries the seeds of its own demise.

The time-non-local evolution and the indefinite time order of measurement events explored here are instances of such a description.

Time-non-locality and indefinite temporal order are, however, not the only possibility.

Coming back to Peres' remarks, if continuous time evolution is operationally ill-defined, perhaps a discrete treatment of time would be the right thing to do. This resonates with our findings: in a timeless setting where the Hamiltonian evolution is generated by the relative degrees of freedom between a clock and the system, unitary evolution and definite time order are recovered if the emergent time model is discrete.

modify the constraint (and therefore the history state |Ψ⟩) or change the way operators are represented. These two approaches exactly correspond to the formalisms studied in the main body of this paper, the purified measurement [START_REF] Giovannetti | Quantum time[END_REF] approach and twirled observable [START_REF] Philipp | Trinity of relational quantum dynamics[END_REF] approach respectively.

Appendix C: Review of GLM and HLS approaches

Review of ideal purified measurement approach

Giovannetti et al. [START_REF] Giovannetti | Quantum time[END_REF] proposed an extension to the Page-Wooters mechanism which overcomes the deficiencies raised by Kuchař . They do so by associating an energy to each measurement, and including it in the constraint as follows: Let us consider an additional purifying ancilla A, which allows the description of a projective measurement at time τ through its Kraus representation as a unitary mapping

V : |ψ(τ )⟩ S ⊗ |r⟩ A → a Ka S |ψ(τ )⟩ S ⊗ |a⟩ A (C1)
acting on H S ⊗ H A , where the ancilla A is initialized in a ready state |r⟩ A and stores the measurement outcome. Here, Ka S are Kraus operators satisfying a ( Ka S ) † Ka S = 1 S . Let ĤSA be the Hamiltonian responsible for the measurement unitary V , i.e. V = e -i ĤSA .

Giovannetti et al. [START_REF] Giovannetti | Quantum time[END_REF] consider the case where ĤA contributes to the total Hamiltonian constraint in the form

Ĥ = ĤC + ĤS + δ( TC -τ ) ⊗ ĤSA , ( C2
)
where δ( TC -τ ) ⊗ ĤSA acts on

H C ⊗ H S ⊗ H A . The solution to the constraint Ĥ |Ψ⟩ = 0,
where Ĥ is given by eq. (C2), can be found by coherent group averaging [START_REF] Marolf | Group averaging and refined algebraic quantization: Where are we now?[END_REF]. This yields the history state

|Ψ⟩ = τ -∞ dt |t⟩ C ⊗ |ψ(t)⟩ S ⊗ |r⟩ A (C3) + ∞ τ dt |t⟩ C ⊗ a ÛS (t -τ ) Ka S |ψ(τ )⟩ S ⊗ |a⟩ A . ( C4
)
The probability that outcome a will be registered by the memory A at a given time t is given by projecting onto the measurement ancilla

P (a|t) = ⟨Ψ| |a⟩⟨a| A ⊗ |t⟩⟨t| C |Ψ⟩ ⟨Ψ| |t⟩⟨t| C |Ψ⟩ . ( C5
)
By construction, Kuchař's first criticism is resolved because the measurement projector now acts on the measurement ancilla and not on the system S. In the special case of projective, non-degenerate von Neumann measurements, the Kraus operators are Ka S = |a⟩⟨a| s and the probability reduces to

P (a|τ ) t = ⟨a| A |ψ(t)⟩ S 2 , (C6)
in accordance with Born's rule if t ≥ τ .

In contrast to the original Page-Wooters proposal, the PM approach allows the discussion of multiple subsequent measurements. Indeed, eq. (C2) can be readily extended by a second measurement ancilla

A ′ Ĥ = ĤC + ĤS + δ( TC -τ ) ⊗ ĤSA + δ( TC -τ ′ ) ⊗ ĤSA ′ . (C7)
Group averaging results in the history state

|Ψ⟩ = τ -∞ dt |t⟩ C ⊗ |ψ(t)⟩ S ⊗ |r⟩ A ⊗ |r⟩ A ′ (C8) + τ ′ τ dt |t⟩ C ⊗ a ÛS (t -τ ) Ka S |ψ(τ )⟩ S ⊗ |a⟩ A ⊗ |r⟩ A ′ (C9) + ∞ τ ′ dt |t⟩ C ⊗ a,b ÛS (t -τ ′ ) Kb S ÛS (τ ′ -τ ) Ka S |ψ(τ )⟩ S ⊗ |a⟩ A ⊗ |b⟩ A ′ . (C10)
It is important to note that the probability P (a|t) in eq. ( C5) is not affected by the presence of the second ancilla. However, the probability of observing the measurement result |b⟩ A ′ at time τ ′ conditioned on outcome |a⟩ A at time τ can now be expressed as

P ((b|τ ′ )|(a|τ )) t = ⟨t| C ⊗ ⟨b| A ′ ⊗ ⟨a| A |Ψ⟩ 2 (⟨t| C ⊗ ⟨a| A |Ψ⟩ 2 , ( C11
)
which results in

P ((b|τ ′ )|(a|τ )) t = Πb S Û (τ ′ -τ ) Πa S |ψ(τ )⟩ 2 Πa S |ψ(τ )⟩ 2 . (C12) if t ≥ τ ′ > τ
. This is the correct expression for two point measurements, in agreement with ordinary quantum mechanics. Kuchař's second criticism is thus resolved as well.

In eq. ( C4) and (C10), the measurement outcome is entirely encoded in the record kept by the measurement ancillas. From a philosophical viewpoint, Giovannetti et al. [START_REF] Giovannetti | Quantum time[END_REF] therefore adopt Wheeler's operationalist stance [START_REF] Archibald | The "past" and the "delayed-choice" double-slit experiment[END_REF] that "the past has no existence except as it is recorded in the present".

Review of the twirled observable approach

Höhn et al. [START_REF] Philipp | Trinity of relational quantum dynamics[END_REF] propose a different extension of the Page-Wootters mechanism, which we call the twirled observable approach or TO approach. As in the Page-Wootters formalism evolution of S is described by a time independent state |Ψ⟩ solving the constraint is invariant under Û (t) and as a result, [ FSC (τ ), Ĥ] = 0, resolving Kuchař's first criticism.

Ĥ |Ψ⟩ = 0, Ĥ = ĤC + ĤS . ( C13
The states |Ψ⟩, solutions to the constraint eq. ( C13) are normalized with respect to the so-called physical scalar product ⟨•|•⟩ phys , which is defined as

⟨Ψ|Φ⟩ phys = ⟨ψ(t)| S ⊗ ⟨t| C ds Û (s) |t⟩ C ⊗ |ϕ(t)⟩ S , (C15)
for any time t. It is straight-forward to verify that this scalar product is well-defined and in particular independent of the choice of t.

The probability to find the system in state |a⟩ S at time τ is defined by

P (a|τ ) = ⟨Ψ| Πa SC (τ ) |Ψ⟩ phys , ( C16
)
where Π a S = |a⟩⟨a| S . This coincides with the TO probability introduced in section III A in the main text. This indeed gives the correct measurement statistics, as readily verified (setting t=0):

P (a|τ ) = ⟨Ψ| dt e -it( ĤC + ĤS ) (|τ ⟩⟨τ | C ⊗ Πa S )e it( ĤC + ĤS ) |Ψ⟩ phys (C17) = ⟨ψ(0)| S ⊗ ⟨0| C ds e -is( ĤC + ĤS ) dt e -it( ĤC + ĤS ) (|τ ⟩ ⟨τ | C ⊗ Πa S )ê it( ĤC + ĤS ) |0⟩ C ⊗ |ψ(0)⟩ S (C18) = ⟨ψ(0)| S ⊗ ⟨0| C ds e -i(s-τ )( ĤC + ĤS ) ⊗ Π a S |τ ⟩ C ⊗ |ψ(τ )⟩ S (C19) = ⟨ψ(τ )| Πa S |ψ(τ )⟩ S , ( C20
)
in agreement with Born's rule.

Because measurements in the TO picture commute with Ĥ, they do not throw |Ψ⟩ out of the constrained Hilbert space. Consequently, eq. (C16) can be readily extended to more than one measurement. Furthermore, one can define conditional probabilities of the form

P ((b|τ ′ )|(a|τ )) = ⟨Ψ| Πa SC (τ ) Πb SC (τ ′ ) Πa SC (τ ) |Ψ⟩ phys ⟨Ψ| Πa SC (τ ) |Ψ⟩ phys . ( C21
)
As {|ϕ t ⟩ C } t∈R is a over-complete basis, if

⟨ϕ t | C Ĥ |Ψ⟩ CSA = 0 ∀t, (E2)
then this is equivalent to

Ĥ |Ψ⟩ CSA = 0. (E3)
Therefore, we consider the following problem, where for the sake of simplicity we absorbed the Hamiltonian of the system into the interaction term

⟨ϕ t | C ĤC + N C dτ |ϕ τ ⟩⟨ϕ τ | C ⊗ KSA (τ ) |Ψ⟩ CSA = 0. (E4)
As

|ϕ t ⟩ C = e -i ĤC t |ϕ 0 ⟩ C is
|Ψ⟩ CSA = N C dτ |ϕ τ ⟩ C ⊗ |ψ(τ )⟩ SA . (E7)
and then contract with ⟨ϕ t |, we will find

ψ(t) SA = N C ⟨ϕ t | C dτ |ϕ τ ⟩ C ⊗ |ψ(τ )⟩ SA . (E8)
where ψ(t)

SA

will not necessarily be a solution of eq. (E6), even if |ψ(t)⟩ SA is.

In the case where |ψ(t)⟩ SA = ⟨ϕ t | C |Ψ⟩ CSA , with |Ψ⟩ CSA defined as in eq. (E7). We find

ψ(t) SA = N C ⟨ϕ t | C dτ |ϕ τ ⟩ C ⊗ |ψ(τ )⟩ SA (E9) = N C ⟨ϕ t | C dτ |ϕ τ ⟩ C ⊗ ⟨ϕ τ | C |Ψ⟩ CSA (E10) = ⟨ϕ t | C |Ψ⟩ CSA = |ψ(t)⟩ SA . (E11)
For such a |ψ(t)⟩ SA we have that |Ψ⟩ CSA , as defined in eq. (E7), fulfills the constraint eq. (E1). As we shall see in the next section, the solution |ψ(t)⟩ SA does satisfy |ψ(t)⟩ SA = ⟨ϕ t | C |Ψ⟩ CSA , with |Ψ⟩ CSA defined according to eq. (E7), thereby being a solution to eq. (E1).

Solving the evolution equation

To solve eq. (E6) we use a procedure inspired by the Born approximation. We then show that in the case where the approximation converges the solution |ψ(t)⟩ SA fulfills the evolution equation and also that

⟨ϕ t | C dτ |ϕ τ ⟩ C |ψ(τ )⟩ SA = |ψ(t)⟩ SA .
Integrating the evolution equation, leads us to the following equation

|ψ(t)⟩ SA = -iN C t -∞ dσ dτ ⟨ϕ σ |ϕ τ ⟩ C ⊗ KSA (τ ) |ψ(τ )⟩ SA + |ψ 0 ⟩ SA . (E12)
To solve this equation, we make the approximation that the system does not change over time

|ψ(t)⟩ 0 SA := |ψ 0 ⟩ SA (E13)
and then use the RHS of eq. (E12) to obtain a (hopefully) better approximation to to the solution

|ψ(t)⟩ 1 SA = |ψ 0 ⟩ SA -iN C t -∞ dσ dτ ⟨ϕ σ |ϕ τ ⟩ C ⊗ KSA (τ ) |ψ 0 ⟩ SA . (E14)
We can simplify this expression, let us define

F (t) = N C t -∞ dσ ⟨ϕ σ |ϕ 0 ⟩ C . (E15)
Using the fact that

F (t -τ ) = N C t-τ -∞ dσ ⟨ϕ σ |ϕ 0 ⟩ C = N C t -∞ dσ ⟨ϕ σ-τ |ϕ 0 ⟩ C = N C t -∞ dσ ⟨ϕ σ |ϕ τ ⟩ C (E16)
we find that

|ψ(t)⟩ 1 SA = |ψ 0 ⟩ SA -i dτ F (t -τ ) KSA (τ ) |ψ 0 ⟩ SA . (E17)
We then use this new approximation as an Ansatz and repeat the procedure, to obtain approximations in higher orders of KSA (t)

|ψ(t)⟩ 2 SA = |ψ(t)⟩ 1 SA + (-i) 2 dτ 1 dτ n F (t -τ 1 )F (τ 1 -τ 2 ) KSA (τ 1 ) KSA (τ 2 ) |ψ 0 ⟩ SA . . . |ψ(t)⟩ n SA = |ψ(t)⟩ n-1 SA + (-i) n dτ 1 • • • dτ n F (t -τ 1 ) . . . F (τ n-1 -τ n ) KSA (τ 1 ) . . . KSA (τ n ) |ψ 0 ⟩ SA (E18)
If this recursive procedure converges we obtain the following expression

|ψ(t)⟩ SA = ∞ n=0 (-i) n dτ 1 • • • dτ n F (t -τ 1 ) . . . F (τ n-1 -τ n ) KSA (τ 1 ) . . . KSA (τ n ) |ψ 0 ⟩ SA . (E19)
In the case when the clock is ideal, i.e. ⟨ϕ t |ϕ t ′ ⟩ C = δ(t -t ′ ), this reduces to the solution to the usual time-dependent Schrödinger equation, as

F (t) = N C t -∞ dσ ⟨ϕ σ |ϕ 0 ⟩ C = Θ(t)
, the Heaviside theta function.

Let us show that |ψ(t)⟩ SA is actually a solution to the evolution equation

i d dt |ψ(t)⟩ SA = ∞ n=1 (-i) (n-1) dτ 1 . . . dτ n N C ⟨ϕ t |ϕ τ 1 ⟩ C F (τ 1 -τ 2 ) . . . F (τ n-1 -τ n ) KSA (τ 1 ) . . . KSA (τ n ) |ψ 0 ⟩ SA = N C dτ ⟨ϕ t |ϕ τ ⟩ C KSA (τ ) ∞ n=1 (-i) (n-1) × dτ 1 . . . dτ n-1 F (τ -τ 1 ) . . . F (τ n-2 -τ n-1 ) KSA (τ 1 ) . . . KSA (τ n-1 ) |ψ 0 ⟩ SA = N C dτ ⟨ϕ t |ϕ τ ⟩ C KSA (τ ) |ψ(τ )⟩ SA . ( E20
⟨ϕ t | C |Ψ⟩ CSA = N C dτ ⟨ϕ t |ϕ τ ⟩ C |ψ(τ )⟩ SA = N C dτ ⟨ϕ t |ϕ τ ⟩ C |ψ 0 ⟩ SA + ∞ n=1 (-i) n N C dτ dτ 1 • • • dτ n ⟨ϕ t |ϕ τ ⟩ C F (τ -τ 1 ) . . . F (τ n-1 -τ n ) × KSA (τ 1 ) . . . KSA (τ n ) |ψ 0 ⟩ SA . (E21)
For the n = 0 term, using the expression of |ϕ τ ⟩ in the Fourier basis, we find

N C dτ ⟨ϕ t |ϕ τ ⟩ C |ψ 0 ⟩ SA = N C dτ ⟨ϕ 0 |ϕ τ ⟩ C |ψ 0 ⟩ SA = 1 2π dτ σ dϵ e -iτ ϵ |ψ 0 ⟩ SA = |ψ 0 ⟩ SA (E22)
For terms with n > 0, we consider the following expression, defining f

(t) = N C ⟨ϕ t |ϕ 0 ⟩ C = 1 2π σ de e iet : N C dτ ⟨ϕ t |ϕ τ ⟩ C F (τ -τ 1 ) = dτ f (t -τ )F (τ -τ 1 ) = dτ f (t -τ 1 -τ )F (τ ) = (f * F )(t -τ 1 ). (E23)
Using the relationship of convolution with integration, we find

(f * F )(t -τ 1 ) = t-τ 1 -∞ (f * f )(τ )dτ (E24)
By definition of the function f , its Fourier transform is F(f ) = χ σ , which implies that

F(f * f ) = F(f ) 2 = F(f ) and thus (f * f )(τ ) = f (τ ). Together this implies that N C dτ ⟨ϕ t |ϕ τ ⟩ C F (τ -τ 1 ) = F (t -τ 1 ). (E25)
Plugging this back into our expression for ⟨ϕ t | C |Ψ⟩ CSA we find

⟨ϕ t | C |Ψ⟩ CSA = |ψ(t)⟩ SA , (E26)
which proves that the solution we found satisfies the constraint.

The evolution equation and its solution for periodic universes

If we want to consider a finite dimensional clock we need to consider a periodic time, where for the clock there exists a time T such that |ϕ T ⟩ C = |ϕ 0 ⟩ C [START_REF] Philipp | Trinity of relational quantum dynamics[END_REF]. From integrating evolution equation

|ψ(t)⟩ SA = (-i)N C t 0 dσ T 0 dτ ⟨ϕ σ |ϕ τ ⟩ C KSA (τ ) |ψ(τ )⟩ SA + |ψ 0 ⟩ SA . (E27)
Applying the same technique as for the non-periodic case we find the following evolution if the recursive procedure converges

|ψ(t)⟩ SA = ∞ n=0 (-i) n t 0 dσ 1 T 0 dτ 1 • • • τ n-1 0 dσ n T 0 dτ n ⟨ϕ σ 1 |ϕ τ 1 ⟩ C . . . ⟨ϕ σn |ϕ τn ⟩ C × KSA (τ ) 1 . . . Kτ n |ψ 0 ⟩ SA (E28)
For simplicity of notation we define

F τ (t) := N C t 0 dσ ⟨ϕ σ |ϕ τ ⟩ C . (E29)
With this definition we can write the solution in a more compact form

|ψ(t)⟩ = ∞ n=0 (-i) n T 0 dτ 1 • • • T 0 dτ n F τ 1 (t) . . . F τn (τ n-1 ) KSA (τ 1 ) . . . KSA (τ n ) |ψ 0 ⟩ SA . (E30)
As in the non-periodic case, we find a candidate for the solution to eq. (E27) by appending the clock and integrating over time

|Ψ⟩ CSA = N C dτ |ϕ τ ⟩ C ⊗ |ψ(τ )⟩ SA . ( E31
⟨ϕ t | C |Ψ⟩ CSA = N C T 0 dτ ⟨ϕ t |ϕ τ ⟩ C |ψ(τ )⟩ SA . (E32)
We derive this expression by t and find

d dt ⟨ϕ t | C |Ψ⟩ CSA = N C T 0 dτ ⟨ϕ t |ϕ τ ⟩ C d dτ |ψ(τ )⟩ SA -N C ⟨ϕ t |ϕ 0 ⟩ C (|ψ(T )⟩ SA -|ψ(0)⟩ SA ) . (E33)
Using that |ψ(t)⟩ SA is a solution to the evolution equation we find We can express the solution as |Ψ⟩ = ΠPM |ψ 0 ⟩, where ΠPM is the "projector" onto the space of solutions to eq. ( 15) and |ψ 0 ⟩ can be interpreted as an initial condition. Now extend the TO approach to obtain the ancillary system A, and assume that A contributes trivially to the TO constraint of eq. ( 7). Then F3) is explicitly a conditional probability in the TO approach. In general, the operators in eq. (F3) are clearly not of the form of eq. ( 9), which represents time-local measurements. Unless we are in the limit where clocks can be approximated as perfect ones, these operators represent measurements that are delocalised in time in the TO approach.

Appendix G: Discrete time model

In the case the spectrum of the clock is the interval [-E, E], even though, in general, clock states are not orthogonal, there still exists times {t k } k , where the clock states are mutually orthogonal ⟨t k |t k ′ ⟩ = δ k,k ′3 . Using the explicit form of the spectrum, we find

⟨t k |t k ′ ⟩ = 1 2π E -E de e i(t k -t k ′ )e = E π sinc(E(t k -t k ′ )) (G1)
with sinc(x) := sin(x)

x . So the clock states are orthogonal ⟨t k |t k ′ ⟩ = 0, if t k -t k ′ = N π E for N ∈ Z. Therefore, we can choose the following orthonormal clock basis

|t k ⟩ t k = kπ E k ∈ Z . ( G2
)
The unitary

Û = k |t k+1 ⟩⟨t k | ⊗ Û (k) SA (G3)
can be interpreted as the unitary evolving the system and the clock for one time step, such that the system undergoes the evolution U k when the clock shows time t k . By choosing the unitary U (k * ) for some integer k * to be an interaction for a von Neumann unitary, then we can interpret this as the system being measured at time t k * .

In this discrete model, we no longer have a Hamiltionian, but the evolution of the system for a single time step. The constraint we considered in the continuous case, is equivalent to the total system not evolving in time:

e i ĤCSA t |Ψ⟩ CSA = |Ψ⟩ CSA ∀ t. (G4)
So for the discrete case we impose the following constraint: We then see that, in this model, the evolution of the system relative to the clock is unitary as long as we consider time steps of size π E . Moreover, for discrete times, it follows immediately that the PM approach just constructed and the discrete version of the TO approach, defined in the obvious way, are physically equivalent.

Û n |Ψ⟩ CSA = |Ψ⟩ CSA ∀n ∈ Z. ( G5

  which is the probability of measuring a at time τ , given that we measured b at time τ ′ . If we omit the subscript t in P ((a|τ )|(b|τ ′ )) t , then mean that we evaluate P ((a|τ )|(b|τ ′ )) t at a time t >> τ, τ ′ . Both the TO and the PM approaches yield the same conditional probabilities in the case of perfect clocks. More precisely, consider two subsequent measurements on a system and let P I ((b|τ ′ )|(a|τ )) denote the probability of observing the measurement result |b⟩ at time τ ′ conditioned on measuring outcome |a⟩ at time τ . Here, I ∈ {PM , TO } is an index determining whether the probability was computed in the PM or TO approach. Then P P M ((b|τ ′ )|(a|τ )) = P T O ((b|τ ′ )|(a|τ )), showing that the TO and PM approaches are entirely equivalent in the case of ideal clocks, even if their treatment of constraints and operators is different. However, as we will see in the next sections, these approaches diverge if non-ideal clocks are considered.

( a )FIG. 3 :

 a3 FIG. 3:The heat map shows the expression F (t -t 1 )F (t 1 -t 2 ) occuring in eq. (19) in the case of non-ideal (left) and ideal (right) clocks. The red square illustrates a region where the contribution of the

)

  The constraint once again recovers the Schrödinger equation for |ψ(t)⟩ S = ⟨t| C |Ψ⟩ S . However, the requirement eq. (C13) results in a different representation of operators acting on |Ψ⟩: An operator f S acting on |ψ(τ )⟩ S is now represented by the incoherent group twirl FSC (τ ) = R dt e -it ĤC |τ ⟩⟨τ | C e it ĤC ⊗ e -it ĤS FS e it ĤS . (C14) acting on |Ψ⟩, where again Û (t) = e -i Ĥt is the free time evolution. By construction, FSC (τ )

  d dt ⟨ϕ t | C |Ψ⟩ CSA = d dt |ψ(τ )⟩ SA -N C ⟨ϕ t |ϕ 0 ⟩ C (|ψ(T )⟩ SA -|ψ(0)⟩ SA ) . (E34)If we assume that the solution |ψ(t)⟩ SA is periodic, this impliesd dt ⟨ϕ t | C |Ψ⟩ CSA = d dt |ψ(τ )⟩ SA . (E35)Integrating both sides of the equation we find⟨ϕ t | C |Ψ⟩ CSA = |ψ(t)⟩ SA + |ψ(0)⟩ SA -N C T 0 dτ ⟨ϕ 0 |ϕ τ ⟩ C |ψ(τ )⟩ SA . (E36) Therefore ⟨ϕ t | C |Ψ⟩ CSA satisfies the constraint if |ψ(0)⟩ SA -N C T 0 dτ ⟨ϕ 0 |ϕ τ ⟩ C |ψ(τ )⟩ SA = 0. (E37)add the denominator Tr(|t⟩⟨t| C |Ψ⟩⟨Ψ| CSA ) to cancel out divergent terms. Both forms of the probability rule are equivalent and we use eq. (F1) in this case for convenience.

  , a solution |η⟩ CSA to the TO constraint now reads ( ĤC + ĤS ) |η⟩ CSA = 0, (F2) with identities on A left implicit. We choose |η⟩ CSA such that |ψ 0 ⟩ = ⟨ϕ 0 | C |η⟩ CSA . Then, using the cyclicity of the trace and the fact that |η⟩⟨η| CSA is left unchanged under incoherent G-twirling T (with respect to the TO time translation), we can rewrite eq. (F1)asP (a|t, τ ) = Tr T [|ϕ 0 ⟩ C Π † PM |a⟩⟨a| A ⊗ |t⟩⟨t| C ΠPM ⟨ϕ 0 | C ] |η⟩⟨η| CSA Tr T [|ϕ 0 ⟩ C Π † PM |t⟩⟨t| C ΠPM ⟨ϕ 0 | C ] |η⟩⟨η| CSA , (F3)where we used that T is self adjoint.Both operators T [|ϕ 0 ⟩ C Π † PM |a⟩⟨a| A ⊗ |t⟩⟨t| C ΠPM ⟨ϕ 0 |] and T [|ϕ 0 ⟩ C Π † PM |t⟩⟨t| C ΠPM ⟨ϕ 0 |]are explicitly invariant under the action generated by the TO constraint. Therefore, eq. (

  a coherent system of states it holds that -id dt ⟨ϕ t | C = ⟨ϕ t | C ĤC . C |Ψ⟩ CSA + N C dτ ⟨ϕ t |ϕ τ ⟩ C KSA (τ ) ⟨ϕ τ | C |Ψ⟩ CSA = 0. (E5) Defining ⟨ϕ t | C |Ψ⟩ CSA = |ψ(t)⟩ SA we find the following evolution equation: SA + N C dτ ⟨ϕ t |ϕ τ ⟩ C KSA (τ ) |ψ(τ )⟩ SA = 0. (E6) When considering solutions to this equation we have to be careful, as in the case of non-ideal clocks the basis {|ϕ t ⟩ C } t is over-complete. As a consequence, not all solutions of eq. (E6) |ψ(t)⟩ SA , are the result of contracting |Ψ⟩ CSA with ⟨ϕ t |, ⟨ϕ t | C |Ψ⟩ CSA , where |Ψ⟩ CSA is a solution to the constraint, eq. (E1). If we define |Ψ⟩ CSA by

	Therefore, we find
	-i	d dt ⟨ϕ t | -i d dt	|ψ(t)⟩

)

  Note that |Ψ⟩ CSA not necessarily need to fulfill the constraint. In this case, we will see that⟨ϕ t | C |Ψ⟩ CSA isa solution to the evolution and, thus, |Ψ⟩ CSA satisfies the constraint if the solution |ψ(t)⟩ SA is periodic and ⟨ϕ 0 | C |Ψ⟩ CSA = |ψ(0)⟩ SA . Let us calculate ⟨ϕ t | C |Ψ⟩ CSA :

)

  This is equivalent toÛ |Ψ⟩ CSA = |Ψ⟩ CSA . (G6)Expanding the state |Ψ⟩ CSA in the discrete clock basis we find the condition

				(G8)
	Applying this relation recursively we find
	|ψ(t k+1 )⟩ SA =	k	Û (j) SA |ψ(t 0 ⟩ SA .	(G9)
		j=0

k |t k+1 ⟩ C ⊗ Û (k) SA |ψ(t k )⟩ SA = k |t k ⟩ C ⊗ |ψ(t k )⟩ SA , (

G7

)

from which we find that the states |ψ(t k )⟩ SA need to fulfill the relation

|ψ(t k+1 )⟩ SA = Û (k) SA |ψ(t k )⟩ SA .

Througout the paper, we use the term measurement event, or simply event, to refer to the notion of measuring a system at a certain time.

This approach generalizes also to other clocks which have times at which the clock states are mutually orthogonal.

ACKNOWLEDGEMENTS L.H. acknowledges support from the Swiss National Science Foundation (SNSF) via project grant No. 200021_188541 and the Quantum Center of ETH Zürich. E.C.-R. is supported by an ETH Zurich Postdoctoral Fellowship and acknowledges financial support from the Swiss National Science Foundation (SNSF) via the National Centers of Competence in Research QSIT and SwissMAP, as well as the project No. 200021_188541. This publication was made possible through the support of the ID# 61466 grant and ID# 62312 grant from the John Templeton Foundation, as part of the project 'The Quantum Information Structure of Spacetime' (QISS). The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

We will show in the next section that this expression indeed recovers the correct measurement statistics for two-point measurements by relating it to the PM approach. Hence the TO approach also resolves Kuchař's second criticism.

Appendix A: Review of the Page-Wootters Formalism

Here we give a pedagogical introduction to the Page-Wootters formalism [START_REF] Page | Evolution without evolution: Dynamics described by stationary observables[END_REF][START_REF] William | time" replaced by quantum correlations[END_REF]. The relational nature and reparametrization-invariance of general relativity suggest that the physical states |Ψ⟩ describing a theory of quantum gravity should satisfy a Wheeler-DeWitt equation

where Ĥ is the total Hamiltonian constraint. An immediate consequence is that states do not evolve according to the standard Schrödinger equation

but appear frozen in time. For this reason, theories satisfying eq. (A1) are often called timeless. Don Page and William Wootters showed that one can nevertheless recover the correct dynamics using a mechanism that is now known as Page-Wootters formalism. In the following, we give a concise summary of this approach.

In the Page-Wootters formalism, the evolution of a system S is described by considering states |Ψ⟩ not on the Hilbert space H S of the system, but on an extended Hilbert space

Here, C is an ancilla system referred to as clock. The clock is equipped with a canonical pair of time and energy coordinates TC , ĤC , satisfying

The Hamiltonian constraint eq. (A1) translates to

which is why it is often referred to as the history state.

One-time measurements of an observable ÔS = a O a |a⟩⟨a| at time τ can be defined in the PaW mechanism by first conditioning on finding the clock in state |τ ⟩ C : The probability to find the system S in state |a⟩ S at time τ is thus given by

This probability is in agreement with Born's rule and thus recovers the correct dynamics of the system S for one-time measurements; however, for consecutive measurements one should be more careful, as we will see below. This gives rise to the different approaches to measurements in the Page-Wootters mechanism discussed in this paper.

Appendix B: Review of Kuchař's criticisms

While eq. ( A7) is in agreement with Born's rule for one-point measurements, it immediately leads to issues when multiple subsequent measurements are performed. These issues were first summarized by Kuchař [START_REF] Kuchař | Time and Interpretations of Quantum Gravity[END_REF] as follows:

1. Violation of constraints: In the PaW mechanism, the measurement of an observable ÔS at time τ is described by a projective operator

acting on |Ψ⟩ CS , where |a⟩ S is an eigenstate of ÔS . However, this operator in general does not commute with the constraint Ĥ. Indeed, as the constraint is the total Hamiltonian, any operator commuting with it must be stationary, while there is obvious experimental evidence that the above projector is time dependent. In that case, applying the measurement operator throws |Ψ⟩ out of the constrained Hilbert space.

Wrong propagators:

As a consequence, the PaW mechanism does not give the correct propagators for multiple subsequent measurements. For example, the dynamical question "If the system is in state |a⟩ S at time τ , what is the probability to find it in state |b⟩ S at time τ ′ > τ ?" cannot be answered by applying the conditional probability in equation (A7) twice. This would yield

which is non-zero only for τ ′ = τ and not in accordance with Born's rule.

Kuchař's first criticism concerns the fact that projective operators and the constraint H do not commute in general. In order to resolve this issue, it seems natural to either The PM and TO approach presented above are seemingly different from both a mathematical and philosophical perspective. Giovannetti et al. [START_REF] Giovannetti | Quantum time[END_REF] overcome Kuchař's criticisms by considering ancillas storing the measurement outcomes and by modifying the constraint operator Ĥ. Their procedure corresponds to an operationalist viewpoint. Höhn et al. [START_REF] Philipp | Trinity of relational quantum dynamics[END_REF] overcome them by representing observables in a way that is invariant under the flow generated by the constraint, without modifying the constraint itself. We now show that these two a priori different approaches are equivalent for multiple-time measurements in the case of perfect clocks, meaning they result in the same measurement outcome probabilities. Note that both approaches agree for single-time measurements already by construction.

To prove this, we want to show that the conditional probabilities eq. ( C11) and (C21) of measuring System S in state |b⟩ S at time τ ′ , given that the system is measured to be in state |a⟩ S at time τ < τ ′ , are equal for both the PM and TO approach. To simplify the calculation, let us define the reduction map R(τ ), which projects |Ψ⟩ onto |ψ(τ )⟩, and its inverse on the constrained Hilbert space:

Indeed, a change of variable yields

With this notation at hand, and using Û † (t) |Ψ⟩ = |Ψ⟩ the operator eq. ( 9) acting on the constrained Hilbert space can be written as

This allows us to expand the conditional probability eq. (C21) in terms of ordinary pro-jection operators Π a S , Π b S :

where we have used equation eq. (D5) and R(τ ′ ) R-1 (τ ) = ÛS (τ ′ -τ ). Eq. (D10) is indeed the correct two time conditional measurement probability as postulated by Born's rule.

We now show that the PM approach yields the same result: Substituting the state in eq. (C8) (with Ka = Πa S ) into equation eq. (C11) yields ) identical to eq. (D10). It is straightforward to generalize this result to more than two subsequent measurements. The PM and TO approach hence result in the same physical predictions for perfect clocks, despite constructing a different physical picture! Appendix E: Calculations for the purified measurement approach with non-ideal clocks

The evolution equation for non-ideal clocks

In a similar manner to how the Schrödinger equation was derived in the ideal clocks case we can also derive the Schrödinger equation in the non-ideal case. Let us remember that the Hamiltonian we consider in the non-ideal case is given by

where N C is a normalization constant such that {N C |ϕ t ⟩⟨ϕ t | C } t∈R forms a POVM.

Appendix F: From purified measurements to twirled observables

The probability of obtaining measurement outcome a at time t for a measurement event at time τ in the PM approach can be written as

where |Ψ⟩ is a solution to the constraint of eq. ( 15). The way we write the probability rule in eq. (F1) is different to the one used in the main text (see eq. ( 14)) in that here we express the rule in quotient form. We "sandwich" with the ket |Ψ⟩ in the numerator and