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Time from quantum entanglement: an experimental illustration

Ekaterina Moreva1,2, Giorgio Brida1, Marco Gramegna1, Vittorio Giovannetti3, Lorenzo Maccone4, Marco Genovese1

1INRIM, strada delle Cacce 91, 10135 Torino, Italy

2International Laser Center of M.V.Lomonosov Moscow State University, 119991, Moscow, Russia

3NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, piazza dei Cavalieri 7, I-56126 Pisa, Italy

3Dip. Fisica “A. Volta”, INFN Sez. Pavia, Univ. of Pavia, via Bassi 6, I-27100 Pavia, Italy

In the last years several theoretical papers discussed if time can be an emergent propertiy deriving

from quantum correlations. Here, to provide an insight into how this phenomenon can occur, we

present an experiment that illustrates Page and Wootters’ mechanism of “static” time, and Gambini

et al. subsequent reﬁnements. A static, entangled state between a clock system and the rest of the

universe is perceived as evolving by internal observers that test the correlations between the two

subsystems. We implement this mechanism using an entangled state of the polarization of two

photons, one of which is used as a clock to gauge the evolution of the second: an “internal” observer

that becomes correlated with the clock photon sees the other system evolve, while an “external”

observer that only observes global properties of the two photons can prove it is static.

“Quid est ergo tempus? si nemo ex me quaerat, scio; si

quaerenti explicare velim, nescio.” [1]

The “problem of time” [2–6] in essence stems from the

fact that a canonical quantization of general relativity

yields the Wheeler-De Witt equation [7, 8] predicting a

static state of the universe, contrary to obvious everyday

evidence. A solution was proposed by Page and Wootters

[9, 10]: thanks to quantum entanglement, a static system

may describe an evolving “universe” from the point of

view of the internal observers. Energy-entanglement be-

tween a “clock” system and the rest of the universe can

yield a stationary state for an (hypothetical) external ob-

server that is able to test the entanglement vs. abstract

coordinate time. The same state will be, instead, evolv-

ing for internal observers that test the correlations be-

tween the clock and the rest [9–14]. Thus, time would be

an emergent property of subsystems of the universe de-

riving from their entangled nature: an extremely elegant

but controversial idea [2, 15]. Here we want to demys-

tify it by showing experimentally that it can be naturally

embedded into (small) subsystems of the universe, where

Page and Wootters’ mechanism (and Gambini et al. sub-

sequent reﬁnements [12, 16]) can be easily studied. We

show how a static, entangled state of two photons can

be seen as evolving by an observer that uses one of the

two photons as a clock to gauge the time-evolution of the

other photon. However, an external observer can show

that the global entangled state does not evolve.

Even though it revolutionizes our ideas on time, Page

and Wootters’ (PaW) mechanism is quite simple [9–11]:

they provide a static entangled state |Ψiwhose subsys-

tems evolve according to the Schr¨odinger equation for

an observer that uses one of the subsystems as a clock

system Cto gauge the time evolution of the rest R.

While the division into subsystems is largely arbitrary,

the PaW model assumes the possibility of neglecting in-

teraction among them writing the Hamiltonian of the

global system as H=Hc⊗11r+11c⊗ Hr, where Hc,Hr

are the local terms associated with Cand R, respectively

FIG. 1: Gate array representation of the PaW mechanisms [9–

11] for a CR non interacting model. Here Ur(t) = e−iHrt/~

and Uc(t) = e−iHct/~are the unitary time evolution opera-

tors of the clock Cand of the rest of universe Rrespectively.

|Ψiis the global state of the system which is assumed to

be eigenstate with null eigenvalue of the global Hamiltonian

H=Hc+Hr(see text).

[10]. In this framework the state of the “universe” |Ψiis

then identiﬁed by enforcing the Wheeler-De Witt equa-

tion H|Ψi= 0, i.e. by requiring |Ψito be an eigenstate

of Hfor the zero eigenvalue. The rational of this choice

follows from the observation that by projecting |Ψion

the states |φ(t)ic=e−iHct/~|φ(0)icof the clock, one gets

the vectors

|ψ(t)ir:= chφ(t)|Ψi=e−iHrt/~|ψ(0)ir,(1)

that describe a proper evolution of the subsystem Run-

der the action of its local Hamiltonian Hr, the initial

arXiv:1310.4691v1 [quant-ph] 17 Oct 2013

2

state being |ψ(0)ir=chφ(0)|Ψi(see Fig. 1). Therefore,

despite the fact that globally the system appears to be

static, its components exhibits correlations that mim-

ics the presence of a dynamical evolution [9–11]. Two

main ﬂaws of the PaW mechanisms have been pointed

out [2, 15]. The ﬁrst is based on the (reasonable) skepti-

cism to accept that quantum mechanics may describe a

system as large as the universe, together with its internal

observers [11, 12]. The second has a more practical char-

acter and is based on the observation that in the PaW

model the calculations of transition probabilities and of

propagators appears to be problematic [2, 11]. An at-

tempt to ﬁx the latter issue has been discussed by Gam-

bini et al. (GPPT) [12, 16] by extending a proposal by

Page [11] and invoking the notion of ‘evolving constants’

of Rovelli [17] (a brief overview of this approach is given

in the appendix).

In this work we present an experiment which allows

reproducing the basic features of the PaW and GPPT

models. In particular the PaW model is realized by iden-

tifying |Ψiwith an entangled state of the vertical Vand

horizontal Hpolarization degree of freedom of two pho-

tons in two spatial modes c, r, i.e. (see following section)

|Ψi=1

√2(|Hic|Vir− |Vic|Hir),(2)

and enforcing the Wheeler-De Witt equation by taking

Hc=Hr=i~ω(|HihV|−|VihH|) as local Hamiltonians

of the system (ωbeing a parameter which deﬁnes the

time scale of the model). For this purpose rotations of

the polarization of the two photons are induced by forc-

ing them to travel through identical birefringent plates

as shown in Fig. 2. This allows us to consider a set-

ting where everything can be decoupled from the “ﬂow

of time”, i.e. when the photons are traveling outside the

plates. Nonetheless, the clock photon is a true (albeit

extremely simple) clock: its polarization rotation is pro-

portional to the time it spends crossing the plates.

Although extremely simple, our model captures the

two, seemingly contradictory, properties of the PaW

mechanism: the evolution of the subsystems relative to

each other, and the staticity of the global system. This

is achieved by running the experiment in two diﬀerent

modes (see Fig. 2a): (1) an “observer” mode, where the

experimenter uses the readings of the clock photon to

gauge the evolution of the other: by measuring the clock

photon polarization he becomes correlated with the sub-

systems and can determine their evolution. This mode

describes the conventional observers in the PaW mech-

anism: they are, themselves, subsystems of the universe

and become entangled with the clock systems so that

they see an evolving universe; (2) a “super-observer”

mode, where he carefully avoids measuring the properties

of the subsystems of the entangled state, but only global

properties: he can then determine that the global system

is static. This mode describes what an (hypothetical) ob-

server external to the universe would see by measuring

global properties of the state |Ψi: such an observer has

access to abstract coordinate time (namely, in our ex-

Super−observer

mode

2

1

Observer

(b)(a)

mode

A

H

V

H

V

1

2

PBS1

H

V

|ψ>

3

V

4

H

PBS

|ψ>

BV

H

4

3

PBS2

A

BS

PBS

Tomo−

graphy

FIG. 2: Details of the experiment. (a) “Observer” and

“super-observer” mode in the PaW mechanism: one subsys-

tem (polarization of the upper photon) evolves with respect

to a clock constituted by the other subsystem (polarization of

the lower photon). The experimenter in observer mode (pink

box) can prove the time evolution of the ﬁrst photon using

only correlation measurements between it and the clock pho-

ton without access to an external clock. The super-observer

mode (yellow box) proves through state tomography that the

global state of the system is static. (b) Two-time measure-

ments in the GPPT mechanism: the two time measurements

are represented by the two polarizing beam splitters PBS1

and PBS2respectively. The blue boxes (A) represent diﬀer-

ent thicknesses of birefringent plates which evolve the pho-

tons by rotating their polarization: diﬀerent thicknesses rep-

resent diﬀerent time evolutions. The PaW mechanism (a) is

completely independent of the thickness, whereas the GPPT

mechanism (b) allows it to be measured by the experimenter

only through the clock photon (the abstract coordinate time

is unaccessible and averaged away); the dashed box (B) rep-

resents a (known) phase delay of the clock photon only; PBS

stands for polarizing beam splitter in the H/V basis; BS for

beam splitter.

perimental implementation he can measure the thickness

of the plates) and he can prove that the global state is

static, as it will not evolve even when the thickness of

the plates is varied.

In observer mode (Fig. 2a, pink box) the clock is the

polarization of a photon. It is an extremely simple clock:

it has a dial with only two values, either |Hi(detec-

tor 1 clicked) corresponding to time t=t1, or |Vi(de-

tector 2 clicked) corresponding to time t=t2. [Here

t2−t1=π/2ω, where ωis the polarization rotation rate

of the quartz plate, since the polarization is ﬂipped in this

time interval.] The experimenter also measures the po-

larization of the ﬁrst photon with detectors 3 and 4. This

last measurement can be expressed as a function of time

(he has access to time only through the clock photon)

by considering the correlations between the results from

the two photons: the time-dependent probability that

the ﬁrst photon is vertically polarized (i.e. that detector

3 ﬁres) is p(t1) = P3|1and p(t2) = P3|2, where P3|xis the

conditional probability that detector 3 ﬁred, conditioned

on detector xﬁring (experimental results are presented

in Fig. 3a). This type of conditioning is typical of ev-

ery time-dependent measurement: experimenters always

condition their results on the value they read on the lab’s

clock (the second photon in this case). The experimenter

has access only to physical clocks, not to abstract coordi-

nate time [10, 17, 18]. In our experiment this restriction

3

is implemented by employing a diﬀerent phase plate A

(of random thickness unknown to the experimenter) in

every experimental run.

In super-observer mode (Fig. 2a, yellow box) the ex-

perimenter takes the place of a hypothetical observer ex-

ternal to the universe that has access to the abstract

coordinate time and tests whether the global state of the

universe has any dependence on it. Hence, he must per-

form a quantum interference experiment that tests the

coherence between the diﬀerent histories (wavefunction

branches) corresponding to the diﬀerent measurement

outcomes of the internal observers, represented by the

which-way information after the polarizing beam splitter

PBS1. In our setup, this interference is implemented by

the beam splitter BS of Fig. 2b. It is basically a quantum

erasure experiment [19, 20] that coherently “erases” the

results of the time measurements of the internal observer:

conditioned on the photon exiting from the right port

of the beam splitter, the information on its input port

(i.e. the outcome of the time measurement) is coherently

erased [21]. The erasure of the time measurement by the

internal observers is necessary to avoid that the exter-

nal observer (super-observer) himself becomes correlated

with the clock. However, the super-observer has access

to abstract coordinate time: he knows the thickness of

the blue plates, which is precluded to the internal ob-

servers, and he can test whether the global state evolves

(experimental results are presented in Fig. 3b).

In addition, we also test the GPPT mechanism show-

ing that our experiment can also account for two-time

measurements (see Fig. 2b). These are implemented by

the two polarizing beam splitter PBS1and PBS2. PBS1

represents the initial time measurement that determines

when the experiment starts: it is a non-demolition mea-

surement obtained by coupling the photon polarization

to its propagation direction, while the initialization of the

system state is here implemented through the entangle-

ment. PBS2together with detectors 1 and 2 represents

the ﬁnal time measurement by determining the ﬁnal po-

larization of the photon. Between these two time mea-

surements both the system and the clock evolve freely

(the evolution is implemented by the birefringent plates

A). In the GPPT mechanism, the abstract coordinate

time (the thickness of the quartz plates A) is unaccessi-

ble and must be averaged over [11, 12, 16]. This restric-

tion is implemented in the experiment by avoiding to

take into account the thickness of the blue quartz plates

A when extracting the conditional probabilities from the

coincidence rates: the rates obtained with diﬀerent plate

thickness are all averaged together. The formal mapping

of the GPPT mechanism to our experiment is detailed in

the appendix.

As before, the time dependent probability of ﬁnding

the system photon vertically polarized is p(t1) = P3|1

and p(t2) = P3|2. However, a clock that returns only

two possible values (t1and t2) is not very useful. To ob-

tain a more interesting clock, the experimenter performs

the same conditional probability measurement introduc-

(a) (b)

t1

t0

FIG. 3: PaW experimental results. (a) Observer mode: plot

of the clock-time dependent probabilities of measurement out-

comes as a function of the of the plate thickness (correspond-

ing to abstract coordinate time T): circles and squares rep-

resent p(t1) = P3|1and p(t2) = P3|2respectively, namely the

probabilities of measuring Von the subsystem 1 as a func-

tion of the clock time t1,t2; circles and triangles represent

P4|1and P4|2, the probabilities of measuring Hon the sub-

system 1 as a function of the clock time. As expected from

the PaW mechanism, these probabilities are independent of

the abstract coordinate time T, represented by diﬀerent phase

plate A thicknesses (here we used a 957µm thick quartz plate

rotated by 15 diﬀerent equiseparated angles). The inset shows

the graph that the observer himself would plot as a function

of clock-time: circles representing the probabilities of ﬁnding

the system photon Vat the two times t1,t2, the triangles of

ﬁnding it H. (b) Super-observer mode: plot of the conditional

ﬁdelity between the tomographic reconstructed state and the

theoretical initial state |Ψiof Eq. (2) as a function of the ab-

stract coordinate time T. The ﬁdelity F=hΨ|ρout|Ψi(which

measures the overlap between the theoretical initial state |Ψi

and the ﬁnal state ρout after its evolution through the plates)

is conditioned on the clock photon exiting the right port of

the beam splitter BS. The fact that the ﬁdelity is constant

and close to one (up to experimental imperfections) proves

that the global entangled state is static.

ing varying time delays to the clock photon, implemented

through quartz plates of variable thickness (dashed box

B in Fig. 2b). [Even though he has no access to abstract

coordinate time, he can have access to systems that im-

plement known time delays, that he can calibrate sepa-

rately.] Now, he obtains a sequence of time-dependent

values for the conditional probability: p(t1+τi) = Pτi

3|1

and p(t2+τi) = Pτi

3|2, where τi=δi/ω is the time delay of

the clock photon obtained by inserting the quartz plate

B with thickness δiin the clock photon path. The exper-

imental results are presented in Fig. 4, where each colour

represents a diﬀerent delay: the yellow points refer to τ0;

the red points to τ1, etc. They are in good agreement

with the theory (dashed line) derived in the appendix.

The reduction in visibility of the sinusoidal time depen-

dence of the probability is caused by the decoherence

eﬀect due to the use of a low-resolution clock (our clock

outputs only two possible values), a well known eﬀect

[10, 16, 22, 23].

In summary, by running our experiment in two dif-

ferent modes (“observer” and “super-observer” mode)

we have experimentally shown how the same energy-

4

FIG. 4: GPPT experimental results: probability p(t) that

the upper photon is V(namely that detector 3 clicked) as a

function of the time trecovered from the lower photon. The

points with matching colors represent p(t1+τi) and p(t2+τi):

yellow, red, blue, etc., for i= 0,1,2,· · · , respectively. Here

nine diﬀerent values of τiare obtained from a 1752µm thick

quartz plate rotated by nine diﬀerent angles from the vertical

(14,16,18,20,21.5,23,25,27,29 degrees). The dashed line is the

theoretical value. Its reduced visibility is an expected eﬀect

of the use of imperfect clocks [10, 16, 22].

entangled Hamiltonian eigenstate can be perceived as

evolving by the internal observers that test the correla-

tions between a clock subsystem and the rest (also when

considering two-time measurements), whereas it is static

for the super-observer that tests its global properties.

Our experiment is a practical implementation of the PaW

and GPPT mechanisms but, obviously, it cannot discrim-

inate between these and other proposed solutions for the

problem of time [2–6]. In closing, we note that the time-

dependent graphs of Fig. 4 have been obtained without

any reference to an external time (or phase) reference,

but only from measurements of correlations between the

clock photon and the rest: they are an implementation of

a ‘relational’ measurement of a physical quantity (time)

relative to an internal quantum reference frame [24, 25].

Experimental setup

The experimental setup (Fig. ??) consists of two

blocks: “preparation” and “measurement”. The prepa-

ration block produces a family of biphoton polarization

entangled states of the form:

|Ψi= cos θ|HHi+eiϕ sin θ|V V i(3)

by exploiting the standard method of coherently super-

imposing the emission of two type I crystals whose optical

axes are rotated of 90o[26].

The measurement block can be mounted in diﬀerent

conﬁgurations corresponding to “observer” and “super-

observer” ones of PaW and GPPT scheme (Fig.1). In

general, each arm of the measurement block contains in-

terference ﬁlters (IF) with central wavelength 702 nm

(FWHM 1 nm) and a polarizing beam splitter (PBS).

Before the PBS the polarization of both photons evolves

in the birefringent quartz plates A (blue boxes in Fig. 2)

as |Vi → |Vicos δ+i|Hisin δ, where δis the material’s

optical thickness. “Observer” mode in PaW scheme

(Fig. 2, block a): In this mode, the polarization of the

photon in the lower arm is used as a clock: the ﬁrst polar-

izing beam splitter PBS1acts as a non-demolition mea-

surement in the H/V basis of the polarization of the sec-

ond photon, ﬁnally detected by single-photon avalanche

diodes (SPAD) 1, 2. In this mode, the experimenter

has no access to an external clock, he can only use the

correlations (coincidences) between detectors: the time-

dependent probability of ﬁnding the ﬁrst photon in |Viis

obtained from the coincidence rate between detectors 1-3

(corresponding to a measurement at time t1), or 2-3 (cor-

responding to a measurement at time t2): appropriately

normalized, these coincidence rates yield the conditional

probabilities P3|x. The impossibility to directly access

abstract coordinate time (the thickness of the plates) is

implemented by averaging the coincidence rates obtained

for all possible thicknesses of the birefringent plates A:

the plate thickness does not enter into the data process-

ing in any way.

“Super-observer” mode in PaW scheme (Fig.1b):

This mode is employed to prove that the global state

is static with respect to abstract coordinate time, rep-

resented by the thickness of the quartz plates A. The

50/50 beam splitter (BS) in block b performs a quan-

tum erasure of the polarization measurement (performed

by the polarizing beam splitter PBS1) conditioned on

the photon exiting its right port. For temporal stabil-

ity, the interferometer is placed into a closed box. The

output state is reconstructed using ququart state tomog-

raphy [27–29] (the two-photon polarization state lives in

a four-dimensional Hilbert space), where the projective

measurements are realized with polarization ﬁlters con-

sisting of a sequence of quarter- and half-wave plates and

a polarization prism which transmits vertical polariza-

tion (Fig.4). The ﬁdelity between the tomographically

reconstructed state and the theoretical state |Ψiis re-

ported in Fig. 3b. GPPT two-time scheme Here a

second PBS preceding detectors allows a two-time mea-

surement. To obtain a more interesting time dependence

than the probability at only two times, we delay the clock

photon with an additional birefringent plate B (dashed

box in Fig. 2), a 1752µm-thick quartz plate rotated at

5

l

2

f

V

M

Ar laser

BBO

TUWF

BS

45

O

l

2@

l

4

V

l

2

IF

SPAD

APD

CC

SPAD

PaW

GPPT

Tomography

Preparation block

FIG. 5: Preparation block: Pairs of degenerate entangled

photons are produced by pumping two orthogonally oriented

type I BBO (β−BaB2O4) crystals (placed into a temperature-

stabilized closed box T) pumped by a 700 mW Ar laser, later

eliminated by a ﬁlter (UWF). The basic state amplitudes are

controlled by a Thompson prism (V), oriented vertically, and

a half-wave plate λ/2 at angle θ. Two 1 mm quartz plates,

that can be rotated along the optical axis, introduce a phase

shift ’ between horizontally and vertically polarized photons.

The beam splitter (BS) is used to split the initial (collinear)

biphoton ﬁeld into distinct spatial modes. It prepares the

singlet Bell state Ψ of Eq. (1) (using θ= 45o; ’φ= 0o, and

an additional half-wave plate λ/2 at 45oin the transmitted

arm). Measurement block: We implement PaW or GPPT

as in Fig.1. In PaW superobserver mode the ﬁnal state is

checked by quantum state tomography [27–29], realised by

registering the coincidence rate for 16 diﬀerent projections

achieved through half and quarter wave plates and a ﬁxed

analyzer (V).

nine diﬀerent angles, placed in the lower arm, and we

repeat the same procedure described above for diﬀerent

thicknesses of the plate B. This represents an internal ob-

server that introduces a (known) time delay to his clock

measurements. The results are shown in Fig. 3.

Appendix

In this appendix we detail how our experiment imple-

ments the Gambini et al. (GPPT) proposal [12, 16] for

extending the PaW mechanism [9–11] to describe multi-

ple time measurements. We also derive the theoretical

curve of Fig. 4.

Time-dependent measurements performed in the lab

typically require two time measurements: they establish

the times at which the experiment starts and ends, re-

spectively. The PaW mechanism can accommodate the

description of these situations by supposing that the state

of the universe will contain records of the previous time

measurements [11]. However, this observation in itself

seems insuﬃcient to derive the two-time correlation func-

tions (transition probabilities and time propagators) with

their required properties, a strong criticism directed to

the PaW mechanism [2, 11]. The GPPT proposal man-

ages to overcome this criticism. It is composed of two

main ingredients: the recourse to Rovelli’s ‘evolving con-

stants’ to describe observables that commute with global

constraints, and the averaging over the abstract coordi-

nate time to eliminate any dependence on it in the ob-

servables. Our experiment tests the latter aspect of the

GPPT theory.

Measurements of a physical quantity at a given clock

time, say t, are described by the conditional probability

of obtaining an outcome on the system, say d, given that

clock time-measurement produces the outcome t. This

conditional probability is given by [12, 16]

p(d|t) = RdT Tr[Pd,t(T)ρ]

RdT Tr[Pt(T)ρ],(4)

where ρis the global state, Pt(T) is the projector rela-

tive to a result tfor a clock measurement at coordinate

time Tand Pd,t(T) is the projector relative to a result d

for a system measurement and tfor a clock measurement

at coordinate time T(working in the Heisenberg picture

with respect to coordinate time T). Clearly, such expres-

sion can be readily generalized to arbitrary POVM mea-

surements. (A similar expression, but in the Schr¨odinger

picture, already appears in [11].) The integral that aver-

ages over the abstract coordinate time Tin (4) embodies

the inaccessibility of the time Tby the experimenter: he

can access only the clock time t, an outcome of measure-

ments on the clock system.

A generalization of this expression to multiple time

measurements is expressed by [12]

p(d=d0|tf, di, ti) (5)

=RdT RdT 0Tr[Pd0,tf(T)Pdi,ti(T0)ρ Pdi,ti(T0)]

RdT RdT 0Tr[Ptf(T)Pdi,ti(T0)ρ Pdi,ti(T0)] ,

which gives the conditional probability of obtaining d0

on the system given that the ﬁnal clock measurement

returns tfand given that a “previous” joint measurement

of the system and clock returns di,ti. (This expression

can also be formulated as a conventional state reduction

driven by the ﬁrst measurement [16].)

In our experiment to implement the GPPT mechanism

(Fig. 2b) we must calculate the conditional probability

that the system photon is V(namely detector 3 clicks)

given that the clock photon is Hafter the ﬁrst polariz-

ing beam splitter PBS1(initial time measurement) and

is Hor Vafter the second polarizing beam splitter (ﬁnal

time measurement). The initial time measurement suc-

ceeds whenever one of photodetectors 1 or 2 click: this

means that the clock photon chose the Hpath at PBS1.

(Our experiment discards the events where the ﬁrst time

measurement at PBS1ﬁnds V, although in principle one

could easily take into account these cases by adding a po-

larizing beam splitter and two photodetectors also in the

Voutput mode of PBS1.) The ﬁnal time measurement is

given by the click either at photodetector 1 or 2: the clock

dial shows tf=t1and tf=t2=t1+π/2ω, respectively.

Using the GPPT mechanism of Eq. (5), this means that

the time dependent probability that the system photon

is vertical (detector 3 clicks) is given by

p(d= 3|tf=tk, di, ti) (6)

=RdT RdT 0Tr[Pd=3,tf=tk(T)Pdi,ti(T0)ρPdi,ti(T0)]

RdT RdT 0Tr[Ptf=tk(T)Pdi,ti(T0)ρPdi,ti(T0)] ,

6

where Pd=3,tf=tkis the joint projector connected to de-

tector 3 and detector k= 1 or k= 2 and Pdi,tiis the

projector connected to the ﬁrst time measurement. The

latter projector is implemented in our experiment by con-

sidering only those events where either detector 1 or de-

tector 2 clicks, this ensures that the clock photon chose

the Hpath at PBS1(namely the initial time is ti) and

that the system photon was initialized as |Viat time

ti. (In principle, we could consider also a diﬀerent ini-

tial time t0

iby employing also the events where the clock

photons choose the path Vat PBS1.) Introducing the

unitary abstract-time evolution operators, UT, the nu-

merator of Eq. (6) becomes

ZdT ZdT 0Tr[Pd=3,tf=tkUT−T0Pdi,tiUT0ρU †

T0Pdi,ti×

U†

T−T0] = ZdT Tr[Pd=3,tf=tkUTPdi,tiρPdi,tiU†

T],

where we use the property UTUT0†=UT−T0and we

dropped one of the two time integrals by taking advan-

tage of the time invariance of the global state ρ(which

has been also tested experimentally in the super-observer

mode). Gambini et al. typically suppose that the clock

and the rest are in a factorized state [16], but this hy-

pothesis is not strictly necessary for their theory [12]: we

drop it so that we can use the same initial global state

that we used for testing the PaW mechanism.

Using the same procedure also to calculate the denom-

inator of Eq. (6), we can rewrite this equation as

p(d= 3|tf=tk, di, ti) = Tr[Pd=3,tf=tk¯ρ]

Tr[Ptf=tk¯ρ],(7)

where ¯ρis the time-average of the global state after the

ﬁrst projection, namely

¯ρ∝ZdT UTρti,diU†

T, ρti,di≡Pdi,tiρPdi,ti,(8)

where the averaging over the abstract coordinate time T

is used to remove its dependence from the state. In our

experiment such average is implemented by introducing

random values of the phase plates A (unknown to the

experimenter) in diﬀerent experimental runs.

In our GPPT experiment there are two possible values

for the initial projector Pdi,ti: either the clock photon

is projected on the Hpath after PBS1(corresponding

to an initial time ti) or it is projected onto the Vpath

(corresponding to an initial time ti+π/2ω). We will

consider only the ﬁrst case, which corresponds to a click

of either detector 1 or 2: we are post-selecting only on

the experiments where the initial time is ti. In this case,

the global initial state will be |Hic|Virwhich is evolved

into the vector |Ψ(T)i= [cos(ω(T+τ))|Hic−sin(ω(T+

τ))|Vic][cos ωT |Vir+ sin ωT |Hir] where His the global

Hamiltonian deﬁned in the main text and τis the time

delay introduced by the plate B of Fig. 2b. Moreover,

the projectors in Eq. (7) are

Pd=3,tf=tk≡ |kichk|⊗|VirhV|,and

Ptf=tk≡ |kichk| ⊗ 11r,(9)

where |k= 0ic≡ |Hicand |k= 1ic≡ |Vic. The projec-

tor Pd=3,tf=tkcorresponds to the joint click of detectors

kand 3, while Ptf=tkcorresponds to the click of detec-

tor kand either one of detectors 3 or 4. In other words,

Eq. (7) can be written as

p(d= 3|tf=tk, di, ti) = P3k/(P3k+P4k),(10)

where Pjk is the joint probability of detectors jand k

clicking. For example, P32 is the joint probability that

detector 3 and 2 click, namely that both the clock and the

system photon were V. Considering only the component

|Vic|Virof the state |Ψ(T)i, this is given by

P32 =1

2πZ2π

0

dϕ sin2(ϕ+ωτ ) cos2ϕ=1 + 2 cos2ωτ

8(11)

where we have calculated the integral over Tof Eq. (8)

using a change of variables ωT =ϕ. Proceeding analo-

gously for all the other joint probabilities, namely replac-

ing the projectors (9) into (7), we ﬁnd the probability for

detector 3 clicking (namely the system photon being V)

conditioned on the time tfread on the clock photon as

p(3|tf=t1) = (1 + 2 cos2ωτ)/4 (12)

p(3|tf=t2) = (1 + 2 sin2ωτ)/4,(13)

which is plotted as a function of τin Fig. 3b (dashed

line). Since t2=t1+π/2ω, we have plotted the points

relative to p(3|t2) as displaced by π/2 with respect to the

points relative to p(3|t1), so that the two curves (12) and

(13) are superimposed in Fig. 3.

Acknowledgments

We thank A. Ashtekar for making us aware of Ref. [12].

We acknowledge the Compagnia di San Paolo for partial

support.E.V.Moreva acknowledges the support from the

Dynasty Foundation and Russian Foundation for Basic

Research (project 13-02-01170-D)

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