Time from quantum entanglement: An experimental illustration

Abstract

In the last years several theoretical papers discussed if time can be an
emergent property 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.

Full text

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)
[1] A. Augustinus Hipponensis, Confessiones, XI, 14 (400).
[2] K.V. Kucha˘r, in “Proceedings of the 4th Canadian Con-
ference on General Relativity and Relativistic Astro-
physics” ed. G. Kunstatter, D. Vincent and J. Williams

7
(World Scientific, Singapore 1992).
[3] C.J. Isham, in “Integrable Systems, Quantum Groups
and Quantum Field Theories” ed. L.A. Ibort and M.A.
Rodr´ıguez (Kluwer, Dordrecht 1993), gr-qc/9210011.
[4] A. Ashtekar, New J. Phys. 7, 198 (2005).
[5] E. Anderson, “The Problem of Time in Quantum Grav-
ity”, in “Classical and Quantum Gravity: Theory, Anal-
ysis and Applications”, Ed. V.R. Frignanni (Nova, New
York 2012), arXiv:1009.2157v3.
[6] Z. Merali, Nature 500, 516-519 (2013).
[7] B.S. DeWitt, Phys. Rev. 160, 1113-1148 (1967).
[8] J. B. Hartle, S. W. Hawking, Phys. Rev. D 28, 2960-2975
(1983).
[9] D.N. Page and W.K. Wootters, Phys. Rev. D, 27, 2885
(1983).
[10] W.K. Wootters, Int. J. Theor. Phys. 23, 701 (1984).
[11] D.N. Page, “Clock time and entropy”, in Physical Ori-
gins of Time Asymmetry, eds. J.J. Halliwell, et al., (Cam-
bridge Univ. Press, 1993), arXiv:gr-qc/9303020.
[12] R. Gambini, R.A. Porto, J. Pullin, and S. Torterolo,
Phys. Rev. D 79, 041501(R) (2009).
[13] A. Peres, Am. J. Phys. 48, 552 (1980).
[14] C. Rovelli, Int. J. of Theor. Phys. 35, 1637 (1996),
arXiv:quant-ph/9609002v2.
[15] W.G. Unruh, R.M. Wald, Phys. Rev. D 40, 2598 (1989).
[16] R.Gambini, L.P.G. Pintos, J. Pullin, Studies His. Phil.
Mod. Phys. 42, 256-263 (2011).
[17] C. Rovelli, Phys. Rev. D 43, 442 (1991).
[18] A. Einstein, Ann. Phys. 17, 891, (1905), translated in
The Principle of Relativity (Methuen & Co., London,
1923).
[19] M. O. Scully, B.-G. Englert, H., Walther , Nature 351,
111 (1991).
[20] D¨urr S., Nonn T., Rempe G., Nature 395, 33 (1998).
[21] J. Preskill, lecture notes at http://www.theory.cal-
tech.edu/people/preskill/ph229, pg. 68.
[22] R. Gambini, L.P.G. Pintos, J. Pullin, Found. Phys. 40,
93-115 (2010).
[23] I.L. Egusquiza, L.J. Garay, J.M. Raya, Phys. Rev. A 59,
3236 (1999).
[24] S.D. Bartlett, T. Rudolph, W. Spekkens, Rev. Mod.
Phys. 79, 555 (2007).
[25] H.M. Wiseman, J. Opt. B: Quantum Semiclass. Opt. 6,
S849 (2004).
[26] M.Genovese, Phys. Rep. 413, 319 (2005).
[27] D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G.
White, Phys. Rev. A, 64, 052312 (2001).
[28] Yu.I.Bogdanov , G.Brida, M. Genovese, A.P. Shu-
rupov,E.V. Moreva,S.P. Kulik , Phys. Rev. Lett. 105,
010404 (2010).
[29] Yu. I. Bogdanov, E. V. Moreva, G. A. Maslennikov, R.
F. Galeev, S. S. Straupe and S. P. Kulik, Phys. Rev. A
73, 063810 (2006).