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Probing the Non-Classicality of Temporal Correlations

Martin Ringbauer

1 2 3

and Rafael Chaves

Centre for Engineered Quantum Systems, School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia

Centre for Quantum Computer and Communication Technology, School of Mathematics and Physics, University of Queensland, Brisbane,

QLD 4072, Australia

Institute of Photonics and Quantum Sciences, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14

4AS, UK

International Institute of Physics, Federal University of Rio Grande do Norte, 59070-405 Natal, Brazil

November 15, 2017

Correlations between spacelike separated

measurements on entangled quantum systems

are stronger than any classical correlations and

are at the heart of numerous quantum tech-

nologies. In practice, however, spacelike sepa-

ration is often not guaranteed and we typically

face situations where measurements have an

underlying time order. Here we aim to pro-

vide a fair comparison of classical and quan-

tum models of temporal correlations on a sin-

gle particle, as well as timelike-separated cor-

relations on multiple particles. We use a causal

modeling approach to show, in theory and

experiment, that quantum correlations out-

perform their classical counterpart when al-

lowed equal, but limited communication re-

sources. This provides a clearer picture of the

role of quantum correlations in timelike sepa-

rated scenarios, which play an important role

in foundational and practical aspects of quan-

tum information processing.

Quantum predictions are fundamentally incompat-

ible with the intuitive notion of cause and eﬀect that

underpins all of classical empirical science, as well

as everyday experience. Most famously, Bell’s the-

orem [1] shows that the correlations revealed by mea-

surements on distant parts of entangled quantum sys-

tems cannot be explained in causal terms, under the

assumptions that local measurement outcomes are not

inﬂuenced by spacelike separated events—local causal-

ity—and that measurement settings can be chosen

freely. This phenomenon has become known as quan-

tum nonlocality, and has been extensively studied for

variants of the scenario above, where spacelike sepa-

rated parties perform measurements on a composite

quantum system [2].

Yet, in practice we often face situations where

spacelike separation between observers is not guar-

anteed and instead there is some time order underly-

ing the observed physical events. Indeed, it has been

shown that a single quantum system measured at dif-

ferent points in time exhibits Bell-like correlations [3–

8]. However, in such a scenario one cannot appeal to

relativity to rule out communication between the time

steps. This can be problematic, since a classical model

with suﬃcient communication can simulate any cor-

relations. Recent research has thus been devoted to

clarifying to what extent the incompatibility between

classical and quantum descriptions encountered in the

spacelike case carries over to the timelike scenario.

Speciﬁcally, one research direction has been to

study how limited resources such as entropy [9], mem-

ory [10–12] or dimension [13–15] can lead to a quan-

tum advantage over classical models. On the other

hand, one can relax the local causality assumption in

Bell’s theorem, aiming to explain quantum correla-

tions by classical models augmented with communi-

cation [16–25]. However, communication is typically

only allowed for the classical system, leading to an un-

fair comparison between classical and quantum mod-

els, and it remains unclear to what extent these re-

sults hold when equal communication power is given

to quantum mechanics [23].

Here we use a causal modeling approach to allow

a fair comparison of classical and quantum models

of timelike separated correlations. First, we show

that for two time-ordered spatially separated mea-

surements, augmented with a limited amount of clas-

sical communication, quantum correlations outper-

form their classical counterpart. Second, we show

that, in contrast to spatial Bell-type scenarios [26],

there are faithful classical causal models reproducing

all the temporal correlations obtained from a series of

projective quantum measurements on a single quan-

tum system. However, we ﬁnd that non-classical cor-

relations can arise in this scenario, when considering a

slightly weaker classical causal model, that is nonethe-

less strictly stronger than previous results [27], which

are contained as a special case. Finally we derive Bell-

type inequalities for the above scenarios and demon-

strate that they are violated in a photonic experiment.

arXiv:1704.05469v2 [quant-ph] 14 Nov 2017

1 Causal modeling and timelike Bell-

scenarios

In the following we employ the formalism of Bayesian

networks [28], which provides a natural framework for

classical causal modeling. A central concept in this

framework is that of a directed acyclic graph (DAG),

which consists of a set of nodes, representing the rel-

evant random variables

in the considered situation,

and directed edges, representing the causal relations

between those variables. A set of variables X

, . . . , X

forms a Bayesian network with respect to some DAG

if and only if the probability distribution p(x

, . . . , x

)

can be decomposed as

p(x

, . . . , x

) =

i=1

p(x

|pa

) (1)

where P A

stands for the set of graph-theoretical par-

ents of the variable X

(i.e. all variables that have a

direct causal inﬂuence over X

). Without loss of gen-

erality each variable can be understood as a determin-

istic function of its parents plus local noise U

that

supplies potential randomness, x

= f

(pa

, u

). This

formalism thus enables a distinction between simple

statistical correlations and actual causation by explic-

itly specifying the underlying mechanism generating

the data.

Here we are interested in DAGs containing so-called

latent variables, which are empirically inaccessible. In

the context of Bell’s theorem [1] these are also known

as hidden variables. For any set of observed corre-

lations, there are in general many DAGs with hid-

den variables that could have produced these obser-

vations. Among these, causal inference is particularly

interested in those fulﬁlling the conditions of minimal-

ity and faithfulness. Minimality requires that, given

two possible causal models, we choose the simplest

one, capable of generating the smallest set of correla-

tions (including the observed one). In turn, faithful-

ness, requires the causal model to be able to explain

the observed data without resorting to ﬁne-tuning of

the causal-statistical parameters. In other words, any

observed (conditional) independence should be a con-

sequence of the causal structure itself, rather than a

speciﬁc choice of parameters. Faithful (i.e. non-ﬁne-

tuned) models are therefore robust against changes in

the causal parameters and thus the preferred choice.

To illustrate the last point, consider the paradig-

matic causal structure of Bell’s theorem in Fig. 1a.

This structure intuitively reﬂects the causal assump-

tions of Bell’s theorem, leading to the so-called local

hidden-variable (LHV) models. First, the two parties

are assumed to be spacelike separated, such that the

correlations between the measurements outcomes A

We adopt the standard convention that uppercase letters

label random variables while their values are denoted in lower

case.

and B can only be mediated via a common source

Λ, implying that p(a, b|x, y, λ) = p(a|x, λ)p(b|y, λ).

Second, it is assumed that the experimenters can

freely choose which observables to measure (repre-

sented by the random variables X and Y ), inde-

pendently of how the system was prepared, that is,

p(x, y, λ) = p(x, y)p(λ). Note that these constraints

implied by the causal model appear at an unobserv-

able level since they explicitly involve the hidden vari-

ables Λ. Yet, they also imply observable constraints

in the form of no-signaling conditions, expressed as

p(a|x, y) = p(a|x) and p(b|x, y) = p(b|y), and Bell

inequalities [1, 29].

Figure 1: Causal structure underlying Bell’s theorem. (a)

Two observers, Alice and Bob, each have the choice of two

measurements represented by the random variables X and Y ,

respectively. The correlation between their measurement out-

comes, modeled as random variables A and B, respectively,

are mediated solely by a common cause in their past—the

hidden variable Λ. (b) Bell’s causal model augmented with

one-way communication from Alice to Bob. The initial state

of the joint system is speciﬁed by the ontic state Λ. First,

Alice performs a measurement with setting x, obtaining out-

come a. She then sends a message m to Bob, who performs

a measurement with setting y, obtaining outcome b.

While quantum correlations obey the no-signaling

conditions, they violate Bell inequalities [1, 29] and

are thus in conﬂict with the assumptions behind the

causal structure in Fig. 1a. In order to maintain a

classical causal explanation, the model in Fig. 1a must

therefore be augmented with additional resources;

something that can only be done at the cost of in-

troducing ﬁne-tuning [26]. For instance, the causal

structure in Fig. 1b can reproduce all quantum corre-

lations, but at the same time allows, in principle, for

non-local correlations between X and B. Hence, in

order to satisfy the no-signaling condition p(b|x, y) =

p(b|y) the causal parameters must be chosen from a

set of measure zero [28], a signature of ﬁne-tuning.

Studying such non-local classical models can pro-

vide valuable insights into the relation between clas-

sical and quantum theory, and their applications [2].

However, at the same time such models lead to an

unfair comparison, since allowing for communication

makes not only classical, but also quantum models

more powerful. In practice, it is more natural to

assume a certain underlying causal structure, and

ask what can be achieved with classical and quan-

tum resources? Bell’s theorem is a particular case

of this broader question, referring to spacelike sep-

arated events. However, there are often situations

where the events are timelike rather than spacelike

ordered. Examples include central quantum infor-

mation tasks, such as teleportation [30], superdense

coding [31], and measurement-based quantum compu-

tation [32], as well as prepare-and-measure scenarios

[13], sequential Bell scenarios [33, 34] and a sequence

of measurements on a single quantum system [27].

2 Non-classicality of timelike correla-

tions augmented by communication

Consider the scenario in Fig. 1b, where two distant

parties, Alice and Bob, share pre-established cor-

relations (represented by Λ) and are allowed one-

way communication (the message M). As shown in

Ref. [16], a classical model of this form (for a large

enough message M) is enough to reproduce all the

correlations obtainable from local measurements on

two-qubit entangled states as in Fig. 1a, which are

described by

p(a, b|x, y) = Tr



⊗ M

)ρ



, (2)

where M

and M

are measurement operators for Al-

ice and Bob, respectively, and ρ is the density matrix

describing the shared quantum state. If, in contrast,

we impose the causal structure of Fig. 1b also to the

quantum case—that is, Bob’s measurement may de-

pend on Alice’s measurement setting and outcome—

then the set of correlations is described by

p(a, b|x, y) =



⊗ M

y,m

)ρ



. (3)

Note that Bob’s measurement operator now explicitly

depends on the values of X and A (via the message

. Clearly, if there are no restrictions on the di-

mensionality of the message, every distribution of the

form above can be also obtained by the classical hid-

den variable model in Fig. 1b, that is, by a model

respecting the decomposition

p(a, b|x, y) =

m,λ

p(λ)p(m|x, a)p(a|x, λ)p(b|y, m, λ).

(4)

In fact, it is enough to choose m = x to reproduce

all possible one-way signalling correlations. To see

this, note that the quantum correlations arising from

Eq. (3) are of the form p(a, b|x, y) = p(b|x, y, a)p(a|x),

and that a can be made a deterministic function a =

(x, λ). Hence a does not carry any information that

is not already contained in λ and x.

Notwithstanding, this picture changes if we impose

restrictions on the message sent from Alice to Bob.

Consider that each party measures three dichotomic

In fact, Bob could apply to his share of the joint quantum

state any completely-positive trace-preserving map dependent

on the message m.

observables (i.e. x, y = 0, 1, 2 and a, b = 0, 1) and

that Alice is bound to send a binary message (m =

0, 1). In this case, every classical model must obey

the inequality [16]

1bit

= hA

i + hA

i (5)

+ hA

i − hA

i + hA

i − hA

i ≤ 6.

Furthermore, it was shown in Ref. [16] that this in-

equality also holds for correlations from local mea-

surements on entangled quantum states, while it

can be violated by more powerful no-signalling cor-

relations. Hence, while one bit of communication

is in this scenario suﬃcient for a classical model

to simulate quantum correlations (without commu-

nication), it is not suﬃcient to simulate all pos-

sible no-signalling correlations. However, just like

communication-augmented classical models become

more powerful, so do quantum models. Speciﬁcally,

local measurements on entangled quantum states aug-

mented with one bit of communication can indeed

violate inequality (5) [23], thus showing that under

fair comparison, quantum advantage persists in such

a timelike-separated Bell scenario.

state preparation

Alice

Bob

HWP

APD

Source

QWP

PPBS

Figure 2: Experimental setup for studying the non-

classicality of timelike correlations. Pairs of single photons

are produced via spontaneous parametric downconversion in

a β-barium borate crystal (not shown). The two photons

are entangled to arbitrary degree using a non-deterministic

controlled-NOT gate (CNOT), based on nonclassical interfer-

ence in a partially polarizing beam splitter (PPBS) [35]. Alice

and Bob then perform local projective measurements on their

share of the entangled state, which are implemented using a

set of half- (HWP) and quarter-waveplates (QWP), a Glan-

Taylor polarizer (GT) and single-photon counters (APD).

As an example, consider that Alice and Bob share a

maximally entangled state |Ψ

i = (|00i + |11i)/

√

2).

Alice performs local measurements with settings A

X, A

Z and encodes her measurement set-

ting in a message m to Bob. For x = 0 she sends

m = 0 and for x = 1 or x = 2 she sends m = 1. As-

suming that all inputs are equally likely this message

has an entropy of H(m) ∼ 0.92 and thus contains less

than 1 bit of information. If Bob receives m = 0, he

measures B

= B

X, while for m = 1 he

measures B

= (

X +

Z)/

√

2, B

= (

X −

Z)/

√

2 and

= −

X. This protocol achieves S

1bit

= 4

√

2, thus

violating inequality (5).

Experimentally we can test inequality (5) with

qubits encoded in the polarization of single photons,

see Fig. 2. Two single photons are ﬁrst entangled

using a non-deterministic controlled-NOT gate, and

then distributed to Alice and Bob. By varying the in-

put states this conﬁguration can produce states with

arbitrary degree of entanglement, quantiﬁed by the

concurrence C [36], see Appendix B for more details.

For simplicity, the message m has been directly taken

into account in Bob’s measurement basis. The ex-

perimental results in Fig. 3 show a clear violation of

inequality (5), up to S

1bit

= 6.66

+0.02

−0.02

fixed

optimized

Figure 3: Experimental test of inequality (5) for

temporally-separated spatial correlations. The inequality

is tested for a family of states with varying degree of en-

tanglement as quantiﬁed by the concurrence. Data points

include 3σ statistical error bars and the theory prediction is

shown as solid lines. Data points in the grey region are com-

patible with inequality (5), while it is violated for data in

the white region. Blue data and theory corresponds to the

ﬁxed measurement scheme outlined in the text, whereas the

orange data and theory is obtained when the measurement

settings are optimized for a given non-maximally entangled

state, see Appendix B for more details.

3 Classical causal models for a se-

quence of projective measurements

Above we have analyzed the correlations between two

timelike separated parties, whose measurements fol-

low an underlying time order. We will now focus on

a series of projective measurements performed on a

single quantum system. A measurement with setting

x, obtaining outcome a, is described by a set of pro-

jective operators M

, such that after the measure-

ment the system, initially in state ρ

, is left in a state

given by ρ

= ρ

= M

. In a next time step a mea-

surement with setting y, producing outcome b is per-

formed, leaving the system in a state ρ

= ρ

= M

and so forth.

Similar to a standard Bell experiment, the clas-

sical causal description of this scenario, illustrated

in Fig. 4, involves a random variable Λ

—the ontic

state [37]—which fully speciﬁes the initial state of

the system. The probability that a measurement x

produces the outcome a is then given by p(a|x) =

p(a|x, λ

)p(λ

). Here we have explicitly used

the measurement independence assumption [22, 38]

p(λ

|x) = p(λ

), that the measurement setting can

be chosen independently of how the system is pre-

pared. After the measurement, the system will be in

a potentially diﬀerent state Λ

. For a projective mea-

surement, Λ

is fully speciﬁed by the setting and out-

come of the preceding measurement, and does not de-

pend directly on the pre-measurement state Λ

, that

is p(λ

|x, a, λ

) = p(λ

|x, a). This implies that all

correlations between Λ

and Λ

are mediated via the

measurement. To see this, note that the quantum

probability distribution after a series of three sequen-

tial measurements is given by

(a, b, c|x, y, z) = p(c|b, y, z)p(b|a, x, y)p(a|x) , (6)

where p(c|b, y, z) = tr [M

], p(b|a, x, y) = tr [M

]

and p(a|x) = tr [M

]. In particular, any poten-

tial correlation between the measurement outcome

c in the third time step with the measurement set-

ting and outcome in the ﬁrst time step (x and a, re-

spectively), is screened-oﬀ by the intermediate mea-

surement setting and outcome (y and b), that is,

p(c|a, b, x, y, z) = p(c|b, y, z). This is similar to the

no-signalling constraints arising in a Bell scenario and

thus imposes restrictions to the classical causal mod-

els describing such a scenario. Speciﬁcally, a causal

model that reproduces this independence without re-

sorting to ﬁne-tuning cannot contain a causal link of

the form Λ

→ Λ

, since such a link can generate un-

wanted correlations between the variables X, A and

C (not mediated by Y, B). In fact, any model that

contains such a link, can only satisfy the condition

p(c|a, b, x, y, z) = p(c|b, y, z) by virtue of causal pa-

rameters chosen from a set of measure zero [28], that

is, the parameters are ﬁne-tuned in such a way that

these correlations are hidden from the observational

data [26].

Following the above description, the case of 3 se-

quential projective measurements on a single qubit

can be represented in terms of the causal structure

in Fig. 4. The temporal correlations p(a, b, c|x, y, z)

compatible with this causal structure can then be de-

composed as (with straightforward generalization to

more time steps)

p(a, b, c|x, y, z) =

,λ

p(a|x, λ

)p(b|y, λ

) (7)

p(c|z, λ

)p(λ

|x, a)p(λ

|y, b).

Note that without any restrictions on the dimension-

ality the hidden states Λ

can contain the full infor-

mation about the measurement settings and outcomes

of the previous time steps. This implies that, in con-

trast to the spacelike Bell scenario, all temporal cor-

relations of the form of Eq. (6) can be faithfully re-

produced by the classical causal model in Fig. 4.

This naturally raises the question whether further

restrictions on the classical causal model, might re-

veal a quantum advantage. Similar to the so-called

prepare-and-measure scenarios [13], one might expect

that quantum systems of a given dimension give rise

to measurement statistics that cannot be reproduced

Time

x a

y b z

-1

Figure 4: A sequence of three projective measurement on

a single qubit. (a) A single quantum system is subject to a

sequence of projective measurements at times t

, t

, and t

with settings x, y, z, and obtaining outcomes a, b, c, respec-

tively. (b) A general classical causal model for the scenario

in (a). Note that, although causal graphs are formulated

without reference to any spacetime structure, here we have

drawn the graph such that the horizontal direction can be

identiﬁed with time.

by classical systems Λ

of the same dimension. Since

restriction on the dimension of Λ

in models of the

form of Fig. 4 would lead to non-convex sets that are

technically very challenging to characterize [39, 40],

one typically considers convex relaxations. The re-

sulting models contain the model of interest as a spe-

cial case, but allow for shared randomness between

the parties. For example, using the results of Ref. [17]

we show in the Appendix A that classical hidden

states of dimension 4 (two bits of classical informa-

tion), together with shared randomness between the

parties, are enough to reproduce all correlations from

a series of projective measurements on a single qubit.

For a fair comparison to a qubit, we then considered

hidden states of dimension two and could not ﬁnd a

diﬀerence between quantum and classical correlations.

In light of these results it would be very interesting to

test the model of Fig. 4 with two-dimensional hidden

states and no shared randomness. Due to the com-

plexity of characterizing such non-convex sets, how-

ever, this remains an open question.

Besides restrictions on the dimension of the hid-

den state, certain physical constraints and assump-

tions might naturally lead to weaker causal models

for a sequence of projective measurements on a sin-

gle qubit. A well-known example is the Leggett-Garg

(LG) model [27] for testing macroscopic realism. This

model is based on the assumption of noninvasive mea-

surability, stating that it should be possible, in prin-

ciple, to determine (measure) the state of a system

without perturbing it. This is a special case of Eq. (7),

where the hidden variable state is unchanged by the

measurements and constant throughout the experi-

ment, that is, λ

= λ, which implies that

p(a, b, c|x, y, z) =

p(a|x, λ)p(b|y, λ)

p(c|z, λ)p(λ).

(8)

Note that this is the usual local hidden variable de-

scription encountered in Bell’s theorem. Further,

since the expectation values of a sequence of projec-

tive measurements on a single qubit are the same as

for local measurements on a pair of entangled parti-

cles [6], quantum correlations also violate macroscopic

realism, manifest as violations of so-called Leggett-

Garg inequalities [41]. The result, however, relies

critically on the assumption of noninvasive measura-

bility, which is diﬃcult to justify for a single quantum

system. In fact, quite generally, the correlations ob-

tained by sequential measurements on a quantum sys-

tem will display signaling (e.g p(b|x, y) 6= p(b|x

, y))

as opposed to the model in Eq. (8) that only allow

for non-signaling correlations. In other terms, the

Leggett-Garg model implies independence relations

that are not observed in the experiment. In this sense,

in order to test incompatibility with the Leggett-Garg

model we do not need to take the strength of the cor-

relations into account and simply look for violations

of independence relations implied by the model [42].

This raises the question of whether one can ﬁnd

examples of temporal quantum experiments to which

classical faithful causal models can reproduce all in-

dependence relations while at the same time being

incompatible with the generated correlations. Next

we show that this is indeed the case.

4 Quantum incompatibility with a

weaker classical model for sequential

projective measurements

From a causal inference perspective, given some ob-

served probability distribution the goal is to ﬁnd a

faithful causal model reproducing all the (conditional)

independence relations implied by this distribution.

As a concrete example, consider a sequence of three

projective measurements (see Fig. 4a) on an initial

maximally mixed qubit state ρ

= /2. It is easy to

verify that for arbitrary projective measurements, the

measurement outcome of the third measurement is in-

dependent of the setting of the ﬁrst measurement, i.e.

p(x, c) = p(x)p(c). In this case, the causal model in

Fig. 4b is not faithful any longer because it allows for

correlations between the variables X and C. Instead,

the most general causal model reproducing such inde-

pendence relation is shown in Fig. 5 where, in com-

parison with the causal model in Fig. 4, the causal

link between the variable B and Λ

is removed. Any

distribution compatible with this model has a decom-

position given by

p(a, b, c|x, y, z) =

,λ

p(a|x, λ

)p(b|y, λ

) (9)

p(c|z, λ

)p(λ

|a, x)p(λ

|y).

Crucially, this classical model faithfully captures the

observed independence p(x, c) = p(x)p(c) that holds

Alice

Bob Charlie

HWP

APD

Source

QWP

Time

CNOT

Figure 5: The causal model of Eq. (9) for a sequence

of projective measurements. (a) Experimentally, the ﬁrst

and last parties, Alice and Charlie, measure the polarization

of the photon directly, while the intermediate party, Bob,

implements a non-destructive measurement by coupling the

system to a meter photon using the CNOT gate in Fig. 2.

Note that, this experiment naturally satisfy the independence

p(c|x) = p(c). (b) The causal graph corresponding to Eq. (9)

faithfully captures this independence. This model is slightly

weaker than Fig. 4, since the ontic state Λ

depends only on

the previous measurement setting Y , but not on the outcome

for a wide range of relevant experimental scenarios.

This includes arbitrary measurements on an initially

maximally mixed state, as well as arbitrary initial

states in the xy-plane of the Bloch-sphere for the mea-

surements in the experimental implementation below.

As detailed in Appendix A any correlations com-

patible with Eq. (9) must respect the inequality

= hA

i + hA

i − hA

i (10)

− hBC

001

i − hBC

101

i − hBC

010

i − hBC

110

i ≤ 6,

where the joint expectation values are deﬁned as

i =

a,b

(−1)

a+b

p(a, b|x, y) and hBC

xyz

i =

a,b,c

(−1)

b+c

p(a, b, c|x, y, z). This inequality can,

however, be violated by a sequence of projective mea-

surements on any initial qubit state. For instance,

choosing measurement settings A

Z, A

= −C

= (

Z +

X)/

√

2, and B

= −C

(

Z −

X)/

√

2 (where

X and

Z are the Pauli operators)

obtains a value of S

= 4

√

2 > 6. Furthermore, for

any initial state in the xy-plane of the Bloch sphere,

the resulting probability distribution p(a, b, c|x, y, z)

respects the independence relation p(c|x) = p(c) im-

plied by the model under test. Through unitary ro-

tations this implies that for any ﬁxed initial quantum

state, one can generate temporal correlations that

cannot be explained by non-ﬁne-tuned models.

Experimentally we test inequality (10) with pho-

tonic polarization qubits for an initial maximally

mixed state, see Fig. 5a. For the intermediate mea-

surement the system is coupled to a meter in the

state |0i. A measurement of the meter in the com-

putational basis {|0i, |1i} achieves a projective mea-

surement of the system in a basis that is chosen by

appropriate single-qubit unitaries applied to the sys-

tem before and after the interaction. Notably, this

measurement design can be straightforwardly general-

ized to more than three parties by replicating the von

Neumann measurement. The experimental results in

Fig. 6 demonstrate a clear violation of inequality (10)

by a series of three projective measurements on a sin-

gle qubit, achieving a value of S

= 6.65

+0.01

−0.01

Figure 6: Experimental results for testing Inequality (10)

with a sequence of measurements on a single quantum

systems. The top part shows the experimental value of S

(vertical black bar) together with the 3σ statistical conﬁ-

dence region (grey shaded area). It also shows the individual

contributions from the 8 expectation values (alternate blue

and orange shadings), which are compared to their theoreti-

cal expectation in the bottom part of the ﬁgure. Here, blue

shaded bars correspond to theoretical expectation values for

ideal states and measurement and the horizontal black bars

represent the experimental values with 3σ statistical error

bars shown as grey shaded areas (almost within the black

bars).

5 Discussion

Our results show, both in theory and experiment, that

the discrepancy between the classical and quantum

descriptions typically associated with spatial correla-

tions extends to temporal correlation scenarios. The

latter arise naturally in situations where spacelike sep-

aration cannot be practically guaranteed, or when a

sequence of measurements is performed on the same

quantum system. In the case of spatial correlations

with an underlying time order, we have shown that

a Bell-type inequality, designed to test classical mod-

els augmented with limited communication, displays

a quantum violation if and only if the quantum pro-

tocol is also augmented with communication power.

This highlights that a quantum advantage persists in

a fair comparison of classical and quantum resources.

In a purely temporal-correlations scenario we have

shown that, as opposed to spatial Bell scenarios [26],

there is a faithful (non-ﬁne-tuned) classical model ca-

pable of simulating all correlations from projective

measurements on quantum states. It remains an

open question whether these models are equivalent

when imposing the same dimensionality to quantum

and classical models. Notwithstanding, we identiﬁed

quantum violations of inequalities associated with a

slightly less powerful classical model that includes the

Leggett-Garg model as special case [27]. We have

experimentally observed such a quantum violation,

which demonstrates a stronger form of non-classicality

of the correlations arising from a temporal sequence

of projective measurements. Furthermore, our exper-

imental design allows for tuning of the intermediate

measurement strength, which will enable future stud-

ies of non-Markovian models with some residual cor-

relations between the ontic states Λ

over multiple

time-steps [43].

Our theoretical results are based on a causal mod-

eling approach and thus formulated without reference

to a background spacetime structure. The direction

of time is only implicitly deduced from the ﬂow of

information between the parties or sequence of mea-

surements. Experimentally, the considered scenarios

are not subject to the locality loophole. However, the

results rely on the related assumption that there is no

hidden communication channel—other than the ones

implied by the models in Figs. 1b and 4—between

the diﬀerent time steps of the scenario under consid-

eration. Practically, we also rely on a fair-sampling

assumption to contend with imperfect detection eﬃ-

ciencies.

Temporal correlation scenarios play an important

role in communication complexity problems [44] and

in the search for a physical principle behind quan-

tum nonlocality, such as information causality [45].

Our results thus provide an avenue towards a more

systematic understanding of the quantum advantage

arising in such scenarios, that not only may lead to

new ways of processing information but also to new

insights into the nature of quantum correlations.

Acknowledgments

We thank C. Budroni, F. Costa, A. Fedrizzi,

and A.G. White for helpful discussions and feed-

back, and T. Vulpecula for experimental assistance.

This work was supported in part by the Centres

for Engineered Quantum Systems (CE110001013)

and for Quantum Computation and Communica-

tion Technology (CE110001027), the Engineering and

Physical Sciences Research Council (grant number

EP/N002962/1), the Brazilian ministries MEC and

MCTIC, the FQXi Fund, and the Templeton World

Charity Foundation (TWCF 0064/AB38).

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A Classical models for a sequence of

projective measurements on a qubit

A.1 Classical simulation with hidden states of

dimension 4

Note that the sequential measurement scenario can

be mapped to an equivalent sequence of quantum

teleportations [30]. In the ﬁrst time step Alice mea-

sures the state in her possession generating some lo-

cal probability distribution p(a|x), which she uses to

prepare diﬀerent states ρ

that she sends to Bob via

the usual quantum teleportation protocol. Then Bob

measures the teleported system generating a proba-

bility distribution p(b|x, a, y) and prepares states ρ

that are teleported to Charlie. Clearly, any classical

(hidden variable) protocol simulating the statistics of

the measurements performed on the teleported states

will immediately lead to a simulation of the equiva-

lent sequence of projective measurements. As shown

in Ref. [17], this can be achieved using only two bits

of classical communication between the parties and

an arbitrary amount of shared randomness, implying

that hidden states of dimension 4 and shared random-

ness are enough to obtain all quantum correlation ob-

tained by a sequence of projective measurements on

a single qubit.

For a fair comparison of classical and quantum re-

sources we have also considered whether a classical

message of dimension 2 (1 bit of communication) is

enough to simulate all projective measurements on a

qubit state. To that aim we considered a scenario

with two time steps. If Alice has the choice between

two measurements (i.e. x = 0, 1) then one classical bit

can carry all the information about the input X and,

using the argument in the main text, all the one-way

signalling correlations in this case can be simulated

using a classical message of dimension 2. We thus

considered the case were X assumes at least 3 diﬀer-

ent values and fully characterized the set of classical

correlations for dichotomic measurements (a, b = 0, 1)

and with x = 0, 1, 2 and y = 0, 1. The polytope cor-

responding to this scenario is described by 864 in-

equalities (many being equivalent under the allowed

symmetries given by party, input and output permu-

tations). Considering qubit states and arbitrary pro-

jective qubit measurements we could not ﬁnd any vi-

olation of these inequalities.

It is interesting to note that among these inequal-

ities we ﬁnd the dimension-witness inequality from

Ref. [13], given by

i + hB

i − hB

i ≤ 3, (A1)

where hB

i =

b=0,1

(−1)

p(b|x, y). As shown in

Ref. [13], this inequality can be violated by measure-

ments on a qubit. There, however, a slightly diﬀer-

ent situation is considered, the so-called prepare-and-

measure scenario, where the variable X uniquely iden-

tiﬁes the state ρ

being prepared and to be measured

in the second time step. In our case the states ρ

to be measured in the second time step will depend

on X, but also on the measurement outcome A, a

feature that seems to be enough to preclude any vio-

lation of the inequality above (or any other deﬁning

the scenario). It would be interesting to derive in-

equalities for more general scenarios including more

measurement settings or time steps to see whether

any violations can be found.

A.2 Derivation of Bell-type inequalities bound-

ing classical models for a sequence of projective

measurements

In order to derive Bell-type inequalities for the tem-

poral scenario in Fig. 5, ﬁrst note that all correla-

tions compatible with such a model are also compat-

ible with a model implying that

p(a, b, c|x, y, z) =

p(λ)p(a|x, λ) (A2)

p(b|a, x, y, λ)p(c|y, z, λ).

This follows from the fact that the arrow between

the hidden variable at a given time step and the

next measurement outcome (e.g. Λ

→ B) can be

replaced by directed arrows from the measurement

choices (or measurement outcomes) of the previous

step to the next one (e.g, X → B and A → B)

plus a local noise variable (Λ

→ B). Note that

these noise terms are implicitly present in the model

in Fig. 5, where they have been absorbed into Λ

, Λ

. When making the above replacement, how-

ever, these local noise terms have to be introduced

explicitly. They can then be combined into a vari-

able Λ = (Λ

, Λ

) that acts as a common ances-

tor and source of randomness for all the measurement

outcomes, see Fig. A1. In principle, one would further

have to impose the independence of the local noise

terms, that is p(λ

, λ

) = p(λ

)p(λ

This, however, would deﬁne a non-convex set that

is very diﬃcult to characterize [39, 40]. Instead, we

consider a more general convex relaxation of this set

which contains the case of independent noise variables

as a special case.

As detailed in Ref. [25], the probability distribu-

tions compatible with Eq. (A2) deﬁne a convex poly-

tope that can be characterized in terms of ﬁnitely

many extremal points. Given the list of extremal

points one can resort to standard convex optimiza-

tion software [46] to ﬁnd the dual description in terms

of linear (Bell-type) inequalities. For quantum cor-

relations arising from a sequence of three projective

measurements it follows that hABCi = hAihBCi. In

other words, the full (three-point) correlator does not

carry any information that is not already contained in

the bipartite and single correlators. For this reason

we focus our attention on inequalities involving the

Y Z

Figure A1: A DAG for the tripartite temporal correlations

scenario that contains the DAG in Fig. 4 of the main text as

a special case. As shown in Ref. [25] the correlations com-

patible with this DAG deﬁne a convex set such that Bell-type

inequalities can be derived using standard convex optimiza-

tion software [46].

expectation values hABi and hBCi, which, due to the

structure of the problem, can be deﬁned in diﬀerent

ways. In the following we use

hAB

x,y

i =

a,b

(−1)

a+b

p(a, b|x, y) (A3)

hBC

x,y,z

i =

a,b,c

(−1)

b+c

p(a, b, c|x, y, z). (A4)

Since there is a causal link between X and B, the

correlations between B and C can explicitly depend

on X. We then compute the Bell-type inequalities in

this subspace, one of which is inequality (10) in the

main text.

A.3 Quantum violation of inequality (10)

In order to search for a possible quantum violation

of inequality (10), we consider a single qubit in the

initial pure state

|Ψi = cos θ

|0i + e

iφ

sin θ

|1i. (A5)

At each time step we measure observables

parametrized by θ

and φ

= cos φ

sin θ

X + sin φ

sin θ

Y + cos θ

Z. (A6)

The full correlator can then be expressed, as

hABCi = hAihBCi with hAi = cos θ

cos θ

cos (φ

− φ

) sin θ

sin θ

and hBCi = cos θ

cos θ

cos (φ

− φ

) sin θ

sin θ

. Hence, as expected, the

projective measurement at the second time step (cor-

responding to the outcome b) destroys any correla-

tions between the ﬁrst and third time steps (outcomes

a and c, respectively). Similar to hBCi, the correla-

tions between the ﬁrst and second time steps are given

by hABi = cos θ

cos θ

+ cos (φ

− φ

) sin θ

sin θ

For projective measurements on a qubit state we

thus obtain hBC

x=0,y,z

i = hBC

x=1,y,z

i. Since the

ﬁrst four terms of inequality (10) are nothing else than

CHSH, this part can be maximized using the stan-

dard settings: A

Z, A

X, B

= (

X +

Z)/

√

= (−

X +

Z)/

√

2. The last four terms are max-

imized by setting C

= −B

and C

= −B

, which

obtains a value of 4. In summary, quantum correla-

tions can obtain S

= 4 + 2

√

2 > 6, thus violating

inequality (10). Curiously, this holds for any initial

single-qubit state.

B Experimental details

To test inequality (5) in the main text we prepare

a single-parameter family of two-qubit states of the

form

|Ψi =



1 + κ

(|HHi + |V V i)

1 − κ

(|HV i + |V Hi)



(A7)

We generate this family of states by subjecting the

separable state |Di⊗(

√

1 + κ|Hi+

√

1 − κ|V i)/

√

2 to

a controlled-NOT gate, as shown in Fig. 2 in the main

text. The parameter 0 ≤ κ ≤ 1 turns out to be equal

to the concurrence C of the resulting bipartite state,

such that the generated amount of entanglement can

be easily controlled by the initial state of the meter

photon.

Using the ﬁxed measurement protocol in the main

text, the states of Eq. (A7) achieve a value of S

1bit

4 + (1 + κ)

√

2 in inequality (5). It is, however, pos-

sible to optimize the measurement settings such that

ideally every pure entangled quantum state violates

inequality (5). Speciﬁcally, this amounts to mod-

ifying the protocol in the main text such that for

m = 0 Bob measures B

= B

X, while

for x = 1 he measures B

= (

X + κ

Z)/

√

1 + κ

= (

X − κ

Z)/

√

1 + κ

and B

= −

X. Using

this protocol, states of the form of Eq. (A7) achieve

1bit

= 4 + 2

√

1 + κ

≥ 6.

This is reminiscent of the observation that, us-

ing optimized measurement settings, the CHSH in-

equality [47] can be violated by every pure two-

qubit entangled quantum state [48]. In fact, inequal-

ity (5) contains a CHSH inequality for the settings

, A

, B

, which can be violated in the usual way,

while the additional communication in our protocol

can be used to maximize the remaining four terms up

to the maximal value of 4 for every state of the form

Eq. (A7). Hence, the expectation value of inequal-

ity (5) can be written as S

1bit

= 4 + S

chsh

, where

chsh

is the CHSH parameter corresponding to the

ﬁrst four terms of inequality (5).

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