Verification of Indefinite Causal Order. COPY with permission

Toward an Experimental Device-Independent Verification of Indefinite

Causal Order

Carla M.D. Richter ,

1,*,† Michael Antesberger ,

1,* Huan Cao ,

1 Philip Walther ,

1,2 and

Lee A. Rozema1,‡

1 University of Vienna, Faculty of Physics, Vienna Center for Quantum Science and Technology (VCQ) &

Research platform TURIS, Boltzmanngasse 5, 1090 Vienna, Austria

2 Christian Doppler Laboratory for Photonic Quantum Computer, Faculty of Physics, University of Vienna,

1090 Vienna, Austria

(Received 9 September 2025; revised 15 December 2025; accepted 6 January 2026; published 17 March 2026)

In classical physics, events follow a definite causal order: the past influences the future, but not the

reverse. Quantum theory, however, permits superpositions of causal orders—the so-called indefinite causal

orders (ICOs)—which can provide operational advantages over classical scenarios. Verifying such phe-

nomena has sparked significant interest, much like earlier efforts devoted to refuting local realism and

confirming quantum entanglement. To date, demonstrations of ICO have all been based a process called

the quantum switch and have relied on device-dependent or semi-device-independent protocols. Achiev-

ing a device-independent verification of ICO would imply that nature allows for correlations that do not

respect causality, independent of any experimental assumptions or underlying theoretical description of

the experiment. To this end, a recent theoretical development introduced a Bell-like inequality that allows

for fully device-independent verification of ICO in a quantum switch. Here we implement this verifica-

tion by experimentally violating this inequality. In particular, we measure a value of 1.8328 ± 0.0045,

which is 18 standard deviations above the definite causal order bound of 1.75. Our work presents the first

implementation of a device-independent protocol to verify ICO, albeit in the presence of experimental

loopholes. This represents an important step toward the device-independent verification of an ICO and

provides a context in which to identify loopholes specifically related to the verification of ICO.

DOI: 10.1103/5t2y-ddmt

I. INTRODUCTION

In a classical understanding of causality, events have

a well-defined order in time, meaning that events in the

past can only influence those in the future. Any process

with a well-defined causal order will satisfy the so-called

causal inequalities, which impose constraints on temporal

correlations generated by causality-respecting processes

[1]. Quantum mechanics appears to allow for events to

occur in a superposition of orders, such processes are said

to have an indefinite causal order (ICO) [1–5], which is

required to violate a causal inequality. Causal inequalities

are device independent (DI), meaning that their viola-

tion would prove, independent of the quantum theory, that

*These two authors contributed equally to this work.

†Contact author: carla.richter@univie.ac.at

‡Contact author: lee.rozema@univie.ac.at

Published by the American Physical Society under the terms of

the Creative Commons Attribution 4.0 International license. Fur-

ther distribution of this work must maintain attribution to the

author(s) and the published article’s title, journal citation, and

DOI.

nature allows for correlations that do not respect our clas-

sical notion of causality. However, not all processes with

an ICO can violate a causal inequality [6].

For example, the quantum switch [7], which can be

experimentally implemented [8–23], does not violate a

causal inequality [24,25]. Nevertheless, its ICO has been

experimentally confirmed in different device dependent

ways, such as demonstrating advantages over causally

ordered process [9], using causal witnesses [10], and even

performing full process tomography [22]. Moreover, the

quantum switch may also be interesting for applications as

it has been shown that it can outperform causally ordered

processes at a wide variety of tasks such as channel dis-

crimination [26], promise problems [27], communication

complexity [28], noise mitigation [29], various thermo-

dynamic applications [30–32], quantum metrology [33],

quantum key distribution [34], entanglement generation

[35], and distillation [36], among others. Thus, both for

foundational interest and to put the many proposed appli-

cations on a solid footing, an unambiguous confirmation

that ICO is a physically real phenomenon is essential. In

other words, we wish to treat the quantum process as an

untrusted adversary and perform a DI test to definitively

2691-3399/26/7(1)/010354(10) 010354-1 Published by the American Physical SocietyCARLA M.D. RICHTER et al. rule out the possibility that the observed correlations result

from a classical causal order.

All current demonstrations of ICO in the quantum

switch have been device dependent or semidevice inde-

pendent [12,15]. Using such an experiment to claim ICO is

akin to claiming a violation of local realism using device-

dependent techniques such as quantum state tomography

or entanglement witnesses: this is valid only if all assump-

tions hold, but it is open to loopholes that could void the

experimental conclusions. In the case of entanglement, this

led to a decades-long push to realize a loophole-free vio-

lation of local realism via a Bell inequality [37–39]. In

the context of ICO, violating a causal inequality is a DI

technique which would take the place of a Bell inequal-

ity. When a process has an ICO, one event can generate

correlations with other events which occur before and

after it. In all device-dependent experiments carried out

so far, assumptions are essentially made about when and

where the events occur. Since it is difficult to ascribe

an event to a photon (which always exists in a superpo-

sition of different times) traversing an optic, there is a

certain amount of ambiguity about what can be claimed

with respect to ICO in existing experimental demonstra-

tions. The device-dependent assumptions in current exper-

iments are made explicit in recent discussions regarding

the validity of quantum switch experiments [40–44], show-

ing the relevance of a loophole-free device-independent

demonstration of ICO.

Since the quantum switch is the only ICO process to be

experimentally studied but it cannot violate causal inequal-

ities, we would like to perform a different DI experiment to

prove that there is no hidden variable description in which

the causal order of the quantum switch is well-defined.

To this end, we present an experimental violation of an

inequality, introduced by van der Lugt, Barret, and Chiri-

bella (VBC) [45], which correlates a hidden variable with

both a fixed causal order in the quantum switch and a sec-

ond observable that is then used to violate a Bell inequality.

A successful violation of the Bell inequality thus implies

that a hidden variable cannot be assigned to the causal

order. This rules out the possibility that any physical the-

ory, consistent with a causally ordered structure, could

explain the observations, even when analyzed without

assuming a specific theory. In other words, we leverage DI

concepts developed for Bell inequalities to provide a DI

certification of ICO using the quantum switch. Although

our experiment does not close the usual Bell loopholes (or

other loopholes specific to ICO), it presents a significant

experimental step toward a loophole-free confirmation of

indefinite causal order (ICO).

II. THEORY

The VBC inequality concerns an experiment carried out

by four parties: Alice 1 and Alice 2 (who may or may not

PRX QUANTUM 7, 010354 (2026)

act in an indefinite causal order), Charlie who acts in the

future light cone of the Alices, and Bob, who is space-

like separated from the other parties [see Fig. 1(a)]. We

then consider DI data generated by these parties. In partic-

ular, on each run of the experiment Alice 1, Alice 2, Bob,

and Charlie choose their settings (x1, x2, y, z) ∈ {0, 1} and

generate outcomes (a1, a2, b, c) ∈ {0, 1}, respectively.

The VBC inequality follows from three assumptions:

definite causal order, relativistic causality, and free inter-

vention.

The definite causal order assumption introduces a hid-

den variable λ, which takes a definite value on every run

of the experiment, with each value of λ corresponding to

a fixed causal order between the parties. One does not

need precisely specify how λ acts, rather, we can simply

assume that its causal influence exists a priori. The rela-

tivistic causality assumption states that the causal order,

influenced by λ, must conform to the light-cone structure

in which the four parties operate. In particular, since Bob

is spacelike separated from all other parties, his settings

and outcomes are independent of the other three parties,

and vice versa. Moreover, since Alice 1 and Alice 2 act

before Charlie, their outcomes and settings are indepen-

dent of Charlie’s actions. With this in mind, the hidden

variable λ can take only two values λ ∈ {1, 2}. For λ= 1,

the order is Alice 1, Alice 2, and then Charlie. While when

(a) (b)

FIG. 1. Causal arrangement for the inequality test. (a) Space-

time arrangement of the four parties, Alice 1 (A1), Alice 2 (A2),

Bob (B), and Charlie (C), who will attempt to violate the inequal-

ity. Alice 1 and Alice 2 act before Charlie, while Bob (B) is

spacelike separated all other participants. (b) Switch-based pro-

tocol to violate the inequality. A Bell state is shared between the

control qubit of the quantum switch and Bob. Alice 1 and Alice

2 perform measurements on a target qubit in the quantum switch.

Notice that our notation, which was introduced in Ref. [45], for

Alice 1 and 2’s settings and outcomes is not the standard Bell

test notation. In particular, ai represents the Alices measurement

outcomes as usual. However, they always measure in the com-

putational basis. Thus no notation is used for their measurement

settings. Instead, xi denotes which state they prepare the post-

measurement system in before it leaves their laboratory. Charlie

and Bob’s notation is more standard: Charlie measures the con-

trol qubit in basis determined by z and receives outcome c, and

Bob’s measurement basis is given by y with outcome b.

010354-2TOWARD AN EXPERIMENTAL DEVICE-INDEPENDENT. . . PRX QUANTUM 7, 010354 (2026)

λ= 2, the overall order is Alice 2, Alice 1, and then Char-

lie. Finally, the free-intervention assumption states has two

parts. First, it states that all parties can choose their settings

freely, independent of variables outside their future light

cone, and independent of λ. Formally this means that the

input–output correlations can be represented as

p(a1a2bc | x1x2yz)=

p(λ)p(a1a2bc | x1x2yzλ).

λ∈{1,2}

(1)

The second part of the free-intervention assumptions

is that, for a given λ, all the settings are statistically

independent of any outcomes of parties outside their

causal order given by λ. This can be formally defined

by defining the following additional statistical indepen-

dencies. First, we require no signaling between Bob

and the other parties as a1a2c ⊥ ⊥p y and b ⊥ ⊥p x1x2z.

Where the symbol ⊥ ⊥p defines statistical independence;

for example, a1a2c ⊥ ⊥p y means b p(a1a2bc | x1x2yz)=

b p(a1a2bc | x1x2y′z). Next, as Charlie is timelike sepa-

rated from Alice 1 and Alice 2, there are two more relevant

independencies to define. For λ= 1, Alice 1 is placed in

the causal past of Alice 2, which means a1b ⊥ ⊥p x2 and

a1a2b ⊥ ⊥p z. On the other hand, λ= 2 gives us causal

order with the Alices reversed, which requires a2b ⊥ ⊥p x1

and a1a2b ⊥ ⊥p z.

With this in place, VBC proved that the correlations

between the four parties are bounded by

p(b= 0, a2 = x1 | y= 0) + p(b= 1, a1 = x2 | y= 0)

+ p(b ⊕ c = yz | x1 = x2 = 0) ≤

7

4. (2)

We can understand VBC’s inequality by considering

it to be two separate games: the “causal order game,”

quantified by the first two terms, and the usual Clauser-

Horne-Shimony-Holt (CHSH) game, given by the third

term. The causal order game is played whenever Bob

chooses the setting y= 0. The game is won if his outcomes

are correlated with the causal order of the Alices. In par-

ticular, the first term quantifies the causal order with λ= 1

in which Alice 1 acts first. This is achieved by checking if

a2 = x1, such that Alice 1 can signal to Alice 2 (Alice 2’s

outcome a2 is correlated with Alice 1’s setting x1) and cor-

relating this with Bob’s outcome b= 0. The second term

is similar, but now it correlates Bob’s other measurement

outcome b= 1 to the causal order where Alice 2 acts first

and λ= 2. Note that mathematically, these two terms are

bounded by 1. Moreover, when they sum to one p(b=

0, a2 = x1 | y= 0) + p(b= 1, a1 = x2 | y= 0)= 1, then

there are perfect correlations between Bob’s measurement

outcomes and the hidden variable λ defining the causal

order. In this case, Bob and the Alices win the causal order

game.

When the parties win the causal order game it means

that λ is correlated with Bob’s measurement outcomes. We

know from Bell’s theorem that if λ is indeed a classical

variable, then Bob’s measurement outcomes cannot violate

a CHSH inequality with Charlie. Thus to check the nature

of λ Bob play the CHSH game with Charlie.

The CHSH game is played whenever the settings of the

Alices agree (i.e., when they both attempt to send each

other 0). This is quantified by the third term of Eq. (2),

which evaluates at the probability of Bob and Charlie to

win the CHSH game, i.e., a referee gives Bob and Char-

lie the bits y and z they need to generate outputs b and c,

respectively, that satisfy the following relation b ⊕ c = yz.

If λ is a classical variable Bob and Charlie should be able

to win the CHSH game with a probability of at most 3

4.

So in total the VBC inequality is bounded by ≤1 + 3

7

4 =

4.

This bound must hold for any process with a definite causal

order. In this case, even if Charlie and Bob share an entan-

gled state and violate the CHSH game in the third term,

then the first two terms must decrease to compensate for

this. For Bob to simultaneously win the causal order game

with the Alices and the CHSH game with Charlie both

entanglement and ICO are required: the causal order of the

Alices must be truly indefinite such that they can win the

causal order game without destroying the quantum entan-

glement required to win the CHSH game with Charlie. This

bound can be formalized and generalized for imperfect

correlations in a fully DI manner [45].

To see how the quantum switch can be used to violate

VBC’s inequality, consider the schematic shown in Fig.

1(b). The quantum switch, represented as the red shaded

areas, takes two input quantum operations (A1 and A2) and

applies them to a target system dependent on the state of

a control qubit: if the control qubit is in |0⟩ the gates are

applied in the order A2A1, while they are applied in the

order A1A2 when the control qubit is in |1⟩. The control

qubit of the quantum switch is entangled in a | +⟩-Bell-

state with an ancilla qubit that is sent to Bob. Alice 1

and Alice 2 are placed inside the quantum switch, and

Charlie performs measurements on the control qubit after

the switch. The target qubit is prepared in |0⟩ and is dis-

carded after the switch. Bob and Charlie then play the

CHSH game: Bob measures in the Z basis if y= 0 and

in the X basis if y= 1; Charlie measures in X + Z basis

if z = 0 and in X− Z if z = 1. Inside the switch, both

Alices always measure in computational basis to produce

outputs ai. They then prepare their outgoing qubit accord-

ing to their randomly chosen setting xi: when xi = 0 Alice

i prepares the state |0⟩ and when xi = 1 they prepare |1⟩.

In this way they can attempt to signal to each other, and

we can check for successful signaling when a2 = x1 or

a1 = x2. Note, the notation for Alice 1 and 2 differs from

a standard CHSH notation. In our case, xi are not the mea-

surement setting but preparation settings; i.e., xi dictates

which state Alice tries to send to the other Alice. ai are

010354-3CARLA M.D. RICHTER et al. PRX QUANTUM 7, 010354 (2026)

(b)

Time bin

delay

Input

delay

Target

preparation

Fiber Fiber

spool spool

Pol. Pol.

filter filter

(d)

(a)

Quantum switch

Fibre Fibre

coupler coupler

Signal

generator

(c)

FIG. 2. Experimental schematic: A type-0 spontaneous parametric down-conversion (SPDC) source generates polarization-

entangled photon pairs in the |φ+⟩ state. (a) Photon 1 of the pair is sent to Bob’s measurement stage (B), represented by the

orange-shaded area. Therein it is measured in a standard polarization measurement stage consisting of a quarter-wave plate (QWP),

half-wave plate (HWP), and a polarizing beamsplitter (PBS). Its detection signal initiates the electrical trigger signal that allows the

ultrafast optical router (UFOR) to switch at the correct time. (b) Photon 2 passes an optical circulator before entering the green area,

where the polarization qubit is deterministically converted into a time-bin qubit using an imbalanced Mach-Zhender-like interferome-

ter opening on a PBS and closing on an UFOR. Additionally, to further refine the polarization, photon 2 passes through a PBS in the

target-preparation stage. (c) The time-bin qubit then serves as the control for the time-bin quantum switch indicated by the pink shaded

region. Here Alice 1 and Alice 2 (A1, A2) perform measurement on the target qubit a1, a2 and reprepare the state in the settings x1, x2 in

the polarization degree of freedom using a linear polarizer and a half-wave plate. After passing through the quantum switch, the photon

travels in the opposite direction, ensuring that both time bins arrive simultaneously at the PBS. (d) Depending on their spatial path and

polarization, the photons are guided from here to one of Charlie’s measurement stages, C0 or C1 (shaded blue region). Charlie’s two

polarization measurement devices allow him to implement a complete set of measurements on the control and target qubits. * Two

half-wave plates in B allow the preparation of any Bell state as the input state, enabling additional measurements to test the effect of

noise on the VBC inequality.

measurement outcomes, like usual but the measurements

are always performed in the computational basis. In this

configuration, when Bob measures the ancillary qubit in

the computational basis, y= 0, he will collapse the con-

trol qubit of the quantum switch such that if b= 0, Alice

1 acts before Alice 2, whereas if b= 1, Alice 2 acts before

Alice 1. Thus, when y= 0, with the Alices’ measurements

and preparations described above, Alice 1 and 2 can signal.

In particular, each of the first two terms of Eq.(2) will be

1

2 . For the last term, when x1 = x2, the switch effectively

acts as an identity channel on the control and target qubits.

This means that Bob and Charlie will share the input

state | +⟩ after the switch. Thus they can win the CHSH

game with a probability given by the Tirelson bound of

1

2 + √2

4 ≈ 0.8536. The net effect is that quantum mechan-

ics (QM) predicts that the setup shown in Fig. 1(b) will

violate the VBC inequality with a value of 1.8536 > 1.75.

III. EXPERIMENT

To experimentally violate VBC’s inequality, we use a

photonic implementation of the quantum switch. The pho-

tonic switch has been built in various forms, using different

degrees of freedom (DOFs) of single photons to encode

the control and target systems [8–23]. Here, we extend

a recent realization of the quantum switch that uses the

temporal DOF for the control qubit and the polarization

DOF for the target qubit [22]. This configuration is partic-

ularly advantageous as it enables the use of simple, wave

plate-based gate operations by Alice 1 and Alice 2 in the

switch, and the time-bin encoding allows for passive phase

stability in Charlie’s interferometric measurement of the

control qubit. To generate entanglement between the con-

trol qubit and Bob’s ancillary qubit we start by generating

polarization-entangled photon pairs at telecom wavelength

(λ= 1550 nm) in the | +⟩ state using a Type-0 sponta-

neous parametric down-conversion (SPDC) source in a

Sagnac interferometer. We measure a fidelity to the target

state of 0.971 97 ± 0.000 66 [Fig. 3(a)], with a coinci-

dence rate of ≈7 kHz, measured directly from the source.

One photon of the pair is directly sent to Bob [photon 1,

Fig. 2(a)], who will perform arbitrary polarization mea-

surements y to receive output b. The detection signal of

photon 1 is then used as a trigger for the synchronization of

the 2 × 2 ultrafast optical routers (UFOR). Meanwhile the

other photon (photon 2) is sent toward the quantum switch.

[Note that this does not follow the spacetime diagram of

Fig. 1(a), and thus opens a loophole which we will discuss

later.] To create entanglement between Bob’s polarization

qubit (encoded in photon 1) and the control qubit of the

010354-4TOWARD AN EXPERIMENTAL DEVICE-INDEPENDENT. . . PRX QUANTUM 7, 010354 (2026)

switch (encoded in photon 2), we transfer the polarization

DOF of photon 2 to a time-bin DOF. We accomplish this

by sending photon 2 to an imbalanced Mach-Zehnder-like

interferometer setup [Fig. 2(b)] that opens on a polarizing

beam splitter (PBS) and closes on an UFOR [46].

If photon 2 is horizontally polarized |H ⟩, it is transmit-

ted through the PBS and takes the short path and is then

routed into the switch by the UFOR resulting in the early

time-bin state |E⟩. When it is vertically polarized |V⟩, on

the other hand, it reflects into the long path. Therein a half-

wave plate (HWP) rotates the polarization state from |V⟩

to |H ⟩ before the photon experiences a τ ≈ 150 ns fiber

delay and is routed into the switch in the late time-bin state

|L⟩. The HWP ensures both time-bin components share the

same polarization and disentangles the polarization of pho-

ton 2 from the polarization of photon 1. This completes

the transfer of entanglement into the time DOF, and sets

photon 2’s polarization to |H ⟩ in both time bins so that it

can be used as the target qubit. More precisely, the state

at this point in the experiment is | ⟩ = 1

√2 (|HE⟩1B,2C +

|VL⟩1B,2C ) ⊗ |H ⟩2T , where the subscript 1B indicates the

polarization encoded ancillary qubit in photon 1, 2C refers

to the time-bin encoded control qubit and 2T denotes the

polarization-encoded target qubit of photon 2.

Upon entering the quantum switch, the two switch [Fig.

2(c)] UFORs are synchronized to route the early time bin to

Alice 1 and then Alice 2, while the late time bin is routed to

Alice 2 and then Alice 1. A comprehensive description of

the quantum switch is provided in the Supplemental Mate-

rial [47] and in Ref. [22]. Within the quantum switch, Alice

1 and Alice 2, perform projective measurements ai in the

computational basis and subsequently reprepare the target

system in |H ⟩ or |V⟩, depending on their setting xi. Exper-

imentally, the measurements are achieved by using linear

polarizers and checking to see if the photon is transmitted

or not. After the polarizer, the state is reprepared with suit-

able wave plates dictated by xi. Note that the final target

state is only read out after the switch operation is complete

(i.e., we must check if the photon arrives at Charlie’s mea-

surement station), which introduces a second loophole by

not enforcing the spacetime structure of Fig. 1(a).

After the switch, Charlie measures the time-bin con-

trol qubit, which requires him to interfere the early and

late time-bin states. We do so using the same Mach-

Zhender-like interferometer in reverse; i.e., the UFOR now

sends the early time bin into the path with the delay

and the late time bin into the shorter arm such that the

two time bins meet on the polarizing beamsplitter. Using

the same interferometer results in passive interferometric

phase stability, see Ref. [22] for more details. Since the

polarization state is in general changed in the switch, we

must consider how the joint state of the control and tar-

get qubits is mapped onto a four-dimensional Hilbert space

spanned by two path modes and two polarization modes.

By placing independent polarization tomography modules

in each output path of the PBS where the time bins inter-

fere—each consisting of a PBS, QWP, and HWP, with four

detectors [one detector at each output PBS output port,

Fig. 2(d)]. Depending on the former time-bin state, that

is now encoded in the path, polarization, and Charlie’s

settings zi, the photon ends up in one of the four detec-

tors. This allows a complete set of measurements on both

qubits to be performed, yielding information about ci and

xi. A detailed description of this measurement scheme is

provided in Supplemental Material [47].

To measure a violation of VBC’s inequality, we now

measure the individual terms of Eq. (2). For the first two

terms, we only need to consider cases where y= 0, which

corresponds to Bob performing measurements in the com-

putational basis. In the first term, we consider outcomes

where he obtains b= 0, which means that the photon is

horizontally polarized, while in the second term Bob finds

his photon to be vertically polarized which represents b=

1. We therefore set Bob to measure in both of these settings

and for each we iterate over all combinations of the Alices’

settings. We effectively trace over Charlie by having him

perform measurements in the Z + X and Z− X bases and

summing the results. For each possible combination of set-

tings, we record the coincidence counts between Charlie’s

four detectors and Bob’s detector in the transmitted output

of his PBS. Note, we discard the reflected outcomes since

we only trigger our UFORs from the transmitted detector.

Since x1, x2, and z take two possible values {0, 1}, each of

the first two terms is constructed from eight probabilities

formed by the possible combinations of these settings. We

then compute individual probabilities as

p1 =

1

8

x1,x2,z∈{0,1}

p(b= 0, a2 = x1|x1x2z, y= 0) (3)

and [45]

1

p2 =

8

x1,x2,z∈{0,1}

p(b= 1, a1 = x2|x1x2z, y= 0). (4)

For the last term in the inequality, Alice 1 and Alice 2

reprepare the target qubit in |0⟩, which means their polar-

izers are set to transmit |H ⟩ and the wave plate after

the polarizers are set to 0◦. This ensures that the target

qubit exits each of their setups with the same polarization

|H ⟩2T it had upon entering. We estimate Bob and Char-

lie’s ability to win the CHSH game by iterating over their

measurements constructing their winning probability as

p3 =

1

4

y,z∈{0,1}

p(b ⊗ c = yz|yz). (5)

Again, the normalization factor arises from the number

of possible combinations of measurement settings. For

010354-5CARLA M.D. RICHTER et al. (a)

(b)

(c)

Tsirelson’s bound cos

Classical bound

(d)

ICO bound

DCO bound

VBC value

Bell state purity

FIG. 3. Experimental violation of a local causal inequality. (a)

Density matrix of the two photon input state as it is generated

by our spontaneous parametric down-conversion-source with a

purity of 0.980 36 ± 0.000 34. (b) Maximally mixed input state

with a purity of 0.257 77 ± 0.000 15 created by generating depo-

larizing noise from experimental data via Eq. (6). (c) Effect of

depolarizing noise, applied on the two photon input state, on the

values of the individual term probabilities p1, p2, p3. (d) Exper-

imental violation of the VBC inequality and simulation of the

maximum possible VBC violation as a function of the purity

of the input Bell states created by applying different amounts of

depolarizing noise. Here, the defined causal order bound (DCO)

and the indefinite causal order (ICO) bound are marked in color,

with the latter representing the maximum achievable value.

each measurement setting we measure the coincidences for

3 min, resulting in an average total counts of ≈7000 per

setting. The individual probabilities are plotted in Fig. 3(b),

wherein the close agreement between the ideal values and

our experimentally measured probabilities is apparent.

Summing these probabilities together, we observe a

clear violation of VBC’s inequality of 1.8328 ± 0.0045,

which is 18 standard deviations above the definite causal

order bound of 1.75. We attribute the small deviation

between our measured violation and the theoretical quan-

tum maximum of (6 + √2)/4 ≈ 1.8536 to imperfections

in the entangled state and a slightly reduced process

fidelity of the quantum switch. To confirm this, we

PRX QUANTUM 7, 010354 (2026)

model a reduced process fidelity of the quantum switch

by introducing “causally separable” noise to the ideal

quantum switch process matrix Wswitch. In other words,

we create the noisy process matrix W= (1− ϵ)Wswitch +

ϵ WA1<A2 + WA2<A1 and vary ϵ to fit our data. When also

including reduced purity of our input state, measured to be

0.980 36 ± 0.000 34 by performing quantum state tomog-

raphy [Fig. 3(a)], we find that a process fidelity of FSwitch =

0.9816 ± 0.0069 almost perfectly describes our measure-

ment result [see the inset of Fig. 3(d)], highlighting our

high-fidelity implementation of the switch.

To provide additional insight into the VBC inequality

we next study how different types of noise affect its vio-

lation. To induce this noise in a controllable manner, we

prepared each of the four Bell states and then carried out

the original measurements with each of the Bell states as

the input state. We then summed these different data sets

together with different weights to mimic different noise.

We first studied the depolarizing channel, which can be

written as

ερ + = (1− η)| +⟩ ⟨ +| + η

I, (6)

4

where I is the identity matrix [48]. This destroys all corre-

lations between Charlie and Bob as well as any coherence

inside the quantum switch.

Our results for applying depolarizing noise, pictured in

Fig. 3 panels (c) and (d), show that the ability to violate the

VBC inequality is strongly influenced by noise, reducing

all three terms of the inequality [Fig. 3(c)]. Since increas-

ing depolarizing noise destroys the entanglement between

Bob and Charlie, it comes as no surprise, that the third term

p3 [Eq. (5)], which represents a CHSH game [red line in

Fig. 3(c)], falls below the definite causal order bound of

0.75. That it also reduces the probabilities p1 and p2 can be

understood from the fact that these terms represent correla-

tions between the ability of Alice 1 to signal to Alice 2 [p1

Eq. (3)] and vice versa [p2 Eq. (4)] to Bobs outcomes in

the computational basis. The depolarizing noise not only

removes the quantum coherence but also these classical

correlations.

The simulation of our experiment with FSwitch =

0.9816 ± 0.0069 agrees well our experimentally obtained

VBC value. The same holds true for the value of pexp

3 =

0.8404 ± 0.0036. However, there is a discrepancy for p1

and p2, which deviate in opposite directions: pexp

2 rises

slightly above the ideal value and pexp

1 below it. This devi-

ation is caused by imperfections in the Bell state generated

by our photon source. Although the state has a high purity,

it has a slight imbalance in the |VV⟩ and |HH ⟩ terms, which

results in an asymmetry p1 and p2 since Alice 2 and signals

to Alice 1 slightly more often.

We also studying dephasing noise defined as

ζρ + = (1− ϑ)| +⟩ ⟨ +| + ϑ |−⟩⟨−| (7)

010354-6TOWARD AN EXPERIMENTAL DEVICE-INDEPENDENT. . . PRX QUANTUM 7, 010354 (2026)

Tsirelson’s bound cos

Classical

bound

Bell state purity

FIG. 4. The effect of dephasing noise on the probabilities p1,

p2, and p3 of the van der Lugt, Barret, and Chiribella-inequality

experimental data modeled with Eq. (7) and simulated curves

with a switch process fidelity of 0.9816 ± 0.0069.

with a phase flip-probability ϑ [48,49]. Where ϑ= 0.5

corresponds to the maximum phase noise. The VBC

inequality behaves differently in this case. Again it reduces

the purity of the joint state and thereby destroys the entan-

glement and decreasing p3, the winning probability of the

CHSH game [red line in Fig. 4(a)]. However, since it

only removes coherence between Charlie and Bob, they

maintain classical correlations. Therefore, the correlations

between Bobs’ results and the causal order inside the quan-

tum switch remains intact. As a result psim

1 and psim

2 remain

constant for any value of ϑ [blue and yellow lines in Fig.

4(b)]. The experiment agrees well with our simulations,

with the slight deviations in p1 and p2 again attributed to

imperfections in the generated Bell states.

IV. DISCUSSION

While the protocol that we have implemented is device

independent, our experimental implementation contains

loopholes that must be closed to achieve a fully device-

independent verification of ICO. Since Bob and Charlie’s

CHSH game lies heart of VBC’s inequality, the standard

Bell loopholes [37–39] must be closed. These include mea-

surement independence (freedom of choice), fair-sample

(detection), and the locality loophole. Since our exper-

iment is a proof of principle, we did not close these

loopholes. Nonetheless, we will briefly mention how they

manifest in our setup. To ensure measurement indepen-

dence, the measurement settings must be either random or

freely chosen [50,51]. Our implementation does not satisfy

this requirement, as the measurement order was prede-

fined in our code. The fair-sampling loophole arises when

a potentially unrepresentative subset of events is detected,

which may falsely suggest a Bell violation. In the present

implementation, our setup experiences significant loss,

mainly due to multiple passes through the UFORs and the

relatively low transmission of the polarizers used for the

Alices’ measurements. In total, the net detection efficiency

through the entire experiment is ≈1%, far from closing this

loophole. Closing the locality loophole requires spacelike

separation between Bob and Charlies measurement events

[50]. Our experiment is built on a single optical table with

measurement stages less than 1 m apart. Moreover, Bob’s

photon is detected first and used as a timing reference for

the UFORs. To close this loophole, we would need to work

with a pulsed source of entangled photons and synchronize

the UFORs to the source. This would allow us to separate

the other parties from Bob by a much greater distance and

perform the measurements without any time delay. Using

this timing method would further strengthen the exper-

iment by allowing the measurement order to match the

theoretical VBC scenario, as registering Bob’s outcome

earlier introduces a potential causal link from Bob to the

other parties.

While we know how to close the standard Bell loop-

holes, the verification of ICO opens new loopholes. Here

we point out two loopholes related the definition of time-

delocalized events [44,52]. The first is specifically related

to VBC’s inequality. As we sketched in Fig. 1(a) Alice

1 and Alice 2 must act in the past light cone of Char-

lie. While the photon certainly passes through the Alices’

labs before reaching Charlie, in our implementation (and in

all implementations of measurements in a switch [10,15])

the measurement results are not read out “locally” in their

labs. However, this is, in principle, possible to achieve in

a purely quantum optical setting. For example, one could

implement a quantum nondemolition measurement of the

photon’s polarization by coupling the photon to an auxil-

iary probe system that is stored locally in each lab using

a single-photon level nonlinearity [53]. If this is not done

properly it will introduce which-path information, deco-

hering the switch. However, by coupling the probe system

to each mode in the local labs one can realize read out

the information locally without yielding which-path infor-

mation. This could be done similar to proposed gedanken

experiments that count the gate uses in the quantum switch

[8,9].

A related but distinct loophole is enforcing the closed

lab assumption [1]. In the closed lab assumption, each

party is imagined to act in an isolated lab with an input

door and output door. Each door is opened exactly once

to ensure the party acts once. Since our photon’s coher-

ence time is shorter than the time required for the photon

to propagate between Alice 1 and Alice 2’s labs there are

two distinct times at which each party might act. This fol-

lows from the fact that the photonic switch is often said

to have four distinct spacetime events [8,19]. Therefore, it

would be possible for Alice 1 to open and close her lab

to let the photon in at time 1 and then again at a second

time. This would not have any observable effect on the

experiment, which could allow us to conclude the pho-

ton entered the lab twice. The situation is different when

the photon is temporally delocalized. For example, if the

photon has a coherence time longer than the propagation

010354-7CARLA M.D. RICHTER et al. time between the labs, then doors could be only opened

once without disturbing the photon and changing the out-

comes. One experiment [19] has realized this, albeit with

unitary operations in the switch. Although using tempo-

rally long photons may be a step in the right direction for

the closed lab assumption, enforcing this in a loophole-

free experiment is another matter, that we leave for future

discussion.

We have used a time-bin implementation of the quan-

tum switch to violate a device-independent inequality,

indicating the presence of ICO between two parties in

our experiment and showed how it responds to different

types of experimentally relevant noise. Our strong viola-

tion of 1.8328 ± 0.0045, close to the theoretical maximum

bound of 1.8563 is made possible by our passively sta-

ble high-fidelity experiment, showing that it is a promising

implementation of the quantum switch both for founda-

tional tasks such as that discussed here and to implement

ICO-based advantages. This represents an important step

toward a loophole-free verification of ICO, which is cru-

cial for supporting photonic quantum switch experiments,

as there currently is not a consensus as to whether such

experiments realize a genuine ICO or merely simulate it.

For instance, some argue that a genuine ICO can arise

only from the so-called gravitational quantum switch [54],

asserting that ICO requires superpositions of gravitational

fields [43]. The other side, in contrast, supports existing

experiments, maintaining that the ICO in photonic quan-

tum switch experiments originates from delocalized events

which is equivalent to a superposition of space-times that

depends solely on the choice of quantum coordinates [55].

Closing the loopholes in the violation of the VBC inequal-

ity presented here would finally settle this debate. Further-

more, it would confirm that ICO is a new quantum resource

distinct from entanglement and would provide a footing for

the many recently proposed protocols exploiting ICO to

accomplish tasks that cannot be carried out with standard

quantum processes.

ACKNOWLEDGMENTS

This project has received funding from the European

Union (ERC, GRAVITES, No. 101071779), the Euro-

pean Union’s Horizon 2020 research and innovation pro-

gramme under Grant Agreement No. 899368 (EPIQUS),

the European Union’s Horizon 2020 research and innova-

tion programme under the Marie Skłodowska-Curie Grant

Agreement No. 956071 (AppQInfo), and the European

Union (HORIZON Europe Research and Innovation Pro-

gramme, EPIQUE, No. 101135288). This research was

funded in whole or in part by the Austrian Science Fund

(FWF) (No. 10.55776/COE1) (Quantum Science Austria),

(No. 10.55776/F71) (BeyondC), and (No. 10.55776/FG5)

(Research Group 5). For open access purposes, the author

has applied a CC BY public copyright license to any

PRX QUANTUM 7, 010354 (2026)

author accepted manuscript version arising from this sub-

mission. This material is based upon work supported

by the Air Force Office of Scientific Research under

Award Nos. FA9550-21-1-0355 (Q-Trust) and FA8655-

23-1-7063 (TIQI). The financial support by the Austrian

Federal Ministry of Labour and Economy, the National

Foundation for Research, Technology and Development,

and the Christian Doppler Research Association is grate-

fully acknowledged.

Views and opinions expressed are however those of

the author(s) only and do not necessarily reflect those of

the European Union or the European Research Council

Executive Agency.

DATA AVAILABILITY

The data that support the findings of this article are

openly available [56].

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