что такое квантовая физика
Квантовая физика доказала, что объективной реальности не ...
Квантовая физика доказала, что объективной реальности не существует. Ученые впервые воспроизвели в реальности парадокс друга Вигнера. В ...
провели
Теперь физики впервые <anchor>провели</anchor> этот мысленный эксперимент в реальном мире. Для этого они использовали квантовый компьютер и три пары запутанных фотонов. Первая пара представляет собой монеты, а две другие используются для их «подбрасывания» — измерения поляризации. При этом каждая «монета» находится в своем замкнутом сосуде, где, помимо нее, присутствует «бросающий» фотон. Снаружи этих двух коробок расположены еще два фотона, которые выполняют функцию «друзей-наблюдателей».
Experimental test of local observer independence | Science Advances
Experimental test of local observer independence \| Science Advances

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# Experimental test of local observer independence

[MassimilianoProietti](#con1) [https://orcid.org/0000-0002-6180-0699](https://orcid.org/0000-0002-6180-0699), [AlexanderPickston](#con2), \[...\], [FrancescoGraffitti](#con3) [https://orcid.org/0000-0002-8802-8360](https://orcid.org/0000-0002-8802-8360), [PeterBarrow](#con4), \[...\], [DmytroKundys](#con5), [CyrilBranciard](#con6) [https://orcid.org/0000-0001-9460-825X](https://orcid.org/0000-0001-9460-825X), [MartinRingbauer](#con7) [https://orcid.org/0000-0001-5055-6240](https://orcid.org/0000-0001-5055-6240), and [AlessandroFedrizzi](#con8) [https://orcid.org/0000-0002-5089-3148](https://orcid.org/0000-0002-5089-3148) [\[email protected\]](/cdn-cgi/l/email-protection#cbaae5adaeafb9a2b1b1a28ba3bce5aaa8e5bea0)+5 authors+3 authorsfewer [Authors Info & Affiliations](#tab-contributors)

Science Advances

20 Sep 2019

Vol 5, Issue 9

[DOI: 10.1126/sciadv.aaw9832](https://doi.org/10.1126/sciadv.aaw9832)

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- Contents

- [Abstract](#abstract)
- [INTRODUCTION](#sec-1)
- [RESULTS](#sec-2)
- [DISCUSSION](#sec-3)
- [MATERIALS AND METHODS](#sec-4)
- [Acknowledgments](#acknowledgments)
- [Supplementary Material](#supplementary-materials)
- [REFERENCES AND NOTES](#bibliography)

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## Abstract

The scientific method relies on facts, established through repeated measurements and agreed upon universally, independently of who observed them. In quantum mechanics the objectivity of observations is not so clear, most markedly exposed in Wigner’s eponymous thought experiment where two observers can experience seemingly different realities. The question whether the observers’ narratives can be reconciled has only recently been made accessible to empirical investigation, through recent no-go theorems that construct an extended Wigner’s friend scenario with four observers. In a state-of-the-art six-photon experiment, we realize this extended Wigner’s friend scenario, experimentally violating the associated Bell-type inequality by five standard deviations. If one holds fast to the assumptions of locality and free choice, this result implies that quantum theory should be interpreted in an observer-dependent way.

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## INTRODUCTION

The observer’s role as final arbiter of universal facts ( [_1_](#R1)) was imperiled by the advent of 20th century science. In relativity, previously absolute observations are now relative to moving reference frames; in quantum theory, all physical processes are continuous and deterministic, except for observations, which are proclaimed to be instantaneous and probabilistic. This fundamental conflict in quantum theory is known as the measurement problem, and it originates because the theory does not provide a precise cut between a process being a measurement or just another unitary physical interaction.

This is best illustrated in the seminal “Wigner’s friend” thought experiment ( [_2_](#R2)), whose far-reaching implications are only starting to become clear ( [_3_](#R3)– [_5_](#R5)). Consider a single photon in a superposition of horizontal ∣ _h_〉 and vertical polarization ∣ _v_〉, measured in the {∣ _h_〉, ∣ _v_〉}-basis by an observer—Wigner’s friend—in an isolated laboratory (see [Fig. 1, A and B](#F1)). According to quantum theory, the friend randomly observes one of two possible outcomes in every run of the experiment. The friend’s record, _h_ or _v_, can be stored in one of two possible orthogonal states of some physical memory, labeled either ∣“photon is _h_ ”〉 or ∣“photon is _v_”〉, and constitutes a “fact” from the friend’s point of view. Wigner, who observes the isolated laboratory from the outside, has no information about his friend’s measurement outcome. According to quantum theory, Wigner must describe the friend’s measurement as a unitary interaction that leaves the photon and the friend’s record in the entangled state (with implicit tensor products)

12(∣h〉±∣v〉)→12(∣h〉∣“photon ish”〉±∣v〉∣“photon isv”〉)≕∣Φphoton/record±〉

(1)

Wigner can now perform an interference experiment in an entangled basis containing the states of [Eq. 1](#E1) to verify that the photon and his friend’s record are indeed in a superposition—a fact from his point of view. From this fact, Wigner concludes that his friend cannot have recorded a definite outcome. Concurrently, however, the friend does always record a definite outcome, which suggests that the original superposition was destroyed and Wigner should not observe any interference. The friend can even tell Wigner that she recorded a definite outcome (without revealing the result), yet Wigner and his friend’s respective descriptions remain unchanged ( [_6_](#R6)). This calls into question the objective status of the facts established by the two observers. Can one reconcile their different records, or are they fundamentally incompatible, so that they cannot be considered objective, observer-independent “facts of the world” ( [_3_](#R3), [_4_](#R4))?

![](/cms/10.1126/sciadv.aaw9832/asset/46c69cd3-19a0-4280-932c-8bd4933b292a/assets/graphic/aaw9832-f1.jpeg)

Fig. 1 Wigner’s friend experiment.

( **A**) A quantum system in an equal superposition of two possible states is measured by Wigner’s friend (inside the box). According to quantum theory, in each run, she will randomly obtain one of two possible measurement outcomes. This can be verified by directly looking into her laboratory and reading which result she recorded. ( **B**) From outside the closed laboratory, however, Wigner must describe his friend and her quantum system as a joint entangled state. Wigner can also verify this state assignment through an interference experiment, concluding that his friend cannot have seen a definite outcome in the first place. ( **C**) We consider an extended version of that experiment, where an entangled state is sent to two different laboratories, each involving an experimenter and their friend.

It was recently shown ( [_4_](#R4)) that this question can be addressed formally, by considering an extension of the Wigner’s friend scenario as follows. Consider a pair of physical systems, shared between two separate laboratories controlled by Alice and Bob, respectively (see [Fig. 1C](#F1)). Inside these laboratories, Alice’s friend and Bob’s friend measure their respective system nondestructively and record the outcomes in some memory. Outside these laboratories, in each run of the experiment, Alice and Bob can choose to either measure the state of their friend’s record—i.e., to attest the facts established by their friend, and whose results define the random variables _A_ 0 (for Alice’s friend) and _B_ 0 (for Bob’s friend), or to jointly measure the friend’s record and the system held by the friend—to establish their own facts, defining variables _A_ 1 (for Alice) and _B_ 1 (for Bob). After comparing their results, Alice and Bob can estimate the probability distributions _P_( _Ax_, _By_) for all four combinations of _x_, _y_ = 0,1. As in the original Wigner’s friend Gedankenexperiment, the facts _A_ 1, _B_ 1 attributed to Alice and Bob and _A_ 0, _B_ 0 attributed to their friends’ measurements may be inconsistent.

This raises the question whether a more general framework exists in which all observers can reconcile their recorded facts. We shall call this assumption O, observer-independent facts, stating that a record or piece of information obtained from a measurement should be a fact of the world that all observers can agree on—and that such facts take definite values even if not all are “co-measured” ( [_7_](#R7), [_8_](#R8)). Under the additional assumptions of locality (L)—Alice’s and Bob’s choices do not influence each others’ outcome, and free choice (F)—Alice and Bob can freely choose their measurements _A_ 0, _A_ 1 and _B_ 0, _B_ 1, it should then be possible to construct a single probability distribution _P_( _A_ 0, _A_ 1, _B_ 0, _B_ 1) for the four individual facts under consideration, whose marginals match the probabilities _P_( _Ax_, _By_) ( [_3_](#R3), [_4_](#R4)).

Any joint probability distribution satisfying these assumptions must then satisfy Bell inequalities ( [_9_](#R9)). More specifically, when the variables _Ax_, _By_ take values _a_, _b_ ∈ { − 1, + 1}, then the average values 〈 _AxBy_〉 = ∑_a_, _b_ _abP_( _Ax_ = _a_, _By_ = _b_) must obey the Clauser-Horne-Shimony-Holt inequality ( [_10_](#R10))

S=〈A1B1〉+〈A1B0〉+〈A0B1〉−〈A0B0〉≤2

(2)

As shown in ( [_3_](#R3), [_4_](#R4)), a violation of the inequality above is, however, possible in a physical world described by quantum theory. Such a violation would demonstrate that the observed probability distributions _P_( _Ax_, _By_) are incompatible with assumptions F, L, and O. Therefore, if we accept F and L, it follows that the pieces of information corresponding to facts established by Alice, Bob, and their friends cannot coexist within a single, observer-independent framework ( [_3_](#R3), [_4_](#R4)). Notably, this is the case, even though Alice and Bob can acknowledge the occurrence of a definite outcome in their friend’s closed laboratory.

We note that, although Bell’s mathematical machinery ( [_11_](#R11)) is used to show the result, the set of assumptions considered here—and therefore the conclusions that can be drawn from a violation of inequality ( [_2_](#R2))—is different from those in standard Bell tests. While they share assumptions L and F, the third assumption of predetermination (PD) in the original Bell theorem ( [_12_](#R12)), differs from our assumption O in that it is only concerned with the deterministic (or otherwise) nature of measurement outcomes, not with their objectivity as in O. A Bell test is indifferent to both the observables used and the underlying system, such that any violation suffices to rule out the conjunction of L, F, and PD. In contrast, a Bell-Wigner test is based on very specific observables that satisfy the definition of an observation given below and thus represent facts relative to different observers. Formally, any Bell-Wigner violation implies a Bell-violation, but not the other way round.

Before we describe our experiment in which we test and indeed violate inequality ( [_2_](#R2)), let us first clarify our notion of an observer. Formally, an observation is the act of extracting and storing information about an observed system. Accordingly, we define an observer as any physical system that can extract information from another system by means of some interaction and store that information in a physical memory.

Such an observer can establish facts, to which we assign the value recorded in their memory. Notably, the formalism of quantum mechanics does not make a distinction between large (even conscious) and small physical systems, which is sometimes referred to as universality. Hence, our definition covers human observers, as well as more commonly used nonconscious observers such as (classical or quantum) computers and other measurement devices—even the simplest possible ones, as long as they satisfy the above requirements. We note that the no-go theorem formulated in ( [_5_](#R5)) requires observers to be “agents,” who “use” quantum theory to make predictions based on the measurement outcomes. In contrast, for the no-go theorem we tested here ( [_4_](#R4)), it is sufficient that they perform a measurement and record the outcome. The enhanced capabilities required of agents were recently discussed in ( [_13_](#R13)).

## RESULTS

Our experiment makes use of three photon-pair sources optimized for brightness and purity ( [_14_](#R14), [_15_](#R15)) based on Sagnac-type ( [_16_](#R16)) design _S_ 0, _SA_, and _SB_ (see [Fig. 2](#F2)), which generate pairs of 1550-nm single photons, entangled in the polarization degree of freedom in the state ∣Ψ−〉=(∣h〉∣v〉−∣v〉∣h〉)/2. We confirmed the almost ideal quality of the prepared states via quantum state tomography, with typical fidelity F=99.62−0.04+0.01%, purity P=99.34−0.09+0.01%, and entanglement as measured by the concurrence C=99.38−0.10+0.02% (see Materials and Methods for details). The photon pair from source _S_ 0 is rotated to

∣Ψ˜〉=1⊗U7π16∣Ψ−〉

(3)

using a half-wave plate (HWP) at an angle 7π/16, given by U7π16=cos(7π8)σz+sin(7π8)σx (where 1 is the identity and σ _z_ and σ _x_ are the Pauli operators). This state maximizes the violation of inequality ( [_2_](#R2)) for our choice of measurement settings (see [Eq. 4](#E4)).

![](/cms/10.1126/sciadv.aaw9832/asset/dd919107-0352-4ffe-91ff-6b15955aa0b9/assets/graphic/aaw9832-f2.jpeg)

Fig. 2 Experimental setup.

Pairs of entangled photons from the source _S_ 0, in modes _a_ and _b_, respectively, are distributed to Alice’s and Bob’s friend, who locally measure their respective photon in the {∣h〉, ∣v〉}-basis using entangled sources _SA_, _SB_ and type I fusion gates. These use nonclassical interference on a polarizing beam splitter (PBS) together with a set of half-wave (HWP) and quarter-wave plates (QWP). The photons in modes α′ and β′ are detected using superconducting nanowire single-photon detectors (SNSPDs) to herald the successful measurement, while the photons in modes α and β record the friends’ measurement results. Alice (Bob) then either performs a Bell-state measurement via nonclassical interference on a 50/50 beam splitter (BS) on modes _a_ and α ( _b_ and β) to measure _A_ 1 ( _B_ 1) and establish her (his) own fact or removes the BS to measure _A_ 0 ( _B_ 0) to infer the fact recorded by their respective friend (see the Supplementary Materials for details).

Source _S_ 0 provides the quantum systems on which Alice’s friend and Bob’s friend perform their measurements. Recalling the above definition of an observer, we use the entangled photon pairs from sources _SA_ and _SB_ as the physical systems that, through interaction in a type-I fusion gate ( [_17_](#R17), [_18_](#R18)) between modes _a_, α′ and _b_, β′, respectively (see [Fig. 2](#F2)), are able to extract information and thereby establish their own facts. When successful, the fusion gate realizes a nondestructive polarization measurement of a photon from _S_ 0 in the {∣ _h_〉, ∣ _v_〉}-basis, whose results ∣“photon is _h_”〉 or ∣“photon is _v_”〉 represent the friend’s record. Via the ancillary entanglement, the extracted information is then stored in the polarization state of the other photon from _SA_ ( _SB_)—in mode α (β)—that acts as a memory, while the photon in mode α′ (β′) is absorbed in a single-photon counter to herald the success of the measurement (see Materials and Methods for details). Note that this detection could be delayed until the end of the experiment as it carries no information about the measurement outcome, akin to the observer in the box communicating that an observation took place ( [_3_](#R3), [_4_](#R4)). From Alice’s and Bob’s perspective, the yet undetected photons from _S_ 0, _SA_, and _SB_ are now in a joint four-photon entangled state (see [Eq. 11](#E11) in Materials and Methods).

To test inequality ( [_2_](#R2)), Alice and Bob then measure the following observables on their respective joint photon/friend’s record systems

A0=B0=1⊗(∣“photon ish”〉〈“photon ish”∣−∣“photon isv”〉〈“photon isv”∣),A1=B1=∣Φphoton/record+〉〈Φphoton/record+∣−∣Φphoton/record−〉〈Φphoton/record−∣

(4)

(with ∣Φphoton/record±〉 as defined in [Eq. 1](#E1)). The observables _A_ 0 and _B_ 0 directly unveil the records established by Alice’s and Bob’s friend, respectively. The observables _A_ 1 and _B_ 1, on the other hand, correspond to Alice’s and Bob’s joint measurements on their friend’s photon and record, and define their own facts in the same way as Wigner in the original thought experiment confirms his entangled state assignment.

We estimate the four average values 〈 _AxBy_〉 in inequality [_2_](#R2) via projection onto each of the 4 × 4 eigenstates of the observables _Ax_ and _By_ (see Materials and Methods for details). For the corresponding 64 settings, we collect 1794 six-photon coincidence events over a total measurement time of 360 hours, from which we calculate the probabilities shown in [Fig. 3](#F3). We achieve a value of Sexp=2.416−0.075+0.075, thus violating inequality ( [_2_](#R2)) by more than five standard deviations. This result is primarily limited by the higher-order photon emissions from our probabilistic photon sources. Statistical uncertainties are independently estimated using an error propagation approach and a Monte Carlo method. Details are discussed in Materials and Methods.

![](/cms/10.1126/sciadv.aaw9832/asset/a1528c93-0d6e-4200-ae8f-2fb595b4bc11/assets/graphic/aaw9832-f3.jpeg)

Fig. 3 Experimental data.

The outcome probabilities comprising each of the four expectation values 〈 _A_ 0 _B_ 0〉, 〈 _A_ 0 _B_ 1〉, 〈 _A_ 1 _B_ 0〉, and 〈 _A_ 1 _B_ 1〉 are obtained from the measured sixfold coincidence events for each set of 4 × 4 eigenvectors during a fixed time window. Shown here are only the data corresponding to nonzero eigenvalues labeled on the horizontal axes + and − for +1 and −1, respectively, with the full data shown in the Supplementary Materials. The theoretical predictions are shown as orange bars, and each measured expectation value is given above the corresponding subfigure. Uncertainties on the latter and error bars on the data represent 1σ statistical confidence intervals assuming Poissonian counting statistics (see the Supplementary Materials).

## DISCUSSION

In principle, “Bell-Wigner tests” like ours are subject to similar loopholes as tests of conventional Bell inequalities ( [_19_](#R19)). To address the detection and space-time loopholes, we make the physically reasonable assumption of fair sampling and rely on the empirical absence of signaling between our measurement devices (which we experimentally verified to be in agreement with the expectation from Poissonian statistics). Another loophole may arise if the observables _A_ 0, _B_ 0 that are measured in practice do not strictly correspond to a measurement of the friends’ records. Here, we assume (with reasonable confidence, up to negligible experimental deviations) that the measured observables indeed factorize as in [Eq. 4](#E4), with the identity on the photon system, so that the above interpretation for _A_ 0, _B_ 0 can be trusted. As discussed in the Supplementary Materials, closing all loopholes in full will be considerably more challenging than for Bell tests.

One might further be tempted to deny our photonic memories the status of “observer.” This, however, would require a convincing revision of our minimal definition of what qualifies as an observer, which typically comes at the cost of introducing new physics that is not described by standard quantum theory. Wigner, for example, argued that the disagreement with his hypothetical friend could not arise due to a supposed impossibility for conscious observers to be in a superposition state ( [_2_](#R2)). However, the lack of objectivity revealed by a Bell-Wigner test does not arise in anyone’s consciousness, but between the recorded facts. Because quantum theory does not distinguish between information recorded in a microscopic system (such as our photonic memory) and in a macroscopic system, the conclusions are the same for both: The measurement records are in conflict regardless of the size or complexity of the observer that records them. Implementing the experiment with more complex observers would not necessarily lead to new insights into the specific issue of observer independence in quantum theory. It would, however, serve to show that quantum mechanics still holds at larger scales, ruling out alternative (collapse) models ( [_20_](#R20)). However, this is not the point of a Bell-Wigner test—less demanding experiments could show that.

Modulo the potential loopholes and accepting the photons’ status as observers, the violation of inequality ( [_2_](#R2)) implies that at least one of the three assumptions of free choice, locality, and observer-independent facts must fail. The related no-go theorem by Frauchiger and Renner ( [_5_](#R5)) rests on different assumptions, which do not explicitly include locality. While the precise interpretation of ( [_5_](#R5)) within nonlocal theories is under debate ( [_21_](#R21)), it seems that abandoning free choice and locality might not resolve the contradiction ( [_5_](#R5)). A compelling way to accommodate our result is then to proclaim that facts of the world can only be established by a privileged observer—e.g., one that would have access to the “global wavefunction” in the many worlds interpretation ( [_22_](#R22)) or Bohmian mechanics ( [_23_](#R23)). Another option is to give up observer independence completely by considering facts only relative to observers ( [_24_](#R24)), or by adopting an interpretation such as QBism, where quantum mechanics is just a tool that captures an agent’s subjective prediction of future measurement outcomes ( [_25_](#R25)). This choice, however, requires us to embrace the possibility that different observers irreconcilably disagree about what happened in an experiment. A further interesting question is whether the conclusions drawn from Bell or Bell-Wigner tests change under relativistic conditions with non-inertial observers ( [_26_](#R26)).

## MATERIALS AND METHODS

### Setup details

A 775-nm, 1.6 ps–pulsed Ti:sapphire laser was focused into a 22-mm periodically poled potassium titanyl phosphate (ppKTP) crystal in a Sagnac-type interferometer ( [_16_](#R16), [_27_](#R27)), where it generated pairs of 1550-nm single photons through collinear type-II parametric down-conversion. The 80-MHz repetition rate of the pump laser was quadrupled through temporal multiplexing ( [_28_](#R28)) to suppress higher-order emissions (see fig. S1). We thereby achieved a signal-to-noise ratio (i.e., photon pairs versus higher-order contributions) of 140 ± 10 in each photon source, generating ∼8000 photon pairs mW−1 s−1 with a typical heralding efficiency η=(cc/s1s2) of ∼50%, where cc is the number of coincidence counts, and _s_ 1 and _s_ 2 are the numbers of singles in the first and second output, respectively. Single photons passed through 3-nm band-pass filters to guarantee high spectral purity and were detected with superconducting nanowire single-photon detectors (SNSPDs) with a detection efficiency of ∼80%. Detector clicks were time-tagged using a field-programmable gate array and processed to detect coincidences within a temporal window of 1 ns.

To benchmark the three required two-qubit states, we performed maximum-likelihood quantum state tomography directly at each source. From the reconstructed density matrices, we computed the fidelity, concurrence, and purity quoted in the main text. Further transmission of the photon pairs to the fusion gates slightly degrades the fidelities of the three entangled pairs to F0=98.79−0.03+0.03%, FA=98.70−0.03+0.03%, and FB=98.59−0.03+0.03% for sources _S_ 0, _SA_, and _SB_, respectively (see [Fig. 2](#F2)). This indicates that the optical circuit preserves the excellent quality of the initial states.

### Measurement protocol

We now describe in detail the measurement procedure sketched in [Fig. 2](#F2). Source _S_ 0 and the HWP on its right output arm produce an entangled pair of photons in the state of [Eq. 3](#E3). This photon pair is distributed to the laboratories of Alice’s friend and Bob’s friend, who measure their photon using type-I fusion gates ( [_17_](#R17)). Each fusion gate is implemented with a polarizing beam splitter (PBS), where horizontally and vertically polarized photons are transmitted and reflected, respectively (by convention collecting a phase _i_ for the latter). Two photons entering the PBS from two different inputs with opposite polarization, ∣ _h_〉 ∣ _v_〉 or ∣ _v_〉 ∣ _h_〉, will exit from the same output port and will therefore not lead to coincident detection. Only the coincident ∣ _h_〉 ∣ _h_〉 and ∣ _v_〉 ∣ _v_〉 components will be recorded in post-selection. For these post-selected photons, the fusion gate induces the following transformations

∣h〉∣h〉→PBS∣h〉∣h〉→Q/HWP∣h〉∣h〉+i∣v〉2,∣v〉∣v〉→PBS−∣v〉∣v〉→Q/HWP−∣v〉∣h〉−i∣v〉2

(5)

where Q/HWP refers to the combination of a quarter-wave plate at π/4 and a half-wave plate at π/8 behind the PBS (see [Fig. 2](#F2)). The second (heralding) photon in the above equation is then projected onto the state ∣ _h_〉 via another PBS. The type-I fusion gate thus implements the operation

FGI=12(∣h〉〈h∣〈h∣−∣v〉〈v∣〈v∣)

(6)

where the factor 12 indicates the success probability of the gate of 12.

To use the fusion gate to measure photon _a_ (see [Fig. 2](#F2)) nondestructively, Alice’s friend uses an ancilla from the entangled pair created by _SA_, prepared as ∣Ψ−〉α′α. Depending on the state of the incoming photon, the operation performed by Alice’s friend transforms the overall state as

∣h〉a∣Ψ−〉α′α=12(∣h〉a∣h〉α′∣v〉α−∣h〉a∣v〉α′∣h〉α)→FGI12∣h〉a∣v〉α,∣v〉a∣Ψ−〉α′α=12(∣v〉a∣h〉α′∣v〉α−∣v〉a∣v〉α′∣h〉α)→FGI12∣v〉a∣h〉α

(7)

Hence, the state ∣ _h_〉 _a_ or ∣ _v_〉 _a_ of the external photon in mode _a_ is copied, after being flipped ( _h_ ↔ _v_), onto Alice’s friend’s photon in mode α. This corresponds to a measurement of the incoming photon in the {∣ _h_〉, ∣ _v_〉}-basis, with the outcome being recorded in the state of photon α such that we can write

∣“photon ish”〉α=∣v〉α,∣“photon isv”〉α=∣h〉α

(8)

The amplitudes 12 in [Eq. 7](#E7) indicate the total success probability of 14 for this procedure.

Consider now the central source _S_ 0 together with Alice’s and Bob’s friend laboratories. According to [Eq. 3](#E3), the state generated by _S_ 0 is, after the unitary U7π16

∣Ψ˜〉ab=12cosπ8(∣h〉a∣v〉b+∣v〉a∣h〉b)+12sinπ8(∣h〉a∣h〉b−∣v〉a∣v〉b)

(9)

The transformations induced by Alice’s and Bob’s friend are then, according to [Eq. 7](#E7)

∣Ψ˜〉ab∣Ψ−〉α′α∣Ψ−〉β′β→FGI⊗214∣Ψ˜′〉aαbβ

(10)

with a global success probability of 116. The state

∣Ψ˜′〉aαbβ=12cosπ8(∣hv〉aα∣vh〉bβ+∣vh〉aα∣hv〉bβ)+12sinπ8(∣hv〉aα∣hv〉bβ−∣vh〉aα∣vh〉bβ)

(11)

is the four-photon state shared by Alice and Bob when both fusion gates are successful.

Recalling from [Eq. 8](#E8) how the friends’ measurement results are encoded in their polarization states, the observables of [Eq. 4](#E4) to be measured on ∣Ψ˜′〉aαbβ are

A0=B0=1⊗(∣v〉〈v∣−∣h〉〈h∣),A1=B1=∣Ψ+〉〈Ψ+∣−∣Ψ−〉〈Ψ−∣

(12)

with ∣Ψ±〉=12(∣hv〉±∣vh〉). To obtain 〈 _AxBy_〉, we projected these states onto all combinations of eigenstates of _Ax_ and _By_ individually and recorded six-photon coincidence events for a fixed duration. More specifically, to measure _A_ 0 (similarly _B_ 0), we projected onto ∣ _hv_〉_a_ α and ∣ _vv_〉_a_ α (eigenvalue +1), and ∣ _hh_〉_a_ α and ∣ _vh_〉_a_ α (eigenvalue −1) using QWP and HWP to implement local rotations before the final PBS, not using the beam splitter (BS) in [Fig. 2](#F2). Note that _A_ 0 cannot be simply measured by ignoring photon _a_ due to the probabilistic nature of the photon source. Hence, this photon has to be measured in a polarization-insensitive way, which, due to the polarization-sensitive nature of the photon-detectors, is best achieved by summing over the projections onto both orthogonal polarizations. To measure _A_ 1 ( _B_ 1), we used a 50/50 BS followed by projection onto ∣ _vh_〉. Because of nonclassical interference in the BS, this implements a projection onto the singlet state ∣Ψ−〉_a_ α with success probability 12. Using quantum measurement tomography, we verified this Bell-state measurement with a fidelity of Fbsm=96.84−0.05+0.05. Projections on the other Bell states are possible via local rotations using the same QWP and HWP as before. Here, ∣Ψ+〉_a_ α takes eigenvalue +1, ∣Ψ−〉_a_ α takes eigenvalue −1, and ∣Φ±〉aα=12(∣hh〉±∣vv〉)aα takes eigenvalue 0. Probabilities are obtained from normalizing the measured counts with respect to the total of the 16 measurements for each pair of observables (see fig. S2). The theoretically expected values for the various probabilities are 14(1+12)≃0.427, 14(1+12)≃0.073, or 0. In addition to this result, an alternative measurement protocol for _A_ 0 and _B_ 0 is presented in the Supplementary Materials.

### Error analysis

As described previously, each average value 〈 _AxBy_〉 was calculated from 16 measured sixfold coincidence counts _ni_. These numbers follow a Poisson distribution with variance σni2=ni. The uncertainty on 〈 _AxBy_〉 = _f_( _n_ 1, …, _n_ 16) can then be computed using

σf2(n1,…,n16)=∑i=116(∂f∂ni)2σni2

(13)

Because the four averages 〈 _A_ 1 _B_ 1〉, 〈 _A_ 1 _B_ 0〉, 〈 _A_ 0 _B_ 1〉, and 〈 _A_ 0 _B_ 0〉 are statistically independent, the uncertainties can be calculated independently and combined to estimate the uncertainty on _S_. To take into account potentially asymmetric errors in the limit of small count rates, we computed the uncertainty on the Bell-Wigner parameter _S_ using a Monte Carlo routine with 100,000 samples. The values obtained through these two methods agree to within 0.0032.

Note that in the results shown in fig. S3 with the observables of eq. S1, errors are correlated due to normalization with a common total. Accounting for this in the error propagation results in slightly larger statistical uncertainty.

The Bell-Wigner value _S_ exp that can be achieved experimentally is primarily limited by multipair emissions from our probabilistic photon pair sources. We first note that any emission of three pairs from any subset of our three sources occurs with roughly similar probability. To exclude unwanted terms, we used six-fold coincidence detection, which can only be successful for an emission of one pair each in _S_ 0, _SA_, and _SB_, or three pairs in _S_ 0. The latter would amount to noise but is excluded by our cross-polarization design and can thus not lead to a coincidence detection. This leaves higher-order contributions, where at least four photon pairs are produced as the main source of errors. Because such events scale with a higher exponent of the pump power, they are suppressed in our experiment by working with a relatively low pump power of 100 mW.

## Acknowledgments

We thank Č. Brukner, R. Renner, and F. Shahandeh for useful discussions. **Funding:** This work was supported by the UK Engineering and Physical Sciences Research Council (grant numbers EP/N002962/1 and EP/L015110/1) and the French National Research Agency (grant number ANR-13-PDOC-0026). This project has received funding from the European Union’s Horizon 2020 research and innovation program under Marie Skłodowska-Curie grant agreement number 801110 and the Austrian Federal Ministry of Education, Science and Research. **Author contributions:** M.R., A.F., and C.B. conceived the project and designed the experiment. M.P., A.P., F.G., P.B., and D.K. performed the experiment and collected the data. M.P. and F.G. analyzed the data. C.B. developed the theory results. All authors contributed to writing the manuscript. **Competing interests:** The authors declare that they have no competing interests. **Data and materials availability:** All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

## Supplementary Material

### Summary

Supplementary Text

Fig. S1. Detailed experimental setup.

Fig. S2. Full experimental data.

Fig. S3. Alternative protocol experimental data.

References ( [_29_](#R29)– [_31_](#R31))

### Resources

File(aaw9832\_sm.pdf)

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Nov. 9, 2020

###### RE:THE ROLE OF TIME IN RELATIONAL QUANTUM MECHANICS

Arturo Tozzi

- Pediatrician
- University of North Texas

Quantum mechanics has been interpreted by influential scientists as a theory assessing observer-dependent relational properties/reference frames (Smerlak and Rovelli 2007; Yang 2018). Relational formulations of quantum mechanics have been experimentally supported by recent papers that contradict the tenet of local realism: "the properties of the physical world are independent from the observer" (The BIG Bell Test Collaboration 2018; Proietti 2019).

Here a philosophical concept from our medioeval past comes into play. In the question 19 of the second quodlibet (ed. Ockham 1991), William of Ockham asks whether the propositon "hoc est corpus meum" ("this is my body") is literally true as uttered by the priest at mass. According to Ockham, to every spoken proposition corresponds some mental proposition. When the priest utters the first words: "hoc est...", the body of Christ does not exist yet in the Eucharist, because the words "hoc est" are not themselves determinate enough to fix the meaning intended by the speaker. Indeed, the priest's utterance in the middle of the phrase is true as mental proposition, but it is not yet true as spoken proposition, because the words "corpus meum" have not yet been pronounced.

A mental proposition is true at the beginning and at the end, because it exists as a whole all at once. This means that a mental proposition can be termed "ens permanens" (a permanent entity). In turn, a spoken proposition exists just successively and not all at once, so that it is true at the end but not at the beginning of the utterance. This means that a spoken proposition can be termed "ens successivum" (a successive entity).

Therefore, the proposition is true not at the beginning, rather at the end of the utterance. In terms of relational quantum mechanics, we may state that reality itself is different at the beginning and at the end of the utterance, or, in non-linguistic terms, reality itself is different at the beginning and at the end of the experimental procedures to detect quantum phenomena. To provide an example, at the beginning of the experiments that assess quantum entanglement, it is not certain whether the quantum state detected at the end will exist or not.

Therefore, quantum reality might stand for an ens successivum (successive entity) such that only a proper part of it exists at any given time. This means that the role of time, underrated and almost dispised in quantum mechanics, must be kept into account: a deep difference does exist in a quantum state before and after the intervention of the observer.

A further leap is required to proceed towards a novel approach to quantum mechanics: from relationism to interactionism.

1) Ockham W. 1991. Quodlibetal questions. Eds. Freddoso AJ, Kelley FE. Yale University Press, New Haven & London.

2) Proietti M, Pickston A,  Graffitti F, Barrow P, Kundys D, et al. 2019. Experimental test of local observer independence. Science Advances, Vol. 5, no. 9, eaaw9832, DOI: 10.1126/sciadv.aaw9832.

3) Smerlak, M.; Rovelli, C. Relational EPR. Found. Phys.2007, 37, 427–445.

4) The BIG Bell Test Collaboration. Challenging local realism with human choices. Nature. 2018, 557, 212–216.

5) Yang, J.M. A Relational Formulation of Quantum Mechanics. Sci. Rep.2018, 8, 13305.

Arturo Tozzi

M.D., Pediatrician, ASL Na2Nord, Italy

phD, Computational Intelligence Lab, University of Manitoba, Canada

Adjunct Assistant Professor in Physics, University of North Texas, USA

[\[email protected\]](/cdn-cgi/l/email-protection#cd99a2b7b7a4acbfb9b8bfa28da1a4afa8bfa2e3a4b9)

Nov. 21, 2019

###### RE: quantum wave function algebra

Suresh Kumar.S

- Chief Scientist \[rt\]
- CSIR

Topological spaces as geometric manifolds have algebraic structures which can be manipulated by the observers and measurement devices or processes.The classical phase space has Hamiltonian vector fields as flows of differentials of the Hamiltonian.The wave forms as complex subspaces can intersect with convergence to indicate the observable as a measured single value,upon observation,in a transition to classicality from quantum space.

The exterior algebra involving wedge product can induce cup product of the cohomology group associated with the symplectic form geometry of classical mechanics.The cup product is dual to subspace intersection,in a mathematical sense,and assigns richer algebraic variants to geometry of manifolds of topological spaces.Measurement as a case of subspace intersections can involve exterior algebra that distinguish between cohomology groups associated with vector spaces. This implies that observer based variations in quantum reality,as measured,becomes topolgically possible

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#### Acknowledgments

We thank Č. Brukner, R. Renner, and F. Shahandeh for useful discussions. **Funding:** This work was supported by the UK Engineering and Physical Sciences Research Council (grant numbers EP/N002962/1 and EP/L015110/1) and the French National Research Agency (grant number ANR-13-PDOC-0026). This project has received funding from the European Union’s Horizon 2020 research and innovation program under Marie Skłodowska-Curie grant agreement number 801110 and the Austrian Federal Ministry of Education, Science and Research. **Author contributions:** M.R., A.F., and C.B. conceived the project and designed the experiment. M.P., A.P., F.G., P.B., and D.K. performed the experiment and collected the data. M.P. and F.G. analyzed the data. C.B. developed the theory results. All authors contributed to writing the manuscript. **Competing interests:** The authors declare that they have no competing interests. **Data and materials availability:** All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

### Authors

#### Affiliations

MassimilianoProietti [https://orcid.org/0000-0002-6180-0699](https://orcid.org/0000-0002-6180-0699)

Scottish Universities Physics Alliance, Institute of Photonics and Quantum Sciences, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK.

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AlexanderPickston

Scottish Universities Physics Alliance, Institute of Photonics and Quantum Sciences, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK.

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FrancescoGraffitti [https://orcid.org/0000-0002-8802-8360](https://orcid.org/0000-0002-8802-8360)

Scottish Universities Physics Alliance, Institute of Photonics and Quantum Sciences, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK.

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PeterBarrow

Scottish Universities Physics Alliance, Institute of Photonics and Quantum Sciences, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK.

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DmytroKundys

Scottish Universities Physics Alliance, Institute of Photonics and Quantum Sciences, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK.

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CyrilBranciard [https://orcid.org/0000-0001-9460-825X](https://orcid.org/0000-0001-9460-825X)

Université Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France.

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MartinRingbauer [https://orcid.org/0000-0001-5055-6240](https://orcid.org/0000-0001-5055-6240)

Scottish Universities Physics Alliance, Institute of Photonics and Quantum Sciences, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK.

Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria.

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AlessandroFedrizzi[\*](#cor1) [https://orcid.org/0000-0002-5089-3148](https://orcid.org/0000-0002-5089-3148) [\[email protected\]](/cdn-cgi/l/email-protection#e687c8808382948f9c9c8fa68e91c88785c8938d)

Scottish Universities Physics Alliance, Institute of Photonics and Quantum Sciences, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK.

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#### Funding Information

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#### Notes

\*

Corresponding author. Email: [\[email protected\]](/cdn-cgi/l/email-protection#3a5b145c5f5e48534040537a524d145b59144f51)

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Experimental test of local observer independence._Sci. Adv._**5**,eaaw9832(2019).DOI: [10.1126/sciadv.aaw9832](https://doi.org/10.1126/sciadv.aaw9832)

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