Quantum erasing the memory of Wigner's friend
Cyril Elouard, Philippe Lewalle, Sreenath K. Manikandan, Spencer Rogers, Adam Frank, Andrew N. Jordan
QQuantum erasing the memory of Wigner’s friend
Cyril Elouard, ∗ Philippe Lewalle, Sreenath K. Manikandan, Spencer Rogers, Adam Frank, and Andrew N. Jordan
1, 2 Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA Institute for Quantum Studies, Chapman University, Orange, CA 92866, USA (Dated: September 22, 2020)The Wigner’s friend paradox concerns one of the most puzzling concepts of quantum mechanics:the consistent description of multiple nested observers. Recently, a variation of Wigner’s gedanken-experiment, introduced by Frauchiger and Renner, has lead to new debates about the self-consistencyof quantum mechanics. We propose a simple single-photon interferometric setup implementing theirscenario, and use our reformulation to shed a new light on the assumptions leading to their paradox.From our description, we argue that the three apparently incompatible properties used to questionthe consistency of quantum mechanics correspond to two logically distinct contexts: either assumingthat Wigner has full control over his friends’ lab, or conversely that some part of the labs remainunaffected by Wigner’s subsequent measurements. The first context may be seen as the quantumerasure of the memory of Wigner’s friend. We further show these properties are associated withobservables which do not commute, and therefore cannot take well-defined values simultaneously.Consequently, the three contradictory properties never hold simultaneously.
I. INTRODUCTION
In his famous gedankenexperiment, Wigner analyzesa setup in which a whole lab, containing a human be-ing (a friend of his) is measured by a “super-observer”assumed to have full control over all the lab degrees offreedom [1]. Paradoxical conclusions may emerge fromsuch situations due to the tension between the rules ofevolution for arbitrary isolated quantum systems, whichin principle can be applied at any scale, and the need forthe projection postulate to describe measurements per-formed by observers. While in most practical situations,it is clear whether or not an entity should be consid-ered as a quantum system or an observer—and thereforewhether its interaction with the system should be de-scribed with a unitary evolution (Schr¨odinger equation)or via the projection postulate—the transition betweenthese two behaviors continues to cause much debate. Inparticular it remains unclear where such a “cut” [2] ex-ists between systems that can exist in superposition andthose which cannot either in principle or in practice.In Ref. [3], Frauchiger and Renner (FR) present an ex-tended Wigner friend scenario involving two observers,the friends of Wigner, and two “super-observers”, W andW. The latter are assumed to have perfect control of thetwo friends and their labs, and be able to measure themin arbitrary bases. FR key result is to formulate appar-ently natural assumptions about this setup which theyargue lead to an inconsistency. This study has triggereda large number of comments and articles re-examiningthe scenario. These new papers have identified hiddenassumptions [4, 5], gathered different arguments againstFR’s surprising conclusion [6–9], and generated new dis-cussions of the quantum formalism and its interpretations ∗ [email protected] [10–14].Below, we propose a simple reformulation of this sce-nario based on an optical interferometer. Crucially, inour setup, the observers and their labs are replaced bysmall quantum systems, that we refer to as memories .Provided they remain untouched after interacting withthe system to be measured, each memory behaves asan observer, in the sense that from the point of viewof any other observer, the quantum statistics of the sys-tem matches that of the state randomly collapsed in themeasurement basis according to Born’s rule. Althoughour simplification marks a philosophical departure fromFR’s discussion of a “friend”, our model, in fact, leads tothe same mathematical description.We can then revisit the assumptions made in Ref. [3]and the properties leading to the alleged inconsistency.We will show that, taken together, these assumptions andproperties either require the memories to be erased by thesuper-observer, preventing them from behaving as properobservers, or require the memories to remain untouchedeliminating the main feature of the super-observer. In theend, the inconsistency only appears if one compares prop-erties from two logically different contexts that we makeexplicit. We also demonstrate that these three proper-ties are associated with the value taken by observableswhich do not commute, and therefore are forbidden byquantum mechanics to take simultaneously well-definedvalues. Thus we find that the argument of Ref. [3] doesnot lead to any of the claimed inconsistency within quan-tum mechanics.Our work does more than deflate the claims of such aninconsistency however. By focusing on the role of mem-ory, projection, and Unitary evolution, we make explicitlimits in the Wigner’s Friend paradox. This discussion isparticularly useful in light of the recent Wigner’s Friendrelated No-Go theorem of [15] that sought to illuminatekey issues in quantum interpretation. a r X i v : . [ qu a n t - ph ] S e p II. RESULTSA. Observers, Super-observers, Memory andEffective Collapse
We begin with a short discussion of the difference be-tween observers and super-observers, as this issue lies atthe heart of the argument of the Wigner’s Friend paradox[1] and of Ref. [3]. In this context an observer begins witha quantum system that has been prepared in a superpo-sition of states. When the observer measures the systemthe projection postulate asserts that it collapses into oneof the states composing the original superposition.The results of the measurement reside in the memoryof the observer. For specificity we can consider that theobserver writes down the measurement results on a pieceof paper, which can also be considered the memory ofthe collapse. Another observer, as opposed to a super-observer, only has access to the result of the measure-ment either by asking the first observer, looking at thepiece of paper, or measuring the quantum system again.A super-observer, however, has full access to the wholequantum system comprising the memory and the mea-sured system. This larger quantum system is a highlyinteracting collection of > atoms. Thus the mean-ing of “full access” is that super-observers can manipu-late all > atoms in their full entangled, superposedstates. As pointed out by Schr¨odinger and his famouscat gedankenexperiment [16], we never see macroscopicobjects like a piece of paper with writing on it in a quan-tum superposition. Beyond the difficulty related to thesize and complexity of such a large system itself, a puresuperposition state is posited on the system being com-pletely closed off from any “environment”. Clearly thisbecomes increasingly difficult to achieve, in practice, asthe system size grows. This issue is, essentially, at theheart of the Wigner’s Friend paradox. We address thisquestion by reducing its complexity to its essence in or-der to understand what would it mean for a macroscopicsystem (such as Wigner’s Friend) to act as a “memory”that could be manipulated as an entangled, superposedquantum state.In order to investigate the transition between a be-havior of observer inducing (at least effectively) a col-lapse of the wavefunction, and that of a quantum sys-tem that should definitely be described by a wavefunc-tion undergoing a unitary evolution, we consider thefollowing model of measurement: the quantum system S being measured is a qubit admitting the basis ofstates {| (cid:105) S , | (cid:105) S } and initially described by state | ψ (cid:105) S = c | (cid:105) S + c | (cid:105) S . During the measurement, this qubit in-teracts with a memory, which is another qubit M admit-ting the basis of states {| (cid:105) M , | (cid:105) M } . The outcome of themeasurement on S , which is assumed to be a measure-ment in basis {| (cid:105) S , | (cid:105) S } , is encoded into the state ofqubit M . After the interaction, which corresponds to ajoint unitary evolution of the two-qubit system, the two qubits are in the entangled state | Ψ (cid:105) SM = c | (cid:105) S ⊗ | (cid:105) M + c | (cid:105) S ⊗ | (cid:105) M . (1)The memory qubit now serves as a retreivable record thatthe system was found either in state | (cid:105) S or | (cid:105) S . In hisseminal model for quantum measurement [17], von Neu-mann then terminates the process by assuming that thememory qubit is then read by an observer and thereforecollapses to either | (cid:105) M or | (cid:105) M . This last steps precludesus from analyzing the case of a super-observer, we there-fore stop the measurement process here.We are now interested in describing sequential mea-surements by independent observers. We therefore as-sume that we do not have access to the memory qubit M ,and that it remains untouched after its interaction withqubit S . Then, the statistics of any additional measure-ment performed afterwards on qubit S are equivalentlydescribed (see Methods) by the state | Ψ (cid:105) above, or bythe statistical mixture state: ρ S = | c | | (cid:105) S (cid:104) | + | c | | (cid:105) S (cid:104) | . (2)This mixed state ρ S has been obtained from | Ψ (cid:105) by ap-plying the projection postulate to describe the first mea-surement (assuming we do not know its result).This equivalence principle has an importance conse-quence when trying to understand the foundations ofquantum measurement. Indeed, one could postulate thata quantum measurement always corresponds to a unitaryevolution with some stable degree of freedom playing therole of the memory, and that the wavefunction collapse ismerely a practical way to get rid of such degrees of free-dom and reduce our description to the system degreesof freedom only. In our example description the pieceof paper on which an observer writes down their resultsrepresents stable degrees of freedom. This is becauseit is comprised of so many interacting systems (atomsof paper and ink) that there is no reasonable measure-ment mechanism by which it, as a whole, can be ma-nipulated after the fact. The same, we argue, can besaid for brains. Further motivation for this approach canbe found by noting that when a quantum system inter-acts with a complex system (the measuring apparatus),the information about the state of the quantum systemis effectively copied multiple time in various degrees offreedom [18] . The reader familiar with quantum information theory might fearthis information replication could be in contradiction with theno-cloning theorem which forbids the perfect copying of unknownquantum states [19–21]. This is not the case however, as the in-formation about a single observable only (characterized by eigen-basis {| (cid:105) S , | (cid:105) S } ) is copied and not the full quantum state. Themechanism behind this replication process is an evolution drivenby the interaction of the system with a large number of degreesof freedom in the apparatus, or by interactions among these de-grees of freedom, which generates entanglement with the systemjust as in Eq. (1). In the context of quantum measurement ex-periments, this process is termed quantum–limited amplification[22–24] and is realized with an increasing level of control. We emphasize however that Eq. (1) does not explainhow a single outcome is obtained from a readout of thememory and only provides the probabilistic correlationsthat can be observed if such a readout is made. The tran-sition to a single definite outcome can only be describedusing the projection postulate. Within the unitary de-scription of the measurement, this last projection canhowever be postponed to anytime after the interactionbetween the system and the memory. Thus the memoryholds the information about the system’s state, and canbe retrieved at the end of the desired quantum mechan-ical treatment that can include further entangling inter-actions with other memories, as in the scenario presentedbelow.
B. Interferometric setup
1. Extended Wigner’s Friend scenario
We now consider the following reformulation of the sit-uation considered by FR in Ref. [3]. For a more detailedexact mapping of notations, see Table I. The scenario in-volves two labs, each containing a qubit and an observer(referred to as a “friend” of Wigner), and two observersable to measure the two labs, W and W. For the sake ofsimplicity, we merge W and W into a single observer ex-ternal to the labs, that we call Wigner. A crucial assump-tion of the proposal is that Wigner is a “super-observer”,able in particular to measure the two labs in bases in-compatible with (rotated with respect to) those in whichthe friends are measuring their own qubits.The mere possibility of Wigner’s full control over thelabs prevents us from using the usual paradigm for mea-surement in quantum mechanics, involving the wavefunc-tion collapse, to describe the measurements done by hisfriends. For this reason, we will use as model for themeasurements performed by the friends unitary interac-tions with a memory as introduced in Section II A. Onthe other hand, as Wigner’s measurement marks the endof the experiment, and it is assumed that nobody elsewill manipulate Wigner and his environment, we canconversely describe Wigner’s measurements via the usualprojection postulate.
2. Degrees of freedom and roles
In our formulation, the role of the Wigner’s two friends,their labs, and the qubits they measure, are all played bydifferent degrees of freedom of a single photon travel-ing through a Mach-Zender interferometer (see Fig. 1).The state of the photon is characterized by several de-grees of freedom which for our purpose can all be mod-eled by qubits (three in total). First, the path takenby the photon (the arm of the interferometer it travelsthrough) plays the role of one of the two qubits. Thisqubit initially belongs to friend F who is able to measure ab
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PBSPBS
Friends' measurement setup Wigner's measurement setup c Friends' measurement setup
Context 1Context 2
PBSPBS
Wigner's measurement setup
FIG. 1. a : Extended Wigner’s friend scenario. Wigner’sfriend F (whose lab is described by states 1 ,
2) measures afirst qubit in basis { l, r } , and depending on the outcome pre-pares another qubit and sends it to the F’s lab. F measuresthe second qubit in basis { H, V } . Finally, Wigner measuresboth labs in the basis { ( A, ok) , ( A, fail) , ( D, ok) , ( D, fail) } . b :Proposed inteferometric setup to implement the extendedWigner’s friend scenario, when Wigner is indeed a super-observer (context 1). c : Setup corresponding to context 2,where Wigner is not a super-observer. The difference is theabsence of the second mode-shaper inside the dotted-blue box.In b , c , PBS refers to polarizing beam-splitter. it. The role of the whole closed lab they live in is playedby the spatiotemporal shape of the photon wavepacket(or equivalently the mode of the waveguide the photon isin). Finally, the polarization of the photon plays therole of the other qubit, the other friend F of Wignerable to measure it and the lab in which they are lo-cated, all together. While one could think of a moresophisticated version of the setup allowing us to distin-guish these roles, it will be enough for our purpose towork with these three qubits. We can specify the photonstate using the orthogonal basis {| α, n, s (cid:105)} . The states | α, n, s (cid:105) ≡ | α (cid:105) pol ⊗ | n (cid:105) path ⊗ | s (cid:105) shape are labeled by thepolarization α = H, V , the path taken by the photon n = l, r and the two possible orthogonal spatiotemporalshapes of its wavepacket s = 1 , | D (cid:105) pol = √ (cid:16) | H (cid:105) pol + | V (cid:105) pol (cid:17) , (3a) | A (cid:105) pol = √ (cid:16) | H (cid:105) pol − | V (cid:105) pol (cid:17) . (3b)
3. Evolution of the photon state
Following the scenario of FR, and as illustrated inFig. 1 a , the photon is initially prepared in polarizationstate | D (cid:105) pol and a wavepacket of shape s = 2. Moreover,the beam-splitter is assumed to have 1 / | Ψ (cid:105) = 1 √ (cid:16) | D, r, (cid:105) + √ | D, l, (cid:105) (cid:17) . (4)F measures the which-path information. In our setup,this corresponds to a correlation between the mode shapeof the photon and the path, that can be generated e.g.if an optical element causing the transition from state | (cid:105) shape to the orthogonal state | (cid:105) shape , is inserted in thearm r , yielding the photon state: | Ψ (cid:105) = 1 √ (cid:16) | D, r, (cid:105) + √ | D, l, (cid:105) (cid:17) . (5)Then, in the scenario of Ref [3], F prepares the stateof the other qubit depending on its outcome. This canbe achieved assuming the presence of a chiral crystal (orwaveplate) in path r rotating the polarization of the pho-ton by 45 ◦ , transforming the diagonal polarization stateinto the vertical polarization: | Ψ (cid:105) = 1 √ (cid:16) | V , r, (cid:105) + √ | D, l, (cid:105) (cid:17) . (6)Finally, the other friend, F, measures this qubit inthe basis H, V , the result being encoded into a memorypresent in the second lab. We already mentioned that,for the sake of simplicity, we do not detail the compo-sition of F’s lab and consequently, nothing happens tothe photon state at this stage (the polarization state as-sumes perfect correlation between the memory and thequbit states).We have now implemented the effect of all the elementsin the red box in Fig. 1, and | Ψ (cid:105) describes the pho-ton exiting the friends’ lab. At this point, Wigner could PBS
Friends' measurement setup
PBS
FIG. 2. Measurement setup allowing Wigner to test the cor-relations in state | Ψ (cid:105) right after the interaction with thefriends. use photon-counters to measure whether the photon tookarm r or l . Furthermore, he could use a PBS placed be-fore the photodetectors to test the correlations betweenthe polarization and the path, as shown in Fig. 2. Eq. (6)implies that Wigner would find perfect correlations be-tween the photon taking arm r and having polarization V , and between the photon taking arm l and having po-larization D , which can be summarized in the properties. Property 1.
The probability for the photon to travelthrough arm r (and therefore having shape ) and havingpolarization H is zero, and Property 2.
The probability for the photon to travelthrough arm l (and therefore having shape ) and hav-ing polarization A is zero.
4. Wigner’s “super-measurements”
The end of the protocol corresponds to Wigner’s mea-surements. Rather than doing the measurement de-scribed above, made to check properties 1 and 2, Wignerwants to measure the whole labs in bases different fromthose in which the friends did their own measurements.The first qubit was measured in the l, r basis, and theoutcome was copied in the shape 1,2. Wigner decides tomeasure the joint qubit-lab state in a basis containingthe states {| ok (cid:105) , | fail (cid:105)} , with | fail (cid:105) = 1 √ | r (cid:105) path ⊗ | (cid:105) shape + | l (cid:105) path ⊗ | (cid:105) shape ) , (7a) | ok (cid:105) = 1 √ | r (cid:105) path ⊗ | (cid:105) shape − | l (cid:105) path ⊗ | (cid:105) shape ) . (7b)The second lab-qubit state is encoded in the polar-ization that was measured in the basis {| H (cid:105) pol , | V (cid:105) pol } .We assume Wigner chooses to measure in the basis {| D (cid:105) pol , | A (cid:105) pol } . In our setup, these two measurements Present notations Frauchiger-Renner articleFirst qubit Path of the photon Quantum coin RMeasurement basis of F {| r (cid:105) path , | l (cid:105) path } {| tails (cid:105) R , | heads (cid:105) R } Lab states of F {| l, (cid:105) , | r, (cid:105) , ... } {| t (cid:105) L , | h (cid:105) L , ... } Basis of Wigner’s measurement on F {| ok (cid:105) , | fail (cid:105)} where | fail (cid:105) = √ (cid:16) | r, (cid:105) + | l, (cid:105) (cid:17) {| ok (cid:105) L , | fail (cid:105) L } where | fail (cid:105) L = √ (cid:16) | t (cid:105) L + | h (cid:105) L (cid:17) Second qubit Polarization Spin 1 / {| H (cid:105) pol , | V (cid:105) pol } {| ↑(cid:105) S , | ↓(cid:105) S } Lab states of F {| H (cid:105) pol , | V (cid:105) pol } {| (cid:105) L , | − (cid:105) L } Basis of Wigner’s measurement on F {| D (cid:105) pol , | A (cid:105) pol } where | D (cid:105) pol = √ (cid:16) | H (cid:105) pol + | V (cid:105) pol (cid:17) {| ok (cid:105) L , | fail (cid:105) L } TABLE I. Correspondance of notations between our setup and Ref. [3]. can be performed simultaneously using the elementsgathered in the blue box in Fig. 1, namely, a mode-shaperturning the photon wavepacket in arm r from state s = 1to s = 2, a balanced beam-splitter (which acts only onthe path degree of freedom), two polarized beam-splitters(which transmit diagonally-polarized photons and reflectantidiagonally-polarized ones), and four photon counters. The net effect of all these elements is that a click atone of the four detectors corresponds to one of the fourpossible outcomes (D,ok), (A,ok), (D,fail) and (A,fail) ofWigner’s measurement and projects the photon onto thecorresponding state.It is useful to express the photon state in the { ok , fail } basis: | Ψ (cid:105) = 1 √ (cid:16) r (cid:122) (cid:125)(cid:124) (cid:123) | V, fail (cid:105) + | V, ok (cid:105) + l (cid:122) (cid:125)(cid:124) (cid:123) | V, fail (cid:105) − | V, ok (cid:105) + | H, fail (cid:105) − | H, ok (cid:105) (cid:17) (8a)= 1 √ (cid:16) | D, fail (cid:105) − | D, ok (cid:105) − | A, fail (cid:105) − | A, ok (cid:105) (cid:17) . (8b)From the state above, we can compute the probabilityof all four outcomes. In particular, the probability offinding (A,ok) is |(cid:104) A, ok | Ψ (cid:105)| = 1 / r (resp. l ) for later discussion. This allows us to seethat the interference between the two arms is responsiblefor the cancellation of the terms proportional to | V, ok (cid:105) .It is important for later to record the following property: Property 3.
Due to the interference between the twophoton paths, the amplitude of the photon reaching oneof the ports labeled by “ok” and having polarization V is zero.
C. The paradox
In Ref. [3], FR use a set of assumptions to point outthree properties of the measurement outcome statistics,that they claim paradoxical. Their intention is to showthat no physical theory can satisfy these three assump-tions simultaneously. In our words and notations, theassumptions are: • (U): The measurements performed by the twofriends of Wigner can be described by Wigner asa unitary (entangling) evolution of the qubits andthe friend’s labs . This is not written explicitly inRef. [3], but is used in their analysis, as pointed outby some comments [4, 5]. It corresponds to describ-ing the friends’ measurements as interaction withmemories rather than using the projection postu-late, as we did above. • (Q): If a quantum system is in a state orthogonalto one of the eigenstates of an observable, then ameasurement of this observable has zero probabilityto yield the corresponding outcome . This assump-tion is formulated differently in Ref. [3], but is usedwith this meaning in their analysis, as pointed outby some comments [4]. This assumption is includedin Born’s rule to compute measurement statistics. • (S): For any measurement performed by a given ob-server, a single definite outcome is obtained. • (C): There is consistency between expectations formeasurement outcomes predicted based on the out-comes of different observers/entities, even whenone observer is actually able to measure anotherone. This can be stated simply as: Different ob-servers should not find different results for obser-vations on the same system.
From these assumptions, FR deduce in Ref. [3] thattheir setup should verify Properties 1, 2 and 3 at thesame time, and that the measurement statistics shouldbe captured by state | Ψ (cid:105) . However, these three propertiescombined seem to rule out the possibility to obtain theoutcome ( A, ok ), according to the following reasoning:1. Property 2 forbids the outcome A to be obtainedwhen the first qubit is found in state | l (cid:105) path , whichmeans that ( A, ok) is only compatible with thefirst qubit being found in | r (cid:105) path .2. Then Property 1 implies that when the firstqubit is found in | r (cid:105) path , the photon must be inpolarization state V .3. Finally, Property 3 can be used to state that if thephoton is in state V , it cannot be in state | ok (cid:105) , suchthat finally ( A, ok) should be forbidden.This conclusion is paradoxical because the statistics ofoutcomes computed from state | Ψ (cid:105) predicts a non-zeroprobability of obtaining the outcome ( A, ok). D. Insights from the interferometric setup
Discussions of this paradox have often involvedquestioning the assumptions made in Ref. [3], see e.g. Refs. [4–7]. Several arguments have been used to showthat quantum mechanical setups do not verify all ofthem. For instance, it is stated in Ref. [6] that theviolation of a special variation of Bell inequalities, whoseexperimental verification is reported in Ref. [25], rulesout assumption (C).Here we take another approach by studying how theparadox would arise in a realistic setup, involving systemswhose dynamics are known to obey to quantum mechan-ics. As an advantage, we therefore do not have to makeassumptions (U), (Q) (S), (C) to predict the outcomesof measurements made on the system. As in the case ofFR, we find that the statistics of Wigner’s measurementoutcomes are given by state | Ψ (cid:105) . At the same time, weexpressed the three paradoxical properties 1, 2 and 3 asa function of the interferometer degrees of freedom.The latter connection allows us to stress a crucial pointconcerning properties 1 to 3, which is that they belongto two different contexts , i.e. two different incompatiblechoices of experimental setups. Indeed, Properties 1, 2refer to the which-path information { r, l } , while Property3 refers to a different basis involving coherent superposi-tions of the path states {| l (cid:105) , | r (cid:105)} . In the absence of thesecond beam-splitter closing the interferometer, the pathtaken by the photon can be measured (see Fig. 2): a clickat one of the photon detectors causes the photon to takea definite path | l (cid:105) or | r (cid:105) , and the validity of Properties1, 2 can be checked. Conversely, when the beam-splitteris present, it ensures that the which-path information re-mains unavailable, i.e. does not take any definite value after a click at any of the detectors. As known sinceYoung’s double slit experiments, this unvailability of thewhich-path information is necessary for the interferencebetween the two paths to take place, which in turn isneeded for Property 3 to hold (see Section II B 4). Inthis context, the validity of properties 1 and 2 cannot bechecked. E. Varying the context
We can expand this discussion by considering varia-tions on the setting just described, in a way that ensuresthe validity of properties 1 and 2. We have just describeda “super-measurement” by Wigner, which affects all de-grees of freedom, including the memories. We refer tothis situation as Context 1. We may instead consider acontrasting scenario in which Wigner performs a moreregular measurement, leaving the some degrees of free-dom untouched. Without loss of generality, we identifythese degrees of freedom with the photon wavefunctionshape s . This new situation therefore corresponds to amodified measurement setup which is depicted in Fig. 1 c and we call Context 2. The net effect of the setup is a si-multaneous measurement of the polarization in the D, A basis, and of the which-path basis: | fail (cid:48) (cid:105) path = 1 √ | r (cid:105) path + | l (cid:105) path ) , (9a) | ok (cid:48) (cid:105) path = 1 √ | r (cid:105) path − | l (cid:105) path ) , (9b) Note that this new measurement does not act on theshape space which plays the role of the memory of thewhich-path measurement. In the new measurement ba-sis, the state | Ψ (cid:105) reads: | Ψ (cid:48) (cid:105) = 1 √ (cid:16) r (cid:122) (cid:125)(cid:124) (cid:123) | V, fail (cid:48) , (cid:105) + | V, ok (cid:48) , (cid:105) + l (cid:122) (cid:125)(cid:124) (cid:123) | V, fail (cid:48) , (cid:105) − | V, ok (cid:48) , (cid:105) + | H, fail (cid:48) , (cid:105) − | H, ok (cid:48) , (cid:105) (cid:17) (10a)= 1 √ (cid:16) | D, fail (cid:48) (cid:105) ⊗ ( | (cid:105) shape + 2 | (cid:105) shape ) − | D, ok (cid:48) (cid:105) ⊗ ( | (cid:105) shape − | (cid:105) shape ) −| A, fail (cid:48) (cid:105) ⊗ | (cid:105) shape − | A, ok (cid:48) (cid:105) ⊗ ( | (cid:105) shape − | (cid:105) shape ) (cid:17) . (10b)When the setting of Fig. 1 c is used, the information aboutthe path taken by the photon is preserved in the shapedegree of freedom. It is therefore possible to check thevalidity of Properties 1, 2. On the other hand, we cansee that the interference between the two paths does notoccur anymore . In particular, the probability of outcome | V, ok (cid:48) (cid:105) does not vanish (cid:107) √ ( | V, ok (cid:48) , (cid:105) − | V, ok (cid:48) , (cid:105) ) (cid:107) = 1 / , which means that Property 3 is automatically violated.Therefore, neither of these two contexts allow all of prop-erties 1 through 3 to hold simultaneously. F. Paradoxical properties as incompatibleobservables
One may consider whether another measurementsetup, that does allow all three properties to remain si-multaneously valid, exists. The answer is no, and this canbe proven by noting that Properties 1 to 3 correspond toassertions on the values taken by a set of non-commuting observables, which are consequently forbidden by quan-tum mechanics to simultaneously all accept a well-definedvalue. To prove this, we introduce the following photonobservables: O = | H, r, (cid:105)(cid:104) H, r, |O = | A, l, (cid:105)(cid:104) A, l, |O = | V, ok (cid:105)(cid:104) V, ok | . (11)These three-qubit observables involve all three degrees offreedom of the photon. They are projectors admittingeigenvalues 0 and 1. It is easy to check that:[ O , O ] = [ O , O ] = 0 , [ O , O ] = ( | A, l, (cid:105)(cid:104) V, ok | + | V, ok (cid:105)(cid:104) A, l, | ) (cid:54) = 0 . (12)Meanwhile, saying properties 1, 2 and 3 hold is equiva-lent to say that, respectively, O , O and O takes the value 0. The fact that O and O do not commute thenrules out the possibility for these three properties to holdsimultaneously in any measurement setup. G. The interferometer and the assumptions ofFrauchiger and Renner
Eventually, it is enlightening to look at how the presentsetup fits within the assumptions of FR. We first notethat for this setup Assumption (Q) is fulfilled as the sys-tem under investigation is clearly a quantum system. Wealso presented the rationale for using Assumption (U) inII A, describing the measurement process as a unitaryinteraction between a system and memory and applyingthis idea to the cases of interest. Interestingly, the fate ofthe two other assumptions depends on whether Wigneris chosen to be a “super-observer” (Context 1 leading tostate | Ψ (cid:105) ) or a regular observer (Context 2 leading tostate | Ψ (cid:48) (cid:105) ), i.e. by inserting, or not, the second mode-shaper in arm r .If Wigner has full control over his friends’ labs (sec-ond mode-shaper present), his measurement statistics arecaptured by state | Ψ (cid:105) . However, one can see explicitlythat assumption (S) is violated. Indeed, the memory ofF’s lab is erased during Wigner’s measurement process,and the branch of the wavefunction | Ψ (cid:105) correspondingto F finding path l interferes with that correspondingto finding path r . The situation is reminiscent of the“quantum eraser” where a measurement is used to erasethe photon which-path information after it was recorded,thereby restoring interference [26]. As said above, thefact that the which-path information is made unavailableis necessary to the validity of property 3. Consequently,the first friend does not have a definite measurement out-come (which would be either r or l ), which contradictsassumption (S). Note that the validity of (C) is hard toanalyze in this case, as pretending we could take the pointof view of the friend, who is in a superposition state, ishazardous. The consequence is that properties 1 and 2do not hold anymore, such that the paradox does notarise.Conversely, if we assume that there exists some pre-served record of the which-path information (secondmode-shaper absent), assumptions (S) and (C) are veri-fied. However, the state | Ψ (cid:48) (cid:105) is now the proper descrip-tion for the outcome statistics, and Property 3 does nothold anymore. Once again, the paradox does not arise.In summary, the paradox appears when comparingproperties associated to two different contexts, i.e. twodifferent experimental setups shown in Fig. 1. Context1 is shown in Fig. 1 b when the mode shaper 1 → l . In this context the quantum stateis described by | Ψ (cid:105) . Context 2 is shown in Fig. 1 c whereno mode shaper is inserted into path l . In this contextthe quantum state is described by | Ψ (cid:48) (cid:105) .Finally, our results can be connected to a recently in-troduced No-go theorem Ref. [15], motivated by extendedWigner’s friend scenarios. The theorem expresses the in-compatibility of the possibility to have a super-observerwith three assumptions: (i) what they call Absolutenessof Observed Events (AOE), which encompasses assump-tions (S) and (C) and is violated in Context 1; (i) free-dom of choice for observers’ actions and (ii) locality inthe sense that the measurements performed on one sys-tem do not affect the statistics of future measurementsperformed on another system. Our setup illustrates theincompatibility of the assumption AOE with the super-observer assumption, when the two other assumptionsare verified.
III. DISCUSSION
We have reformulated an extended Wigner’s friend sce-nario introduced in Ref. [3] as an interferometric setup,involving three different degrees of freedom of a singlephoton to play the roles of two qubits and the agentsmeasuring them. By analyzing this setup, we have shownthat the three properties highlighted to be paradoxicalcorrespond to two different contexts, in which the which-path information takes well defined values or not. Thesetwo contexts correspond to two different measurementsetups, which access the values of different sets of ob-servables. The observables involved do not commute witheach other and are then forbidden by quantum mechanicsto all simultaneously take well-defined values. As a con-sequence, the paradox never arises in any physical setupobeying quantum mechanics.The fact that the three properties considered to formu-late the paradox cannot hold simultaneously was alreadyargued in Ref. [7]. Our setup illustrates the transitionbetween the validity regime of Properties 1 and 2 andthat of Property 3. In doing so, we can specifically iden-tify the transition between the coherent (quantum) andincoherent (observer) behaviors of the friends. This tran-sition is dictated by the presence, or not, of an untouchedmemory degree of freedom. Our work is relevant however to more than just theFR thought experiment. Our results speak to unresolvedissues in the Wigner’s Friend paradox by identifying thestate specific role a “memory” must play for an observer(rather than a superobserver) in the paradox. Thus, thetransition between the two experimental contexts (cap-tured by states | Ψ (cid:105) and | Ψ (cid:48) (cid:105) respectively) can be inter-preted as an assumption about the location of Heisen-berg’s cut [2]. More precisely, it allows us to under-stand which entities/apparatuses must be considered asobservers—the ‘ultimate measuring instruments’ of Bohr[27]—or included in the system described by quantum-mechanics [10, 28]. Indeed, for the friends to be con-sidered as observers, their memory must, by definition,be preserved leading to state | Ψ (cid:48) (cid:105) , while | Ψ (cid:105) requires theability to alter their memory.Interpretations of quantum mechanics treating ob-servers and quantum systems on different footing are of-ten criticized because of the apparent flexibility in theposition where such cut should be placed. However, oursetup, and the notion of memory state, can be used toargue that the place of this cut is actually imposed by thepractical (and objective) resolution of the experimental-ist’s apparatuses. Degrees of freedom (objectively) out ofthe control of the experimentalist’s apparatuses can playthe role of memories which should always be consideredto be beyond the cut to accurately predict measurementstatistics.Because different experimentalists with different appa-ratuses may be able to control different degrees of free-dom, this approach can also be naturally related to a“QBist” interpretation [29–31] where the quantum stateis ascribed by a given observer to a quantum system andmay therefore be observer-dependent. A second point ofcontact with QBist approaches come with the emphasison memory and its quantum accessibility. QBism holdsthat quantum states are epistemological rather than on-tological. In particular, given the fundamental nature ofprobabilities in quantum states, QBism stresses that theyshould be seen as bets, conditioned by priors, placed onthe results of an experiment. The priors are then updatedonce the experimental results are obtained. Priors are,essentially, memories. They are information about pre-vious states of the world held by an observer and used tocalculate quantum states. Thus by showing how mem-ory, as a manipulable quantum state, must function inWigner’s Friend argument, our setup may help articu-late the ways in which quantum states in general shouldbe viewed for macroscopic observers such as ourselves.Further comments regarding a QBist treatment of theproblem can be found in Ref. [13, 32]. IV. METHODSEquivalence of measurement descriptions
The equivalence of the entangled state Eq. (1) and thecollapsed state Eq. (2) for subsequent measurements canbe showed by computing the probability to obtain aneigenstate | a (cid:105) S of an arbitrary observable ˆ A S of qubit S : SM (cid:104) Ψ | a (cid:105) S (cid:104) a | Ψ (cid:105) SM = | c | | S (cid:104) a | (cid:105) S | + | c | | S (cid:104) a | (cid:105) S | = S (cid:104) a | ρ S | a (cid:105) S . (13) This implies that the statistics of subsequent measure-ment on the qubit S is captured identically by bothstates. V. ACKNOWLEDGEMENTS
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