Quantum Erasing the Memory of Wigner's Friend

Quantum erasing the memory of Wigner’s friend

Cyril Elouard1,2, Philippe Lewalle1, Sreenath K. Manikandan1, Spencer Rogers1, Adam Frank1, and Andrew N. Jordan1,3

1Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA 2QUANTIC lab, INRIA, 2 Rue Simone IFF, 75012 Paris, France 3Institute for Quantum Studies, Chapman University, Orange, CA 92866, USA subdate

The Wigner’s friend paradox concerns 1 Introduction one of the most puzzling concepts of quantum mechanics: the consistent description In his famous gedankenexperiment, Wigner ana- of multiple nested observers. Recently, lyzes a setup in which a “super-observer” is as- a variation of Wigner’s gedankenexperi- sumed to be able to measure a whole lab, con- ment, introduced by Frauchiger and Ren- taining a human being (a friend of his), in any ner, has lead to new debates about the self- basis [1]. Paradoxical conclusions may emerge consistency of quantum mechanics. At the from such situations due to the tension between core of the paradox lies the description of the rules of evolution for isolated quantum sys- an observer and the object it measures as tems, which in principle can be applied at any a closed system obeying the Schrödinger scale, and the need for the projection postulate to equation. We revisit this assumption to describe measurements performed by observers. derive a necessary condition on a quan- While in most practical situations, it is clear tum system to behave as an observer. We whether or not an entity should be considered as then propose a simple single-photon inter- a quantum system or an observer – and therefore ferometric setup implementing Frauchiger whether its interaction with the system should be and Renner’s scenario, and use the de- described with a unitary evolution (Schrödinger rived condition to shed a new light on equation) or via the projection postulate – the the assumptions leading to their paradox. transition between these two behaviors continues From our description, we argue that the to cause much debate. In particular it remains three apparently incompatible properties unclear where such a “cut” [2] exists between sys- used to question the consistency of quan- tems that can be in quantum superposition and tum mechanics correspond to two logically those which cannot either in principle or in prac- distinct contexts: either one assumes that tice. Wigner has full control over his friends’ In Ref. [3], Frauchiger and Renner (FR) lab, or conversely that some parts of the present an extended Wigner friend scenario in- labs remain unaffected by Wigner’s sub- volving two observers, the friends of Wigner, and sequent measurements. The first context two “super-observers”, W and W. The latter are may be seen as the quantum erasure of the assumed to be able to measure their friends and arXiv:2009.09905v4 [quant-ph] 29 Jun 2021 memory of Wigner’s friend. We further their labs in arbitrary bases of states. FR’s key show these properties are associated with result is to formulate apparently natural assump- observables which do not commute, and tions about this setup which they argue lead to therefore cannot take well-defined values an inconsistency. This study has triggered a large simultaneously. Consequently, the three number of comments and articles re-examining contradictory properties never hold simul- the scenario. These new papers have identified taneously. hidden assumptions [4,5], gathered different arguments questioning FR’s surprising conclusion [6,7,8,9], and generated new discussions of Cyril Elouard: [email protected] the quantum formalism and its interpretations

Accepted in Quantum 2021-06-24, click title to verify. Published under CC-BY 4.0. 1 [10, 11, 12, 13, 15, 14]. Below, we start from the assumption (made by Wigner and FR) that an observer can be described as a quantum system that becomes entangled with the system is measures. We then derive a necessary condition for such quantum system to behave as an observer, from the point of view of any other observer: the existence of a stable degree of freedom that we refer to as a memory, which becomes entangled with the measured system and retains a record of the measure- Figure 1: Extended Wigner’s friend scenario. Wigner’s ment outcome until the end of the experiment. friend F (whose lab is described by states 1, 2) mea- We then use this result to introduce a sim- sures a first qubit in basis {l, r}, and depending on the outcome prepares another qubit and sends it to ple reformulation of FR’s scenario, based on an the F’s lab. F measures the second qubit in basis optical interferometer, which allows us to an- {H,V }. Finally, Wigner measures both labs in the basis alyze the alleged paradox. Crucially, in our {(A, ok), (A, fail), (D, ok), (D, fail)}. setup, only small systems that can be unam- biguously treated quantum-mechanically are entangled. Although our reformulation arguably has been examined [17] to introduce an exper- marks a philosophical departure from FR’s dis- iment distinguishing between Everett’s and the cussion of a “friend” reaching an entangled state, Copenhagen formulations of quantum mechan- our model, in fact, leads to the same mathemati- ics, and recently to shed new light on the original cal description. We then revisit the assumptions Wigner’s friend scenario [18]. and properties used in Ref. [3] to formulate their paradox. We show that some of them require the memories to be erased by the super-observer, 2 Results preventing the corresponding agents from behaving as observers. Conversely, others of these 2.1 Observers and memory properties and assumptions require the memo- In most situations, it is easy to identify enti- ries to remain untouched, eliminating the main ties that behave as observers. The measurement feature of the super-observer. In the end, the an observer O performs on a quantum system inconsistency only appears if one compares prop- is then well described by the measurement pos- erties from two logically different contexts that tulate, which attributes a well-defined outcome we make explicit. We also demonstrate that the to the measurement and asserts that the mea- three properties involved in the paradox are asso- sured system is projected into the corresponding ciated with the values taken by observables which eigenstate of the measured observable. The re- do not commute, and therefore are forbidden by sult of the measurement resides in the observer’s quantum mechanics to take simultaneously well- memory. For specificity we can consider that O defined values. Thus, we find that the argument writes down the measurement results on a piece of Ref. [3] does not lead to any inconsistency of paper, which also serves as a memory that the within quantum mechanics. projection occurred. Another observer O0 only By focusing on the role of memory, projection, has access to the result of the measurement ei- and unitary evolution, we make explicit limits in ther by asking O, looking at the piece of paper, the Wigner’s Friend paradox and the mechanism or immediately repeating O0s measurement. of quantum measurement, and provide an opera- Wigner’s friend and FR’s paradoxes arise when tional prescription to identify devices behaving as assuming the existence of a super-observer. In an observer. This discussion is particularly useful contrast with a regular observer, a super-observer in light of the recent related No-Go theorem of has full access to the whole quantum system com- [16] that sought to illuminate key issues in quan- prising the memory of another observer and the tum interpretation. We also note that the role measured system. This larger quantum system of the material record of an observer’s outcome is potentially a highly interacting collection of

Accepted in Quantum 2021-06-24, click title to verify. Published under CC-BY 4.0. 2 > 1026 atoms. Thus the meaning of “full access” tonian between S and M switched on only dur- is that super-observers can manipulate all > 1026 ing a finite time. As long as the system SM is atoms in their full entangled, superposed states in state |ΨiSM , the memory qubit serves as a re- to measure them in an arbitrary way. In par- trievable record that the system was found either ticular, such measurement can create quantum in state |0iS or |1iS. Such entangled state Eq. (1) superposition of macroscopically different states indeed captures the perfect correlation between of the observer. As pointed out by Schrödinger the memory and system’s state |0iS and |0iM on and his famous cat gedankenexperiment [19], we one hand, |1iS and |1iM on the other hand. never see macroscopic objects like a piece of pa- To derive a necessary condition for such a uni- per with writing on it in a quantum superposi- tary description of the memory and system evolution. Beyond the difficulty related to the size and tion to faithfully model a quantum measurement, complexity of such a large system itself, a pure we now consider the point of view of a second in- superposition state is posited on the system being dependent observer O0 measuring the qubit after completely closed off from any “environment”, O. Contrary to Wigner, O0 is a regular observer, which becomes increasingly difficult to achieve, who does not have access to the whole state of S in practice, as the system size grows. Apart and the memory M of O. Therefore, the statis- from this practical limitation however, there is tics of the measurement performed by O0 should no evidence that closed systems should not be de- not depend on whether we described the mea- scribed by a unitary evolution (i.e. Schrödinger’s surement by O with the measurement postulate equation) beyond a given size or complexity. This or with a unitary evolution, for the latter to be fact may seem in contradiction with the non- valid. One can indeed explicitly check that any unitary projection predicted by the measurement measurement on qubit S yields the same statis- postulate, and Wigner’s friend gedankenexper- tics (see Appendix A) when computed from state iment (and extensions) precisely intend to in- |ΨiSM and from the state resulting of the appli- vestigate this tension. To do so, one generally cation of the measurement postulate (assuming assumes that (i) Wigner’s technological means O0 does not know the result of the measurement grants him such degree of control on another ob- by O): server O which has performed a measurement of a quantum system S, and (ii) O and S consti- ρ = |c |2|0i h0| + |c |2|1i h1|. (2) tute a closed quantum system whose evolution is S 0 S 1 S ruled by Schrödinger’sequation. In this context, the measurement postulate is therefore assumed On the other hand, it may be enough to alter to be an effective evolution for the measured sys- the state of S and M before the measurement tem, arising because one does not have access to of O0, to cause the statistics of some measure- the full state of O and S. ments performed on qubit S to differ from that As Wigner’s technology allows him to perform obtained from Eq. (2) (see Appendix A for a di- arbitrary measurements on the memory of O, its rect proof). A striking situation corresponds to complexity does not matter, and we can think the case where one is able to totally erase the of some properties of the measurement process memory M, i.e. apply the inverse of the mea- with the following toy model: Let the system S surement unitary to prepare back the initial state being measured be a qubit admitting the basis |ψiS|0iM . In this case, there is no evidence in any of states {|0iS, |1iS} and initially described by future operation done on the qubit that the mea- state |ψiS = c0|0iS + c1|1iS, and the memory surement has ever been made. In other words, of O be another qubit M admitting the basis of the unitary description of the measurement re- states {|0iM , |1iM }. We assume that O measures quires the memory to be stable (its joint state the qubit in the basis {|0iS, |1iS}. The unitary with the system must be unchanged) until the evolution of the memory and the system prepares end of the experiment. One can make this condi- an entangled state: tion more precise by considering that the memory is composed on many degrees of freedom (as |Ψi = c |0i ⊗ |0i + c |1i ⊗ |1i . (1) SM 0 S M 1 S M the piece of paper certainly is) in which the mea- Such evolution can e.g. be generated by the surement outcome is redundantly copied multiple Schrödingerequation assuming a coupling Hamil- times [20]. The memory-system state at the end

Accepted in Quantum 2021-06-24, click title to verify. Published under CC-BY 4.0. 3 of the interaction would then rather look like: 2.2 The Extended Wigner’s friend scenario as an interferometer

|ΨiSM1....MN = c0|0iS ⊗ |0iM1 ... ⊗ |0iMN 2.2.1 Interferometric setup +c1|1iS ⊗ |1iM ... ⊗ |1iMN . (3) We now consider the following reformulation of In principle, it is then sufficient that one single the situation considered by FR in Ref. [3]. For degree of freedom of the memory remains entan- a more detailed exact mapping of notations, see gled with the measured qubit for the statistics of Table1. The scenario involves two labs, each any later measurement on S to be the same as containing a qubit and an observer (referred to computed from state ρ . It is then clear that for S as a “friend” of Wigner), and two super-observers a macroscopic memory, composed of many de- able to measure the two labs, W and W. For the grees of freedom able to become entangled with sake of simplicity, we merge W and W into a the system, it is practically impossible to per- single observer external to the labs, that we call fectly inverse the unitary leading to Eq. (3), and Wigner. evidence of the measurement will always remain in future measurement statistics. These observa- The mere possibility of Wigner’s full control tions allow us to formulate the following neces- over the labs prevents us from using the mea- sary condition: surement postulate directly to describe the measurements done by his friends. We rather use Condition 1. For an entity O to behave as an the description presented in Section 2.1 based on observer, there must exist at least one degree of a unitary evolution of quantum systems playing freedom in which O can encode the results of mea- the role of the friends’ memories. On the other surement, and which is untouched for the dura- hand, as Wigner’s measurement marks the end of tion of the considered experiment. the experiment, and it is assumed that nobody else will measure Wigner and his environment, This condition can also be obtained by con- we can describe Wigner’s measurements via the sidering that a defining property of a measure- usual projection postulate. ment is to generate an outcome. The Condition In our formulation, the role of the two friends’ 1 ensures that the operation performed by O re- memories, and the qubits they measure, are all sults in registering an outcome that will survive played by different degrees of freedom of a single subsequent stages of the experiment. In particu- photon traveling through a Mach-Zender inter- lar, to be used in Wigner’s friend type scenarios, ferometer (see Fig.1). These degrees of freedom this condition applies to the friends solely if their can all, for our purpose, be modeled by qubits memory remain stable even when Wigner makes (three in total). First, the path taken by the his own measurements. photon (the arm of the interferometer it trav- As a final remark for this section, we emphasize els through) plays the role of one of the two that entangled states as Eq. (1) or Eq. (3) do not qubits. This qubit initially belongs to friend F explain how a single outcome is obtained from a who is able to measure it. The memory of F is readout of the memory. These states merely pro- represented by the spatiotemporal shape of the vides the probabilistic correlations that can be photon wavepacket (or equivalently the mode of observed if such readout is made. The transition the waveguide the photon is in). Finally, the to a single definite outcome can only be described polarization of the photon plays the role of the using the projection postulate to model the read- other qubit and the memory of the other friend out of the memory, as done for instance in von F of Wigner, all together. While one could think Neumann’s seminal model of quantum measure- of a more sophisticated version of the setup al- ment [21]. This last projection can however be lowing us to distinguish these roles, it will be postponed to anytime after the interaction be- enough for our purpose to work with these three tween the system and the memory without affect- qubits. We can specify the photon state using the statistics of measurements performed by ing the orthogonal basis {|α, n, si}. The states other observers. Meanwhile, the effect of all op- |α, n, si ≡ |αipol ⊗ |nipath ⊗ |sishape are labeled erations involving the memory (as in the scenario by the polarization α = H,V , the path taken by investigated below) can be taken into account. the photon n = l, r and the two possible orthog-

Accepted in Quantum 2021-06-24, click title to verify. Published under CC-BY 4.0. 4 a Context 1 Wigner's measurement setup b Context 2 Wigner's measurement setup

Friends' measurement setup PBS Friends' measurement setup PBS

PBS PBS

50% 50%

33% 33%

Figure 2: a: Proposed inteferometric setup to implement the extended Wigner’s friend scenario, when Wigner is indeed a super-observer (context 1). The photon is initially prepared in polarization state |Dipol and a wavepacket of shape s = 2. The red box corresponds to the two friends’ measurements. The first beam-splitter is assumed to have 1/3 transmission probability, such that after the first beam-splitter of√ the interferometer (which does not to affect the wave-form), it is described by the state |Ψ i = √1 (|D, r, 2i + 2|D, l, 2i). F measures the which-path 0 3 information. In our setup, this corresponds to a correlation between the mode shape of the photon and the path, that can be generated e.g. if an optical element causing the transition from state√ |2ishape to the orthogonal state |1i , is inserted in the arm r, yielding the photon state |Ψ i = √1 (|D, r, 1i + 2|D, l, 2i). Then, in the scenario shape 1 3 of Ref [3], F prepares the state of the other qubit, encoded in the polarization, depending on its outcome. This step does not really requires F to read out the memory, and can be rather done by correlating directly the memory (the shape of the photon) with the polarization. This can be achieved assuming the presence of a chiral crystal (or waveplate) in path r rotating the polarization of the photon by 45◦, transforming the diagonal polarization state into the vertical polarization. This yields state |Ψ2i expressed in Eq. (4). Finally, Wigner performs a measurement in the basis {(D, ok), (A, ok), (D, fail), (A, fail)}. In our setup, this can be achieved using the elements gathered in the blue box, namely, a mode-shaper turning the photon wavepacket in arm r from state s = 1 to s = 2, a balanced beam-splitter (which acts only on the path degree of freedom), two polarized beam-splitters (PBS) which transmit diagonally-polarized photons and reflect antidiagonally-polarized ones and four photon counters. b: Setup corresponding to context 2, where Wigner is not a super-observer. The difference is the absence of the second mode-shaper inside the blue box.

Accepted in Quantum 2021-06-24, click title to verify. Published under CC-BY 4.0. 5 Wigner would ﬁnd perfect correlations between the photon taking arm r and having polarization Friends' measurement setup V , and between the photon taking arm l and hav-

PBS ing polarization D, which can be summarized in the properties.

PBS Property 1. The probability for the photon to 33% travel through arm r (and therefore having shape 1) and having polarization H is zero,

and Figure 3: Measurement setup allowing Wigner to test Property 2. The probability for the photon to the correlations in state |Ψ2i right after the interaction with the friends. travel through arm l (and therefore having shape 2) and having polarization A is zero. onal shapes of its wavepacket s = 1, 2. 2.2.2 Wigner’s “super-measurements” The unitary evolutions associated with the two friends’ measurements can be implemented owing The end of the protocol corresponds to Wigner’s to the optical elements in the red box of Fig.2 a, measurements. Rather than doing the measure- preparing state: ment described above, made to check properties 1 and 2, Wigner wants to probe bases diﬀerent 1 √ |Ψ2i = √ |V, r, 1i + 2|D, l, 2i . (4) from those in which the friends did their own 3 measurements. The ﬁrst qubit was measured in We have introduced the diagonal D and antidi- the l, r basis, and the outcome was copied in the agonal A polarization states shape 1,2. Wigner decides to measure the joint qubit-lab state in a basis containing the states |Di = √1 |Hi + |V i , (5a) pol 2 pol pol {|oki, |faili}, with 1 |Aipol = √ |Hipol − |V ipol . (5b) 1 2 |faili = √ (|ripath ⊗ |1ishape + |lipath ⊗ |2ishape), (6a) 2 |Ψ2i describes the photon exiting the friends’ lab. 1 |oki = √ (|ripath ⊗ |1ishape − |lipath ⊗ |2ishape). (6b) At this point, Wigner could use photon-counters 2 to measure whether the photon took arm r or l. Furthermore, he could use a polarizing beam- The second memory-qubit state is encoded in splitter placed before the photodetectors to test the polarization that Wigner chooses to measure the correlations between the polarization and the in the basis {|Dipol, |Aipol}. It is useful to express path, as shown in Fig.3. Eq. (4) implies that the photon state in the {ok, fail} basis:

r l 1 z }| { z }| { |Ψi = √ |V, faili + |V, oki + |V, faili − |V, oki + |H, faili − |H, oki (7a) 6 1 = √ 3|D, faili − |D, oki − |A, faili − |A, oki . (7b) 12

From the state above, we can compute the prob- red) the terms coming from the photon traveling ability of all four outcomes. In particular, the through arm r (resp. l) for later discussion. We probability of ﬁnding (A,ok) is |hA, ok|Ψi|2 = see that the interference between the two arms 1/12. is responsible for the cancellation of the terms proportional to |V, oki. It is important for later In Eq. (7a), we have indicated in blue (resp.

Accepted in Quantum 2021-06-24, click title to verify. Published under CC-BY 4.0. 6 Present notations Frauchiger-Renner article

First qubit Path of the photon Quantum coin R

Measurement basis of F {|ripath, |lipath} {|tailsiR, |headsiR}

Lab states of F {|l, 1i, |r, 2i, ...} {|tiL, |hiL, ...}

Basis of Wigner’s measurement on F {|oki, |faili} where {|okiL, |failiL} where |faili = √1 |r, 1i + |l, 2i |faili = √1 |ti + |hi 2 L 2 L L

Second qubit Polarization Spin 1/2 S

Measurement basis of F {|Hipol, |V ipol} {| ↑iS, | ↓iS}

1 1 Lab states of F {|Hipol, |V ipol} {| 2 iL, | − 2 iL}

Basis of Wigner’s measurement on F {|Dipol, |Aipol} where {|okiL, |failiL} |Di = √1 |Hi + |V i pol 2 pol pol

Table 1: Correspondance of notations between our setup and Ref. [3].

Accepted in Quantum 2021-06-24, click title to verify. Published under CC-BY 4.0. 7 to record the following property: even when one observer is actually able to measure another one (or another’s one Property 3. Due to the interference between memory). This can be stated simply as: Dif- the two photon paths, the amplitude of the ferent observers should not find different re- photon reaching one of the ports labeled by “ok” sults for observations on the same system. and having polarization V is zero. Note that FR’s argument solely requires consistency between outcomes that are certain (to which Born’s rule assign a unit probabil- 2.3 The paradox ity). In Ref. [3], FR use a set of assumptions to point out three properties of the measurement outcome From these assumptions, FR deduce in Ref. [3] statistics, that they claim are paradoxical. Their that their setup should verify Properties1,2 and intention is to show that no physical theory can 3 at the same time, and that the measurement satisfy these three assumptions simultaneously. statistics should be captured by state |Ψi. How- In our words and notations, the assumptions are: ever, these three properties combined seem to rule out the possibility to obtain the outcome • (U): The measurements performed by the (A, ok), according to the following reasoning: two friends of Wigner can be described by 1. Property2 forbids the outcome A to be Wigner as a unitary (entangling) evolution obtained when the first qubit is found in of the qubits and the friend’s labs. This is state |li , which means that (A, ok) is not written explicitly in Ref. [3], but is used path only compatible with the first qubit being in their analysis, as pointed out by some found in |ri . comments [5,4]. It corresponds to describ- path ing the friends’ measurements as interaction with memories rather than using the projec- 2. Then Property1 implies that when the first tion postulate, as we did above. This as- qubit is found in |ripath, the photon must sumption requires than the memory and the be in polarization state V . measured system form a closed system. It also implies than the memory can be manip- ulated after the interaction with the system, 3. Finally, Property3 can be used to state that and that in principle future evidence of the if the photon is in state V , it cannot be in measurement can be erased. state |oki, such that finally (A, ok) should be forbidden. • (Q): If a quantum system is in a state orthogonal to one of the eigenstates of an observ- This conclusion is paradoxical because the statis- able, then a measurement of this observable tics of outcomes computed from state |Ψi pre- has zero probability to yield the correspond- dicts a non-zero probability of obtaining the out- ing outcome. This assumption is formulated come (A, ok)1. differently in Ref. [3], but is used with this meaning in their analysis, as pointed out by 2.4 Insights from the interferometric setup some comments [4]. This assumption is included in Born’s rule to compute measure- Discussions of this paradox have often involved ment statistics. questioning the assumptions made in Ref. [3], as in e.g. Refs. [6,7,4,5]. Several arguments have • (S): For any measurement performed by a been used to show that quantum mechanical given observer, a single definite outcome is setups do not verify all of them. For instance, obtained. Note that this assumption requires it is stated in Ref. [6] that the violation of that at the end of the experiment, the mem- a special variation of Bell inequalities, whose ory is read out. experimental verification is reported in Ref. [22],

• (C): There is consistency between expecta- 1In FR’s article, one considers that the experiment is tions for measurement outcomes predicted repeated multiple time until outcome (A, ok) is obtained, based on the outcomes of diﬀerent observers, which corresponds to an occurrence of the paradox.

Accepted in Quantum 2021-06-24, click title to verify. Published under CC-BY 4.0. 8 rules out assumption (C). tion 2.2.2), but the validity of properties1 and2 cannot be checked. Here we take another approach by studying Remembering that the shape corresponds to how the paradox would arise in a realistic setup, the memory in which F registers the outcome of involving systems whose dynamics are known to the which-path measurement, one can also give obey to quantum mechanics. As an advantage, the following interpretation: the mode-shaper re- we therefore do not have to make assumptions quired to implement Wigner’s measurement uni- (U), (Q) (S), (C) to predict the outcomes of mea- tarily erases the memory of F. As explained in surements made on the system. As in the case of Section 2.1, this means that Condition1 is vio- FR, we find that the statistics of Wigner’s mea- lated for F and therefore no trace that the which- surement outcomes are given by state |Ψi. Mean- path measurement has been made remains. while, we expressed the three paradoxical proper- ties1,2 and3 as a function of the interferometer’s 2.5 Varying the context degrees of freedom. We can expand this discussion by considering The latter connection allows us to stress a cru- variations on the setting just described, in a way cial point concerning properties1 to3, which is that ensures the validity of properties1 and2 that they belong to two different contexts, i.e. two (and that F behaves as an observer). Referring different incompatible choices of experimental se- to the “super-measurement” by Wigner we have tups. Indeed, Properties1,2 refer to the which- described in previous section as Context 1, we path information {r, l}, while Property3 refers instead consider a contrasting scenario in which to a different basis involving coherent superposi- Wigner performs a regular measurement, leav- tions of the path states {|li, |ri}. In the absence ing some degrees of freedom in the memory un- of the second beam-splitter closing the interfer- touched. Without loss of generality, we identify ometer, the path taken by the photon can be these degrees of freedom with the photon wave- measured (see Fig.3): a click at one of the pho- function shape s. This new situation corresponds ton detectors causes the photon to take a definite to a modified measurement setup (see Fig.1 b) path |li or |ri, and the validity of Properties1, that we call Context 2 whose net effect is a si- 2 can be checked. Conversely, when the beam- multaneous measurement of the polarization in splitter is present, it ensures that the which-path the D,A basis, and of the path in basis: information remains unavailable, i.e does not take 0 1 any definite value after a click at any of the detec- |fail ipath = √ (|ripath + |lipath), (8a) 2 tors. As known from single-photon interferome- 0 1 ter experiments, this unvailability of the which- |ok ipath = √ (|ripath − |lipath), (8b) 2 path information is necessary for the interference between the two paths to take place, which in In the new measurement basis, the state |Ψ2i turn is needed for Property3 to hold (see Sec- reads:

r l 1 z }| { z }| { |Ψ0i = √ |V, fail0, 1i + |V, ok0, 1i + |V, fail0, 2i − |V, ok0, 2i + |H, fail0, 2i − |H, ok0, 2i (9a) 6 1 0 0 = √ |D, fail i ⊗ (|1ishape + 2|2ishape) − |D, ok i ⊗ (|1ishape − 2|2ishape) 12 0 0 −|A, fail i ⊗ |1ishape − |A, ok i ⊗ (|1ishape − 2|2ishape) . (9b)

When the setting of Fig.1 b is used, the infor- erties1 and2 can be checked. However, we can mation about the path taken by the photon is see that the interference between the two paths preserved in the shape and the validity of Prop- does not occur anymore and the probability of

Accepted in Quantum 2021-06-24, click title to verify. Published under CC-BY 4.0. 9 outcome |V, ok0i does not vanish thereby restoring interference [23, 24, 25]. As said above, the fact that the which-path informa- k √1 (|V, ok0, 1i − |V, ok0, 2i)k2 = 1/3, 6 tion is made unavailable is necessary to the validity of property3. Consequently, the first friend which means that Property3 is automatically vi- does not have a definite measurement outcome olated. Therefore, neither of these two contexts (which would be either r or l), which contradicts allow all of properties 1 through 3 to hold simul- assumption (S). Note that the validity of (C) is taneously. hard to analyze in this case, as pretending we One may consider whether another measure- could take the point of view of the friend, who is ment setup, that does allow all three properties in a superposition state, is hazardous. The con- to remain simultaneously valid, exists. The an- sequence is that properties1 and2 do not hold swer is no, and this can be proven by noting anymore, such that the paradox does not arise. that Properties1 to3 correspond to assertions Conversely, if we assume that there exists some on the values taken by a set of non-commuting preserved record of the which-path information observables, which are consequently forbidden by (second mode-shaper absent), assumptions (S) quantum mechanics to simultaneously all accept and (C) are verified. However, the state |Ψ0i a well-defined value (see Appendix B)2. is now the proper description for the outcome statistics, and Property3 does not hold any- 2.6 The interferometer and the assumptions of more. Once again, the paradox does not arise. In Frauchiger and Renner summary, the paradox appears when comparing Eventually, it is enlightening to look at how the properties associated to two different contexts, present setup fits within the assumptions of FR. i.e. two different experimental setups shown in We first note that for this setup Assumption (Q) Fig.1. is fulfilled as the system under investigation is Finally, our results can be connected to a re- clearly an isolated quantum system. We also pre- cently introduced No-go theorem Ref. [16], invali- sented the rationale for using Assumption (U) in dating assumptions (S) and (C) in the presence of 2.1. Interestingly, the fate of the two other as- a super-observer, when assuming observers’ free- sumptions depends on whether Wigner is chosen dom of choice and locality of measurements. to be a “super-observer” (Context 1 leading to state |Ψi) or a regular observer (Context 2 leading to state |Ψ0i), i.e. by inserting, or not, the 3 Discussion second mode-shaper in arm r. If Wigner has full control over his friends’ labs We have identified a condition for a quantum sys- (second mode-shaper present), his measurement tem to behave as an observer, which is to possess statistics are captured by state |Ψi. However, a stable memory. Then, we have reformulated one can see explicitly that assumption (S) is vi- an extended Wigner’s friend scenario introduced olated. Indeed, the memory of F’s lab is erased in Ref. [3] as an interferometric setup, involving during Wigner’s measurement process before be- three different degrees of freedom of a single photon to play the roles of two qubits and the mem- ing read, and the branch of the wavefunction |Ψ2i corresponding to F finding path l interferes with ory of the friends measuring them. By analyzing that corresponding to finding path r. The sit- this setup, we have shown that the three prop- uation is reminiscent of the “quantum eraser” erties highlighted to be paradoxical correspond where a measurement is used to erase the pho- to two different contexts, in which the which- ton which-path information after it was recorded, path information takes well defined values or not. These two contexts correspond to two different 2In the language of quantum contextuality, one can measurement setups, which access the values of say that Properties 1, 2 and 3 are associated with the rays different sets of observables, which we show do defined by P1 = |H, rihH, r|, P2 = |A, lihA, l| and P3 = not commute. They are then forbidden by quan- |V, okihV, ok|, respectively, taking the value 0. However, tum mechanics to all simultaneously take well- ray P1 can be jointly measured with P2 or with P3 (in the sense that one can form sets of orthogonal projectors defined values. As a consequence, the paradox summing to one which include both P1 and P2 or P1 and never arises in any physical setup obeying quan- P3, but not with both simultaneously. tum mechanics.

Accepted in Quantum 2021-06-24, click title to verify. Published under CC-BY 4.0. 10 The fact that the three properties considered ori given) experimental resolution, one can find to formulate the paradox cannot hold simulta- systems belonging to this class, which can there- neously was already argued in Ref. [7]. We fore be defined as observers’ memory, and their here illustrate the transition between the validity action modeled through the measurement pos- regime of Properties1 and2 and that of Prop- tulate. From their mathematical treatment of erty3. In doing so, we can specifically iden- the friends and qubits’ state, it is clear than FR tify the transition between the coherent (quan- make the assumption of technological means far tum) and incoherent (observer) behaviors of the beyond current technology level, such that mea- friends. This transition is dictated by the pres- surements can be made in arbitrary bases even on ence, or not, of an untouched memory degree of systems such as human beings. In our prescrip- freedom. We also note that our interferometric tion, making this assumption implies that, just setup has similarities with Hardy’s paradox [26], as single photons, the ability of Wigner’s friend with an additionnal degree of freedom (the shape to behave as an observer is not granted, but con- of the photon) to control the interference. ditioned to the fact their memory is not erased during the experiment we consider. While this Our work is relevant however to more than just holds true in FR’s scenario for the two agents W the FR thought experiment. Our results speak to and W (that we have merged into Wigner), this unresolved issues in the Wigner’s Friend paradox condition is clearly violated for F and F which by identifying the state specific role a “memory” are therefore forbidden to behave as observers by must play for an observer in the paradox. Thus, construction. the transition between the two experimental contexts (captured by states |Ψi and |Ψ0i respec- More broadly, interpretations of quantum me- tively) can be interpreted as an assumption about chanics treating observers and quantum systems the location of Heisenberg’s cut [2]. More pre- on different footing are often criticized because of cisely, it allows us to understand which appara- the apparent flexibility in the position where such tuses must be considered as observers – the ‘ulti- this cut should be placed. However, our setup, mate measuring instruments’ of Bohr [27], which and the notion of memory, can be used to ar- verify Condition1 – or included in the system gue that the place of this cut is actually imposed described by quantum-mechanics [14, 28]. by the practical (and objective) resolution of the It may seem from Condition1 that the status experimentalist’s apparatuses. Because different of a device (observer or not) depends on future experimentalists with different apparatuses may choices made by super-observers. One can actu- be able to control different degrees of freedom, ally distinguish two cases: First, simple enough this approach can also be naturally related to systems, such as single photons, over which one a “QBist” interpretation [29, 30, 31] where the can have a large degree of control and deliber- quantum state is ascribed by a given observer to a ately choose to alter or not the state. In this quantum system and may therefore be observer- case, it is indeed a choice to have them behave dependent. A second point of contact with QBist as observers, and this choice can be delayed until approaches come with the emphasis on memory after they interacted with the system. One could and its quantum accessibility. QBism holds that argue that rather than being observers, those sys- quantum states are epistemological rather than tems can behave as such in specific situations. ontological. In particular, given the fundamental On the other hand, it is easy to identify a class nature of probabilities in quantum states, QBism of systems complex enough that when they are stresses that they should be seen as bets, con- used as memories, the unitary control required ditioned by priors, placed on the results of an to erase their content is definitely out of reach. experiment. The priors are then updated once Those systems satisfy Condition1 at any time the experimental results are obtained. Priors and can be qualified as observer without ambi- are, essentially, memories. They are information guity. In this discussion, the recent technolog- about previous states of the world held by an ical advances allowing the unitary manipulation observer and used to calculate quantum states. of larger and larger quantum systems can be seen Thus by showing how memory, as a manipula- as a (slight) shift of which systems belong to this ble quantum system, must function in Wigner’s class. An important point is that for any (a pri- Friend argument, our setup may help articulate

Accepted in Quantum 2021-06-24, click title to verify. Published under CC-BY 4.0. 11 the ways in which quantum states in general itself. Sci. Rep. 9, 1–8 (2019). URL https: should be viewed. Further comments regarding //doi.org/10.1038/s41598-018-37535-1. a QBist treatment of the problem can be found [8] Fortin, S. & Lombardi, O. Wigner and in Refs. [12, 32]. his many friends: A new no-go result? arXiv (2019). URL https://arxiv.org/ 4 Acknowledgements abs/1904.07412v1. 1904.07412. [9] Waaijer, M. & van Neerven, J. Relational We thank Joseph Eberly, Zachery Brown, analysis of the Frauchiger–Renner paradox Etienne´ Jussiau, Tathagata Karmakar, Joseph and existence of records from the past. Murphree, John Steinmetz and Jing Yang for arXiv (2019). URL https://arxiv.org/ stimulating discussions that led to this paper. We abs/1902.07139v1. 1902.07139. acknowledge funding from NSF grant no. DMR- 1809343 and John Templeton Foundation, award [10] Baumann, V. & Wolf, S. On Formalisms number 61835. and Interpretations. Quantum 2, 99 (2018). 1710.07212v6. URL https://doi.org/10. . References 22331/q-2018-10-15-99 [11] Krismer, R. Representation Lost: The Case [1] Wigner, E. P. Remarks on the Mind- for a Relational Interpretation of Quantum Body Question. In Symmetries and Re- Mechanics. Entropy 20, 975 (2018). URL ﬂections, 171–184 (Indiana University Press, https://doi.org/10.3390/e20120975. 1967). URL https://doi.org/10.1007/ [12] Stacey, B. C. On QBism and Assumption 978-3-642-78374-6_20. (Q). arXiv (2019). URL https://arxiv. [2] Heisenberg, W. The Physical Principles of org/abs/1907.03805v2. 1907.03805. the Quantum Theory (University of Chicago [13] Losada, M., Laura, R. & Lombardi, O. Press, Chicago, 1930). URL https://doi. Frauchiger-Renner argument and quantum org/10.1007/978-3-642-61742-3_10. histories. Phys. Rev. A 100, 052114 [3] Frauchiger, D. & Renner, R. Quantum theory (2019). URL https://doi.org/10.1103/ cannot consistently describe the use of itself. PhysRevA.100.052114. Nat. Commun. 9, 1–10 (2018). URL https: //doi.org/10.1038/s41467-018-05739-8 [14] Bub, J. Understanding the Frauchiger–Renner Argument. Found. [4] Sudbery, A. The hidden assumptions of Phys. 51, 36–9 (2021). URL https: Frauchiger and Renner (2019). URL https: //doi.org/10.1007/s10701-021-00420-5. //arxiv.org/abs/1905.13248v2. arXiv: 1905.13248. [15] Nurgalieva, Nuriya & Renner, Renato, Testing quantum theory with thought ex- [5] Nurgalieva, N. & del Rio, L. Inadequacy of periments. Contemp. Phys. 61, 193–216 modal logic in quantum settings. In Selinger, (2020). URL https://doi.org/10.1080/ P. & Chiribella, G. (eds.) Proceedings of the 00107514.2021.1880075. 15th International Conference on Quantum Physics and Logic, Halifax, Canada, 3-7th [16] Bong, K.-W. et al. A strong no-go theorem June 2018, vol. 287 of Electronic Proceedings on the Wigner’s friend paradox. Nat. Phys. 1– in Theoretical Computer Science, 267–297 7 (2020). URL https://doi.org/10.1038/ (Open Publishing Association, 2019). URL s41567-020-0990-x. https://doi.org/10.4204/EPTCS.287.16 [17] Deutsch, D. Quantum theory as a universal [6] Healey, R. Quantum Theory and the Limits physical theory. Int. J. Theor. Phys. 24, 1– of Objectivity. Found. Phys. 48, 1568–1589 41 (1985). URL https://doi.org/10.1007/ (2018). URL https://doi.org/10.1007/ BF00670071. s10701-018-0216-6. [18] Matzkin, A. & Sokolovski, D. Wigner’s [7] Lazarovici, D. & Hubert, M. How Quantum friend, Feynman’s paths and material Mechanics can consistently describe the use of records. EPL 131, 40001 (2020). URL

Accepted in Quantum 2021-06-24, click title to verify. Published under CC-BY 4.0. 12 https://doi.org/10.1209/0295-5075/ [27] Bohr, N. The causality problem in atomic 131/40001. physics. In New Theories in Physics, 11–30 [19] Schrödinger, E. Die gegenwärtige Situa- (International Institute of Intellectual Coop- tion in der Quantenmechanik. Naturwis- eration, Warsaw, 1939). URL https://doi. senschaften 23, 807–812 (1935). URL https: org/10.1016/S1876-0503(08)70376-1. //doi.org/10.1007/BF01491987. [28] Bub, J. ‘Two Dogmas’ Redux. In Orly [20] Zurek, W. H. Quantum Darwinism. Nat. Shenker and Meir Hemmo (eds), Quantum, Phys. 5, 181–188 (2009). URL https://doi. Probability, Logic: The Work and Influ- org/10.1038/nphys1202. ence of Itamar Pitowsky, 199-–215 (Springer, 2020). URL https://doi.org/10.1007/ [21] von Neumann, J. Mathematical Foundations 978-3-030-34316-3. 1907.06240. of Quantum Mechanics (Princeton University [29] Bub, J. & Pitowsky, I. Two dogmas Press, Princeton, New Jersey, 1955). about quantum mechanics. In Simon [22] Proietti, M. et al. Experimental test of local Saunders, Jonathan Barrett, Adrian Kent, observer independence. Sci. Adv. 5, eaaw9832 and David Wallace (eds), Many Worlds?: (2019). URL https://doi.org/10.1126/ Everett, Quantum Theory, and Real- sciadv.aaw9832. ity (Oxford Scholarship Online, 2010). [23] Scully, M. O. & Drühl, K. Quantum eraser: URL https://doi.org/10.1093/acprof: A proposed photon correlation experiment oso/9780199560561.003.0016. 0712.4258. concerning observation and ”delayed choice” [30] Fuchs, C. A. & Schack, R. Quantum- in quantum mechanics. Phys. Rev. A 25, Bayesian Coherence. arXiv (2009). URL 2208–2213 (1982). URL https://doi.org/ https://arxiv.org/abs/0906.2187v1. 10.1103/PhysRevA.25.2208. 0906.2187. [24] Kim, Y.-H., Yu, R., Kulik, S. P., Shih, [31] Fuchs, C. A. & Stacey, B. C. QBism: Y. & Scully, M. O. Delayed ”Choice” Quantum Theory as a Hero’s Handbook. Quantum Eraser. Phys. Rev. Lett. 84, 1– In Ernst M. Rasel, Wolfgang P. Schleich, 5 (2000). URL https://doi.org/10.1103/ Sabine Wölk(eds), Proceedings of the Inter- PhysRevLett.84.1. national School of Physics ”Enrico Fermi”: [25] Walborn, S. P., Terra Cunha, M. O., Pádua, Foundations of Quantum Theory, 133-–202 S. & Monken, C. H. Double-slit quan- (IOS Press, 2019). URL https://doi.org/ tum eraser. Phys. Rev. A 65, 033818 10.3254/978-1-61499-937-9-133. 1612. (2002). URL https://doi.org/10.1103/ 07308. PhysRevA.65.033818. [32] DeBrota, J. B., Fuchs, C. A. & Schack, [26] Hardy, L. Quantum mechanics, local re- R. Respecting One’s Fellow: QBism’s alistic theories, and Lorentz-invariant realis- Analysis of Wigner’s Friend. arXiv tic theories. Phys. Rev. Lett. 68, 2981–2984 (2020). URL https://arxiv.org/abs/ (1992). URL https://doi.org/10.1103/ 2008.03572v1. 2008.03572. PhysRevLett.68.2981.

Accepted in Quantum 2021-06-24, click title to verify. Published under CC-BY 4.0. 13 A Appendix: Equivalence of measurement descriptions

The equivalence of the entangled state Eq. (1) and the collapsed state Eq. (2) for subsequent measurements can be showed by computing the probability to obtain an eigenstate |aiS of an arbitrary observable AˆS of qubit S:

2 2 2 2 SM hΨ|aiSha|ΨiSM = |c0| |Sha|0iS| + |c1| |Sha|1iS|

= Sha|ρS|aiS. (10) This implies that the statistics of subsequent measurement on the qubit S is captured identically by both states. This property is a direct consequence of the perfect correlation of the memory states |0iM and |0iM with the system states |0iS and |0iS, respectively. A unitary operation on the memory gen P and the system will in general generate a state |Ψ iSM = i,j=0,1 dij|iiS ⊗ |jiM that will yield probability:

gen gen X ∗ SM hΨ |aiSha|Ψ iSM = di,jdk,jSha|iSha|jiS i,j,k=0,1 (11)

gen of finding eigenstate |aiS. It is now easy to check that only the states |Ψ iSM which verify the P 2 2 ∗ ∗ specific conditions (1) j=0,1 |dij| = |ci| and (2) d00d10 + d01d11 = 0 preserve the probability dis- tribution (10) for any |aiS. This corresponds to unitary transformations which acts on the memory state and performs a basis rotation, transforming |0iM and |1iM to two orthogonal states. Any unitary transformation that does not respect this specific property, for instance that acts on the memory in a way conditionned to the system’s state, will change the measurement statistics. In particular, the unitary transformation involved in Wigner’s “super-measurement” (see Fig.1 b) erases the er memory, yielding a state of the form |Ψ iSM = (c0|0iS + c1|0iS) ⊗ |0iM which implies a probability er er P ∗ SM hΨ |aiSha|Ψ iSM = i,j ci cjShi|aiSha|jiS which is not equal to Sha|ρS|aiS as long as ci and cj both differ from 0.

B Appendix: Paradoxical properties as incompatible observables

We introduce the following photon observables:

O1 = |H, r, 1ihH, r, 1|

O2 = |A, l, 2ihA, l, 2|

O3 = |V, okihV, ok|. (12) These three-qubit observables involve all three degrees of freedom of the photon. They are projectors admitting eigenvalues 0 and 1. It is easy to check that:

[O1, O2] = [O1, O3] = 0, 1 [O2, O3] = 2 (|A, l, 2ihV, ok| + |V, okihA, l, 2|) 6= 0. (13)

Meanwhile, saying properties1,2 and3 hold is equivalent to say that, respectively, O1, O2 and O3 takes the value 0. The fact that O2 and O3 do not commute then rules out the possibility for these three properties to hold simultaneously in any measurement setup.

Accepted in Quantum 2021-06-24, click title to verify. Published under CC-BY 4.0. 14