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 quan- tum 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 Schr¨odinger 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 (Schr¨odinger 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 ar- guments 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 de- scribed as a quantum system that becomes en- tangled 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 mea- sured 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 en- tangled. 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 behav- ing 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 Schr¨odinger 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 evolu- tion. 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.