Wigner Friend scenarios with non-invasive weak measurements

Wigner Friend scenarios -- in which an external agent describes quantum mechanically a laboratory in which a Friend is making a measurement -- give rise to possible inconsistencies due to the ambiguous character of quantum measurements. In this work, we investigate Wigner Friend scenarios in which the external agents can probe in a non-invasive manner the dynamics inside the laboratories. We examine probes that can be very weakly coupled to the systems measured by the Friends, or to the pointers or environments inside the laboratories. These couplings, known as Weak Measurements, are asymptotically small and do not change the outcomes obtained by the Friends nor their probabilities. Within our scheme, we show that the weakly coupled probes indicate to the external agents how to obtain consistent predictions, irrespective of the possible inconsistencies of quantum measurement theory. These non-invasive couplings could be implemented with present-day technologies.


I. INTRODUCTION
Measurements are special in Quantum Mechanics. Indeed quantum systems evolve unitarily, except when a measurement takes place: a "special rule" needs to be enforced in order to account for the appearance of a definite measurement outcome out of the linear superposition of pointer-system states. Technically, this rule appears as a projection of the quantum state right before measurement to an eigenstate of the measured observable. Physically, there is no unambiguous and consistent manner of accounting for this rule [1]. Logically, this makes quantum theory inconsistent [2]. Several options are on the table, but none of them is entirely satisfactory and there is unsurpisingly no consensus on these issues, overall known as the celebrated Measurement problem.
Could there be an underlying basis that would account for the projection that takes place when a measurement is completed -the "collapse" of the wavefunction ? In a series of papers [3][4][5], Wigner assumed unitary evolution to be universal -implying it also applies to macroscopic bodies -but that linear superposition cannot be applied to conscious observers making a measurement. Hence if a Stern-Gerlach experiment takes place in a sealed laboratory L, an agent W outside the laboratory would describe the state of the atom having passed through the inhomogeneous magnetic field by a linear superposition of up and down spin along the field direction. Now if there is a friend F inside the sealed lab running the experiment, she would take the superposition to "collapse" upon completion of her measurement. But how should W describe the experiment?
If unitary evolution is universal, the answer would be that as long as W is unaware of the outcome, he will take the quantum state of the laboratory to be in superposition even after F completed her measurement. Wigner was dubious that a superposition of different states of consciousness would make sense 1 , but concluded (at least in his 1962 paper [3])) that an experimental test attempting to distinguish a linear superposition of L from a mixed state would in practice be impossible. Still it is worthwhile to investigate whether such assumptions can be taken to be consistent.
Remarkably, in two recent works Brukner [7] and Frauchiger and Renner [8] have shown by employing an extended Wigner friend scenario composed of two sets i = 1, 2 each involving a Friend F i inside a Laboratory L i and an agent W i external to the laboratories L 1 and L 2 that assuming the Friends get a definite outcome while the agents W i describe these processes by unitary evolutions leads to a contradiction. The implications of this has sparked a wild interest [9][10][11][12][13][14][15][16], though its relevance depends on a set of specific assumptions that are explicitly or implicitly being made.
In this work we will not be interested in these specific assumptions -in particular in those made in [7,8] -but focus instead on the basic hypothesis at the root of the Wigner Friend Scenarios according to which a completed measurement in a closed laboratory can be described by an external agent as a unitary evolution. We will introduce a procedure by which the agents W i can acquire information on the state of the system or the state of the pointer inside the laboratories L i without perturbing in any essential way the measurements taking place in L i , in particular without modifying any outcome probabilities. This information will be acquired with non-invasive probes through a minimal disturbing scheme known as weak measurements [17]. In general, a weak measurement involves a unitary interaction between a system and a very weakly coupled probe. The weakly coupled probe becomes entangled with the system, and it is only when the system is subsequently subjected to a measurement of another observable that the information can be retrieved from the probe.
Here our strategy will be to connect a non-invasive probe available to the agents W i to elements inside the otherwise sealed laboratories L i . For instance such probes may be coupled through a weak interaction to the spins inside L i , to the pointers measuring the spins, or event to the environment inside L i by which the Friends acquire information on the spin measurement outcomes. We will see that even in the original Wigner setup, assuming unitary evolution leads to a contradiction as W would read from the weakly coupled probe that the spin (or the pointer that measured the spin, or the environment inside L) is in a state of superposition, while F announces she obtained a definite outcome. This contradiction subsists in the extended scenario: the Friends announce they obtained a definite outcome but if the external agents assume unitary evolution, this would not be reflected in the state of the probes. Actually if the agents W i are allowed to communicate, they would realize their probe states act as a witness of entanglement between the spins (or the pointers, or the environments) of each laboratory. This contradiction is of course the counterpart of the contradiction uncovered in Refs. [7,8]. As expected, no contradiction occurs if the Friends' measurements result in a global collapse: in this case the weakly coupled probes are in perfect agreement with the outcomes announced by F 1 and F 2 . In this paper we will take "collapse" to represent the second step of the measurement process, recalled in Sec.
II, without any commitment to an underlying picture nor specific interpretation (we will sometimes recall this point by writing "possibly effective collapse").
In Sec. II we will introduce the original (WFS) and the extended Wigner Friend (EWFS) setups we will be working with. Sec. III will deal with implementing different non-invasive measurements with weakly coupled probes in the laboratories for both the WFS and EWFS.
We will assume the external agents W can compute the quantum state evolution either by following full unitary evolution or by applying globally the projection postulate, each case leading to different predictions for the weakly coupled probes. We will discuss our results in Sec. IV; in particular we will argue that although connecting a weak non invasive probe to a laboratory breaks the independence of L from a logical viewpoint, it does not change the physical situation in the laboratories. We close with our conclusions in Sec. V.

A. WFS: Single laboratory and observers
In the original Wigner Friend scenario (WFS) Wigner [3] introduces a laboratory L in which a friend F performs a Stern-Gerlach experiment on an atomic spin, while an agent W is outside the isolated laboratory and ultimately measures the quantum state of the laboratory (see Fig. 1). We will introduce the following quantum states: |s , |m and |e label respectively the spin, the pointer measuring the spin, and the environment of the isolated laboratory, whose overall quantum state is denoted by |L . |M is the quantum state of W's pointer, that is assumed to measure L.
A measurement in quantum theory is represented in two steps. First a unitary evolution couples the system observableÂ that is being measured to a pointer observableP (typically the momentum). The coupling Hamiltonian is H int = g(t)ÂP where g(t) the coupling strength. Before the measurement the overall quantum state can be put as ψ(t = 0) = k c k |a k |m , where the system is expanded over the eigenstates |a k ofÂ. The pointer state |m is typically a tightly localized function (eg, a Gaussian), initially centered on X = 0. Unitary evolution leads to where we have X| m k = m(X −ga k ) and set g ≡ dt g(t ) assumingÂ is time independent and recalling that exp −iga kP / is a translation operator. Each term of the sum in Eq. In the Wigner Friend scenario, F measures the spin component along some direction, say x, henceÂ is given by σ x and there are 2 pointer states m(X ∓ g) correlated resp. with the spin s x = ±1. The second step of the measurement is the projection that, as was first axiomatized by von Neumann [18], is necessary in order to account for the appearance of a definite outcome. Indeed including additional unitary interactions, eg between the pointer and the environment of the pointer (that could include the photons having interacted with the pointer traveling toward the Friend's eye) would only create a growing chain of entangled states, the von Neumann chain, defined in the measurement basis; eg, including the quantum states of the environment |e is tantamount to replacing Eq. (5) by k c k |a k |m k |e k .
So as far as F measuring the spin is concerned, the inital state |L(t = 0) introduced in Eq. (2) goes into |L(t) = ± c ± |± m ± |e ± in the first unitary step but once the measurement is completed at t = t f and an outcome, say + (resp. −) is obtained, the final state is the projection |L+ ≡ |+ +| L(t f ) (resp. |L− ≡ |− −| L(t f ) ). According to the "universality" assumption, W should treat L with F inside performing a measurement just as he would treat any other unmeasured quantum system, in which case the evolution in L is considered to be fully unitary and before W's measurement the quantum state is where we assumed W's pointer has no evolution previous to the coupling with L. Alternatively, if W discards the universality assumption and assumes instead that measurements are always "special" (as per von Neumann's rule, irrespective of where and how the measurement was made), he will take the Friend's measurement to result in a (possibly effective) collapse of the global quantum state. In this case W would consider the state of L to be given not by |Ψ(t) but by a statistical mixture of density matrices |L+ L+| and |L− L−|. Wigner in his paper [3] concludes that in practice, no experiment could discriminate between both alternatives. Indeed, for a genuine macroscopic size laboratory the interferences that need to be detected are so small that they become effectively undetectable [19,20], an argument that goes back to Bohm [21].

B. EWFS: Two sets of laboratories
Drawing on a version of the WFS proposed by Deutsch [2], Brukner [7] and Frauchiger and Renner [8] introduce an Extended Wigner Friend Setup (EWFS) composed of two sets of Wigner Friend setups sharing spins 1/2 in an entangled state (see Fig. 2). Our scheme will follow the latter proposal [8], in which a pair of spin 1/2 atoms is prepared in the non-maximally entangled state where ↑↓ refer to the eigenstates of the spin compoment along z, σ z |↑ = |↑ , σ z |↓ = − |↑ and |± = (|↓ ± |↑ ) / √ 2 are the eigenstates of σ x . Employing the entangled state (7) leads to a Bell-type inequality with 0 as the upper bound [22,23], instead of the usual bound of 2 obtained with maximally entangled states that Brukner uses in his scheme [7].
Conceptually, the advantage of having 0 as an upper bound is that the violation of the inequality by the quantum version appears as logically impossible for a classical model in which all the measured observables have joint values (recall that when observables do not commute, the quantum formalism precludes the existence of joint probability distributions [24]).
In the narrative of Ref. [8], F 1 measures her spin first and then sends the other atom to L 2 in state |↓ 2 if the outcome was + and in the state |+ 2 if F 1 obtained −. This is equivalent to dealing with Eq. (7) from the outset but has the advantage of starting with a product state for the atoms, which at t = 0 are both inside L 1 . In this case however L 1 and L 2 are not isolated from each other, though our main requirement is that they should be isolated from the external agents W 1 and W 2 .
The initial state of L 1 is thus where and |s 2 can be any spin state. The first step of F 1 's measurement involves the unitary coupling (3) leading to Upon completion of the measurement at t = t 1 , an observer inside L 1 reads a definite outcome projecting the pre-measurement state to we have assumed a gate coupling the 2 spins brings |s 2 to either |↓ 2 or |+ 2 depending on whether |+ 1 or |− 1 were obtained. Spin 2 is sent to L 2 at t 1 whose state becomes where |m 2 is F 2 's pointer in the ready state. F 2 measures her spin in the |↑↓ basis, again through the interaction (3) and after her measurement is complete at F 2 has at that point obtained a definite outcome.
Now if we assume that for the external agents W 1 and W 2 , L 1 and L 2 are ordinary (though macroscopic) quantum systems, the laboratories evolution is described unitarily and instead of Eq. (11) we have Upon receiving atom 2, L 2 gets entangled with L 1 and the overall state at t 2 is where we have used the notation introduced in Eqs. (11) and (13). Finally W 1 measures L 1 in the basis |L 1 ↑↓ and W 2 measures L 2 in the basis |L 2 ± defined as per the spin basis by It is useful to rewrite Eq. (15) in the Ws' measurement basis, The contradiction obtained by Frauchiger and Renner [8] takes here the following form.
Statements (iv) and (v) are in direct contradiction. Technically, this follows from the fact that taken together, points (ii)-(iv) implicitly assume the existence of a joint probability distribution for the outcomes of σ x and σ z within each set {L i ,W i }. This is well-known to be impossible, given that σ x and σ z do not commute: if only unitary evolution is assumed, the outcome probabilities are obtained by adding amplitudes that interfere as per (v). But here agents W i still need to account for the outcomes obtained by the Friends which is the reason points (ii)-(iv) appear to hold. Note that if global collapse occurs, then neither (ii) nor (iii) (and therefore (iv)) hold, since the correlations between the outcomes of W 1 and F 2 as those between W 2 and F 1 follow from Eqs. (11) and (13). We are only adding probabilities and no contradiction appears.

DISTURBING MEASUREMENTS
We work out here non-invasive minimally disturbing schemes that allow an external observer to gain information on the quantum states of laboratory systems or of a laboratory itself. In order to be non-invasive, the readout of the measurement cannot change the outcomes nor the outcome probabilities of the Friends or the external agents W. Any type of fuzzy meter [25] with a weak coupling strength could be considered for our scheme.
For definiteness we will use weak measurements, a minimally disturbing scheme that has been widely investigated recently. The underlying conceptual issue -whether any type of interaction, be it non-invasive, fundamentally changes the closed character of the laboratories L i will be discussed below in Sec. IV.

A. Weak measurements
The scheme we use is based on Weak measurements (WM), originally introduced by Aharonov, Albert and Vaidman [17] (see eg [26,27] for an introduction to the main concepts).
The idea is to extract information about a given property, represented by an observableÂ on a system that evolves from a prepared initial state |ζ(t = 0) towards the final eigenstate |b f obtained after measuring a different observableB. This is done by creating at some time t w a von Neumann interaction as per Eq. (3), but with a very weak coupling constant (that will be labeled γ instead of g) so that the ensuing evolution operator can be expanded to first order as 1 − iγÂP / (as above,P is an observable of the probe to whichÂ is coupled to). Then we let the system, now effectively entangled with the weakly coupled probe, evolve up to t f when a standard projective measurement ofB is made.
Up to some time t w < t < t f , the system-probe state evolves according to where |ϕ denotes the quantum state of the probe and U is the system evolution operator (for simplicity we assume no evolution for the probe). Assume that the result of theB measurement at t f is b f and |b f denotes the corresponding eigenstate. It is straightforward to show (see, eg [27]) that the weakly coupled probe state after b f has been obtained is where is known as the weak value ofÂ. A w f can be complex. If |ϕ(t = 0) is a Gaussian in position space andP is the momentum Eq. (20) states that the initial probe state is shifted by Since γ is small the shift will be very small -the initial and final pointer states substantially overlap and in practice the measurements must be repeated in order to gather enough statistics. Note that through the weak value, the final probe state depends on the initial quantum state of the system |ζ but also on the final projective measurement that is made on the system. This is a property that we will see to be useful in the Wigner Friend scenarios.
Note also that the weak interaction modifies the evolution of the system state to first order in γ (compare Eq. (19) with the free evolution |ζ(t) = U (t, 0) |ζ(t = 0) ). In particular the probabilities to obtain an outcome for any subsequent measurement are not modified to first order (they depend on γ 2 ). In this sense the weak measurement ofÂ is minimally disturbing.
Let us consider the original Wigner setup, with a laboratory L 1 , a friend F 1 and an external agent W 1 , and for definiteness assume L 1 is initially in state |L 1 (t = 0 as given by Eq. (8), except for the spin s 2 that plays here no role and will be left out. We further assume W 1 measures L 1 in the σ z basis as in Sec. II B above, with a pointer initially in the ready state |M 1 (t = 0) . In addition W 1 has a probe with momentum variableP W 1 , initially in state ϕ 1 (X W 1 ) that will be weakly coupled to the spin through an interaction Hamiltonian γ(t)ω 1PW 1 (see Fig. 1), whereω 1 is an observable related to the spin (eg, a spin component along a convieniently chosen axis, or a spin projection operator). The weak interaction takes place before F measures her spin in the σ x basis, and the initial state evolves for t < t 1 to where |s 1 is given by Eq. (9). The unitary with the weak coupling is expanded to first order and the standard von Neumann term is expanded in the |± 1 basis, yielding We have not included the pointer state |M 1 that plays no role at this stage, and we have set γ ≡ γ(t )dt as in the usual von Neumann measurement [see below Eq. (5)]. The quantities are the weak values. In most instances the exact choice ofω 1 is not important, as long as it is known to W 1 so he can infer the weak value by measuring the probe. Indeed, the dependence of the weak value on the initial and final states |s 1 and |± is what will matter most in the present context. However when the choice ofω 1 becomes significant we will mention it. Note that some specific choices can be particularly useful in some circumstances -for instanceω 1 = Π + ≡ |+ +| will leave the probe untouched when the spin outcome is −, since by Eq. (24) we then have −| Π + |s 1 = 0.
If we assume global collapse upon F's measurement then Eq. (23) leads to with |L 1 ± = |± 1 |m 1 ± |e 1 ± . W 1 finally measures L 1 in the σ z basis by coupling his pointer in the ready state |M 1 (t = 0) . This is a standard projective measurement that leaves the pointer in state |M 1 ↑ or |M 1 ↓ . The probe states are shifted by the weak values ω ± 1 correlated with the states |L 1 ± . W 1 therefore obtains one of the following final states: .
If instead unitary evolution is assumed, Eq. (23) becomes Note that the laboratory has become slightly entangled with the non-invasive probe (slightly in the sense that γ is small so that the zeroth order term is the product state . Finally W 1 measures L 1 in the σ z basis, so writing Eq. (27) in that basis yields Hence when W 1 obtains ↑ (or ↓) the probe will be found in the states given by the term between [.] and {.} respectively, indicating in both cases a superposition of the spin in states ±.
The first important result we have obtained is that a weakly coupled probe allows us to empirically discriminate the mixed pointer states obtained from Eq. (26)

EWFS
We now apply the same method to the Extended Wigner Friend scenario of Sec. II B.
The external agents W 1 and W 2 couple a probe with a weak interaction to the atoms in laboratories L 1 and L 2 respectively (see Fig. 2). For W 1 , the process is exactly the one we have described immediately above. W 2 also has a probe in state ϕ 2 (X W 2 ) that couples to the spin s 2 of atom 2 once it is received by F 2 . For definiteness assume the coupling Hamiltonian to be given by γ(t)ω 2PW 2 (where the notation follows the pattern introduced above).
a. Global collapse upon the Friends measurement After F 1 completes her measurement, a collapse of her meter m 1 (X) leads to a weakly coupled probe in states ϕ 1 (X W 1 − γω + 1 ) or ϕ 1 (X W 1 − γω − 1 ). Concurrently L 2 is in either of the states given by Eq. (12). With the weakly coupled probe initially in state ϕ 2 (X W 2 ), the same analysis 2 leading to Eq. (25) can be done for L 2 yielding with |L 2 ↑ = |↑ 2 |m 2 ↑ |e 2 ↑ and where ω 2 is a spin operator or projector related to atom 2, |s c = |↓ or |+ is the spin state of the atom received by F 2 and |s f = |↓ or |↑ is the outcome of F 2 's measurement. It can be checked that ω ↓↓ 2 is the expectation value ↓|ω 2 |↓ . Finally the external observers W i make their measurements. For W 1 the outcomes were given in Eq. (26). W 2 measures L 2 and the outcomes are |L 2 ± as given by Eq. (17). His pointer |M 2 is thus correlated with .
The joint outcomes for W 1 and W 2 are independent and can be obtained from Eqs. b. Unitary evolution of the laboratories If L 1 and L 2 are asumed to evolve unitarily (as per the "universality" assumption) the weak couplings with W 1 and W 2 's non-invasive probes are taken into account by the evolution operator e −iγω 2PW 2 / e −iγω 1PW 1 / so that the overall wavefunction given by Eq. (15) becomes with this coupling Let us look at the correlation between outcomes as by points (i)-(v) below Eq. (18): for which the leading term vanishes. We thus neglect it and we are left with the sole term |L 1 + |L 2 − ϕ 1 (X W 1 − γω + 1 )ϕ 2 (X W 2 − γω ↓↓ 2 ) coming from the upper line. This implies that (ii) holds [W 2 = − ⇒F 1 = +] and the weak probe ϕ 2 further indicates to W 2 that F 2 received and measured the atom spin in state ↓.
(iii ) We write Eq. (32) in the basis |L 1 |L 2 ; to leading order, we are left with . Hence (iii) holds [W 1 =↑⇒F 2 =↑] and we have the information through the weak probes that the outcome for F 1 was −.
The probes do not indicate to the external observers W i that a specific outcome has been obtained.
We will see below that if the probes are coupled to the pointer or the environment inside each L i (hence after each Friend's measurement has occurred), the behavior represented by Eq. (34) is the generic one. We therefore defer its discussion to the paragraph immediately below. We wish to emphasize here that we are combining projective measurements and weak measurements. The projective measurements carried out in points (ii ), (iii ) and (v ) are not compatible (since they involve non-commuting observables). The weak measurements are non-invasive unitary interactions that give information on an observable. Therefore the problem of compatibility with the final projective measurement does not arise -this is actually one of the reasons weak measurements have found to be useful -but it should be stressed that weak values are conditioned on how the amplitudes interfere upon a final projective measurement.

C. Monitoring inside the laboratory
We investigate the case in which the external agents W i weakly couple an external probe to an element inside L i rather than to the atoms after F i s measurement has occurred. By "element" we mean a system that carries information about the measurement, such as the pointers, or the environment inside L i . In the version in which the Friend's measurement induces an overall collapse a weak probe of this type would be redundant, as it would simply confirm the outcome obtained by each of the Friends' projective measurements. So we will only consider weak coupling in the unitary evolution scheme that would follow from Wigner's "universality" assumption. We will simplify the notation by using to stand for "↑ or ↓".

WFS
For definiteness, assume the weakly coupled probe, placed outside L 1 interacts, through a hole, with the measurement apparatus in the otherwise sealed laboratory. This interaction takes place after the spin interacted with the Friend's pointer. Hence Eq. (22) is slightly modified to whereω 1 is now an obervable related to the measurement apparatus whose state after the spin measurement is |m 1 ± . W 1 measures L 1 in the |L 1 basis, leading to Since L 1 | L 1 ± = ±1/ √ 2, we have essentially the same result as when the probe was weakly coupled to the spin, see Eq. (28). For both outcomes , W 1 would find according to unitary evolution his weakly coupled probe ϕ 1 as indicating a superposition of the spin in states ±. So W 1 can again conclude on the inconsistency between the information given by ϕ 1 and the definite outcome F 1 announced she obtained; and discriminate empirically the assumed unitary evolution from the mixed state expected if the Friend's spin measurement induces a global collapse.

EWFS
Now both W 1 and W 2 couple a probe to the measuring apparatus in L 1 and L 2 respectively. Assuming unitary evolution, the quantum state reads where |L 12 is given by Eqs. (15) or (18). The same projective measurements as in Sec.
III B 2 b are performed by W 1 and W 2 on |Ψ(t 2 ) . The weak values are now defined relative to the pointer states |m 1 or |m 2 : where α labels the state in the measurement basis of the Friend F i and β a state in the basis in which W i measures. Points (i )-(iv ) are the same as in the case in which the probe was coupled to the spins. Point (v ) is slightly different: (ii ) If we take the scalar products L 1 ±| L 2 −| to leading order only the term L 1 +| L 2 −| Ψ is non-zero with the probes left in states ϕ 1 (X W − γω ++ 1 )ϕ 2 (X W 2 − γω −↓ 2 ), implying [W 2 = − ⇒F 1 = + and F 2 =↓], exactly as in (ii').
(iii ) Computing the scalar products L 1 ↑| L 2 | Ψ leaves the leading term with the (iv ) We have the same contradiction as in (iv').
(v ) The projection L 1 ↑| L 2 −| leads to To lowest order in γ, this can be put under the form This is similar to Eq. Let us start by summarizing the results we have obtained.
In the original WFS, if the external agent W assumes L evolves unitarily, he will run into a contradiction: a non-invasive probe coupled (and therefore entangled with) the Friend's atom or her pointer would display a signature of the macroscopic superposition. Indeed W can measure the probe before or after F tells him she obtained a definite outcome; but if unitary evolution is assumed, the probe would instead indicate a superposition of the atom spin or pointer states, incompatible with F having obtained a definite outcome.
Instead if global collapse is assumed the weakly coupled probes would consistently indicate the outcome obtained by F. Note that while measuring the state of an entire macroscopic laboratory in order to discriminate a mixed state from unitary evolution is experimentally out of reach, measuring non invasive weakly coupled probes could be achieved with present day technologies.
In the EWFS, assuming unitary evolution results in entangled probes. This is a signature that the atoms or the pointers to which the probes were coupled are entangled. But this is impossible since F 1 and F 2 have announced they have obtained definite outcomes. This inconsistency when unitary evolution is assumed is similar to the one for the original WFS, except that here the superposition takes the specific form of entanglement. This inconsistency is similar to the logical contradiction obtained by Frauchiger and Renner [8] [see points (i)-(v) below Eq. (18)], except that here this contradiction appears as information the agents could read on their probes. The agents W i would therefore conclude by reading their probes that they were wrong in assuming unitary evolution, as the probes are consistent with the Friends' announcements only if global collapse is assumed.

B. Closed laboratories and non-invasive measurements
According to the summary we have just given, an external agent should apply the projection postulate ("global collapse") and not assume unitary evolution for a closed laboratory in which a measurement takes place, as this will lead him in to an observable contradiction.
One may question however whether the laboratories can still be qualified as closed when a very weak interaction is allowed with the external world.
From a purely logical viewpoint, this objection makes sense: an interaction, however small, breaks the purported independence of the closed laboratory from the external world.
The very idea of attempting to discriminate observer-dependent from observer independent facts [7] seems indeed ruined. This is a crucial point in Deutsch's approach [2] that employs a notion called "kinematic independence" to assume two subsystems can evolve as independent worlds.
From the physical viewpoint we adopt here however, this independence is a fiction: a laboratory can never be perfectly isolated. Operationally, a laboratory is said to be closed if the inevitable interactions with the rest of the Universe can be neglected relative to the effects that are investigated. Here, the non-invasive interaction is asymptotically small and does not change in any essential way the dynamical evolution inside the laboratories: in particular the predictions made inside the laboratories and the outcome probabilities are not modified. From this perspective, adding non-invasive probes does not fundamentally alter the physical interactions and processes inside the laboratories. Our operational assumption is that such a probe simply records the local value of the operator states of the system it is interacting with, with minimal change to the system's state . This is how the effect of weak measurements are generally interpreted [27].
Another way to tackle this question, that we examined very recently [28], is to associate a measurement outcome with the existence of a stable material record. For practical or fundamental reasons, on which we will not speculate here, a stable material record is immune to linear superpositions (in Feynman's words, amplitudes are lost and play no more role [29]).
We have argued [28] that the material record includes F's memory (or any printout that would record the measurement), so that if the stable records produced by F's measurement could be erased (through unitary evolution) by W's subsequent measurement, there is no way to ensure F's measurement has even existed. This is precisely what the weakly coupled probes indicate when unitary evolution is assumed. Hence within this perspective, the question is not whether an ideally sealed laboratory evolves unitarily, but whether a noninvasive interaction can prevent the formation of a stable material record following F's measurement inside L (admittedly, this sounds quite unlikely given the properties of weak measurements).

C. Measurements: Quantum Mechanics is consistent in its inconsistency
Wigner Friend scenarios are at the crossroad of two distinct but related problems: what constitutes a measurement, and whether quantum mechanics holds in the macroscopic limit.
Any account of the WFS and the problems that it might raise will hinge on one's stance on these issues, which to a large extent depends on the interpretation one has in mind (this includes a set assumptions some of which might be clearly expressed explicitly, along with others that might be fuzzily implicitly presupposed).
Wigner, shortly after he published the WFS argument, endorsed more clearly the idea that quantum mechanics applied to macroscopic bodies but not to conscious observers, for whom unitary evolution "is not credible" (Sec. 12 of [5]). Obviously, this position conflicts with the idea that quantum mechanics should apply in a "universal" manner to macroscopic systems, the assumption that is at the root of the WFS and more particularly in the EWFS of Refs [7,8]: the agents W i would discard that the Friends evolved unitarily once they completed their measurements (and therefore no contradiction will be obtained between the agents).
Purely unitary approaches (such as the Many-Worlds Interpretation) uphold that the projection postulate makes quantum theory non-universal and logically inconsistent [2]. In such approaches, whether the Friends and the agents W evolve or not in different worlds, observing or not different facts and how different facts can be reconciled when all the agents meet depends on subtle assumptions concerning the branching mechanism (for a recent discussion detailing on how branching issues in EPR or GHZ setups might lead agents evolving independently to reconcile their viewpoints, see [30]).
In most other approaches, the projection postulate for all the agents must be implicitly or explicitly employed. It is then straightforward to see [10] that, while the application of the projection postulate remains ambiguous, all the agents involved in the EWFS can agree beforehand on how to apply it and avoid reaching contradictions. It has also been argued [16] that the sole unitary coupling is insufficient to define a measurement -unitary evolution leads at best to the premeasurement step of the measurement process, but this does not suffice in order to obtain recorded outcomes. Then all the agents will agree on their measurements. Note that the collapse can be implicit, like when taking the partial trace over the environment in a decoherence based approach (a detailed decoherence based account of the WFS and EWFS has recently been given [15]), or in the projectors defining consistent histories (such an approach for the EWFS has also been given [13]).
For approaches in which the state vector is considered to embody an objective physical reality, global collapse upon measurement for all the agents is the only option. This is obvious in dynamical collapse theories [31] in which a non-linear term is added to to the Schrödinger equation, preventing unitary evolution. While we do not know of any detailed rendering of the WFS with dynamical collapse, once the measurement is completed our results assuming global collapse in Sec. III will be recovered. In the de Broglie-Bohm approach, the dual particle-wave ontology renders the effective collapse somewhat less obvious, as the particle is detected but the wavefunction -assumed to be unique for all the agents -presents (in most versions of this approach) empty branches that still evolve, creating an ambiguity over when they remain dynamically relevant [32,33]. While slightly different, the recent de Broglie-Bohm accounts of the WFS and EWFS [9,11,12] all agree that particle detection by the Friends changes the Bohmian particle configurations and trajectories for the external agents W, so that no contradiction is obtained.
The conclusion we can draw is that Quantum Mechanics is consistent in its inconsistency: even if we have good reasons to believe that quantum mechanics should apply to macroscopic bodies, a special rule, involving a global effective collapse relying on an ambiguous "shifty split" [34] remains necessary. Quantum measurements are special and the theoretical accounts remain unsatisfactory, if not logically inconsistent [2,34], but operationally there is no inconsistency in accounting for the observed outcomes.

V. CONCLUSION
In this paper, we have investigated Wigner Friend scenarios in which the external agents make non-invasive measurements inside the otherwise sealed laboratories. In the original WFS, an external agent W who would want to test the Friend's unitary evolution would not be able to infer anything by directly measuring the quantum states of the macroscopic laboratory. With a non-invasive probe weakly coupled to the Friend's atom or measuring apparatus -with a coupling so weak that it does not affect the outcomes nor their probabilities inside the laboratory -W will be able to infer the state of the laboratory after the Friend's measurement. Within the limitations of our scheme -in which the Friend's laboratory cannot said to be perflectly closed -unitary evolution can be empirically tested, and probably invalidated, since if this hypothesis is assumed, we are lead to a contradiction between the Friend having obtained a definite outcome and the state of the probe.
For the extended version of the WFS recently introduced by Brukner [7] and Frauchiger and Renner [8], we have shown that the non-invasive probes given to each external agent would be entangled if the hypothesis held. But we have seen that if the Friends obtained definite outcomes, the probes cannot be entangled (this forms the basis of the Bell-type inequality that leads to the contradiction put forward in Ref [8]). Again, in our framework with non-invasive probes, this becomes a question that can be given an experimental confirmation.
To conclude, we have investigated Wigner Friend scenarios with non-invasive probes.
While there is room to support the idea that the measurement problem makes quantum theory logically inconsistent, we can rely on the non-invasive probes to avoid inconsistency at the operational level. This approach could moreover be confirmed by experiments.