Weak versus Deterministic Macroscopic Realism, and Einstein–Podolsky–Rosen’s Elements of Reality

The violation of a Leggett–Garg inequality confirms the incompatibility between quantum mechanics and the combined premises (called macro-realism) of macroscopic realism (MR) and noninvasive measurability (NIM). Arguments can be given that the incompatibility arises because MR fails for systems in a superposition of macroscopically distinct states—or else, that NIM fails. In this paper, we consider a strong negation of macro-realism, involving superpositions of coherent states, where the NIM premise is replaced by Bell’s locality premise. We follow recent work and propose the validity of a subset of Einstein–Podolsky–Rosen (EPR) and Leggett–Garg premises, referred to as weak macroscopic realism (wMR). In finding consistency with wMR, we identify that the Leggett–Garg inequalities are violated because of failure of both MR and NIM, but also that both are valid in a weaker (less restrictive) sense. Weak MR is distinguished from deterministic macroscopic realism (dMR) by recognizing that a measurement involves a reversible unitary interaction that establishes the measurement setting. Weak MR posits that a predetermined value for the outcome of a measurement can be attributed to the system after the interaction, when the measurement setting is experimentally specified. An extended definition of wMR considers the “element of reality” defined by EPR for system A, where one can predict with certainty the outcome of a measurement on A by performing a measurement on system B. Weak MR posits that this element of reality exists once the unitary interaction determining the measurement setting at B has occurred. We demonstrate compatibility of systems violating Leggett–Garg inequalities with wMR but point out that dMR has been shown to be falsifiable. Other tests of wMR are proposed, the predictions of wMR agreeing with quantum mechanics. Finally, we compare wMR with macro-realism models discussed elsewhere. An argument in favour of wMR is presented: wMR resolves a potential contradiction pointed out by Leggett and Garg between failure of macro-realism and assumptions intrinsic to quantum measurement theory.


I. INTRODUCTION
The interpretation of the quantum superposition of two macroscopically distinguishable states has been a topic of interest for decades [1][2][3][4][5][6][7][8][9].Schrödinger considered a superposition |ψ M ⟩ = 1 √ 2 (|a⟩ + |d⟩) where |a⟩ and |d⟩ are macroscopically distinct quantum states, giving outcomes a and d for a measurement M [1].The outcomes are associated with macroscopically distinct physical properties, analogous to a cat alive or dead.Schrödinger explained how the standard interpretation given to a quantum superposition introduces a paradox when applied to the macroscopic system.The system is interpreted as being in neither state |a⟩ or |d⟩ prior to measurement M , suggesting it is somehow simultaneously in both states − which would be "ridiculous" [1].
Leggett and Garg proposed concrete tests of macroscopic realism versus quantum mechanics [10].They introduced macroscopic realism (MR) as the premise that "a system with two macroscopically distinct states available to it will at all times be in one or other of those states".Specifically, they assumed the system to be described by a hidden variable λ M , which takes the value +1 or −1 depending on which of the two states the system is in.The variable is hidden, because quantum mechanics does not give such an interpretation for the state |ψ M ⟩.In the work of Leggett and Garg, the variable λ M specifies the outcome (a or d) for the measurement M , but not the outcomes for other more microscopic measurements.
Hence, macroscopic realism (MR) posits that the system is in a state with a predetermined value λ M for the measurement M , but does not imply the stronger assumption that the system is in one or other of any quantum state, prior to measurement M .In order to test MR, Leggett and Garg introduced the additional assumption of macroscopic noninvasive measurability (NIM).This assumption however is generally challenging to justify [11][12][13][14][15][16].The combined assumptions of MR and NIM are referred to as macrorealism.Assuming macrorealism, Leggett and Garg derived inequalities, which are predicted by quantum mechanics to be violated for certain dynamically evolving systems involving macroscopic superposition states.
There have been many predictions and demonstrations of violation of the Leggett-Garg inequalities [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].However, many of these tests rely on microscopic realisations, which therefore involve the stronger assumption of microscopic realism and cannot test macroscopic realism.The argument can be put forward that the violation of the Leggett-Garg inequalities is due to failure of realism at a microscopic (not macroscopic) level.Macroscopic tests exist [12,13,16,22,23], but these are susceptible to the criticism that the local measurements used are in fact invasive [30], or else involve auxiliary assumptions e.g. that the "states" the system is in, according to MR, can be prepared in the laboratory, and are therefore describable as quantum states [13].
Motivated by the need to rigorously test macroscopic arXiv:2101.09476v2[quant-ph] 25 Aug 2023 realism, we examine in this paper a recently proposed test of macrorealism involving superpositions of coherent states, namely, entangled cat states [32,33].Here, a measurement of the sign Ŝ of a quadrature phase amplitude X distinguishes between two coherent states |α⟩ and |−α⟩ where α → ∞, and macroscopic realism implies the outcome of Ŝ to be predetermined as either positive or negative.In this proposal, the question of there being an invasive measurement is partly resolved, because the measurement of Ŝ is made by a spatially separated system B, so that the NIM premise is justified by the Bell's assumption of locality [26,[34][35][36][37][38].The proposal may also be regarded as a macroscopic Bell test, in which a Bell inequality involving the hidden variables λ M is violated.Such tests violate Bell inequalities for macroscopically coarse-grained measurements, where it is not necessary to fully resolve the amplitude X.While other macroscopic Bell-nonlocality tests have been put forward [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58], the distinction here is that macroscopically distinct states can be readily identified that allow application of Leggett-Garg's definition of macroscopic realism (MR).Similar tests involving coarse-grained measurements have been proposed [33,[59][60][61][62].
In this paper, our motivation is to examine whether and how it is possible to obtain consistency with MR, despite that the Leggett-Garg-Bell inequalities are violated for a macroscopic proposal with a rigorous justification of non-invasive measurability.In the proposal, it is clear that the dynamics U θ associated with the choice of measurement setting θ in the Bell experiment plays a key role in understanding how it is possible to find consistency with MR.The work of Thenabadu and Reid [32] has pointed out that MR can hold for Bell violations, if defined appropriately to take into account this dynamics.
Two definitions of MR exist.The definitions depend on whether the term "measurement" includes the unitary interaction U θ that determines the measurement setting or not.Deterministic macroscopic realism (dMR) posits that the value for the outcome of the measurement M ≡ Ŝ is specified for the system prior to the entire measurement process, including that determining the measurement setting.Here, we consider that the macroscopically distinct states giving a definite outcome for Ŝθ can be identified for the system prior to the entire measurement process, including U θ .The analogy is in classical mechanics, where states with a definite x and p are defined for the system at any point of time, prior to measurement of either.It has been shown that dMR is falsifiable, according to quantum predictions [32,33].
On the other hand, weak macroscopic realism (wMR) is a set of weaker assumptions that are not negated by the macroscopic Bell inequalities [32].The premise of wMR posits that [32]: (1) The predetermined value λ M describes the system as it exists after the unitary dynamics U θ , when the measurement setting is established in the experiment.This means the predetermination is given for the system prepared with respect to the measurement basis only.It is assumed also that: (2) The value λ M is not changed retrocausally by any future measurement or interaction, and is not changed by any spacelike-separated interactions U ϕ or events that might occur at a spatially separated site.
Recently, Fulton et al [63] have extended the definition of wMR to the situation of the Einstein, Podolsky and Rosen (EPR) paradox [64], where one can predict with certainty the result of a measurement at A by making a measurement at a space-like separated system B. Here, wMR posits that: (3) The value for the outcome Ŝθ at A is specified by an "element of reality" given by the hidden variable λ Fulton et al point out that wMR can be generalised to apply to the standard microscopic Bell systems, in which case the premise is referred to as weak local realism (wLR) [63].This premise is not negated by Greenberger-Horne-Zeilinger (GHZ) experiments [65], and is not inconsistent with the EPR-Bohm paradox [66].The EPR paradoxes and GHZ contradictions arise from the strong version of local realism, where the predetermination is prior to the unitary interactions, U (A) θ and U (B) ϕ [63].Joseph et al have applied the wMR and wLR premises to examine realism in Wigner's friend paradoxes [67].
In this paper, we demonstrate compatibility of the macroscopic Leggett-Garg-Bell test proposed in [32], with the extended wMR premises.In Sections II-V, we review the arguments put forward in [32], and then focus on the wMR premise (3) involving the assumption of an "element of reality", showing consistency with that premise.Four tests of wMR are proposed, all of which show consistency between wMR and quantum predictions.In Section VI, we explain how the wMR premise can be implemented in quantum measurement theory, where a meter is coupled to a system to allow a readout of a final value.It was raised by Leggett and Garg that a system violating macrorealism may not be a good measurement device [10].Extending arguments put forward in [32], we show how the premise of wMR resolves this paradox of quantum measurement.We also explain in Section VII how the premise is connected to models of MR put forward by Maroney and Timpson [30,31].

II. CAT STATES AND WEAK MACROSCOPIC REALISM
We begin by considering the Schrödinger cat state [5,6,68,69]  of a single-mode field A.Here | ± α⟩ are coherent states with α large and real.These states becomes macroscopically distinguishable in phase space for large α, in analogy with the "alive and dead" states, |a⟩ and |d⟩, of the "cat".Quadrature phase amplitude measurements ) are defined (in a rotating frame) where â † , â are mode boson operators (ℏ = 1) [5].The states | ± α⟩ can be distinguished by the measurement M given as Ŝ(A) , which has a value +1 if the outcome of XA is positive, and −1 otherwise.The outcomes +1 and −1 are analogous to the spin outcomes in a Bell experiment.We refer to Ŝ(A) as the pointer measurement, since the value for the outcome can be amplified as in a homodyne measurement scheme, to give a readout on a macroscopic meter.Here, we say that the system in the state |ψ M ⟩ has been prepared in a superposition of pointer eigenstates, where the measurement basis is for the pointer measurement Ŝ(A) , since α real.The coherent states become effective pointer eigenstates for large α, since the outcome for Ŝ(A) is given as 1 or −1 for |α⟩ and | − α⟩ respectively.The state |ψ M ⟩ is prepared for the pointer measurement Ŝ(A) (or XA ), the measurement basis being eigenstates of Ŝ(A) (or XA ).
Before examining whether wMR can be compatible with quantum mechanics, we present an argument against wMR [32].This concerns the apparent incompleteness of quantum mechanics.Weak macroscopic realism postulates the system to have a definite "spin" outcome, +1 or −1, for Ŝ(A) , which implies it must be in a state with a sufficiently localised outcome X A , for XA .If the state is to be a quantum state, we arrive at a constraint on the outcomes P A of measurement PA .The distribution for X A gives two Gaussian hills each with variance 1/2 (Figure 1).Supposing the system to be in a classical mixture of two states, one for each Gaussian, in accordance with wMR, then for each we specify ∆ 2 X A = 1/2.If the two states are quantum states, then the uncertainty relation ∆X A ∆P A ≥ 1/2 for each implies that the overall variance in P A satisfies ∆ 2 P A ≥ 1/2 [32].The observation of ∆ 2 P A < 1/2 leads to an EPR-type paradox, where, since the state consistent with wMR cannot also be consistent with the uncertainty principle, one argues either failure of wMR or else an incompleteness of quantum mechanics.For the cat state (1), a fringe distribution is observed for PA (Figure 1), and The paradox is obtained for all α, albeit by a vanishingly small amount for larger α [32,[70][71][72].Similar paradoxes include [73,74] but are not so directly based on wMR.The apparent inconsistency between the macroscopic quantum state and the completeness of quantum mechanics was raised by Schrödinger in his famous essay [1].
While the original EPR paradox revealed inconsistency between local realism and the completeness of quantum mechanics [64], Bell later proved local realism could be negated [34], giving a resolution of the paradox.The above argument however is based on weak macroscopic realism (wMR), which motivates the question of whether wMR can also be negated.

III. A STRONG TEST OF MACROREALISM USING ENTANGLED CAT STATES
To examine this question, we first follow [23] to demonstrate how Leggett and Garg's macrorealism [10] can be violated for the cat state (1).Macrorealism is defined as the combined assumptions of wMR and noninvasive measurability (NIM) − that one may determine the value of λ M without a subsequent macroscopic disturbance to the future dynamics [10].
At time t 1 = 0, we consider that system A is prepared in |α⟩.The system then evolves according to the nonlinear Hamiltonian H N L = Ωn 4 where Ω is a constant and n = â † â.After a time t 2 = π/4Ω, it can be shown that the system is in the state [23,32] where we write U π/8 = U A (t 2 ) = e −iH (A) N L t2/ℏ .After further evolution, at time t 3 = π/2Ω, the system is in the cat state where U π/4 = U A (t 3 ).At each time t i , we define S i to be the outcome of the measurement ŜA) .Assuming the system satisfies wMR, the value for S i is determined by a macroscopic hidden variable λ M , which we denote by λ i , with values +1 and −1.Algebra reveals that ⟨λ 1 λ 2 ⟩ − ⟨λ 1 λ 3 ⟩ + ⟨λ 2 λ 3 ⟩ ≤ 1 [10,17].The two-time correlations are given as ⟨S i S j ⟩ = ⟨λ i λ j ⟩.The assumption NIM implies these could be measured, since an ideal measurement of S i at times t i determines the value of λ i without subsequent disturbance to the system.Macrorealism therefore implies the Leggett-Garg inequality [10,17] Quantum mechanics predicts ⟨S 1 S 2 ⟩ = cos(π/4) and ⟨S 1 S 3 ⟩ = 0, since the outcome for S 1 is known to be 1 from preparation.Establishing ⟨S 2 S 3 ⟩ is not so clear, because one may argue that a realistic measurement at time t 2 will affect the future dynamics.However, assuming the system is actually in one of the states |α⟩ or | − α⟩ at t 2 , the system at the later time t 3 will evolve to U π/4 |α⟩ or U π/4 | − α⟩ [10].This implies ⟨S 2 S 3 ⟩ = cos(π/4).The inequality ( 5) is violated, the left side being √ 2. Of course, one sees from the paradox (2) that the system cannot actually quite be in either state |α⟩ or | − α⟩ at time t 2 , prior to measurement.
The failure of macrorealism is demonstrated more convincingly, if one is able to perform the measurement at the time t 2 without direct disturbance.To this end, we note the mapping that leads to the proposal of a macroscopic version of the Bell experiment [32].As α → ∞, |α⟩ and | − α⟩ are orthogonal, and we map the system onto spin-qubits | ↑⟩ and | ↓⟩, defined as eigenstates of Pauli spin σ(A) z .The rotations U π/8 , U π/4 and U 3π/8 (defined below) become precisely the spin rotations required in the Bell experiments, realised by Stern-Gerlach analysers or polarising beam splitters [34][35][36].
Consider two space-like separated systems A and B prepared at time t 1 = 0 in the state [8,32] where |β⟩ B is a coherent state for system B, N = 2 )} −1/2 and we will take α = β with α real.We define the operators X, P , Ŝ, n, H N L , U π/8 , U π/4 and hidden variables λ i as above for each system A and B, denoting by a superscript A or B in each case.The systems A and B evolve independently for times t a and t b , respectively, according to local interaction Hamiltonians If both systems evolve for a time t 3 = π/2Ω, the system is in the similar Bell state π/4 |ψ Bell ⟩.The premise wMR assigns to A and B after interaction times t a = t i and t b = t j (i, j = 1, 2 or 3) the macroscopic hidden variables λ i .It is argued that this measurement is noninvasive to the system A, based on the assumption of macroscopic Bell locality (ML).ML asserts that for spacelike-separated events at A and B, the events at B cannot change the value of the hidden variable λ (A) M at A, and vice versa.This is assumed for all events over the time interval t 1 to t 3 , implying no macroscopic changes to the outcomes at A at any time t i due to measurement at B [75].Assuming macrorealism, the inequality (5) becomes the Bell inequality [34] −⟨S The predictions based on the measurements XA and XB are calculated by evaluating P (X A , X B ) (Figure 2).
Where α, β > 1, the predictions for −⟨S (B) i S (A) j ⟩ are indistinguishable from those of ⟨S i S j ⟩ given above for the predictions of the Leggett-Garg inequality (5).Violation of ( 8) is predicted, as for ( 5), the left side being √ 2. The violations are valid for arbitrarily large α, β and falsify the combined assumptions of wMR and ML.Hence, one cannot conclude violation of wMR directly.

IV. FALSIFYING DETERMINISTIC MACROSCOPIC REALISM
However, one may falsify deterministic macroscopic realism (dMR).The inequality given by ( 8) is seen to be a macroscopic version of Bell's original inequality [34,35], applied to macroscopic spin observables Ŝ(A) j and Ŝ(B) j .The choice between two times of evolution for each system A and B (e.g.t 2 and t 3 for A, and t 1 and t 2 for B) corresponds to a choice between two measurement settings (e.g.θ 2 and θ 3 for A, and ϕ 1 and ϕ 2 for B).This choice of unitary rotation t j maps in the microscopic Bell experiment to a choice of analyzer setting θ j .The Bell inequality (8) can be derived assuming deterministic macroscopic realism (dMR) [32]: each system A and B is simultaneously predetermined to be in one or other of two macroscopically distinct states, prior to the choice of measurement setting, so that two macroscopic hidden variables (e.g λ for B) are ascribed to each system at the time t 1 .This assumption naturally incorporates ML, since it is specified that λ j cannot change over the course of the unitary dynamics associated with the adjustment of measurement setting, at either site.Violation of (8) falsifies dMR.
We may also consider evolution of (6) for the time t 4 = 3π/4Ω, in which case the evolved state is 3π/8 |α⟩ = e i3π/8 cos 3π/8|α⟩ + i sin 3π/8| − α⟩ (9) This allows evaluation of the familiar Clauser-Horne-Shimony-Holt Bell inequality [35,36] 4 ⟩| ≤ 2 (10) which can also be derived from dMR.A violation is predicted, the left side being 2 √ 2. The system prepared in the Bell state (6) thus evolves after a time t 4 to the Bell state , and a violation of (10), the left side being 2 √ 2. Eq. ( 10) can be viewed as the Leggett-Garg inequality (11) derived in [10] .Similar to (8), to obtain (10) we justify the NIM premise using ML, and put S (B) i = −S (A) i for times t 1 and t 3 , based on the anti-correlation of the spins for the Bell states.Alternatively, Eq. ( 10) is seen to be a macroscopic Bell inequality, where one measures the correlation

V. FINDING CONSISTENCY OF THE LEGGETT-GARG-BELL VIOLATIONS WITH WEAK MACROSCOPIC REALISM
We now ask whether one can reconcile the violations of macrorealism and deterministic macroscopic realism (dMR) with the validity of weak macroscopic realism (wMR).For the Leggett-Garg-Bell tests, it is clear that the systems are indeed prepared in the pointer superposition |ψ Bell ⟩ i at the time t i .If weak macroscopic realism (wMR) holds, then the value of the S (A/B) i are given by λ (A/B) i at the time t i , in which case the violations must arise because the non-invasive measurability (NIM), as justified by locality (ML), premise breaks down.Consistency with wMR is possible, because the unitary dynamics (which in the Bell test gives the choice of measurement setting) has a finite time duration.This is evident in the Figure 2, which plots the dynamics given by U .The dynamics transforms the Bell state |ψ Bell ⟩ 1 prepared in the pointer basis of σz at time t 1 = 0, into a different Bell state |ψ Bell ⟩ 2 at time t 2 = π/4 (prepared with respect to different basis ), and then into a different state |ψ Bell ⟩ 3 at t 3 = π/2 (prepared in the basis of σy ).The system given by the state |ψ Bell ⟩ 1 is not viewed to be simultaneously in all three pointer superpositions.One is therefore able to postulate wMR, without requiring to assume dMR, which fails, by violation of ( 5), ( 8) and (10).
Examination of the dynamics associated with the measurement settings for the Leggett-Garg-Bell tests reveals features consistent with wMR.We first summarise two tests that were explained in [32,85].

A. Test 1: Unitary rotations are required at both sites to display the macroscopic nonlocality
Any theory for which wMR is valid predicts that it is the dynamics involving a unitary rotation at both sites that yields the violation of the inequalities ( 5) and (8).A similar analysis holds for the violation of (10).To show this, we examine the top sequence of Figure 2. The system is prepared in the pointer-measurement basis at time t 1 .A unitary rotation giving a change of measurement basis then takes place at A but not B. According to wMR, the system at time t 2 given by snapshot (π/4, 0) can be specified by two variables λ of measurement Ŝ(A) at time t 1 (given by (0, 0)).The outcome for S (A) 1 can be determined at time t 2 without further unitary rotation, because this is given by the  2, we see that there is no visual difference between the plots at the initial time t = 0.The top sequence involving a rotation at one site only remains visually indistinguishable from that of the entangled state shown in Figure 2.However, we see that the final state of the lower sequence involving a change of basis at each site becomes macroscopically different.pointer measurement at B at time t 2 .Hence, wMR (3) applies.We obtain a description for measurements made at times t 1 and t 2 that is consistent with macrorealism.
By contrast, the lower sequence of Figure 2 has rotations giving a change of measurement basis for both A and B. For the system given by (π/2, π/4) at time t 3 , wMR asserts validity of variables λ cannot be specified simultaneously for system A at the times t i that allows violation of the Leggett-Garg inequality.This is not possible, because for the bipartite system it is only possible to prepare the systems in pointer bases for two measurements simultaneously (one at each site).
Test 1 provides a way to test wMR.The dynamics for the mixed state for which a macrorealistic model holds, can be experimentally compared with that of |ψ Bell ⟩ 1 .Here, |α⟩ and |β⟩are coherent states for systems A and B, and we take α = β.Weak macroscopic realism predicts that the dynamics between the two will diverge where there are unitary rotations at both sites.According to wMR, a violation would not arise where there are rotations at single sites only.Quantum mechanics predicts for such an experiment consistency with wMR. Figure 3 shows the dynamics of the measurements required to test the Leggett-Garg-Bell inequality for the system prepared initially at time t 1 in ρ mix .The difference between the plots is of order e −α 2 which vanishes for the macroscopic case, where α = β → ∞[? ].By contrast, for the lower sequences where there is a rotation (change of basis) at both sites, the P (X A , X B ), while indistinguishable at t 1 = 0, become macroscopically different at the later times t 2 and t 3 .This is seen when comparing the final plots of the lower sequences: the contour plot for the evolution of ρ (AB) mix (Figure 3) is clearly different to that of |ψ Bell ⟩ 1 (Figure 2).
In a model where wMR is valid, it is the dynamics that occurs over the time intervals of the combination of both unitary rotations involving a change of measurement basis at each site that results in the violation of the macroscopic Leggett-Garg-Bell inequalities ( 8) and ( 10).This is consistent with calculations for violations of the Bell inequalities for microscopic spin Bell states, where it is well known that the quantum interference arising from a nonzero angles θ and ϕ is necessary to create the violation of the inequality (10).

B. Test 2: Delaying the collapse stage of the measurement makes no difference
The second test of wMR concerns the timing of the irreversible "collapse" stage of measurement, when the system s coupled to a detector to read out the value of X.We address how the results are affected by the irreversible "collapse" stage of measurement, when the system B is coupled to a detector.The unitary evolution U , which precedes the collapse stage of the measurement, prepares the system for the pointer measurement at the time t i , by establishing the measurement setting (i.e.measurement basis).To calculate the measurable probabilities, we note that in quantum mechanics, the state for the system is written as a superposition of pointer eigenstates.
In a model where wMR is valid, the hidden variable λ (A) i is fixed in value (+1 or −1) at the time t i , after U (A) .This gives a record of λ (A) i at the time t i − which cannot be changed by the future collapse at B, nor by a future unitary evolution at A.
Quantum mechanics predicts consistency with wMR: It is possible to delay the collapse stage of the measurement Ŝ(B) j at B by any time after the measurement at A i.e. after t 3 , and it makes no detectable difference to P (X A , X B ), any corrections being of order e −|α| 2 [32].As above, the macroscopic nonclassical effects only arise where there is unitary rotation at both sites.

C. Test 3: Considering delayed-choice, no-retrocausality implies extra dimensions
The premise of wMR specifies a given value λ i for the outcome of the measurement Ŝi at the given time t i .This cannot be changed by any future event.One might hence expect that delayed-choice experiments would falsify wMR.
We note that the joint probabilities P (X A , X B ) depend on the local interaction times t a and t b , not the relative timing.Hence, one can delay the choice t b to measure until after the final detection at system A, at time t 3 .This might suggest therefore that the measurement at B is noninvasive of the dynamics at A, and hence that violation of macrorealism is due to failure of wMR.However, as with delayed-choice experiments for spin-qubits [76,[78][79][80][81], this interpretation can be countered [82]: Analysis reveals unitary evolution U occurs at both sites after time t 2 : the violation of macrorealism can be then explained by failure of dMR [32,85].
On the other hand, the delayed-choice Wheeler-Chaves-Lemos-Pienaar experiment [82][83][84] falsifies all two-dimensional non-retrocausal models for a two-state system, described by qubits {| ↑⟩, | ↓⟩}.Using the mapping (3) onto macroscopic qubits {|α⟩, | − α⟩} and generalising to rotations U (A) θ where θ is a multiple of π/8, one may falsify all two-dimensional non-retrocausal models based on the macroscopic qubits [85].This contradicts wMR − but only if we restrict to two-dimensional models.We avoid conclusions of retrocausality however, by noting the extra dimensions associated with the continuous-variable phase-space representation of the cat-states, which are measurable.This is explained in [85].

D. Test 4: EPR's elements of reality are justified after the setting dynamics
We now consider the postulate (3) of wMR in the setup of the EPR experiment.This extends the earlier work of [32].
Examining the state (6), we see that the Bohm-EPR paradox for spin applies.At the given time t i , the outcome of the measurement Ŝ(A) i at A can be predicted with certainty by the measurement of Ŝ(B) i on system B. We see that S .EPR's original premises posit that there exists an element of reality λ (A) i for system A at this time [64].This value predetermines the outcome of the measurement S (A) i if measured directly at A, regardless of whether the measurement at B is performed or not, because the outcome at A can be predicted in principle by establishing the measurement at B and nothing at B can influence the system at A, according to locality.The value for the element of reality is λ i , and can be determined by finalizing the measurement at B i.e. making readout at B.
The original EPR premises can be falsified, because the assumption of the λ (A) i can be applied to (noncommuting) measurements at the different times, t i and t j , as explained in Section IV.Hence, the EPR premises leads to the premise of dMR which is falsified by the Bell test violating inequalities (8) or (10) [34].
However, the weaker postulate of wMR ( 3) is not falsified, because it refers to the system B at the time t i after any appropriate unitary interaction U (B) has taken place at B, to finalise the measurement setting at B i.e. to prepare the system for the final pointer measurement Ŝ(B) .This suggests that the "elements of reality" do apply, in certain circumstances.As explained by Clauser and Shimony [36], the importance of the dynamics associated with the choice of measurement setting was commented on by Bohr, in his reply to Einstein, Podolsky and Rosen [86].
Figure 2 depicts the dynamics showing consistency of the quantum predictions with the premise wMR (3).At the time t 1 = t 0 = 0, the system B has been prepared for the final detection and readout of spin Ŝ(B) 1 , which takes place by the measurement XB and a detection.The wMR postulate is that the system B has the predetermined value λ B i for the outcome of that measurement.This also gives the value for the measurement Ŝ(A) 1 at A, regardless of any further unitary interactions U can be obtained by a final measurement readout at the space-like-separated system B.This gives the prediction for Ŝ(A) 1 .The value for Ŝ(A) 1 can be confirmed correct, both without and then with the unitary rotation U (A) π/4 followed by its reversal.

VI. WEAK MACROSCOPIC REALISM, WEAK LOCAL REALISM AND QUANTUM MEASUREMENT
The wMR-model can elucidate the nature of the measurement.This can be put forward as an argument in favour of wMR.A fundamental question is how to understand the connection between "realism" and the states such as formed at a time t k after a macroscopic measurement device interacts with a microsystem A prepared in Here, |β⟩ and | − β⟩ are coherent states.The readout of Ŝ(B) gives the measured value of σ(A) z .A model for an interaction which evolves ( 15) into ( 14) has been presented [87][88][89].In that model, the meter system is prepared initially in a coherent state |γ⟩.The phase of β is determined by the phase γ of the initial coherent state.
A fundamental question arises: At what point in the measurement process does the value emerge?A deeper question is: How is realism connected to measurement [90,91]?

A. Weak local realism
It becomes apparent that the wMR premises can also be applied to spin systems such as {| ↑⟩, | ↓⟩}, even though the states are not macroscopically distinct.This is because the systems violating Bell and Leggett-Garg inequalities for {| ↑⟩, | ↓⟩} are a direct mapping of those considered in this paper for {|α⟩, | − α⟩}.In this case, we refer to the premises of weak macroscopic realism as simply weak local realism (wLR).An explanation has been given in [63].These premises are weaker (less restrictive) than those of local realism defined by Bell and are not negated by violations of Bell inequalities, as we have seen for wMR.We note in view of Section V.C, this suggests a model in which the spin states be completed by extra dimensions.

B. Schrödinger's paradox
Understanding the entangled state ( 14) was the paradox put forward by Schrödinger in his essay [1].It is often supposed that the value of the outcome is not determined prior to measurement, but in the wMR and wLR models, this is overstated.In these models, the value for the outcome of spin σ(A) z of A is specified at or by the time t k of the creation of the entangled systemmeter state (14), since the measurement basis (setting) for the meter is specified by the interaction, through the phase of the coherent field.Only direct amplification and detection of XB of the meter is required to complete the measurement.Hence, according to wMR, there is a predetermined value λ M for the macroscopic meter B at this time.According to wMR (3), this value is an "element of reality" for the outcome of σ(A) z at A, since it gives the value if it were to be measured directly.Hence, in the wMR and wLR models, the value for the outcome of the measurement can be assigned to the system A at time t k , prior to the final detection and readout, since the measurement setting for A has been established.
A local unitary interaction at A can be further applied, to change the measurement setting for the spin measurement at A. However, we have seen that in the wMR-wLR models, this makes no difference to the outcome λ to actually be measured, a further unitary interaction giving a reversal then takes place.
Similarly, if a local unitary interaction at B is implemented, while keeping the system A unchanged, it does not change the element of reality for the system B that is implied by the fact the outcome XB can be inferred by the spin measurement at A. On the other hand, if unitary interactions are implemented to change the measurement settings at both A and B, then in the wMR-wLR models, we can no longer suppose that the value of λ M applies to a future measurement.

C. Leggett and Garg's paradox
In their paper [10], Leggett and Garg consider states such as (14).They explain that the violations of macrorealism should "not be formally in conflict with the arguments so often given in discussions of the quantum theory of measurement to the effect that once a microsystem has interacted with a realistic measuring device, the device (and, if necessary, the microsystem) behave as if it were in a definite (and noninvasively measurable) macroscopic state".
They also suggest that a system |ψ M ⟩ if violating macrorealism would not be a suitable measuring device, by continuing: "The macroscopic systems suitable for a macroscopic quantum coherence experiment are certainly not able to be measuring devices, at least under the conditions specified.But such a result might cause us to think a great deal harder about the significance of "as if"!" We extend the analysis of the statements of Leggett and Garg for this system, given in [32].We examine the first statement of Leggett and Garg.The premise of wMR does imply a definite "state" for the device, given by system B in (14), in the sense that there is a predetermination of the outcome of Ŝ(B) .This is because the measurement setting Ŝ(B) is specified by the phase of the coherent-state amplitude β.
Assuming wLR, the microsystem A also has a definite value for the outcome of σ(A) z − but only when prepared (after the choice of measurement setting) in a superposition with respect to the pointer bases of σ(A) z and Ŝ(B) .When we write the original state |ψ A ⟩, it is not specified whether or not the measurement basis has been determined experimentally.However, we see as explained above that once entangled with the meter as in the state (14), there is a definite value for the outcome of σ(A) z .This is because the measurement setting for the microsystem is specified.
Hence, there is no direct conflict with the statement "that the device behaves as if it were in a definite macroscopic state".The basis for the spin at A is determined fixed as σ(A) z because the coherent states that act as the meter (when XB is measured) have a definite fixed phase, and no further rotation U (B) is necessary.The value λ M .According to the analysis of the previous section, there is an element of reality λ M for the result of spin σ(A) z of A, for the system in the entangled meter-system state, and this value is not changed by any further unitary interactions that may occur at system A. Now we turn to examine Leggett and Garg's second statement.By mapping {| − α⟩, |α⟩} onto {| ↑⟩, | ↓⟩} for (6) in Section III, we see that macrorealism is indeed violated for the macroscopic measuring device B. Yet, contrary to what may be suggested by Leggett and Garg's statements, for theories where wMR (or wLR) is valid, we have argued that there is no conflict with the arguments of quantum measurement theory.This is because system B has a definite value λ (B) M for the outcome of Ŝ(B) .The system A of |ψ M ⟩ also has a definite value λ (A) M for the outcome of σz when prepared in (14).
In short, contrary to what might be supposed, the argument that the systems ( 14) can be considered to have a definite real property λ (B)

M and λ (A)
M does not contradict the Bell violations (for example, of (10)), since we have shown consistency with wMR (and also with wLR) for such violations.The values however refer only to systems prepared appropriately at a given time in a superposition with respect to pointer-bases of σ(A) z and Ŝ(B) .Suppose one could specify that A (prior to the measurement interaction) were prepared appropriately in |ψ⟩ A of (15), for the pointer-basis of σ(A) z : According to wLR, the system A has a definite value λ (A) for the outcome of σ(A) z .Provided λ (A) = λ M , it can then be argued that the system |ψ M ⟩ is a suitable measuring device.
There is however a "conflict" as referred to by Leggett and Garg, who refer to a "definite macroscopic state".The conflict arises when we consider the consequence of the wMR and wLR assumptions, summarised in Section II, concerning the completeness of quantum mechanics: The systems cannot be considered to be in either quantum state | ↑⟩ A or | ↓⟩ A , or in |β⟩ or | − β⟩.If wMR is valid, it is unclear what 'state' each of the systems are in?An analysis of ontological states defining macroscopic realism has been given by Maroney and Timpson [30,31], which motivates the following section.

VII. COMPARISON WITH OTHER MODELS OF MACROSCOPIC REALISM
Our conclusions are consistent with those of Maroney [30] and Maroney and Timpson [31], who in analysing tests of macrorealism have argued that violations of the Leggett-Garg inequalities arise from a nonclassical form of measurement disturbance and do not necessarily imply failure of macroscopic realism.Maroney and Timpson considered three models of macroscopic realism, which they refer to as macrorealism models.First, they defined operational eigenstates of a property as "those preparations [of the system] which determinately fix the value of the property".In our context, these are preparations of the system for which there is a predetermined value for the outcome of the measurement Ŝθ .
The three models of macroscopic realism considered are: operational eigenstate mixture macrorealism (OEM-MR); operational eigenstate support macrorealism (OES-MR); and supra eigenstate support macrorealism (SES-MR).Maroney and Timpson argued that only OEM-MR gives the strict form of macrorealism that necessarily leads to the derivation of the Leggett-Garg inequality.We next examine each of these models for consistency with weak macroscopic realism as defined in this paper.

A. Operational eigenstate mixture macrorealism
The OEM-MR specifies that the system after preparation (after the unitary interactions U θ ) is in a mixture of the operational eigenstates.This model is negated by the violation of Leggett-Garg inequalities, and is compatible with wMR, but is a stronger model than required by wMR.
An example of an OEM-MR model is the mixed state (13), where the system A prior to the measurement Ŝ can be considered to be with some probability either in the state |α⟩ or | − α⟩.The coherent states become quantum eigenstates of Ŝ (for large α), and are also operational eigenstates.Here, Ŝ distinguishes between the two coherent states (for large α).The measurement can be shown to not change the system placed in one or other coherent states.
Maroney analyses the three-box paradox, where would imply that a ball placed in a superposition of being in one of three boxes is always actually in one or other box [30].A Condition (III) is satisfied that a measurement made on the system where a ball is placed in a box is confirmed to be non-disturbing to the state of the system.This confirms that the measurement is non-invasive for operational eigenstates.Maroney claims that "An intuition lurking alongside the idea that the ball is always in one, and only in one, of the boxes, is that whenever the ball is in a given box, it behaves exactly as it appears to behave when it is observed to be in that box.This runs into difficulties, for when the ball's location is observed, it is in an operational eigenstate.This rather natural idea of macrorealism would lead to operational eigenstate mixture macrorealism ..." Weak macroscopic realism (wMR) does not imply OEM-MR, since it is not assumed that the state of the system before and after the measurement are the same.This is evident from the analysis of Section II, where it is proved for the cat state (1) that, if wMR holds, the system prior to the measurement Ŝ cannot be in one or other quantum state that is an eigenstate of Ŝ.The states of the cat-system satisfying wMR are necessarily different before and after the measurement.

B. Operational eigenstate support macrorealism
The OES-MR and SES-MR models consider the system to be, prior to measurement Ŝ, in a mixture of ontic states which have definite predetermined values for the measurement Ŝ.For the three-box paradox, the measurement Ŝ corresponds to observing whether the ball is found in a given box.Operational eigenstate support macrorealism (OES-MR) constrains the ontic states to be in the support of the operational eigenstates.
For OES-MR, Maroney comments about the application to the three-box paradox [30]: "The unobserved ball's ontic state is always one that can occur when the ball is being observed.However, the price is that those ontic states must now be behaving differently to their appearances.Neither positive-nor negative-result noninvasiveness will be possible, even for operational eigen-states.While the observed behaviour of the ball, determinately placed in one box while Bob checks Condition (III), is showing no detectable disturbance, something must nevertheless be undergoing change, below the level of appearances, as a result of Bob's measurements.This change takes place even when Bob is only interacting with a different box: placing the ball in Box 1, then opening the empty Box 2, somehow disturbs the ball in Box 1 in an unobservable way.But when the system is prepared as in a quantum superposition, and the ball is not being directly observed, these same disturbances emerge and lead to observable consequences." Positive-result and negative-result non-invasiveness refers to the measurement having no disturbance to the system when the system is directly measured as a ball being observed in a Box, and indirectly measured, as in a ball not being observed in a Box.The OES-MR model allows for nonlocality, since there can be a disturbance to the 'state' of the ball in Box 1, when an empty Box 2 is observed.
Our work expands the analysis of Maroney for the OES-MR model.Here, wMR posits that the observation of a ball not being in Box 2 would not change the variable λ (1) M that predetermines the outcome of the measurement on the Box 1.However, the state of the system can change.If there is a further unitary interaction at Box 1, and also at Box 2, so that measurement settings change, observable paradoxes can occur.
We note that wMR counters OES-MR, since it is not true that "the unobserved ball's ontic state is always one that can occur when the ball is being observed".The "observed" state of the system is identifiable as a quantum state, and for the cat state (1), we have seen that the assumption of wMR implies the system cannot be in a quantum state prior to measurement.

C. Supra eigenstate support macrorealism
The supra eigenstate support macrorealism (SES-MR) model also considers the system to be, prior to measurement Ŝ, in a mixture of ontic states which have definite predetermined values for the measurement Ŝ. Different to the OES-MR model however, the SES-MR model allows novel ontic states that cannot be prepared quantum mechanically.
Maroney in examining the third SES-MR model states that: "Supra eigenstate support macrorealism takes the opposite route.Operational eigenstates do not appear to be disturbed by Bob's measurements, and it may be maintained that the ontic states in their support are not, in fact disturbed.However, when the ball is prepared through a quantum superposition, it may now be in an ontic state that does not appear in any operational eigenstate.When it is not being observed, the ball can behave differently." The premise of wMR gives support to the SES-MR model of macroscopic realism proposed by Maroney and Timpson.These authors also present the de Broglie-Bohm model [66] as an example of an SES-MR model [30].In a recent paper [93], the wMR premises have been shown consistent with a model for realism based on the Q function [94][95][96].Analysis of that model suggests ontic states that cannot be compatible with "prepared" or "observed" states [97].

VIII. DISCUSSION AND CONCLUSIONS
In this paper, we have examined a macroscopic version of a Leggett-Garg and Bell test presented earlier [32], in which the spin states | ↑⟩ and | ↓⟩ are realised by coherent states |α⟩ or | − α⟩, with α → ∞, and the unitary interactions determining the measurement settings θ in the Bell test, normally realised by polarising beam splitters or Stern-Gerlach apparatuses, are realised by local nonlinear interactions U θ = e −iH N L t/ℏ .In particular, the set-up allows the noninvasive measurability premise of the Leggett-Garg inequalities to be replaced by that of Bell's locality assumption.The corresponding Bell test is macroscopic, meaning that the Bell premises combine the assumptions of macroscopic realism (MR) and locality at a macroscopic level (ML).
Earlier work showed how MR if defined deterministically can be falsified [32,61].Macroscopic realism applies to a system with two or more macroscopically distinct states available to it, and assumes the system is in one of those states, to the extent that a measurement Ŝθ distinguishing between the states has a predetermined outcome.Deterministic macroscopic realism posits a predetermination of the outcome prior to the entire measurement dynamics, including the implementation of U θ , and is a stronger (more restrictive) assumption.In such a model, as in classical mechanics, it is assumed there are a set of macroscopically distinct states giving a definite outcome for Ŝθ , which can be identified for the system prior to the time at which U θ is implemented.
Violations of Bell inequalities are explained generally as a failure of "local realism", or of "local hidden variables" [34].The violations exclude that there can be hidden variables satisfying the Einstein-Podolsky-Rosen (EPR) premises.EPR's "elements of reality" are negated by Bell violations.The macroscopic version of the Bell test motivates a deeper consideration of the meaning of local realism and, in particular, of the EPR premises, since any rejection of macroscopic realism would be a more startling conclusion than the rejection of local realism at the microscopic level.
Our conclusion is that MR is not contradicted by the Bell violations, and can be viewed consistently with the violations, if defined in a less restrictive way, as weak macroscopic realism (wMR).Weak macroscopic realism has been proposed earlier, and recent work gives an ex-tension of the definition to the bipartite set-up of EPR [63].The earlier work showed consistency of the macroscopic Bell violations with a subset of the wMR premises [32].Here, we show consistency of the macroscopic Bell violations with the full definition of wMR.Ref. [32] proposed three tests of wMR, where the results would be consistent with wMR according to quantum mechanics.We extend to present a fourth test, involving EPR's "elements of reality".
The consequence of our work is a model consistent with quantum mechanics, in which there is an understanding of when the EPR "elements of reality" can be considered to apply.The "elements of reality" will apply to the system defined after the unitary interaction U θ has been carried out in the experiment.We see that the violation of the Bell inequalities occurs due to a combination of a failure of realism and locality.On the other hand, both a weaker version of realism and a weaker version of locality apply: The system has a real property for the outcome of the measurement Ŝθ after the implementation of U θ .Also, the system has an "element of reality" for the outcome of Ŝθ , if the outcome of Ŝθ at A can be predicted with certainty by a measurement Ŝϕ on a second system B − but this applies only once the implementation of the unitary interaction U ϕ at B has taken place.
A justification for wMR is given on considering the nature of quantum measurement.Consider a system A for which Ŝθ is being measured, by a coupling to a macroscopic meter, system B.This is a situation for which EPR's "element of reality" apply, because one can predict with certainty the outcome of the measurement on system A by performing a measurement on the meter B. While EPR's traditional "elements of reality" can be negated, this particular "element of reality" is justified by wMR, because the coupling interaction is such that the measurement basis θ has already been specified.Hence, wMR resolves paradoxes about macroscopic realism and measurement, as highlighted by Leggett and Garg [10].
While the motivation for proposing that wMR is valid is to arrive at a model allowing some form of macroscopic reality, the mapping between the microscopic and macroscopic Bell tests ensures that a similar definition, weak local realism (wLR) [63], can be applied to the original set-up involving spin states | ↑⟩ and | ↓⟩.The original Bell violations can be explained consistently with wLR.A justification for wLR can also be given based on the argument that in the microscopic tests at the time after the unitary dynamics U θ establishing the measurement setting θ, there will be some form of amplification, such as a coupling to a meter [63].Hence, wMR can be applied at this time.
It is interesting to consider the possibility of an experiment.While the predictions of wMR are consistent with those of quantum mechanics, four tests of wMR have been presented, which motivates an experiment.Twomode entangled cat states have been experimentally re-alised [8].However, it is challenging to realise U θ .The Bell example with cat states was presented because of the strength of the conclusions that follow from a Bell violation, and because of the simplicity of the argument from a theoretical viewpoint.Other macroscopic realisations of quantum correlations can be considered however [98].This includes the continuous variable correlations of the Einstein-Podolsky-Rosen (EPR) paradox which are measured by homodyne detection, the measurement setting θ being a phase shift [99,100].Here, set-ups are possible where amplification takes place prior to the implementation of the phase shift θ [101][102][103], so that macroscopic states can be defined and both dMR and wMR posited for the system.A study of wMR for such an experiment would lead to the possibility of tests of wMR, along the lines proposed for Bohm's version of the EPR paradox in [63].We also note that mesoscopic quantum correlations have been achieved for atomic systems [45,104,105,[107][108][109][110][111][112][113].In particular, EPR correlations involving atomic clouds have been measured, including where the measurement setting is adjustable locally [114].This has led to a realisation of Schrödinger's description of the EPR paradox, in which there is a simultaneous measurement of two non-commuting observables, x and p [115].An analysis of wMR showing consistency for such EPR correlations would seem possible.
In conclusion, we have outlined how a weak form of local realism can be consistent with realism at a macroscopic level, despite violations of macroscopic Bell inequalities.Yet, the argument presented by Schrödinger is that there is inconsistency between (weak) macroscopic realism and the completeness of quantum mechanics [1]: for a macroscopic superposition, if there is macroscopic realism, then what state is the system in prior to detection − the system cannot be viewed as being in any quantum state?This motivates analysis of deeper models or interpretations of quantum mechanics [92,116,117].As one example, a model for field amplitudes based on the Q function shows how consistency between quantum mechanics and weak macroscopic realism might be achieved using microscopic retrocausal fields [94][95][96].
surement setting at B has taken place.

Figure 2 .
Figure 2. Violation of the macroscopic Leggett-Garg-Bell inequality (8) using cat states: The violation occurs for all α,β → ∞.Contour plots of P (XA, XB) show the dynamics as the system prepared in the state |ψ Bell ⟩1 at time t1 = 0 evolves through the three measurement sequences of the Leggett-Garg-Bell test violating (8).The local systems evolve according to H (A/B) N L for times ta and t b given by (ta, t b ) in units of Ω −1 , for the systems A and B. Top: Snapshots for the measurement of ⟨S (B) 1 S (A) 2 ⟩ and ⟨S (B) 1 S (A) 3 ⟩, where evolution is stopped at B at time t1, so that t b = 0.This corresponds to a series of successive unitary rotations occurring at A. The interactions realise U A π/8 , and hence preparation for measurement of Ŝ(A) 2 , at time ta = π/4; and U (A) π/4 , and hence preparation for measurement Ŝ(A) 3 , at time ta = π/2.Lower: Snapshots for measurement of ⟨S (B) 2 S (A) 3 ⟩, where evolution is stopped at B at time t2, so that t b = π/4.This measurement involve two further unitary rotations, and hence a change of measurement basis at each site A and B. Here, t1 = 0, t2 = π/4 and t3 = π/2.α = β = 3.

Figure 3 .
Figure 3. Testing weak macroscopic realism by comparing the dynamics with that of a mixed state: The contours show the sequences associated with the measurements needed to test the Leggett-Garg-Bell inequality (8) as described for Figure 2, except here the initial state is taken to be the non-entangled mixed state ρ (AB) mix where no violation is possible.The top sequence is for measurement of ⟨S (B) 1 S (A) 2 ⟩ and ⟨S (B) 1 S (A) 3 ⟩, where a unitary rotation creating a change of measurement basis takes place at system A only.The lower sequence shows the measurement dynamics for ⟨S (B) 2 S (A) 3 ⟩ involving unitary rotations and hence a change of measurement basis for both systems, A and B. Comparing with Figure2, we see that there is no visual difference between the plots at the initial time t = 0.The top sequence involving a rotation at one site only remains visually indistinguishable from that of the entangled state shown in Figure2.However, we see that the final state of the lower sequence involving a change of basis at each site becomes macroscopically different.

2 ,
but there is no determination of the outcome of Ŝ(A) 1 .It is the fact that the three variables λ mix .The top sequence involving a rotation (change of measurement basis) at one site only is visually unaltered between the cat state |ψ Bell ⟩ 1 (Figure 2) and the mixture ρ (AB)

1 at
depicts the situation where the system at A is prepared for the measurement of spin Ŝ(A) 1 , without a unitary rotation being required at A. It would be possible to rotate the measurement basis at A to prepare for the measurement Ŝ(A) 2 , by evolving the system A according to U (A) π/4 , but keeping B unchanged.This creates a new state.However, this does not change the "element of reality" λ and performing the final part of the measurement, XA .In other words, after a further time, the system evolves according to the dynamics (U (A) π/4 ) −1 , and the prediction according to quantum mechanics is that the results of the measurements at A and B remain anti-correlated.The value λ A, regardless of any local reversible unitary interactions, such as U (A) π/4 at A. An experimental test of this premise wMR (3) (and of quantum mechanics) can be performed.The value for the element of reality λ (B) 1 as given by the "element of reality" defined at B. The result for spin σ(A) z is specified by the meter, and would be verified if the measurement σ(A) z at A is actually performed.If a local unitary interaction has changed the measurement setting at A, then for the spin σ(A) z