From quantum to classical by numbers

We follow Kofler and Brukner (2007 Phys. Rev. Lett. 99 180403) in studying the conditions under which a classical picture emerges from the results of not too accurate quantum measurements made on large macroscopic objects. We show that for such objects, consisting of a large number of microscopic elements obeying quantum laws, the Central Limit theorem guarantees the existence of classical values for collective variables, even if the corresponding operators do not commute. Owing to localisation of the overall wave function in any chosen representation, these values can be measured to a small relative error without significantly altering the state of the object. We study a simple model, which includes a rudimentary observer capable of detecting in the coordinate space the position of a macroscopic pointer. The pointer can be employed to measure such quantities, not directly accessible to the observer, as linear or angular momenta. A purely classical picture arises provided the measurements are made on macroscopic objects. Results of measurements, made on small quantum objects, cannot be predicted with certainty, but acquire certain objectivity when encoded in macroscopic pointers’ positions accessible to all observers. Our estimates show that the classical conditions could, in principle, be realised for systems with number of constituent parts of the order of the Avogadro constant. It is possible that the approach captures the essential features of the quantum-to-classical behaviour, although its extension to more realistic systems is likely to be required.


Introduction
The old question of how, and where, quantum properties of the micro-world turn into classical-like experiences of an experimentalist continues to date without a definite answer. In quantum physics, one can observe certain outcomes whose probabilities are supplied by the theory. Calculation of these probabilities (frequencies) requires a recourse to complex valued probability amplitudes, or wave functions, whose precise status is still debated in the literature (see, for example, [1]). Seen by some as a purely mathematical tool, the amplitudes are pervasive in the theory, and their values can sometimes be divulged from the observed frequencies [2]. Moreover, measurement of a particular quantity perturbs the measured system, so that two quantities, whose operators do not commute, cannot have well defined simultaneous values.
Classical physics, by its very nature a limiting case of quantum theory, knows nothing of the mentioned difficulties. It postulates a unique verifiable outcome for each observation and, since classical observations do not disturb the monitored system, ascribes definite simultaneous values to all quantities.
Different schools approach the problem from different perspectives. For example, the standard Copenhagen interpretation of quantum mechanics (for a review see [3], and references therein) relies on the 'Heisenberg cut', a hypothetical interface between quantum events and the Observer's information. The consistent histories approach (see [4] and references therein) purports to define probabilities for a closed quantum system, and reserves no special role for an Observer. Observer's role is at best passive [5] in the Everett's many-world interpretation ( [6], and references therein). Similarly, Bohmian quantum mechanics denies the Observer any role in the formulation of the physical picture of the world (see [7], and, more relevant to our discussion, [8]). Detailed arguments for and against these suggestions are beyond the scope of this paper, and can be found elsewhere in the literature. It is fair, however, to say that none of the above approaches have yet provided a definite answer to the quantum-to-classical question.
In 2007 Kofler and Brukner demonstrated that macro realism and the classical laws of motion emerge from the standard quantum formalism, provided the measurements, made on large systems, are coarse-grained, i.e. limited in their accuracy [9]. Their work was preceded by the one of Lloyd and Slotine, who showed that imprecise (weak) measurements made on a set of identical systems can determine the properties of an individual system while affecting it only slightly [10]. It is therefore plausible, that a classical picture could emerge from quantum description, provided the Observer limits him/her/itself to studying only a certain class of classical phenomena, using only certain types of classical instruments. Let us mention that measurements and their classical limit have also been discussed in several works, see for instance [11,12] including semiclassical approaches that show that the fine structure of quantum probability distributions cannot be resolved for a macroscopic object and can therefore be discarded [13,14].
In this paper we will follow both of the previous references in looking for a way to recover the classical picture from quantum properties, by analysing the observation (measurement) procedures offered by quantum mechanics. More precisely, the classicality we want to study relies in the following assumptions: (A) Provided a system is large enough, there exist measurements yielding a unique classical value, if not exactly, then with a vanishing relative error.
(B) All macroscopic quantities should have well defined classical values, in the sense of the above.
(C) The disturbance produced on the measured system should be if not exactly zero, then small enough for the next measurement of any different quantity to produce its own unique classical result.
(D) The result of any measurement can be encoded into the spatial position of a classical pointer, which can be 'read' by any number of Observers, without altering its state, or altering it only by a negligible amount.
With this in mind, we will carefully examine different situations involving a quantum system or systems, one or several quantum pointers and, eventually, rudimentary external Observers. In particular, we will evoke a simple model similar to that used in [9,10], and study what happens if an Observer restricts him/her/itself to monitoring only large conglomerates of non-interacting elementary quantum systems, by means of instruments, whose pointers are composites of quantum particles considered in the coordinate representation. We will also consider the case where an Observer can access the information about an elementary quantum system, encoded into the position of a large macroscopic pointer. In most cases we evaluate the change, produced by an observation on the system's state, and estimate its effect on the result on a subsequent measurement of a different variable.
A more detailed layout of the paper as follows. In section 2 we briefly review the basic elements of Quantum measurement theory, used throughout this work. In section 3 we look at the classical picture, which emerges when a large set of equally polarised spin-1/2 particles, is monitored by an equally large set of quantum pointers. In section 4 we apply the Central Limit theorem (CLT) to measurements of collective additive quantities, and relate the emergence of 'classical values', to the localisation of the composite's wave function in the chosen representation. As an illustration, in section 5 we examine the case in which a large number of spin-1/2 particles are monitored by a single quantum pointer. We evaluate the damage to the composite's quantum state done by a measurement, and its consequence for a follow up measurement of a different total spin's component. Section 6 describes a similar study, this time of a large number of non-relativistic quantum particles, prepared in the same state. A classical picture is recovered for measurements resolutions, which ensure sufficiently accurate simultaneous values of the composite's centre of mass (COM) and its total momentum. In section 7 we let the cloud of particles be split after a collision with a potential barrier, and obtain a similar classical picture for its transmitted part only. In section 8 we add a 'rudimentary Observer', who's primitive sensor allows him/her/it detect ('see', see the assumption D of the Introduction) position of the COM of a large cloud of particles, which plays the role of a classical pointer. The sensor is taken to be a single pointer, capable of coupling to macroscopic objects, and whose displacement encodes the value of the measured quantity. We need to model the Observer of the system-device-observer sequence in one way or another, and this way is certainly the simplest. In section 9 such a pointer is used to perform a measurement of a component of a single spin-1/2. In section 10 we ask what would be the Observer's experience, after looking at a classical pointer, previously prepared in a superposition of spatially separated macroscopic states. In section 11 we recover the classical limit of the measurement theory, by considering a large angular momentum, monitored by a macroscopic pointer. In section 12 we look for observations, which might convince an Observer that the classical picture, hitherto perceived, is, after all, only an approximate one. We present our conclusions in section 13. Appendices A and B discuss certain technical points.

Quantum measurements
Our main tool will be a standard von Neumann measurement [15][16][17], which we will now briefly review. Suppose we set out to measure an operator Â, with eigenstates ñ n | , and eigenvalues A n , in a Hilbert space of dimension N. If only  J N eigenvalues A j are distinct, we can write å å p p p p d where p ĵ projects onto the subspace corresponding to an A j . At t=0, we couple the system in a state Y ñ 0 | to a pointer, a massive particle with position f and momentum λ, via a brief yet strong interaction (we put =  1), | , and the resulting probability distribution of the pointer's readings, ρ( f ), take particularly simple forms å å where we have chosen the units so as to put the coupling strength g in equation (2) is just the total probability of finding the result A j in an ideally accurate measurement of Â, ). This will let us apply the CLT [18] in the most interesting for us case of a measurement made on a large set of identical quantum systems.
The results are readily generalised to measuring an operator with a continuous spectrum (see appendix A), where we have (δ (z) is the Dirac delta) ò n n n n n d n n Finally, the results of this section are easily generalised to a system, initially in a mixed state given by a convex sum of one-dimensional projectors. We would only need to apply the above analysis to each term, and then add the results, as appropriate.

A follow up measurement
Immediately after obtaining a reading f for the operator Â in (7) we may decide to measure a different operator using a second pointer with position ¢ f , prepared in a state ¢ñ G | . For the final state of the system + two pointers, . Now the joint probability distribution for the readings f and ¢ f is given by We will always assume that the pointers are prepared in real valued Gaussian states of a width Δ f, and D ¢ f , respectively. For example, for the first pointer we write where Δ f determines the accuracy (resolution) of the measurement, which is accurate ('strong') when Δ f is small, and inaccurate ('weak') when it is large.
The first measurement cannot be affected by the second one (see, for example, [19]), and its readings are distributed according to ò ò In general, the second measurement's results are not what they would have been, had the first measurement not been made and would reduce to ò , or if the exponential in the rhs of (15) can be put to unity. This illustrates the well known fact that measurements of quantities such as different components of a spin, or of the particle's position and momentum, must perturb each other. If we wish to recover the classical picture, avoiding this perturbation should be our first priority.

Many spins, and as many quantum pointers
Our first attempt at recovering the classical picture, outlined in the Introduction, will involve K 1  spin-1/2 systems, all polarised along the z-axis, so the initial state of the whole set is given by the product Together, the spins amount to an angular momentum so large (in units of ), that we expect it to exhibit certain classical properties. In particular, we should be able to measure its component on any chosen axis to a good accuracy. In addition, successive measurements along various axis should not be affected by their predecessors. We will also assume that we dispose of K von Neumann pointers, with the positions f k , k=1, K, K, all prepared in identical Gaussian states of a width Δ f, each coupled to one of the spins, and all enacted at the same time, as illustrated in figure 1.
Let the first measurement be along a direction f q = n , ( )  , making angles f and θ with the x-and z-axes, respectively. The kth pointer measures (up to a factor of 1/2) the kth spin's component along the chosen axis, the measured operator A k has eigenvalues A 1,2 =±1, and is given by ( ) are the mean and the standard deviation (SD) of each individual measurement, respectively. Thus, the distribution ρ( f tot ) is centred at K times the average of Â in the state  ñ z | , and has a SD K times larger than it would be for just one of the f k ʼs. Since the SD grows with K much slower than the largest possible value of the projection, K, we can have a good 'classical' measurement, provided

The follow up measurement
In order for the classical picture to emerge we need to show that it is possible to choose an accuracy of each individual pointer, Δ f, good enough to obtain the expected classical result, q K cos , yet poor enough to allow for a subsequent evaluation of the total spin's projection along a different direction, , yielding the correct result q¢ K cos with a negligible error. Let operators be measured after the A k ʼs in equation (17) to a new accuracy D ¢ f . We need to perturb each spin only slightly, and should choose Df 1  . Then, expanding the exponentials in equation (15), for the mean reading of each pointer we have Figure 1. A set of K 1  spin-1/2, all polarised along the z-axis, amount to a large angular momentum K/2 (in units of ), directed along the axis. Coupling quantum pointers, one to each spin, allows one to determine any projection of the total spin to a negligible relative error, and almost without perturbing the spins' state.
allows us to have two good measurements of the total spin components along n  and ¢ n  , such that Note that the second measurement will leave the system only slightly perturbed, and ready for the next measurement along some new direction ¢¢ n  . In this sense the classical picture is recovered.

An estimate
As an illustration, we evaluate the odds on measuring the x-component, (f=0, θ=π/2), of the total spin to a good accuracy, and still find it polarised along the z-axis, (f q ¢ = ¢ = 0, 0), if the second measurement is made. Evaluation of the scalar products in equation (15), for the distribution of the second pointer's readings yields For a macroscopic sample we expect the number of spins to be of order of the Avogadro number, K≈10 24 . Choosing Δ f=10 3 guarantees that the first measurement would yield 15 , which is 10 9 times smaller than the typical total spin size K. The second measurement will yield a mean value which differs from K=10 24 by a factor of the order 10 18 , or only by about 0.0001% of the measured value. The spread of the readings ¢ f around the mean, not affected by the first measurement, is determined only by the accuracy of the second set of pointers, D ¢ f .

A brief summary
So far, we have described a procedure which transfers the information about the total spin (magnetic moment) of a simple multi-spin system to an ensemble of quantum pointers. The value of the total spin's projection onto an arbitrary axis can then be accurately deduced from the pointers' readings without seriously affecting the state of the spins. Please note certain similarity with the so-called weak measurements [2,20]. Having achieved the goals (A), (B) and (C) of our wish list in the Introduction, we have made little progress on the (D). It is not clear how an individual pointer could be 'read', as this might require another measuring device to observe the pointer, and yet another device to watch the first device, and so on. Also, the necessity to have 10 24 individual pointers, is in itself prohibitive. These problems can be remedied, at least to some extent, by considering the so-called collective measurements [10], where the information about a macroscopic property of a system is passed to a single pointer.

Collective measurements. Localisation of the wave function
Next we follow [9,10], and consider a composite of K 1  N-dimensional quantum systems, all prepared in the same state yñ |  y Our aim is to measure the total value of a quantity represented, for each system, by the same operator Â (see equation (1)) where the subscript k refers to a particular system. The eigenstates of A tot are the products ñ º ñ ñ ñ n n n n ... K A max are the smallest and the largest eigenvalues of Â, respectively. As shown in section 2, the state of the composite after obtaining a reading f is We are interested in the structure of the coefficients multiplying the states ñ j | , and recall that is just the probability that the sum of K independent variables equals A j tot . Thus, in the limit K 1  the CLT predicts that are the mean and variance of the operator Â in the individual state yñ | . Notably, since the CLT holds for an arbitrary individual distribution, equation (33) is valid for any choice of Â and yñ | . The case of an operator with a continuous spectrum can be treated similarly, where we obtain ò ò å p s s p p p n n d n | . The mean á ñ A and the variance s A 2 are still given by equations (34), but with the sums replaced by integrals, Equations (31)-(35) are our main result so far. With the states ñ j | and ñ a | appropriately normalised, we can say that the wave function is 'localised' in the region of width~K around a 'macroscopic' value á ñ K A , with negligible contributions from the ñ j | and ñ a | outside this region. Thus, it is possible to choose a measurement's accuracy Δ f small enough for the error, relative to the typical large value of the measured quantity, to be small, . At the same time, it is possible to have Δ f large enough for G( f ) to be practically constant for all important states in the decompositions (31) or (35), . With the state of the composite system barely changed, the system is ready for the next collective measurement, not affected by the previous ones. In summary, we can have a good (accurate) classical measurement, provided The measurement is 'classical', firstly, because its single realisation yields the classical value with a negligible relative error and, secondly, because its back action on the measured system is negligible as well. The localisation property of a wave function, describing a large conglomerate of non-interacting components, must hold in every representation, and for all additive quantities. Our ability to assign to a macroscopic system in a quantum state a set of 'objectively existing' classical values signals, therefore, return to the classical picture.

Many spins, and only one quantum pointer
Next we apply the approach of the previous section to a measurement of the component of the total spin along a direction n  , for a system in the state (16), see figure 2. The corresponding operator (up to a factor of 1/2) is given projects onto a subspace spanned by all possible products of the states  ñ . Using well known properties of the binomial distribution [21], we obtain for . Note that equation (39) can be obtained directly from equations (31)-(33), by noting that For a Gaussian pointer (13), evaluation of Gaussian integrals yields for the distribution of the readings Thus, for the number of spins sufficiently large, there are many possibilities to realise a 'classical' measurement, as described in the previous section. Indeed, with D~+  f K 1 2 ( ) , 0<ò<1 we achieve, as  ¥ K , More precisely, if one writes K=10 n then n and ò satisfy For instance, considering a number K=10 8 (n=8) of particles, one can set ò=(α+n)/2n, that yields Δ f∼10 6 andK 2 10 4 . Hence, all quantities involved in equation (42) differ by two order of magnitude.

Damage to the initial state
It is easy to assess the damage done to the state Y ñ 0 | , provided the pointer reads f. A convenient measure of the change produced by the measurement is the norm of the difference between Y ñ 0 | and the properly normalised Replacing the sum over j by an integral ò -¥ ¥ A d tot , evaluating several Gaussian integrals, and taking the limit q Df K 2 sin  , we find with the second Gaussian in equations (45) much broader than the first one, and Err( f ) will stay close to zero for all readings f, which are likely to occur in the measurement. Thus, would be a suitable choice, provided the number of spins, K, is sufficiently large. Indeed, we obtain Err( f r )=r<1, provided the pointer reads q q =  D - . Now the probability to have an error greater than r is (adding a factor of 2 for the two tails of the Gaussian) ò r

The follow up measurement
Next we want to look for a regime in which a measurement of the total spin's component will not be affected by previously measuring it along f q = n , ( )  . The corresponding distribution r ¢ f ( )is now given by a discrete sum (see equation (15) , and the projectors p m K ( ), p ¢ m K ( ) and p j K ( ), for A tot and B tot , respectively, are defined in equations (31). Calculation of r ¢ f ( )with the help of equation (15) would require evaluation of numerous scalar products, and we will ask a simpler question instead. As in section 3.3, we will try to measure the z-component of the total spin, with and without measuring of A tot first. With no measurement of A tot made, only one Gaussian will be present in the sum (49) . We, therefore, have a condition for a good classical measurement of any projection of the system's total spin (see equations (42))

An estimate
To make our arguments plausible, we need to check whether this section's simple model is at least in the right ballpark. The Avogadro constant, N A ≈6×10 23 is a reasonable estimate for the number of constituent parts of a 'macroscopic' object, and we will use it throughout the rest of the paper.
In the previous example, for K∼10 24 , the maximum size of the total spin (in units of ) is of order of 10 24 . For the 'characteristic size' of the spin state from (39) we have The measurement is not likely to change the state of the spins. From equation (47) the probability to incur an error of more than 1% is negligible. Finally, equation (51) shows that the coefficient multiplying )in the expression for the distribution of the second pointer's readings (15), is very close to unity. With all coefficients multiplying the Gʼs in the sum (15) non-negative, and all  B K j tot | | we find the mean reading of the second pointer close to its unperturbed value We have, therefore, two good non-perturbing 'classical' measurements along two non-collinear axes, which leave the system of the chosen size ready for more measurements of this type.

Many quantum particles, and only one quantum pointer
Our next example involves K 1  non-interacting free particles, all in the same quantum state yñ | , with a mean momentum p 0 , We will be interested in determining the position of the system's COM, as well as its total momentum, see figure 4. Thus, we consider operators as well as two Gaussian pointers, with positions f and ¢ f .

Position of the COM
With the help of equation (35), we find the corresponding wave function to be The precise form of the states ñ a | , d á ¢ñ = -¢ a a a a | ( ) , defined in equation (35), is of no importance to us. It is sufficient to note that the wave function (59) is localised in a region of a size s D = a K 2 x around = á ñ a K x , and that a measurement to an accuracy Df K  will yield á ñ = á ñ f K x , and leave the state of all K particles virtually unchanged. For example, for the particles in identical Gaussian states with a mean momentum p 0 , for the meter readings we find figure 5. To describe the COM of the cloud of particles, we need to rescale f by a factor of K, thus introducing We can also evaluate the final mixed state of the particles, R part  x COM ( )  the measurement yields the position of the COM of the composite system to an accuracy Df COM , without seriously affecting its quantum state. Next we want to see whether such a measurement would still allow us to accurately determine the cloud's momentum as well.

Follow up measurement of the total momentum
Rather than evaluate the damage to the cloud's state which the measurement of the COM is likely to produce, we will go straight to the distribution of the second pointer's reading. From (15) we have Evaluating the Gaussian integrals, after some algebra we obtain Thus, on average, the pointer points towards the correct value of the total momentum of the cloud, á ¢ñ = f Kp 0 . As was expected, choosing the accuracy of the first measurement in such a way that D K f K   guarantees the it does not affect the measurement of the momentum, since d ¢ f tends to its unperturbed value, d ¢ »  provides for a good classical measurement of the momentum, which leaves the state of the cloud practically unperturbed, and ready for the next classical observation.

The classical trajectory
Since the first measurement of the position of the COM (more precisely, of the operator X , see equation (57)) at t=0 appears to perturb the state of the cloud only slightly, we should be able to make a second measurement at some t>0, and find the COM where the classical mechanics would put it, in our case, displaced by vt=p 0 t/m, m being the particle's mass. Taking x 0 =0 in equation (61), from (15) we have is the propagator for the cloud of non-interacting particles. After evaluation of the Gaussian integrals involved we obtain We note that we are in the 'classical regime', provided the first two terms in the rhs of equation (72) are dominant. As expected, for a given time t, the effect of the first measurement disappears for Df K  , and the second measurement becomes a 'good' classical one, for Remaining within these limits, one will always stay in the classical regime, where measurements describe a large quantum cloud of particles as a classical point-sized object, which follows a well defined trajectory.

Splitting a cloud of particles by scattering
To extend the discussion beyond free motion, we assume that our cloud of particles, travelling from left to right, meets with a potential barrier which is non-zero only between x=−d and x=0, see figure 6. If one waits long enough, each particle's state will be split into the transmitted (T) and reflected (R) parts y y y ñ = ñ + ñ, localised far to the right and to the left of the barrier, respectively. Now the COM of the system lies somewhere between y ñ T | and y ñ R | , where no particles are found, and its position is of little interest. We can, however, specify to the transmission channel, by considering two commuting operators (Θ(x)=1 for x>0, and 0 otherwise) for a measurement of X T . Denoting the corresponding pointer readings as f N T and f X T , from equation (35) we obtain where the states ñ a | in equations (77) and (78) are defined as in equation (35) for the operators N T and X T , respectively. Thus, with ,a n d , it is possible to have good classical measurements of both the number of transmitted particles, and the position of the COM of the transmitted cloud, for K 1  . The same can be repeated for the reflected part of the cloud, by replacing Θ(x k ) by Θ(−x k −d) in equations (74). Such measurements, performed before and after the particles interact with the barrier, yield a picture of a point sized object of a mass mK being divided into two parts of masses P T mK and (1−P T )mK, moving to the right and to the left, respectively.

Quantum measurements with an observer
Now we can consider an Observer, existing in hugely oversimplified world of objects, made up from small parts which obey quantum mechanical rules. Some of the objects are large, due to the large number of their constituent parts. For reasons unknown to us, the Observer can only gain information about the collective properties of objects, such as the total spin, position of the COM, or the total momentum, by applying a singlepointer measurement procedure, described above. The accuracy of the measurements is always good enough to ensure an error small relative to the large value measured, yet sufficiently poor so as not to perturb the quantum state of a sufficiently large conglomerate. We must conclude that such an Observer, dealing with large objects, would perceive an essentially classical world in which all components of magnetic moments can be measured simultaneously. He/she/it would also visualise a cloud of quantum particles as a single small object, possessing a well defined position and momentum at all times, and moving along a classical trajectory, prescribed by classical mechanics.
This suggests using the COM of the cloud as a pointer. It was shown in section 7, that the COM's position can be determined to a sufficient accuracy, without significantly altering its quantum state. Thus the result of a measurement, encoded in its position, can be verified by other independent Observers, thus becoming, in Einstein's words [22], an 'element of reality'.

Macroscopic pointer in a 'grotesque' state
Quantum mechanics allows macroscopic superpositions although in practice creating such superpositions may be difficult. Below we question what Observer would see when looking at a 'grotesque' [24] state, where all K particles, which make up the pointer, are prepared in one of the two Gaussian states, ψ(x±d), well separated by a distance 2d. The overall state is, therefore, similar to (83), except for the absence of the spin states, since this time no spin is involved As before, the Observer detects the position of the pointer's COM. Noting that áY , and tracing out the pointer's variables, for the Observer's density matrix we obtain Choosing, as in section 10 9 4 x COM ( )   and sending  ¥ K , in the COM variables we find Thus, the Observer's density matrix is nearly diagonal, and the two possible measured values are ±d. We must, therefore, conclude that presented with a pointer in a macroscopic superposition, the Observer will find it either at d, or -d, with the probabilities given by the absolute squares of the amplitudes α and β in equation (92).

Classical limit of the measurement theory
Our discussion would be incomplete without mentioning the purely classical measurement, where a classical pointer is employed to determine a projection of a large classical angular momentum onto a given direction n  . Thus, we consider . Now there are three Gaussian widths, one associated with the localisation of the wave function of the K spins, q K 2 sin , another σ x , describing the states of the particles which form the pointer, and D = D f N f COM , which describes the Observer's own resolution. Our goal is to choose them in such a way that the Observer gets an accurate result without disturbing neither the pointer, nor the spins. Performing the Gaussian integrals, for the distribution of the Observer's readings we obtain we can consider the set of particles to be shifted by q K cos as a whole. The next step, observation of the cloud by an Observer, whose intrinsic resolution is Δ f, was discussed in section 7. In particular, if s Df N x COM  , the Observer will register the COM of the particles at q = f K cos , and leave the state of the classical pointer almost unchanged. We note also that in equation (102) the difference -¢ j j 2 ( ) must be of order of q K sin 2 2 , and with s q NK sin 2 x 2 2  , the initial pure state of the spin system will also remain largely unchanged. In summary, with K N , 1  , we find ourselves in the classical regime if which can be satisfied, for instance, for σ x ∼K 5/6 , N∼K 1/6 , and Df K COM 5 6 . Now we have classicality as defined in the Introduction. A classical pointer couples to a large classical angular momentum (spin) in such a manner that the Observer obtains the classical value of its projection with a negligible error. A different Observer can then measure the spin's projection on a different direction, and obtain the corresponding classical value, also with a negligible error. Finally, the second Observer can 'look' at the first Observer's pointer, and verify the result obtained by his predecessor. All the mentioned values 'exist' in the sense that they can be obtained any number of times without altering the states of either the spin or the pointer.
We conclude this section by writing down the classical equations of motion for the case where a classical pointer monitors a system, composed of a large number of particles. By the CLT (see section 5), to a good accuracy, we can ascribe to each of the two its position as well as its momentum, say, (p, x) and (X, P), respectively. The measured operator Â is replaced by a classical dynamical variable, A(x, p), and the coupling with the pointer takes a form g(t)PA(x, p), where g(t) is some switching function. Now the Hamilton's equations read , , where H 0 (x, p) is the system's own energy in a state, characterised by x and p. The pointer's momentum is conserved, P=const, and its displacement after a time T is which for a special choice g(t)=1/T=const coincides with a time average of the A. Note, however, that the pointer does not perturb the observed system if, and only if, the pointer's momentum is zero, P=0. Quantally, an accurate determination of the initial position X(0), leads to a large spread in the pointer's momenta around P=0, which inevitably disturb the observed system. In the classical limit, it is possible to keep the last terms in the first and second of equations (102) small, compared to the large H 0 (x, p), and non-invasive monitoring of a classical system is, in principle, possible.

From classical back to quantum
Finally, we briefly return to the Observer, able to monitor macroscopic objects, by following their COM, to an accuracy Δ f, as discussed in section 5. How can he/she/it learn that the classical description is, in fact, an approximation? There are at least two possible ways. One consists in studying small individual objects, such as a single spin-1/2. As discussed in the previous section, it is possible to prepare K 1  spins all polarised along the z-axis, then obtain, practically with certainty, a zero value for the total spin's projection on the x-axis, and finally obtain the original value K, in another measurement along the z-axis. For a single spin, this is, of course, no longer true, as was discussed in section 10. Aligning it with the z-axis, one then obtains the value of 1/2 or −1/2 along the x-axis, and has only a 50% chance to recover the original polarisation if a second measurement along the z-direction is made. Thus, the classical picture of an angular momentum having well defined components along all directions breaks down, and the Observer becomes aware of the different laws governing the behaviour of microscopic objects.

Conclusions and discussion
In summary, we have followed [9,10] in asking whether it is possible, in principle, to obtain a classical picture by observing, not too scrupulously, macroscopic objects, composed of a large number of constituent parts, which individually obey known quantum laws. In particular, we studied a scheme in which a rudimentary Observer has a direct access to the position of a macroscopic pointer, which, in turn may be coupled to elementary quantum systems, such as a qubit, or to other classical-like systems, like large collections of polarised spins. We found it plausible that such an observer would be able to have an optimal resolution, which would allow him/her obtain a classical result to a high relative accuracy, while dealing with 'large' macroscopic systems, whose number of constituent parts, K, is of order of the Avogadro constant. Within these resolution limits, large conglomerates of quantum particles would be perceived as point-sized objects, moving along trajectories prescribed by the laws of classical mechanics. Such an object would appear to possess a well defined momentum, which can be measured to a good accuracy, without affecting the object's position. In a similar way, a large angular momentum would also be seen as classical quantity, all of whose components could be determined simultaneously, and with a small relative error. Should the measurements of this type be the only possible way to access physical reality, this reality would have an essentially classical aspect, with the underlying quantum nature revealed only through more accurate measurements, or through studying individual particles, or spins. Such classical-like description is made possible by the tendency of the wave function, describing a macroscopic object, to become concentrated in a relatively small region of the spectrum of an operator representing a particular collective variable. This property is repeated in different representations, and for various non-commuting operators. Consequently, it is possible to devise a quantum measurement [9], which would not destroy the coherence between the essential components of a macroscopic quantum state, and yet yield only an error small compared to the large overall value of the measured quantity. With the state perturbed only slightly, each new measurement is no longer affected by its predecessors, and the measured values, associated with the peaks of the wave function, acquire certain objectivity, reminiscent of Einstein's 'elements of reality' [22].
Needless to say, our model is vastly oversimplified. Macroscopic object are not, in general, composed of noninteraction particles in Gaussian states. Spins do not usually all point in the same direction. A human eye observing the needle of a metre cannot be expected to perform a von Neumann measurement of the needle's COM. To estimate the COM's position an Observer would need to know also the number of particles, which make up the pointer. However, it retains some of the features, associated with everyday experiences. An Observer cannot directly 'see' a spin, or an angular momentum, yet is able to perceive large macroscopic objects in the coordinate space, clearly preferred by the human eye. A macroscopic pointer is perceived as a single object, rather than a point in a K-dimensional configuration space, as was first proposed in section 3. Our analysis requires no special role for human conscience [25] beyond what has been just said. Neither have we attempted to resolve the vexed problem of the 'wave function collapse' (see, for example, [26]), as we never tried to follow the fate of the parts of the quantum state, which correspond to the outcomes, not materialised in the course of the experiment. Rather we note that the Observer is involved only with the 'pointer' part of the spin+pointer composite. The state of the pointer in the entangled state, formed after the interaction with the spin, is a mixed one, which, already contains the probabilities. According to standard quantum mechanics, these probabilities are passed on to the Observer, who has no difficulty in seeing the COM of the pointer moved either to the left, or to the right, in each run of the experiment. In this imperfect way, we are able to place the divide between quantum and classical behaviour at the stage where the measured value becomes encoded in the displacement of a macroscopic object with classical properties. It remains to be seen whether the model, despite of its numerous shortcomings, captures the essential features of the quantum-to-classical behaviour. Yet, as was claimed in [9], it offers an alternative way towards its understanding.
We conclude with an attempt at a more general excuse for leaving outside any mention of Observer's conscience. In our discussion, an Observer has access to only a fraction of the complex world, through his/her/ its ability to perceive certain 'physical phenomena'. The 'physical laws', established by the Observer, are, therefore, but connections and relations between Observer's accessible experiences. As such, they may not amount to an exhaustive objective picture of the world, but are destined to reflect particular properties and limitations of the Observer itself. Such is, for example, the rule (see section 7) that the COM of a large cluster of free particles always moves with a constant velocity. The very notion of COM is particular to the manner in which the Observer perceives this macroscopic object. The 'unseen' part of the far more complex world will remain outside the remit of Observer's physics, unless or until a better experiment provides for a new experience. It is possible then that new physics, based on this experience (e.g. quantum theory), will still be incomplete, yet leave room for further improvement. It may also be possible that Observer's own abilities to perceive will have been exhausted, without providing a complete understanding of physical reality. In this latter case, a theory is likely to descend, like it seems to happen with quantum mechanics, to the level where 'no one has found any machinery behind the law' [27]. An Observer, aware of own limitations, will almost certainly consider him/her/itself not a suitable object of a study, as has often been claimed in modern literature (see, for example, [28]). With the Observer excluded from the list of the studied physical phenomena, quantum mechanics acquires internal consistency. Outcomes of the experiments an Observer may perform on the parts of the outside world occur with certain probabilities, or frequencies. Quantum theory provides the recipes for calculating those probabilities for all eventualities.