Whenever a quantum environment emerges as a classical system, it behaves like a measuring apparatus

We consider a quantum principal system $\Gamma$, with an environment $\Xi$ made of $N$ elementary quantum components. By using properties of generalized coherent states for the environment we demonstrate that whenever $\Xi$ behaves according to an effective classical theory such theory emerges from the quantum description of a measuring apparatus in the large-$N$ limit, regardless of the actual interaction between $\Gamma$ and $\Xi$. In fact, we show how to construct a Hamiltonian model for the dynamical process that rules the coexistence of any two quantum systems whenever one of them is macroscopic and behaves classically. This result wears two hats: on the one hand it clarifies the physical origin of the formal statement that, under certain conditions, any channel from $\rho_\Gamma$ to $\rho_\Xi$ takes the form of a measure-and-prepare map; on the other hand, it formalizes the qualitative argument that the reason why we do not observe state superpositions is the continual measurement performed by the environment.


Introduction
There exist two closely-related questions about the quantum mechanical nature of our universe that keep being intriguing after decades of thought processing: how is it that we do not experience state superpositions, and why we cannot even see them when observing quantum systems. As for the latter question, it is somehow assumed Caterina Foti: caterina.foti@unifi.it that this is due to the continual measurement process acted upon by the environment. However, despite often being considered as an acceptable answer, this argument is not a formal result, and attempts to make it such have been only recently proposed [1,2]. In fact, the current analysis of the quantum measurement process [3], its Hamiltonian description [4,5], as well as its characterization in the framework of the open quantum systems (OQS) dynamics [6] has revealed the qualitative nature of the above argument, thus making it ever more urgent to develop a rigorous approach to the original question. This is the main goal of our work.
Getting back to the first question, the answer offered by the statement that microscopic systems obey quantum rules while macroscopic objects follow the classical ones, is by now considered unsatisfactory. Macroscopic objects, indeed, may exhibit a distinctive quantum behaviour (as seen for instance in superconductivity, Bose-Einstein condensation, magnetic properties of large molecules with S = 1/2), meaning that the large-N condition is not sufficient persé for a system made of N quantum particles to behave classically. In fact, there exist assumptions which single out the minimal structure any quantum theory should possess if it is to have a classical limit [7]. Although variously expressed depending on the approach adopted by different authors (see the thorough discussion on the relation between large-N limits and classical theories developed in Sec.VII of Ref. [7]), these assumptions imply precise physical constraints on the quantum theory that describes a macroscopic quantum system if this has to behave classically. In what follows, these assumptions will formally characterize the quantum environment, in order to guarantee that the environment, and it alone, behaves classically. The relevance of the sentence "and it alone" must be stressed: indeed, the work done in the second half of the last century on the N → ∞ limit of quantum theories is quite comprehensive but it neglects the case when the large-N system is the big partner of a principal quantum system, that only indirectly experiences such limit. This is, however, an exemplary situation in quantum technologies and OQS, hence the questions asked at the beginning of this Introduction have recently been formulated in the corresponding framework [1,2,[8][9][10][11][12][13][14][15][16].
In this work, we develop an original approach which uses results for the large-N limit of quantum theories in the framework of OQS dynamics. This allows us to show that details of the interaction between a quantum principal system Γ and its environment Ξ are irrelevant in determining the main features of the state of Ξ at any time τ in the large-N limit, as long as such limit implies a classical behaviour for Ξ itself. If this is the case, indeed, such state can always be recognized as that of an apparatus that measures some observable of the principal system. The relation between our findings and the two questions that open this section is evident.
The paper is structured as follows. In the first section we define the dynamical maps characterizing the two evolutions that we aim at comparing. We do so through a parametric representation introduced in Sec. 3. In Sec. 4, we focus on a peculiar property of generalized coherent states, particularly relevant when the large-N limit is considered. As the environment is doomed to be macrocopic and behave classically, we then implement such limit in Sec. 5, being finally able to show what we were looking for. In Sec. 6 we comment on the assumptions made, while the results obtained are summed up in the concluding section.

Schmidt decomposition and dynamical maps
We consider the unitary evolution of an isolated bipartite system Ψ = Γ + Ξ, with Hilbert space where = 1 andĤ is any Hamiltonian, describing whatever interaction between Γ and Ξ. The state |Ψ is assumed separable meaning that we begin studying the evolution at a time t = 0 when both Γ and Ξ are in pure states. This is not a neutral assumption, and we will get back to it in Sec. 6. At any fixed time τ , there exists a Schmidt decomposition of the state (1), , and the symbol ⊗ understood (as hereafter done whenever convenient). The states {|γ } H Γ , and {|ξ j } H Ξ with j = 1, ... dim H Ξ , form what we will hereafter call the τ -Schmidt bases, to remind that the Schmidt decomposition is state-specific and therefore depends on the time τ appearing in the LHS of Eq.(3). Consistently with the idea that Ξ is a macroscopic system, we take γ max < dimH Ξ : therefore, the states {|ξ γ } H Ξ entering Eq.(3) are a subset of the pertaining τ -Schmidt basis. Given that |Γ is fully generic, the unitary evolution (1) defines, via ρ Ξ = Tr Γ ρ Ψ , the CPTP linear map (from Γ-to Ξ-states) Being the output ρ Ξ a convex sum of orthogonal projectors, Eq.(4) might describe a projective measurement acted upon by Ξ on the principal system Γ, by what is often referred to as measure-and-prepare (m&p) map. However, for this being the case, the probability reproducibility condition [17] must also hold, meaning that, it should also be c 2 γ = |a γ | 2 , ∀γ. This condition, however, cannot be generally true, if only for the τ -dependence of the Schmidt coefficients {c γ } which is not featured by the set {a γ }. In fact, there exists a dynamical model (the Ozawa's model [4] for projective von Neumann measurement described in Appendix A) for which c 2 γ = |a γ | 2 , ∀γ and ∀τ . Such model is defined by a Hamiltonian where the operators acting on Γ must commute with each other, a condition that identifies what we will hereafter dub a measurelike Hamiltonian,Ĥ M , with the apex M hinting at the corresponding measurement process. The evolution defined by exp{−itĤ M } will be consistently dubbed measure-like dynamics 1 .
Once established that Eq.(4) does not define a m&p map, we can nonetheless use the elements provided by the Schmidt decomposition as ingredients to construct a measure-like Hamilto-nianĤ M whose corresponding m&p map, E M : |Γ Γ| → ρ M Ξ is the "nearest" possible to the actual E, Eq.(4).
To this aim, we first use the τ -Schmidt bases {|γ } H Γ and {|ξ j } H Ξ to define the hermitian operatorŝ (6) with ε γ , E j arbitrary real numbers; we then write the interaction Hamiltonian with g some coupling constant, which has the form prescribed by the Ozawa's model (see Appendix A for more details). Further using the Schmidt coefficients, we construct the separable state where |Γ is the same as in Eq.(2), while |Ξ M = γ c γ |ξ γ , with c γ and |ξ γ as in Eq.(3). Finally we define that reads, with ϕ γγ ≡ τ gε γ E γ ∈ R.
1 Giving a Hamiltonian description of more general quantum measurement processes, i.e., identifying the appropriate propagator for the dynamics of such processes up to the output production, is a very relevant problem that has recently attracted the interest of several authors, including some of us.
Given that |Γ is fully generic, Equation (9) defines, via ρ Ξ = Tr Γ ρ Ψ , the CPTP map from Γto Ξ-states Comparing Eqs. (4) and (11) we see that E M has the right coefficients {|a γ | 2 } but the wrong form, i.e., it is not a sum of orthogonal projectors, while E has the correct form but with the wrong coefficients, {c 2 γ }. In fact, were these two maps equal in some limit, it would mean the following: for each time τ , there exists an observable for Γ such that the state into which Ξ has evolved due to its true interaction with Γ, is the same state, in such limit, in which Ξ would evolve as if Ξ itself were some measuring apparatus proper to that observable, which is quite a statement. Since E and E M are linear, they are the same map iff the output states ρ Ξ and ρ M Ξ are equal for whatever input |Γ . We can therefore concentrate upon the structure of such output states, which we will do in the next section by introducing a proper parametric representation.

Parametric representation with environmental coherent states
The parametric representation with environmental coherent states (PRECS) is a theoretical tool that has been recently introduced [18,19] to specifically address those bipartite quantum systems where one part, on its own made by N elementary components, shows an emerging classical behaviour in the large-N limit [5,[20][21][22][23]. The method makes use of generalized coherent states (GCS) for the system intended to become macroscopic.
The construction of GCS, sometimes referred to as group-theoretic, goes as follows [24]. Associated to any quantum system there is a Hilbert space H and a dynamical group G, which is the group containing all the propagators that describe possible evolutions of the system (quite equivalently, G is the group corresponding to the Lie algebra g to which all the physical Hamiltonians of the system belong). Once these ingredients are known, a reference state |0 is arbitrarily chosen in H and the subgroup F of the propagators that leave such state unchanged (apart from an irrelevant overall phase) is determined. This is usually referred to as the stability subgroup. Elementsω of G that do not belong to such subgroup,ω ∈ G/F, generate the GCS upon acting on the reference state,ω |0 = |ω , and are usually dubbed "displacement" operators. The GCS construction further entails the definition of an invariant 2 measure dµ(ω) on G/F such that a resolution of the identity on H is provided in the form One of the most relevant byproduct of the GCS construction is the definition of a differentiable manifold M via the chain of one-to-one correspondenceŝ so that to any GCS is univoquely associated a point on M, and viceversa. A measure dµ(ω) on M is consistently associated to the above introduced dµ(ω), so that requiring GCS to be normalized, ω|ω = 1, implies notice that GCS are not necessarily orthogonal. One important aspect of the GCS construction is that it ensures the function ω|ρ|ω for whatever state ρ (often called Husimi function in the literature 3 ) is a well-behaved probability distribution on M that uniquely identifies ρ itself. As a consequence, studying ω|ρ|ω on M is fully equivalent to perform a state-tomography of ρ on the Hilbert space, and once GCS are available one can analyze any state ρ of the system by studying its Husimi function on M, which is what we will do in the following. We refer the reader to Refs. [24,25] for more details. When GCS are relative to a system Ξ which is the environment of a principal system Γ, we call them Environmental Coherent States (ECS).
Getting back to the setting of section 2, we first recognize that, if they were to represent different evolutions of the same physical system, the propagators exp{−iĤτ } and exp{−iĤ M τ } must belong to the same dynamical group, as far as their action on H Ξ is concerned. This is the group to be used for constructing the ECS, according to the procedure briefly sketched above. Once ECS are constructed, the PRECS of any pure state |ψ of Ψ is obtained by inserting an identity resolution in the form (12) into any decomposition of |ψ as linear combination of separable (w.r.t. the partition Ψ = Γ + Ξ) states. Explicitly, one has where |Γ(ω) is a normalized state for Γ that parametrically depends on ω, while χ(ω) is a real function on M whose square is the environmental Husimi function relative to ρ Ξ = Tr Γ |ψ ψ|, i.e., the normalized distribution on M that here represents the probability for the environment Ξ to be in the GCS |ω when Ψ is in the pure state |ψ . The explicit form of χ(ω) and |Γ(ω) is obtained from any decomposition of |ψ into a linear combination of separable (w.r.t. the partition Γ + Ξ) states.
In particular, for the states (3) and (10), it is and respectively.
Comparing χ(ω) 2 and χ M (ω) 2 is equivalent to compare ρ Ξ and ρ M Ξ , and hence the maps (4) and (11). However, despite the very specific construction leading to |Ψ M τ , we cannot yet make any meaningful specific comparison between χ(ω) 2 and χ M (ω) 2 at this stage. Indeed, we still have to exploit the fact that the environment is doomed to be big and behave classically, which is why ECS turn out to be so relevant to the final result, as shown in the next section.

Large-N and classical limit
As mentioned in the Introduction, a physical system which is made by a large number N of quantum constituents does not necessarily obey the rules of classical physics. However, several authors [7,24,26,27] have shown that if GCS exist and feature some specific properties, then the structure of a classical theory C emerges from that of a quantum theory Q. In particular, the existence of GCS establishes a relation between the Hilbert space of Q and the manifold M that their construction implies, which turns out to be the phase-space of the classical theory that emerges as the large-N limit of Q. In fact, one should rather speak about the k → 0 limit of Q, with k the real positive number, referred to as "quanticity parameter", such that all the commutators of the theory (or anticommutators, in the fermionic case) vanish with k. However, all known quantum theories for systems made by N components have k ∼ 1 N p with p a positive number: therefore, for the sake of clarity, we will not hereafter use the vanishing of the quanticity parameters but rather refer to the large-N limit (see Appendix B for more details).
Amongst the above properties of GCS, that are thoroughly explained and discussed for instance in Ref. [7], one that plays a key role in this work regards the overlaps ω|ξ , whose square modulus represents the probability that a system in some generic pure state |ξ be observed in the coherent state |ω . These overlaps never vanish for finite N , due to the overcompleteness of GCS: as a consequence, if one considers two orthonormal states, say |ξ and |ξ , there might be a finite probability for a system in a GCS |ω to be observed either in |ξ or in |ξ . This formally implies that, defined S ξ the set of points on M where | ω|ξ | > 0, it generally is S ξ ∩ S ξ = ∅.
On the other hand, the quantity features some very relevant properties. First of all, if |ξ is another GCS, say |ω , the square modulus | ω|ω | 2 exponentially vanishes with |ω − ω | 2 in such a way that the limit (19) converges to the Dirac distribution δ(ω − ω ), thus restoring a notion of distinguishability between different GCS in the large-N limit. Moreover, in Appendix C we demonstrate that meaning that orthonormal states are put together by distinguishable sets of GCS. In other terms, the large-N limit enforces the emergence of a oneto-one correspondence between elements of any orthonormal basis {|ξ } and disjoint sets of GCS, in such a way that the distinguishability of the former is reflected into the disjunction of the latter. Given the relevance of Eq. (20) to this work, let us discuss its meaning with two explicit examples.
As for the overlaps entering Eq.(18), let us first consider the case when the states {|ξ γ } are Fock states. In Fig. 1 we show | α|n | 2 as a function of |α| 2 , for n = 1, 2 and different values of N . It is clearly seen that S n ∩ S n → ∅ as N → ∞, meaning that the product of overlaps in Eq. (18) vanishes unless γ = γ , i.e. n = n in this specific example. In order to better visualize S n and S n on M, in Fig. 2 we contour-plot the sum | α|1 | 2 + | α|2 | 2 : indeed we see that, as N increases, S 1 and S 2 do not intersect. Notice that increasing N does not squeeze S n to the neighbourghood of some point on M, as is the case for lim N →∞ | α|α | 2 = δ(α − α ), but rather to that of the circle |α| 2 = n. In other terms, more field coherent states overlap with the same Fock state, but different Fock states overlap with distinct sets of field coherent states, in the large-N limit. This picture holds not only for Fock states but, as expressed by Eq. (20), for any pair of orthonormal states. In Fig. 3, for instance, we contour-plot the sum | α|+ | 2 + α|− | 2 with |± ≡ (|1 ± |2 )/ √ 2: in this case S + and S − are disjoint already for N = 1, and keep shrinking as N increases.

Spin Coherent States
A very similar scenario appears when studying a system Ξ whose Lie algebra is su(2), i.e., the vector space spanned by {Ŝ + ,Ŝ − ,Ŝ z }, with Lie brackets [Ŝ + ,Ŝ − ] = 2Ŝ z , [Ŝ z ,Ŝ ± ] = ±Ŝ ± , and |Ŝ| 2 = S(S + 1); in this case the quanticity parameter is identified by noticing that the normalized operatorsŝ * ≡ 1 SŜ * , * = z, ±, have vanishing commutators in the large-S limit. Further taking S ∝ N it is easily found that k ∼ 1/N . As for the GCS , they are the so-called spin dinates. As for the overlaps entering Eq.(18), the analytical expression for Ω|m is available (see for instance Ref. [24]), which allows us to show, in Fig. 4, the square modulus | Ω|m | 2 for m /S = 0.8 and m /S = 0.4, for different values of N . Again we see that S m ∩ S m → ∅ as N → ∞, implying that the product in Eq. (18) vanishes unless γ = γ , i.e., m = m in this specific example. In Fig. 5 we show the sum | Ω|m | 2 + | Ω|m | 2 as density-plot on part of the unit sphere: besides the expected shrinking of the regions where the overlaps are finite, we notice that, as seen in the bosonic case, the support of lim N →∞ | Ω|m | 2 does not shrink into the neighbourghood of a point on the sphere, as is the case for lim N →∞ | Ω|Ω | 2 = δ(Ω−Ω ), but rather into that of the parallel cos θ = m/S.

A macroscopic environment that behaves classically
Let us now get back to the general case and to Eq.
Using γ |a γ | 2 = 1, and the swap γ ↔ γ, we finally obtain which is what we wanted to prove, namely that the the dynamical maps (4) and (11) are equal when Ξ is a quantum macroscopic system whose behaviour can be effectively described classically.

Discussion
Aim of this section is to comment upon some specific aspects of our results, with possible reference to the way other authors have recently tackled the same subject. Let us first consider the assumption that the initial state (2) of the total system Ψ = Γ + Ξ be separable. If this is not the case, as it may happen, one must look for the different partition Ψ = A + B, such that |Ψ = |A ⊗ |B . If this partition is still such that the subsystem B is macroscopic and behaves classically, the change is harmless and the whole construction can be repeated with A the quantum system being observed and B its observing environment. On the other hand, if the new partition is such that neither A nor B meet the conditions for being a classical environment, then the problem reduces to the usual one of studying the dynamics of two interacting quantum systems, for which any approach based on effective descriptions is incongrous, as details of the true Hamiltonian will always be relevant. Notice that this analysis is fully consistent with the results presented in Ref. [1], which are embodied into inequalities whose meaning wears off as dimH B diminishes. The case when Ψ is not initially in a pure state is similarly tackled by enlarging Ψ → Ψ as much as necessary for Ψ to be in a pure state: a proper choice of a new partition of Ψ will follow. We then want to clarify in what sense the Hamiltonian (7) is said to induce a "measure-like dynamics" or, which is quite equivalent, the channel (11) to define a m&p map: the quotes indicate that the actual output production, which happens at a certain time according to some process whose nature we do not discuss, is not considered and it only enters the description via the requirement that the probability for each output is the one predicted by Born's rule. To this respect, one might also ask what is the property of Γ which is observed by Ξ: this is the one represented, in the Ozawa's model, by the operatorÔ Γ , and it therefore depends on the true evolution via the Schmidt decomposition of the evolved state. To put it another way, details of the interaction do not modify the measure-like nature of the dynamics in the large-N limit, but they do affect what actual measurement is performed by the environment.
We finally close this section discussing the connection between our results and Quantum Darwinism [2,1]. As mentioned at the end of Appendix B, a sufficient condition for a quantum theory to have a large-N limit which is a classical theory is the existence of a global symmetry, i.e., such that its group-elements act nontrivially upon the Hilbert space of each and every component of the total system Ξ that the theory describes. In fact, few simple examples show that quantum theories with different global symmetries can flow into the same classical theory in the large-N limit: in other words, echoing L. G. Yaffe in Ref. [7], different quantum theories can be "classically equivalent". If one further argues that amongst classically equivalent quantum theories there always exists a free theory, describing N non-interacting subsystems, it is possible to show that each macroscopic fragment of Ξ can be effectively described as if it were the same measurement apparatus. Work on this point is in progress, based on the quantum de Finetti theorem, results from Refs. [9,1], and the preliminary analysis reported in Ref. [28].

Conclusions
The idea that the interaction with macroscopic environments causes the continual statereduction of quantum systems is crucial for making sense of our everyday experience w.r.t. the quantum description of nature. However, the formal analysis of this idea has been unsatisfactory for decades, due to several reasons, amongst which we underline the following. First of all such analysis requires a clean procedure for taking the large-N limit of the quantum theory that describes the environment in a way such that it formally transforms into a proper classical one. Moreover it must be possible to implement such limit only upon the environment, without affecting the microscopic nature of the principal quantum system. Finally, the analysis must not imply assumptions on the state of the observed system before the interaction starts, or on the form of the interaction itself.
In this work the above three issues have been addressed combining approaches from quantum field theory, formally describing the conditions for a classical theory to emerge as the large-N limit of a quantum one, with tools of open quantum systems theory, such as the dynamical-map description of the environmental evolution. In particular, using the group-theoretic construction of coherent states for the environment, we have defined an exact parametric representation that allows one to take the large-N limit as described above. In this limit, a comparison between different evolutions of the environment, totally independent on the initial state of the principal sys-tem, becomes possible in terms of environmental dynamical maps.
Our approach allows us to tackle the so-called quantum to classical crossover [29] by a rigorous mathematical formulation that provides a physically intuitive picture of the underlying dynamical process. In fact, exploiting the most relevant fact that not every theory has a classical limit, we have shown that any dynamics of whatever OQS defines a Hamiltonian model that characterizes its environment as a measuring apparatus if the conditions ensuring that the above classical limit exists and corresponds to a large-N condition upon the environment itself are fulfilled. In other words, if some dynamics emerges in the classical world, it necessarily is a measure-like one.
We already mentioned the phenomenon known as Quantum Darwinism, introduced in [2] and recently considered in [1] from an information theoretic viewpoint. Our work provides a way of understanding Quantum Darwinism as a dynamical process, and its generality as deriving from the versatilility of the Hamiltonian model for the quantum measurement process, and the loss of resolution inherent in the classical description.
Acknowledgments CF acknowledges M. Piani and M. Ziman for useful and stimulating discussions. SM and TH acknowledge financial support from the Academy of Finland via the Centre of Excellence program (Project no. 312058) as well as Project no. 287750. CF and PV acknowledge financial support from the University of Florence in the framework of the University Strategic Project Program 2015 (project BRS00215). PV acknowledges financial support from the Italian National Research Council (CNR) via the "Short term mobility" program STM-2015, and declares to have worked in the framework of the Convenzione Operativa between the Institute for Complex Systems of CNR and the Department of Physics and Astronomy of the University of Florence. Finally, CF and PV warmly thank the Turku Centre for Quantum Physics for the kind hospitality.