Whenever a Quantum Environment Emerges As a Classical System, It Behaves Like a Measuring Apparatus

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

Caterina Foti1,2, Teiko Heinosaari3, Sabrina Maniscalco3, and Paola Verrucchi4,1,2

1Dipartimento di Fisica e Astronomia, Università di Firenze, I-50019, Sesto Fiorentino (FI), Italy 2INFN, Sezione di Firenze, I-50019, Sesto Fiorentino (FI), Italy 3QTF Centre of Excellence, Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FIN-20014, Turku, Finland 4ISC-CNR, at Dipartimento di Fisica e Astronomia, Università di Firenze, I-50019, Sesto Fiorentino (FI), Italy August 6, 2019

We study the dynamics of a quantum malizes the qualitative argument that the system Γ with an environment Ξ made of reason why we do not observe state su- N elementary quantum components. We perpositions is the continual measurement aim at answering the following questions: performed by the environment. can the evolution of Γ be characterized by some general features when N becomes very large, regardless of the specific form 1 Introduction of its interaction with each and every component of Ξ? In other terms: should we There exist two closely-related questions about expect all quantum systems with a macro- the quantum mechanical nature of our uni- scopic environment to undergo a somehow verse that keep being intriguing after decades of similar evolution? And if yes, of what thought processing: how is it that we do not ex- type? In order to answer these questions perience state superpositions, and why we cannot we use well established results from large- even see them when observing quantum systems. N quantum field theories, particularly re- As for the latter question, it is somehow assumed ferring to the conditions ensuring a large- that this is due to the continual measurement pro- N quantum model to be effectively de- cess acted upon by the environment. However, scribed by a classical theory. We demon- despite often being considered as an acceptable strate that the fulfillment of these con- answer, this argument is not a formal result, and ditions, when properly imported into the attempts to make it such have been only recently framework of the open quantum systems proposed [1–3]. In fact, the current analysis of dynamics, guarantees that the evolution of the quantum measurement process [4], its Hamil- Γ is always of the same type of that ex- tonian description [5,6], as well as its characteri- pected if Ξ were a measuring apparatus, zation in the framework of the open quantum sys- no matter the details of the actual inter- tems (OQS) dynamics [7] has revealed the qual- action. On the other hand, such details itative nature of the above argument, thus mak- are found to determine the specific basis ing it ever more urgent to develop a rigorous ap- arXiv:1810.10261v3 [quant-ph] 5 Aug 2019 w.r.t. which Γ undergoes the decoherence proach to the original question. This is the main dictated by the dynamical description of goal of our work. the quantum measurement process. This Getting back to the first question, the answer result wears two hats: on the one hand it offered by the statement that microscopic sys- clarifies the physical origin of the formal tems obey quantum rules while macroscopic ob- statement that, under certain conditions, jects follow the classical ones, is by now considered unsatisfactory. Macroscopic objects, in- any channel from ρΓ to ρΞ takes the form of a measure-and-prepare map, as recently deed, may exhibit a distinctive quantum be- shown in Ref. [1]; on the other hand, it for- haviour (as seen for instance in superconductiv- ity, Bose-Einstein condensation, magnetic prop- Caterina Foti: caterina.foti@unifi.it erties of large molecules with S = 1/2), meaning

1 that the large-N condition is not sufficient per- plement such limit in Sec.5, being finally able sé for a system made of N quantum particles to to show what we were looking for. In Sec.6 we behave classically. In fact, there exist assump- comment on the assumptions made, while the re- tions which single out the minimal structure any sults obtained are summed up in the concluding quantum theory should possess if it is to have a section. classical limit [8]. Although variously expressed depending on the approach adopted by different authors (see the thorough discussion on the re- 2 Schmidt decomposition and dynam- lation between large-N limits and classical the- ical maps ories developed in Sec.VII of Ref. [8]), these assumptions imply precise physical constraints on We consider the unitary evolution of an isolated the quantum theory that describes a macroscopic bipartite system Ψ = Γ + Ξ, with Hilbert space quantum system if this has to behave classically. HΓ ⊗ HΞ ; being Ψ isolated, it is In what follows, these assumptions will formally ˆ characterize the quantum environment, in order |Ψ(t)i = e−iHt |Ψi , (1) to guarantee that the environment, and it alone, behaves classically. The relevance of the sentence where ~ = 1 and Hˆ is any Hamiltonian, describ- "and it alone" must be stressed: indeed, the work ing whatever interaction between Γ and Ξ. The done in the second half of the last century on state |Ψi is assumed separable the N → ∞ limit of quantum theories is quite comprehensive but it neglects the case when the |Ψi = |Γi ⊗ |Ξi , (2) large-N system is the big partner of a principal quantum system, that only indirectly experiences meaning that we begin studying the evolution at such limit. This is, however, an exemplary situ- a time t = 0 when both Γ and Ξ are in pure ation in quantum technologies and OQS, hence states. This is not a neutral assumption, and we the questions asked at the beginning of this In- will get back to it in Sec.6. troduction have recently been formulated in the At any fixed time τ, there exists a Schmidt corresponding framework [1–3,9–16]. decomposition of the state (1), X In this work, we develop an original approach |Ψ(τ)i = cγ |γi |ξγi , (3) which uses results for the large-N limit of quan- γ tum theories in the framework of OQS dynamics. + This allows us to show that details of the interac- with γ = 1, ..., dimHΓ, cγ ∈ R for γ ≤ γmax ≤ P 2 tion between a quantum principal system Γ and dimHΓ, cγ = 0 for γ > γmax, γ cγ = 1, its environment Ξ are irrelevant in determining and the symbol ⊗ understood (as hereafter done the main features of the state of Ξ at any time whenever convenient). The states {|γi}HΓ , and

τ in the large-N limit, as long as such limit im- {|ξji}HΞ with j = 1, ... dim HΞ, form what we plies a classical behaviour for Ξ itself. If this is will hereafter call the τ-Schmidt bases, to remind the case, indeed, such state can always be recog- that the Schmidt decomposition is state-specific nized as that of an apparatus that measures some and therefore depends on the time τ appearing observable of the principal system. The relation in the LHS of Eq.(3), in whose RHS we have between our findings and the two questions that instead understood the τ-dependence of cγ, |γi, open this section is evident. and |Rγi, for the sake of a lighter notation. Con- sistently with the idea that Ξ is a macroscopic The paper is structured as follows. In the first system, we take γmax < dimHΞ: therefore, the section we define the dynamical maps character- states {|ξγi}HΞ entering Eq.(3) are a subset of izing the two evolutions that we aim at compar- the pertaining τ-Schmidt basis. Given that |Γi ing. We do so through a parametric representa- is fully generic, the unitary evolution (1) defines, tion introduced in Sec.3. In Sec.4, we focus on a via ρΞ = TrΓ ρΨ , the CPTP linear map (from Γ- peculiar property of generalized coherent states, to Ξ-states) particularly relevant when the large-N limit is considered. As the environment is doomed to be X 2 E: |ΓihΓ| → ρΞ = cγ |ξγihξγ| . (4) macrocopic and behave classically, we then im- γ

2 Being the output ρΞ a convex sum of orthogo- Further using the Schmidt coefficients, we con- nal projectors, Eq.(4) might describe a projec- struct the separable state tive measurement acted upon by Ξ on the prin- M M cipal system Γ, by what is often referred to as |Ψ i = |Γi ⊗ |Ξ i , (8) measure-and-prepare (m&p) map. However, for where |Γi is the same as in Eq.(2), while |ΞMi = this being the case, the probability reproducibil- P c |ξ i , with c and |ξ i as in Eq.(3). Finally ity condition [17] must also hold, meaning that, γ γ γ γ γ we define given X ˆ M |Γi = aγ |γi , (5) |ΨMi = e−iH τ |ΨMi , (9) γ τ 2 2 ˆ ˆ it should also be cγ = |aγ| , ∀γ. This condi- that reads, using OΓ |γi = εγ |γi, OΞ |ξγi = P tion, however, cannot be generally true, if only Eγ |ξγi, and |Γi = γ aγ |γi, for the τ-dependence of the Schmidt coefficients M −iHˆ Mτ X X {cγ} which is not featured by the set {aγ}. In |Ψτ i =e aγ |γi cγ0 |ξγ0 i fact, there exists a dynamical model (the Ozawa’s γ γ0 model [5] for projective von Neumann measure- X −iϕγγ0 = aγ |γi cγ0 e |ξγ0 i , (10) 2 ment described in AppendixA) for which cγ = γ,γ0 2 |aγ| , ∀γ and ∀τ. Such model is defined by a Hamiltonian where the operators acting on Γ with ϕγγ0 ≡ τgεγEγ0 ∈ R. Do notice the differ- must commute with each other, a condition that ent notation for the time-dependence in Eqs. (3) identifies what we will hereafter dub a measure- and (9): this is to underline that while the for- like Hamiltonian, Hˆ M, with the apex M hinting mer indicates how the state |Ψi of a system with ˆ at the corresponding measurement process. The Hamiltonian H evolves into |Ψ(t)i at any time t, evolution defined by exp{−itHˆ M} will be consis- the latter represents a state whose dependence on tently dubbed measure-like dynamics 1. τ not only enters as a proper time in the prop- Once established that Eq.(4) does not define a agator, but also, as a parameter, in the defini- ˆ M M m&p map, we can nonetheless use the elements tion of H and |Ξ i, via the τ-dependence of provided by the Schmidt decomposition as in- the Schmidt decomposition (3). Nonetheless, the M gredients to construct a measure-like Hamilto- state |Ψτ i can still be recognized as that in which M nian Hˆ M whose corresponding m&p map, E M : Ψ would be at time τ, were its initial state |Ψ i M and its evolution ruled by the measure-like inter- |ΓihΓ| → ρΞ is the "nearest" possible to the actual E, Eq.(4). action Eq. (7). To this aim, we first use the τ-Schmidt bases Given that |Γi is fully generic, Equation (9)

defines, via ρΞ = TrΓ ρΨ , the CPTP map from Γ- {|γi}HΓ and {|ξji}HΞ to define the hermitian operators to Ξ-states ˆ X ˆ X E M : |ΓihΓ| → ρM = OΓ = εγ |γihγ| , OΞ = Ej |ξjihξj| , (6) Ξ γ j X 2 i(ϕ 00 −ϕ 0 ) = |aγ| cγ0 cγ00 e γγ γγ |ξγ0 ihξγ00 | . γγ0γ00 with εγ,Ej arbitrary real numbers; we then write the interaction Hamiltonian (11) M Hˆ M = gOˆ ⊗ Oˆ , (7) Notice that E depends on τ directly, via ϕγγ0 ∝ Γ Ξ τ, and indirectly, via the τ-dependence of the with g some coupling constant, which has the Schmidt decomposition, that is of the coefficients form prescribed by the Ozawa’s model (see Ap- cγ and the states |ξγi. Comparing Eqs.(4) and pendixA for more details). (11) we see that E M has the right coefficients 2 {|aγ| } but the wrong form, i.e., it is not a sum 1 Giving a Hamiltonian description of more general of orthogonal projectors, while E has the correct quantum measurement processes, i.e., identifying the ap- 2 propriate propagator for the dynamics of such processes form but with the wrong coefficients, {cγ}. In up to the output production, is a very relevant problem fact, were these two maps equal in some limit, it that has recently attracted the interest of several authors, would mean the following: for each time τ, there including some of us. exists an observable for Γ, (depending on τ itself)

3 such that the state into which Ξ has evolved due 10 to its true interaction with Γ is the same, in such limit, as if Ξ itself were some measuring appara- 8 tus proper to that observable, which is quite a 6 statement. Since E and E M are linear, they are the same map iﬀ the output states ρ and ρM Ξ Ξ 4 are equal for whatever input |Γi. We can therefore concentrate upon the structure of such out- 2 put states, which we will do in the next section by introducing a proper parametric representation. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 √ Figure 1: |hα|ni|2 as a function of αα∗, with n = 1 3 Parametric representation with envi- (left) and n = 4 (right), for N = 1, 10, 1000 (bottom to ronmental coherent states top).

The parametric representation with environmen- One of the most relevant byproduct of the GCS tal coherent states (PRECS) is a theoretical tool construction is the definition of a differentiable that has been recently introduced [18, 19] to manifold M via the chain of one-to-one corre- specifically address those bipartite quantum sys- spondences tems where one part, on its own made by N elementary components, shows an emerging classical ωˆ ⊂ G/F ⇔ |ωi ∈ H ⇔ ω ⊂ M , (13) behaviour in the large-N limit [6, 20–23]. The method makes use of generalized coherent states so that to any GCS is univoquely associated a (GCS) for the system intended to become macro- point on M, and viceversa. A measure dµ(ω) on scopic. M is consistently associated to the above intro- The construction of GCS, sometimes referred duced dµ(ˆω), so that requiring GCS to be nor- to as group-theoretic, goes as follows [24]. Asso- malized, hω|ωi = 1, implies ciated to any quantum system there is a Hilbert "Z # space H and a dynamical group G, which is the hω|ωi = hω| dµ(ˆω) |ωihω| |ωi group containing all the propagators that de- G/F scribe possible evolutions of the system (quite Z = dµ(ω)|hω|ωi|2 = 1 ; (14) equivalently, G is the group corresponding to the M Lie algebra g to which all the physical Hamilto- nians of the system belong). Once these ingredi- notice that GCS are not necessarily orthogonal. ents are known, a reference state |0i is arbitrarily One important aspect of the GCS construction chosen in H and the subgroup F of the propaga- is that it ensures the function hω|ρ|ωi for what- tors that leave such state unchanged (apart from ever state ρ (often called Husimi function in the 3 an irrelevant overall phase) is determined. This literature ) is a well-behaved probability distri- is usually referred to as the stability subgroup. bution on M that uniquely identifies ρ itself. As Elements ωˆ of G that do not belong to such sub- a consequence, studying hω|ρ|ωi on M is fully group, ωˆ ∈ G/F, generate the GCS upon act- equivalent to perform a state-tomography of ρ on ing on the reference state, ωˆ |0i = |ωi, and are the Hilbert space, and once GCS are available one usually dubbed "displacement" operators. The can analyze any state ρ of the system by study- GCS construction further entails the definition of ing its Husimi function on M, which is what we an invariant2 measure dµ(ˆω) on G/F such that a will do in the following. We refer the reader to resolution of the identity on H is provided in the Refs. [24, 25] for more details. form Z 3In fact, a "Husimi function" is in principle defined on ˆ dµ(ˆω) |ωihω| = IH . (12) a classical phase-space, while M is a differential manifold G/F with a simplectic structure that should not be considered a phase-space, yet, i.e., before the large-N limit is taken; 2The measure dµ(ˆω) is called invariant because it is however, it is quite conventional to extend the term to the left unchanged by the action of G. expectation value of ρ on GCS.

4 When GCS are relative to a system Ξ which is respectively. the environment of a principal system Γ, we call Comparing χ(ω)2 and χM(ω)2 is equivalent to M them Environmental Coherent States (ECS). compare ρΞ and ρΞ , and hence the maps (4) and Getting back to the setting of section2, we (11). However, despite the very specific construc- M first recognize that, if they were to represent dif- tion leading to |Ψτ i, we cannot yet make any ferent evolutions of the same physical system, the meaningful specific comparison between χ(ω)2 propagators exp{−iHτˆ } and exp{−iHˆ Mτ} must and χM(ω)2 at this stage. Indeed, we still have to belong to the same dynamical group, as far as exploit the fact that the environment is doomed their action on HΞ is concerned. More explicitely, to be big and behave classically, which is why this group is identified as follows: i) consider all ECS turn out to be so relevant to the final result, the operators acting on HΞ in the total Hamilto- as shown in the next section. nians Hˆ and Hˆ M ; ii) find the algebra to which they all belong (notice that, as both Hamiltoni- ans refer to the same physical system, the above 4 Large-N and classical limit operators must belong to the same algebra g; iii) recognize the dynamical group as that associated As mentioned in the Introduction, a physical sys- to the above algebra g via the usual exponential tem which is made by a large number N of quan- Lie map (for several examples see for instance tum constituents does not necessarily obey the Refs. [6, 18, 22, 23]). This is the group to be rules of classical physics. However, several au- used for constructing the ECS, according to the thors [8, 24, 26, 27] have shown that if GCS ex- procedure briefly sketched above. Once ECS are ist and feature some specific properties, then the constructed, the PRECS of any pure state |ψi of structure of a classical theory C emerges from that Ψ is obtained by inserting an identity resolution of a quantum theory Q. In particular, the exis- in the form (12) into any decomposition of |ψi as tence of GCS establishes a relation between the linear combination of separable (w.r.t. the parti- Hilbert space of Q and the manifold M that their tion Ψ = Γ + Ξ) states. Explicitly, one has construction implies, which turns out to be the Z phase-space of the classical theory that emerges |ψi = dµ(ω)χ(ω) |ωi |Γ(ω)i , (15) as the large-N limit of Q. In fact, one should M rather speak about the k → 0 limit of Q, with k where |Γ(ω)i is a normalized state for Γ that the real positive number, referred to as "quantic- parametrically depends on ω, while χ(ω) is a real ity parameter", such that all the commutators of function on M whose square the theory (or anticommutators, in the fermionic case) vanish with k. However, all known quan- χ(ω)2 = hω|ρ |ωi , (16) Ξ tum theories for systems made by N components 1 is the environmental Husimi function relative to have k ∼ N p with p a positive number: therefore,

ρΞ = TrΓ |ψihψ|, i.e., the normalized distribution for the sake of clarity, we will not hereafter use on M that here represents the probability for the the vanishing of the quanticity parameters but environment Ξ to be in the GCS |ωi when Ψ is in rather refer to the large-N limit (see AppendixB the pure state |ψi. The explicit form of χ(ω) and for more details). |Γ(ω)i is obtained from any decomposition of |ψi Amongst the above properties of GCS, that are into a linear combination of separable (w.r.t. the thoroughly explained and discussed in Ref. [8] as partition Γ + Ξ) states. the assumptions guaranteeing the large-N limit In particular, for the states (3) and (10), it is to define a classical theory, one that plays a key 2 X 2 2 role in this work regards the overlaps hω|ξi, whose χ(ω) = cγ |hω| ξγi| , (17) γ square modulus represents the probability that a system in some generic pure state |ξi be observed and in the coherent state |ωi. These overlaps never χM(ω)2 = vanish for finite N, due to the overcompleteness of GCS: as a consequence, if one considers two X 2 i(ϕ 00 −ϕ 0 ) = |a | c 0 c 00 e γγ γγ hω| ξ 0 i hξ 00 | ωi , γ γ γ γ γ orthonormal states, say |ξ0i and |ξ00i, there might γγ0γ00 be a finite probability for a system in a GCS |ωi to (18) be observed either in |ξ0i or in |ξ00i. This formally

5 , ’

Figure 2: Sum |hα|n0i|2 + |hα|n00i|2 with n0 = 1 and n00 = 4 for N = 1, 10, 1000 (left to right): Contourplot on part of M, which is now the complex plane (values increase from blue to red).

q implies that, defined S the set of points on M (†) 2~ (†) ξ aˆ → Mω aˆ , and observe that all the com- where |hω|ξi| > 0, it generally is Sξ0 ∩ Sξ00 6= ∅. mutators vanish in the large-M limit. Further On the other hand, the quantity taking M ∝ N, meaning that the total mass of Ξ is the sum of the masses of the elemen- lim |hω|ξi|2 (19) N→∞ tary components, which are assumed to have the same mass for the sake of simplicity, it is easily features some very relevant properties. First of found that k ∼ 1/N. As for the GCS , they are 0 all, if |ξi is another GCS, say |ω i, the square the well known field coherent states {|αi}, with 0 2 modulus |hω|ω i| exponentially vanishes with |0i :a ˆ |0i = 0 the reference state, and M the 0 2 |ω − ω | in such a way that the limit (19) con- complex plane. The eigenstates of nˆ are the Fock 0 verges to the Dirac distribution δ(ω − ω ), thus states {|ni}, and exp{αaˆ − α∗aˆ†} ≡ αˆ is the dis- restoring a notion of distinguishability between placement operator such that |αi =α ˆ |0i. different GCS in the large-N limit. Moreover, in AppendixC we demonstrate that As for the overlaps entering Eq.(18), let us first consider the case when the states {|ξγi} are Fock 0 00 states. In Fig.1 we show |hα|ni|2 as a function hξ |ξ i = δξ0ξ00 ⇔ lim Sξ0 ∩ Sξ00 = ∅ , (20) N→∞ of |α|2, for n = 1, 2 and different values of N. It meaning that orthonormal states are put together is clearly seen that Sn0 ∩ Sn00 → ∅ as N → ∞, by distinguishable sets of GCS. In other terms, meaning that the product of overlaps in Eq.(18) 0 00 0 00 the large-N limit enforces the emergence of a one- vanishes unless γ = γ , i.e. n = n in this to-one correspondence between elements of any specific example. In order to better visualize Sn0 orthonormal basis {|ξi} and disjoint sets of GCS, and Sn00 on M, in Fig.2 we contour-plot the sum 2 2 in such a way that the distinguishability of the |hα|1i| + |hα|2i| : indeed we see that, as N in- former is reflected into the disjunction of the lat- creases, S1 and S2 do not intersect. Notice that ter. Given the relevance of Eq.(20) to this work, increasing N does not squeeze Sn to the neigh- let us discuss its meaning with two explicit exam- bourghood of some point on M, as is the case for 0 2 0 ples. limN→∞ |hα|α i| = 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, 4.1 Field Coherent States but different Fock states overlap with distinct sets Consider a system Ξ whose Lie algebra is h4, i.e., of field coherent states, in the large-N limit. This the vector space spanned by {a,ˆ aˆ†, nˆ ≡ aˆ†a,ˆ Î}, picture holds not only for Fock states but, as ex- with Lie brackets [â, aˆ†] = 1, and [â(†), nˆ] = pressed by Eq.(20), for any pair of orthonormal (†) (−)â . In order to identify the quanticity pa- states. In Fig.3, for instance, we contour-plot the√ rameter k, i.e., the parameter whose vanishing sum |hα|+i|2 +hα|−i|2 with |±i ≡ (|1i±|2i)/ 2: makes the Lie brackets of the theory go to zero, in this case S+ and S− are disjoint already for one can restore dimensionful ladder operators, N = 1, and keep shrinking as N increases.

6 , ’ √ Figure 3: Sum |hα|+i|2 + |hα|−i|2 with |±i = (|1i ± |2i)/ 2, for N = 1, 10, 1000 (left to right): Contourplot on M, which is now the complex plane (values increase from blue to red).

4.2 Spin Coherent States 14

A very similar scenario appears when studying 10 a system Ξ whose Lie algebra is su(2), i.e., the 8 vector space spanned by {Sˆ+, Sˆ−, Sˆz}, with Lie brackets [Sˆ+, Sˆ−] = 2Sˆz, [Sˆz, Sˆ±] = ±Sˆ±, and 6 2 |Sˆ| = S(S + 1), with S fixed and constant; 4 in this case the quanticity parameter is identi- 2 fied by noticing that the normalized operators ∗ 1 ˆ∗ sˆ ≡ S S , ∗ = z, ±, have vanishing commuta- 0.0 0.5 1.0 1.5 tors in the large-S limit. Further taking S ∝ N, Figure 4: |hΩ|mi|2 as a function of θ, for m/S = 0.8 meaning that the total spin of Ξ is a conserved (left) and 0.4 (right), for N = 10, 100, 1000 (bottom to quantity, whose value is the sum of the spins top). of each individual component, it is easily found that k ∼ 1/N. As for the GCS , they are the 0 2 0 so-called spin (or atomic) coherent states {|Ωi}, case for limN→∞ |hΩ|Ω i| = δ(Ω−Ω ), but rather with the reference state |0i : Sˆz |0i = −S |0i, into that of the parallel cos θ = m/S. and M the unit sphere. The eigenstates of Sˆz are {|mi} : Sˆz |mi = (−S + m) |mi, and the displacement operators are Ωˆ = exp{ηSˆ− − η∗Sˆ+}, 5 A macroscopic environment that be- with η = θ eiφ, and θ ∈ [0, π], φ ∈ [0, 2π) the 2 haves classically spherical coordinates. As for the overlaps entering Eq.(18), the analytical expression for hΩ|mi is Let us now get back to the general case and to available (see for instance Ref. [24]), which allows Eq.(18): the states |ξ 0 i and |ξ 00 i are othonormal us to show, in Fig.4, the square modulus |hΩ|mi|2 γ γ by definition, being elements of the τ-Schmidt for m0/S = 0.8 and m00/S = 0.4, for different val- basis {|ξji}H introduced in Sec.2. Therefore ues of N. Again we see that S 0 ∩ S 00 → ∅ as Ξ m m Eq.(20) holds, meaning N → ∞, implying that the product in Eq.(18) vanishes unless γ0 = γ00, i.e., m0 = m00 in this 2 lim hω|ξγ0 ihξγ00 |ωi = lim |hω|ξγ0 i| δγ0γ00 , specific example. In Fig.5 we show the sum N→∞ N→∞ |hΩ|m0i|2 + |hΩ|m00i|2 as density-plot on part of (21) the unit sphere: besides the expected shrinking and hence of the regions where the overlaps are finite, we notice that, as seen in the bosonic case, the sup- M 2 X 2 2 2 lim χ (ω) = |aγ| cγ0 lim |hω|ξγ0 i| . 2 N→∞ N→∞ port of limN→∞ |hΩ|mi| does not shrink into the γγ0 neighbourghood of a point on the sphere, as is the (22)

7 ,

Figure 5: Sum |hΩ|m0i|2 + |hΩ|m”i|2 with m0/S = 0.8 and m00/S = 0.4, for N = 10, 100, 1000 (left to right): Densityplot on part of M, which is now the unit sphere (values increase from blue to red).

P 2 0 Using γ |aγ| = 1, and the swap γ ↔ γ, we when Ψ is not initially in a pure state is similarly finally obtain tackled by enlarging Ψ → Ψe as much as necessary for Ψe to be in a pure state: a proper choice of a lim χM(ω)2 = lim χ(ω)2 , (23) N→∞ N→∞ new partition of Ψe will follow. which is what we wanted to prove, namely that We then want to clarify in what sense the the the dynamical maps (4) and (11) are equal Hamiltonian (7) is said to induce a "measure-like when Ξ is a quantum macroscopic system whose dynamics" or, which is quite equivalent, the chan- behaviour can be effectively described classically. nel (11) to define a m&p map: the quotes indicate that the actual output production, which hap- 6 Discussion pens at a certain time according to some process whose nature we do not discuss, is not considered Aim of this section is to comment upon some spe- and it only enters the description via the require- cific aspects of our results, with possible reference ment that the probability for each output is the to the way other authors have recently tackled the one predicted by Born’s rule. To this respect, one same subject. Let us first consider the assump- might also ask what is the property of Γ which tion that the initial state (2) of the total system is observed by Ξ: this is the one represented, in ˆ Ψ = Γ + Ξ be separable. If this is not the case, the Ozawa’s model, by the operator OΓ, and it as it may happen, one must look for the different therefore depends on the true evolution via the partition Ψ = A + B, such that |Ψi = |Ai ⊗ |Bi. Schmidt decomposition of the evolved state. To If this partition is still such that the subsystem B put it another way, details of the interaction do is macroscopic and behaves classically, the change not modify the measure-like nature of the dynam- is harmless and the whole construction can be ics in the large-N limit, but they do affect what repeated with A the quantum system being ob- actual measurement is performed by the environ- served and B its observing environment. On the ment. other hand, if the new partition is such that nei- Let us now discuss possible connections be- ther A nor B meet the conditions for being a tween our results and Quantum Darwinism [2,1]. classical environment, then the problem reduces As mentioned at the end of AppendixB, a suf- to the usual one of studying the dynamics of two ficient condition for a quantum theory to have a interacting quantum systems, for which any ap- large-N limit which is a classical theory is the proach based on effective descriptions is incon- existence of a global symmetry, i.e., such that its grous, as details of the true Hamiltonian will al- group-elements act non-trivially upon the Hilbert ways be relevant. Notice that this analysis is fully space of each and every component of the to- consistent with the results presented in Ref. [1], tal system Ξ that the theory describes. In fact, which are embodied into inequalities whose mean- few simple examples show that quantum theo- ing wears off as dimHB diminishes. The case ries with different global symmetries can flow

8 into the same classical theory in the large-N tively, we have addressed the above three issues limit: in other words, echoing L. G. Yaffe in as follows. As for the first point, the analysis is Ref. [8], different quantum theories can be "clas- developed by comparing CPTP linear maps from sically equivalent". If one further argues that Γ- to Ξ-states, that do not depend on the initial amongst classically equivalent quantum theories state of Γ by definition. The considered maps, there always exists a free theory, describing N Eqs. (4) and (11), are defined using ingredients non-interacting subsystems, it is possible to show provided by the Schmidt decomposition of the that each macroscopic fragment of Ξ can be ef- system-plus-environment evolved state, Eq. (1), fectively described as if it were the same mea- that exists at any time, and whatever the form of surement apparatus. Work on this point is in the interaction between Γ and Ξ is. Regarding the progress, based on the quantum de Finetti theo- second issue, we have used a parametric represen- rem, results from Refs. [10,1], and the prelimi- tation of the overall system state, Eq. (15), that nary analysis reported in Refs. [28, 29]. We close resorts to generalized coherent states (i.e., coher- this section by mentioning the possible connec- ent states as defined via the group-theoretical ap- tion between our description and the way the no- proach) for describing Ξ. This representation, tion of "objective information" is seen to emerge both for its parametric nature and the peculiar in Ref. [3]: in fact, the idea that there may be properties of coherent states when the quantum- no quantum-to-classical transition involved in the to-classical crossover is considered, allows us to perception of the world around us, that might implement the large-N limit for Ξ without mak- rather emerge just as a reflection of some spe- ing assumptions on Γ or affecting its quantum cific properties of the underlying quantum states, character. The third point has been tackled by seems to be consistent with the discussion re- using results from large-N quantum field theories: ported above, and we believe that further inves- these results provided us with formal conditions tigation on this point might be enlightening. that generalized coherent states must fulfill, particularly Eq. (26), in order to describe a macroscopic system that behaves classically. 7 Conclusions After this elaboration, we have managed to The idea that the interaction with macro- compare the map defined by the true evolu- scopic environments causes the continual state- tion of Γ + Ξ, Eq. (4), with that correspond- reduction of any quantum system is crucial for ing to a measure-and-prepare dynamical process, making sense of our everyday experience w.r.t. Eq. (11), in terms of the difference between prob- the quantum description of nature. However, the ability functions entering the parametric repre- formal analysis of this idea has been unsatisfac- sentation, Eqs. (17) and (18). These functions tory for decades, due to several reasons, amongst have been demonstrated to become equal when which we underline the following. the large-N limit defines a classical dynamics for Firstly the generality of the above idea implies Ξ. that assumptions on the initial state of the quan- Overall, our approach allows one to tackle the tum system, and the specific form of the interac- so-called quantum to classical crossover [30] by a tion with its environment, should not be made. rigorous mathematical formulation that provides Secondly, formal tools must be devised to allow a physically intuitive picture of the underlying the study of the system-plus-environment dynam- dynamical process. In fact, exploiting the most ics in a way that guarantees a genuinely quan- relevant fact that not every theory has a classi- tum description of the system throughout the cal limit, we have shown that any dynamics of crossover of the environment towards a classical whatever OQS defines a Hamiltonian model that behaviour. Finally, a clean procedure is required characterizes its environment as a measuring ap- to ensure that the above crossover takes place paratus if the conditions ensuring that the above when the environment becomes macroscopic, i.e., classical limit exists and corresponds to a large- in the large-N limit of the quantum theory that N condition upon the environment itself are ful- describes it. filled. In other words, if some dynamics emerges In this work, reminding that principal system in the classical world, it necessarily is a measure- and environment are dubbed Γ and Ξ, respec- like one.

9 Let us conclude by briefly commenting upon 2016. ISBN 3319433873, 9783319433875. the already mentioned phenomenon known as DOI: 10.1007/978-3-319-43389-9. Quantum Darwinism, introduced in [2] and re- [5] M. Ozawa. Quantum measuring processes cently considered in [1] from an information the- of continuous observables. Journal of Math- oretic viewpoint. Our work suggests that Quan- ematical Physics, 25(1):79–87, 1984. DOI: tum Darwinism might emerge as a dynamical pro- 10.1063/1.526000. cess, with its generality due to the versatilility of [6] P. Liuzzo Scorpo, A. Cuccoli, and P. Ver- the Hamiltonian model for the quantum measure- rucchi. Parametric description of the quan- ment process, and the loss of resolution inherent tum measurement process. EPL (Euro- in the classical description. physics Letters), 111(4):40008, 2015. DOI: 10.1209/0295-5075/111/40008. Acknowledgments [7] T. Heinosaari and M. Ziman. The Math- ematical Language of Quantum Theory: CF acknowledges M. Piani and M. Ziman for From Uncertainty to Entanglement. Cam- useful and stimulating discussions. SM and bridge University Press, 2012. DOI: TH acknowledge financial support from the 10.1017/CBO9781139031103. Academy of Finland via the Centre of Excel- [8] Laurence G. Yaffe. Large n limits as classi- lence program (Project no. 312058) as well as cal mechanics. Rev. Mod. Phys., 54:407–435, Project no. 287750. CF and PV acknowledge 1982. DOI: 10.1103/RevModPhys.54.407. financial support from the University of Florence [9] D. Braun, F. Haake, and W.T. Strunz. Uni- in the framework of the University Strategic versality of decoherence. Phys. Rev. Lett., Project Program 2015 (project BRS00215). 86:2913–2917, 2001. DOI: 10.1103/Phys- PV acknowledges financial support from the RevLett.86.2913. Italian National Research Council (CNR) via the "Short term mobility" program STM-2015, and [10] G. Chiribella and G.M. D’Ariano. Quan- declares to have worked in the framework of the tum information becomes classical when Convenzione Operativa between the Institute for distributed to many users. Phys. Rev. Complex Systems of CNR and the Department Lett., 97:250503, 2006. DOI: 10.1103/Phys- of Physics and Astronomy of the University of RevLett.97.250503. Florence. Finally, CF and PV warmly thank the [11] F. Galve, R. Zambrini, and S. Maniscalco. Turku Centre for Quantum Physics for the kind Non-markovianity hinders quantum darwin- hospitality. ism. Scientific Reports, 6:19607, 2016. DOI: 10.1038/srep19607. [12] G.L. Giorgi, F. Galve, and R. Zambrini. Quantum darwinism and non-markovian dis- References sipative dynamics from quantum phases of the spin-1/2 xx model. Phys. Rev. [1] F.G.S.L. Brandao, M. Piani, and A, 92:022105, 2015. DOI: 10.1103/Phys- P. Horodecki. Generic emergence of RevA.92.022105. classical features in quantum darwinism. Nature Communications, 6:7908, 2015. DOI: [13] L. Rigovacca, A. Farace, A. De Pasquale, 10.1038/ncomms8908. and V. Giovannetti. Gaussian discriminat- [2] W. H. Zurek. Quantum darwinism. ing strength. Phys. Rev. A, 92:042331, 2015. Nature Physics, 5:181, 2009. DOI: DOI: 10.1103/PhysRevA.92.042331. 10.1038/nphys1202. [14] P.A. Knott, T. Tufarelli, M. Piani, and [3] R. Horodecki, J.K. Korbicz, and G. Adesso. Generic emergence of objec- P. Horodecki. Quantum origins of ob- tivity of observables in infinite dimensions. jectivity. Phys. Rev. A, 91:032122, 2015. Phys. Rev. Lett., 121:160401, 2018. DOI: DOI: 10.1103/PhysRevA.91.032122. 10.1103/PhysRevLett.121.160401. [4] P. Busch, P. Lahti, J. P. Pellonp, and K. Yli- [15] J. K. Korbicz, E. A. Aguilar, P. Ćwik- nen. Quantum Measurement. Springer Pub- liński, and P. Horodecki. Generic appear- lishing Company, Incorporated, 1st edition, ance of objective results in quantum mea-

10 surements. Phys. Rev. A, 96:032124, 2017. pani, P. Verrucchi, and M.G.A. Paris. Ef- DOI: 10.1103/PhysRevA.96.032124. fective description of the short-time dynam- [16] G. Pleasance and B.M. Garraway. Applica- ics in open quantum systems. Phys. Rev. tion of quantum darwinism to a structured A, 96:032116, 2017. DOI: 10.1103/Phys- environment. Phys. Rev. A, 96:062105, 2017. RevA.96.032116. DOI: 10.1103/PhysRevA.96.062105. [24] W.M. Zhang, D.H. Feng, and R. Gilmore. [17] P. Busch, P.J. Lahti, and P Mittel- Coherent states: Theory and some applica- staedt. The quantum theory of measure- tions. Rev. Mod. Phys., 62:867–927, 1990. ment. Springer-Verlag, Berlin, 1996. DOI: DOI: 10.1103/RevModPhys.62.867. 10.1007/978-3-540-37205-9. [25] A.M. Perelomov. Coherent states for arbi- [18] D. Calvani, A. Cuccoli, N. I. Gidopoulos, trary Lie group. Communications in Mathe- and P. Verrucchi. Parametric representa- matical Physics, 26(3):222–236, 1972. ISSN tion of open quantum systems and cross- 0010-3616. DOI: 10.1007/BF01645091. over from quantum to classical environ- [26] E. H. Lieb. The classical limit of quantum ment. Proceedings of the National Academy spin systems. Communications in Mathe- of Sciences, 110(17):6748–6753, 2013. DOI: matical Physics, 31(4):327–340, 1973. DOI: 10.1073/pnas.1217776110. 10.1007/BF01646493. [19] D. Calvani. The Parametric Representation [27] S. Gnutzmann and M. Kus. Coherent states of an Open Quantum System. PhD thesis, and the classical limit on irreducible su(3) Università degli Studi di Firenze, 2012. representations. Journal of Physics A: Math- [20] D. Calvani, A. Cuccoli, N. I. Gidopoulos, ematical and General, 31(49):9871, 1998. and P. Verrucchi. Dynamics of open quan- DOI: 10.1088/0305-4470/31/49/011. tum systems using parametric representa- [28] L. Querini. How quantum dynamics shape tion with coherent states. Open Systems & macroscopic evidences. Master thesis, Uni- Information Dynamics, 20(3):1340002, 2013. versity of Florence, 2016. DOI: 10.1142/S1230161213400027. [29] C. Foti. On the macroscopic limit of quan- [21] P. Liuzzo Scorpo, A. Cuccoli, and P. Verruc- tum systems. PhD thesis, Università degli chi. Getting information via a quantum mea- Studi di Firenze, 2019. surement: The role of decoherence. Int. J. Theor. Phys., 54(12):4356–4366, 2015. ISSN [30] M. Schlosshauer. Decoherence and the 1572-9575. DOI: 10.1007/s10773-015-2548- Quantum-To-Classical Transition. The 8. Frontiers Collection. Springer, 2007. DOI: [22] C. Foti, A. Cuccoli, and P. Verrucchi. Quan- 10.1007/978-3-540-35775-9. tum dynamics of a macroscopic magnet op- [31] F. A. Berezin. Models of gross-neveu erating as an environment of a mechanical type are quantization of a classical mechan- oscillator. Phys. Rev. A, 94:062127, 2016. ics with nonlinear phase space. Comm. DOI: 10.1103/PhysRevA.94.062127. Math. Phys., 63(2):131–153, 1978. DOI: [23] M.A.C. Rossi, C. Foti, A. Cuccoli, J. Tra- 10.1007/BF01220849.

11 A From Ozawa’s model to the measure-and-prepare map

Given a projective measurement with measurement operators {|πihπ|} acting on HΓ, its dynamical description according to the Ozawa’s model is deﬁned by the propagator exp{−itHˆ M}, with

M Hˆ = gOˆΓ ⊗ OˆΞ , (24) ˆ P ˆ where OΓ = π ωπ |πihπ| is the measured observable, while OΞ is the operator on HΞ conjugate to the pointer observable [30]. The resulting, measure-like, dynamics is such that decoherence of P 2 π π π π0 ρΓ(t) w.r.t. the basis {|πi} implies ρΞ(t) = π |aπ| |Ξt ihΞt | with hΞt |Ξt i = δππ0 and aπ such P that |Γ(0)i = π aπ |πi, and viceversa. Here t indicates any time prior the output production when decoherence has already occurred. This dynamics deﬁnes a CPTP map E M via

B Large-N as classical limit

In order to define the classical limit of a quantum theory Q it is first necessary to identify a parameter k, usually dubbed "quanticity parameter", such that Q transforms into a classical theory C as k vanishes. By "transform" it is meant that a formal relation is set between Hilbert and phase spaces, Lie and Poisson brackets, Hamiltonian operators and functions. Consequently, the large-N limit of Q implies a classical behaviour of the macroscopic system it describes IF N → ∞ implies k → 0. On the other hand, in order for this being the case it proves sufficient that GCS {|ωi}for Q exist and feature some specific properties [31,8]. Amongst these, particularly relevant to this work is that h i lim k ln |hω0|ωi| ≤ 0 , (26) k→0 where the equality holds iff ω = ω0, and the property implies the limit exists. From the above property it follows4 1 lim |hω|ω0i|2 = δ(ω − ω0) , (27) k→0 k which is a most relevant properties of GCS, namely that they become orthogonal in the classical limit. It is worth mentioning that if Q features a global symmetry (also dubbed "supersymmetry" in the literature), GCS can be explicitly constructed and shown to feature the properties ensuring that the large-N limit is indeed a classical one [8]. However, whether the existence of one such symmetry be a necessary condition for a system to behave classically in the large-N limit is not proven, although all of the known physical theories, be they vector-, matrix-, or gauge-theories, confirm the statement (see Sec.VII of Ref. [8] for a thorough discussion about this point). Incidentally, we believe the above supersymmetry be essential in defining what a macroscopic observer should actually be in order for Quantum Darwinism to occur, in a way similar to that discussed in Ref. [10] in the specific case of a quantum theory for N distinguishable particles with permutation global symmetry.

C Overlap between GCS and elements of an orthonormal basis in the large-N limit

One of the output of the GCS construction, and key-ingredient for their use, is the invariant measure dµ(ˆω) entering the identity resolution Eq.(12). It is demonstrated [8] that in order for such resolution

4 2 We use the Dirac-δ representation δ(x − y) = lim→0(1/) exp{(x − y) /}.

12 to keep holding for whatever value of the quanticity parameter k it must be dµ(ω) = ckdm(ω), with ck a constant on M that depends on the normalization of the group-measure dµ(ˆω) and should be computed on a case-by-case basis. However, normalization of GCS is guaranteed by construction, and hence, via Eq.(12), Z 0 0 2 hω|ωi = ckdm(ω )|hω|ω i| = 1 , ∀ |ωi ; (28) M Furthermore, from Eq.(27) it follows |hω|ω0i|2 → kδ(ω − ω0) as k vanishes, and hence Z 0 0 lim ckk dm(ω )δ(ω − ω ) = 1 , (29) k→0 M

1 which implies ck = k , as readily verified in those cases where an explicit form of GCS is available. The 1 fact that ck is independent of ω and goes like k for vanishing k, enforces Z 1 0 00 lim dm(ω)hξ |ωihω|ξ i = δξ0 ξ00 (30) k→0 M k to hold for whatever pair (|ξ0 i , |ξ00i) of orthonormal states: as neither dm(ω) nor M depend on k, this is only possible if the two overlaps entering the integral are never simultaneously finite on M or, more precisely, on a set of finite measure. In other terms, Eq.(30) implies Eq.(20), and viceversa (which is trivial).