Bulk-boundary asymptotic equivalence of two strict deformation quantizations

The existence of a strict deformation quantization of $X_k=S(M_k({\mathbb{C}}))$, the state space of the $k\times k$ matrices $M_k({\mathbb{C}})$ which is canonically a compact Poisson manifold (with stratified boundary) has recently been proven by both authors and K. Landsman \cite{LMV}. In fact, since increasing tensor powers of the $k\times k$ matrices $M_k({\mathbb{C}})$ are known to give rise to a continuous bundle of $C^*$-algebras over $I=\{0\}\cup 1/\mathbb{N}\subset[0,1]$ with fibers $A_{1/N}=M_k({\mathbb{C}})^{\otimes N}$ and $A_0=C(X_k)$, we were able to define a strict deformation quantization of $X_k$ \`{a} la Rieffel, specified by quantization maps $Q_{1/N}: \tilde{A}_0\rightarrow A_{1/N}$, with $\tilde{A}_0$ a dense Poisson subalgebra of $A_0$. A similar result is known for the symplectic manifold $S^2\subset\mathbb{R}^3$, for which in this case the fibers $A'_{1/N}=M_{N+1}(\mathbb{C})\cong B(\text{Sym}^N(\mathbb{C}^2))$ and $A_0'=C(S^2)$ form a continuous bundle of $C^*$-algebras over the same base space $I$, and where quantization is specified by (a priori different) quantization maps $Q_{1/N}': \tilde{A}_0' \rightarrow A_{1/N}'$. In this paper we focus on the particular case $X_2\cong B^3$ (i.e the unit three-ball in $\mathbb{R}^3$) and show that for any function $f\in \tilde{A}_0$ one has $\lim_{N\to\infty}||(Q_{1/N}(f))|_{\text{Sym}^N(\mathbb{C}^2)}-Q_{1/N}'(f|_{_{S^2}})||_N=0$, were $\text{Sym}^N(\mathbb{C}^2)$ denotes the symmetric subspace of $(\mathbb{C}^2)^{N \otimes}$. Finally, we give an application regarding the (quantum) Curie-Weiss model.

Abstract. The existence of a strict deformation quantization of X k = S(M k (C)), the state space of the k ×k matrices M k (C) which is canonically a compact Poisson manifold (with stratified boundary) has recently been proven by both authors and K. Landsman [15]. In fact, since increasing tensor powers of the k × k matrices M k (C) are known to give rise to a continuous bundle of C * -algebras over I = {0}∪ 1/N ⊂ [0, 1] with fibers A 1/N = M k (C) ⊗N and A 0 = C(X k ), we were able to define a strict deformation quantization of X kà la Rieffel, specified by quantization maps Q 1/N :Ã 0 → A 1/N , withÃ 0 a dense Poisson subalgebra of A 0 . A similar result is known for the symplectic manifold S 2 ⊂ R 3 , for which in this case the fibers A ′ 1/N = M N +1 (C) ∼ = B(Sym N (C 2 )) and A ′ 0 = C(S 2 ) form a continuous bundle of C * -algebras over the same base space I, and where quantization is specified by (a priori different) quantization maps Q ′ 1/N :Ã ′ 0 → A ′ 1/N . In this paper we focus on the particular case X 2 ∼ = B 3 (i.e the unit three-ball in R 3 ) and show that for any function f ∈Ã 0 one has lim N →∞ ||(Q 1/N (f ))| Sym N (C 2 ) − Q ′ 1/N (f | S 2 )|| N = 0, were Sym N (C 2 ) denotes the symmetric subspace of (C 2 ) N ⊗ . Finally, we give an application regarding the (quantum) Curie-Weiss model.

Introduction
An important field of research within mathemetical physics concerns the relation between classical theories viewed as limits of quantum theories. For example, classical mechanics of a particle on the phase space R 2n versus quantum mechanics on the Hilbert space L 2 (R n ), or classical thermodynamics of a spin system versus statistical mechanics of a quantum spin system on a finite lattice [14].
In these examples the relation between both (different) theories can be described by a continuous bundle of algebras of observables equipped with certain quantization maps. A modern way establishing a link between both theories is based on the concept of strict deformation quantization, i.e., the mathematical formalism that describes the transition from a classical theory to a quantum theory [21,22,13] in terms of deformations of (commutative) Poisson algebras (representing the classical theory) into non-commutative C * algebras characterizing the quantum theory.

Strict deformation quantization maps
Let us focus to the first known example starting from the familiar classical phase space R 2n . For convenience, we only consider the Poisson algebra of smooth compactly supported functions f ∈ C ∞ c (R 2n ) where the Poisson structure is the one associated to the natural symplectic form n j=1 dp j ∧dq j . In order to relate C ∞ c (R 2n ) to a quantum theory described on some Hilbert space, one needs to deform C ∞ c (R 2n ) into non-commutatative C * -algebras exploiting a family of quantization maps. Weyl proposed the quantization maps [21,22,7,14] Q : where ∈ (0, 1]; B 0 (H) is the C * -algebra of compact operators on the Hilbert space H = L 2 (R n ), and for each point (p, q) ∈ R 2n the (projection) operator defining the well-known (Schrodinger) coherent states. Inspired by Dixmier's concept of a continuous bundle [7], Rieffel showed that [21,22] 1. The fibers A 0 = C 0 (R 2n ) and A = B 0 (L 2 (R n )), h ∈ (0, 1], can be combined into a (locally non-trivial) continuous bundle A of C * -algebras over I = [0, 1]; 3. Each quantization map Q :Ã 0 → A is linear, and if we also define Q 0 :Ã 0 ֒→ A 0 as the inclusion map, then the ensuing family Q = (Q ) ∈I satisfies: (b) For each f ∈Ã 0 the following cross-section of the bundle is continuous: (1.5) (c) Each pair f, g ∈Ã 0 satisfies the Dirac-Groenewold-Rieffel condition: This led to the general concept of a strict deformation of a Poisson manifold X [21,13], which we here state in the case of interest to us in which X is compact, or more generally in which X is a manifold with stratified boundary [15,19]. In that case the space I in which takes values cannot be all of [0, 1], but should be a subspace I ⊂ [0, 1] thereof that at least contains 0 as an accumulation point. This is assumed in what follows. Furthermore, the Poisson bracket on X is denoted, as usual, by {·, ·} : is suitably defined when X is a more complicated object thatn a compact smooth manifold as we shall say shortly).
Definition 1.1. A strict deformation quantization of a compact Poisson manifold X consists of an index space I ⊂ [0, 1], including 0 as accumulation point, for as detailed above, as well as: • A continuous bundle of unital C * -algebras (A ) ∈I over I with A 0 = C(X) equipped with the standard commutative C * -algebra structure with respect to the norm · ∞ ; • A family Q = (Q ) ∈I of linear maps Q :Ã 0 → A indexed by ∈ I (called quantization maps) such that Q 0 is the inclusion mapÃ 0 ֒→ A 0 , and the above conditions (a) -(c) hold, as well as Q (1 X ) = 1 A (the unit of A ).
It follows from the definition of a continuous bundle of C * -algebras that two continuity properties holds hold automatically [13,14].

Spin systems and generalizations
Mean-field quantum spin systems 1 fit into this framework. There, the index set I is given by (0 / ∈ N := {1, 2, 3, . . .}) with the topology inherited from [0, 1]. That is, we put = 1/N, where N ∈ N is interpreted as the number of sites of the model; our interest is the limit N → ∞. In the framework of C * -algebraic quantization theory, the analogy between the "classical" limit → 0 in typical examples from mechanics (see e.g. our first example [10]) and the "thermodynamic" limit N → ∞ in typical quantum spin systems (see e.g. [16,15]) is developed in detail in [14]. We remark that the limit N → ∞ can be taken in two entirely different ways, which depends on the class of observables one considers, namely either quasi-local observables or macroscopic observables. The former are the ones traditionally studied for quantum spin systems, but the latter relate these systems to strict deformation quantization, since macroscopic observables are precisely defined by (quasi-) symmetric sequences which form the continuous cross sections of a continuous bundle of C * -algebras. This continuous bundle of C * -algebras is defined over base space I given by (1.9) with fibers (1.10) 11) and continuity structure specified by continuous cross-sections which are thus given by all quasi-symmetric sequences [15] [14, Ch.10]. 2 We refer to the appendix for some useful definitions, or to [15] for a more comprehensive explanation. The space X k = S(M k (C)) ⊂ R k 2 −1 has the structure of a compact Poisson manifold with stratified boundary. The space C ∞ (X k ) is here made of the restrictions to X k of the smooth functions in R k 2 −1 and the Poisson bracket is the restriction for f, g ∈ C ∞ (R k 2 −1 ) and where C c ab are the structure constants of SU(k) (see Sect. 2.3 of [15] for details). In turn, the Poisson algebraÃ 0 dense in A 0 = C(X k ) is made of the restrictions to X k of the polynomials in the k 2 −1 coordinates of R k 2 −1 Let us pass to describe Q 1/N . Each polynomial p of degree L uniquely corresponds to a polynomial of symmetric elementary tensors of the form b j 1 ⊗ s · · · ⊗ s b j L , where ib 1 , . . . , ib k 2 −1 form a basis of the Lie algebra of SU(k). That is the image of p according to Q 1/N . More precisely, if the quantization maps Q 1/N :Ã 0 → M k (C) N act as (see the appendix for S L,N ) and more generally they are defined as the unique continuous and linear extensions of the written maps. It has been shown in [15] that the quantization maps Q 1/N satisfy all the axioms of Definition 1.1. 3 These data together imply the existence of a strict deformation quantization of the Poisson manifold X k = S(M k (C)) (see [15,Theorem 3.4] for a detailed proof). We specialize these models to the case k = 2. One-dimensional quantum spin systems arising in that way are widely studied in (condensed matter) physics, but also in mathematical physics they form an important field of research, especially in view of spontaneous symmetry breaking (SSB). One tries to calculate quantities like the free energy, or the entropy of the system in question and considers their thermodynamic limit as the number of sites N increas to infinity [16]. For this reason the case k = 2 is already of huge interest, since each site of such a spin chain is exactly described by the algebra of (2 × 2)-matrices. On the other hand the Bloch sphere S 2 acting as a classical phase space which describs a physical system may be a spin system of total spin j, but it can also be a collection of n two-level atoms [2] corresponding to a spin chain of n sites, which is for example the case for the quantum Curie-Weiss model [15]. Inspired by that model, which admits a classical limit 4 on S 2 (i.e. the smooth boundary of X 2 = S(M 2 (C)) ∼ = B 3 , where B 3 denotes the closed unit ball in R 3 ), we asked ourselves if the quantization maps Q 1/N quantizing X 2 could in general be related to another well-known strict deformation quantization of S 2 whose details are explained in what follows. 5 From the mathematical side, we observe that k = 2 is the unique case where X k admits a smooth boundary, as said X 2 = B 3 and ∂X 2 = S 2 . Furthermore S 2 is a Poisson submanifold of B 3 , when the latter is equipped with the Poisson structure (1.16) specialized to k = 2, so that C a bc = ǫ abc . This is because S 2 (and also B 3 ) is invariant under the flow of the Hamilton vector fields of R k 2 −1 constructed out of the Poisson bracket (1.16). For k = 2, we precisely have (1.15) with obvious notation. In particular, This paper therefore only concerns the case k = 2. In fact, under the maps (1.13)-(1.14) one can consider the quantization ofÃ 0 ⊂ C(B 3 ) when the image of its elements viewewd as operators invariant subspace of C N ⊗ which might correspond to the domain of the operators in the images of another quantization map, quantizating an a priori different Poisson manifold. We studied the particular case for the symmetric subspace 6 Sym N (C 2 ) ⊂ (C 2 ) N ⊗ , for which the corresponding algebras B(Sym N (C 2 )) exactly correspond to the fibers (for N = 0) of another continuous bundle of C * -algebras given by (1.17) -(1.18) below. It is a well-known fact that these fibers together with quantization maps (1.21) -(1.22) below give rise to a strict deformation quantization of S 2 [4,13,18]   5 Of course, one can always try to restrictÃ 0 toÃ ′ 0 but in that case the same manifolds are quantized which is not of particular new interest. 6 This space is clearly invariant under the maps (1.13) -(1.14).
Indicating the algebra of bounded operators by B(Sym N (C 2 )), it is known [14, Theorem 8.1] that (1.18) are the fibers of a continuous bundle of C * -algebras over the same base space I as in (1.9) whose continuous cross-sections are given by all sequences Here, the symbol Q ′ 1/N denotes the quantization maps is the dense Poisson subalgebra made of polynomials in three real variables restricted the sphere S 2 . The maps Q ′ 1/N are defined by 7 the integral computed in weak sense where p denotes an arbitrary polynomial restricted to S 2 , dΩ indicates the unique SO(3)-invariant Haar measure on S 2 with S 2 dΩ = 4π, and |Ω Ω| N ∈ B(Sym N (C 2 )) are so-called N coherent spin states defined in Appendix B.
In particular, if 1 is the constant function 1(Ω) = 1, (Ω ∈ S 2 ), and 1 N is the identity on A ′ 1/N = B(Sym N (C 2 )), the provious definition implies Indeed, it can be shown that the quantization maps (1.21) -(1.22) satisfy the axioms of Definition 1.1, which implies the existence of a strict deformation quantization of S 2 . 8 These quantization maps, constructed from a family coherent states (as opposed to the maps (1.13) -(1.14) which are defined in a complete different way), also define a so-called Berezin quantization [13] for which (B.3) typically holds as well as positivity, in that The main result of this work is an asymptotic relation connecting the bulk and the boundary quantization maps: We stress that the validity of the Dirac-Groenewold-Rieffel condition (1.6) for both maps is possible just thank to (1.15).
The plan of this paper is as follows. In section 2 we state and prove our main theorem (Theorem 2.3) establishing a connection between the strict deformation quantization of X 2 and the one of S 2 defined above. We show that the quantization maps Q 1/N defined by (1.13) -(1.14) whose images are restricted to Sym N (C 2 ) satisfy the identity above with respect to the other quantization map Q ′ 1/N . In section 3 we apply our theorem to the Curie-Weiss model which links the corresponding quantum Hamiltonian to its classical counter part on the sphere. In the appendix we provide a comprehensive overview of useful definitions.
2 Interplay of bulk quantization map Q 1/N and boundary quantization map Q ′

1/N
In order to arrive at the main thereom of this paper we first introduce some vector spaces. We let P N to be the complex vector space of polynomials in the variables x, y, z ∈ R 3 of degree ≤ N where N ≥ 1, and let P N (S 2 ) be the vector space made of the restrictions to S 2 of those polynomials.

Preparatory results on Q ′ 1/N and harmonic polynomials
Definition (1.21) can actually be stated replacing the polynomial p by a generic f ∈ C(S 2 ), though its meaning as a quantization map is valid for the domain of the polynomials restricted to S 2 as indicated in (1.21). The map is well-defined and it is surjective on B(Sym N (C 2 )) since, for every A : , there exists a function p ∈ P N (S 2 ) such that where Ω ∈ S 2 and Ω → ∆ N ∈ Sym N (C 2 ) is defined by Definition (2.6) in [12], defines a polynomial on the sphere, i.e.

tr(A∆
In particular, we realize that the map (2.1) cannot be injective on the domain C(S 1 ) since, if starting from f ∈ C(S 2 ) such that f ∈ P N (S 2 ) and constructing the associated Q ′ 1/N (f ), we can find p ∈ P N (S 2 ) such that Q ′ 1/N (f ) = Q ′ 1/N (p). Nevertheless, if restricting the domain to P N (S 2 ), the said map turns out to be bijective.
Proof. The said map is obviously surjective, as already observed, because, by defining p(Ω) := tr(A∆ where ζ and ζ are as usual interpreted as real independent variables and ζ := tan θ 2 e iφ ∈ C , ζ := tan θ 2 e −iφ ∈ C with θ ∈ (0, π), φ ∈ (−π, π) being the polar spherical angles of the unit vector Ω ∈ S 2 parallel to (x, y, z). With a lengthy computation, it is possibly to prove that every function (2.5) can always be re-written as the restriction to S 2 of a polynomial in the variables (x, y, z) ∈ R 3 of the form if N > 1, so that the maps (2.5) which just amount to (N + 1) 2 linearly independed functions must also form a vector basis of P N (S 2 ). In summary, the surjective map (2.1) is also injective on the domain P N (S 2 ).
Going back to Weyl, let us recall a few results of the theory of SO (3) representations on polynomials restricted to the unit sphere. The group SO (3) admits a natural representation on P N (S 2 ) given by In turn, the space P N (S 2 ) admits a direct decomposition into invariant and irreducible subspaces under the action of ρ, viz. Each subspace P (j) N (S 2 ) consists of the restrictions to S 2 of the homegeneous polynomials of order j that are also harmonic functions. P (j) N (S 2 ) has dimension 2j + 1.
2 (S 2 ) . In the right-hand side, the first subaspace is the span of the restriction to S 2 of the constant polynomial p(x, y, z) := 1, the second one is the span of the restrictions of the three polynomials p j (x, y, z) := x j , j = 1, 2, 3 where x 1 = x, x 2 = y, x 3 = z, and the third one is the the span of the restrictions to S 2 of five elements suitably chosen 9 of the six polynomials p ij (x, y, z) := If ρ (j) is the restriction of ρ to P Each class of matrices {D (j) (R)} R∈SO(3) defines an irreducible representation of SO(3) in C 2j+1 . These representations are completely fixed by their dimension i.e., by j, up to equivalence given by similarity transformations, and different j correspond to similarity inequivalent representations. Every irreducible representation of SO (3) is unitarily equivalent to one of the D (j) .

The main theorem
Before arriving at the main theorem of this paper, we recall that by construction the spaceÃ 0 is the complex vector space of polynomials in three variables on the closed unit ball B 3 which in particular contains all polynomials of P M (M ∈ N) restricted to B 3 [15]. In the proof of the theorem we occasionally use the spaceÃ 0 as well as P N , where the former is the domain of the quantization maps Q 1/N , whereas the latter is used to underline the degree of the polynomial in question.
Theorem 2.3. If p ∈Ã 0 , then the (operator) norm being the one on B(Sym N (C 2 )), Remark 2.4. We stress that the result does not automatically imply that the cross-sections (1.13) -(1.14) whose images are resticted to Sym N (C 2 ) are also continous cross-sections of the fibers defined in (1.17) -(1.18), since . Proof. We start the proof by discussing the interplay between the action of SO(3) and the quantization maps Q 1/N , defined in (1.13). We first focus on a homogeneous polynomial of order M < N. 10 If k 1 , . . . , k M are taken in {1, 2, 3} and p k 1 ···k M (x, y, z) := x k 1 · · · x k M , (2.8) the representation (2.6) implies that We stress that when restricting to S 2 , every p (j) m is a linear combination of the restricitions of the polynomials p k 1 ···k M so that, by extending (2.9) by linearity and working on p (j) m , (2.9) must coincide with (2.7) Since both sides are restrictions of homegeneous polynomials of the same degree j, this identity is valid also removing the contraint (x, y, z) ∈ S 2 : where now the p (j) m are homegeneous polynomials in P M whose restrictions are the basis elements of P (j) M (S 2 ) with the same name. 11 By definition of the quantization maps Q 1/N we know that Let us indicate by R U ∈ SO(3) the image of U ∈ SU(2) through the universal covering homomorphism Π : SU(2) → SO(3). This covering homomorphism as is well known satisfies (using the summation convention on repeated indices) 11 For our quantization maps Q 1/N we need p (j) m to be a polynomial inÃ 0 , rather than in P M . However, sinceÃ 0 contains all polynomials of P M restricted to B 3 which has non-empty interior, polynomials of P M are in one-to-one correspondence with those ofÃ 0 . Therefore, in view of (2.10) the same statement holds when we replace (x, y, z) ∈ R 3 by (x, y, z) ∈ B 3 . =   U ⊗ · · · ⊗ U N times Let us consider linear combinations p (j) m of polynomials p k 1 ···k M whose restriction to S 2 define the basis element, indicated with the same symbol, p Let us now pass to the other quantization map Q ′ 1/N observing that (2.12) and Proposition 2.1 entail for some q (j) m ∈ P N (S 2 ) (where N > M in general) is the unknown restriction to S 2 of a polynomial in P N . Exploiting (2.12) and linearity we find (2.14) Again, from (2.1) we have the general relation where the phase is irrelevant as it disappears in view of later computations, hence V A (N ) Since the map (2.1) is bijective on P N (S 2 ) it must be Since the set of the (restrictions of to the sphere of the) p (j) is a basis, Since the representation D (j) is irreducible, Schur's lemma implies that there are complex numbers C (j ′ ,j) such that In summary, (2.19) reduces to However, since the elements in the left-hand side are 2j ′ + 1 whereas, for every j in the right-hand side we have 2j + 1 elements and the spaces of these representations transform separately, the only possibility is that C (j ′ ,j) = 0 if j = j ′ . In other words, Let us examine what happens to C (j) N at large N. First observe that (2.20) immediately implies Taking the expectation value Ω ′ | · |Ω ′ , we find In Lemma 2.5 below we prove that lim N →+∞ C (j) N exists and is finite. Hence, where we exploited Proposition 4.2 of [15], so that lim N →+∞ C (j) N = 1 . This reasonig implies the claim for the considered special polynomials since, for N → +∞, The found result immediately extends to every polynomial of given degree M which can be written as a linear combination of the p (j) m viewed as polynomials. To pass to a generic polynomal inÃ 0 (say of degree M) we observe that, as a consequence of known results [23], the map has a kernel made of all possible polynomials of the form q(x, y, z)(x 2 + y 2 + z 2 − 1) with q ∈ P M −2 . Furthermore, Proposition 2.7 below proves that, for every q ∈ P M −2 , ||Q 1/N (q(x, y, z)(x 2 + y 2 + z 2 − 1))| Sym N (C 2 ) || N → 0 as N → +∞. (2.23) So, if p ∈Ã 0 is a polynomial of degree M, then we can write for a finite number of coefficients C (j,m) and some polynomial q ∈ P M −2 , p = j,m C (j,m) p (j) m + q(x, y, z)(x 2 + y 2 + z 2 − 1), (2.24) where the p (j) m and q are here intepreted as elements of P M and P M −2 respectively, restricted to B 3 . Hence, The former term on the right-hand side tends to Q ′ 1/N (p| S 2 ), the latter vanishes as N → +∞ proving the thesis. Lemma 2.5. lim N →+∞ C (j) N exists and is finite.

Subsidiary technical results
Proof. Since the left-hand side of (2.21) does not depend on N and the integral in the right-hand side tends to p  Proof. As is well known (see [15] for a summary of those properties and technical references), Ω · σ|Ω 1 = |Ω 1 .
Proof. We use the canonical (Dicke) basis [18,15] |n, N − n for Sym N (C 2 ) (n = 0, ..., N) and first show that the matrix elements with respect to this basis are zero: Consider now a basis vector |k, N − k . We first expand |k, N − k in the standard basis vectors β i (i = 1, ..., 2 N ) spanning the Hilbert space N C 2 . We denote by O k the orbit consisting of N k -basis vectors β i with the same number of occurrence of the vectors e 2 and e 1 , the two basis vectors of C 2 . By convention, we take e 1 such that σ 3 e 1 = e 1 , and σ 3 e 2 = −e 2 . It is not difficult to show that [26,27] |k, where the subindex l in β k,l labels the basis vector β k,l ∈ β within the same orbit O k . Since we have N k such vectors per orbit, the sum in the above equation indeed is from l = 1, ..., N k . By definition Q 1/N (x 2 i ) = S 2,N (σ i ⊗ σ i ) for i = 1, 2, 3. Using a combinatorial argument and the fact that all |k are symmetric it follows that Similarly, In view of Definition 1.1 (property 3(b)) the cross-section 0 → f and 1/N → Q 1/N (f ) defines a continuous section of the bundle implying that the following condition (see also the remark below Definition 1.1) is automatically satisfied: We apply this with f = q(x, y, z) and g(x, y, z) = x 2 + y 2 + z 2 − 1. We first show that n|Q 1/N (q(x, y, z))Q 1/N (x 2 + y 2 + z 2 − 1))|k = 0, for all basis vectors |n and |k in Sym N (C 2 ). Indeed, using the above identities one finds Since this holds for all basis vectors and Sym N (C 2 ) is invariant under Q 1/N (q(x, y, z)) and Q 1/N (x 2 + y 2 + z 2 − 1), we conclude Therefore, for any symmetric unit vector φ ∈ Sym N (C 2 ) we compute ||Q 1/N (q(x, y, z)(x 2 + y 2 + z 2 − 1))φ|| N = Q 1/N (q(x, y, z)(x 2 + y 2 + z 2 − 1)) − Q 1/N (q(x, y, z))Q 1/N (x 2 + y 2 + z 2 − 1) φ N ≤ ||Q 1/N (q(x, y, z)(x 2 + y 2 + z 2 − 1)) − Q 1/N (q(x, y, z))Q 1/N (x 2 + y 2 + z 2 − 1)|| N .
As a consequence of (2.26), for every ǫ > 0 there is N ǫ such that the crucial observation is that due to (2.26) the number N ǫ does not depend on the unit vector φ ∈ Sym N (C 2 ). Therefore the above bound is uniform, and which means This closes the proof of the proposition.

Application to the quantum Curie-Weiss model
We apply the previous theorem to the (quantum) Curie-Weiss model 12 , which is an exemplary quantum mean-field spin model. We recall that the quantum Curie Weiss defined on a lattice with N sites is Here σ k (j) stands for I 2 ⊗ · · · ⊗ σ k ⊗ · · · ⊗ I 2 , where σ k occupies the j-th slot, and J, B ∈ R are given constants defining the strength of the spin-spin coupling and the (transverse) external magnetic field, respectively. Note that where Sym(M 2 (C) ⊗N ) is the range of the symmetrizer. Our interest will lie in the limit N → ∞. As such, we rewrite h CW 1/N as where B 3 = {x ∈ R 3 | x ≤ 1} is the closed unit ball in R 3 . Using these observations we now show that the quantum Curie-Weiss Hamiltonian restricted to the symmetric space is asymptotically norm-equivalent also to the other quantization map Q ′ 1/N applied to h CW 0 | S 2 . Proof. Using (3.4) and Theorem (2.3), This in particular establishes a link between the (compressed) quantum Curie-Weiss spin Hamiltonian and its classical counterpart on the sphere.