Acta Polytechnica https://doi.org/10.14311/AP.2022.62.0008 Acta Polytechnica 62(1):8–15, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague QUANTUM DESCRIPTION OF ANGLES IN THE PLANE Roberto Beneducia, Emmanuel Frionb, Jean-Pierre Gazeauc, ∗ a Università della Calabria and Istituto Nazionale di Fisica Nucleare, Gruppo c. Cosenza, 87036 Arcavacata di Rende (Cs), Italy b University of Helsinki, Helsinki Institute of Physics, P. O. Box 64, FIN-00014 Helsinki, Finland c Université de Paris, CNRS, Astroparticule et Cosmologie, 75013 Paris, France ∗ corresponding author: gazeau@apc.in2p3.fr Abstract. The real plane with its set of orientations or angles in [0, π) is the simplest non trivial example of a (projective) Hilbert space and provides nice illustrations of quantum formalism. We present some of them, namely covariant integral quantization, linear polarisation of light as a quantum measurement, interpretation of entanglement leading to the violation of Bell inequalities, and spin one-half coherent states viewed as two entangled angles. Keywords: Integral quantization, real Hilbert spaces, quantum entanglement. 1. Introduction The formulation of quantum mechanics in a real Hilbert space has been analyzed by Stueckelberg in 1960 [1] in order to show that the need for a com- plex Hilbert space is connected to the uncertainty principle. Later, Solèr [2] showed that the lattice of elementary propositions is isomorphic to the lattice of closed subspaces of a separable Hilbert space (over the reals, the complex numbers or the quaternions). In other words, the lattice structure of propositions in quantum physics does not suggest the Hilbert space to be complex. More recently, Moretti and Oppio [3] gave stronger motivation for the Hilbert space to be complex which rests on the symmetries of elementary relativistic systems. In this contribution, we do not address the ques- tion of the physical validity of the real Hilbert space formulation of quantum mechanics but limit ourselves to use the real 2-dimensional case, i.e. the Euclidean plane, as a toy model for illustrating some aspects of the quantum formalism, as quantization, entangle- ment and quantum measurement. The latter is nicely represented by the linear polarization of light. This real 2-dimensional case relies on the manipulation of the two real Pauli matrices σ1 = ( 0 1 1 0 ) , σ3 = ( 1 0 0 −1 ) , (1) and their tensor products, with no mention of the third, complex matrix σ2 = ( 0 −i i 0 ) . As a matter of fact, many examples aimed to illustrate tools and concepts of quantum information, quantum measure- ment, quantum foundations, ... (e.g., Peres [4]) are illustrated with manipulations of these matrices. In [5], it was shown that the set of pure states in the plane is represented by half of the unit circle and the set of mixed states by half the unit disk, and also that rotations in the plane rule time evolution through Majorana-like equations, all of this using only real quantities for both closed and open systems. This paper is a direct extension of our previous paper [6], and for this reason we start the discussion by recalling some key elements of the mathematical formalism. 2. Background 2.1. Definition of POVMs We start with the definition of a normalized Positive- Operator Valued measure (POVM) [7]. It is defined as a map F : B(Ω) → L+s (H) from the Borel σ-algebra of a topological space Ω to the space of linear positive self-adjoint operators on a Hilbert space H such that F ( ∞⋃ n=1 ∆n ) = ∞∑ n=1 F (∆n) F (Ω) = 1 . (2) In this definition, {∆n} is a countable family of dis- joint sets in B(Ω) and the series converges in the weak operator topology. If Ω = R, we have a real POVM. If F (∆) is a projection operator for every ∆ ∈ B(Ω), we recover the usual projection-valued measure (PVM). A quantum state is defined as a non-negative, bounded self-adjoint operator with trace 1. The space of states is a convex space and is denoted by S(H). A quantum measurement corresponds to an affine map S(H) 7→ M+(Ω) from quantum states to probability measures, ρ 7→ µρ. There is [8] a one-to-one correspon- dence between POVMs F : B(Ω) → L+s (H) and affine maps S(H) 7→ M+(Ω) given by µρ(∆) = Tr(ρF (∆)), ∆ ∈ B(Ω). 2.2. Integral quantization Quantum mechanics is usually taught in terms of pro- jection operators and PVM, but measurements usually 8 https://doi.org/10.14311/AP.2022.62.0008 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 62 no. 1/2022 Quantum angles give a statistical distribution around a mean value, incompatible with the theory. We recall here a gen- eralization of a quantization procedure, the integral quantization, based on POVMs instead of PVM. The basic requirements of this programme are the follow- ing: the quantization of a classical function defined on a set X must respect (1.) Linearity. Quantization is a linear map f 7→ Af : Q : C(X) 7→ A(H) , Q(f ) = Af , (3) where • C(X) is a vector space of complex or real-valued functions f (x) on a set X, i.e. a “classical” math- ematical model, • A(H) is a vector space of linear operators in some real or complex Hilbert space H, i.e., a “quantum” mathematical model, notwithstanding the ques- tion of common domains in the case of unbounded operators. (2.) Unity. The map (3) is such that the function f = 1 is mapped to the identity operator 1 on H. (3.) Reality. A real function f is mapped to a self- adjoint or normal operator Af in H or, at least, a symmetric operator (in the infinite-dimensional case). (4.) Covariance. Defining the action of a symmetry group G on X by (g, x) ∈ G × X such as (g, x) 7→ g · x ∈ X, there is a unitary representation U of G such that AT (g)f = U (g)Af U (g−1), with (T (g)f )(x) = f ( g−1 · x ) . Performing the integral quantization [9] of a func- tion f (x) on a measure space (X, ν) boils down to the linear map: f 7→ Af = ∫ X M(x) f (x) dν(x) , (4) where we introduce a family of operators M(x) solving the identity. More precisely, we have X ∋ x 7→ M(x) , ∫ X M(x) dν(x) = 1 . (5) If the M(x) are non-negative, they provide a POVM. Indeed, the quantization of the characteristic function on the Borel set ∆, A(χ∆), F (∆) := A(χ∆) = ∫ ∆ M(x) dν(x) . (6) is a POVM which provides a quantization procedure f 7→ Af = ∫ X f (x) dF (x). 3. Euclidean plane as Hilbert space of quantum states 3.1. Mixed states as density matrices Density matrices act as a family of operators which can be used to perform covariant integral quantization. - 6 ı̂ = |0⟩ ≡ ( 1 0 ) |ϕ⟩ = ( cos ϕ sin ϕ ) ↔ Eϕ = |ϕ⟩⟨ϕ| O ⟨0|0⟩ = 1 = 〈 π 2 ∣∣ π 2 〉 , ⟨0 ∣∣π 2 〉 = 0 ȷ̂ = | π2 ⟩ ≡ ( 0 1 ) 1 ϕ � � � � � � � � �� Figure 1. The Euclidean plane and its unit vectors viewed as pure quantum states in Dirac ket nota- tions. In the context of the Euclidean plane and its rotational symmetry, one associates the polar angle ϕ ∈ [0, 2π) with the unit vector ûϕ to define the pure state |ϕ⟩ := |ûϕ⟩. As shown in Figure 1, two orthogonal pure states ı̂ = |0⟩ and ȷ̂ = ∣∣∣π 2 〉 are readily identified with the unit vectors spanning the plane. In this configuration, the pure state |ϕ⟩ is defined by an anticlockwise rotation of angle ϕ of the pure state |0⟩. Denoting the orthogonal projectors on ı̂ and ȷ̂ by |0⟩⟨0| and ∣∣π 2 〉〈 π 2 ∣∣ respectively, we visualize the resolution of the identity as follows 1 = |0⟩⟨0| + ∣∣∣π 2 〉〈π 2 ∣∣∣ ⇕( 1 0 0 1 ) = ( 1 0 0 0 ) + ( 0 0 0 1 ) . (7) Recalling that a pure state in the plane, equivalently an orientation, can be decomposed as |ϕ⟩ = cos ϕ |0⟩ + sin ϕ ∣∣π 2 〉 , with ⟨0|ϕ⟩ = cos ϕ and 〈 π 2 ∣∣ ϕ〉 = sin ϕ, it is straightforward to find the orthogonal projector corresponding to the pure state |ϕ⟩, Eϕ = ( cos2 ϕ cos ϕ sin ϕ cos ϕ sin ϕ sin2 ϕ ) , (8) from which we can construct the density matrix cor- responding to all the mixed states ρ = ( 1 + r 2 ) Eϕ + ( 1 − r 2 ) Eϕ+π/2 , 0 ≤ r ≤ 1 . (9) In this expression, the parameter r represents the degree of mixing. Hence the upper half-disk (r, ϕ), 0 ≤ r ≤ 1, 0 ≤ ϕ < π is in one-to-one correspondence 9 R. Beneduci, E. Frion, J.-P. Gazeau Acta Polytechnica with the set of density matrices ρ ≡ ρr,ϕ written as ρr,ϕ = 1 2 1 + r 2 R(ϕ)σ3R(−ϕ) = (1 2 + r 2 cos 2ϕ r 2 sin 2ϕ r 2 sin 2ϕ 1 2 − r 2 cos 2ϕ ) = 1 2 (1 + rσ2ϕ) , (10) where R(ϕ) = ( cos ϕ − sin ϕ sin ϕ cos ϕ ) is a rotation matrix in the plane, and σϕ := cos ϕ σ3 + sin ϕ σ1 ≡ −→σ · ûϕ = ( cos ϕ sin ϕ sin ϕ − cos ϕ ) = R(ϕ) σ3 . (11) The observable σϕ has eigenvalues {±1} and eigenvec- tors ∣∣∣ϕ2 〉 and ∣∣∣ϕ+π2 〉 respectively. It plays a crucial rôle since, as we show right after, it is at the core of both the non-commutative character and the en- tanglement of two quantum states of the real space. It is a typical observable used to illustrate quantum formalism [4]. 3.2. Describing non-commutativity and finding Naimark extensions through rotations Let us apply integral quantization with the real density matrices (10). With X = S1, the unit circle, equipped with the measure dν(x) = dϕ π , ϕ ∈ [0, 2π), we obtain the resolution of the identity for an arbitrary ϕ0,∫ 2π 0 ρr,ϕ+ϕ0 dϕ π = 1 . (12) Hence, quantizing a function (or distribution) f (ϕ) on the circle is done through the map f 7→ Af = ∫ 2π 0 f (ϕ)ρr,ϕ+ϕ0 dϕ π = ( ⟨f ⟩ + r2 Cc (Rϕ0 f ) r 2 Cs (Rϕ0 f ) r 2 Cs (Rϕ0 f ) ⟨f ⟩ − r 2 Cc (Rϕ0 f ) ) = ⟨f ⟩ 1 + r 2 [Cc (Rϕ0 f ) σ3 + Cs (Rϕ0 f ) σ1] , (13) with ⟨f ⟩ := 12π ∫ 2π 0 f (ϕ) dϕ the average of f on the unit circle and Rϕ0 (f )(ϕ) := f (ϕ − ϕ0). Here we have defined cosine and sine doubled angle Fourier coefficients of f Cc s (f ) = ∫ 2π 0 f (ϕ) { cos sin 2ϕ dϕ π . (14) In [6], we drew three consequences from this result. The first consequence is that, upon identification of R3 with the subspace V3 = Span { e0(ϕ) := 1√2 , e1(ϕ) := cos 2ϕ, e2(ϕ) := sin 2ϕ } in L2(S1, dϕ/π), the inte- gral quantization map with ρr,ϕ+ϕ0 yields a non- commutative version of R3 : Ae0 = 1 √ 2 , Ae1 = r 2 [cos 2ϕ0 σ3 + sin 2ϕ0 σ1] ≡ r 2 σ2ϕ0 , Ae2 = r 2 [− sin 2ϕ0 σ3 + cos 2ϕ0 σ1] ≡ r 2 σ2ϕ0+π/2 . Now, the commutation rule reads [Ae1 , Ae2 ] = − r2 2 τ2 , τ2 := ( 0 −1 1 0 ) = −iσ2 , which depends on the real version of the last Pauli matrix and on the degree of mixing. A second consequence, typical of quantum- mechanical ensembles, is that all functions f (ϕ) in V3 yielding density matrices through this map imply that ρs,θ = ∫ 2π 0 [ 1 2 + s r cos 2ϕ ] ︸ ︷︷ ︸ f (ϕ) ρr,ϕ+θ dϕ π . (15) If r ≥ 2s, this continuous superposition of mixed states is convex. Therefore, a mixed state is composed of an infinite number of other mixed states. This has consequences in quantum cryptography, for example, since the initial signal cannot be recovered from the output. The third and last consequence we mention here concerns the Naimark extension of a function defined on the circle. In particular, we focus on the Toeplitz quantization of f (ϕ), which is a kind of integral quan- tization. In [6], we used this framework to show there exist orthogonal projectors from L2(S1, dϕ/π) to R2 such that for a function f (ϕ) the multiplication oper- ator on L2(S1, dϕ/π), defined by v 7→ Mf v = f v , (16) maps Mf to Af . They are precisely Naimark’s ex- tensions of POVMs represented by density matrices (see [6] for details). 3.3. Linear polarization of light as a quantum phenomenon In this section, we recall that the polarization tensor of light can be expressed as a density matrix, which allows us to relate the polarization of light to quantum phenomena such as the Malus Law and the incompat- ibility between two sequential measurements [6]. First, remember that a complex-valued electric field for a propagating quasi-monochromatic electromag- netic wave along the z-axis reads as −→ E (t) = −→ E0(t) eiωt = Ex ı̂ + Ey ȷ̂ = (Eα) , (17) in which we have used the previous notations for the unit vectors in the plane. The polarization is deter- mined by −→ E0(t). It slowly varies with time, and can be 10 vol. 62 no. 1/2022 Quantum angles measured through Nicol prisms, or other devices, by measuring the intensity of the light yielded by mean values ∝ EαEβ , EαE ∗β and conjugates. Due to rapidly oscillating factors and a null temporal average ⟨·⟩t, a partially polarized light is described by the 2 × 2 Hermitian matrix (Stokes parameters) [10–12] 1 J ( ⟨E0xE ∗0x⟩t 〈 E0xE ∗0y 〉 t ⟨E0y E ∗0x⟩t 〈 E0y E ∗0y 〉 t ) ≡ ρr,ϕ + A 2 σ2 = 1 + r 2 Eϕ + 1 − r 2 Eϕ+π/2 + i A 2 τ2 . Here, J describes the intensity of the wave. In the second line, it is clear that the degree of mixing r describes linear polarization, while the parameter A (−1 ≤ A ≤ 1) is related to circular polarization. In real space, we have A = 0, so we effectively describe the linear polarization of light. - 6 k̂ ȷ̂ ı̂ � � � �� �� � �� •H H HHY Re (−→ E ) y z x We now wish to describe the interaction between a polarizer and a partially linear polarized light as a quantum measurement. We need to introduce two planes and their tensor product: the first one is the Hilbert space on which act the states ρMs,θ of the po- larizer viewed as an orientation pointer. Note that the action of the generator of rotations τ2 = −iσ2 on these states corresponds to a π/2 rotation : τ2ρ M s,θ τ −1 2 = −τ2ρ M s,θ τ2 = ρ M s,θ+π/2 . (18) The second plane is the Hilbert space on which act the partially linearized polarization states ρLr,ϕ of the plane wave crossing the polarizer. Its spectral decomposition corresponds to the incoherent superposition of two completely linearly polarized waves ρLr,ϕ = 1 + r 2 Eϕ + 1 − r 2 Eϕ+π/2 . (19) The pointer detects an orientation in the plane de- termined by the angle ϕ. Through the interaction pointer-system, we generate a measurement whose time duration is the interval IM = (tM − η, tM + η) centred at tM . The interaction is described by the (pseudo-) Hamiltonian operator H̃int(t) = g η M (t)τ2 ⊗ ρ L r,ϕ , (20) where gηM is a Dirac sequence with support in IM , i.e., lim η→0 ∫ +∞ −∞ dt f (t) gηM (t) = f (tM ) . The interaction (20) is the tensor product of an an- tisymmetric operator for the pointer with an operator for the system which is symmetric (i.e., Hamiltonian). The operator defined for t0 < tM − η as U (t, t0) = exp [∫ t t0 dt′ gηM (t ′) τ2 ⊗ ρLr,ϕ ] = exp [ G η M (t) τ2 ⊗ ρ L r,ϕ ] , (21) with GηM (t) = ∫ t t0 dt′ gηM (t ′), is a unitary evolution operator. From the formula involving an orthogonal projector P , exp(θτ2 ⊗ P ) = R(θ) ⊗ P + 1 ⊗ (1 − P ) , (22) we obtain U (t, t0) =R ( G η M (t) 1 + r 2 ) ⊗ Eϕ + R ( G η M (t) 1 − r 2 ) ⊗ Eϕ+π/2 . (23) For t0 < tM − η and t > tM + η, we finally obtain U (t, t0) = R ( 1 + r 2 ) ⊗ Eϕ + R ( 1 − r 2 ) ⊗ Eϕ+π/2 . (24) Preparing the polarizer in the state ρMs0,θ0 , we obtain the evolution U (t, t0) ρMs0,θ0 ⊗ ρ L r0,ϕ0 U (t, t0)† of the initial state for t > tM + η ρM s0,θ0+ 1+r2 ⊗ 1 + r0 cos 2(ϕ − ϕ0) 2 Eϕ + ρM s0,θ0+ 1−r2 ⊗ 1 − r0 cos 2(ϕ − ϕ0) 2 Eϕ+π/2 + 1 4 (R(r) + s0σ2θ0+1) ⊗ r0 sin 2(ϕ − ϕ0) Eϕτ2 − 1 4 (R(−r) + s0σ2θ0+1) ⊗ r0 sin 2(ϕ − ϕ0) τ2Eϕ . (25) Therefore, the probability for the pointer to rotate by 1+r2 , corresponding to the polarization along the orientation ϕ is Tr [( U (t, t0) ρMs0,θ0 ⊗ ρ L r0,ϕ0 U (t, t0)† ) (1 ⊗ Eϕ) ] = 1 + r0 cos 2(ϕ − ϕ0) 2 , (26) that for the completely linear polarization of the light, i.e. r0 = 1, becomes the familiar Malus law, cos2(ϕ − ϕ0). Similarly, the second term gives the probability for the perpendicular orientation ϕ + π/2 and the pointer rotation by 1−r2 Tr [( U (t, t0) ρMs0,θ0 ⊗ ρ L r0,ϕ0 U (t, t0)† )( 1 ⊗ Eϕ+π/2 )] = 1 − r0 cos 2(ϕ − ϕ0) 2 , (27) corresponding (in the case r0 = 1) to the Malus law sin2(ϕ − ϕ0). 11 R. Beneduci, E. Frion, J.-P. Gazeau Acta Polytechnica 4. Entanglement and isomorphisms In this section, we develop our previous results fur- ther by giving an interpretation in terms of quantum entanglement. Previously, we described the interac- tion between a polarizer and a light ray as the tensor product (20), which is analogous to the quantum en- tanglement of states, since it is a logical consequence of the construction of tensor products of Hilbert spaces for describing quantum states of composite system. In the present case, we are in presence of a remarkable sequence of vector space isomorphisms due to the fact that 2 × 2 = 2 + 2 1 : R2 ⊗ R2 ∼= R2 × R2 ∼= R2 ⊕ R2 ∼= C2 ∼= H , (28) where H is the field of quaternions. Therefore, the description of the entanglement in a real Hilbert space is equivalent to the description of a single system (e.g., a spin 1/2) in the complex Hilbert space C2, or in H. In Section 4.3 we develop such an observation. 4.1. Bell states and quantum correlations It is straightforward to transpose into the present set- ting the 1964 analysis and result presented by Bell in his discussion about the EPR paper [13] and about the subsequent Bohm’s approaches based on the assump- tion of hidden variables [14]. We only need to replace the Bell spin one-half particles with the horizontal (i.e., +1) and vertical (i.e., −1) quantum orientations in the plane as the only possible issues of the observ- able σϕ (11), supposing that there exists a pointer device designed for measuring such orientations with outcomes ±1 only. In order to define Bell states and their quantum correlations, let us first write the canonical, orthonor- mal basis of the tensor product R2A ⊗ R 2 B , the first factor being for system “A” and the other for system “B”, as |0⟩A ⊗ |0⟩B , ∣∣∣π 2 〉 A ⊗ ∣∣∣π 2 〉 B , |0⟩A ⊗ ∣∣∣π 2 〉 B , ∣∣∣π 2 〉 A ⊗ |0⟩B . (29) The states |0⟩ and ∣∣π 2 〉 pertain to A or B, and are named “q-bit” or “qubit” in the standard language of quantum information. Since they are pure states, they can be associated to a pointer measuring the horizontal (resp. vertical) direction or polarisation described by the state |0⟩ (resp. ∣∣π 2 〉 ). There are four Bell pure states in R2A ⊗ R 2 B , namely |Φ±⟩ = 1 √ 2 ( |0⟩A ⊗ |0⟩B ± ∣∣∣π 2 〉 A ⊗ ∣∣∣π 2 〉 B ) , (30) |Ψ±⟩ = 1 √ 2 ( ±|0⟩A ⊗ ∣∣∣π 2 〉 B + ∣∣∣π 2 〉 A ⊗ |0⟩B ) . (31) 1Remind that dim(V ⊗ W ) = dimV dimW while dim(V × W ) = dimV + dimW for 2 finite-dimensional vector spaces V and W We say that they represent maximally entangled quan- tum states of two qubits. Consider for instance the state |Φ+⟩. If the pointer associated to A measures its qubit in the standard basis, the outcome would be perfectly random, with either possibility having a probability 1/2. But if the pointer associated to B then measures its qubit instead, the outcome, al- though random for it alone, is the same as the one A gets. There is quantum correlation. 4.2. Bell inequality and its violation Let us consider a bipartite system in the state Ψ−. In such a state, if a measurement of the component σAϕa := −→σ A · ûϕa (ûϕa is an unit vector with polar angle ϕa) yields the value +1 (polarization along the direction ϕa/2), then a measurement of σBϕb when ϕb = ϕa must yield the value −1 (polarization along the direction ϕa+π2 ), and vice-versa. From a classi- cal perspective, the explanation of such a correlation needs a predetermination by means of the existence of hidden parameters λ in some set Λ. Assuming the two measurements to be separated by a space-like inter- val, the result εA ∈ {−1, +1} (resp. εB ∈ {−1, +1}) of measuring σAϕa (resp. σ B ϕb ) is then determined by ϕa and λ only (locality assumption), not by ϕb, i.e. εA = εA(ϕa, λ) (resp. εB = εB (ϕb, λ)). Given a prob- ability distribution ρ(λ) on Λ, the classical expecta- tion value of the product of the two components σAϕa and σBϕb is given by P(ϕa, ϕb) = ∫ Λ dλ ρ(λ) εA(ϕa, λ) εB (ϕb, λ) . (32) Since ∫ Λ dλ ρ(λ) = 1 and εA,B = ±1 , (33) we have −1 ≤ P(ϕa, ϕb) ≤ 1. Equivalent predictions within the quantum setting then imposes the equality between the classical and quantum expecta- tion values: P(ϕa, ϕb) = 〈 Ψ− ∣∣ σAϕa ⊗ σBϕb ∣∣Ψ−〉 = −ûϕa · ûϕb = − cos(ϕa − ϕb) . (34) In the above equation, the value −1 is reached at ϕa = ϕb. This is possible for P(ϕa, ϕa) only if εA(ϕa, λ) = −εB (ϕa, λ). Hence, we can write P(ϕa, ϕb) as P(ϕa, ϕb) = − ∫ Λ dλ ρ(λ) ε(ϕa, λ) ε(ϕb, λ) , ε(ϕ, λ) ≡ εA(ϕ, λ) = ±1 . (35) Let us now introduce a third unit vector ûϕc . Due to ε2 = 1, we have P(ϕa, ϕb) − P(ϕa, ϕc) = ∫ Λ dλ ρ(λ) ε(ϕa, λ) ε(ϕb, λ) × [ε(ϕb, λ) ε(ϕc, λ) − 1] . (36) 12 vol. 62 no. 1/2022 Quantum angles From this results the (baby) Bell inequality: |P(ϕa, ϕb) − P(ϕa, ϕc)| ≤ ∫ Λ dλ ρ(λ) [1 − ε(ϕb, λ) ε(ϕc, λ)] = 1 + P(ϕb, ϕc) . Hence, the validity of the existence of hidden vari- able(s) for justifying the quantum correlation in the singlet state Ψ−, and which is encapsulated by the above equation, has the following consequence on the arbitrary triple (ϕa, ϕb, ϕc): 1 − cos(ϕb − ϕc) ≥ |cos(ϕb − ϕa) − cos(ϕc − ϕa)| . Equivalently, in terms of the two independent angles ζ and η, ζ = ϕa − ϕb 2 , η = ϕb − ϕc 2 , we have ∣∣sin2 ζ − sin2(η + ζ)∣∣ ≤ sin2 η . (37) It is easy to find pairs (ζ, η) for which the inequality (37) does not hold true. For instance with η = ζ ̸= 0, i.e., ϕb = ϕa + ϕc 2 , we obtain |4 sin2 η − 3| ≤ 1 , (38) which does not hold true for all |η| < π/4, i.e., for |ϕa − ϕb| = |ϕb − ϕc| < π/2. Actually, we did not follow here the proof given by Bell, which is a lot more elaborate. Also, Bell considered unit vectors in 3-space. Restricting his proof to vectors in the plane does not make any difference, as it is actually the case in many works devoted to the foundations of quantum mechanics. 4.3. Entanglement of two angles Quantum entanglement is usually described by the complex two-dimensional Hilbert space C2. As a com- plex vector space, C2, with canonical basis (e1, e2), has a real structure, i.e., is isomorphic to a real vector space which makes it isomorphic to R4, itself isomor- phic to R2 ⊗ R2. A real structure is obtained by considering the vector expansion C2 ∈ v = z1e1 + z2e2 = x1e1 + y1 (ie1) + x2e2 + y2 (ie2) , (39) which is equivalent to writing z1 = x1 + iy1, z2 = x2 + iy2, and considering the set of vectors {e1, e2, (ie1) , (ie2)} (40) as forming a basis of R4. Forgetting about the sub- scripts A and B in (29), we can map vectors in the Euclidean plane R2 to the complex “plane” C by |0⟩ 7→ 1 , ∣∣∣π 2 〉 7→ i , (41) which allows the correspondence between bases as |0⟩ ⊗ |0⟩ = e1 , ∣∣∣π 2 〉 ⊗ ∣∣∣π 2 〉 = −e2 , |0⟩ ⊗ ∣∣∣π 2 〉 = (ie1) , ∣∣∣π 2 〉 ⊗ |0⟩ = (ie2) . (42) Also, the spin of a particle in a real basis, given by the “up” and “down” states, are defined by e1 ≡ | ↑ ⟩ ≡ ( 1 0 ) , e2 ≡ | ↓ ⟩ ≡ ( 0 1 ) . (43) Finally, we obtain an unitary map from the Bell basis to the basis of real structure of C2( |Φ+⟩ |Φ−⟩ |Ψ+⟩ |Ψ−⟩ ) = ( e1 e2 (ie1) (ie2) ) 1 √ 2   1 1 0 0 −1 1 0 0 0 0 1 −1 0 0 1 1   . In terms of respective components of vectors in their respective spaces, we have  x1 x2 y1 y2   = 1√2   1 1 0 0 −1 1 0 0 0 0 1 −1 0 0 1 1     x+ x− y+ y−   . (44) In complex notations, with z± = x± + iy±, this is equivalent to( z+ z− ) = 1 √ 2 ( 1 −C C 1 )( z1 z2 ) ≡ C@ ( z1 z2 ) , (45) in which we have introduced the conjugation operator Cz = z̄, i.e., the mirror symmetry with respect to the real axis, −C being the mirror symmetry with respect to the imaginary axis. Let us now see what is the influence of having real Bell states on Schrödinger cat states. The operator “cat” C@ can be expressed as C@ = 1 √ 2 (1 + F) , F := Cτ2 = ( 0 −C C 0 ) . (46) Therefore, with the above choice of isomorphisms, Bell entanglement in R2 ⊗ R2 is not represented by a simple linear superposition in C2. It involves also the two mirror symmetries ±C. The operator F is a kind of “flip” whereas the “cat” or “beam splitter” operator C@ builds, using the up and down basic states, the two elementary Schrödinger cats F | ↑ ⟩ = | ↓ ⟩ , C@ | ↑ ⟩ = 1 √ 2 (| ↑ ⟩ + | ↓ ⟩) , (47) F | ↓ ⟩ = −| ↑ ⟩ , C@ | ↓ ⟩ = 1 √ 2 (−| ↑ ⟩ + | ↓ ⟩) . (48) The flip operator also appears in the construction of the spin one-half coherent states |θ, ϕ⟩, defined in 13 R. Beneduci, E. Frion, J.-P. Gazeau Acta Polytechnica terms of spherical coordinates (θ, ϕ) as the quantum counterpart of the classical state n̂(θ, ϕ) in the sphere S2 by |θ, ϕ⟩ = ( cos θ 2 | ↑ ⟩ + eiϕ sin θ 2 | ↓ ⟩ ) ≡( cos θ2 eiϕ sin θ2 ) = ( cos θ2 − sin θ 2 e −iϕ sin θ2 e iϕ cos θ2 )( 1 0 ) ≡ D 1 2 ( ξ−1n̂ ) | ↑ ⟩ . (49) Here, ξn̂ corresponds, through the homomorphism SO(3) 7→ SU(2), to the specific rotation Rn̂ mapping the unit vector pointing to the north pole, k̂ = (0, 0, 1), to n̂. The operator D 1 2 ( ξ−1n̂ ) represents the element ξ−1n̂ of SU(2) in its complex two-dimensional unitary irreducible representation. As we can see in matrix (49), the second column of D 1 2 ( ξ−1n̂ ) is precisely the flip of the first one, D 1 2 ( ξ−1n̂ ) = ( |θ, ϕ⟩ F|θ, ϕ⟩ ) . (50) Actually, we can learn more about the isomorphisms C2 ∼= H ∼= R+ × SU(2) through the flip and matrix representations of quaternions. In quaternionic alge- bra, we have the property ı̂ = ȷ̂k̂ + even permutations, and a quaternion q is represented by H ∋ q = q0 + q1ı̂ + q2ȷ̂ + q3k̂ = q0 + q3k̂ + ȷ̂ ( q1k̂ + q2 ) ≡ ( q0 + iq3 q2 + iq1 ) ≡ Zq ∈ C2 , (51) after identifying k̂ ≡ i as both are roots of −1. Then the flip appears naturally in the final identification H ∼= R+ × SU(2) as q ≡ ( q0 + iq3 −q2 + iq1 q2 + iq1 q0 − iq3 ) = ( Zq FZq ) . (52) Let us close this article with a final remark on spin- 1/2 coherent states as vectors in R2A ⊗ R 2 B . The “cat states” in C2 given by (49) and equivalently viewed as 4-vectors in H ∼ R4 as |θ, ϕ⟩ 7→   cos θ2 − sin θ2 cos ϕ sin θ2 sin ϕ 0   , (53) are represented as entangled states in R2A ⊗ R 2 B by |θ, ϕ⟩ = cos θ 2 |0⟩A ⊗ |0⟩B − sin θ 2 cos ϕ ∣∣∣π 2 〉 A ⊗ ∣∣∣π 2 〉 B + sin θ 2 sin ϕ|0⟩A ⊗ ∣∣∣π 2 〉 B + 0 ∣∣∣π 2 〉 A ⊗ |0⟩B . Therefore, we can say that two entangled angles in the plane can be viewed as a point in the upper half-sphere S2/Z2 in R3 shown in Figure 2. Figure 2. Each point in the upper half-sphere is in one-to-one correspondence with two entangled angles in the plane. 5. Conclusions Integral quantization is a quantization scheme con- structed on Positive Operator-Value Measures. When applied to a two-dimensional real space, it allows for a description of quantum states as pointers in the real unit half-plane. We recalled in this paper that in this case, a family of density matrices is sufficient to perform this kind of quantization as it describes all the mixed states in this space. Furthermore, a density matrix in a two-dimensional real space depends on the usual observable σϕ = ( cos ϕ sin ϕ sin ϕ − cos ϕ ) , which captures the essence of non-commutativity in real space. As a consequence, commutation relations are expressed in terms of the real matrix τ2, which serves as the basis to the description of quantum measure- ment. We provide an illustration considering linearly- polarized light passing through a polarizer. The pointer, associated with τ2, can rotate by an angle (1±r)/2 with r the degree of mixing of the density ma- trix, with a probability given by the usual Malus’ laws (26) and (27). We extended the analysis by showing that the interaction between a polarizer and a light ray is equivalent to the quantum entanglement of two Hilbert spaces. Orientations in the plane have only two outcomes (±1), which are the possible issues of σϕ. We showed that for a general bipartite system, the classical and quantum measurement of σϕ deny the existence of local hidden variables, resulting in the well-known violation of Bell inequalities, here given by (37). Finally, we demonstrated that the isomorphism C2 ≃ R4 allows to write Bell states in real space, with the introduction of the “flip” operator (46). This operator is necessary for constructing spin one-half coherent states, that we can fully describe by a set of orientations in R3, as shown in (53). Acknowledgements R.B. The present work was performed under the auspices of the GNFM (Gruppo Nazionale di Fisica Matematica). 14 vol. 62 no. 1/2022 Quantum angles EF thanks the Helsinki Institute of Physics (HIP) for their hospitality. References [1] E. C. G. Stueckelberg. Quantum theory in real Hilbert space. Helvetica Physica Acta 33(8):727–752, 1960. [2] M. P. Solèr. 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Physical Review 85(2):166–179, 1952. https://doi.org/10.1103/PhysRev.85.166. 15 https://doi.org/10.1080/00927879508825218 https://doi.org/10.1142/S0129055X19500132 https://doi.org/10.1007/BF01883517 https://doi.org/10.1007/s13538-019-00652-x https://arxiv.org/abs/2108.04086 https://doi.org/10.1016/S0034-4877(17)30035-6 https://doi.org/10.1007/978-88-7642-378-9 https://doi.org/10.1016/j.aop.2014.02.008 https://doi.org/10.1119/1.1933744 https://doi.org/10.1119/1.2386162 https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195 https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195 https://doi.org/10.1103/PhysRev.85.166 Acta Polytechnica 62(1):8–15, 2022 1 Introduction 2 Background 2.1 Definition of POVMs 2.2 Integral quantization 3 Euclidean plane as Hilbert space of quantum states 3.1 Mixed states as density matrices 3.2 Describing non-commutativity and finding Naimark extensions through rotations 3.3 Linear polarization of light as a quantum phenomenon 4 Entanglement and isomorphisms 4.1 Bell states and quantum correlations 4.2 Bell inequality and its violation 4.3 Entanglement of two angles 5 Conclusions Acknowledgements References