\(^{1}\) Centro Brasileiro de Pesquisas Fı́sicas
\(^{2}\) National Institute of Science and Technology for Complex Systems
Rua Xavier Sigaud 150, Rio de Janeiro, Brazil
\(^{3}\)Université Paris Cité, CNRS, Astroparticule et Cosmologie, 75013 Paris, France
\(^{4}\)Faculty of Mathematics, University of Białystok, 15-245 Białystok, Poland

1 Introduction↩︎

The appearance of the imaginary unit \(\mathsf{i}=\sqrt{-1}\) in the Schrödinger equation has been a source of conceptual unease since the inception of quantum mechanics. Dyson captured this vividly: “One of the most profound jokes of nature is the square root of minus one that Schrödinger put into his equation…Suddenly it became a wave equation instead of a heat conduction equation” [1]. Concretely, every measurable quantity in classical mechanics is real, and complex numbers appeared there only as a computational aid. Their apparent necessity in quantum theory - to encode interference, entanglement, and unitary evolution - has therefore demanded explanation.

Early attempts at a real formulation, pioneered by Stueckelberg in the 1960s [2], established a standard recipe: replace a complex \(d\)-dimensional Hilbert space by a real \(2d\)-dimensional one via the substitution \[\label{eq:doubling} 1 \leftrightarrow \mathbb{1}_2 = \begin{pmatrix}1&0\\0&1\end{pmatrix}, \qquad \mathsf{i}\leftrightarrow \tau = \begin{pmatrix}0&-1\\1&0\end{pmatrix},\tag{1}\] where \(\tau^2 = -\mathbb{1}_2\) mirrors the defining relation \(\mathsf{i}^2 = -1\), and \(\tau\) is concretely the matrix of counterclockwise rotation by \(\pi/2\) in the plane. This doubling map reproduces single-system statistics faithfully. For composite systems, however, one must form the tensor product of two such doubled spaces. Using the standard Kronecker product \(\otimes_{\mathbb{R}}\) on the doubled spaces on real numbers does not recover the complex tensor product \(\otimes_{\mathbb{C}}\): the dimension count alone shows \[\dim(\mathbb{R}^{2m}\otimes_{\mathbb{R}}\mathbb{R}^{2n})=4mn,\] whereas \[\dim(\mathbb{C}^m \otimes_{\mathbb{C}} \mathbb{C}^n)=mn\] over \(\mathbb{C}\) (equivalently \(2mn\) over \(\mathbb{R}\)). The two theories diverge precisely in multi-partite scenarios.

It is perhaps remarkable that, nearly a century after the mathematical foundations of quantum mechanics were laid, the correct notion of tensor product for composite systems continues to generate confusion. The tensor product \(\otimes\) is arguably the most conceptually subtle operation in the quantum formalism: familiar in its matrix guise as the Kronecker product, yet deceptively simple notation concealing the passage from independent to genuinely entangled degrees of freedom: unlike direct sums or operator products, it encodes the correlation structure of compound systems in a way that has no classical counterpart. Its definition is not merely a notational choice but carries deep algebraic content - specifically, it must be compatible with every piece of structure that the constituent spaces carry. In a complex Hilbert space, that structure includes the field \(\mathbb{C}\) itself: the tensor product \(\otimes_{\mathbb{C}}\) is a tensor product over \(\mathbb{C}\), enforcing \(\mathbb{C}\)-linearity across subsystems. When one passes to a real description via a doubling map, this \(\mathbb{C}\)-linearity does not disappear; it re-emerges as compatibility with the complex structure \(J\). Replacing \(\otimes_{\mathbb{C}}\) directly by the Kronecker product \(\otimes_{\mathbb{R}}\) on doubled spaces discards precisely this constraint, producing a strictly larger - and physically spurious - space. The algebraic incompatibility of \(\otimes_{\mathbb{R}}\) with the complex structure \(J\) of the doubled space becomes evident once made explicit, yet its consequences for no-go arguments were not previously noted [3].

Renou et al. [3] exploited this divergence to claim an experimental falsification of real quantum theory, demonstrating that the \(\mathrm{CHSH}_3\) inequality - tailored for a tripartite entanglement-swapping network - admits a maximal quantum violation of \(6\sqrt{2}\) that, they argued, is unreachable by any real quantum theory (in their sense). A subsequent experiment confirmed the prediction [4].

It is important to note that the no-go theorem derived by Renou et al. [3] is internally consistent within their axiomatic framework of a very specific “real quantum physics” (RQP). However, here we show that the incompatibility identified by Renou et al.is not a property of real numbers as such, but of the (inadequate) specific composition rule \(\otimes_{\mathbb{R}}\) that, we argue, is incompatible with a faithful real representation of quantum mechanics. The core observation, already implicit in the Stueckelberg programme, is that the matrices \(\mathbb{1}_2\) and \(\tau\) in Eq. 1 are not just a notational convenience - they constitute a complex structure \(J=\tau\otimes\mathbb{1}_N\) on the real doubled space. This structure promotes the doubled space to a Kähler space \((\mathbb{R}^{2N},g,\omega,J)\) [5], [6], and the compatible composition rule for Kähler spaces - the symplectic tensor product \(\otimes^{\mathcal{K}}\) - is distinct from \(\otimes_{\mathbb{R}}\) (which justifies the subscript notation).

Our main result is the following isomorphism theorem (proved in Appendix):

This result - which we substantiate with explicit matrix calculations, including the full \(\mathrm{CHSH}_{3}\) computation - directly contradicts the Renou et al.no-go theorem. It is consistent with, and provides the explicit constructive counterpart to the recent independent results of Hoffreumon and Woods [7], [8], who showed from an operational/postulational perspective that a real quantum theory with representation locality is possible, and that the key assumption of Renou et al.(product-state independence of sources) is experimentally untestable [8]. Related works appeared simultaneously [5], [6], [9].

It is only fair to acknowledge that complex structure is not an accidental feature of quantum mechanics: it is deeply woven into the formalism from the outset. The Schrödinger equation is intrinsically complex; unitary evolution, the superposition principle, and interference phenomena all rely on the full algebraic richness of \(\mathbb{C}\). What our analysis clarifies, however, is that this complexity is not fundamental in the ontological sense - it is not a primitive ingredient that must be postulated independently of the real structure of the theory. Rather, \(\mathbb{C}\) encodes a real geometric datum: the complex structure \(J\) of a Kähler manifold, satisfying \(J^2 = -\mathbb{1}\) and compatible with both the metric \(g\) and the symplectic form \(\omega\). In this light, the debate between “real” and “complex” quantum mechanics is somewhat misleading: the two descriptions are not rival theories but dual languages for the same geometry. The question is not whether complex numbers appear, but whether their appearance is irreducible or whether it reflects an underlying real structure that can be made explicit. Our isomorphism theorem answers this question unambiguously in favour of the latter.

2 Kähler space quantum mechanics↩︎

2.1 Definition of a Kähler space↩︎

A Kähler space \(\mathcal{K}\) [5], [6] is a quadruplet \((\mathbb{V},g,\omega,J)\) where \(\mathbb{V}\) is a real vector space, \(g\) is a positive-definite inner product (metric), \(\omega\) is a non-degenerate skew-symmetric bilinear form (symplectic form), and \(J:\mathbb{V}\to\mathbb{V}\) is a complex structure satisfying \(J^2=-\mathbb{1}\). These are linked by the fundamental compatibility relations \[\label{eq:kahler} g(x,y) = \omega(x,Jy), \qquad \omega(Jx,Jy) = \omega(x,y), \qquad x,y\in\mathbb{V}.\tag{2}\] Together, \((g,\omega,J)\) identifies \(\mathbb{V}\) with a complex Hilbert space \(\mathcal{H}\) over \(\mathbb{C}\) through the bijection \((a\mathbb{1}+bJ)v_\mathcal{K}\leftrightarrow (a+\mathsf{i}b) v_\mathcal{H}\). The Kähler space \(\mathcal{K}\) and the complex Hilbert space \(\mathcal{H}\) thus encode identical physics in different but mutually translatable languages.

2.2 Realification and complexification maps↩︎

For an \(N\)-dimensional complex Hilbert space \(\mathcal{H}\), the corresponding Kähler space is \(\mathcal{K}= \mathbb{R}^{2N}\) with the complex structure \[\label{eq:J} J = \tau \otimes \mathbb{1}_N, \qquad \tau = \begin{pmatrix}0&-1\\1&0\end{pmatrix},\tag{3}\] where \(\otimes\equiv \otimes_\mathbb{R}\).

Any linear operator \(L=X+\mathsf{i}Y\) on \(\mathcal{H}\) (with \(X,Y\) real \(N\times N\) matrices) has a Kähler space counterpart \[\label{eq:gammainv} \mathcal{L} = \gamma^{-1}(L) = \mathbb{1}_2\otimes\operatorname{Re}(L) + \tau\otimes\operatorname{Im}(L) = \begin{pmatrix}X&-Y\\Y&X\end{pmatrix}.\tag{4}\] The inverse map \(\gamma:\mathcal{K}\to\mathcal{H}\) extracts the complex operator from a Kähler block matrix: \[\label{eq:gamma} L = \gamma(\mathcal{L}) = \frac{1}{2}\!\left( \mathcal{T}[\mathcal{L}] + \mathcal{T}[(-\sigma_y\otimes\mathbb{1}_N)\,\mathcal{L}] \right),\tag{5}\] where \(\require{physics} \mathcal{T}[A\otimes C]=(\Tr A)C\) is the tensor contraction (see Appendix). The same prescription applies to state vectors: for \(\ket{\psi}_{\mathcal{H}}=\ket{R}+\mathsf{i}\ket{I}\) with real component vectors \(\ket{R},\ket{I}\in \;subspace\;\sim \mathbb{R}^N\), \[\label{eq:statevec} \ket{\psi}_{\mathcal{K}} = \gamma^{-1}(\ket{\psi}_{\mathcal{H}}) = \begin{pmatrix}\ket{R}&-\ket{I}\\\ket{I}&\ket{R}\end{pmatrix},\tag{6}\] a \(2N\times 2\) real matrix. Bra-states map as \(\bra{\psi}_{\mathcal{K}} = \mathbb{1}_2\otimes\bra{R} - \tau\otimes\bra{I}\).

2.3 Inner product and symplectic form↩︎

In Kähler space the physical inner product is the Riemannian metric \(g\), not the formal matrix product \(\braket{\psi_2}{\psi_1}_{\mathcal{K}}\) (which is not a real scalar). Explicitly, \[\require{physics} \label{eq:metric} g\!\left(\ket{\psi_2}_{\mathcal{K}},\ket{\psi_1}_{\mathcal{K}}\right) = \tfrac{1}{2}\,\Tr\!\left[\braket{\psi_2}{\psi_1}_{\mathcal{K}}\right] = \operatorname{Re}\!\braket{\psi_2}{\psi_1}_{\mathcal{H}},\tag{7}\] while the symplectic form reads \[\require{physics} \omega\!\left(\ket{\psi_2}_{\mathcal{K}},\ket{\psi_1}_{\mathcal{K}}\right) = \tfrac{1}{2}\,\Tr\!\left[-J\braket{\psi_2}{\psi_1}_{\mathcal{K}}\right] = \operatorname{Im}\!\braket{\psi_2}{\psi_1}_{\mathcal{H}}.\] All expectation values and probabilities computed via \(g\) in \(\mathcal{K}\) agree precisely with those computed via \(\langle\cdot|\cdot\rangle_{\mathcal{H}}\) in \(\mathcal{H}\) (see Section S3).

3 Symplectic composition rule and isomorphism↩︎

The failure of literal quantum mechanics on real numbers in multipartite settings originates entirely in the composition rule for subsystems. To see why, note that the algebraic rule for complex multiplication is \((X_A+\mathsf{i}Y_A)\otimes(X_B+\mathsf{i}Y_B) = (X_A\otimes X_B - Y_A\otimes Y_B) + \mathsf{i}(X_A\otimes Y_B + Y_A\otimes X_B)\). The standard Kronecker product on doubled real spaces does not respect this rule; instead it mixes real and imaginary sectors incoherently.

3.1 The symplectic tensor product↩︎

This rule encodes the complex multiplication law \((A\otimes C-B\otimes D) +\mathsf{i}(A\otimes D + B\otimes C)\) directly in the block-matrix structure. The commutative diagram in Fig. 1 captures the equivalence: going round either path - first composing in \(\mathcal{K}\) or first lifting to \(\mathcal{H}\) and then projecting back - yields the same result.

3.2 Proof sketch of the isomorphism theorem↩︎

The full proof occupies Section S4; we outline the key steps.

Bijection. Computing \(\gamma(\gamma^{-1}(L))\) and \(\gamma^{-1}(\gamma(\mathcal{L}))\) using Eqs. 4 –5 directly yields \(L\) and \(\mathcal{L}\) respectively, establishing the bijection.

Multiplicativity. Both \(\gamma\) and \(\gamma^{-1}\) satisfy \(\gamma( \mathcal{L}_A\,\mathcal{L}_B)=\gamma( \mathcal{L}_A)\gamma(\mathcal{L}_B)\), i.e.they are ring homomorphisms (appendix).

Monoidal structure. The two key identities \[\label{eq:mono} \gamma(\mathcal{L}_A\otimes^{\mathcal{K}}\mathcal{L}_B) = \gamma(\mathcal{L}_A)\otimes_{\mathbb{C}}\gamma(\mathcal{L}_B), \qquad \gamma^{-1}(A\otimes_{\mathbb{C}} B) = \gamma^{-1}(A)\otimes^{\mathcal{K}}\gamma^{-1}(B),\tag{9}\] are established in appendix. Together they constitute the isomorphism of monoidal quantum theories. For the connection with the balanced tensor product (see section S8)

4 Why Renou et al.’s construction fails↩︎

The realification introduced in Eq. (1) of Ref. [3] associates to each complex density matrix \(\rho\) a real matrix \(\rho^{\mathbb{R}}\) on a space of doubled dimension via the same substitution \(\mathsf{i}\mapsto\tau\). This is the map \(\gamma^{-1}\) restricted to density matrices. The crucial point, however, concerns composite systems. For a bipartite state \(\rho_{AB}\), Renou et al. form the composite real space as \(\mathcal{H}_A^{\mathbb{R}}\otimes_{\mathbb{R}}\mathcal{H}_B^{\mathbb{R}}\), i.e.they use the standard Kronecker product on the doubled spaces. This is not the symplectic product \(\otimes^{\mathcal{K}}\); it is an incompatible operation that does not commute with the complex structure. It is important to stress that this divergence is not confined to the choice of tensor product in the abstract sense: because \(\otimes_{\mathbb{R}}\) and \(\otimes^{\mathcal{K}}\) act on the same single-system blocks but combine them differently, the resulting composite-system matrices and state vectors are themselves different objects, living in real spaces of different dimension (\(4mn\) versus \(2mn\)), and obeying different algebraic relations. The composition rule and the representation of the composite state are thus two faces of the same failure, not independent choices. Algebraically, \[\begin{align} \rho^{\mathbb{R}}_{AB} &= \gamma^{-1}(\rho_A)\otimes_{\mathbb{R}}\gamma^{-1}(\rho_B) \;\neq\; \gamma^{-1}(\rho_A)\otimes^{\mathcal{K}}\gamma^{-1}(\rho_B) = \gamma^{-1}(\rho_A\otimes_{\mathbb{C}}\rho_B). \end{align}\]

The inequality is strict as soon as \(\rho_A\) or \(\rho_B\) has a non-zero imaginary part. As a result, the real composite space of Renou et al. is a strict enlargement of the physically relevant one: \(\dim(\mathcal{H}_A^{\mathbb{R}}\otimes_{\mathbb{R}}\mathcal{H}_B^{\mathbb{R}}) = 4\dim(\mathcal{H}_A)\dim(\mathcal{H}_B)\), versus \(\dim(\mathcal{K}_A\otimes^{\mathcal{K}}\mathcal{K}_B) = 2\dim(\mathcal{H}_A)\dim(\mathcal{H}_B)\) (matching standard quantum mechanics). The extra dimensions carry no physical content; they arise from tensor-mixing the mathematical encoding structure (the \(\mathbb{1}_2/\tau\) block) with the physical degrees of freedom.

5 Bell-inequality violations↩︎

5.1 CHSH inequality↩︎

Renou et al.showed [3] that the standard real formulation can maximally violate the CHSH inequality, and this remains true in our framework. The Kähler space Pauli matrices are (Appendix, Eqs.(S9)) \[\sigma_x^{\mathcal{K}} = \mathbb{1}_2\otimes\sigma_x, \quad \sigma_y^{\mathcal{K}} = \tau\otimes\tau, \quad \sigma_z^{\mathcal{K}} = \mathbb{1}_2\otimes\sigma_z.\] These satisfy the Kähler space anti-commutation relations \(\{\sigma_a^{\mathcal{K}},\sigma_b^{\mathcal{K}}\}=2\delta_{ab}\mathbb{1}_4\), inherited from the complex algebra via \(\gamma\). A Bell state \(\ket{\psi^-}_{\mathcal{H}}\) maps to \(\ket{\psi^-}_{\mathcal{K}}=\gamma^{-1}(\ket{\psi^-}_{\mathcal{H}})\) (Eq. 6 ), and the standard CHSH combination yields \(g(\ket{\psi^-}_{\mathcal{K}},\hat{C}_{\mathrm{CHSH}}^{\mathcal{K}}\ket{\psi^-}_{\mathcal{K}}) =2\sqrt{2}\), the Tsirelson bound.

5.2 CHSH\(_3\) inequality and the \(6\sqrt{2}\) violation↩︎

The \(\mathrm{CHSH}_3\) functional for the tripartite entanglement-swapping scenario (Alice, Bob, Charlie; two independent sources) was constructed in Ref. [3] precisely to separate real and complex quantum theories. It is defined as \[\mathrm{CHSH}_3 = \mathrm{CHSH}(1,2;1,2)+\mathrm{CHSH}(1,3;3,4)+\mathrm{CHSH}(2,3;5,6),\] with a sign adaptation functional \(\mathscr{T}_b\) conditioned on Bob’s Bell-measurement outcome \(b=(b_1b_2)\) (see Section S6) for the complete definition). The maximum value achievable by local-realist theories is \(6\); standard real quantum theory (with \(\otimes_{\mathbb{R}}\)) attains at most \(\approx 7.66\); complex quantum theory reaches \(6\sqrt{2}\approx 8.49\).

In our Kähler space framework, for Bob’s outcome \(b=00\) and the Bell state \(\ket{\phi^+}\), the relevant operator in Kähler space is \[\label{eq:T00} \hat{\mathscr{T}}_{00}^{\mathcal{K}} = 2\sqrt{2}\,\bigl( \sigma_z^{\mathcal{K}}\otimes^{\mathcal{K}}\sigma_z^{\mathcal{K}} + \sigma_x^{\mathcal{K}}\otimes^{\mathcal{K}}\sigma_x^{\mathcal{K}} - \sigma_y^{\mathcal{K}}\otimes^{\mathcal{K}}\sigma_y^{\mathcal{K}} \bigr),\tag{10}\] with the Bell state mapped as \(\ket{\phi^+}_{\mathcal{K}}=\gamma^{-1}(\ket{\phi^+})=\mathbb{1}_2\otimes\ket{\phi^+}\). A direct computation (see Section S6) gives \[\label{eq:chsh3result} \bra{\phi^+}_{\mathcal{K}}\, \hat{\mathscr{T}}_{00}^{\mathcal{K}}\, \ket{\phi^+}_{\mathcal{K}} = 6\sqrt{2}\,\mathbb{1}_2,\tag{11}\] so the metric expectation value is \(g(\ket{\phi^+}_{\mathcal{K}},\hat{\mathscr{T}}_{00}^{\mathcal{K}}\ket{\phi^+}_{\mathcal{K}}) =\tfrac{1}{2}\mathcal{T}[6\sqrt{2}\,\mathbb{1}_2]=6\sqrt{2}\). This is the maximum quantum value, achieved using only real arithmetic. The critical algebraic ingredient is the Kähler space identity \(\sigma_x^{\mathcal{K}}\sigma_y^{\mathcal{K}}=J\sigma_z^{\mathcal{K}}\), where \(J=\tau\otimes\mathbb{1}\) replaces the imaginary unit, together with the strict anti-commutation \(\{\sigma_x^{\mathcal{K}},\sigma_y^{\mathcal{K}}\}=0\) (see Section S6). Both relations are forbidden in the Renou et al.real formalism (which restricts to real symmetric operators and \(\otimes_{\mathbb{R}}\)) but hold in \(\mathcal{K}\).

6 Discussion↩︎

Our results establish that the Renou et al.no-go theorem [3] does not apply to Kähler space quantum mechanics. The theorem is a theorem about a particular real formulation - one that uses the standard Kronecker product on doubled spaces - not about all possible real formulations. Our isomorphism theorem shows that a perfectly adequate real formulation exists, provided one uses the algebraically compatible composition rule \(\otimes^{\mathcal{K}}\).

6.0.0.1 Connection to Hoffreumon–Woods.

Our results are strongly consonant with those of Hoffreumon and Woods [7], [8]. Their 2025 paper shows operationally that a real quantum theory with representation locality is possible; our work provides the explicit constructive realisation (the \(\gamma\) map and \(\otimes^{\mathcal{K}}\) rule) and verifies the isomorphism by direct computation. Their 2026 paper additionally shows that the “product-state independence” assumption in Ref. [3] is experimentally untestable—an independent argument for the same conclusion. Remark 2 explains how the two perspectives are related.

6.0.0.2 The “nonlocality” objection.

Feng, Ren, and Vedral [10] argued that any modified tensor product of the Hoffreumon–Woods type introduces a “fundamental nonlocal map”. In our framework, the \(\mathbb{1}_2/\tau\) block structure is the complex structure \(J\) of the Kähler space—a fixed geometric attribute of the space, not a dynamical physical system. No physical ancilla is introduced; the apparent nonlocality is an artefact of interpreting the encoding structure as a physical degree of freedom. Local operations in \(\mathcal{H}\) (unitary evolution \(U_A\otimes\mathbb{1}_B\), CPTP maps \(\Phi_A\otimes\mathcal{I}_B\)) map directly to local operations in \(\mathcal{K}\): \(U_A^{\mathcal{K}}\otimes^{\mathcal{K}}\mathbb{1}_B^{\mathcal{K}}\) acting on \(\rho_{AB}^{\mathcal{K}}\) (see Section S7). Crucially, this identical geometric logic applies to the mapping developed by Hoffreumon and Woods [7], [8]; because their formulation also relies on a fixed encoding structure rather than a physical ancilla, the nonlocality critique is equally invalid for both results.

6.0.0.3 Geometric interpretation.

Writing \(\psi=q+\mathsf{i}p\), the Kähler space is \(\mathbb{R}^{2N}\) with the triple \((g,\omega,J)\) satisfying \(g(\cdot,\cdot)=\omega(\cdot,J\cdot)\). The symmetry group is \(U(N)=O(2N)\cap Sp(2N,\mathbb{R})\). Complex numbers in quantum mechanics are therefore not fundamental: they encode the real geometric structure of a Kähler manifold governing phase and composition. This view harmonises with Volovich’s independent analysis [5], [6].

7 Conclusions↩︎

We have demonstrated: (i) a bijection \(\gamma\) between complex Hilbert space \(\mathcal{H}\) and Kähler space \(\mathcal{K}\); (ii) an explicit symplectic composition rule \(\otimes^{\mathcal{K}}\) making the bijection monoidal; (iii) the maximal \(\mathrm{CHSH}_3\) violation \(6\sqrt{2}\) in purely real arithmetic; (iv) the precise reason why the Renou et al.construction fails (using \(\otimes_{\mathbb{R}}\) in place of \(\otimes^{\mathcal{K}}\) is not a monoidal equivalence of theories). Together these results establish that complex numbers are not necessary for quantum mechanics: they encode a deeper real geometric structure governing phase and composition.

To conclude, our approach should on no account be read as an attempt to expel complex numbers from quantum formalism. The complex Hilbert space \(\mathcal{H}\), or equivalently the real Kähler space \(\mathcal{K}\), carries a fundamental involutive symmetry \(\mathsf{i}\mapsto -\mathsf{i}\) (complex conjugation), whose geometric avatar in \(\mathcal{K}\) is the involution \(J\mapsto -J\) of the compatible complex structure. The physical ramifications of this single structural feature are far-reaching: Wigner’s antiunitary time-reversal operator, charge conjugation, the CPT theorem, and the canonical decomposition of any operator into its self-adjoint and skew-adjoint parts all flow from it. A bare real Hilbert space, lacking \(J\) altogether, is entirely devoid of this structure and therefore cannot support these foundational symmetries of nature. What our framework establishes, rather, is that the indispensable role of \(\mathsf{i}\) is already fully encoded in the symplectic geometry of \(\mathcal{K}\): complex structure is not imposed from without but is a derived feature of the real, geometrically natural formulation. The imaginary unit is not discarded - it is explained.

8 Acknowledgments↩︎

We thank the financial support from the Brazilian scientific agencies Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).

9 Author contributions↩︎

A.C.M., E.M.F.C., and J.-P.G. conceptualized the framework. A.C.M. developed the mathematical proofs and carried out the CHSH\(_3\) calculations. All authors discussed the results, interpreted the geometric framework, and contributed to writing the manuscript.

10 Overview↩︎

This Appendices presents the detailed mathematical framework supporting the main text. We prove the isomorphism between complex quantum mechanics, i.e., QM formulated on a complex Hilbert space \(\mathcal{H}\)) and our real formulation (formulated on a Kähler space \(\mathcal{K}\)), and we carry out the explicit Bell-inequality calculations.

Structure. Section 10 revisits the concept of tensor product Section 11 introduces tensor contraction. Section 12 develops the Kähler space framework: definition, maps \(\gamma\) and \(\gamma^{-1}\), bijection proof, fundamental relations, and multiplicativity. Section 13 defines the symplectic composition rule \(\otimes^{\mathcal{K}}\) and proves the two isomorphism lemmas. Section 14 treats the CHSH inequality. Section 15 treats the \(\mathrm{CHSH}_3\) inequality. Section 16 discusses local maps. Section 17 is about the balanced tensor product.

Notation. Throughout: \(\mathbb{1}\) denotes the identity operator (subscript indicates dimension when needed); \(\tau=\begin{pmatrix}0&-1\\1&0\end{pmatrix} = -\mathsf{i}\sigma_y\); \(\sigma_x,\sigma_y,\sigma_z\) are the usual Pauli matrices; \(\mathcal{T}\) denotes tensor contraction (Section 11); \(\mathsf{i}=\sqrt{-1}\); superscript \(\mathcal{K}\) on an operator or state denotes its Kähler space representative.

Tensor product. In essence, the Cartesian product, traditionally employed to model two systems independently in classical physics, undergoes a transformative process known as quantization. This metamorphosis results in the emergence of the tensor product of two vector spaces, a fundamental framework indispensable for understanding the intricacies of quantum mechanical systems.

A vector space \(V\) over a field \(\mathbb{F}\), e.g., \(\mathbb{Q}\) or \(\mathbb{R}\) or \(\mathbb{C}\), is a set whose elements or vectors, may be added together and multiplied (“scaled”) by elements (“scalars”) in \(\mathbb{F}\). Two essential properties must be satisfied: the distributivity of scalar multiplication with respect to the vector addition and the distributivity of scalar multiplication with respect to field addition. Then, the tensor product \(V \otimes_\mathbb{F} W\) of two vector spaces \(V\) and \(W\) (over the same field) is the vector space over \(\mathbb{F}\) consisting of all bilinear forms from \(V \times W\) to \(\mathbb{F}\).

As a consequence, \[\mathrm{dim}(V\otimes_\mathbb{F} W)= \mathrm{dim}V\mathrm{dim}W\] while \[\mathrm{dim}(V\times W)= \mathrm{dim}V+\mathrm{dim}W.\]

Usually, one simplifies \(\otimes= \otimes_\mathbb{F}\) when there is no risk of confusion. On the other hand we will introduce the symbol \(\otimes^{\mathcal{K}}\) within the context of Kähler spaces defined in this material.

The tensor product extends naturally to linear operators. If \(A: V\to V\) and \(B: W\to W\) are linear maps, their tensor product \(A\otimes_\mathbb{F} B: V\otimes_\mathbb{F} W \to V\otimes_\mathbb{F} W\) is the unique linear map defined on simple tensors by \[(A\otimes_\mathbb{F} B)(v\otimes w) = (Av)\otimes(Bw),\] and extended by linearity. In matrix terms, if \(A\in M_m(\mathbb{F})\) and \(B\in M_n(\mathbb{F})\), then \(A\otimes_\mathbb{F} B\) is the \(mn\times mn\) Kronecker product. Crucially, the field \(\mathbb{F}\) must be the same for both factors: this is precisely the point at issue in the Renou et al.experiment, where \(\mathbb{R}\)-linear and \(\mathbb{C}\)-linear tensor products are conflated. Our Kähler framework resolves this by working exclusively over \(\mathbb{R}\) while encoding the complex structure in the automorphism \(J\), so that \(\otimes^{\mathcal{K}}\) (Definition 5) plays the role of \(\otimes_\mathbb{C}\) without ever leaving \(\mathbb{R}\).

11 Tensor contraction↩︎

In traditional quantum mechanics the partial trace is the canonical reduction operation for density matrices. We generalise it to rectangular objects via the tensor contraction \(\mathcal{T}\).

When \(C\) is square, \(\mathcal{T}\) reduces to the standard partial trace \(\require{physics} \Tr_A\) over subsystem \(A\). When \(C\) is rectangular, \(\mathcal{T}\) provides the linear extraction rule used in the complexification map \(\gamma\).

Key instances used below. \(\mathcal{T}[\mathbb{1}_2\otimes C] = 2C\); \(\mathcal{T}[\tau\otimes C] = 0\).

12 Kähler space framework↩︎

12.1 Definition of a Kähler space↩︎

12.2 Realification: from \(\mathcal{H}\) to \(\mathcal{K}\)↩︎

Let \(L = X + \mathsf{i}Y\) be a linear operator on \(\mathcal{H}\cong\mathbb{C}^N\), where \(X = \operatorname{Re}(L)\) and \(Y = \operatorname{Im}(L)\) are real \(N\times N\) matrices. Its Kähler space representative is \[\label{eq:sm:gammainv} \mathcal{L} = \gamma^{-1}(L) = \mathbb{1}_2\otimes X + \tau\otimes Y = \begin{pmatrix}X & -Y \\ Y & X\end{pmatrix}.\tag{15}\] Equivalently, using \(L^* = X - \mathsf{i}Y\): \[\gamma^{-1}(L) = \frac{1}{2}\!\left[ \mathbb{1}_2\otimes(L+L^*) - \sigma_y\otimes(L-L^*) \right].\] The same formula applies to state vectors. For \(\ket{\psi}_{\mathcal{H}} = \ket{R}+\mathsf{i}\ket{I}\) with \(\ket{R},\ket{I}\in\mathbb{R}^N\): \[\label{eq:sm:psikahler} \ket{\psi}_{\mathcal{K}} = \gamma^{-1}(\ket{\psi}_{\mathcal{H}}) = \mathbb{1}_2\otimes\ket{R} + \tau\otimes\ket{I} = \begin{pmatrix}\ket{R} & -\ket{I} \\ \ket{I} & \ket{R}\end{pmatrix},\tag{16}\] a \(2N\times 2\) real matrix (\(\ket{\psi}_{\mathcal{H}}\) is an \(N\times 1\) column vector, so \(\ket{\psi}_{\mathcal{K}}\) has dimensions \(2N\times 2\)). The conjugate bra-vector is \[\bra{\psi}_{\mathcal{K}} = \bigl(\ket{\psi}_{\mathcal{K}}\bigr)^\dagger = \mathbb{1}_2\otimes\bra{R} - \tau\otimes\bra{I} = \begin{pmatrix}\bra{R} & \bra{I} \\ -\bra{I} & \bra{R}\end{pmatrix}.\]

12.2.0.1 Worked example: Kähler–Pauli matrices.

The Pauli matrices \(\sigma_x,\sigma_y,\sigma_z\) map as follows. Since \(\sigma_x\) and \(\sigma_z\) are purely real, \[\label{eq:sm:KahPaux} \sigma_x^{\mathcal{K}} = \mathbb{1}_2\otimes\sigma_x = \begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}, \quad \sigma_z^{\mathcal{K}} = \mathbb{1}_2\otimes\sigma_z = \begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&1&0\\0&0&0&-1\end{pmatrix}.\tag{17}\] For \(\sigma_y = \mathsf{i}\tau\) (so \(\operatorname{Re}(\sigma_y)=0\), \(\operatorname{Im}(\sigma_y)=\tau\)): \[\label{eq:sm:KahPauy} \sigma_y^{\mathcal{K}} = \tau\otimes\tau = \begin{pmatrix}0&-1\\1&0\end{pmatrix} \otimes \begin{pmatrix}0&-1\\1&0\end{pmatrix} = \begin{pmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{pmatrix}. '}\tag{18}\] The Kähler complex structure \(J\) (Eq. (4) of the main text) acts as \[\label{eq:sm:Jact} \sigma_x^{\mathcal{K}}\sigma_y^{\mathcal{K}} = J\sigma_z^{\mathcal{K}}, \qquad J = \tau\otimes\mathbb{1}_2,\tag{19}\] so \(J\) replaces the imaginary unit \(\mathsf{i}\) in the Kähler-Pauli algebra.

12.2.0.2 Worked example: Kähler space qubit.

Consider a general qubit \(\ket{\psi}_{\mathcal{H}}=\cos(\theta/2)\ket{0}_{\mathcal{H}} + e^{\mathsf{i}\phi}\sin(\theta/2)\ket{1}_{\mathcal{H}}\). Writing \(e^{\mathsf{i}\phi}=\cos\phi+\mathsf{i}\sin\phi\) we identify \(\ket{R}=\cos(\theta/2)\ket{0}_{\mathcal{H}}+\cos(\phi)\sin(\theta/2)\ket{1}_{\mathcal{H}}\) and \(\ket{I}=\sin(\phi)\sin(\theta/2)\ket{1}_{\mathcal{H}}\). Then \(\ket{\psi}_{\mathcal{K}}\) is the \(4\times 2\) real matrix \[\ket{\psi}_{\mathcal{K}} = \begin{pmatrix} \cos(\theta/2) & 0 \\ \cos\phi\sin(\theta/2) & -\sin\phi\sin(\theta/2) \\ 0 & \cos(\theta/2) \\ \sin\phi\sin(\theta/2) & \cos\phi\sin(\theta/2) \end{pmatrix}.\] The matrices \(\mathbb{1}_2\) and \(\tau\) carry the complex structure of the Kähler space; they are not a physical qubit.

12.3 Complexification: from \(\mathcal{K}\) to \(\mathcal{H}\)↩︎

Worked example. Applying \(\gamma\) to \(\sigma_y^{\mathcal{K}}=\tau\otimes\tau\): \[\begin{align} \gamma(\sigma_y^{\mathcal{K}}) &= \tfrac{1}{2}\Bigl\{ \mathcal{T}[\tau\otimes\tau] + \mathcal{T}[(-\sigma_y\otimes\mathbb{1})(\tau\otimes\tau)] \Bigr\} \\ &= \tfrac{1}{2}\Bigl\{ 0 + \mathcal{T}[(-\sigma_y\tau)\otimes\tau] \Bigr\}. \end{align}\]

Using \(\sigma_y = \mathsf{i}\tau\), so \(-\sigma_y\otimes\mathbb{1}= -\mathsf{i}\tau\otimes\mathbb{1}\). Then \((-\sigma_y\otimes\mathbb{1})(\tau\otimes\tau) = (-\mathsf{i}\tau\cdot\tau)\otimes(\mathbb{1}\cdot\tau) = (-\mathsf{i}\cdot(-\mathbb{1}_2))\otimes\tau = \mathsf{i}\mathbb{1}_2\otimes\tau\). Thus \(\require{physics} \mathcal{T}[\mathsf{i}\mathbb{1}_2\otimes\tau] = \mathsf{i}\cdot(\Tr\mathbb{1}_2)\cdot\tau = 2\mathsf{i}\tau\), and \(\gamma(\sigma_y^{\mathcal{K}})=\tfrac{1}{2}\cdot 2\mathsf{i}\tau=\mathsf{i}\tau=\sigma_y\). ✔

12.4 Bijection theorem↩︎

12.5 Fundamental Kähler relations↩︎

Define two states \(\ket{\psi_j}_{\mathcal{H}}=\ket{R_j}+\mathsf{i}\ket{I_j}\), \(j=1,2\). The Hilbert-space inner product is \[\braket{\psi_2}{\psi_1}_{\mathcal{H}} = \bigl(\braket{R_2}{R_1}+\braket{I_2}{I_1}\bigr) + \mathsf{i}\bigl(\braket{R_2}{I_1}-\braket{I_2}{R_1}\bigr),\] while its Kähler space version is \(\braket{\psi_2}{\psi_1}_{\mathcal{K}} = \mathbb{1}_2\otimes(\braket{R_2}{R_1}+\braket{I_2}{I_1}) + \tau\otimes(\braket{R_2}{I_1}-\braket{I_2}{R_1})\). The metric (real-valued inner product) in \(\mathcal{K}\) is \[\label{eq:sm:metric} g\!\bigl(\ket{\psi_2}_{\mathcal{K}},\ket{\psi_1}_{\mathcal{K}}\bigr) = \tfrac{1}{2}\,\mathcal{T}\!\bigl[\braket{\psi_2}{\psi_1}_{\mathcal{K}}\bigr] = \operatorname{Re}\braket{\psi_2}{\psi_1}_{\mathcal{H}},\tag{21}\] and the symplectic form is \[\label{eq:sm:omega} \omega\!\bigl(\ket{\psi_2}_{\mathcal{K}},\ket{\psi_1}_{\mathcal{K}}\bigr) = \tfrac{1}{2}\,\mathcal{T}\!\bigl[-J\braket{\psi_2}{\psi_1}_{\mathcal{K}}\bigr] = \operatorname{Im}\braket{\psi_2}{\psi_1}_{\mathcal{H}},\tag{22}\] with complex structure \(J=\tau\otimes\mathbb{1}_N\). One directly verifies Eqs. 13 –14 .

12.6 Multiplicativity of \(\gamma\)↩︎

It is worth mentioning that \(\mathcal{L}_A\) and \(\mathcal{L}_B\) are not restricted to the same Kähler space, the only requirement is that the operation \(\mathcal{L}_A\mathcal{L}_B\) must be well-defined. For instance \(\mathcal{L}_A\) can be a linear operator and \(\mathcal{L}_B\) a physical state. Also, the requirement for \(L_A\) and \(L_B\) is analogous.

13 Symplectic composition rule and isomorphism↩︎

13.1 Definition of \(\otimes^{\mathcal{K}}\)↩︎

Mnemonic. The rule \(\otimes^{\mathcal{K}}\) mimics matrix multiplication on the \(2\times 2\) block structure: \[\begin{pmatrix}X_A&-Y_A\\Y_A&X_A\end{pmatrix} \otimes^{\mathcal{K}} \begin{pmatrix}X_B&-Y_B\\Y_B&X_B\end{pmatrix} = \text{``matrix product of blocks''},\] where each “product” of blocks uses the Kronecker product \(\otimes\) of the constituent real matrices.

Commutative diagram. The equivalence \(\gamma^{-1}\circ\otimes_{\mathbb{C}} = \otimes^{\mathcal{K}}\circ(\gamma^{-1}\times\gamma^{-1})\) is depicted in Fig. 1.

13.2 Isomorphism lemmas↩︎

Together with Theorem 2 (bijection) and Lemma 1 (multiplicativity), Lemmas 2 and 3 establish Theorem 1 of the main text: \((\mathcal{H},\otimes_{\mathbb{C}})\) and \((\mathcal{K},\otimes^{\mathcal{K}})\) are isomorphic monoidal quantum theories.

14 CHSH inequality↩︎

We verify the CHSH inequality in Kähler space. Choose Alice’s operators \(A_0=\sigma_z\), \(A_1=\sigma_x\) and Bob’s operators \(B_0=-(\sigma_x+\sigma_z)/\sqrt{2}\), \(B_1=(-\sigma_x+\sigma_z)/\sqrt{2}\) (all real-coefficient). Because the coefficients are real, the Kähler maps are block-diagonal: \[C_{a,b}^{\mathcal{K}} = \gamma^{-1}(A_a\otimes_{\mathbb{C}} B_b) = \begin{pmatrix}A_a\otimes B_b & 0 \\ 0 & A_a\otimes B_b\end{pmatrix} = \mathbb{1}_2\otimes(A_a\otimes B_b),\] a real \(8\times 8\) matrix (\(a,b\in\{0,1\}\)).

The Bell state \(\ket{\psi^-}_{\mathcal{H}}=(\ket{10}-\ket{01})/\sqrt{2}\) maps to \[\ket{\psi^-}_{\mathcal{K}} = \gamma^{-1}(\ket{\psi^-}_{\mathcal{H}}) = \mathbb{1}_2\otimes\ket{\psi^-},\] a \(8\times 2\) real matrix (since \(\ket{\psi^-}_{\mathcal{H}}\) is \(4\times 1\) over \(\mathbb{R}\), and the overall Kähler state is \(8\times 2\); we write \(\mathbb{1}_2\otimes\ket{\psi^-}\) in the \(8\times 2\) sense).

The metric expectation values are \(\langle C_{0,0}^{\mathcal{K}}\rangle = \langle C_{1,0}^{\mathcal{K}}\rangle = \langle C_{1,1}^{\mathcal{K}}\rangle = 1/\sqrt{2}\) and \(\langle C_{0,1}^{\mathcal{K}}\rangle = -1/\sqrt{2}\). Hence the CHSH combination reads \[\langle C_{0,0}^{\mathcal{K}}\rangle +\langle C_{1,0}^{\mathcal{K}}\rangle -\langle C_{0,1}^{\mathcal{K}}\rangle +\langle C_{1,1}^{\mathcal{K}}\rangle = 2\sqrt{2},\] achieving the Tsirelson bound [11].

15 CHSH\(_3\) inequality↩︎

15.1 Setup↩︎

The \(\mathrm{CHSH}_3\) inequality is designed for an entanglement-swapping network with three parties (Alice, Bob, Charlie) and two independent sources. Source 1 distributes particles to Alice and Bob; Source 2 to Bob and Charlie. Alice uses three measurement settings \(x=1,2,3\); Charlie uses six settings \(z=1,\ldots,6\); Bob performs a Bell-state measurement with four outcomes \(b=b_1b_2\in\{00,01,10,11\}\), corresponding to \(\{\ket{\phi^-},\ket{\psi^-},\ket{\phi^+},\ket{\psi^+}\}\) respectively, where \(\ket{\phi^\pm}=(\ket{00}\pm\ket{11})/\sqrt{2}\) and \(\ket{\psi^\pm}=(\ket{10}\pm\ket{01})/\sqrt{2}\).

15.2 Operator definitions↩︎

For Bob’s outcome \(b=(b_1,b_2)\), the functional is \[\begin{align} \mathscr{T}_b(P) &= (-1)^{b_2}(S_{11}^b+S_{12}^b) + (-1)^{b_1}(S_{21}^b-S_{22}^b) \notag\\ &\quad + (-1)^{b_2}(S_{13}^b+S_{14}^b) - (-1)^{b_1+b_2}(S_{33}^b-S_{34}^b) \notag\\ &\quad + (-1)^{b_1}(S_{25}^b+S_{26}^b) - (-1)^{b_1+b_2}(S_{35}^b-S_{36}^b), \end{align}\] where \(S_{xz}^b=\sum_{a,c}ac\,P(a,b,c|x,z)\). This defines an operator \(\hat{\mathscr{T}}_b\) via \(\mathscr{T}_b(P)=\bra{\psi}\hat{\mathscr{T}}_b\ket{\psi}\). With Charlie’s diagonal observables \(D_{ij}^C=(\sigma_i+\sigma_j)/\sqrt{2}\) and \(E_{ij}^C=(\sigma_i-\sigma_j)/\sqrt{2}\) (\(i,j\in\{x,y,z\}\)): \[\begin{align} \label{eq:sm:opchsh3} \hat{\mathscr{T}}_b &= (-1)^{b_2}Z^A(D_{zx}^C+E_{zx}^C) + (-1)^{b_1}X^A(D_{zx}^C-E_{zx}^C) \notag\\ &\quad + (-1)^{b_2}Z^A(D_{zy}^C+E_{zy}^C) - (-1)^{b_1+b_2}Y^A(D_{zy}^C-E_{zy}^C) \notag\\ &\quad + (-1)^{b_1}X^A(D_{xy}^C+E_{xy}^C) - (-1)^{b_1+b_2}Y^A(D_{xy}^C-E_{xy}^C), \end{align}\tag{27}\] and Alice’s operators \(Z^A, X^A, Y^A\) corresponds to Pauli matrices \(\sigma_z, \;\sigma_x,\;\sigma_y\), respectively. Bounds: local-realist \(\leq 6\); real QT of Ref. [3] \(\leq 7.66\); complex QT \(= 6\sqrt{2}\approx 8.49\).

15.3 Key algebraic obstruction in the Renou formalism↩︎

The Renou et al.proof uses self-testing of local Pauli operators \(X^C, Y^C, Z^C\) from measurement statistics, that are defined as \(X^C=(D_{zx}^C-E_{zx}^C)/\sqrt{2}\;\), \(Y^C=(D_{zy}^C-E_{zy}^C)/\sqrt{2}\;\), and \(Z^C=(D_{zx}^C+E_{zx}^C)/\sqrt{2}\). The crucial anti-commutation \(\{X^C,Y^C\}=0\) and the product rule \(X^CY^C=\mathsf{i}Z^C\) cannot simultaneously hold for real symmetric matrices (the only matrices allowed in their formalism with \(\otimes_{\mathbb{R}}\)).

In our framework these relations hold via the Kähler-space replacements: \[\label{eq:sm:ksalg} \sigma_x^{\mathcal{K}}\sigma_y^{\mathcal{K}} = (\mathbb{1}_2\otimes\sigma_x)(\tau\otimes\tau) = \tau\otimes(\sigma_x\tau) = J\cdot(\mathbb{1}_2\otimes\sigma_z^{\phantom{\mathcal{K}}}) = J\sigma_z^{\mathcal{K}},\tag{28}\] and \[\{\sigma_x^{\mathcal{K}},\sigma_y^{\mathcal{K}}\} = J\sigma_z^{\mathcal{K}}+(-J\sigma_z^{\mathcal{K}}) = 0,\] where \(J=\tau\otimes\mathbb{1}\) is the complex structure. These are real matrix equations; no complex numbers appear.

15.4 Explicit computation: \(b=00\)↩︎

For \(b=00\), Eq. 27 reduces to \[\begin{align} \hat{\mathscr{T}}_{00} &= \sqrt{2}\bigl( \sigma_z\otimes\sigma_z + \sigma_x\otimes\sigma_x + \sigma_z\otimes\sigma_z - \sigma_y\otimes\sigma_y + \sigma_x\otimes\sigma_x - \sigma_y\otimes\sigma_y \bigr)\notag\\ &= 2\sqrt{2}\bigl( \sigma_z\otimes\sigma_z + \sigma_x\otimes\sigma_x - \sigma_y\otimes\sigma_y \bigr).\label{eq:sm:T00} \end{align}\tag{29}\] In complex quantum mechanics, \(\bra{\phi^+}\hat{\mathscr{T}}_{00}\ket{\phi^+}=6\sqrt{2}\).

In Kähler space, the operator maps as \[\begin{align} \hat{\mathscr{T}}_{00}^{\mathcal{K}} &= 2\sqrt{2}\bigl( \sigma_z^{\mathcal{K}}\otimes^{\mathcal{K}}\sigma_z^{\mathcal{K}} + \sigma_x^{\mathcal{K}}\otimes^{\mathcal{K}}\sigma_x^{\mathcal{K}} - \sigma_y^{\mathcal{K}}\otimes^{\mathcal{K}}\sigma_y^{\mathcal{K}} \bigr),\label{eq:sm:T00ks} \end{align}\tag{30}\] with the Bell state mapped as \(\ket{\phi^+}_{\mathcal{K}} = \gamma^{-1}(\ket{\phi^+}) = \mathbb{1}_2\otimes\ket{\phi^+}\).

Since \(\sigma_z\), \(\sigma_x\), \(\sigma_y\) all have real coefficients in the Hilbert-space sense (or purely imaginary for \(\sigma_y\)), we can apply Lemma 2 directly. Each term transforms as \(\gamma^{-1}(\sigma_a\otimes\sigma_a)=\sigma_a^{\mathcal{K}}\otimes^{\mathcal{K}}\sigma_a^{\mathcal{K}}\), so \(\bra{\phi^+}_{\mathcal{K}}\hat{\mathscr{T}}_{00}^{\mathcal{K}}\ket{\phi^+}_{\mathcal{K}} = \gamma^{-1}(\bra{\phi^+}\hat{\mathscr{T}}_{00}\ket{\phi^+}) = \gamma^{-1}(6\sqrt{2}) = 6\sqrt{2}\,\mathbb{1}_2\).

The metric expectation value (Eq. 21 ) is therefore \[\require{physics} \label{eq:sm:chsh3result} g\!\bigl(\ket{\phi^+}_{\mathcal{K}},\hat{\mathscr{T}}_{00}^{\mathcal{K}}\ket{\phi^+}_{\mathcal{K}}\bigr) = \tfrac{1}{2}\Tr[6\sqrt{2}\,\mathbb{1}_2] = 6\sqrt{2},\tag{31}\] the maximum quantum value, achieved with purely real arithmetic. 0◻

Additionally, the \(\mathrm{CHSH}_3\) violation can be explicitly visualized for both the standard complex and real Kähler framework using the provided Mathematica supplement.

16 Local maps in Kähler space↩︎

In complex quantum mechanics, a local unitary acting on subsystem \(A\) is \(\rho'_{AB}=(U_A\otimes\mathbb{1}_B)\rho_{AB}(U_A^\dagger\otimes\mathbb{1}_B)\). The same operation in Kähler space is \[\label{eq:sm:localmap} {\rho'}^{\mathcal{K}}_{AB} = (U_A^{\mathcal{K}}\otimes^{\mathcal{K}}\mathbb{1}_B^{\mathcal{K}})\,\rho^{\mathcal{K}}_{AB}\, ({U_A^{\mathcal{K}}}^\dagger\otimes^{\mathcal{K}}\mathbb{1}_B^{\mathcal{K}}),\tag{32}\] where \(U_A^{\mathcal{K}}=\gamma^{-1}(U_A)\), \(\mathbb{1}_B^{\mathcal{K}}=\mathbb{1}_2\otimes\mathbb{1}_B\), and \(\rho^{\mathcal{K}}_{AB}=\gamma^{-1}(\rho_{AB})\). By Lemma 2, \(\gamma(\rho'^{\mathcal{K}}_{AB})=\rho'_{AB}\), confirming that local unitary evolutions are represented locally in \(\mathcal{K}\) with respect to \(\otimes^{\mathcal{K}}\).

More generally, for a CPTP map \(\Phi_A\) with Kraus operators \(\{M_i\}\) satisfying \(\sum_i M_i^\dagger M_i=\mathbb{1}_A\), the Kähler space action is \[{\rho'}^{\mathcal{K}}_{AB} = \sum_i (M_i^{\mathcal{K}}\otimes^{\mathcal{K}}\mathbb{1}_B^{\mathcal{K}})\,\rho^{\mathcal{K}}_{AB}\, ({M_i^{\mathcal{K}}}^\dagger\otimes^{\mathcal{K}}\mathbb{1}_B^{\mathcal{K}}),\] and the completeness relation becomes \(\sum_i {M_i^{\mathcal{K}}}^\dagger M_i^{\mathcal{K}}=\mathbb{1}_2\otimes\mathbb{1}_A=\mathbb{1}_A^{\mathcal{K}}\) in \(\mathcal{K}\). This confirms that local maps in \(\mathcal{H}\) correspond exactly to local maps in \(\mathcal{K}\): there is no nonlocal overhead, contrary to the claim of Ref. [10]. The \(\mathbb{1}_2/\tau\) block is part of the fixed geometric structure (the complex structure \(J\)) and not a dynamical ancillary system.

17 Connection to the balanced tensor product↩︎

From the perspective of algebraic quantum theory, the symplectic product \(\otimes^{\mathcal{K}}\) is the balanced tensor product over the complex structure [5]: \[\mathcal{H}_A\otimes_{(J_A,J_B)}\mathcal{H}_B \mathrel{\vcenter{:}}= (\mathcal{H}_A\otimes_{\mathbb{R}}\mathcal{H}_B)\big/ \langle J_A x\otimes y - x\otimes J_B y\rangle.\] Quotienting by this relation enforces \((J_A x)\otimes y = x\otimes(J_B y)\), i.e.\(\mathsf{i}\)-linearity across subsystems. In block-matrix language this quotient is precisely Definition 1.

References↩︎