Quantum mechanics over real numbers fully reproduces standard quantum theory

Abstract: Standard quantum mechanics employs complex Hilbert spaces, but whether complex numbers are fundamental or merely convenient has long been debated. For decades, real-valued equivalents were considered mathematically possible but cumbersome. However, a landmark 2021 result claimed that any quantum theory based on real numbers is experimentally falsifiable via network Bell experiments. Yet, it remains an open question whether this falsification applies to all real-valued theories. Here we show that this conclusion rests on an incomplete real formulation, and we present a rigorous real-valued framework that perfectly reproduces all predictions of standard quantum mechanics, i.e. standard quantum mechanics. We demonstrate that the standard real tensor product ($\otimes_{\mathbb{R}}$) used in previous no-go theorems is algebraically incompatible with the rich structure of standard quantum mechanics. We present a real framework based on \ka space and prove that it is exactly isomorphic to standard quantum mechanics via an explicit bijection $γ$. The isomorphism extends to composite systems through a symplectic composition rule $\otimes^{\ks}$ that replaces the Kronecker product. Consequently, our formulation achieves the maximal $\mathrm{CHSH}_{3}$ violation of $6\sqrt{2}$ using purely real variables, directly contradicting previous falsification claims. These results demonstrate that complex numbers are not fundamentally required by nature; rather, they encode a deeper real geometric structure that governs quantum interference and entanglement, settling this long debate.

• The paper's main contribution is demonstrating that a real-valued Kähler space framework reproduces standard quantum mechanics, including maximal Bell-inequality violations.
• It introduces a novel symplectic tensor product (⊗ᴷ) that enforces J-linearity and establishes an explicit isomorphism with complex Hilbert spaces.
• The work refutes prior no-go theorems, providing new insights into the geometric role of complex numbers and alternatives for quantum computation.

Quantum Mechanics Over Real Numbers: Complete Reproduction of Standard Quantum Theory

Motivation and Context

The canonical formulation of quantum mechanics employs complex Hilbert spaces, with the imaginary unit $i$ integral to key phenomena such as interference and unitary evolution. Historically, attempts to reconstruct quantum theory over real Hilbert spaces have been perceived as mathematically feasible but physically incomplete, especially concerning multipartite systems. The prevailing consensus, cemented by results such as Renou et al. [Nature 600, 2021], was that real-number quantum theory is experimentally distinguishable from standard quantum mechanics due to limitations in Bell-inequality violations and entanglement-swapping scenarios.

This paper demonstrates that such conclusions stem from using an incomplete real formulation, particularly the standard Kronecker product $\otimes_{\mathbb{R}}$ for the tensor product of doubled real spaces. The authors introduce a rigorous real-valued framework based on Kähler spaces, equipped with a symplectic composition rule $\otimes^K$, establishing an explicit isomorphism with complex quantum mechanics. This construction reproduces all predictions of standard quantum theory, including maximal Bell-inequality violations, thus disproving previous claims that the real formulation is experimentally falsifiable.

Kähler Space Formalism and Isomorphism

Realification via Complex Structure

A Kähler space $K = (\mathbb{R}^{2N}, g, \omega, J)$ is specified by a real vector space endowed with a metric $g$, symplectic form $\omega$, and complex structure $J$ ($J^2 = -1$), satisfying compatibility relations $g(x, y) = \omega(x, Jy)$ and $\omega(Jx, Jy) = \omega(x, y)$. This structure allows identification with a complex Hilbert space $\otimes_{\mathbb{R}}$0 via the bijection $\otimes_{\mathbb{R}}$1:

• $\otimes_{\mathbb{R}}$2,
• State vectors and operators are mapped as block matrices in $\otimes_{\mathbb{R}}$3 corresponding to their complex representations in $\otimes_{\mathbb{R}}$4.

Linear operators $\otimes_{\mathbb{R}}$5 in $\otimes_{\mathbb{R}}$6 are represented in $\otimes_{\mathbb{R}}$7 as $\otimes_{\mathbb{R}}$8, where $\otimes_{\mathbb{R}}$9 is the matrix representing $\otimes^K$0 (counterclockwise rotation), ensuring that the imaginary unit has a geometric counterpart in the real framework.

Symplectic Composition Rule

For composite systems, the symplectic tensor product $\otimes^K$1 is defined to preserve the complex multiplication law within the real Kähler space:

$\otimes^K$2

This composition replaces the standard Kronecker product, ensuring the algebraic compatibility required for proper entanglement structure and violation of multipartite Bell inequalities.

Isomorphism Theorem

The authors prove (via explicit calculations) that $\otimes^K$3 and $\otimes^K$4 are isomorphic as monoidal quantum theories. The bijection $\otimes^K$5 between the Hilbert space and Kähler space representations commutes with the tensor product:

$\otimes^K$6

This result ensures that the real-number Kähler space quantum mechanics reproduces all predictions of the standard formulation—including those involving composite and entangled systems—thereby preserving the operational and algebraic structure.

Bell Inequality Violations

Standard CHSH

The framework yields maximal CHSH violations identical to standard quantum mechanics, with the Kähler-space Pauli matrices satisfying the required algebraic relations for entangled states.

CHSH$\otimes^K$7 and Maximal Tripartite Violation

A critical benchmark is the maximal $\otimes^K$8 violation ($\otimes^K$9), previously claimed to be unattainable in real-number quantum theory. The paper demonstrates that the Kähler-space framework, with its symplectic tensor product and complex structure encoding, achieves this maximal quantum value using exclusively real arithmetic. This result directly refutes the assertion of experimental distinguishability made by Renou et al., and by extension, the standard no-go theorems relying on the Kronecker product.

Analysis of Previous No-Go Theorems

The authors meticulously show that the failure of previous real formulations in multipartite scenarios is due solely to the use of the Kronecker product, which does not respect the complex structure across subsystems. This results in an enlarged (and spurious) state space devoid of physical content. By enforcing $K = (\mathbb{R}^{2N}, g, \omega, J)$0-compatibility in the tensor product, the authors restore operational equivalence with complex quantum mechanics.

Further, the connection with the balanced tensor product from algebraic quantum theory is established, demonstrating that the Kähler-space composition is precisely the quotient enforcing $K = (\mathbb{R}^{2N}, g, \omega, J)$1-linearity across subspaces.

Implications and Theoretical Significance

The paper settles a longstanding debate: complex numbers are not fundamental ingredients required a priori for quantum theory, but encode a deeper real geometric structure—specifically the complex structure $K = (\mathbb{R}^{2N}, g, \omega, J)$2 of a Kähler manifold—that governs quantum phase and composition. Symmetries such as Wigner's antiunitary transformations and the CPT theorem are revealed to be the geometric avatars of complex conjugation, already present in the Kähler framework.

Practically, this opens the possibility for alternative formulations and implementations of quantum theory (e.g., quantum computation, simulation) over real numbers, potentially simplifying architectures or enabling new classes of encodings. The geometric insight may also inform future developments in quantum foundations and the search for alternative representations of quantum theory, including operational and postulate-based approaches.

Comparison With Concurrent Work

The construction aligns with operational/postulational results such as those by Hoffreumon and Woods (Hoffreumon et al., 3 Apr 2025, Hoffreumon et al., 19 Mar 2026), who argue for the existence of real quantum theories meeting all experimental benchmarks. The present paper provides the explicit constructive realization and algebraic proof supporting these perspectives. The critique of the "product-state independence" assumption and alleged fundamental nonlocality is addressed, showing that no extra physical ancilla is introduced; the block structure arises from the complex geometry of the real space.

Conclusion

The authors establish:

• A bijection between complex Hilbert space and real Kähler space quantum mechanics.
• The symplectic composition rule $K = (\mathbb{R}^{2N}, g, \omega, J)$3 as the essential ingredient for operational equivalence.
• Reproduction of all Bell-inequality violations, including the maximal $K = (\mathbb{R}^{2N}, g, \omega, J)$4 result, using real variables.
• Clarification of the geometric role of complex numbers in quantum mechanics.

Complex numbers, instead of being imposed from outside, are fully encoded and explained by the symplectic geometry of Kähler space—the imaginary unit is not discarded but realized as a derived feature in a mathematically natural framework. This result settles the debate regarding the necessity of complex numbers in quantum theory and rigorously validates a real-number formulation that is fully congruent with the standard complex approach.