Where did the imaginary unit go?

Every experimental prediction in quantum mechanics is a real probability. Yet the standard theory places states in complex Hilbert spaces, represents observables by complex Hermitian operators, and evolves a closed state through the Schrödinger equation iℏ∂_t|ψ⟩ = H|ψ⟩. Complex numbers do more here than shorten a calculation. Their multiplication rule organizes continuous phase rotation and interference.

Still, complex coordinates are not the only way to carry that organization. A complex vector space of dimension d can always be viewed as a real vector space of dimension 2d. The real and imaginary parts become separate coordinates. Multiplication by i becomes an ordinary real linear operator J that rotates each coordinate pair by a quarter turn.

This already gives the article’s central answer. The imaginary unit can leave the scalar arithmetic. Its defining work does not vanish. It becomes an operator on a doubled real space.

Let |ψ⟩ have complex components c_j. The standard realification sends it to a vector containing the real parts and imaginary parts separately. The 2026 Barrios Hita and colleagues write this as a physical system accompanied by a two-dimensional “flag”:

The flag is not an extra observable qubit. Allowed real observables commute with rotations of the flag, so a measurement cannot simply ask whether a coefficient came from a real or imaginary component. Expectation values are preserved exactly: 𝒮(|ψ⟩)^T𝒯(A)𝒮(|ψ⟩) = ⟨ψ|A|ψ⟩.

Global phase also survives in translated form. Multiplying |ψ⟩ by e^iαdoes not change a physical state in ordinary quantum mechanics. In the real description, it rotates the flag by the corresponding SO(2) angle. Two real vectors can therefore be orthogonal as vectors and still represent the same physical state because the allowed measurements identify them.

None of this is new in spirit. Real-coordinate simulations of individual quantum systems have been known for decades. The hard question is whether the translation remains local and economical when independently prepared systems are brought together.

Suppose two complex systems have dimensions m and n. Their composite has complex dimension mn, or real dimension 2mn. If each system is first doubled and the resulting real spaces are combined with the ordinary real tensor product, the dimension becomes 4mn. The naive construction has introduced twice as many real coordinates as the complex composite requires.

The mismatch has a physical interpretation. A product state has only one unobservable global phase, but independently realifying the subsystems creates one local flag for each. The same complex product can be written after moving phase from one factor to another: |0⟩_A⊗|1⟩_B =i|0⟩_A⊗(−i)|1⟩_B. Applying the one-system map to each factor gives different real vectors unless those different phase assignments are explicitly identified.

This is why the debate cannot be settled by saying “split every amplitude into real and imaginary parts.” The one-system recipe is straightforward. A theory must also specify which real composite vectors represent the same physical state, how local operations embed into the composite, and what “independent preparation” means in a theory that may not be locally tomographic.

In 2021, Renou and collaborators compared standard complex quantum theory with a specific real foil theory. In that foil, Hilbert spaces are real, composite systems use the ordinary real tensor product, and independently prepared sources are represented by product states. The authors constructed a network Bell scenario in which the complex theory reaches correlations that this real theory cannot.

The network matters. A single system can always be realified. Ordinary bipartite Bell correlations can also be reproduced in a real description. The separation appears when two independent sources distribute systems into an entanglement-swapping network. The combination of local measurements and source independence exposes composition structure that a single-source test does not.

Optical experiments subsequently violated the real-theory bounds, including a strict-locality test reported in 2022 by Wu and collaborators. Those experiments are genuine tests of the restricted real theory used to derive the bounds. They do not prove that every mathematical formulation using real scalars must fail. The 2021 paper itself was careful about this boundary: a real reformulation can recover the standard predictions only by changing at least one of the postulates under examination.

The June 2026 Physical Review Letters paper does not find an algebraic mistake in the 2021 network inequality. It chooses a different principle for composition. Instead of postulating that the composite Hilbert space must be the ordinary tensor product of the local real spaces, it postulates a physical locality condition: an operation on subsystem A must act trivially on subsystem B, and local operations on separated subsystems must commute.

The construction starts with local real flags, then takes a quotient of their real tensor product. Vectors that differ only by redistributing an unobservable global phase among the local flags belong to the same equivalence class. The quotient removes the redundant coordinates and leaves one effective composite flag. The authors prove that local actions and expectation values are well defined on these equivalence classes.

The result is an exact real-coordinate representation of finite-dimensional standard quantum mechanics, including multipartite experiments. This is not the same theory that the 2021 inequality separated from complex quantum mechanics. It is a reformulation engineered to be one-to-one with the complex theory.

A second 2026 route reaches a related conclusion from a different direction. Hoffreumon and Woods retain ordinary real quantum theory but distinguish two meanings of independent sources. Product-state independence constrains the matrix form of the source state. Operational independence asks only whether all allowed local tests produce factorized statistics. These notions coincide in standard complex quantum theory, but not in real quantum theory because the latter is not locally tomographic.

Once independence is imposed operationally, they prove that every finite network correlation allowed by complex quantum theory has a real model with the same locality structure for the measurements. They extend the result to finite sequential multipartite protocols involving channels and measurements. The price is interpretive: the real model may contain correlations in its state representation that no allowed local experiment can reveal.

A third route, developed in the 2026 Kähler-space preprint by Maioli, Curado, and Gazeau, makes the displaced structure especially explicit. The state space is real, but it carries a metric g, a symplectic form ω, and a complex-structure operator J. Their balanced symplectic composition replaces the naive real tensor product. In this language, complex numbers are compressed notation for a compatible real geometry.

At the level of scalar arithmetic, yes. The PRL construction uses real vectors and real matrices. At the level of mathematical structure, no. A real two-dimensional flag carries the original distinction between real and imaginary components. Its quarter-turn operator obeys the same defining relation as i. The quotient relation determines how phase is shared across systems. The Kähler formulation retains J and ω explicitly.

This is why “complex numbers are merely convenient” should not be read as “phase structure is dispensable.” A more accurate statement is that complex scalars are one economical representation of a structure that can be unpacked into real geometry and more elaborate composition rules.

Locality remains the sharpest interpretive pressure point. In a preprint revised in April 2026, Feng, Ren, and Vedral argue that real-number reconstructions compatible with the independent-source assumption require a nonlocal map. In their reading, complex quantum theory is equivalent to a real theory with hidden nonlocal degrees of freedom, so the complex formalism is what keeps the composition of independent entangled systems manifestly local.

The PRL authors frame their result differently: local operators act trivially on remote subsystems, and the quotient handles only redundant phase assignments. These positions do not use “locality” at exactly the same descriptive level. One concerns the operational action of local operators; the other concerns whether the map that assembles the real composite can itself be factorized into independent local maps. The literature has therefore clarified the tradeoff more than it has erased it.

The newest caution comes from an April 2026 comment by Moradi Kalarde, Xu, and Renou. They ask whether operational independence can serve as a universal physical definition of independent preparation. Their counterexample uses fermionic information theory, where parity superselection restricts the local states and measurements that are physically allowed.

In that setting, a bipartite state can produce factorized statistics for every allowed local measurement and still fail to be independently preparable as a product of valid local fermionic states. Operational independence and preparation independence therefore come apart for reasons that have nothing to do with replacing complex numbers by real ones.

This does not undo the explicit algebraic equivalence in the quotient-space construction, nor does it restore the claim that no real formulation can reproduce complex quantum mechanics. It challenges a stronger philosophical move: treating one operational criterion as a general principle that uniquely tells every physical theory what independent preparation must mean. Superselection rules make that principle too broad without further qualification.

Quantum mechanics does not require complex numbers as its scalar field if one is willing to carry equivalent structure elsewhere. A finite-dimensional complex Hilbert-space model can be translated into a real one with the same observable predictions.

That statement does not make the complex structure accidental. The imaginary unit packages a quarter-turn operator, phase symmetry, and a compatible rule for composition. In real formulations, those ingredients return as flags, quotient relations, symplectic forms, balanced products, or hidden correlations that the available measurements cannot resolve.

The live scientific question is therefore not simply “real or complex?” It is which structural principles should be considered primitive: scalar field, local tomography, tensor-product composition, operational independence, preparation independence, or manifest locality of the translation itself. Different answers define different theories, even when all their matrix entries are real.