Real Quantum Theory: Foundations & Implications

• Real Quantum Theory is the formulation of quantum mechanics using real Hilbert spaces, where states, operators, and composite systems are defined with real numbers.
• It alters key structural properties such as local tomography and entanglement, leading to bilocal reconstruction and different multi-system correlations.
• The theory challenges the necessity of complex numbers by affecting reversible dynamics and experimental network scenarios, prompting new tests in quantum foundations.

Searching arXiv for recent and foundational papers on real quantum theory, including the supplied target paper and related work. Real Quantum Theory denotes the formulation of quantum mechanics over real Hilbert spaces rather than complex Hilbert spaces. In this technical sense, pure states are vectors with real amplitudes, mixed states are real symmetric density matrices, observables are represented by real operators, and composite systems are usually taken to compose by the tensor product over real vector spaces. The subject occupies a distinctive place in quantum foundations because it is close enough to standard complex quantum theory to reproduce many familiar phenomena, yet different enough to alter core structural properties such as local tomography, entanglement sharing, and, depending on the assumptions imposed on source independence and composition, the operational status of complex numbers themselves (Finkelstein, 2021).

1. Definition and formal scope

In the standard contrast used in the foundations literature, Complex Quantum Theory (CQT) employs complex Hilbert spaces, whereas Real Quantum Theory (RQT) restricts states, operators, and amplitudes to real numbers while keeping the usual quantum postulates as far as possible (Renou et al., 2021). In the formulation tested in recent no-go results, the standard axioms are retained except that Hilbert spaces are real: a pure system is a unit vector in Hilbert space, closed-system dynamics is unitary, observables are Hermitian operators with Born-rule probabilities, and composite systems are formed by the tensor product (Chen et al., 2021).

A central subtlety is that the phrase “real quantum theory” is used in more than one sense across the literature. In the technical sense relevant here, it refers to real-Hilbert-space quantum mechanics. This differs from works on “real world interpretations,” “the real meaning of quantum mechanics,” or “real path quantum theory,” where “real” is ontological or interpretational rather than algebraic (Kent, 2007). That distinction is essential because debates about whether quantum states are real, whether one world is realized, or whether path ontologies can be defined do not address the question of whether quantum theory can be formulated over $\mathbb{R}$ instead of $\mathbb{C}$.

The standard realification construction explains why the issue remained open for so long. A complex density matrix $\rho$ can be embedded into a larger real formalism by adjoining an ancilla qubit prepared in the $\ket{\pm i}$ basis,

$$\tilde{\rho} = \frac{1}{2}\left(\rho\otimes \ket{+i}\!\bra{+i}+\rho^*\otimes \ket{-i}\!\bra{-i}\right), \qquad \ket{\pm i}=\frac{1}{\sqrt{2}}(\ket{0}\pm i\ket{1}),$$

with corresponding embeddings for observables (Renou et al., 2021). This shows that many complex-quantum predictions can be represented inside a larger real state space. The conceptual issue is therefore not whether real simulation is possible in an enlarged space, but whether the real-valued standard formalism itself, with its own tensor-product structure, is empirically adequate (Chen et al., 2021).

2. Structural properties: tomography, composition, and latent degrees of freedom

One of the main structural differences between CQT and RQT concerns tomography. Standard complex quantum theory is locally tomographic: the state of a composite system can be reconstructed from the statistics of measurements on the individual components. In the operational framework of Hardy and Wootters, this corresponds to the parameter-counting identity

$K_{AB}=K_AK_B$

under local tomography (Hardy et al., 2010). For complex Hilbert space of dimension $N$, an unnormalized density matrix has

$K=N^2,$

which fits the locally tomographic form (Hardy et al., 2010).

Real-vector-space quantum theory fails local tomography. For an $N$-dimensional real Hilbert space, an unnormalized density matrix is a real symmetric $N\times N$ matrix with

$\mathbb{C}$0

so in general $\mathbb{C}$1 (Hardy et al., 2010). The theory is therefore more holistic than CQT in the specific operational sense that some global state parameters are not recoverable from purely local statistics. Hardy and Wootters show, however, that RQT is bilocally tomographic: the state of a composite system is determined by the statistics of one-component and two-component measurements (Hardy et al., 2010). In their general bilocal framework, RQT fits

$\mathbb{C}$2

with $\mathbb{C}$3 counting latent parameters inaccessible to local measurements alone (Hardy et al., 2010).

This failure of local tomography is closely tied to later debates about source independence. A 2026 analysis identifies non-local tomography as the “best-known structural difference” between RQT and QT and argues that, in RQT, product-state independence is stronger than operational independence (Hoffreumon et al., 19 Mar 2026). That point matters because earlier network no-go theorems treated independence as a product-state constraint, whereas in RQT there can exist mathematically non-product states that remain operationally independent under all real local POVMs (Hoffreumon et al., 19 Mar 2026). The paper exhibits the explicit state

$\mathbb{C}$4

with $\mathbb{C}$5, as separable in complex QT, entangled in RQT, and still operationally independent for local real measurements because real symmetric POVM elements are orthogonal to the antisymmetric $\mathbb{C}$6 (Hoffreumon et al., 19 Mar 2026).

The composition rule is therefore not a peripheral technicality. It is the fulcrum on which many results about the empirical status of complex numbers turn. Finkelstein’s note emphasizes exactly this point: the cited 2021 experiments do not rule out every real formulation, but only those real quantum theories that continue to obey the standard tensor-product rule in the usual way (Finkelstein, 2021).

3. Entanglement in real Hilbert spaces

RQT differs sharply from CQT in the structure of entanglement. Wootters shows that if one replaces the complex vector space of states by a real vector space while keeping a standard resource-based definition of entanglement, then the familiar limitation on entanglement sharing can disappear entirely (Wootters, 2010). In ordinary complex quantum theory, two qubits that are completely entangled with each other cannot be entangled with a third system; this is the usual monogamy intuition. In the real theory, by contrast, there exist states of arbitrarily many two-level real systems—“rebits”—in which every rebit is maximally entangled with every other rebit (Wootters, 2010).

The contrast appears already for the two-rebit mixed state

$\mathbb{C}$7

where

$\mathbb{C}$8

In CQT, this state has zero concurrence because it admits a decomposition into product states using complex product vectors. In RQT, that decomposition is unavailable, and the real concurrence is

$\mathbb{C}$9

so that

$\rho$0

(Wootters, 2010). The same density matrix is therefore separable in the complex theory and maximally entangled in the real theory.

Wootters then constructs explicit multi-rebit examples. For three rebits,

$\rho$1

$\rho$2

and tracing out any one rebit leaves the remaining pair in the mixed state $\rho$3, which has $\rho$4 (Wootters, 2010). The construction generalizes to arbitrarily many rebits, with pairwise concurrence

$\rho$5

for every pair $\rho$6 (Wootters, 2010).

A further peculiarity is that these maximally pairwise entangled real states need not exhibit pairwise classical correlations in local measurements. For the mixed states $\rho$7, the local-measurement statistics factorize, even though the states are entangled in the resource sense because they cannot be prepared without quantum communication in the real theory (Wootters, 2010). This suggests that in RQT the relation between entanglement and observable local correlation structure is looser than in CQT.

4. Computation and reversible dynamics

RQT also supports a nontrivial theory of reversible quantum computation. In the real setting, elementary systems are rebits, states live in $\rho$8, and reversible transformations form the orthogonal group

$\rho$9

whose connected component is

$\ket{\pm i}$0

(Belenchia et al., 2012). This is immediately different from CQT, where $\ket{\pm i}$1-qubit reversible dynamics is $\ket{\pm i}$2 (Belenchia et al., 2012).

The gate-theoretic result of D’Ariano, Manessi, Perinotti, and Tosini is that C-NOT together with local gates is universal in RQT as well, but not in the same sense as in CQT (Belenchia et al., 2012). The paper distinguishes:

Concept	Meaning in the paper
Strong universality	Every reversible transformation on $\ket{\pm i}$3 elementary systems is simulated using exactly $\ket{\pm i}$4 systems
Weak universality	Every reversible transformation on $\ket{\pm i}$5 elementary systems is simulated using $\ket{\pm i}$6 systems, with the extra systems discarded

In CQT, C-NOT is strongly universal. In RQT, C-NOT plus local gates generates all of $\ket{\pm i}$7, but arbitrary orthogonal transformations with determinant $\ket{\pm i}$8 generally require an extra ancillary rebit (Belenchia et al., 2012). The obstruction comes from the determinant structure of the orthogonal group and the Kronecker-product identity

$\ket{\pm i}$9

If $$\tilde{\rho} = \frac{1}{2}\left(\rho\otimes \ket{+i}\!\bra{+i}+\rho^*\otimes \ket{-i}\!\bra{-i}\right), \qquad \ket{\pm i}=\frac{1}{\sqrt{2}}(\ket{0}\pm i\ket{1}),$$0 is an $$\tilde{\rho} = \frac{1}{2}\left(\rho\otimes \ket{+i}\!\bra{+i}+\rho^*\otimes \ket{-i}\!\bra{-i}\right), \qquad \ket{\pm i}=\frac{1}{\sqrt{2}}(\ket{0}\pm i\ket{1}),$$1-rebit gate with $$\tilde{\rho} = \frac{1}{2}\left(\rho\otimes \ket{+i}\!\bra{+i}+\rho^*\otimes \ket{-i}\!\bra{-i}\right), \qquad \ket{\pm i}=\frac{1}{\sqrt{2}}(\ket{0}\pm i\ket{1}),$$2, one can instead implement

$$\tilde{\rho} = \frac{1}{2}\left(\rho\otimes \ket{+i}\!\bra{+i}+\rho^*\otimes \ket{-i}\!\bra{-i}\right), \qquad \ket{\pm i}=\frac{1}{\sqrt{2}}(\ket{0}\pm i\ket{1}),$$3

which has determinant $$\tilde{\rho} = \frac{1}{2}\left(\rho\otimes \ket{+i}\!\bra{+i}+\rho^*\otimes \ket{-i}\!\bra{-i}\right), \qquad \ket{\pm i}=\frac{1}{\sqrt{2}}(\ket{0}\pm i\ket{1}),$$4 and hence lies in the connected component accessible from local gates and C-NOT (Belenchia et al., 2012). The resulting conclusion is that C-NOT is only weakly universal for reversible computation in RQT.

This matters for reconstruction programs. The paper was written in response to arguments that the universality of an entangling gate, combined with local structure, helps characterize quantum theory. Since RQT also admits a universal bipartite gate scheme, universality alone does not distinguish complex from real quantum theory; local discriminability and the qubit structure remain crucial (Belenchia et al., 2012).

5. Experimental and network-theoretic tests

The modern empirical debate about RQT turns on network scenarios with independent sources. Renou et al. showed that real and complex quantum theory make different predictions in entanglement-swapping networks with independent states and measurements (Renou et al., 2021). Their Bell-like functional $$\tilde{\rho} = \frac{1}{2}\left(\rho\otimes \ket{+i}\!\bra{+i}+\rho^*\otimes \ket{-i}\!\bra{-i}\right), \qquad \ket{\pm i}=\frac{1}{\sqrt{2}}(\ket{0}\pm i\ket{1}),$$5 satisfies

$$\tilde{\rho} = \frac{1}{2}\left(\rho\otimes \ket{+i}\!\bra{+i}+\rho^*\otimes \ket{-i}\!\bra{-i}\right), \qquad \ket{\pm i}=\frac{1}{\sqrt{2}}(\ket{0}\pm i\ket{1}),$$6

for any distribution admitting a real quantum representation in the SWAP scenario, while the ideal complex realization reaches

$$\tilde{\rho} = \frac{1}{2}\left(\rho\otimes \ket{+i}\!\bra{+i}+\rho^*\otimes \ket{-i}\!\bra{-i}\right), \qquad \ket{\pm i}=\frac{1}{\sqrt{2}}(\ket{0}\pm i\ket{1}),$$7

(Renou et al., 2021). The argument is not dimension-dependent and relies on the network’s source-independence structure rather than on ordinary Bell nonlocality.

Chen et al. implemented a closely related entanglement-swapping game on a four-qubit superconducting processor and reported a score

$$\tilde{\rho} = \frac{1}{2}\left(\rho\otimes \ket{+i}\!\bra{+i}+\rho^*\otimes \ket{-i}\!\bra{-i}\right), \qquad \ket{\pm i}=\frac{1}{\sqrt{2}}(\ket{0}\pm i\ket{1}),$$8

exceeding the real-number bound $$\tilde{\rho} = \frac{1}{2}\left(\rho\otimes \ket{+i}\!\bra{+i}+\rho^*\otimes \ket{-i}\!\bra{-i}\right), \qquad \ket{\pm i}=\frac{1}{\sqrt{2}}(\ket{0}\pm i\ket{1}),$$9 by 43 standard deviations (Chen et al., 2021). The experiment used a linear chain $K_{AB}=K_AK_B$0, two EPR pairs, a Bell-state measurement on qubits $K_{AB}=K_AK_B$1 and $K_{AB}=K_AK_B$2, and iSWAP-type interactions with process fidelity 96.7\%; the swapped Bell states between qubits $K_{AB}=K_AK_B$3 and $K_{AB}=K_AK_B$4 had average fidelity $K_{AB}=K_AK_B$5, above the threshold fidelity about $K_{AB}=K_AK_B$6 required to violate the real bound (Chen et al., 2021).

A photonic implementation later adapted the same idea to an optical three-party network with two independent EPR sources based on type-0 SPDC in PPLN crystals. It measured the Bell-type quantity

$K_{AB}=K_AK_B$7

exceeding the semidefinite-programming RQT bound

$K_{AB}=K_AK_B$8

by $K_{AB}=K_AK_B$9 standard deviations (Li et al., 2021). The experiment reported 26,954 four-photon coincidence events over 21,742 switching cycles and found

$N$0

(Li et al., 2021).

These works established that real and complex quantum theory can be separated in network experiments once independent sources are present. A later theoretical extension showed that the gap can become arbitrarily large in multipartite star networks. In a star network with $N$1 external parties and one central party, a conditional Bell inequality yields

$N$2

so the ratio is $N$3, growing linearly with network size (Sarkar et al., 12 Mar 2025). This result shows that, under the same notion of RQT used in the 2021 network arguments, the relative expressive deficit of real Hilbert-space quantum mechanics in complex networks is not merely nonzero but asymptotically unbounded (Sarkar et al., 12 Mar 2025).

6. Controversies, reformulations, and current status

The strongest controversy concerns what exactly the 2021 experiments excluded. Finkelstein’s “In defense of real quantum theory” argues that the experimental and theoretical no-go results do not show that complex numbers are necessary for quantum theory in general (Finkelstein, 2021). Instead, they exclude only those real quantum theories that retain the standard complex-quantum formalism, especially the usual tensor-product composition rule. His counterexample is the standard embedding of a complex state $N$4 into a real operator on an enlarged Hilbert space: $N$5 where

$N$6

Because $N$7 is a real operator, it defines a real quantum theory operationally equivalent to standard complex quantum theory, but it does not preserve the standard tensor-product rule in the same way (Finkelstein, 2021). On this view, the claim “quantum physics needs complex numbers” is too strong unless one explicitly excludes real formulations with altered compositional structure.

A more radical 2026 intervention goes further and argues that RQT cannot be experimentally falsified in any finite network or finite sequential multipartite protocol whose statistics are compatible with QT, provided source independence is imposed operationally rather than as a product-state constraint (Hoffreumon et al., 19 Mar 2026). The paper states:

“For any finite network of independent sources and locally measuring parties, if the sources are required only to be operationally independent, then every outcome distribution predicted by the QT model of the network can also be predicted by an equivalent RQT model.”

It extends this to arbitrary finite protocols of channels and POVMs with prescribed locality structure and concludes:

“Thus, as long as no violation of QT is observed, RQT cannot be experimentally falsified.” (Hoffreumon et al., 19 Mar 2026)

This produces a sharp conceptual distinction. One class of results treats RQT as the real-valued analogue of standard quantum theory with ordinary tensor-product composition and product-state independence; under that definition, network experiments separate it from CQT (Renou et al., 2021). Another class argues that this definition is too restrictive because, in a non-locally-tomographic theory, operational independence need not coincide with product-state independence; under that broader reading, RQT can remain empirically indistinguishable from QT in the operational scenarios studied (Hoffreumon et al., 19 Mar 2026). A plausible implication is that current disagreements are as much about which formal constraints define RQT as about the empirical data themselves.

A final terminological caution is necessary. Papers on “The Real Meaning of Quantum Mechanics,” “Real World Interpretations of Quantum Theory,” “Path Integrals and Reality,” or qr-number approaches use the word “real” in ontological, interpretational, or topos-theoretic senses rather than to denote quantum theory over real Hilbert spaces (Pris, 2021). They belong to adjacent foundational debates, but not to RQT in the technical algebraic sense.

7. Significance in quantum foundations

Real Quantum Theory serves as a controlled alternative to standard quantum mechanics that isolates which features of QT depend specifically on complex structure. The literature shows that replacing $N$8 by $N$9 while otherwise preserving the standard formalism changes at least four major features: the tomographic structure of composites, the geometry of entanglement, the group-theoretic structure of reversible computation, and the behavior of network correlations under source-independence constraints (Hardy et al., 2010).

At the same time, the theory’s relation to standard QT depends strongly on how one formulates composition and independence. The 2021 network results and their experimental implementations support the claim that complex amplitudes have operational content in standard-formalism network scenarios (Chen et al., 2021). Finkelstein’s response and the 2026 equivalence theorem show that this conclusion is not invariant under broader notions of real reformulation (Finkelstein, 2021). The foundational lesson is therefore twofold. First, RQT is not a trivial notational variant of CQT; it has distinctive structural consequences. Second, whether those consequences are experimentally decisive depends on which assumptions—tensor-product structure, product-state independence, tomography, ancilla embeddings, or operational equivalence—are taken to define the admissible real theory.

In that respect, RQT functions less as a single fixed rival theory than as a family of closely related real-number formulations whose differences become visible precisely where the composition of systems, the independence of sources, and the operational meaning of locality are most constrained.