Quantum theory based on real numbers cannot be experimentally falsified

Abstract

Whether the complex numbers of standard quantum theory are experimentally indispensable has remained open for decades. Real quantum theory (RQT), obtained by replacing complex amplitudes with real ones while retaining the usual Kronecker-product composition rule, reproduces all single-party and bipartite Bell correlations of quantum theory (QT), but its lack of local tomography suggested that the two theories might diverge in more general local experiments. This possibility appeared to be confirmed by Renou et al., who argued that a bilocal network experiment can falsify RQT without falsifying QT. Here we show that this conclusion relies on an experimentally untestable assumption. The key distinction is between product-state independence, which constrains the mathematical form of source states, and operational independence, which is defined entirely by the absence of observable cross-source correlations. We prove that, once source independence is imposed operationally, every finite network correlation achievable in QT is also achievable in RQT with the same locality structure of the measurements. We then extend this equivalence to arbitrary finite sequential multipartite protocols involving channels and measurements with prescribed locality structure. Thus, as long as no violation of QT is observed, RQT cannot be experimentally falsified. Our results restore the empirical indistinguishability of QT and RQT, while showing that they support markedly different pictures of the correlation structure underlying the same observed world.

Figures (5)

• Figure 1: General $n$-party network considered in Theorem 1. Independent sources $S_1,\dots,S_L$ distribute physical systems to subsets of the local parties $A_1,\dots,A_n$. Each party performs a local measurement specified by a classical input $\mu_i$ (their setting) and returns a classical output $x_i$ (their outcome). The locality structure of the measurements is trusted and represented explicitly by the separation into local parties, whereas source independence is imposed only operationally, namely through the absence of observable cross-source correlations in the statistics.
• Figure 2: Consider an $n$-partite system undergoing a sequence of rounds. In each round, a quantum channel is applied whose locality structure may be arbitrary but fixed: the channel may act jointly on some chosen subset of subsystems while factorising across the rest. This may be followed by a measurement round, again with a specified locality structure: a local measurement on one subsystem, a joint measurement on a designated block, or more generally any POVM that factorises according to a given partition of the parties. One may then iterate this pattern, allowing a general interleaving of channels and measurements with changing locality structure from round to round. The figure illustrates two such iterations: at round 1, the channel $\mathcal{E}^{(1)}$ is applied, then followed by a measurement in which outcome $x_1$ was obtained, yielding the post-measurement state-update channel (also known as a quantum instrument) $\mathcal{M}^{(1)}_{x_1}$; at round 2, $\mathcal{E}^{(2)}_{|x_1}$ is applied followed by a bipartite local measurement in which outcome $x_2=(x_{2,1},x_{2,2})$ was obtained, yielding the instrument $\mathcal{M}^{(2)}_{x_{2,1}|x_1} \mathbin{\otimes_K}\mathcal{M}^{(2)}_{x_{2,2}|x_1}$. Such protocols encompass not only standard nonlocal games but also sequential network experiments, adaptive tests, and multi-stage information-processing scenarios such as LOCC (Local Operations and Classical Communication).
• Figure 3: Conceptual illustration of the interpretive difference between QT and RQT. The two panels depict the same visible universe and therefore the same observable phenomena. In the RQT description (right), however, the underlying state may contain additional correlations---shown schematically as faint curved links---that remain operationally inaccessible. The figure is purely illustrative: it conveys that RQT can attribute a denser correlation structure to its description of the world while remaining empirically indistinguishable from QT in experiments whose statistics lie within the quantum set.
• Figure 4: Relations between sets of bipartite states in real quantum theory (RQT).\ref{['prop:independent_sources']} shows that in RQT, the sets 2 and 2.1 are disjoint. \ref{['prop:locallyIndistinguishable']} further shows that many states are locally indistinguishable from the product states. An example of such locally equivalent non-product-yet-non-correlated states is provided by set 2.2: the image of QT product states in RNQT (set 2.2) is valid set of RQT states (within set 1) that are operationally independent (within set 2), but distinct from the set of product states (outside of set 2.1), yet locally indistinguishable from it. The red circle, set 3, sketches the relaxation of the product-state assumption considered in Refs. Renou2021Weilenmann2025. Our argument is that the operationally motivated relaxation of set 2.1 should rather be set 2.
• Figure 5: Schematic space--time representation of the alternating channel--measurement process on an $n$-partite system. Vertical lines denote worldlines of subsystems, with time flowing towards the bottom of the page. The overall operations on the $n$ subsystems are represented by dashed boxes. Each operation carries a classical locality label (shown on the left), encoding the partition of subsystems across which the corresponding channel or measurement factorises into non-trivial sub-operations. These sub-operations are represented by grey boxes when they are channels and blue boxes when they are measurements, with the horizontal extent of each box indicating the subset of subsystems on which the corresponding sub-operation acts nontrivially. Measurement rounds additionally produce classical outcomes (shown on the right). Note that when several local measurements occur, as is the case in the last operation (measurement phase of the second round), the measurement outcomes are split into sub-outcomes, $x_t = (x_{t,1},x_{t,2},\dots)$. Each sub-outcome $x_{t,i}$ corresponds to what a local party has observed when measuring their respective set of subsystems $B_i$.

Theorems & Definitions (63)

• Theorem 1
• Theorem 2
• Definition 1
• Definition 2: Independent sources
• Proposition 3
• proof
• Proposition 4: Single-partite QT models are sub-models of single-partite RQT models.
• Proposition 5: QT models are RNQT models.
• Proposition 6: Local QT models are sub-models of local RQT models.
• Proposition 7