A New Perspective on Real-Valued Quantum Theory
Complex numbers appear throughout the mathematical formalism of quantum theory, but physicists have long debated whether they are truly essential for describing nature [1]. Could quantum phenomena instead be described using only real numbers? Over the years, two real-valued formulations of quantum theory have emerged: One preserves the standard mathematical rule for combining independent quantum systems, while the second modifies this rule [2, 3]. The first has been ruled out theoretically and experimentally, because it fails to reproduce certain “multipartite” quantum experiments [4]. The second can reproduce all predictions of standard quantum mechanics; by changing the composition rule for combined quantum systems, it ultimately requires us to rethink the basic notions of quantum information. Now Pedro Barrios Hita at the German Aerospace Center and his collaborators revisit this long-standing debate and show that it can be reframed by introducing an additional assumption—one that’s based on a physically motivated principle—about how independent quantum systems combine [5]. With this extra puzzle piece, the second approach emerges as a compelling alternative (Fig. 1).
In standard quantum theory, every physical system is associated with a complex Hilbert space. The dimension of this space characterizes how much information can reliably be stored and read out from it. For instance, a two-dimensional Hilbert space can encode one bit of information: Either of two perfectly distinguishable states can be prepared, and the knowledge of which one was prepared can later be recovered. Quantum theory also specifies how independent systems combine through a mathematical operation known as the tensor product, under which the dimensions of the combined systems multiply together. For instance, if photons from two distant lasers are represented in Hilbert spaces that have different dimensions, they will jointly occupy a space with dimension given by the product of their respective dimensions. The tensor product therefore determines both how independent systems are represented and the dimensions of the combined Hilbert spaces.
A natural route toward a real-valued formulation of quantum mechanics is to preserve this compositional structure while replacing complex Hilbert spaces by real ones. Since each complex amplitude contains two real components, it is natural to double dimensions. For example, a photon in a two-dimensional Hilbert space could be represented in a four-dimensional space. This can be understood by associating each quantum system with an additional two-level system that carries the missing information from the imaginary component. This auxiliary system is known as the flag qubit. The difficulty appears when independent systems are combined: The tensor product gives a composite space with twice as many real dimensions as are needed to represent the combined system with the corresponding complex representation.
The two above-mentioned approaches address this issue in two different ways. The first approach preserves the standard tensor-product rule while allowing arbitrary real-valued representations of experiments. These new representations could be completely disconnected from standard quantum theory: For example, a system associated with a given dimension could be represented by real-valued states of various dimensions depending on the experimental context.
This formalism might appear plausible, especially as it can reproduce Bell-type experiments [6]. However, a study published in 2021 challenged its validity [4]. It showed that more-general experiments involving independent systems cannot be reproduced by any real-valued formulation that preserves the standard tensor-product structure.
The second approach instead modifies the composition rule itself by sharing the flag qubit globally across subsystems: In the first approach, there are as many flag qubits as there are quantum systems; in the second approach there is only ever one flag. Formulations using the latter strategy reproduce all the predictions of standard quantum mechanics, but at the price of abandoning the usual tensor-product structure for independently prepared systems [2, 3]. This is a problem because this structure is the basis on which all our understanding of quantum information is formed.
Hence, the question of whether quantum theory can be reformulated using only real numbers appeared to have a different answer depending on the mathematical composition of independent systems: Combining them with the tensor-product rule meant that real-valued formulations could be discarded experimentally, but using another rule—if well chosen—made it seem possible to eliminate complex numbers from quantum theory. Perhaps the most unsatisfying aspect of this impasse was that the debate centered on abstract compositional axioms rather than on physically motivated principles [7].
The work by Barrios Hita and collaborators takes a step in a new direction. Rather than postulating a composition rule directly, the researchers begin from a physically motivated principle: A local operation acting on one subsystem should not affect another independently prepared subsystem. Starting from a specific real-valued representation of single-system states and operations, they impose the constraint that independent complex systems remain independent in the real-valued formulation. Together, these requirements lead to a modified composition rule.
The resulting framework restores a redundancy present in standard quantum theory. Since multiplying a state by a complex phase has no physical consequence, a composite system admits several mathematically distinct descriptions corresponding to the same physical situation. The framework proposed by Barrios Hita and colleagues recovers an analogous redundancy in the real-valued setting.
This construction reproduces all the predictions of standard quantum mechanics. In this sense, complex numbers are not fundamentally necessary to reproduce observable quantum phenomena once the tensor-product rule is relaxed—and Barrios Hita and collaborators offer a physically motivated justification for the alternate rule that they use instead. Beyond the specific construction itself, the work shifts the focus toward the physical principles that should determine how independent quantum systems can be combined.
While Barrios Hita and co-workers’ framework is highly appealing, an important conceptual question remains open: Can a real-valued formulation equivalent to standard quantum mechanics be derived entirely from physically motivated principles? Besides the proposed compositional principle, the construction still relies on additional assumptions, such as ones on the nature of the representation of single quantum systems and on the preservation of independent preparation within the real-valued framework. An important next step would therefore be to determine whether a more complete set of operational principles could uniquely determine the resulting theory.
A second open question concerns the proposed compositional principle itself. The principle seems natural for distinguishable particles, since one can identify subsystems within individual particles. This identification makes the definition of local operations on independent subsystems intuitive. For indistinguishable particles, however, locality must instead be formulated in terms of modes that arise through the so-called second quantization formalism. Recent work on indistinguishable particles has highlighted that subsystem composition becomes considerably more nuanced in such settings [8]. It therefore remains unclear to what extent the compositional principle proposed by Barrios Hita and colleagues can be consistently formulated for these systems.
References
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- M.-O. Renou et al., “Quantum theory based on real numbers can be experimentally falsified,” Nature 600, 625 (2021).
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