Recent work by Renou et al. (2021) has led to some controversy concerning the question of whether quantum theory requires complex numbers for its formulation. We promote the view that the main result of that work is best understood not as a claim about the relative merits of different representations of quantum theory, but rather as a claim about the possibility of experimentally adjudicating between standard quantum theory and an alternative theory -- a foil theory -- known as real-amplitude quantum theory (RQT). In particular, the claim is that this adjudication can be achieved given only an assumption about the causal structure of the experiment. Here, we aim to shed some light on why this is possible, by reconceptualizing the comparison of the two theories as an instance of a broader class of such theory comparisons. By recasting RQT as the subtheory of quantum theory that arises by symmetrizing with respect to the collective action of a time-reversal symmetry, we can compare it to other subtheories that arise by symmetrization, but for different symmetries. If the symmetry has a unitary representation, the resulting foil theory is termed a twirled quantum world, and if it does not (as is the case in RQT), the resulting foil theory is termed a swirled quantum world. We show that, in contrast to RQT, there is no possibility of distinguishing any twirled quantum world from quantum theory given only an assumption about causal structure. We also define analogues of twirling and swirling for an arbitrary generalized probabilistic theory and identify certain necessary conditions on a causal structure for it to be able to support a causal compatibility gap between the theory and its symmetrized version. We draw out the implications of these analyses for the question of how a lack of a shared reference frame state features into the possibility of such a gap.
Recent work by Renou et al. (2021) has led to some controversy concerning the question of whether quantum theory requires complex numbers for its formulation. We promote the view that the main result of that work is best understood not as a claim about the relative merits of different representations of quantum theory, but rather as a claim about the possibility of experimentally adjudicating between standard quantum theory and an alternative theory -- a foil theory -- known as real-amplitude quantum theory (RQT). In particular, the claim is that this adjudication can be achieved given only an assumption about the causal structure of the experiment. Here, we aim to shed some light on why this is possible, by reconceptualizing the comparison of the two theories as an instance of a broader class of such theory comparisons. By recasting RQT as the subtheory of quantum theory that arises by symmetrizing with respect to the collective action of a time-reversal symmetry, we can compare it to other subtheories that arise by symmetrization, but for different symmetries. If the symmetry has a unitary representation, the resulting foil theory is termed a twirled quantum world, and if it does not (as is the case in RQT), the resulting foil theory is termed a swirled quantum world. We show that, in contrast to RQT, there is no possibility of distinguishing any twirled quantum world from quantum theory given only an assumption about causal structure. We also define analogues of twirling and swirling for an arbitrary generalized probabilistic theory and identify certain necessary conditions on a causal structure for it to be able to support a causal compatibility gap between the theory and its symmetrized version. We draw out the implications of these analyses for the question of how a lack of a shared reference frame state features into the possibility of such a gap.
Abstract Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces 1,2 . This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural 3 . In fact, previous studies have shown that such a ‘real quantum theory’ can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states 4 . Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.
In our previous paper on the matter, we proved that real quantum theory (RQT) cannot be experimentally falsified without the experiment falsifying quantum theory (QT) as well. Our result is in direct contradiction to the central claim of an article by Renou et al., published in Nature [600, 625--629 (2021)], reporting a bilocal network experiment that could falsify real quantum theory without falsifying quantum theory.
The purpose of this note is to address several seemingly natural attempts to recover the experimental conclusion of Renou et al. We first explain how these results, although based on nonlocality experiments, are far from a Bell-like level of proof. Then we explain why the proposed experiment requires an experimentally untestable premise about independent preparations necessarily being represented as product states. Finally, we discuss some concerns that lie outside the scope of both results.
Quantum mechanics has achieved unparalleled success as an operational theory, describing a wide range of experiments to remarkable accuracy. However, the physical foundations on which it rests remain as puzzling as they were a century ago, and a concise statement of the physical principles that underlie quantum mechanics is still outstanding. One promising approach holds that these principles should be formulated in terms of information, since many of the counter-intuitive effects in quantum theory concern questions such as what one can know about a quantum system and how the information encoded therein can be processed and distributed between parties. Another challenging feature of quantum theory is its incompatibility with the other cornerstone of modern physics, general relativity. In order to reconcile the two, one must identify and retain only the essential concepts and principles of each theory, and in the case of general relativity, causality has been identified as such a concept. This raises the question of how our classical understanding of causality must change when quantum theory is taken into account. In order to address these questions about information, knowledge and causality, I turn to the framework of causal models. In classical statistics, causal models explain the relations among a set of variables in terms of causal influences, which makes them a powerful tool for structuring our knowledge about complex systems and developing strategies for interacting with them. More importantly, the framework provides the conceptual underpinnings and mathematical methods for addressing questions about causation, information and knowledge in a rigorous manner. In this thesis, I develop a version of the classical causal models framework that is compatible with quantum theory and explore its physical implications. \n \n \nA first step is to define quantum versions of fundamental elements such as variables, conditionals and belief propagation rules. This allows one to consider the question of what one can come to know about a quantum variable from the point of view of causal modelling. The conditionals relating quantum variables are found to have a richer structure than their classical counterparts, which can be exploited for the task of discerning causal relations given limited data -- a central problem in classical causal modelling. The mathematical properties of quantum conditionals also establish a correspondence between various classes of two-party correlations, such as bound and distillable entanglement, and the types of causal structures that can give rise to them, which may become a useful tool for entanglement theory and quantum information processing. In the context of open quantum systems dynamics, quantum causal models provide a clear physical explanation of not completely positive maps and, more broadly, of non-Markovian quantum dynamics. Finally, I consider the possibility of non-classical effects in the way that different causal mechanisms are combined -- that is, non-classical causal structures. For the simple case of two causally ordered variables, I propose indicators that witness different classes of combinations of causal mechanisms, including a non-classical mixture, and describe an experiment realizing examples of the different classes. \n \n \nAs these results illustrate, the framework of quantum causal models provides both greater conceptual clarity and a comprehensive mathematical formalism for studying a diverse set of problems, ranging from foundational questions to applications in open systems dynamics and the exploration of non-classical causal structures, which are likely to feature in a future theory of quantum gravity. The material presented in this thesis is intended as a foundation and inspiration for further applications of quantum causal models.
[Note added after submission. After posting the first version of this preprint and corresponding with Ved Kunte and Kuntal Sengupta, we identified an issue with the claimed separation between real quantum theory and ordinary complex quantum theory in the process-matrix framework. This version imposes normalization only for local CPTP maps acting on the parties' process input-output systems, which we call N1. A stronger compositional requirement is to impose normalization also after arbitrary shared ancillary systems are introduced and each party acts jointly on its process system and local share of the ancilla, which we call N2. Under N2, the process matrix used in this version to separate RQT from QT is not valid, and the claimed RQT/QT separation is therefore not established. We are preparing a revised version that clarifies this distinction and revises the affected claims. This temporary update is intended to alert readers to this issue while we work on revising the manuscript.]
The role of complex quantities in quantum theory has been puzzling physicists since the beginnings. It is, thus, natural to ask whether, in order to describe our experiments, the mathematical structure of the complex Hilbert spaces it is built on is really necessary. Recently, it was shown that this structure is inevitable in network scenarios with independent sources. More precisely, Real Quantum Theory cannot explain the predictions of (Complex) Quantum Theory [Renou et al., Nature (London) 600, 625 (2021).NATUAS0028-083610.1038/s41586-021-04160-4]. Here, we revisit the independence assumption underlying this work. We show that assuming partial independence is sufficient for showing the inadequacy of Real Quantum Theory. We derive a tradeoff between source independence and the Bell value achievable in Real Quantum Theory, which also lower bounds the source correlations required to explain previous experiments by means of real quantum systems. We further show that 1 bit of entanglement is necessary and sufficient for recovering the complex quantum correlations by means of Real Quantum Theory in the scenario from [Renou et al., Nature (London) 600, 625 (2021).NATUAS0028-083610.1038/s41586-021-04160-4]. Finally, building on [McKague et al., Phys. Rev. Lett. 102, 2009PRLTAO0031-900710.1103/PhysRevLett.102.020505], we provide a construction to simulate any complex quantum setup with m independent sources by means of Real Quantum Theory by allowing the sources to share an m real-qubit entangled state in the first round of the experiment.
Twirled and swirled worlds describe generalised probabilistic theories (GPTs) restricted by symmetry-imposed superselection rules. We extend these constructions to a fully compositional setting for finite-dimensional GPTs and compact groups, treating states, effects, and multipartite transformations uniformly as processes. We distinguish system actions from process-space actions. We define actions on a theory as a whole by requiring the corresponding process-space actions to respect sequential and parallel composition. Averaging physical theory actions defines twirling, while averaging nonphysical theory actions defines swirling. We show that invariant subtheories might not satisfy process tomography, and establish conditions under which they do, involving invariant dynamically faithful states or internal classical reference frames. For locally tomographic parent theories, we prove that every theory symmetry admits a representation by linear system actions and that the invariant subtheory satisfies process tomography. An invariant multipartite process space fails local tomography if and only if some bipartition has sides sharing a non-trivial irreducible representation. We characterise when ancillary systems are necessary, which representations they must carry, and when entanglement is required for ancilla-free process tomography. We develop a framework for unifying twirled subtheories and extend incoherent simulations to compact groups. Finally, we recover bilocal classical theory as a twirled subtheory of classical theory and real quantum theory as a swirled subtheory of complex quantum theory, deriving their tomographic properties from their representation structure.