Quantum theory does not need complex numbers

TL;DR

The paper addresses whether quantum theory can be formulated over real numbers without losing predictive power, and provides a constructive RNQT that preserves representation locality and reproduces all QT statistics.The approach builds a real embedding using a map $\Gamma$ from complex Hermitian operators to special symmetric real matrices, and defines a real tensor product $\otimes_{\text{r}}$ to replace the standard tensor product while maintaining locality and positivity.Key contributions include proving RNQT exists, establishing a one-to-one correspondence on the embedded sector, and showing RNQT is tomographically local with a consistent dynamics via a real Liouville–von Neumann equation and orthosymplectic representations for unitary evolution.The work implies that complex numbers are not fundamental to QM, offering a conceptual unification with real-valued classical theories and suggesting future directions for multipartite extensions and potential links to gravity.

Abstract

Quantum theory was radically different from the theories of nature which came before it. One key difference was its use of complex numbers. This opened a longstanding debate over whether quantum theory fundamentally requires complex numbers -- or if their use is merely a convenient choice. Until recently, this question was considered open. However, in a 2021 Nature article, a decisive argument was presented asserting that quantum theory needs complex numbers since real-number quantum theory is inconsistent with the postulates of quantum theory. In this work, we show that this conclusion was premature, and in actual fact, a real-number quantum theory is consistent with the postulates of quantum theory. Our theory retains key features such as representation locality (i.e. local physical operations are represented by local changes to the states). A direct consequence of our results is that quantum theory based on real or complex numbers are experimentally indistinguishable.

Figures (1)

• Figure 1: Representation locality. In a characterization of QT which is representation local, the following holds: a) Two non-interacting atoms which are independently prepared, have a mathematical description of the form $\rho \otimes \sigma$. b) Suppose a laser pulse is subsequently applied exclusively to the first atom (prompting it to change state), while the second atom is not affected by the laser pulse (such that it remains in the identical state). The mathematical description is updated to $\rho' \otimes \sigma$, where $\rho'=h(\rho)$ and $h$ is a quantum channel. A representation of QT which is not representation local would require a state update of the form $\rho\otimes\sigma \mapsto h'(\rho \otimes \sigma)$, where the channel $h'$ acts non-trivially on $\sigma$. Representation locality is an important property because it implies that physically local degrees of freedom (i.e. those of the first atom), are locally represented in the mathematical description.

Theorems & Definitions (70)

• Definition 2.1
• Definition 3.1
• Lemma 3.1
• Theorem 3.1
• Definition A.1: Tensor product of vector spaces
• Definition A.2: Kronecker product
• Proposition B.1
• Definition B.2
• Corollary B.3
• proof