Skip to content

Physicists Take the Imaginary Numbers Out of Quantum Mechanics

Source: Quanta Magazine
Author: Daniel Garisto
Date: November 7, 2025

The Core Debate: Is i Essential?

For a century, the imaginary number i (√-1) has been central to the Schrödinger equation. Schrödinger himself had hoped for an "entirely real version," calling the original complex formulation "a certain crudeness at the moment."

In 2021, a team led by Marc-Olivier Renou and Nicolas Gisin devised a three-party Bell test (Alice, Bob, Charlie) with two entanglement sources. When a group at USTC in Hefei ran the experiment, the observed correlations exceeded the ceiling for real-valued quantum theory — strongly suggesting complex numbers were empirically necessary.

The 2025 Counter-Revolution: Three Strikes

The new papers identify the 2021 team's critical flaw: their tensor product assumption (the rule for combining quantum states). The standard tensor product is natural for complex spaces but is a restrictive special case. By adopting a more general rule, real-valued theories can do anything complex ones can.

  1. The German Team (March 2025) — Michael Epping, Dagmar Bruß, Anton Trushechkin, Pedro Barrios Hita, Hermann Kampermann. Produced a real-valued QM exactly equivalent to the standard complex version.

  2. The French Team (April 2025) — Timothée Hoffreumon and Mischa Woods. Paper titled "Quantum theory does not need complex numbers," with a different tensor product yielding identical predictions.

  3. The Quantum Computing Proof (September 2025) — Craig Gidney (Google Quantum AI). Showed that all T gates (logic gates relying on complex-plane rotations) can be eliminated from any quantum algorithm, proving numerically that quantum computing doesn't require complex numbers.

The Ghost of i

While these new theories eliminate i, they don't eliminate the structure of complex arithmetic:

  • Real-valued formulations exist since Ernst Stueckelberg (1960) but are notoriously cumbersome — e.g., 2 particles (4 complex numbers) become 16 real numbers.
  • The new theories largely copy i's ability to rotate vectors.
  • Bill Wootters (Williams): "Even when you translate quantum theory into real numbers, you still see the hallmark of complex-number arithmetic."
  • Anton Trushechkin (HHU Düsseldorf): They "simulate complex numbers by means of real numbers."
  • Vlatko Vedral (Oxford): "You can write them down whichever way you like, but it's unavoidable that they have to multiply exactly as though they were complex numbers."

Why Is the Complex Formulation So Much Simpler?

  • Chao-Yang Lu (USTC): "Complex quantum theory, with its natural tensor product, remains far more concise, elegant and mathematically straightforward."
  • Jill North (Rutgers philosopher): "Even if complex numbers aren't truly necessary, they do give rise to a formulation that seems particularly well suited to quantum mechanics."
  • Vedral: "We really don't have a single alternative to how quantum mechanics was already done 100 years ago. And the question is, why? Why can't we go beyond this?"

Key Takeaways

  • The 2021 claim that i is empirically necessary has been overturned by 2025 work.
  • Real-valued QM is exactly equivalent to standard QM but significantly more complex.
  • The "hallmark" of complex arithmetic (rotation) persists in these real-valued formulations.
  • The search continues for a truly novel, simpler reformulation — and for a deeper understanding of why complex numbers fit quantum mechanics so naturally.