Why Quantum Physics Needs Imaginary Numbers

Multiply a number by itself and the result is positive. Two times two is four. Negative two times negative two is also four.

In 1572, Italian mathematician Rafael Bombelli broke this rule by defining a number, i, whose square is negative one. For centuries, physicists treated this imaginary number as a convenient bookkeeping trick. In classical wave mechanics, engineers use complex numbers—the sum of a real number and an imaginary number—to simplify calculations for alternating currents or radio waves. But at the end of the calculation, they always discard the imaginary part. The physical world, classic intuition dictated, is measured only in real numbers.

Even Erwin Schrödinger, who introduced i into the foundational equation of quantum mechanics in 1925, viewed its presence with unease. He wrote to Hendrik Lorentz expressing hope that the wave function would ultimately prove to be a purely real function. For nearly a century, it seemed he might be right. Physicists found they could mathematically reformulate quantum mechanics using only real numbers by expanding the state space, mimicking the behavior of complex wave functions without needing the square root of negative one.

Now, that mathematical safety net has collapsed. Recent experimental tests of Bell-like inequalities show that a quantum theory restricted to real numbers cannot reproduce the actual correlation measurements of entangled particles. To describe the universe we inhabit, imaginary numbers are a physical necessity.

The Tripartite Game

The breakdown of real-valued quantum mechanics does not occur in simple, isolated systems. A single qubit can be modeled using real numbers by adding an extra dimension to the mathematical space. To expose the inadequacy of real numbers, researchers had to design a more complex arena.

In 2021, a theoretical proposal outlined a way to test the necessity of complex numbers using a network of three parties—traditionally named Alice, Bob, and Charlie. Unlike standard Bell tests, which use a single source to send entangled particles to two observers, this setup uses two independent sources.

The first source sits between Alice and Bob, sending them an entangled pair of particles. The second source sits between Bob and Charlie, sending them a separate entangled pair. Alice and Charlie never interact directly, and the two sources share no prior physical connection.

Bob, positioned in the middle, receives one particle from each source. He performs a joint measurement on his two particles, projecting them into an entangled state. This measurement, known as a joint entanglement swapping operation, glues the system together.

According to reporting on the theoretical work in Physics World, if the universe operated under a version of quantum mechanics restricted to real numbers, the correlations measured among Alice, Bob, and Charlie would be strictly bounded. A mathematical limit exists for how strongly their measurement outcomes can align if they only use real-valued wave functions. If the correlations exceed this limit, a real-valued quantum theory is falsified.

Inside the Laboratories

Two independent experimental teams took up the challenge to build this apparatus, publishing their results in early 2022.

One team, led by Jian-Wei Pan at the University of Science and Technology of China (USTC) in Hefei, built their system using superconducting qubits. They programmed a quantum processor to act as the two independent sources and the three receiving nodes. By executing precise microwave pulses, they generated the entangled states, performed the joint measurement at the central node, and recorded the final states.

The second team, led by Philip Walther at the University of Vienna, used an optoelectronic setup. They generated pairs of entangled photons using non-linear crystals pumped by lasers. The photons traveled through fiber-optic cables to high-speed detectors. Bob’s joint measurement was achieved using a beam splitter and single-photon detectors to perform a Bell-state measurement.

Both experiments required rigorous isolation to ensure the two sources remained entirely independent until the moments of measurement. Any leakage of information or unwanted coupling between the sources would invalidate the test, allowing real numbers to sneak back in as an explanation.

The teams measured the correlation values across thousands of experimental runs.

The real-valued prediction was violated. The USTC superconducting system surpassed the real-number limit by more than 40 standard deviations. The Vienna photonic experiment exceeded the threshold by several standard deviations.

Measured.

The Geometry of Reality

The implications of these experiments go beyond mere academic bookkeeping. They alter our understanding of the relationship between mathematics and physical reality.

In classical physics, mathematics is a language of representation. We use real numbers to denote position, velocity, mass, and charge. Imaginary numbers are occasionally introduced to make the differential equations of wave motion easier to solve, but they can always be factored out. A classical instrument never points to a value containing i.

In quantum mechanics, the wave function itself is not directly observable. Instead, we square its absolute value to calculate the probability of an outcome. Because the squaring of a complex number yields a real number, it was tempting to assume that the underlying complex wave function was just another mathematical convenience—a scaffold that could be dismantled once the building was complete.

The tripartite entanglement swapping experiments prove the scaffold is structural. When we restrict the quantum state space to real numbers, we lose the phase relationships necessary to coordinate entangled states across independent sources. The real-valued theory cannot hold enough information to coordinate the joint measurements of Alice, Bob, and Charlie without violating the principle of local causality.

By forcing quantum mechanics to do away with imaginary numbers, the theory becomes less accurate, failing to predict the very correlations we observe in the laboratory. The square root of negative one is not a human invention to make homework easier. It is a fundamental pixel in the fabric of the cosmos.

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This article is AI-generated (synthetic) content, produced by an automated editorial system with human direction and review. Every claim is traced to published, peer-reviewed sources.