June 2, 2021

# International team including University of Calgary researcher proves 'imaginary' numbers have real function in quantum world

Faculty of Science mathematician on team that showed complex numbers are necessary for distinguishing quantum states

Visualize putting two apples on a table. That number “2” is easy to recognize and understand, because it’s a real number clearly observable in the real world.

Now try putting the square root of negative one, or √(−1), of apple on a table. It’s impossible because this number has very different properties from a real number — nothing in the real world can represent it.

Such numbers are called “imaginary” numbers.

The term was originally coined in the 17th century by French mathematician René Descartes, who regarded such numbers as fictitious or useless.

However, imaginary numbers have actually turned out to be useful. Combining them with real numbers makes “complex” numbers, used in many fields of science and engineering because mathematical calculations can be done faster and in a more practical way using these numbers.

Yet in such general calculations, “complex numbers are not strictly necessary. They’re just a tool,” says Dr. Carlo Maria Scandolo, PhD, assistant professor of mathematical statistics in the Department of Mathematics and Statistics in the Faculty of Science.

In quantum mechanics, complex numbers are used in the equations that describe the behaviour of objects that can behave like particles under some conditions and like waves in others, says Scandolo, a member of UCalgary’s Institute for Quantum Science and Technology.

For example, the most famous equation of quantum mechanics, called the Schrödinger equation, contains the imaginary number i, which is also called the “imaginary unit.”

Carlo Maria Scandolo.

### Imaginary numbers have real consequences

Theoretical physicists have been puzzled for nearly a century by complex numbers in quantum theory. Do these numbers actually have physical consequences in quantum mechanics and are they are necessary to describe the quantum world?

Now, a team of researchers at the University of Science and Technology in China, the University of Warsaw in Poland, and Scandolo at the University of Calgary, has answered that question.

In a new study, the team has shown for the first time that complex numbers — including their imaginary component — carry real information about quantum states that can be observed and measured in the quantum world.

“They are not a mere mathematical artifact,” says study co-author Scandolo. “Complex numbers really do exist and have operational meaning.”

The team’s research increases the fundamental understanding of how the quantum world works and how scientists might better harness quantum resources to, for example, build powerful quantum computers and quantum networks.

Their research is published in two papers, one in Physical Review Letters and the other in the journal Physical Review A.

### Study showed complex numbers are a 'quantum resource'

The research team used an approach known as resource theories to devise a theoretical description, or mathematical framework, for whether complex numbers are a “resource.” In quantum theory, a resource is a property that enables new actions that would otherwise be impossible.

For example, quantum entanglement is a resource because it allows actions such as teleportation, or the transfer of quantum bits of information between separate locations.

The team’s mathematical framework, for which Scandolo did part of the calculations and provided some ideas, suggested that complex numbers are indeed a quantum resource.

To test whether this was the case, the Chinese researchers set up a “game,” a linear optics experiment in which the gamemaster sent a pair of quantum-entangled photons (particles of light) to two researchers, “Alice” and “Bob,” who were in separate locations. Each received one of the entangled protons.

Their task was to identify which quantum states were prepared by the gamemaster. They could do local measurements on their own photon and then compare measurements, to calculate their probability of guessing the correct state.

Alice and Bob were able to identify some of the quantum states with 100-per-cent accuracy, but only if they were allowed to use complex numbers (including imaginary numbers) in their local measurements.

However, if they were constrained to using only real numbers in their measurements, “suddenly they had no clue about which state was prepared,” Scandolo notes.

“This experiment showed that we can associate a task for which only the presence of imaginary numbers allows us to do well,” he says.

### Exploiting complex numbers as a quantum resource

Albert Einstein famously said that quantum mechanics should allow two objects to affect each other’s behaviour instantly across vast distances, which he called “spooky action at a distance.”

The research team’s study indicates that complex numbers limit the amount of non-local behaviour in some sense, because without these numbers scientists wouldn’t be able to discriminate between quantum states locally.

“Complex numbers are a resource because they allow us to do this local discrimination,” Scandolo says.

Gaining a better understanding of this resource would enable scientists to exploit complex numbers to their fullest in situations where using quantum information is advantageous.

The research team’s next step is to explore other situations involving local discrimination (of other real quantum objects, for example) in which complex numbers could be a quantum resource.

Scandolo began this line of research as a postdoctoral scholar at the University of Calgary. His postdoc work was supported by a Faculty of Science Grand Challenges Award and by funding from the Pacific Institute for the Mathematical Sciences.

The University of Calgary’s researchers continue to be leaders in quantum science.

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