Physicists Prove That the Imaginary Part of Quantum Mechanics Really Exists!

An international research team has proven that the imaginary part of quantum mechanics can be observed in action in the real world.

For almost a century, physicists have been intrigued by the fundamental question: why are complex numbers so important in quantum mechanics, that is, numbers containing a component with the imaginary number i? Usually, it was assumed that they are only a mathematical trick to facilitate the description of phenomena, and only results expressed in real numbers have a physical meaning. However, a Polish-Chinese-Canadian team of researchers has proved that the imaginary part of quantum mechanics can be observed in action in the real world.

We need to significantly reconstruct our naive ideas about the ability of numbers to describe the physical world. Until now, it seemed that only real numbers were related to measurable physical quantities. However, research conducted by the team of Dr. Alexander Streltsov from the Centre for Quantum Optical Technologies (QOT) at the University of Warsaw with the participation of scientists from the University of Science and Technology of China (USTC) in Hefei and the University of Calgary, found quantum states of entangled photons that cannot be distinguished without resorting to complex numbers. Moreover, the researchers also conducted an experiment confirming the importance of complex numbers for quantum mechanics. Articles describing the theory and measurements have just appeared in the journals Physical Review Letters and Physical Review A.

Photons Quantum Mechanics Complex Numbers

Photons can be so entangled that within quantum mechanics their states cannot be described without using complex numbers. Credit: QOT/jch

“In physics, complex numbers were considered to be purely mathematical in nature. It is true that although they play a basic role in quantum mechanics equations, they were treated simply as a tool, something to facilitate calculations for physicists. Now, we have theoretically and experimentally proved that there are quantum states that can only be distinguished when the calculations are performed with the indispensable participation of complex numbers,” explains Dr. Streltsov.

Complex numbers are made up of two components, real and imaginary. They have the form a + bi, where the numbers a and b are real. The bi component is responsible for the specific features of complex numbers. The key role here is played by the imaginary number i, i.e. the square root of -1.

There is nothing in the physical world that can be directly related to the number i. If there are 2 or 3 apples on a table, this is natural. When we take one apple away, we can speak of a physical deficiency and describe it with the negative integer -1. We can cut the apple into two or three sections, obtaining the physical equivalents of the rational numbers 1/2 or 1/3. If the table is a perfect square, its diagonal will be the (irrational) square root of 2 multiplied by the length of the side. At the same time, with the best will in the world, it is still impossible to put i apples on the table.

The photon source used to produce quantum states requiring description by complex numbers. Credit: USTC

The surprising career of complex numbers in physics is related to the fact that they can be used to describe all sorts of oscillations much more conveniently than with the use of popular trigonometric functions. Calculations are therefore carried out using complex numbers, and then at the end only the real numbers in them are taken into account.

Compared to other physical theories, quantum mechanics is special because it has to describe objects that can behave like particles under some conditions, and like waves in others. The basic equation of this theory, taken as a postulate, is the Schrödinger equation. It describes changes in time of a certain function, called the wave function, which is related to the probability distribution of finding a system in a specific state. However, the imaginary number i openly appears next to the wave function in the Schrödinger equation.

“For decades, there has been a debate as to whether one can create coherent and complete quantum mechanics with real numbers alone. So, we decided to find quantum states that could be distinguished from each other only by using complex numbers. The decisive moment was the experiment where we created these states and physically checked whether they were distinguishable or not,” says Dr. Streltsov, whose research was funded by the Foundation for Polish Science.

The experiment verifying the role of complex numbers in quantum mechanics can be presented in the form of a game played by Alice and Bob with the participation of a master conducting the game. Using a device with lasers and crystals, the game master binds two photons into one of two quantum states, absolutely requiring the use of complex numbers to distinguish between them. Then, one photon is sent to Alice and the other to Bob. Each of them measures their photon and then communicates with the other to establish any existing correlations.

“Let’s assume Alice and Bob’s measurement results can only take on the values of 0 or 1. Alice sees a nonsensical sequence of 0s and 1s, as does Bob. However, if they communicate, they can establish links between the relevant measurements. If the game master sends them a correlated state, when one sees a result of 0, so will the other. If they receive an anti-correlated state, when Alice measures 0, Bob will have 1. By mutual agreement, Alice and Bob could distinguish our states, but only if their quantum nature was fundamentally complex,” says Dr. Streltsov.

An approach known as quantum resource theory was used for the theoretical description. The experiment itself with local discrimination between entangled two-photon states was carried out in the laboratory at Hefei using linear optics techniques. The quantum states prepared by the researchers turned out to be distinguishable, which proves that complex numbers are an integral, indelible part of quantum mechanics.

The achievement of the Polish-Chinese-Canadian team of researchers is of fundamental importance, but it is so profound that it may translate into new quantum technologies. In particular, research into the role of complex numbers in quantum mechanics can help to better understand the sources of the efficiency of quantum computers, qualitatively new computing machines capable of solving some problems at speeds unattainable by classical computers.


“Operational Resource Theory of Imaginarity” by Kang-Da Wu, Tulja Varun Kondra, Swapan Rana, Carlo Maria Scandolo, Guo-Yong Xiang, Chuan-Feng Li, Guang-Can Guo and Alexander Streltsov, 1 March 2021, Physical Review Letters.
DOI: 10.1103/PhysRevLett.126.090401

“Resource theory of imaginarity: Quantification and state conversion” by Kang-Da Wu, Tulja Varun Kondra, Swapan Rana, Carlo Maria Scandolo, Guo-Yong Xiang, Chuan-Feng Li, Guang-Can Guo and Alexander Streltsov, 1 March 2021, Physical Review A.
DOI: 10.1103/PhysRevA.103.032401

The Centre for Quantum Optical Technologies at the University of Warsaw (UW) is a unit of the International Research Agendas program implemented by the Foundation for Polish Science from the funds of the Intelligent Development Operational Programme. The seat of the unit is the Centre of New Technologies at the University of Warsaw. The unit conducts research on the use of quantum phenomena such as quantum superposition or entanglement in optical technologies. These phenomena have potential applications in communications, where they can ensure the security of data transmission, in imaging, where they help to improve resolution, and in metrology to increase the accuracy of measurements. The Centre for Quantum Optical Technologies at the University of Warsaw is actively looking for opportunities to cooperate with external entities in order to use the research results in practice.


View Comments

  • You can put i apples on a table as follows.
    If we say an apple is represented by complex number a * bi, that is equivalent to an apple with amplitude a and frequency of b. We think of an apple as having a constant value, but if say you ate it and planted the seeds, and waited until another apple grew, you could say it has a frequency, say of five years. So with an apple of size a, and frequency b = 5 years, it makes sense to represent it by a complex number.

  • This is probably because i itself has a repetitive( therefore wavelike) exponential sequence: i⁰ = 1, i¹ = i, i² = -1, i³ = -i, i⁴ = 1, etc..

  • There is absolutely a physical meaning for i. It is a counterclockwise rotation by 90°. Start at (1,0) on the x-y plane. Rotate by 90° twice. You end up at (-1,0). Two rotations are a "multiplication" of imaginary numbers.

  • Imaginery numbers are related to a "measurable, physical quantity." Ask any engineer about real and reactive power in AC circuits.

  • What if we "i"magnined there were 3 apples on the table sitting next to 2 real apples: 2 + 3i.

  • The issue most people have is they confuse mathematical terms like real and imaginary with their general or common meaning. In mathematics, real and imaginary (or complex) numbers actually exist and are used to accurately describe actual physical systems. In the mathematical sense, imaginary or complex means the square root of -1 is needed to describe a system.

  • All complex numbers and functions can be equivalently represented by 2x2 real number matices. So there is nothing necessarily more "real" about imaginary numbers than there has always been about vectors and tensors.

  • Sorry to be a party pooper, but there's a pretty good size group of physicists out there (and not a few computer scientists and engineers as well) who've known for a very long time that imaginary numbers are real — and I mean that literally. The trick lies in an extension of standard 3D vector algebra to higher dimensions and even non-Euclidean spaces, known to mathematicians as Clifford algebras although its aficionados prefer to call them geometric algebras (as Clifford himself did). These algebras are over the real numbers (i.e. they contain only real scalars) but typically contain entities that square to -1 and commute with vectors, just like complex scalars would. The difference is that these geometric imaginary units can be understood as oriented spatial magnitudes, in much the same way that vectors are understood as oriented linear magnitudes. Such geometric interpretations can go a long way towards demystifying quantum mechanics (although the do not, of course, solve the measurement problem).

    For just one of the many introductions to geometric algebra out there, see

  • While the described experiments are informative, the talk that imaginary numbers are "real-world" and it is "first time" is a usual PR talk to gain more publicity and hence grants. I've seen pop-sci talks about "real" imaginary numbers some 40 years ago, only in the context of electromagnetics.

  • Since the square 0f imaginary part i is itself has a repetitive pattern in nature ( therefore wavelike) exponential sequence: i⁰ = 1, i¹ = i, i² = -1, i³ = -i, i⁴ = 1, etc.. it predicts the past, present and future of the probable events at quantum energy states of an events or things.
    Q Sheikh
    Jamshedpur, India
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Faculty of Physics University of Warsaw

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