It was usually assumed that complex numbers, those containing a component with the imaginary number i (i squared equals minus one), are only a mathematical trick. A Polish-Chinese-Canadian team of scientists has proven, however, that the imaginary part of quantum mechanics can be seen in action in the real world, the Centre of New Technologies at the University of Warsaw reports.
We have to significantly reconstruct our intuitive 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. But researchers found quantum states of entangled photons that could not be distinguished without resorting to complex numbers. The researchers also conducted an experiment confirming the importance of complex numbers for quantum mechanics.
Dr. Alexander Streltsov from the Centre for Quantum Optical Technologies (QOT) at the University of Warsaw said: “In physics, complex numbers were considered to be purely mathematical in nature. It is true that while they have a fundamental role in quantum mechanics equations, they were treated simply as a tool, something to facilitate calculations for physicists. Now, we have theoretically and experimentally proven that there are quantum states that can only be distinguished when the calculations are performed with the indispensable participation of complex numbers.”
Complex numbers consist 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. The number i is the square root of -1 (i squared equals minus one).
There is nothing in the physical world that can be directly linked 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 parts, 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 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 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.
Dr. Streltsov, whose research was funded by the Foundation for Polish Science said: “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.”
COMPLEX GAME
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.
Dr. Streltsov said: “Let's assume that Alice and Bob's measurement results can only have the values of 0 or 1. Alice sees a nonsensical sequence of 0s and 1s, and so 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.”
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.
A press release from the Centre of New Technologies at the University of Warsaw said: “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 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.”
Articles describing the theory and measurements have now appeared in the journals Physical Review Letters and Physical Review A.
The research was 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 (UCalgary). Papers describing the theory and measurements have just appeared in the journals Physical Review Letters (https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.090401) and Physical Review A(https://journals.aps.org/pra/abstract/10.1103/PhysRevA.103.032401).
They have been summarized in a release sent to PAP by the Centre of New Technologies at the University of Warsaw.
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