Mermin's Magic Square and the Denial of Reality in Quantum Mechanics

In classical mechanics, it was assumed without basis that for physical quantities, there exist true values independent of measurement, and that accurate measurements of those values exist. However, quantum mechanics possesses the property that the true values of these physical quantities do not exist, and that unavoidable measurement errors and disturbances to the physical quantities occur. These physical quantities include the number of elementary particles such as photons, and the fact that their true values do not exist means that the elementary particles themselves do not exist as realities.

In 2022, the Nobel Prize in Physics was awarded for experiments on the violation of Bell's inequalities, which denied local realism—the idea that "things are there."

Confirming that quantum mechanics does not possess reality in the ordinary sense, as classical mechanics does, can also be done through other methods.

In classical mechanics, it has been thought that different physical quantities each have their own true values, and that the product of the true values of those physical quantities is also a physical reality. However, it is now well known through the Kochen-Specker theorem and others that this property does not hold in quantum mechanics. Here, I would like to introduce what is known as the "magic square" discovered by David Mermin, which is the simplest example of this.

First, we prepare a quantum system consisting of two two-level spins. Then, we consider the list of nine physical quantities shown in Figure 1, called Mermin's magic square. For example, in the measurement of the physical quantity in the first row and first column, we measure the x-component of the first spin. The value of the physical quantity in the first row and second column is obtained by measuring the x-component of the first spin and the x-component of the second spin, respectively, and multiplying the results. The values of the nine physical quantities in the list are either +1 or -1. And it can be shown that these individual values do not constitute a "reality" as hidden variables. If each of these physical quantity values existed before measurement, it can be proven that a contradiction arises.

It can be shown from the properties of Pauli matrices that if you perform ideal measurements of the three physical quantities in each row of the magic square from right to left and take the product of the results obtained, it will always be +1. Also, if you perform ideal measurements of the three physical quantities in each column from bottom to top and take the product of the results obtained, it will always be -1.

However, it is impossible to assign values of +1 or -1 to the nine physical quantities before measurement in a way that satisfies this property. For example, if you do it as shown in Figure 2, the product of the three values in each row is correctly +1, and the product of the first and second columns is -1, but only the product of the third column, which should be -1, becomes +1. With other assignments of +1 and -1, incorrect product values appear somewhere as well. In other words, if we assume that each physical quantity clearly has a value of +1 or -1 before measurement and that it can be measured without disturbance, a contradiction arises. This means that the values of physical quantities as naive realities, like those in classical mechanics, do not exist in quantum mechanics.

For example, the measurement of the physical quantity defined by the product of the x-component and z-component of the spin in the second row and first column causes an unavoidable disturbance to the physical quantity in the second row and second column, which contains the y-component of the spin. In quantum mechanics, the result changes depending on which physical quantities are measured and in what order.

The choice of what to measure and the decision of the order in which to perform those measurements is generally called "context" in quantum mechanics. Mermin's magic square clearly shows that the values of physical quantities before measurement, independent of such context, do not exist.