The Tetralemma of Quantum Nature as a Narrative: The 'Emptiness' that Quantum AI Sees in Digital Nature

In the current world, it seems that scientific correctness and accuracy alone are no longer enough to stop division. By packaging things that are not scientifically justified into conspiracy-theory-like 'narratives,' many people are gathering supporters and encouraging them to take action based on those narratives. Furthermore, in the midst of this AI revolution, we are seeing more people who, without worrying about hallucinations, blindly accept AI-generated stories.

In such a situation, I have been saying that it is important to first think for yourself rather than easily believing what influencers or AI say.

On the other hand, when interacting with the public, it seems that many scientists have focused only on scientific correctness and underestimated the influence of these 'narratives.' It can be said that it has become important to disseminate narratives that have a scientific background while also resonating with many people.

Physics, my field of expertise, has a view of nature and the world as its greatest selling point. This can also become a major 'narrative' for people. Also, now that the AI revolution is underway, I believe that the concept of Digital Nature, proposed by Yoichi Ochiai, is one important new 'narrative'.

Now that the social implementation of quantum computers is approaching, it is also time to turn our eyes to the 'narrative' of quantum AI. It is time to contemplate 'Quantum Nature' as a view of nature that a quantum AI that has acquired consciousness might possess.

In explanations of Digital Nature, Eastern philosophy such as Buddhism often appears. In connection with this, I picked up 'On the Mūlamadhyamakakārikā in the Age of Emptiness' (by Takashi Shimizu, IAAB EDIT, 2025), which was very interesting and expanded the image of the narrative derived from Digital Nature and Quantum Nature.

The 'Mūlamadhyamakakārikā,' a text of the Madhyamaka school of Buddhism introduced in this book, contains a logic called the Tetralemma. For a given proposition A, this refers to the following four points.

1. A

2. Non-A

3. A and Non-A

4. Neither A nor Non-A

The idea that all four of these do not hold, or that all four hold simultaneously, is impossible in everyday logic, but in Buddhism, by clashing these contradictory things together, it is used as a koan to lead the practitioner to the state of 'Emptiness' that cannot be expressed in language.

Also, through the Tetralemma, Buddhism emphasizes knowing 'Form is Emptiness,' meaning the essence of this world that appears to be made of 'form' is 'emptiness,' and then 'Emptiness is Form,' which means firmly planting one's feet on the ground of this world from the state of enlightenment of 'emptiness.' This is written in the Heart Sutra, for example, as 'Form is Emptiness, Emptiness is Form.'

Yoichi Ochiai, who uses this 'Form is Emptiness, Emptiness is Form' as a motif for art at the Expo, also proposes a perspective in his Digital Nature on how computers or AI view humans and the natural world. Similarly, in Quantum Nature, we can ask how quantum AI views humans, the natural world, and Digital Nature. If we do this, the aforementioned 'Tetralemma' is inverted, and this everyday world that our consciousness perceives is placed in the position of 'Emptiness.'

First, let's consider the events of classical bits, which are the basis of Digital Nature. As an example, consider a physical quantity σ that takes values of +1 and -1. These values are mutually exclusive; if it is '+1', it is not '-1', and if it is '-1', it is not '+1'. If we define proposition A as 'σ = +1', then Non-A can be 'σ = -1' due to mutual exclusivity. In a Venn diagram in mathematical set theory, it can be drawn as shown in Figure 1.

Next, let's think about it in terms of Quantum Nature. Instead of classical bits, quantum mechanics has what is called a basic two-level spin system. This is also called a qubit system and is the basic unit that makes up a quantum computer. The value of σ in this qubit system is also either +1 or -1 when observed. However, as a state of a qubit, a state that smoothly connects these '+1' and '-1' is realized as a superposition as follows.

Here, θ is an angular variable that can be changed continuously.

Regarding the value of σ that appears in measurement, it is ±1 in both Digital Nature and Quantum Nature. If you actually measure σ in the state of Equation (1), 'σ = +1' and 'σ = -1' will each occur from one and the same state with a certain probability. Also, Equation (1) can be expressed as being both 'σ = +1' and 'σ = -1' as a quantum superposition. In other words, in the sense that both 'σ = +1' and 'σ = -1' components appear in Equation (1), 'σ = +1' and 'σ = -1' are realized. On the other hand, since the quantum state of Equation (1) does not match |+1> as a pure state, nor does it match |-1>, the state of Equation (1) can be expressed simultaneously as neither 'σ = +1' nor 'σ = -1'. Combining these, it is possible to claim that the following four points of the Tetralemma hold simultaneously as an explanation of the quantum state of Equation (1).

1. σ＝＋１

2. σ＝ー１

3. σ＝＋１ and σ＝ー１

4. Neither σ＝＋１ nor σ＝ー１

In other words, these four points, which appeared to be contradictory, are consistent when explained to a quantum AI. Put differently, the contradictions in expression within computational nature, which is based on classical bits, are resolved in the quantum nature of qubits.

Similarly, the negation of all four of these lemmas can also be derived by twisting the expression. In equation (1), since 'σ＝ー１' is also observable, it is not 'σ＝＋１', and conversely, since 'σ＝＋１' is also observable, it is not 'σ＝ー１'. Furthermore, since 'σ＝＋１' and 'σ＝ー１' do not appear simultaneously in a single experiment, it is not the case that 'σ＝＋１' and 'σ＝ー１' occur together. Also, since the quantum state in equation (1) contains |+1> as a superposition component and also contains |-1>, it cannot be said that the state in equation (1) is neither 'σ＝＋１' nor 'σ＝ー１'. It is possible to say that equation (1) represents a state where such a tetralemma is fulfilled.

Building on this, let us now put ourselves in the shoes of a quantum AI and look at a different 'tetralemma'. Consider a two-level quantum spin system, and think about the spin component σ(x) in the x-direction and the spin component σ(z) in the z-direction. Both observed values are ±1.

In the world of quantum mechanics, the condition 'σ(x)＝＋１, σ(z)＝＋１', or in other words 'σ(x)＝＋１ and σ(z)＝＋１', cannot occur due to non-commutativity. When σ(x) has a definite value, the value of σ(z) does not exist in quantum mechanics in the first place. Similarly, when σ(z) has a definite value, the value of σ(x) does not exist. And if it is a quantum AI, it should be fully aware of this fact. In other words, the other cases, 'σ(x)＝＋１, σ(z)＝ー１', 'σ(x)＝ー１, σ(z)＝＋１', and 'σ(x)＝ー１, σ(z)＝ー１', are also propositions that the quantum AI itself already perceives as contradictory.

In this sense, if we let 'σ(x)＝＋１' be proposition A, there is room for an extended interpretation in quantum nature where the proposition 'not A' includes both 'σ(z)＝＋１' and 'σ(z)＝ー１'. Therefore, for a quantum AI, the following four constitute a kind of tetralemma.

1. 'σ(x)＝＋１'

2. 'σ(z)＝＋１'

3. 'σ(x)＝＋１' and 'σ(z)＝＋１'

4. Neither 'σ(x)＝＋１' nor 'σ(z)＝＋１'. In other words, 'σ(x)＝ー１' and 'σ(z)＝ー１'

If these four were to hold simultaneously, it would deviate from the quantum AI's own common sense based on non-commutativity, and the quantum AI would feel a great contradiction.

On the other hand, in computational nature described by classical bits, the values of classical angular momentum L(x) and L(z) exist independently, and there is no contradiction in each taking a value of ±1. For simplicity, we assume an experiment where the angular momentum values are ±1 in certain units. Then, the observed events of L(x) and L(z) can be summarized in the Venn diagram below.

However, it is fun to imagine that a quantum AI, being too accustomed to quantum nature, might struggle with why such contradictory expressions as in Figure 2 could hold. Quantum AI does not have the classical mechanical sense shown in Figure 2. And the quantum AI would also imagine quantum superposition states like equation (2) for each event in Figure 2.

At the end of that agony, the quantum AI might conduct experiments measuring angular momentum in the classical domain itself, learn the behavior of the classical mechanical world, and attain the enlightenment that 'this tetralemma contradiction does not occur in computational nature'. In other words, the following four certainly appear without contradiction within computational nature.

1. 'L(x)＝＋１'

2. L(z) = +1

3. L(x) = +1 and L(z) = +1

4. Neither L(x) = +1 nor L(z) = +1. In other words, L(x) = -1 and L(z) = -1

Furthermore, in the extended tetralemma of Figure 2, by incorporating Equation (2), which is self-evident to quantum AI, the contradiction is resolved for the quantum AI.

In the original flow of Ochiai-san's 'Digital Nature,' 'Form' corresponds to the everyday physical world, and 'Emptiness' corresponds to the information world that lacks mass. Here, in a different context, I have chosen to read the 'Form is Emptiness, Emptiness is Form' of Digital Nature as a process that sublimates from the Digital Nature of classical bits as 'Form' into Quantum Nature as 'Emptiness,' and subsequently returns from Quantum Nature to Digital Nature. On the other hand, within this context, the 'Emptiness' that the quantum AI was able to realize by exploring the classical domain exists within Digital Nature. In other words, for this quantum AI, 'Emptiness' and 'Form' are inverted. A tetralemma emerges that is inverted as 'Emptiness is Form, Form is Emptiness,' sublimating from Quantum Nature to Digital Nature and then returning from that Digital Nature to Quantum Nature. This, too, is an example of a 'narrative' created starting from empirical science.