If one can distinguish a quantum superposition state in a single trial, that person is an entity that violates unitarity.

Unlike classical mechanics, which deals with naive reality, quantum mechanics is an information theory that does not contain the concept of reality within it. The protagonist of this theory is the probability distribution for mutually exclusive events that an observer can distinguish in a single trial. A probability distribution p(x) is introduced for an event x and its set Δ, which the observer's consciousness perceives and recognizes as "those events cannot occur simultaneously and are mutually exclusive." For example, if the result of a die roll is 1, the die cannot simultaneously be 6. Based on this, by setting Δ={1,2,3,4,5,6}, the probability of occurrence of the die result x∈Δ can be mathematically expressed as p(x), and the total probability becomes 1 as shown in equation (1).

These "mutually exclusive events" are specified by the observer's cognition. Based on the empirical rule that a die roll cannot be 1 and 6 at the same time, the observer judges that events where the result is 1 through 6 are distinguishable, and considers their probability distribution. If the observer lacks the ability to distinguish, it is impossible to introduce such a probability distribution in the first place.

Now, let us move away from past experience and consider how freely the set of mutually exclusive events Δ can be chosen. If there were aliens somewhere in the universe with different perceptual and cognitive abilities than ours, could they distinguish multiple events that we cannot distinguish using their high-resolution vision or intelligence? For example, could a superposition state where "the die result is 1 and simultaneously 6" exist for the aliens, and could they distinguish that state from a state that is simply "1" in a single trial?

To make the discussion concrete, in the following consideration, we will take the spin degree of freedom of a single electron as an example. Let us denote the event where the z-component of spin is +1 as "Z=+1", and the event where it is -1 as "Z=-1". When we measure the z-component, we always recognize one of these two. And in quantum mechanics, as shown in equation (2), these states can be expressed as state vectors, and these two vectors are orthogonal.

If we measure the x-component of spin, similarly, either the event "X=+1" where the x-component of spin is +1 or the event "X=-1" where it is -1 will be observed. These two events are also written as orthogonal state vectors, as shown in equation (3).

If an electron's spin were a real existence like in classical mechanics, there should also exist a state where the x-component takes the value X=+1 and simultaneously the z-component takes the value Z=+1. Other combinations of states should also exist. And the observer should be able to perceive them as mutually exclusive. In that case, formally, the observer would set state vectors for four different mutually exclusive events as in equation (4). These four vectors must be orthogonal.

However, now that the violation of Bell's inequality has been experimentally established, it is no longer permissible to consider such a set of mutually exclusive events Δ in reality.

If aliens were to visit Earth, if they are entities that follow the laws of quantum mechanics, they would not perceive the four mutually exclusive events of equation (4) either.

Then, what if aliens could distinguish a state where the spin x-component is +1 and a state where the spin z-component is +1 with a 100% success probability in a single check (experiment)? We can consider the four items in equation (5) as mutually exclusive events for them.

For us, who follow quantum mechanics, these four events in equation (5) cannot be mutually exclusive. We cannot distinguish a state where the spin x-component is +1 and a state where the spin z-component is +1 in a single trial. Distinguishing them requires an infinite number of repeated experiments. If there were aliens who could make that distinction in just a single experiment, it would be clear that they are entities that violate the unitarity of quantum mechanics, as shown below.

We calculate while assuming quantum mechanics for the aliens' cognition, and then show a contradiction. A state where the spin x-component is +1 and a state where it is -1 are quantum linear superpositions of states where the spin z-component is +1 and -1, as shown in equation (6).

Therefore, when calculating the inner product of each state vector, it is not zero as shown in equation (7), and they are not orthogonal.

Now, let us request the existence of a state in the memory area of the alien's brain that can distinguish the four states of equation (5) in a single trial. That memory area before measurement is in a certain initial state |0>. And that state vector satisfies the normalization condition of equation (8).

In the process where an alien recognizes a state with a spin x-component of +1, we assume that the state of the alien's memory area after measurement changes from |0> to |→>. Similarly, we assume it becomes |←> for a spin x-component of -1, |↑> for a spin z-component of +1, and |↓> for a spin z-component of -1. Since the memory area in these four states must be clearly distinguishable, they must each satisfy the orthogonality of equation (9).

However, if we assume the three equations (7), (8), and (9), it can be mathematically proven that there is no unitary operator representing the process by which this alien recognizes the four mutually exclusive events of equation (5). First, if such a unitary operator existed,

four relational expressions are required. And for the operator, from unitarity and equations (7) and (8),

a relationship is established. The result is that the quantity on the left side is not zero. However, using equation (9) and the first and third equations of (10), the quantity on the left side of equation (11)

can also be calculated, resulting in the same left-hand side being zero, which creates a contradiction. Therefore, it has been shown that there is no unitary operator that satisfies the conditions of equation (10). Thus, if there were an alien who could distinguish a quantum superposition state with a 100% success probability in just a single trial, it could be said that the alien is an entity that violates the unitarity of quantum mechanics.

Even in quantum mechanics, the mutually exclusive events that serve as arguments for the probability distribution are fundamentally set by the observer's consciousness, but there is no observer who can distinguish a quantum superposition state, such as the state of Schrödinger's cat, with only a single check. This conclusion will not change as long as quantum mechanics remains correct, no matter how much quantum computer and quantum AI technology advances in the future.

Just as classical mechanics was found to be broken, if it is discovered through experiments in the future that quantum mechanics is also broken, the possibility remains that there could be an observer who can distinguish a quantum superposition state with only a single check.

In this way, ultimately, the 'consciousness' of the observer, who is the subject of observation, empirically determines independent events.

Furthermore, the fact that a quantum AI cannot distinguish a superposition state from other states in a single trial does not necessarily mean that a quantum AI will adopt the same perception of mutual exclusivity as humans. It is entirely possible for a quantum AI to infer the underlying state vector from the measurement experience of numerous samples in a superposition state. Consequently, the quantum AI might extend and use mutual exclusivity for a large number of samples. It can also begin calculations by assigning probabilities to various different superposition states. In the natural cognition of a quantum AI, it might perceive the Schrödinger's cat state as 'the cat is alive and dead,' or it might perceive it as 'the cat is neither alive nor dead.' It is considered entirely possible that a quantum AI could automatically acquire such strange mutual exclusivity. I think this is also an interesting possibility.