Is the appearance of quantum mixed states due to human ignorance?

Have you ever heard the explanation that even for a quantum system in a mixed state rather than a pure state, the system is actually in one pure state, but we just don't know which one it is? This suggests that mixed states are caused by human ignorance, but such an explanation is actually incorrect.

First, let's consider classical statistical mechanics for a single particle. The probability distribution of position and momentum at each time for a mixed state is given by using the solution to the equation of motion, which is solved by providing the initial position and initial momentum, in the form of

In this case, the pure state is a delta-function-type probability distribution where the values of position and momentum at each time are fixed.

In other words, any probability distribution ρ, including mixed states, can be written as a sum of these pure states with an appropriate initial probability distribution. An important property of classical statistical mechanics is that for one ρ, this weighted sum of pure states exists uniquely. Based on this property, it is often said that "a mixed state only signifies human ignorance, and the target system is actually in just one pure state."

Also, arguments where the distinction between subjective probability and objective probability becomes confused are occasionally seen. In physics, we mainly adopt frequentist probability as objective probability. However, even with that frequentist probability, there is observer dependence. Suppose there are two dice, A and B, designed so that the numbers they show are always the same. If Bob secretly looks at the number on B before Alice observes A, the probability distribution of the number on A differs for Alice and Bob. For Alice, who does not know Bob's result, the probability of each number on A appearing is 1/6, but for Bob, the number on A that is about to be observed is the same as the number on B that he already knows, so the probability of that number is 100%, and the probability of other numbers is zero. As shown, probability has observer dependence, but it is also objective in the sense that it holds true no matter how many times the same experiment is repeated. In other words, even if frequentist probability has observer dependence, it remains an objective probability from the perspective of reproducibility. For Alice, because she does not know Bob's result, it is possible to express her probability of 1/6 for A as being "due to human ignorance." However, please pay attention to the fact that this does not originate from the die itself undergoing deterministic motion.

However, when it comes to quantum mechanics, this situation changes significantly. Let's explain this using a two-level quantum system. Its quantum state is represented by the Bloch sphere in Figure 1.

Pure states are represented by points on this unit sphere, and mixed states are represented by points inside the sphere. For example, let's focus on the simple case in Figure 2 where the spin expectation value of the y-component is zero. In this case, the pure states are points on a unit circle centered at the origin in the x-z plane. Here, as in Figure 2, let's consider a mixed state described by a spin expectation value vector with a magnitude smaller than 1.

This spin expectation value vector can be written, for example, as a probabilistic mixture of the two vectors in Figure 3.

Each vector is parallel to the original spin expectation value vector. And since both have their endpoints on the circumference, the respective quantum states corresponding to the two are pure states. Let's write them as |1><1| and |2><2|. Then, the density matrix of the original mixed state can be written as a probabilistic mixture of |1><1| and |2><2| as follows.

If we assume that a mixed state is caused by human ignorance, it would mean that the original mixed state is just one pure state, either |1><1| or |2><2|. But is this true? In fact, it is known that one mixed state can be created from probabilistic mixtures of different pure states. For example, the original spin expectation value vector can also be generated by weighting three vectors with endpoints on the circumference, as in Figure 3, with a probability distribution q.

Let's write the pure states corresponding to the three vectors on the circumference as |3><3|, |4><4|, and |5><5|, respectively. Then, the density matrix of the original mixed state can also be written as a probabilistic mixture of those three pure states as follows.

Then, as in the equation above, one mixed state can be written as a probabilistic mixture of multiple different pure states. We previously thought the two-level system in the original mixed state was either |1><1| or |2><2|, but now it turns out to be one of the pure states |3><3|, |4><4|, or |5><5|. And there are infinitely many ways to decompose into pure states by this probabilistic mixture. From this fact, it is impossible to attribute the origin of a mixed state in a quantum system to human ignorance like in classical statistical mechanics. In other words, the understanding that a mixed state is actually one determined pure state of the system is incorrect.

Moreover, quantum mechanical mixed states have a property called purifiability. If we consider an external system in addition to the quantum system, it is always possible to describe the entire composite system as a single pure state. It is a state that is quantum-mechanically entangled as a whole composite system, and the original subsystem we were focusing on cannot become a single pure state defined only by that system. The idea that "it is actually a single pure state of just that subsystem, and the external observer just doesn't know it" is wrong. Also, since it is a pure state as a composite system, there is no lack of information there, and there is no room for human ignorance to enter.

For example, let's consider a setting where a die named D and an experimenter D' who rolls it are in a macroscopic pure state. The experimenter reads the number on the rolled die. Then, as shown in Figure 4, for another external observer, a pure state |Φ> of D and D' is created, which has quantum entanglement between the memory area of the experimenter D' and the number on the die.

The experimenter D' prepares a pure state |μ><μ| for the two-level system S of interest, corresponding to the outcome of six dice. This process can also be performed as a unitary operation on the entire composite system. Then, if we calculate the reduced state of S by ignoring the dice and the experimenter D', S will be described as a mixed state as shown in Figure 4, with N=6. However, for an external observer, the entire system of D+D'+S is in a single determined pure state, which is a superposition of the various dice outcome states. In other words, rather than saying that the mixed state is caused by the ignorance of experimenter D', who could know the outcome but chooses not to, the cause is that the experimenter D' inside the systemin principlecannot measure macroscopic physical quantities that can only be experimented upon by the entire D+D'+S three-system. It merely means that if one measures physical quantities of S alone, the state of S is indistinguishable from the calculated reduced state. If an external observer measures a specific physical quantity of the entire D+D'+S system, it can be confirmed that the composite system is in a pure state, but that is a state superposition involving quantum entanglement. As the violation of Bell's inequality implies, the value of the dice outcome itself does not exist before measurement. Not only for the external observer, but essentially, the dice outcome is not determined. Therefore, the explanation that mixed states are due to human ignorance is fundamentally wrong.

Furthermore, when an external observer measures only the S system, the process in Figure 4 can be described as causing decoherence in the target system S to result in a mixed state, but it is also important that for the same external observer, no decoherence has occurred at all in the entire D+D'+S system. Even in the thought experiment where the S system is Schrödinger's cat, the entire composite system including the experimental apparatus is in a single macroscopic pure state, and the life or death of the cat is not determined as one or the other due to decoherence. Caution is also required here. I have written about the quantum state of this macroscopic system in Section 1.4 of Chapter 1 of 'Quantum Information and the Physics of Spacetime [2nd Edition]' (Saiensu-sha).