Quantum Theory – A view from the inside Part III

Our virtual observer sits in a simulated quantum universe and its global state evolves unitarily. The interactions in the simulated universe are local, allowing parts of the system to be isolated by spatial separation. And this observer is very smart. He has somehow figured out that systems, which are isolated reasonably well, evolve in a specific predictable way. It is not important how exactly he did that, let us just assume that it is the case.

Mathematically, he would formulate a law that we would write as

\left|\psi(t)\right\rangle = U(t,t_0)\left|\psi(t_0)\right\rangle

where \left|\psi(t)\right\rangle is the state of the system at the time t and U(t,t_0) is the unitary operator that evolves the state from the time t to the time t_0.

As he develops the theory he also becomes aware of entanglement and that the state of subsystems cannot be generally described by a Hilbert space ray. He will find that there is a more general class of state descriptions, in the form of state operators (I will avoid the term “density operator” for reasons that should become obvious later) which also evolve unitarily for isolated systems, but with a slightly different law:

\rho(t) = U(t,t_0) \rho(t_0) U^\dagger (t,t_0)

The usual quantum states can be embedded naturally in this new state space by mapping them to their projectors \left|\psi\right\rangle\mapsto\rho=\left|\psi\right\rangle \left\langle\psi\right| and the evolution is preserved. As it turns out, the state operator is always a non-negative hermitian operator with finite non-zero trace. This does not require a statistical interpretation of the operator, but cannot be easily seen without going much deeper. For this, I have to refer to the scientific article that I will cite in one of the next posts. Right here it would only distract from the following argument.

Assume that the observer and everything that interacts with him directly (during a certain time span) can be regarded as one system that is approximately isolated. This separates the Universe into the part of the universe that directly contributes to the experience of the observer and the environment unknown to him. The observer could try to reconstruct the state of his observable subsystem and would choose the most general state representation he knows, a state operator. Remember that all his knowledge can only be based on the history of states of the universe. But since all he knows about the universe is contained in this subsystem, all he can possibly know is a result of the sequence of states of this subsystem. In this situation, is the state operator that the virtual observer finds uniquely determined?

This is the same situation as for the cellular automaton, where changing the cell contents did not make a difference for the structure contained in the sequence of its states. Only this time the allowed transformations are different. Generally, the transformation must be reversible so that the contained information in the representations remains the same.

Let us consider a simple example. If \rho(t) is a reconstructed state, we define \bar{\rho}(t):= \rho^2(t). The time evolution law of \bar{\rho}(t) follows as

\bar{\rho}(t) = U(t,t_0) \rho(t_0) U^\dagger (t,t_0) U(t,t_0) \rho(t_0) U^\dagger (t,t_0)

which simplifies with U^\dagger U = 1 to \bar{\rho}(t)=U(t,t_0)\bar{\rho}(t_0)U^\dagger (t,t_0). That means \bar{\rho} has the same time evolution law as \rho. Because state operators are hermitian and non-negative, squaring is a bijection. As a result, both \rho and \bar{\rho} are valid representations of the state history. Interestingly, none of the two descriptions is any simpler or preferable. Of course, squaring appears like an extra operation that makes it seem like a worse option, but that is only something we did as a construction example, the observer would have estimated \rho or \bar{\rho} directly, and to him both seem equally correct.

Clearly, squaring is not the only way to construct a new valid representation. It is easily verified that the same argument applies to any positive integer power of the state operator, they all generate a bijection between non-negative hermitian operators and the unitary evolution is preserved. We can also use linear combinations of positive integer powers, but then we have to be careful with preserving the bijectivity. It turns out that the  class of interesting bijections is defined as the analytic continuation of monotonically increasing analytic functions f:\mathbb{R}^+_0\to\mathbb{R}^+_0 with f(0)=0. In other words, take any such function and apply its power series to a state operator in order to retrieve a new state operator, that is also a valid description of the observer’s state history.

Any choice of a single state operator from this infinite family of possible states would be purely arbitrary. That also means our scientifically working virtual observer cannot deduce any such state operator representation from his state history. He will have to come up with an alternative state representation that eliminates the redundancy encoded in the state operators. This representation must describe something, which all equivalent state operators have in common. And it should also come in a familiar form, allowing to use the same formulation for the dynamics.

The hermitian non-negative operators we used to represent states have real non-negative eigenvalues, and the associated eigensubspaces are mutually orthogonal. From the way they are constructed, degeneracy would be purely accidental  and we may assume, that all eigensubspaces, that are not the nullspace, are in fact one-dimensional. This will simplify our discussion. As a result, we can describe each single state operator with a list of eigenvalues and the associated eigenvectors. The transformations between valid representations as defined above act as maps on the eigenvalues only, by applying the analytic monotonic functions to them. The monotonicity of these function preserves the order of the eigenvalues. That means once sorted they will stay sorted even after the transformation. However, the actual eigenvalues do change.

This implies two things: The eigenvalues do not contain information about the reconstructed state, because they change with the representation. The order of the eigenvalues does contain information, because it is shared by all state representations. That gives us a new less redundant state representation: A list of eigenvectors, sorted by their eigenvalues. The eigenvalues are not listed however.

But is this really a convenient representation that the observer would pick? Even if the observed part of the universe is not very large, the number of eigenvectors with non-zero eigenvalue would be enormous. How could he keep track of them all? How does he even find out how many there are exactly?

The answer is that he cannot count the subspaces, because the unitary dynamics he uses to extract information from his environment is not influenced by that number. But he knows that there is at least one subspace. In order to construct a one-subspace representation we have to send all eigenvalues to zero, except for one. We cannot do that with a bijection, but we can approximate it arbitrarily well using the bijective transformations. Taking the step to non-bijectivity is the price we have to pay, because we cannot count the subspaces. The reconstruction is unique nonetheless, as the monotonicity of the transformation functions forces us to preserve the largest eigenvalue as the only nonzero eigenvalue. For the reasons  mentioned earlier, we can assume the associated eigensubspace to be one-dimensional.

We have just deduced, that the observer will reconstruct the state of the system containing himself and the environment he interacts with as the eigenvector of the objective state operator corresponding to the greatest eigenvalue. This eigenvector evolves unitarily and we get the usual time dependence of a state vector:

\left|\rho_\mathrm{max}(t)\right\rangle=U(t,t_0)\left|\rho_\mathrm{max}(t_0)\right\rangle

Next time we will discuss the consequences of dropping the isolation requirement for the system and the implied non-unitary evolution.