Quantum Theory – A view from the inside Part IV

by aotell

Last time, we have derived how an observer inside a quantum system can only reconstruct a crude approximation to the real quantum state of this system. Specifically, he will observe the system to evolve like the eigenvector belonging to the greatest eigenvalue of the objective state operator describing his subsystem. To get this result however, we had to assume that the subsystem the observer interacts with is isolated and the evolution therefore unitary.

In order to understand how a non-unitary evolution affect the state reconstruction, we have to understand how unitary and non-unitary transformations modify the eigenstructure of an hermitian operator. This structure is given by real eigenvalues and orthogonal eigensubspaces that span the whole space. Both unitary and non-unitary transformations have to preserve this general structure. An additional constraint comes from the time-continuity of the evolution that translates to a continuous evolution of the eigenvalues and eigensubspaces.

Consider a hermitian operator A and its eigenvector \left|v\right\rangle with the eigenvalue v. It then follows that UAU^\dagger has an eigenvector U\left|v \right\rangle with the eigenvalue v, where U is a unitary operator. Or in other words, unitary evolution does not change the eigenvalues, but it rotates the eigensubspaces while preserving their orthogonality.

Now for non-unitary evolution the only possible way to generalize that behavior is to unlock the eigenvalues. This implies that eigensubspaces can fuse when distinct eigenvalues evolve into being equal. We also have the reverse process where they split up instead. But most of the time we expect to see a simple intersection of two eigenvalues with a fused eigensubspace only existing at a single point in time. This really summarizes all that can possibly happen. The Subspaces can also only be rotated because the hermitian constraint enforces orthogonality at all times. Non-unitary evolution is really not that complicated if presented like this!

The implications for the state reconstruction performed by our virtual observer are relatively simple. As long as the non-unitary evolution does not lead to an eigenvalue intersection, the order of the eigenvalue-sorted subspaces will stay the same. We have seen in the last post that the observer can not determine the actual eigenvalues anyway, so nothing changes for him. He cannot distinguish such a non-unitary evolution from a unitary one. If we do allow for eigenvalue intersections, then it depends on which eigenvalues are affected. If the intersection does not involve the greatest eigenvalue then the observer will still not be able to detect it, because the corresponding subspaces are not reconstructable.

The only case, when the intersection has any influence on the observed evolution, is when the greatest eigenvalue intersects with another eigenvalue. Then the eigensubspace associated with the greatest eigenvalue switches instantly. The virtual observer would perceive this event as a discontinuous evolution of the system state.

Have you realized what we have just derived? We have seen, that an observer, who is part of a quantum system, perceives the evolution of this system that includes himself and his nearby environment as the unitary evolution of a pure state. But when the system interacts with some part of the universe that is not included in his description, the pure quantum state can suddenly jump to a completely different pure state. And that can happen in a way that is entirely not unitary.

This description is very close to the postulates of the Copenhagen interpretation. The only missing aspect is a statement about the probability and results of the discontinuous state evolution. We will deal with that in the next post!