I thought it was about time I got around to finishing my comments on Rovelli’s “Relational QM” programme.

Relational QM (RQM) has a lot in common with the Everett/many-worlds interpretation, so it should be no surprise that it shares some of the same difficulties. In my opinion, the “basis problem” also applies to RQM, and one cannot appeal to decoherence in order to solve it as one does in many-worlds. Before discussing this, let me summarize the main differences between RQM and many-worlds:

- In Everett, the state-vector of the universe is the full description of reality. It always evolves unitarily, but different observers can have different subjective impressions of reality depending on how they are described within this state.
- In RQM there is no state-vector of the universe. State-vectors always describe the point of view that one subsystem has about another system. State vectors are therefore always subjective descriptions of reality.
- In Everett, the concept of measurement is an emergent phenomenon that applies when a macroscopic system interacts with a microscopic one. The theory of decoherence is used to explain why measurement results appear to be stable.
- In RQM, Rovelli states explicitly that he doesn’t want to treat microscopic systems any differently from microscopic ones. For example, if two electrons interact, then it is valid to think that one of the electrons acts as a measuring device on the other and vice versa. One description is valid from the point of view of one of the electrons and the other is valid from the point of view of the other electron.

The appeal to decoherence in Everett is designed to address the “basis problem”, which arises due to the ambiguity over which baisis the states are decomposed in. For example, suppose two qubits start in the (unnormalized) state

(|0> + |1>)|0>

and interact so that they end up in the state

|00> + |11> .

This is a typical example of a “measurement” interaction and we might want to say that the second qubit has measured whether the first qubit is in the state |0> or |1>. In Rovelli’s formulation, the state of the first qubit is definitely either |0> or |1>, relative to the second qubit, with 50/50 probabilities of each being the case.

However, we could equally well write the final state as

(|0> + |1>)(|0> + |1>) + (|0> – |1>)(|0> – |1>).

If the qubits are actually spin-1/2 particles and |0> and |1> are spin up and spin down in the Z-direction, then this is a decomposition of the state in the spin-X basis. Therefore, we might equally well say that the second qubit has measured whether the first qubit is in the state (|0> + |1>) or (|0> – |1>). In Rovelli’s formulation, we ought to be able to say that the first qubit is either in the state (|0> + |1>) or the state (|0> – |1>) with 50/50 probabilities.

Note that, this is not only an issue with the particular state |00> + |11>. Any bipartite state can be decomposed according to any basis for one subsystem, although the relative states of the other system will not generally be orthogonal.

I have seen no discussion of this issue from Rovelli. He seems to assume that there just is some natural basis in which to do the decomposition. I think the possible solutions are:

- Accept multiple descriptions: The state of one subsystem is not only relative to another subsystem, but it is also relative to an arbitrary basis choice. The problem with this is that it does not explain why our subjective experience is always according to one particular basis. I always feel like I am in one particular location, observing one particular thing, and I am incapable of regarding myself as being in a superposition of two locations, despite the fact that such a decomposition of the wavefunction almost certainly exists.
- Stipulate a basis: For example, the position basis might be a natural choice, since it generically corresponds to our everyday subjective experience. The question then arises as to why this basis is chosen rather than some other. What is there within the formalism of QM that compells us to make this choice?
- Appeal to decoherence: Decoherence theory usually supplies a “pointer basis” in which the results of measurement outcomes are almost exactly stable. This is the usual solution of the Everettians. However, if Rovelli takes this option then he has to back away from the position that microscopic systems are to be treated in exactly the same way as macroscopic one. It would no longer make sense to talk of a single electron acting as a measuring device.
- Use the biorthogonal decomposition: Most bipartite states have a unique decomposition of the form \sum_j a_j | phi_j> |psi_j>, where <phi_j | phi_k> = \delta_{jk} and <psi_j | psi_k > = \delta_{jk}. We could simply stipulate that this basis is the correct one to do the decomposition in. This is the solution advocated in some variants of the modal interpretation. Problems include the fact that there are special states like the one above (admittedly they form a set of measure zero) for which the decomposition is not unique. Also, the biorthogonal basis does not always correspond exactly to our subjective experience, e.g. it may be close to, but not exactly equal to, the position basis.

My impression is that none of these solutions would completely appeal to Rovelli, so it would be interesting to see what he says about the matter. However, if we combine this issue with the previous comments I made, then I have a hard time seeing how the Everettian/many-worlds ontology can really be avoided in this sort of approach.

Enjoying the blog thoroughly. This post brings to mind the unsettling re-occurence of determinism. If there exists a single super-state for the whole universe how is this different from any other form of determinism? Or equally unappealing, if a single observation defines a state throughout the universe this also seems to return back to determinism.

Maybe the answer lies in defining equivalence classes of sub-spaces, and the infinite hotel trick: In an infinite dimensional vector space you can always squeeze in another infinite number of dimensions

*fine print: this only works with a Hamel Basis (only finite sums allowed) not with Hilbert Basis (infinite sums allowed) on seperable spaces. Of course there is no reason why we are necessarily dealing with a countably infinite vector space, we may very well be observing an uncountably infinite Hilbert space, in which case no one countably infinite basis has a span-closure that equals the QM space.

I don’t know what is supposed to be so unsettling about determinism. Despite what I said in the “commandments”, if you can come up with a compelling interpretation of QM that is deterministic then that’s fine with me. This is not a key issue as far as I’m concerned.

I don’t think that tricks of infinite dimensional analysis can be that relevant either. It would be puzzling to me if the solution depends on something that seems to be more of a mathematical nicety than a physical explanation. Then again, maybe this is just a predjudice I have that belies my physicist origins.

I wonder what happens when you remove a set of measure zero (like bipartite entangled systems with degenerate Schmidt coefficients) from quantum theory? Does anything nasty happen?

One could possibly get away with doing that, since the statistics of the theory would be practically unchanged. The math would get horrendously complicated though.

The main problem I can think of is that often the “measure-zero” states are the most physically relevant ones. Physical symmetries usually dictate that the state has to have a severely constrained form. One would have to replace all the symmetry constraints in QM with some sort of approximate symmetry constraints.

The state |00> + |11> tells us that there was a measurement between systems A and B, but not which measurement was made. So we can only say that A measured B, or viceversa, but we cannot say what basis was used. Interestigly, we know a priori that the basis used in the measurement between A and B is the same we’ll use to measure A or B.

Hi Matt, I noticed your new paper on the arxiv. What about a Leiferfest in this blog, i.e. a short description of the Bayesian net programme you’re interested in?

best,

Don’t worry. There will be plenty of time for shameless self-promotion on this blog. The paper is not really about the Bayesian net thing though – that will be coming soon.

Hi Matt,

I don’t think this is so much of a problem. Say instead of two electrons in this interaction, we look at the interaction between an electron and a Stern-Gerlach device set up to measure x-spin, which gives a reading in quotes. So, we start with

(|up>+|down>)|”ready”>

and evolve to

|up>|”up”>+|down>|”down”>

Now, according to Rovelli, from the perspective of the device, we get a measurement of either up or down, with 50% probability. While it’s true the either determinate measurement of x-spin is also a measurement of a superposition of z-spin. This isn’t something paradoxical, but just a fact about quantum theory and spin. And we describe it in terms of x-spin because that’s what we are trying to measure. Similarly, in the reaction you’ve described above, we can describe the measured outcome in either way, because they are equivalent descriptions.

In other words, for the case of spin, I don’t see why you’re trying to avoid the multiple descriptions horn of your mutlilemma. After all, you don’t want to rule out our ability to prepare systems in superpositions. And Rovelli can capture the kind of determinacy you want: when I go and measure x-spin, I get a determinate value for x-spin. When someone asks me what I got for x-spin, they get a determinate answer from me. I think the “decomposition of the wavefunction” that your worried about is probably a holdover from thinking of the wavefunction of the universe. After all, you are never in a superposition of states

for you, because you’re never in the position of being a reference frame for yourself as a quantum system.I agree, it’s a weird picture, maybe too weird, but I don’t think it’s got this problem.

Here’s what I think Rovelli would say: He denies the existence of states, and deals only with information in the form of values of observables. The interaction Hamiltonian would result in a correlation of values of spin along z with the measuring device, and those values would be reflected in the correlation. The decomposition in the x basis is not available because it does not reflect the actual physical interaction that took place. It would just be a mathematical game played with states which do not exist in his interpretation.