Monday, February 15, 2010

Van Fraassen on "Rovelli's World"

I’ve long been interested in Carlo Rovelli’s Relational Quantum Mechanics (RQM), and had been aware that philosopher of science Bas C. van Fraassen had an unpublished paper discussing RQM.  Recently, I saw that a preprint draft of this paper, called “Rovelli’s World,” had appeared on his website.

In the opening paragraph, van Fraassen calls RQM an “inspiring” original vision, and says “its presentation involves taking sides on a fundamental divide within philosophy itself.”  Unfortunately, he doesn’t return explicitly to this last statement (and there is no conclusion section in the paper), but it is pretty clear that the key controversy of RQM revolves around the issue of realism.  RQM seeks a consistent and complete interpretation of a quantum mechanical world, but this comes at the expense of fully objective realism.  We give up the idea of absolute observer-independent quantum states, likewise observer-independent values of physical quantities; “the theory describes only the information systems have about each other.”

The main content of van Fraassen’s paper is a careful exercise in analyzing RQM to see what higher-order aspects of the world it describes are actually “absolute” (or objectively known) even as the states and measurement outcomes only exist relationally.  He wants to compare what Rovelli - qua the author of the paper on RQM - seems to know about the world, as opposed to what a particular system in the world (playfully denoted “ROV”) can know, assuming the theory is correct.  He looks at length at a specific example, where ROV is a third observer following on a “Wigner’s friend”-style example:  based on his analysis he concludes an additional postulate should be added to RQM to clarify the scheme.

Below the fold are my notes on the paper:  they are somewhat sketchy; please refer to the paper for the real deal.

Section1:  van Fraassen first summarizes the theory and its place in the family of QM interpretations.

Again, there are no observer-independent states.  A system has a state relative to an observer.  An “observer” is just any system (nothing special about macroscopic vs. microscopic, conscious vs. non-conscious, etc).

Each system has its own frame of reference.  It seems that the first absolute or invariant aspect of RQM is that each observer applies the same rules of quantum mechanics.  Also note that observables will be relational in the same manner as states as the theory retains the eigenstate-eigenvalue link.

On the one hand, like in the Copenhagen interpretation, the observer is external to the system under consideration.  On the other hand, observers are ubiquitous and equivalent.  Any system can be a frame of reference.  Like Many-Worlds and unlike Copenhagen, RQM wants to be a complete description of the world with no assignment of special status to any subset of reality.  Of course RQM is in direct opposition to Many-Worlds in that it takes measurements (equivalent in RQM to interactions) as foundational elements, where the MWI tries to dispose of measurements.

Like some other contemporary treatments of quantum theory, RQM uses information theory, where information is correlation established by an interaction.

Van Fraassen sees a predecessor with regard to the use of information theory in the work of H.J. Groenewold, who evidently described states in terms of observer-obtained information.  In recent times, work exemplified by that of Christopher Fuchs use an information-theoretic approach:  “The quantum system represents something real and independent of us; the quantum state represents a collection of subjective degrees of belief about something to do with that system…” (From Fuchs quant-ph/0205039).  Van Fraassen also discusses a paper by Clifton, Bub, and Halvorson which discusses quantum states as “information depositories,” and additional work by Bub.

Section 2.  “Is there a view from nowhere?”
Van Fraassen discusses the key example Rovelli uses in discussing RQM, where a system has different states relative to two observers.  (Rovelli calls this situation the “main observation” motivating elaboration of RQM).  System S has different states relative to observer O (who has performed a measurement on S) and observer P, who (from previous measurements) knows of a correlation between S&O, but only knows S to be in a mixed state.

A first worry when hearing this type of scenario described (as in “Wigner’s friend”) is that we tend to naturally assume that the state of S relative to O is the “real” state, where P is just ignorant of the reality.  But this is not the situation.  Imagine a 3rd observer, ROV.  ROV only has info about S, O, and P on the basis of earlier measurements, plus later states resulting from unitary evolution.  That's all he can know.  The limited information known by this system ROV should not be confused with the story being told by Rovelli when narrating how RQM works. “Narrator Rovelli” is telling us only a story to convey the general form that an observer’s information can take, given that certain measurement interactions have taken place.  There’s no omniscient “view from nowhere” which has primacy.

But, if we accept RQM, van Fraassen stresses we do have some omniscient or “transcendental” knowledge that is about QM as a set of principles constraining the general form information can take.

Next, we note with Rovelli, that the specific info had by one system relative to another, can only be a record of measurement outcomes.  A measurement is a physical interaction: a system delivers a value for some observable; this also serves as a preparation, so the value obtained has probabilistic predictive content.  While the eigenstate-eigenvalue link is retained (unlike in van Fraassens’ original modal interpretation), recall again that the “collapse” of the system is only relative to the particular observing system.  So, the observable is observer-relative.  Because actual, physical measurements are needed, the states assigned would presumably not be pure, so the info available will not be sharp.

Let’s take a closer look at the information available and the information obtained. What in the scheme is absolute in a sense vs. relational?

Each system is characterized in the first place by a set of questions which can be asked about it.  This set would count as one of the “absolutes” in the theory.  This set is the specification of the family of observables that pertain to S.

Also, that an observer who has been in a measurement interaction with a system has a record of the questions asked and the outcomes thus obtained is an absolute fact (we can’t describe the specific info objectively, of course).

Section 3.  States as Observer Information
Rovelli’s RQM paper put forward two postulates constraining information acquisition.

First Postulate (Limited Information):  There is a maximum amount of relevant info that can be extracted from a system.

While answers to questions have (probabilistically) predictive value, earlier answers typically become irrelevant once subsequent answers are had.  This creates a limit on info.  There exists some maximally productive (most non-redundant) question-answer sequence (when asking any further questions reduces info.)  Postulate One says this sequence is finite. One can ask how many questions are needed to extract maximal info, leading to the assignment of a pure state relative to the observer.  This number of questions is an “absolute” fact.

Note that given the same string of past outcomes, the transition probabilities would not differ based on which observer asks the next question, and the past items which become irrelevant also would not differ.  So these are other examples of “absolutes” as interpreted by van Fraassen.

Rovelli says we can say any system has a maximum info. capacity measured in bits, -- this depends on the dimension of the state space.

Postulate 2 (Unlimited Information) It is always possible to acquire new info about a system.

This is possible because new info can make older info irrelevant.  And even with maximal info, there are questions which can be asked where you won’t know the answer before hand (different observables with no joint eigenstate).

Sections 4&5.

Van Fraassen looks at the most puzzling aspects of RQM.  Getting back to a system S and observers O and P:  Can O and P “contradict” each other with regard to outcomes of their measurements on S?

Here van Fraassen takes a careful look at what O and P could know about what each other have measured, with the key point being that one has to treat each question as a quantum mechanical interaction.  P knows that O has measured S, and if P interacts subsequently with O, he will confirm consistent answers based on the fact that a correlation was established between O (the pointer observable of O) and S.

Van Fraassen extends Rovelli’s example by bringing in a third observer, ROV.ROV has made measurements on S, O, &P in the past.  Subsequent to the measurements by O and P, suppose ROV conducts his own measurements of O and P to see what they found.  He can establish an agreement was reached between O and P, but not what the value was they had earlier relative to their own point of view.  Working through the formalism, one can see that the relative state of observable A on S does not change to ROV from O’s measurement to P’s measurement.

To clarify matters, van Fraassen  proposed an additional postulate relating relative states:
For any systems S, O & P, witnessed by ROV:
-         The state of S relative to O (if any), cannot be orthogonal to the state of S relative to (O+P) (if any), and,
-         The state of S relative to P (if any) similarly cannot be orthogonal to the state of S relative to (O+P) (if any), and,
-         And the state of S relative to any of these cannot be orthogonal to the state of S relative to ROV, and,
-         (and so forth for larger composite situations).

These are for pure states; the requirement for non-orthogonality will be less restrictive for mixtures.

So, ROV should indeed know that O and P are in agreement (even as Rovelli did), even without engaging in further measurement, given the facts of QM, including this extra, clarifying, postulate.

Section 6:  Relational EPR

Both Smerlak and Rovelli, and earlier, Laudisa, took a look at the RQM interpretation of the EPR set-up and Bell’s inequalities.  Van Fraassen takes a look, including his additional postulate.

Now S is a two-part system, and ROV will assign a mixture of consistent outcomes for the results as measured by the two EPR observers.  ROV can know this, if not which of the two consistent options they got (absent further measurements).  But even this seems to go beyond what Smerlak and Rovelli discussed (where they stressed the need for the original two to interact before confirming the consistent result was reached – taking away the need to suggest “action at a distance.”)


Anonymous said...

Van Fraassen paper is now published. In "Foundations fo Physics".

Steve said...


Pmer said...

Steve, I wrote all this up in 1996:

Relational Quantum Mechanics from Contextual Ontologies: Notes: leaned up a little

Suppose the x-spin of an electron, e, is going to be measured (in the sense of quantum mechanics) by a measuring apparatus, ma. The electron is represented by a vector

in a Hilbert Space Hma, given by

This represents the maximal amount of physical information there is about the electron's x-spin in the measuring apparatus' frame of reference. Equivalently,

represents the maximal amount of physical information there is about the measuring apparatus' magnetic orientation along x in the electron's frame of reference. The electron necessarily instantiates a frame of reference because there is an object there, that exists, for which the rest of the universe must conform to the physical laws.

In either frame, if the measurement can happen at any time, then

I doubt the amplitudes (as opposed to the probabilities) have to be equal, since all we require is coordination upon observation. In this case there's a symmetry involved. Which group it is depends on the evolution (Schoedinger, Dirac, etc.) and the bases.

There is a transformation between the measuring apparatus' frame of reference and the electron's frame of reference that preserves the evolution equation and Plank's constant, exactly analogous to Lorentz transformations among classical but relatively moving frames of reference. (And it's non-trivial since the mass et. al. of the measuring apparatus is larger than the electron's mass.)

There is no such thing as "the" quantum state of the electron. Its "quantum state" depends on the (quantum) frame of reference.

That's how it can be that, for Wigner's friend, Schroedinger's cat is in the superposition

whilst the cat finds itself to be in one of the two classical states

(5) alive, or else, dead

at all times. (The cat's "times". Time is a parameter describing relatively quantum systems, so there is a different time-variable for each system and they are in a particular sense ontologically independent. See previous posts.)

In the cat's frame, Wigner's friend is in the state

Whilst Wigner's friend finds himself to actually be in one of the two states

(7) happy, or else, sad

at all (of his) times.

That's also how, after Wigner's friend opens the box, he can find the cat to be in one of the classical states

(8) alive, or else, dead

and the cat can find Wigner's friend to be in one of the classical states

(9) happy, or else, sad

whilst in Wigner's frame of reference the combined friend-and-cat system is still ("still" in Wigner's time) in the superposition

Note that a system S remains in a quantum state relative to another system O if they represent the other by a vector in a Hilbert space. This is so even if the vector happens to be an eigenvector of a relevant observable. A person observing the results of a Stern-Gerlach experiment does not represent himself as a vector in a Hilbert space (which he would then have to project onto to find the probability of his being in the state he thought he was in? No.)

I wrote up and tried to promulgate these ideas in 1996, independently of Carlo Rovelli.

[... quantum state is intransitive and what exists... the total amount of physical information... at some moment, in a system's own notion of time, does not form an equivalence class...]

Steve Esser said...

Thanks! Since the formalism portions didn't come through on the comment (for some reason), I'll put a link here back to the material at your blog.