hilbert.md

Who's Afraid of the Church of the Larger Hilbert Space?

Length: ~1500 words, approximately 6 minutes*

Status: Draft

What, exactly, is a measurement? This is a seemly innocuous question. Obviously, you get your tape measure out, wrap it round you waist, and count off how many inches you've added to your girth over the holiday season. But in the world of quantum theory, the theory that seems to be the operating system of our universe, when exactly a measurement occurs and it exactly what it is only very oddly defined. For our waistlines this could be a benefit, but as a scientific theory this is quite disconcerting.

There are many ways to present quantum theory. For example, one approach is to start with wave functions and add the Schrödinger wave equation which governs the time evolution of these wave functions. An equally valid starting point emphasizes linear algebra (which my friend David Hogg likes to remind us is not an algebra). But one of the clearest ways to get at the basics of quantum theory is through a simple set of a basic axioms. Probably my favorite version of these are from John Preskill's lecture notes. Here are rough paraphrases of these axioms, without the measurement axiom.

• Axiom 1 States are rays in Hilbert space.
• Axiom 2 Observables are self-adjoint operators.
• Axiom 4 Dynamics of a state, evolution in time, is a unitary operator.
• Axiom 5 When you take two systems and form a composite, the state space of the composite system is the tensor product of the two constituent systems' Hilbert spaces.

The missing axiom here is Axiom 3 which describes measurement. Here it is in full:

A measurement is a process in which information about the state of a physical system is acquired by an observer. In quantum mechanics, the measurement of an observable $\bf A$ prepares an eigenstate of $\bf A$, and the observer learns the value of the corresponding eigenvalue. If the quantum state just prior to the measurement is $|\psi\rangle$, then the outcome $a_n$ is obtained with a priori probability $${\rm Prob}(a_n) = ||{\bf E}_n|\psi\rangle||^2 = \langle \psi |{\bf E}_n |\psi \rangle$$ (ed note: here ${\bf E}_n$ is the projector onto the eigenvectors of $\bf A$, labeled by the outcome $n$.) if the outcome $a_n$ is attained, then the (normalized) quantum state just after the measurement is $$\frac{{\bf E}_n|\psi\rangle}{||{\bf E}_n |\psi\rangle ||}$$

This axiom takes up more length on the page than the other axioms, but it also throws out a lot of terminology. First of all there is an observer. What is that? And it describes measurement as a process, but what makes it a different process than the normal unitary dynamics of axiom 4? All of this should make you a little uncomfortable. The great thing about quantum theory, however, is that even with these sort of ambiguities, there has always been a way to just make them work, when you hit the real world of experiments.

For the experts in this area, I'm sure what I am saying is naïve, but I believe even they will have a feeling of encountering something that is somehow mysterious when encountering the measurement axiom. The measurement axiom is somehow how things become, how they transition into something more permanent, existing as records of what happened.

There are a group of physicists (or physicist-philosophers), loosely we would call them quantum foundations researchers, who spend a lot of time thinking hard about quantum theory and what it means or what it tells us or how to live with quantum theory and not go insane. What a measurement is and how it meshes with the other axioms of quantum theory is one of the areas in which quantum founds folk have spent a lot of time investigating. I've been lucky enough to work a bit on this back in my younger days. In my spare time, I still think a lot about this. Nothing says crank like being a software engineer thinking about the foundations of quantum theory! People whose papers I will read whenever they come out include Matthew Leifer, Robert Spekkens, Chris Fuchs, and more. Down those links is a lot of head scratching and some well earned moments of clarity.

In reading that literature, and thinking about some of these problems myself, one thing that strikes me is that there is a good deal of work in that arena which works very directly with a sort of variation on the axiom-ized version of quantum theory. In this approach, one considers preparations of states, evolutions, and measurements as separate processes (sometimes measurement and preparation are combined). In this approach one thinks a lot about the measurement results, and how they correspond to the reality or unreality of the quantum state, contextuality, and quantum non-locality. There is a lot to be said for this approach. It leads to Bell inequalities, Kochen-Specker experiments on contextuality, and results about the epistimic or non-epistimic nature of quantum theory. But it does make a particular sleight of hand, in that it does make one think about preparation, dynamics, and measurement as sort of separate.

Enter religion to the debate. In quantum information there is only one church and that church is the Church of the Larger Hilbert Space. This phrase "Going to the Church of the Larger Hilbert Space" was coined by John Smolin to describe a very useful tool. In particular it describes going from the system under study to a larger system that contains the original system as a tool for proving things about the original system. This can be done for both quantum states, but also for quantum channels. And it turns out to be an amazingly useful tool (All hail the Church! Amen.)

For quantum states this tool is used when working with mixed states. A mixed state is described, unlike in axiom 1 above, by a density matrix. The interesting thing about this density matrix is it that it can be purified. That is a mixed state can be thought of as a pure state in a larger system.

Connecting back now to those axioms, it strikes me that when one thinks about preparations, evolutions, and measurements separately, one has to be able to think and make sure that your reasoning is robust to the Church of the Larger Hilbert Space. In particular whatever one is doing for these setups, which delineate into three separate processes (though sometimes measurement and preparation are combined), then one needs to be worried that these in fact are actually working in larger space. That is, when reasoning about the these as separate objects, one needs to be careful that these arises from the Church of the Larger Hilbert Space trick.

As a concrete example, consider measurement of a qubit in the computational basis. One can describe this as the above axiom 3 applied to an observable with different values for $|0\rangle$ and $|1\rangle$ eigenvectors. If one is then reasoning about measurement from within the context of a preparation, unitary, measurement framework, then one will need to handle this measurement. But one can also think about this measurement as a process in which one attaches another qubit to the system, initially in the $|0\rangle$ state, and applies a controlled-NOT from the original qubit to this qubit. If this second qubit is never touched again, then the first system will behave as if it was measured. By thinking about the density matrix after the measurement as a purification in a larger pure state, the Church trick, one has transformed the original setup into something that doesn't really contain measurement (assuming here that the measurement outcome isn't really used, just to keep things simple). So one really then needs to explain what your theory says about the just unitary evolution part of the setup and give this measurement result.

And this, I think, is an important issue which those trying to explain quantum theory or see deeper into its interpretation have to overcome. They need to be able to deal with the case of everything being unitary (or possibly everything being a quantum operation).

So who's afraid of the Church of the Larger Hilbert Space? I think everyone who wants to think about quantum theory in a theory that separates preparation, dynamics, and measurement and studies them separately, should be scared. One needs to respect the Church, that we can get much of what we do by working in a larger state space. And reasoning about foundations should be wary of this, maybe even going so far as to say that we should not make these separations. Down that path we may find the many-world interpretation of quantum theory, but that's another discussion of religion for another day.

-dmb