Statistical Interpretation of Quantum State

Since I have an interest in quantum foundations, I want to try to understand the paper “The Quantum State Cannot be Interpreted Statistically” by Matthew Pusey, Jonathan Barrett and Terry Rudolph (PBR) in my own words. The paper came out on 14 November, and by 17 November, it is featured on Nature News. In fact, it was also submitted by four graduate students for an arXiv review meeting here at UNM that I didnot attend. (People often submit papers not long before the meeting.) Soon several blog posts about this paper popped up and I found comments by Matt Leifer at Scott Aaronson’s blog and Matt’s own blog post to be tremendously informative.

Given the hype, it is not so surprising that it caught the eye of someone that I know outside of the physics community as well. The opportunity to share my enthusiasm about quantum theory with them was the main motivation for this blog post, but soon I realize that this paper is not so interesting if you do not know about the Bohr-Einstein debate, the EPR paper, and the Bell’s theorem before. So I suggest a truly interested one to read those up as well. Personally, my main resources are Quantiki, Scholarpedia, Stanford Encyclopedia of Philosophy, and Asher Peres’ Quantum theory: concepts and methods.

What do they mean by statistical interpretation?

One reason that makes quantum theory difficult to grasp intuitively is its probabilistic aspect. Formal prediction of experimental result (by means of so-called “observables”) are probabilistic (although the probability can be zero or one). Other than the quantity we choose to observe, the probability is also determined by a quantum state (or “wave function” if you prefer) that the system is in via Born’s rule. Max Born was given a Nobel prize in physics precisely because of “his statistical interpretation of the wavefunction.” These are all accepted by every working physicist, and this is not what PBR refuted. Some might think of Ballentine’s ensemble interpretation, but that is not the target of PBR too. The target of PBR is the view that there is some underlying reality and quantum states only reflects our own knowledge about a given physical system; the point of view that was prominently advocated by Einstein as quoted in the paper itself but quoted here again for your convenience

“… I incline to the opinion that the wave function does not (completely) describe what is real, but only a (to us) empirically accessible maximal knowledge regarding that which really exists [...] This is what I mean when I advance the view that quantum mechanics gives an incomplete description of the real state of affairs.” Albert Einstein

Why state of knowledge?

One motivation is that if you are already a classical Bayesian and think that probability is only in the mind (this is not problematic, for example, in classical statistical mechanics), then there is a chance that quantum probability is no different. But this depends on the interpretation of probabilities, which is another story.

A stronger motivation comes from the analogy between the “wave function collapse” and information updating in probability theory. In orthodox (“Copenhagen”) QM, collapse supposes to happen instantaneously and you should start to smell a rat. If two observers Alice and Bob far apart each make a measurement on an EPR pair, in one frame of reference, Alice will collapse a state before Bob does, and in another frame of reference, Bob will collapse a state before Alice does. If collapse is a physical process (implied by the reality of the quantum state), then causality is apparently violated. This problem is dissolved if the quantum state, even a pure state, merely represents an observer’s state of knowledge. That is, we do not have to presume any local cause in this scenario.

Nevertheless, it seems like what we do when we take the epistemic interpretation seriously is just pushing the mystery further. True; Gleason’s theorem gives Born’s rule, but why should we be in this complex Hilbert space in the first place? This is similar to a trivial and natural proof of a mathematical theorem making use of complicated and seemingly ad hoc definitions. However, the epistemic approach has been a motivation for proving important results that shed some light on this issue like the quantum de Finetti theorem by Caves et al, which requires complex numbers in QM.

The ultimate question still remains untouched: “states of knowledge about what?” This leads us to what is traditionally known as hidden variables.

A classification of epistemic views of quantum state

The definitive reference here is Harrigan and Spekkens (HS). In fact, everyone who wants to understand what the PBR result is all about should read it because it provides the context of the definitions that PBR use.

The idea is to take Bell’s approach to hidden variable theories that he used to prove his celebrated inequality. Suppose that there is a set of variables collectively denoted $\lambda$ that we can imagine live in what we might call the ontic state space $\Lambda$ which does not have to be the same as Hilbert space. $\lambda$ determines the probability of every measurement outcome that one can make via a set of positive functions $P_{M_{k}}(\lambda)$ , where $M$ labels a measurement and $k$ labels an outcome. (To get a contextual theory, we demand that $P_{M_{k}}$ depends on other observables that are simultaneously measured as well.)¹ If measurement outcomes are discrete, then for fixed $M$ and $\lambda$

$\sum_{k}P_{M_{k}}(\lambda)=1$

for all $\lambda\in\Lambda$ , since probabilities must add up to one. This simple requirement is the ingredient used in the final step of the PBR proof.

Let $\Pi_{k}$ a projector associated with outcome $k$ of an observable $M$ . The probability of a getting outcome $k$ upon measuring $A$ in state $|\psi\rangle$ is $\langle\psi|\Pi_{k}|\psi\rangle:=\langle\Pi_{k}\rangle$ by quantum theory. But now with the hidden variables

$\langle\Pi_{k}\rangle =\int d\lambda \rho_{\psi} (\lambda) P_{M_k} (\lambda)$

where $\rho_{\psi}(\lambda)$ is the distribution of $\rho$ given a state $\psi$ . A precise classification of hidden variable theories then comes from looking at the map between the ontic state space and quantum state space and measurements in that theory.

Akin to the value-laden word “reality,” whether something is ontic or epistemic is, up to some degree, a matter of convention. So we define an ontic theory and an epistemic theory precisely in order to prove theorems about them. There is and always be a debate going on whether these definitions are justified, but this does not diminish the value of the PBR result. As with everything in life, we frame our concepts to reason about them. We do not fabricate a rule of reason first and then arbitrarily changing the meaning of its elements. And I think that the HS definition is useful enough to be an effective definition.

• In a $\psi$ -complete theory, we want to be able to say that quantum states determine everything that is to be said. Mathematically, this demands the ontic state space to be isomorphic to the standard Hilbert space of quantum theory, and every ontic state $\lambda$ is realizable by means of some preparation method.

• If a theory is not $\psi$ -complete, it is said to be $\psi$ -incomplete. That theory may happen to be a $\psi$ -augmented theory like the de Broglie-Bohm theory.

• An ontic theory is $\psi$ -ontic if $\lambda$ is reflected in quantum states unambiguously i.e. when we compare any two $\psi$ ‘s, their set of $\lambda$ are disjoint.

• If a theory is not $\psi$ -ontic, it is said to be $\psi$ -epistemic.

You can see that these definitions are natural within this framework by trying to come up with definition of $\psi$ -complete, $\psi$ -incomplete, $\psi$ -ontic, and $\psi$ -epistemic theory on your own without looking at the definitions by HS.

By their definitions, a $\psi$ -complete model is ontic. By logic then a $\psi$ -epistemic model is incomplete, so a set of $\psi$ -incomplete model is larger than a set of $\psi$ -epistemic model, for example.

The PBR result

According to PBR, “If the quantum state is statistical in nature (the second view), then a full specification of $\lambda$ need not determine the quantum state uniquely.”

What PBR calls a statistical interpretation of quantum state is precisely our $\psi$ -epistemic model, and PBR shows that such a model is impossible. To prove that, it is sufficient to reach a contradiction by assuming that for any pair of $\psi$ , their sets of $\lambda$ overlap. PBR proves this by generalizing from a special case where $\psi$ are $|0\rangle$ and $|+\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ of a two-level system that I shall call a qubit.

Suppose that we have two qubits labeled 1 and 2. Assume that the distributions $\rho_{|0\rangle}(\lambda)$ and $\rho_{|+\rangle}(\lambda)$ overlap. If we specify $\lambda$ that lies in the support of both distributions of qubit 1, then the qubit is either in the state $|0\rangle$ or $|+\rangle$ . Do the same with $\lambda_{2}$ and qubit 2. The joint state of both qubits is then one of the four possible states:

$|\psi_{1}\rangle =|00\rangle ,$
$|\psi_2 \rangle =|0\rangle \otimes |+\rangle,$
$|\psi_3 \rangle =|+\rangle \otimes |0\rangle,$
$|\psi_4 \rangle =|+\rangle \otimes |+\rangle .$

Now, we want to show that such $\lambda$ does not exist. PBR gives a four-outcome joint measurement such that for each and every outcome, the post-measurement state is orthogonal to one of these four possible pre-measurement state.

$|\xi_{1}\rangle =\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle),$
$|\xi_{2}\rangle =\frac{1}{\sqrt{2}}(|0\rangle\otimes|-\rangle+|1\rangle\otimes|+\rangle)=\frac{1}{2}(|00\rangle-|01\rangle+|10\rangle+|11\rangle),$
$|\xi_{3}\rangle =\frac{1}{\sqrt{2}}(|+\rangle\otimes|1\rangle+|-\rangle\otimes|0\rangle)=\frac{1}{2}(|00\rangle+|01\rangle-|10\rangle+|11\rangle),$
$|\xi_{4}\rangle =\frac{1}{\sqrt{2}}(|+\rangle\otimes|-\rangle+|-\rangle\otimes|+\rangle)=\frac{1}{\sqrt{2}}(|00\rangle-|11\rangle).$

It is easy to see that $\{|\xi_{i}\rangle\}$ forms an orthonormal basis and that $\langle\psi_{i}|\xi_{i}\rangle=0$ for all $i$ since $\{|0\rangle\otimes|+\rangle,|0\rangle\otimes|-\rangle,|1\rangle\otimes|+\rangle,|1\rangle\otimes|-\rangle\}$ is also an orthonormal basis.

But $P_{\xi_{k}}(\lambda)$ is specified by $\lambda$ alone and not the quantum state, so $P_{\xi_{1}}(\lambda)=P_{\xi_{2}}(\lambda)=P_{\xi_{2}}(\lambda)=P_{\xi_{2}}(\lambda)=0$ . (A crucial assumption is that the distribution of $\lambda$ is “well-behaved under a tensor product” e.g. the distribution of the product state $|00\rangle$ is simply $\rho_{|0\rangle}\rho_{|0\rangle}$ , the product of the distributions.) Then it is immediate that $\sum_{k=1}^{4}P_{\xi_{k}}=0\neq1$ , which is a contradiction. $\blacksquare$

Does this come as a surprise? Bell’s theorem and Bell-Kochen-Specker theorem was motivated by the question of whether the peculiar features of the de Broglie-Bohm theory namely nonlocality are necessary or not, and both theorem apply to the class of ontic models we have talked about. So we already know since 1966 that $\psi$ -epistemic ontic models within the Bell framework have to be nonlocal and contextual. But PBR go further and rule out all such $\psi$ -epistemic ontic models. So we may think that this actually shakes the foundations of quantum physics. It turns out, however, that this approach is not seriously taken by the foundations community anymore according to Matt Leifer. Nevertheless, an explicit proof is always a nice thing to have.

Implications of the PBR result

“…what is proved by impossibility proofs is lack of imagination.” John Stewart Bell.

Einstein was known to emphasize the incompleteness of QM by this dilemma:

“[T]he paradox forces us to relinquish one of the following two assertions:

(1) the description by means of the $\psi$ -function is complete

(2) the real states of spatially separate objects are independent of each other

Even if we are convinced that the first option is right, he did not tell us how to complete it. The ontic models discussed above is such an attempt. So the dilemma posed by QM is usually phrased as a conflict between realism and locality since if quantum states are real, the locality condition makes no sense because quantum states are not even separable in space.

For those who wants to hold on to locality (and believe that a measurement has an outcome), the old solution still works: thinking of quantum states as epistemic and being agnostic of anything beyond that. A description (quantum state) may be nonlocal but information is localized. It is just that now if we are excused of our lack of imagination and confine ourselves only to the Bell’s framework, then the “underlying reality” is not just nonlocal, it does not even exist.

1. Spekkens, R. W. (2005). “Contextuality for preparations, transformations, and unsharp measurements.” Phys. Rev. A 71, 052108. arXiv:0406166

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Written by Ninnat Dangniam

12/03/2011 at 5:06 pm

Posted in quantum

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