Tag Archives: foundations

Can the quantum state be interpreted statistically?

Posted on 20 November, 2011 by | 143 comments

A new preprint entitled The Quantum State Cannot be Interpreted Statistically by Pusey, Barrett and Rudolph (henceforth known as PBR) has been generating a significant amount of buzz in the last couple of days. Nature posted an article about it on their website, Scott Aaronson and Lubos Motl blogged about it, and I have been seeing a lot of commentary about it on Twitter and Google+. In this post, I am going to explain the background to this theorem and outline exactly what it entails for the interpretation of the quantum state. I am not going to explain the technicalities in great detail, since these are explained very clearly in the paper itself. The main aim is to clear up misconceptions.

First up, I would like to say that I find the use of the word “Statistically” in the title to be a rather unfortunate choice. It is liable to make people think that the authors are arguing against the Born rule (Lubos Motl has fallen into this trap in particular), whereas in fact the opposite is true.  The result is all about reproducing the Born rule within a realist theory.  The question is whether a scientific realist can interpret the quantum state as an epistemic state (state of knowledge) or whether it must be an ontic state (state of reality). It seems to show that only the ontic interpretation is viable, but, in my view, this is a bit too quick. On careful analysis, it does not really rule out any of the positions that are advocated by contemporary researchers in quantum foundations. However, it does answer an important question that was previously open, and confirms an intuition that many of us already held. Before going into more detail, I also want to say that I regard this as the most important result in quantum foundations in the past couple of years, well deserving of a good amount of hype if anything is. I am not sure I would go as far as Antony Valentini, who is quoted in the Nature article saying that it is the most important result since Bell’s theorem, or David Wallace, who says that it is the most significant result he has seen in his career. Of course, these two are likely to be very happy about the result, since they already subscribe to interpretations of quantum theory in which the quantum state is ontic (de Broglie-Bohm theory and many-worlds respectively) and perhaps they believe that it poses more of a dilemma for epistemicists like myself then it actually does.

Classical Ontic States

Before explaining the result itself, it is important to be clear on what all this epistemic/ontic state business is all about and why it matters. It is easiest to introduce the distinction via a classical example, for which the interpretation of states is clear. Therefore, consider the Newtonian dynamics of a single point particle in one dimension. The trajectory of the particle can be determined by specifying initial conditions, which in this case consists of a position \(x(t_0)\) and momentum \(p(t_0)\) at some initial time \(t_0\). These specify a point in the particle’s phase space, which consists of all possible pairs \((x,p)\) of positions and momenta.

Classical Ontic State

The ontic state space for a single classical particle, with the initial ontic state marked.

Then, assuming we know all the relevant forces, we can compute the position and momentum \((x(t),p(t))\) at some other time \(t\) using Newton’s laws or, equivalently, Hamilton’s equations. At any time \(t\), the phase space point \((x(t),p(t))\) can be thought of as the instantaneous state of the particle. It is clearly an ontic state (state of reality), since the particle either does or does not possess that particular position and momentum, independently of whether we know that it possesses those values ((There are actually subtleties about whether we should think of phase space points as instantaneous ontic states. For one thing, the momentum depends on the first derivative of position, so maybe we should really think of the state being defined on an infinitesimal time interval. Secondly, the fact that momentum appears is because Newtonian mechanics is defined by second order differential equations. If it were higher order then we would have to include variables depending on higher derivatives in our definition of phase space. This is bad if you believe in a clean separation between basic ontology and physical laws. To avoid this, one could define the ontic state to be the position only, i.e. a point in configuration space, and have the boundary conditions specified by the position of the particle at two different times. Alternatively, one might regard the entire spacetime trajectory of the particle as the ontic state, and regard the Newtonian laws themselves as a mere pattern in the space of possible trajectories. Of course, all these descriptions are mathematically equivalent, but they are conceptually quite different and they lead to different intuitions as to how we should understand the concept of state in quantum theory. For present purposes, I will ignore these subtleties and follow the usual practice of regarding phase space points as the unambiguous ontic states of classical mechanics.)). The same goes for more complicated systems, such as multiparticle systems and fields. In all cases, I can derive a phase space consisting of configurations and generalized momenta. This is the space of ontic states for any classical system.

Classical Epistemic States

Although the description of classical mechanics in terms of ontic phase space trajectories is clear and unambiguous, we are often, indeed usually, more interested in tracking what we know about a system. For example, in statistical mechanics, we may only know some macroscopic properties of a large collection of systems, such as pressure or temperature. We are interested in how these quantities change over time, and there are many different possible microscopic trajectories that are compatible with this. Generally speaking, our knowledge about a classical system is determined by assigning a probability distribution over phase space, which represents our uncertainty about the actual point occupied by the system.

A classical epistemic state

An epistemic state of a single classical particles. The ellipses represent contour lines of constant probability.

We can track how this probability distribution changes using Liouville’s equation, which is derived by applying Hamilton’s equations weighted with the probability assigned to each phase space point. The probability distribution is pretty clearly an epistemic state. The actual system only occupies one phase space point and does not care what probability we have assigned to it. Crucially, the ontic state occupied by the system would be regarded as possible by us in more than one probability distribution, in fact it is compatible with infinitely many.

Overlapping epistemic states

Epistemic states can overlap, so each ontic state is possible in more than one epistemic state. In this diagram, the two phase space axes have been schematically compressed into one, so that we can sketch the probability density graphs of epistemic states. The ontic state marked with a cross is possible in both epistemic states sketched on the graph.

Quantum States

We have seen that there are two clear notions of state in classical mechanics: ontic states (phase space points) and epistemic states (probability distributions over the ontic states). In quantum theory, we have a different notion of state — the wavefunction — and the question is: should we think of it as an ontic state (more like a phase space point), an epistemic state (more like a probability distribution), or something else entirely?

Here are three possible answers to this question:

  1. Wavefunctions are epistemic and there is some underlying ontic state. Quantum mechanics is the statistical theory of these ontic states in analogy with Liouville mechanics.
  2. Wavefunctions are epistemic, but there is no deeper underlying reality.
  3. Wavefunctions are ontic (there may also be additional ontic degrees of freedom, which is an important distinction but not relevant to the present discussion).

I will call options 1 and 2 psi-epistemic and option 3 psi-ontic. Advocates of option 3 are called psi-ontologists, in an intentional pun coined by Chris Granade. Options 1 and 3 share a conviction of scientific realism, which is the idea that there must be some description of what is going on in reality that is independent of our knowledge of it. Option 2 is broadly anti-realist, although there can be some subtleties here ((The subtlety is basically a person called Chris Fuchs. He is clearly in the option 2 camp, but claims to be a scientific realist. Whether he is successful at maintaining realism is a matter of debate.)).

The theorem in the paper attempts to rule out option 1, which would mean that scientific realists should become psi-ontologists. I am pretty sure that no theorem on Earth could rule out option 2, so that is always a refuge for psi-epistemicists, at least if their psi-epistemic conviction is stronger than their realist one.

I would classify the Copenhagen interpretation, as represented by Niels Bohr ((Note, this is distinct from the orthodox interpretation as represented by the textbooks of Dirac and von-Neumann, which is also sometimes called the Copenhagen interpretation. Orthodoxy accepts the eigenvalue-eigenstate link.  Observables can sometimes have definite values, in which case they are objective properties of the system. A system has such a property when it is in an eigenstate of the corresponding observable. Since every wavefunction is an eigenstate of some observable, it follows that this is a psi-ontic view, albeit one in which there are no additional ontic degrees of freedom beyond the quantum state.)), under option 2. One of his famous quotes is:

There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature… ((Sourced from Wikiquote.))

and “what we can say” certainly seems to imply that we are talking about our knowledge of reality rather than reality itself. Various contemporary neo-Copenhagen approaches also fall under this option, e.g. the Quantum Bayesianism of Carlton Caves, Chris Fuchs and Ruediger Schack; Anton Zeilinger’s idea that quantum physics is only about information; and the view presently advocated by the philosopher Jeff Bub. These views are safe from refutation by the PBR theorem, although one may debate whether they are desirable on other grounds, e.g. the accusation of instrumentalism.

Pretty much all of the well-developed interpretations that take a realist stance fall under option 3, so they are in the psi-ontic camp. This includes the Everett/many-worlds interpretation, de Broglie-Bohm theory, and spontaneous collapse models. Advocates of these approaches are likely to rejoice at the PBR result, as it apparently rules out their only realist competition, and they are unlikely to regard anti-realist approaches as viable.

Perhaps the best known contemporary advocate of option 1 is Rob Spekkens, but I also include myself and Terry Rudolph (one of the authors of the paper) in this camp. Rob gives a fairly convincing argument that option 1 characterizes Einstein’s views in this paper, which also gives a lot of technical background on the distinction between options 1 and 2.

Why be a psi-epistemicist?

Why should the epistemic view of the quantum state should be taken seriously in the first place, at least seriously enough to prove a theorem about it? The most naive argument is that, generically, quantum states only predict probabilities for observables rather than definite values. In this sense, they are unlike classical phase space points, which determine the values of all observables uniquely. However, this argument is not compelling because determinism is not the real issue here. We can allow there to be some genuine stochasticity in nature whilst still maintaining realism.

An argument that I personally find motivating is that quantum theory can be viewed as a noncommutative generalization of classical probability theory, as was first pointed out by von Neumann. My own exposition of this idea is contained in this paper. Even if we don’t always realize it, we are always using this idea whenever we generalize a result from classical to quantum information theory. The idea is so useful, i.e. it has such great explanatory power, that it would be very puzzling if it were a mere accident, but it does appear to be just an accident in most psi-ontic interpretations of quantum theory.  For example, try to think about why quantum theory should be formally a generalization of probability theory from a many-worlds point of view.  Nevertheless, this argument may not be compelling to everyone, since it mainly entails that mixed states have to be epistemic. Classically, the pure states are the extremal probability distributions, i.e. they are just delta functions on a single ontic state. Thus, they are in one-to-one correspondence with the ontic states. The same could be true of pure quantum states without ruining the analogy ((but note that the resulting theory would essentially be the orthodox interpretation, which has a measurement problem.)).

A more convincing argument concerns the instantaneous change that occurs after a measurement — the collapse of the wavefunction. When we acquire new information about a classical epistemic state (probability distribution) say by measuring the position of a particle, it also undergoes an instantaneous change. All the weight we assigned to phase space points that have positions that differ from the measured value is rescaled to zero and the rest of the probability distribution is renormalized. This is just Bayesian conditioning. It represents a change in our knowledge about the system, but no change to the system itself. It is still occupying the same phase space point as it was before, so there is no change to the ontic state of the system. If the quantum state is epistemic, then instantaneous changes upon measurement are unproblematic, having a similar status to Bayesian conditioning. Therefore, the measurement problem is completely dissolved within this approach.

Finally, if we allow a more sophisticated analogy between quantum states and probabilities, in particular by allowing constraints on how much may be known and allowing measurements to locally disturb the ontic state, then we can qualitatively explain a large number of phenomena that are puzzing for a psi-ontologist very simply within a psi-epistemic approach. These include: teleportation, superdense coding, and much of the rest of quantum information theory. Crucially, it also includes interference, which is often held as a convincing reason for psi-ontology. This was demonstrated in a very convincing way by Rob Spekkens via a toy theory, which is recommended reading for all those interested in quantum foundations. In fact, since this paper contains the most compelling reasons for being a psi-epistemicist, you should definitely make sure you read it so that you can be more shocked by the PBR result.

Ontic models

If we accept that the psi-epistemic position is reasonable, then it would be superficially resonable to pick option 1 and try to maintain scientific realism. This leads us into the realm of ontic models for quantum theory, otherwise known as hidden variable theories ((The terminology “ontic model” is preferred to “hidden variable theory” for two reasons. Firstly, we do not want to exclude the case where the wavefunction is ontic, but there are no extra degrees of freedom (as in the orthodox interpretation). Secondly, it is often the case that the “hidden” variables are the ones that we actually observe rather than the wavefunction, e.g. in Bohmian mechanics the particle positions are not “hidden”.)). A pretty standard framework for discussing such models has existed since John Bell’s work in the 1960’s, and almost everyone adopts the same definitions that were laid down then. The basic idea is that systems have properties. There is some space \(\Lambda\) of ontic states, analogous to the phase space of a classical theory, and the system has a value \(\lambda \in \Lambda\) that specifies all its properties, analogous to the phase space points. When we prepare a system in some quantum state \(\Ket{\psi}\) in the lab, what is really happening is that an ontic state \(\lambda\) is sampled from a probability distribution over \(\mu(\lambda)\) that depends on \(\Ket{\psi}\).

Representation of a quantum state in an ontic model

In an ontic model, a quantum state (indicated heuristically on the left as a vector in the Bloch sphere) is represented by a probability distribution over ontic states, as indicated on the right.

We also need to know how to represent measurements in the model ((Generally, we would need to represent dynamics as well, but the PBR theorem does not depend on this.)).  For each possible measurement that we could make on the system, the model must specify the outcome probabilities for each possible ontic state.  Note that we are not assuming determinism here.  The measurement is allowed to be stochastic even given a full specification of the ontic state.  Thus, for each measurement \(M\), we need a set of functions \(\xi^M_k(\lambda)\) , where \(k\) labels the outcome.  \(\xi^M_k(\lambda)\) is the probability of obtaining outcome \(k\) in a measurement of \(M\) when the ontic state is \(\lambda\).  In order for these probabilities to be well defined the functions \(\xi^M_k\) must be positive and they must satisfy \(\sum_k \xi^M_k(\lambda) = 1\) for all \(\lambda \in \Lambda\). This normalization condition is very important in the proof of the PBR theorem, so please memorize it now.

Overall, the probability of obtaining outcome \(k\) in a measurement of \(M\) when the system is prepared in state \(\Ket{\psi}\) is given by

\[\mbox{Prob}(k|M,\Ket{\psi}) = \int_{\Lambda} \xi^M_k(\lambda) \mu(\lambda) d\lambda, \]
which is just the average of the outcome probabilities over the ontic state space.

If the model is going to reproduce the predictions of quantum theory, then these probabilities must match the Born rule.  Suppose that the \(k\)th outcome of \(M\) corresponds to the projector \(P_k\).  Then, this condition boils down to

\[\Bra{\psi} P_k \Ket{\psi} = \int_{\Lambda} \xi^M_k(\lambda) \mu(\lambda) d\lambda,\]

and this must hold for all quantum states, and all outcomes of all possible measurements.

Constraints on Ontic Models

Even disregarding the PBR paper, we already know that ontic models expressible in this framework have to have a number of undesirable properties. Bell’s theorem implies that they have to be nonlocal, which is not great if we want to maintain Lorentz invariance, and the Kochen-Specker theorem implies that they have to be contextual. Further, Lucien Hardy’s ontological excess baggage theorem shows that the ontic state space for even a qubit would have to have infinite cardinality. Following this, Montina proved a series of results, which culminated in the claim that there would have to be an object satisfying the Schrödinger equation present within the ontic state (see this paper). This latter result is close to the implication of the PBR theorem itself.

Given these constraints, it is perhaps not surprising that most psi-epistemicists have already opted for option 2, denouncing scientific realism entirely. Those of us who cling to realism have mostly decided that the ontic state must be a different type of object than it is in the framework described above.  We could discard the idea that individual systems have well-defined properties, or the idea that the probabilities that we assign to those properties should depend only on the quantum state. Spekkens advocates the first possibility, arguing that only relational properties are ontic. On the other hand, I, following Huw Price, am partial to the idea of epistemic hidden variable theories with retrocausal influences, in which case the probability distributions over ontic states would depend on measurement choices as well as which quantum state is prepared. Neither of these possibilities are ruled out by the previous results, and they are not ruled out by PBR either. This is why I say that their result does not rule out any position that is seriously held by any researchers in quantum foundations. Nevertheless, until the PBR paper, there remained the question of whether a conventional psi-epistemic model was possible even in principle. Such a theory could at least have been a competitor to Bohmian mechanics. This possibility has now been ruled out fairly convincingly, and so we now turn to the basic idea of their result.

The Result

Recall from our classical example that each ontic state (phase space point) occurs in the support of more than one epistemic state (Liouville distribution), in fact infinitely many. This is just because probability distributions can have overlapping support. Now, consider what would happen if we restricted the theory to only allow epistemic states with disjoint support. For example, we could partition phase space into a number of disjoint cells and only consider probability distributions that are uniform over one cell and zero everywhere else.

Restricted classical theory

A restricted classical theory in which only the distributions indicated are allowed as epistemic states. In this case, each ontic state is only possible in one epistemic state, so it is more accurate to say that the epistemic states represent a property of the ontic state.

Given this restriction, the ontic state determines the epistemic state uniquely. If someone tells you the ontic state, then you know which cell it is in, so you know what the epistemic state must be. Therefore, in this restricted theory, the epistemic state is not really epistemic. Its image is contained in the ontic state, and it would be better to say that we were talking about a property of the ontic state, rather than something that represents knowledge. According to the PBR result, this is exactly what must happen in any ontic model of quantum theory within the Bell framework.

Here is the analog of this in ontic models of quantum theory.  Suppose that two nonorthogonal quantum states \(\Ket{\psi_1}\) and \(\Ket{\psi_2}\) are represented as follows in an ontic model:

Psi-epistemic model

Representation of nonorthogonal states in a psi-epistemic model

Because the distributions overlap, there are ontic states that are compatible with more than one quantum states, so this is a psi-epistemic model.

In contrast, if, for every pair of quantum states \(\Ket{\psi_1},\Ket{\psi_2}\), the probability distributions do not overlap, i.e. the representation of each pair looks like this

Psi-ontic model

Representation of a pair of quantum states in a psi-ontic model

then the quantum state is uniquely determined by the ontic state, and it is therefore better regarded as a property of \(\lambda\) rather than a representation of knowledge.  Such a model is psi-ontic.  The PBR theorem states that all ontic models that reproduce the Born rule must be psi-ontic.

Sketch of the proof

In order to establish the result, PBR make use of the following idea. In an ontic model, the ontic state \(\lambda\) determines the probabilities for the outcomes of any possible measurement via the functions \(\xi^M_k\). The Born rule probabilities must be obtained by averaging these conditional probabilities with respect to the probability distribution \(\mu(\lambda)\) representing the quantum state. Suppose there is some measurement \(M\) that has an outcome \(k\) to which the quantum state \(\Ket{\psi}\) assigns probability zero according to the Born rule. Then, it must be the case that \(\xi^M_k(\lambda) = 0\) for every \(\lambda\) in the support of \(\mu(\lambda)\). Now consider two quantum states \(\Ket{\psi_1}\) and \(\Ket{\psi_2}\) and suppose that we can find a two outcome measurement such that that the first state gives zero Born rule probability to the first outcome and the second state gives zero Born rule probability to the second outcome. Suppose also that there is some \(\lambda\) that is in the support of both the distributions, \(\mu_1\) and \(\mu_2\), that represent \(\Ket{\psi_1}\) and \(\Ket{\psi_2}\) in the ontic model. Then, we must have \(\xi^M_1(\lambda) = \xi^M_2(\lambda) = 0\), which contradicts the normalization assumption \(\xi^M_1(\lambda) + \xi^M_2(\lambda) = 1\).

Now, it is fairly easy to see that there is no such measurement for a pair of nonorthogonal states, because this would mean that they could be distinguished with certainty, so we do not have a result quite yet. The trick to get around this is to consider multiple copies. Consider then, the four states \(\Ket{\psi_1}\otimes\Ket{\psi_1}, \Ket{\psi_1}\otimes\Ket{\psi_2}, \Ket{\psi_2}\otimes\Ket{\psi_1}\) and \(\Ket{\psi_2}\otimes\Ket{\psi_2}\) and suppose that there is a four outcome measurement such that \(\Ket{\psi_1}\otimes\Ket{\psi_1}\) gives zero probability to the first outcome, \(\Ket{\psi_1}\otimes\Ket{\psi_2}\) gives zero probability to the second outcome, and so on. In addition to this, we make an independence assumption that the probability distributions representing these four states must satisfy. Let \(\lambda\) be the ontic state of the first system and let \(\lambda’\) be the ontic state of the second. The independence assumption states that the probability densities representing the four quantum states in the ontic model are \(\mu_1(\lambda)\mu_1(\lambda’), \mu_1(\lambda)\mu_2(\lambda’), \mu_2(\lambda)\mu_1(\lambda’)\) and \(\mu_2(\lambda)\mu_2(\lambda’)\). This is a reasonable assumption because there is no entanglement between the two systems and we could do completely independent experiments on each of them. Assuming there is an ontic state \(\lambda\) in the support of both \(\mu_1\) and \(\mu_2\), there will be some nonzero probability that both systems occupy this ontic state whenever any of the four states are prepared. But, in this case, all four functions \(\xi^M_1,\xi^M_2,\xi^M_3\) and \(\xi^M_4\) must have value zero when both systems are in this state, which contradicts the normalization \(\sum_k \xi^M_k = 1\).

This argument works for the pair of states \(\Ket{\psi_1} = \Ket{0}\) and \(\Ket{\psi_2} = \Ket{+} = \frac{1}{\sqrt{2}} \left ( \Ket{0} + \Ket{1}\right )\). In this case, the four outcome measurement is a measurement in the basis:

\[\Ket{\phi_1} = \frac{1}{\sqrt{2}} \left ( \Ket{0}\otimes\Ket{1} + \Ket{1} \otimes \Ket{0} \right )\]
\[\Ket{\phi_2} = \frac{1}{\sqrt{2}} \left ( \Ket{0}\otimes\Ket{-} + \Ket{1} \otimes \Ket{+} \right )\]
\[\Ket{\phi_3} = \frac{1}{\sqrt{2}} \left ( \Ket{+}\otimes\Ket{1} + \Ket{-} \otimes \Ket{0} \right )\]
\[\Ket{\phi_4} = \frac{1}{\sqrt{2}} \left ( \Ket{+}\otimes\Ket{-} + \Ket{-} \otimes \Ket{+} \right ),\]

where \(\Ket{-} = \frac{1}{\sqrt{2}} \left ( \Ket{0} – \Ket{1}\right )\). It is easy to check that \(\Ket{\phi_1}\) is orthogonal to \(\Ket{0}\otimes\Ket{0}\), \(\Ket{\phi_2}\) is orthogonal to \(\Ket{0}\otimes\Ket{+}\), \(\Ket{\phi_3}\) is orthogonal to \(\Ket{+}\otimes\Ket{0}\), and \(\Ket{\phi_4}\) is orthogonal to \(\Ket{+}\otimes\Ket{+}\). Therefore, the argument applies and there can be no overlap in the probability distributions representing \(\Ket{0}\) and \(\Ket{+}\) in the model.

To establish psi-ontology, we need a similar argument for every pair of states \(\Ket{\psi_1}\) and \(\Ket{\psi_2}\). PBR establish that such an argument can always be made, but the general case is more complicated and requires more than two copies of the system. I refer you to the paper for details where it is explained very clearly.

Conclusions

The PBR theorem rules out psi-epistemic models within the standard Bell framework for ontological models. The remaining options are to adopt psi-ontology, remain psi-epistemic and abandon realism, or remain psi-epistemic and abandon the Bell framework. One of the things that a good interpretation of a physical theory should have is explanatory power. For me, the epistemic view of quantum states is so explanatory that it is worth trying to preserve it. Realism too is something that we should not abandon too hastily. Therefore, it seems to me that we should be questioning the assumptions of the Bell framework by allowing more general ontologies, perhaps involving relational or retrocausal degrees of freedom. At the very least, this option is the path less travelled, so we might learn something by exploring it more thoroughly.

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Foundations Mailing Lists

Posted on 7 December, 2010 by | Leave a comment

Bob Coecke has recently set up an email mailing list for announcements in the foundations of quantum theory (conference announcements, job postings and the like). You can subscribe by sending a blank email to quantum-foundations-subscribe@maillist.ox.ac.uk. The mailing list is moderated so you will not get inundated by messages from cranks.

On a similar note, I thought I would mention the philosophy of physics mailing list, which has been going for about seven years and also often features announcements that are relevant to the foundations of quantum theory. Obviously, the focus is more on the philosophy side, but I have often heard about interesting conferences and workshops via this list.

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Job/Course/Conference Announcements

Posted on 26 November, 2010 by | Leave a comment

Here are a few announcements that have arrived in my inbox in the past few days.

Perimeter Scholars International

Canada’s Perimeter Institute for Theoretical Physics (PI), in partnership with the University of Waterloo, welcomes applications to the Master’s level course, Perimeter Scholars International (PSI). Exceptional students with an undergraduate honours degree in Physics, Math, Engineering or Computer Science are encouraged to apply. Students must have a minimum of 3 upper level undergraduate or graduate courses in physics. PSI recruits a diverse group of students and especially encourages applications from qualified women candidates. The due date for applications to PSI is February 1st, 2011. Complete details are available at www.perimeterscholars.org.

Foundations Postdocs

Also a reminder that it is currently postdoc hiring season at Perimeter Institute. Although, the deadline for applications has passed, they will always consider applications from qualified candidates if not all positions have been filled. Anyone looking for a postdoc in quantum foundations should definitely apply. In fact, if you are looking for a foundations job and you have not applied to PI then you must be quite mad, since there are not a lot of foundations positions in physics to be had elsewhere. Details are here.

Quantum Interactions

I will admit that this next conference announcement is a little leftfield, but some of the areas it covers are very interesting and worthwhile in my opinion, particularly the biological and artificial intelligence applications.

————————–

CALL FOR PAPERS

————————–

The Fifth International Symposium on Quantum Interaction (QI’2010, http://www.rgu.ac.uk/qi2011), 27-29 June 2010, Aberdeen, United Kingdom.

Quantum Interaction (QI) is an emerging field which is applying quantum theory (QT) to domains such as artificial intelligence, human language, cognition, information retrieval, biology, political science, economics, organisations and social interaction.

After highly successful previous meetings (QI’2007 at Stanford, QI’2008 at Oxford, QI’2009 at Saarbruecken, QI’2010 at Washington DC), the Fifth International Quantum Interaction Symposium will take place in Aberdeen, UK from 27 to 29 June 2011.

This symposium will bring together researchers interested in how QT addresses problems in non-quantum domains. QI’2011 will also include a half day tutorial session on 26 June 2011, with a number of leading researchers delivering tutorial on the foundations of QT, the application of QT to human cognition and decision making, and QT inspired semantic information processing.

***Call for Papers***

We are seeking submission of high-quality and original research papers that have not been previously published and are not under review for another conference or journal. Papers should address one or more of the following broad content areas, but not limited to:

– Artificial Intelligence (Logic, planning, agents and multi-agent systems)

– Biological or Complex Systems

– Cognition and Brain (memory, cognitive processes, neural networks, consciousness)

– Decision Theory (political, psychological, cultural, organisational, social sciences)

– Finance and Economics (decision-making, mergers, corporate cultures)

– Information Processing and Retrieval

– Language and Linguistics

The post-conference proceedings of QI’2011 will be published by Springer in its Lecture Notes in Computer Science (LNCS) series. Authors will be required to submit a final version 14 days after the conference to reflect the comments made at the conference. We will also consider organizing a special issue for a suitable journal to publish selected best papers.

***Important Dates***

28th March 2011: Abstract submission deadline

1st April 2011: Paper submission deadline

1st May 2011: Notification of acceptance

1st June 2011: Camera-Ready Copy

26th June 2011: Tutorial Session

27th – 29th June 2011: Conference

***Submission***

Authors are invited to submit research papers up to 12 pages. All submissions should be prepared in English using the LNCS template, which can be downloaded from http://www.springer.com/computer/lncs?SGWID=0-164-6-793341-0.

Please submit online at:

http://www.easychair.org/conferences/?conf=qi2011

***Organization***

Steering Committee:

Peter Bruza (Queensland University of Technology, Australia)

William Lawless (Paine College, USA)

Keith van Rijsbergen (University of Glasgow, UK)

Donald Sofge (Naval Research Laboratory, USA)

Dominic Widdows (Google, USA)

General Chair:

Dawei Song (Robert Gordon University, UK)

Programme Committee Chair:

Massimo Melucci (University of Padua, Italy)

Publicity Chair:

Sachi Arafat (University of Glasgow, UK)

Proceedings Chair:

Ingo Frommholz (University of Glasgow, UK)

Local Organization co-Chairs:

Jun Wang and Peng Zhang (Robert Gordon University, UK)

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Quantum Foundations Meetings

Posted on 11 November, 2010 by | Leave a comment

Prompted in part by the Quantum Pontiff’s post about the APS March meeting, I thought it would be a good idea to post one of my extremely irregular lists of interesting conferences about the foundations of quantum theory that are coming up. A lot of my usual sources for this sort of information have become defunct in the couple of years I was away from work, so if anyone knows of any other interesting meetings then please post them in the comments.

  • March 21st-25th 2011: APS March Meeting (Dallas, Texas) – Includes a special session on Quantum Information For Quantum Foundations. Abstract submission deadline Nov. 19th.
  • April 29th-May 1st 2011: New Directions in the Foundations of Physics (Washington DC) – Always one of the highlights of the foundations calendar, but invite only.
  • May 2nd-6th 2011: 5th Feynman Festival (Brazil) – Includes foundations of quantum theory as one of its topics, but likely there will be more quantum information/computation talks. Registration deadline Feb. 1st, Abstract submission deadline Feb. 15th.
  • July 25th-30th 2011: Frontiers of Quantum and Mesoscopic Thermodynamics (Prague, Czech Republic) – Not strictly a quantum foundations conference, but there are a few foundations speakers and foundations of thermodynamics is interesting to many quantum foundations people.

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Time Travel and Information Processing

Posted on 15 September, 2010 by | 9 comments

Lately, the quant-ph section of the arXiv has been aflurry with papers investigating what would happen to quantum information processing if time travel were possible (see the more recent papers here). I am not sure exactly why this topic has become fashionable, but it may well be an example of the Bennett effect in quantum information research. That is, a research topic can meander along slowly at its own pace for a few years until Charlie Bennett publishes an (often important) paper ((Bennett, C. H. et. al. (2009). “Can closed timelike curves or nonlinear quantum mechanics improve quantum state discrimination or help solve hard problems”. Phys. Rev. Lett. 103:170502. eprint arXiv:0908.3023.)) on the subject and then everyone is suddenly talking and writing about it for a couple of years. In any case, there have been a number of counter-intuitive claims that time travel enables quantum information processing to be souped up. Specifically, it supposedly enables super-hard computational problems that are in complexity classes larger than NP to be solved efficiently ((Brun, T. A. and Wilde, Mark M. (2010). “Perfect state distinguishability and computational speedups with postselected closed timelike curves”. eprint arXiv:1008.0433.)) ((Aaronson, S. and Watrous, J. (2009). Closed timelike curves make quantum and classical computing equivalent. Proc. R. Soc. A 465:631-647. eprint arXiv:0808.2669.)) ((Bacon, D. (2004). Quantum Computational Complexity in the Presence of Closed Timelike Curves. Phys. Rev. A 70:032309. eprint arXiv:quant-ph/0309189.)) ((Brun, T. A. (2003). Computers with closed timelike curves can solve hard problems. Found. Phys. Lett. 16:245-253. eprint arXiv:gr-qc/0209061.)) and it supposedly allows nonorthogonal quantum states to be perfectly distinguished ((ref:2)) ((Brun, Todd A., Harrington, J. and Wilde, M. M. (2009). “Localized closed timelike curves can perfectly distinguish quantum states”. Phys. Rev. Lett. 102:210402. eprint arXiv:0811.1209.)). These claims are based on two different models for quantum time-travel, one due to David Deutsch ((Deutsch, D. (1991). “Quantum mechanics near closed timelike lines”. Phys. Rev. D 44:3197—3217.)) and one due to a multitude of independent authors based on post-selected teleportation (this paper ((Lloyd, S. et. al. (2010). “The quantum mechanics of time travel through post-selected teleportation”. eprint arXiv:1007.2615)) does a good job of the history in the introduction).

In this post, I am going to give a basic introduction to the physics of time-travel. In later posts, I will explain the Deutsch and teleportation-based models and evaluate the information processing claims that have been made about them. What is most interesting to me about this whole topic, is that the correct model for time travelling quantum systems, and hence their information processing power, seems to depend sensitively on both the formalism and the interpretation of quantum theory that is adopted ((I should mention that Joseph Fitzsimons (@jfitzsimons) disagreed with this statement in our Twitter conversations on this subject, and no doubt many physicists would too, but I hope to convince you that it is correct by the end of this series of posts.)). For this reason, it is a useful test-bed for ideas in quantum foundations.

Basic Concepts of Time-Travel

Everyone is familiar with the sensation of time-travel into the future. We all do it at a rate of one second per second every day of our lives. If you would like to speed up your rate of future time travel, relative to Earth, then all you have to do is take a space trip at a speed close to the speed of light. When you get back, a lot more time will have elapsed on Earth than you will have experienced on your journey. This is the time-dilation effect of special relativity. Therefore, the problem of time-travel into the future is completely solved in theory, although in practice you would need a vast source of energy in order to accelerate yourself fast enough to make the effect significant. It also causes no conceptual problems for physics, since we have a perfectly good framework for quantum theories that are compatible with special relativity, known as quantum field theory.

On the other hand, time travel into the past is a much more tricky and conceptually interesting proposition. For one thing, it seems to entail time-travel paradoxes, such as the grandfather paradox where you go back in time and kill your grandfather before your parents were born, so that you are never born, so that you cannot go back in time and kill your grandfather, so that you are born, so that you can go back in time and kill your grandfather etc. (see this article for a philosophical and physics-based discussion of time travel paradoxes). For this reason, many physicists are highly sceptical of the idea that time travel into the past is possible. However, General Relativity (GR) provides a reason to temper our skepticism.

Closed Timelike Curves in GR

It has been well-known for a long time that GR admits solutions that include closed timelike curves (CTCs), i.e. world-lines that return to their starting point and loop around. If you happened to be travelling along a CTC then you would eventually end up in the past of where you started from. Actually, it is a bit more complicated than that because the usual notions of past and future do not really make sense on a CTC. However, imagine what it would look like to an observer in a part of the universe that respects causality in the usual sense. First of all, she would see you appear out of nowhere, claiming to have knowledge of events that she regards as being in the future. Some time later she would see you disappear out of existence. From her perspective it certainly looks like time-travel into the past. What things would feel like from your point of view is more of a mystery, as the notion of a CTC makes a mockery of our usual notion of “now”, i.e. it is a fundamentally block-universe construct.

The possibility of CTCs in GR was first noticed by Willem van Stockum in 1937 ((Stockum, W. J. van (1937). “The gravitational field of a distribution of particles rotating around an axis of symmetry”. Proc. Roy. Soc. Edinburgh A 57: 135.)) and later by Kurt Gödel in 1949 ((Kurt Gödel (1949). “An Example of a New Type of Cosmological Solution of Einstein’s Field Equations of Gravitation”. Rev. Mod. Phys. 21: 447.)). Perhaps the most important solution that incorporates CTCs is the Kerr vacuum, which is the solution that describes an uncharged rotating black hole. Since most black holes in the universe are likely to be rotating, there is a sense in which one can say that CTCs are generic. The caveat is that the CTCs in the Kerr vacuum only occur in the interior of the black hole so that the physics outside the event horizon respects causality in the usual sense. Many physicists believe that the CTCs in the Kerr vacuum are mathematical artifacts, which will perhaps not occur in a full theory of quantum gravity. Nevertheless, the conceptual possibility of CTCs in General Relativity is a good reason to look at their physics more closely.

There have been a few attempts to look for solutions of GR that incorporate CTCs that a human being would actually be able to travel along without getting torn to pieces. This is a bit beyond my current knowledge, but, as far as I am aware, all such solutions involve large quantities of negative energy, so they are unlikely to exist in nature and it is unlikely that we can construct them artificially. For this reason, CTCs are currently more of a curiosity for foundationally inclined physicists like myself than they are a practical method of time-travel.

Other Retrocausal Effects in Physics

Apart from GR, other forms of backwards-in-time, or retrocausal, effect have been proposed in physics from time to time. For example, there is the Wheeler-Feynman absorber theory of electrodynamics, which postulates a backwards-in-time propagating field in addition to the usual forwards-in-time propagating field, and Feynman also postulated that positrons might be electrons travelling backwards in time. There is also Cramer’s transactional interpretation of quantum theory ((Cramer, J. G. (1986). “The transactional interpretation of quantum mechanics”. Rev. Mod. Phys. 58:647-687.)), which does a similar thing with quantum wavefunctions, and the distinct, but conceptually similar, two-state vector formalism of Aharonov and collaborators ((Aharonov, Y. and Vaidman, L. (2001). “The Two-State Vector Formalism of Quantum Mechanics: An Updated Review”. in “Time in Quantum Mechanics”, Muga, J. G., Sala Mayato, R. and Egusquiza, I. L. eprint arXiv:quant-ph/0105101.)). Finally, retrocausal influences have been suggested as a mechanism to reproduce the violations of Bell-inequalities in quantum theory without the need for Lorentz-invariance violating nonlocal influences ((For example, see Price, H. (1997). “Time’s Arrow and Archimedes’ Point”. OUP.)).

However, none of these proposals are as compelling an argument for taking the physics of time-travel into the past seriously as the existence of CTCs in General Relativity. This is because, none of these theories gives provides a method for exploiting the retrocausal effect to actually travel back in time. Also, in each case, there is an alternative approach to the same phenomena that does not involve retrocausal influences. Nevertheless, it is possible that the models to be discussed have applications to these alternative approaches to physics.

Consistency Constraints and The Interpretation of Quantum Theory

Any viable theory of time travel into the past has to rule out things like the grandfather paradox. Consistency conditions have to be imposed on any physical model to so that time-travel cannot be used to alter the past. This raises interesting questions about free will, e.g. what exactly stops someone from freely deciding to pull the trigger on their grandfather? Whilst these questions are philosophically interesting, physicists are more inclined to just lay out the mathematics of consistency and see what it leads to. The different models of quantum time travel are essentially just different methods of imposing this sort of consistency constraint on quantum systems.

That is pretty much it for the basic introduction, but I want to leave you with a quick thought experiment to illustrate the sort of quantum foundational issues that come up when considering time-travel into the past. Suppose you prepare a spin-\(\frac{1}{2}\) particle in a spin up state in the z direction and then measure it in the x direction, so that it has a 50-50 chance of giving the spin up or spin down outcome. After observing the outcome you jump onto a CTC, travel back into the past and watch yourself perform the experiment again. The question is, would you see the experiment have the same outcome the second time around?

A consistency condition for time travel has to say something like “the full ontic state (state of things that exist in reality) of the universe must be the same the second time round as it was the first time round”, albeit that your subjective position within it has changed. If you believe, as many-worlds supporters do, that the quantum wavefunction is the complete description of reality then it, and only it, must be the same the second time around. Therefore, it must be the case that the probabilities are still 50-50 and you could see either outcome. This is not inconsistent because the many-worlds supporters believe that both outcomes happened the first time round in any case. If you are a Bohmian then the ontic state includes the positions of all particles in addition to the wavefunction and these, taken together, can be used to determine the outcome of the experiment uniquely. Therefore, a Bohmian must believe that the measurement outcome has to be the same the second time around. Finally, if you are some sort of anti-realist neo-Copenhagen type then it is not clear exactly what you believe, but, then again, it is not clear exaclty what you believe even when there is no time-travel.

There are some subtleties in these arguments. For example, it is not clear what happens to the correlations between you and the observed system when you go around the causal loop. If they still exist then this may restrict the ability of the earlier version of you to prepare a pure state. On the other hand, perhaps they get wiped out or perhaps your memory of the outcome gets wiped. The different models for the quantum physics of CTCs differ on how they handle this sort of issue, and this is what I will be looking at in future posts. If you have travelled along a CTC and happen to have brought a copy of these future posts with you then I would be very grateful if you could email them to me because that would be much easier for me than actually writing them.

‘Till next time!

References

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Quantum Foundations Resources

Posted on 14 May, 2007 by | Leave a comment

Since I get asked a lot, I have added a collection of links to resources on quantum foundations to the About page.  Any suggestions for additions will be gratefully received, especially if you know of any good quality popular talks that can be viewed online.

P.S.  In case you were thinking of asking, neither “The Tao of Physics” or “What The Bleep Do We Know?” are ever going to be added.

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Baez on Quantum Foundations

Posted on 11 May, 2007 by | Leave a comment

I just wrote another post on the fqxi site, but to cut a long story short it gives a link to the latest “This Week’s Finds..” on quantum foundations.

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Foundations at APS, take 2

Posted on 6 March, 2007 by | 3 comments

It doesn’t seem that a year has gone by since I wrote about the first sessions on quantum foundations organized by the topical group on quantum information, concepts and computation at the APS March meeting. Nevertheless it has, and I am here in Denver after possibly the longest day of continuous sitting through talks in my life. I arrived at 8am to chair the session on Quantum Limited Measurements, which was interesting, but readers of this blog won’t want to hear about such practical matters, so instead I’ll spill the beans on the two foundations sessions that followed.

In the first foundations session, things got off to a good start with Rob Spekkens as the invited speaker explaining to us once again why quantum states are states of knowledge. OK, I’m biased because he’s a collaborator, but he did throw us a new tidbit on how to make an analog of the Elitzur Vaidman bomb experiment in his toy theory by constructing a version for field theory.

Next, there was a talk by some complete crackpot called Matt Leifer. He talked about this.

Frank Schroeck gave an overview of his formulation of quantum mechanics on phase space, which did pique my interest, but 10 minutes was really too short to do it justice. Someday I’ll read his book.

Chris Fuchs gave a talk which was surprisingly not the same as his usual quantum Bayesian propaganda speech. It contained some new results about Symmetric Informationally Complete POVMs, including the fact that the states the POVM elements are proportional to are minimum uncertainty states with respect to mutually unbiased bases. This should be hitting an arXiv near you very soon.

Caslav Brukner talked about his recent work on the emergence of classicality via coarse graining. I’ve mentioned it before on this blog, and it’s definitely a topic I’m becoming much more interested in.

Later on, Jeff Tollaksen talked about generalizing a theorem proved by Rob Spekkens and myself about pre- and post-selected quantum systems to the case of weak measurements. I’m not sure I agree with the particular spin he gives on it, especially his idea of “quantum contextuality”, but you can decide for yourself by reading this.

Jan-Ake Larrson gave a very comprehensible talk about a “loophole” (he prefers the term “experimental problem”) in Bell inequality tests to do with coincidence times of photon detection. You can deal with it by having a detection efficiency just a few percent higher than that needed to overcome the detection loophole. Read all about it here.

Most of the rest of the talks in this session were more quantum information oriented, but I suppose you can argue they were at the foundational end of quantum information. Animesh Datta talked about the role of entanglement in the Knill-Laflamme model of quantum computation with one pure qubit, Anil Shaji talked about using easily computable entanglement measures to put bounds on those that aren’t so easy to compute and finally Ian Durham made some interesting observations about the connections between entropy, information and Bell inequalities.

The second foundations session was more of a mixed bag, but let me just mention a couple of the talks that appealed to me. Marcello Sarandy Alioscia Hamma talked about generalizing the quantum adiabatic theorem to open systems, where you don’t necessarily have a Hamiltonian with well-defined eigenstates to talk about and Kicheon Kang talked about a proposal for a quantum eraser experiment with electrons.

On Tuesday, Bill Wootters won a prize for best research at an undergraduate teaching college. He gave a great talk about his discrete Wigner functions, which included some new stuff about minumum uncertainty states and analogs of coherent states.

That’s pretty much it for the foundations talks at APS this year. It’s all quantum information from here on in. That is unless you count Zeilinger, who is talking on Thursday. He’s supposed to be talking about quantum cryptography, but perhaps he will say something about the more foundationy experiments going on in his lab as well.

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What can decoherence do for us?

Posted on 24 January, 2007 by | 12 comments

OK, so it’s time for the promised post about decoherence, but where to begin? Decoherence theory is now a vast subject with an enormous literature covering a wide variety of physical systems and scenarios. I will not deal with everything here, but just make some comments on how the theory looks from my point of view about the foundations of quantum theory. Alexei Grinbaum pointed me to a review article by Maximilian Schlosshauer on the role of decoherence in solving the measurement problem and in interpretations of quantum theory. That’s a good entry into the literature for people who want to know more.

OK, let me start by defining two problems that I take to be at the heart of understanding quantum theory:

1) The Emergence of Classicality: Our most fundamental theories of the world are quantum mechanical, but the world appears classical to us at the everyday level. Explain why we do not find ourselves making mistakes in using classical theories to make predictions about the everyday world of experience. By this I mean not only classical dynamics, but also classical probability theory, information theory, computer science, etc.

2) The ontology problem: The mathematical formalism of quantum theory provides an algorithm for computing the probabilities of outcomes of measurements made in experiments. Explain what things exist in reality and what laws they obey in such a way as to account for the correctness of the predictions of the theory.

I take these to be the fundamental challenges of understanding quantum mechanics. You will note that I did not mention the measurement problem, Schroedinger’s cat, or the other conventional ways of expressing the foundational challenges of quantum theory. This is because, as I have argued before, these problems are not interpretation neutral. Instead, one begins with something like the orthodox interpretation and shows that unitary evolution and the measurement postulates are in apparent conflict within that interpretation depending on whether we choose to view the measuring apparatus as a physical system obeying quantum theory or to leave it unanalysed. The problems with this are twofold:

i) It is not the case that we cannot solve the measurement problem. Several solutions exist, such as the account given by Bohmian mechanics, that of Everett/many-worlds, etc. The fact that there is more than one solution, and that none of them have been found to be universally compelling, indicates that it is not solving the measurement problem per se that is the issue. You could say that it is solving the measurement problem in a compelling way that is the issue, but I would say it is better to formulate the problem in such a way that it is obvious how it applies to each of the different interpretations.

ii) The standard way of describing the problems essentially assumes that the quantum state-vector corresponds more or less directly to whatever exists in reality, and that it is in fact all that exists in reality. This is an assumption of the orthodox interpretation, so we are talking about a problem with the standard interpretation and not with quantum theory itself. Assuming the reality of the state-vector simply begs the question. What if it does not correspond to an element of reality, but is just an epistemic object with a status akin to a probability distribution in classical theories? This is an idea that I favor, but now is not the time to go into detailed arguments for it. The mere fact that it is a possibility, and is taken seriously by a significant section of the foundations community, means that we should try to formulate the problems in a language that is independent of the ontological status of the state-vector.

Given this background viewpoint, we can now ask to what extent decoherence can help us with 1) and 2), i.e. the emergence and ontology problems. Let me begin with a very short description of what decoherence is in this context. The first point is that it takes seriously the idea that quantum systems, particularly the sort that we usually describe as “classical”, are open, i.e. interact strongly with a large environment. Correlations between system and environment are typically established very quickly in some particular basis, determined by the form of the system-environment interaction Hamiltonain, so that the density matrix of the system quickly becomes diagonal in that basis. Furthermore, the basis in which the correlations exist is stable over a very long period of time, which can typically be much longer than the lifetime of the universe. Finally, for many realistic Hamiltonians and a wide variety of systems, the decoherence basis corresponds very well to the kind of states we actually observe.

From my point of view, the short answer to the role of decoherence in foundations is that it provides a good framework for addressing emergence, but has almost nothing to say about ontology.  The reason for saying that should be clear:  we have a good correspondence with our observations, but at no point in my description of decoherence did I find it necessary to mention a reality underlying quantum mechanics.  Having said that, a couple of caveats are in order. Firstly, decoherence can do much more if it is placed within a framework with a well defined ontology. For example, in Everett/many-worlds, the ontology is the state-vector, which always evolves unitarily and never collapses. The trouble with this is that the ontology doesn’t correspond to our subjective experience, so we need to supplement it with some account of why we see collapses, definite measurement outcomes, etc. Decoherence theory does a pretty good job of this by providing us with rules to describe this subjective experience, i.e. we will experience the world relative to the basis that decoherence theory picks out. However, the point here is that the work is not being done by decoherence alone, as claimed by some physicists, but also by a nontrivial ontological assumption about the state-vector. As I remarked earlier, the latter is itself a point of contention, so it is clear that decoherence alone is not providing a complete solution.

The second caveat, is that some people, including Max Schlosshauer in his review, would argue for plausible denial of the need to answer the ontology question at all. So long as we can account for our subjective experience in a compelling manner then why should we demand any more of our theories? The idea is then that decoherence can solve the emergence problem, and then we are done because the ontology problem need not be solved at all. One could argue for this position, but to do so is thoroughly wrongheaded in my opinion, and this is so independently of my conviction that physics is about trying to describe a reality that goes beyond subjective experience. The simple point is that someone who takes this view seriously really has no need for decoherence theory at all. Firstly, given that we are not assigning ontological status to anything, let alone the state-vector, then you are free to collapse it, uncollapse it, evolve it, swing it around your head or do anything else you like with it. After all, if it is not supposed to represent anything existing in reality then there need not be any physical consequences for reality of any mathematical manipulation, such as a projection, that you might care to do. The second point is that if we are prepared to give a privelliged status to observers in our physical theories, by saying that physics needs to describe their experience and nothing more, then we can simply say that the collapse is a subjective property of the observer’s experience and leave it at that. We already have privelliged systems in our theory on this view, so what extra harm could that do?

Of course, I don’t subscribe to this viewpoint myself, but on both views described so far, decoherence theory either needs to be supplemented with an ontology, or is not needed at all for addressing foundational issues.

Finally, I want to make a couple of comments about how odd the decoherence solution looks from my particular point of view as a believer in the epistemic nature of wavefunctions. The first is that, from this point of view, the decoherence solution appears to have things backwards. When constructing a classical probabilistic theory, we first identify the ontological entities, e.g. particles that have definite trajectories, and describe their dynamics, e.g. Hamilton’s equations. Only then do we introduce probabilities and derive the corresponding probabilistic theory, e.g. Liouville mechanics. Decoherence theory does things in the other direction, starting from Schroedinger mechanics and then seeking to define the states of reality in terms of the probabilistic object, i.e. the state-vector. Whilst this is not obviously incorrect, since we don’t necessarily have to do things the same way in classical and quantum theories, it does seem a little perverse from my point of view. I’d rather start with an ontology and derive the fact that the state-vector is a good mathematical object for making probabilistic predictions, instead of the other way round.

The second comment concerns an analogy between the emergence of classicality in QM and the emergence of the second law of thermodynamics from statistical mechanics. For the latter, we have a multitude of conceptually different approaches, which all arrive at somewhat similar results from a practical point of view. For a state-vector epistemist like myself, the interventionist approach to statistical mechanics seems very similar to the decoherence approach to the emergence problem in QM. Both say that the respective problems cannot be solved by looking at a closed Hamiltonian system, but only by considering interaction with a somewhat uncontrollable environment. In the case of stat-mech, this is used to explain the statistical fluctuations observed in what would be an otherwise deterministic system. The introduction of correlations between system and environment is the mechanism behind both processes. Somewhat bizzarely, most physicists currently prefer closed-system approaches to the derivation of the second law, based on coarse-graining, but prefer the decoherence approach when it comes to the emergence of classicality from quantum theory. Closed system approaches have the advantage of being applicable to the universe as a whole, where there is no external environment to rely on. However, apart from special cases like this, one can broadly say that the two types of approach are complimentary for stat mech, and neither has a monopoly on explaining the second law. It is then natural to ask whether closed system approaches to emergence in QM are available making use of coarse graining, and whether they ought to be given equal weight to the decoherence explanation. Indeed, such arguments have been given – here is a recent example, which has many precursors too numerous to go through in detail. I myself am thinking about a similar kind of approach at the moment. Right now, such arguments have a disadvantage over decoherence in that the “measurement basis” has to be put in by hand, rather than emerging from the physics as in decoherence. However, further work is needed to determine whether this is an insurmountable obstacle.

In conclusion, decoherence theory has done a lot for our understanding of the emergence of classicality from quantum theory. However, it does not solve all the foundational queations about quantum theory, at least not on it’s own. Further, its importance may have been overemphasized by the physics community because other less-developed approaches to emergence could turn out to be of equal importance.

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Steane Roller

Posted on 15 December, 2006 by | 3 comments

Earlier, I promised some discussion of Andrew Steane‘s new paper: Context, spactime loops, and the interpretation of quantum mechanics. Whilst it is impossible to summarize everything in the paper, I can give a short description of what I think are the most important points.

  • Firstly, he does believe that the whole universe obeys the laws of quantum mechanics, which are not required to be generalized.
  • Secondly, he does not think that Everett/Many-Worlds is a good way to go because it doesn’t give a well-defined rule for when we see one particular outcome of a measurement in one particular basis.
  • He believes that collapse is a real phenomenon and so the problem is to come up with a rule for assigning a basis in which the wavefunction collapses, as well as, roughly speaking, a spacetime location at which it occurs.
  • For now, he describes collapse as an unanalysed fundamenally stochastic process that achieves this, but he recognizes that it might be useful to come up with a more detailed mechanism by which this occurs.

Steane’s problem therefore reduces to picking a basis and a spacetime location. For the former, he uses the standard ideas from decoherence theory, i.e. the basis in which collapse occurs is the basis in which the reduced state of the system is diagonal. However, the location of collapse is what is really interesting about the proposal, and makes it more interesting and more bizzare than most of the proposals I have seen so far.

Firstly, note that the process of collapse destroys the phase information between the system and the environment. Therefore, if the environmental degrees of freedom could ever be gathered together and re-interacted with the system, then QM would predict interference effects that would not be present if a genuine collapse had occurred. Since Steane believes in the universal validity of QM, he has to come up with a way of having a genuine collapse without getting into a contradiction with this possibility.

His first innovation is to assert that the collapse need not be associated to an exactly precise location in spacetime. Instead, it can be a function of what is going on in a larger region of spacetime. Presumably, for events that we would normally regard as “classical” this region is supposed to be rather small, but for coherent evolutions it could be quite large.

The rule is easiest to state for special cases, so for now we will assume that we are talking about particles with a discrete quantum degree of freedom, e.g. spin, but that the position and momentum can be treated classically. Now, suppose we have 3 qubits and that they are in the state |000> + e^i phi |111>. The state of the first two qubits is a density operator, diagonal in the basis {|00>, |11>}, with a probability 1/2 for each of the two states. The phase e^i phi will only ever be detectable if the third qubit re-interacts with the first two. Whether or not this can happen is determined by the relative locations of the qubits, since the interaction Hamiltonias in nature are local. Since we are treating position and momentum classically at the moment, there is a matter of fact about whether this will occur and Steane’s rule is simple: if the qubits re-interact in the future then there is no collapse, but if they don’t then the then the first two qubits have collapsed into the state |00> or the state |11> with probability 1/2 for each one.

Things are going to get more complicated if we quantize the position and momentum, or indeed if we move to quantum field theory, since then we don’t have definite particle trajectories to work with. It is not entirely clear to me whether Steane’s proposal can be made to work in the general case, and he does admit that further technical work is needed. However, he still asserts that whether or not a system has collapsed at a given point is spacetime is in principle a function of its entire future, i.e. whether or not it will eventually re-interact with the environment it has decohered with respect to.

At this point, I want to highlight a bizzare physical prediction that can be made if you believe Steane’s point of view. Really, it is metaphysics, since the experiment is not at all practical. For starters, the fact that I experience myself being in a definite state rather than a superposition means that there are environmental degrees of freedom that I have interacted with in the past that have decohered me into a particular basis. We can in principle imagine an omnipotent “Maxwell’s demon” type character, who can collect up every degree of freedom I have ever interacted with, bring it all together and reverse the evolution, eliminating me in the process. Whilst this is impractical, there is nothing in principle to stop it happening if we believe that QM applies to the entire universe. However, according to Steane, the very fact that I have a definite experience means that we can predict with certainty that no such interaction happens in the future. If it did, there would be no basis for my definite experience at the moment.

Contrast this with a many-worlds account a la David Wallace. There, the entire global wavefunction still exists, and the fact that I experience the world in a particular basis is due to the fact that only certain special bases, the ones in which decoherence occurs, are capable of supporting systems complex enough to achieve conciousness. There is nothing in this view to rule out the Maxwell’s demon conclusively, although we may note that he is very unlikely to be generated by a natural process due to the second law of thermodynamics.

Therefore, there is something comforting about Steane’s proposal. If true, my very existence can be used to infer that I will never be wiped out by a Maxwell’s demon. All we need to do to test the theory is to try and wipe out a conscious being by constructing such a demon, which is obviously impractical and also unethical. Needless to say, there is something troubling about drawing such a strong metaphysical conclusion from quantum theory, which is why I still prefer the many-worlds account over Steane’s proposal at the moment. (That’s not to say that I agree with the former either though.)

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