# Thinking about Robert Wald's take on the loss, or not, of information into black holes

A warning to readers: As far as physics goes, I tend to use this blog to muse out loud about things I am trying to understand better, rather than to provide lapidary intuitive summaries for the enlightenment of a general audience on matters I am already expert on. Musing out loud is what's going on in this post, for sure. I will try, I'm sure not always successfully, not to mislead, but I'll be unembarassed about admitting what I don't know.

I recently did a first reading (so, skipped and skimmed some, and did not follow all calculations/reasoning) of Robert Wald's book "Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics".  I like Wald's style --- not too lengthy, focused on getting the important concepts and points across and not getting bogged down in calculational details, but also aiming for mathematical rigor in the formulation of the important concepts and results.

Wald uses the algebraic approach to quantum field theory (AQFT), and his approach to AQFT involves looking at the space of solutions to the classical equations of motion as a symplectic manifold, and then quantizing from that point of view, in a somewhat Dirac-like manner (the idea is that Poisson brackets, which are natural mathematical objects on a symplectic manifold, should go to commutators $[p_j, q_k] = \delta_{qk} i \hbar$ between generalized positions and momenta, but what is actually used is the Weyl form $e^{i \sigma p_j} e^{i \tau q_k} = e^{i \delta_{qk} \sigma \tau} e^{i \tau q_k} e^{i \sigma p_j}$ of the commutation relations), doing the Minkowski-space (special relativistic, flat space) version before embarking on the curved-space, (semiclassical general relativistic) one.   He argues that this manner of formulating quantum field theory has great advantages in curved space, where the dependence of the notion of "particle" on the reference frame can make quantization in terms of an expansion in Fourier modes of the field ("particles") problematic.  AQFT gets somewhat short shrift among mainstream quantum field theorists, I sense, in part because (at least when I was learning about it---things may have changed slightly, but I think not that much) no-one has given a rigorous mathematical example of an algebraic quantum field theory of interacting (as opposed to freely propagating) fields in a spacetime with three space dimensions.  (And perhaps the number of AQFT's that have been constructed even in fewer space dimensions is not very large?).  There is also the matter pointed out by Rafael Sorkin, that when AQFT's are formulated, as is often done, in terms of a "net" of local algebras of observables (each algebra associated with an open spacetime region, with compatibility conditions defining what it means to have a "net" of algebras on a spacetime, e.g. the subalgebra corresponding to a subset of region R is a subalgebra of the algebra for region R; if two subsets of a region R are spacelike separated then their corresponding subalgebras commute), the implicit assumption that every Hermitian operator in the algebra associated with a region can be measured "locally"  in that region actually creates difficulties with causal locality---since regions are extended in spacetime, coupling together measurements made in different regions through perfectly timelike classical feedforward of the results of one measurement to the setting of another, can create spacelike causality (and probably even signaling).  See Rafael's paper "Impossible measurements on quantum fields".   (I wonder if that is related to the difficulties in formulating a consistent interacting theory in higher spacetime dimension.)

That's probably tangential to our concerns here, though, because it appears we can understand the basics of the Hawking effect, of radiation by black holes, leading to black-hole evaporation and the consequent worry about "nonunitarity" or "information loss" in black holes, without needing a quantized interacting field theory.  We treat spacetime, and the matter that is collapsing to form the black hole, in classical general relativistic terms, and the Hawking radiation arises in the free field theory of photons in this background.

I liked Wald's discussion of black hole information loss in the book.  His attitude is that he is not bothered by it, because the spacelike hypersurface on which the state is mixed after the black hole evaporates (even when the states on similar spacelike hypersurfaces before black hole formation are pure) is not a Cauchy surface for the spacetime.  There are non-spacelike, inextensible curves that don't intersect that hypersurface.  The pre-black-hole spacelike hypersurfaces on which the state is pure are, by contrast, Cauchy surfaces---but some of the trajectories crossing such an initial surface go into the black hole and hit the singularity, "destroying" information.  So we should not expect purity of the state on the post-evaporation spacelike hypersurfaces any more than we should expect, say, a pure state on a hyperboloid of revolution contained in a forward light-cone in Minkowski space --- there are trajectories that never intersect that hyperboloid.

Wald's talk at last year's firewall conference is an excellent presentation of these ideas; most of it makes the same points made in the book, but with a few nice extra observations. There are additional sections, for instance on why he thinks black holes do form (i.e. rejects the idea that a "frozen star" could be the whole story), and dealing with anti de sitter / conformal field theory models of black hole evaporation. In the latter he stresses the idea that early and late times in the boundary CFT do not correspond in any clear way to early and late times in the bulk field theory (at least that is how I recall it).

I am not satisfied with a mere statement that the information "is destroyed at the singularity", however.  The singularity is a feature of the classical general relativistic mathematical description, and near it the curvature becomes so great that we expect quantum aspects of spacetime to become relevant.  We don't know what happens to the degrees of freedom inside the horizon with which variables outside the horizon are entangled (giving rise to a mixed state outside the horizon), once they get into this region.  One thing that a priori seems possible is that the spacetime geometry, or maybe some pre-spacetime quantum (or post-quantum) variables that underly the emergence of spacetime in our universe (i.e. our portion of the universe, or multiverse if you like) may go into a superposition (the components of which have different values of these inside-the-horizon degrees of freedom that are still correlated (entangled) with the post-evaporation variables). Perhaps this is a superposition including pieces of spacetime disconnected from ours, perhaps of weirder things still involving pre-spacetime degrees of freedom.  It could also be, as speculated by those who also speculate that the state on the post-evaporation hypersurface in our (portion of the) universe is pure, that these quantum fluctuations in spacetime somehow mediate the transfer of the information back out of the black hole in the evaporation process, despite worries that this process violates constraints of spacetime causality.  I'm not that clear on the various mechanisms proposed for this, but would look again at the work of Susskind, and Susskind and Maldacena ("ER=EPR") to try to recall some of the proposals. (My rough idea of the "ER=EPR" proposals is that they want to view entangled "EPR" ("Einstein-Podolsky-Rosen") pairs of particles, or at least the Hawking radiation quanta and their entangled partners that went into the black hole, as also associated with miniature "wormholes" ("Einstein-Rosen", or ER, bridges) in spacetime connecting the inside to the outside of the black hole; somehow this is supposed to help out with the issue of nonlocality, in a way that I might understand better if I understood why nonlocality threatens to begin with.)

The main thing I've taken from Wald's talk is a feeling of not being worried by the possible lack of unitarity in the transformation from a spacelike pre-black-hole hypersurface in our (portion of the) universe to a post-black-hole-evaporation one in our (portion of the) universe. Quantum gravity effects at the singularity either transfer the information into inaccessible regions of spacetime ("other universes"), leaving (if things started in a pure state on the pre-black-hole surface) a mixed state on the post-evaporation surface in our portion of the universe, but still one that is pure in some sense overall, or they funnel it back out into our portion of the universe as the black hole evaporates. It is a challenge, and one that should help stimulate the development of quantum gravity theories, to figure out which, and exactly what is going on, but I don't feel any strong a priori compulsion toward one or the other of a unitary or a nonunitary evolution on from pre-black-hole to post-evaporation spacelike hypersurfaces in our portion of the universe.

# Quantum imaging with entanglement and undetected photons, II: short version

Here's a short explanation of the experiment reported in "Quantum imaging with undetected photons" by members of Anton Zeilinger's group in Vienna (Barreta Lemos, Borish, Cole, Ramelow, Lapkiewicz and Zeilinger).  The previous post also explains the experiment, but in a way that is closer to my real-time reading of the article; this post is cleaner and more succinct.

It's most easily understood by comparison to an ordinary Mach-Zehnder interferometry experiment. (The most informative part of the wikipedia article is the section "How it works"; Fig. 3 provides a picture.)  In this sort of experiment, photons from a source such as a laser encounter a beamsplitter and go into a superposition of being transmitted and reflected.  One beam goes through an object to be imaged, and acquires a phase factor---a complex number of modulus 1 that depends on the refractive index of the material out of which the object is made, and the thickness of the object at the point at which the beam goes through.  You can think of this complex number as an arrow of length 1 lying in a two-dimensional plane; the arrow rotates as the photon passes through material, with the rate of rotation depending on the refractive index of the material. (If the thickness and/or refractive index varies on a scale smaller than the beamwidth, then the phase shift may vary over the beam cross-section, allowing the creation of an image of how the thickness of the object---or at least, the total phase imparted by the object, since the refractive index may be varying too---varies in the plane transverse to the beam.  Otherwise, to create an image rather than just measure the total phase it imparts at a point, the beam may need to be scanned across the object.)  The phase shift can be detected by recombining the beams at the second beamsplitter, and observing the intensity of light in each of the two output beams, since the relative probability of a photon coming out one way or the other depends on the relative phase of the the two input beams; this dependence is called "interference".

Now open the homepage of the Nature article and click on Figure 1 to enlarge it.  This is a simplified schematic of the experiment done in Vienna.  Just as in ordinary Mach-Zehnder interferometry, a beam of photons is split on a beamsplitter (labeled BS1 in the figure).  One can think of each photon from the source going into a superposition of being reflected and transmitted at the first beamsplitter.  The transmitted part is downconverted by passing through the nonlinear crystal NL1 into an entangled pair consisting of a yellow and a red photon; the red photon is siphoned off by a dichroic (color-dependent) beamsplitter, D1, and passed through the object O to be imaged, acquiring a phase dependent on the refractive index of the object and its thickness.  The phase, as I understand things, is associated with the photon pair even though it is imparted by the passing only the red photon through the object.  In order to observe the phase via interferometry, one needs to involve both the red and yellow photon, coherently.  (If one could observe it as soon as it was imparted to the pair by just interacting with the yellow photon, one could send a signal from the interaction point to the yellow part of the beam instantaneously, violating relativity.)   The red part of the beam is then recombined (at dichroic beamsplitter D2) with the reflected portion of the beam (which is still at the original wavelength), and that portion of the beam is passed through another nonlinear crystal, NL2.  This downconverts the part of the beam that is at the original wavelength into a red-yellow pair, with the resulting red component aligned with --- and indistinguishable from---the red component that has gone through the object.  The phase associated with the photon pair created in the transmitted part of the beam whose red member went through the object is now associated with the yellow photons in the transmitted beam, since the red photons in that beam have been rendered indistinguishable from the ones created in the reflected beam, and so retain no information about the relative phase.  This means that the phase can be observed siphoning out the red photons (at dichroic beamsplitter D3), recombining just the yellow photons with a beamsplitter BS2, and observing the intensitities at the two outputs of this final beamsplitter, precisely as in the last stage of an ordinary Mach-Zehnder experiment.  The potential advantage over ordinary Mach-Zehnder interferometry is that one can image the total phase imparted by the object at a wavelength different from the wavelength of the photons that are interfered and detected at the final stage, which could be an advantage for instance if good detectors are not available at the wavelength one wants to image the object at.

# Quantum imaging with entanglement and undetected photons in Vienna

[Update 9/1:  I have been planning (before any comments, incidentally) to write a version of this post which just provides a concise verbal explanation of the experiment, supplemented perhaps with a little formal calculation.  However, I think the discussion below comes to a correct understanding of the experiment, and I will leave it up as an example of how a physicist somewhat conversant with but not usually working in quantum optics reads and quickly comes to a correct understanding of a paper.  Yes, the understanding is correct even if some misleading language was used in places, but I thank commenter Andreas for pointing out the latter.]

Thanks to tweeters @AtheistMissionary and @robertwrighter for bringing to my attention this experiment by a University of Vienna group (Gabriela Barreto Lemos, Victoria Borish, Garrett D. Cole, Sven Ramelo, Radek Lapkiewicz and Anton Zeilinger), published in Nature, on imaging using entangled pairs of photons.  It seems vaguely familiar, perhaps from my visit to the Brukner, Aspelmeyer and Zeilinger groups in Vienna earlier this year;  it may be that one of the group members showed or described it to me when I was touring their labs.  I'll have to look back at my notes.

This New Scientist summary prompts the Atheist and Robert to ask (perhaps tongue-in-cheek?) if it allows faster-than-light signaling.  The answer is of course no. The New Scientist article fails to point out a crucial aspect of the experiment, which is that there are two entangled pairs created, each one at a different nonlinear crystal, labeled NL1 and NL2 in Fig. 1 of the Nature article.  [Update 9/1: As I suggest parenthetically, but in not sufficiently emphatic terms, four sentences below, and as commenter Andreas points out,  there is (eventually) a superposition of an entangled pair having been created at different points in the setup; "two pairs" here is potentially misleading shorthand for that.] To follow along with my explanation, open the Nature article preview, and click on Figure 1 to enlarge it.  Each pair is coherent with the other pair, because the two pairs are created on different arms of an interferometer, fed by the same pump laser.  The initial beamsplitter labeled "BS1" is where these two arms are created (the nonlinear crystals come later). (It might be a bit misleading to say two pairs are created by the nonlinear crystals, since that suggests that in a "single shot" the mean photon number in the system after both nonlinear crystals  have been passed is 4, whereas I'd guess it's actually 2 --- i.e. the system is in a superposition of "photon pair created at NL1" and "photon pair created at NL2".)  Each pair consists of a red and a yellow photon; on one arm of the interferometer, the red photon created at NL1 is passed through the object "O".  Crucially, the second pair is not created until after this beam containing the red photon that has passed through the object is recombined with the other beam from the initial beamsplitter (at D2).  ("D" stands for "dichroic mirror"---this mirror reflects red photons, but is transparent at the original (undownconverted) wavelength.)  Only then is the resulting combination passed through the nonlinear crystal, NL2.  Then the red mode (which is fed not only by the red mode that passed through the object and has been recombined into the beam, but also by the downconversion process from photons of the original wavelength impinging on NL2) is pulled out of the beam by another dichroic mirror.  The yellow mode is then recombined with the yellow mode from NL1 on the other arm of the interferometer, and the resulting interference observed by the detectors at lower right in the figure.

It is easy to see why this experiment does not allow superluminal signaling by altering the imaged object, and thereby altering the image.  For there is an effectively lightlike or timelike (it will be effectively timelike, given the delays introduced by the beamsplitters and mirrors and such) path from the object to the detectors.  It is crucial that the red light passed through the object be recombined, at least for a while, with the light that has not passed through the object, in some spacetime region in the past light cone of the detectors, for it is the recombination here that enables the interference between light not passed through the object, and light passed through the object, that allows the image to show up in the yellow light that has not (on either arm of the interferometer) passed through the object.  Since the object must be in the past lightcone of the recombination region where the red light interferes, which in turn must be in the past lightcone of the final detectors, the object must be in the past lightcone of the final detectors.  So we can signal by changing the object and thereby changing the image at the final detectors, but the signaling is not faster-than-light.

Perhaps the most interesting thing about the experiment, as the authors point out, is that it enables an object to be imaged at a wavelength that may be difficult to efficiently detect, using detectors at a different wavelength, as long as there is a downconversion process that creates a pair of photons with one member of the pair at each wavelength.  By not pointing out the crucial fact that this is an interference experiment between two entangled pairs [Update 9/1: per my parenthetical remark above, and Andreas' comment, this should be taken as shorthand for "between a component of the wavefunction in which an entangled pair is created in the upper arm of the interferometer, and one in which one is created in the lower arm"], the description in New Scientist does naturally suggest that the image might be created in one member of an entangled pair, by passing the other member through the object,  without any recombination of the photons that have passed through the object with a beam on a path to the final detectors, which would indeed violate no-signaling.

I haven't done a calculation of what should happen in the experiment, but my rough intuition at the moment   is that the red photons that have come through the object interfere with the red component of the beam created in the downconversion process, and since the photons that came through the object have entangled yellow partners in the upper arm of the interferometer that did not pass through the object, and the red photons that did not pass through the object have yellow partners created along with them in the lower part of the interferometer, the interference pattern between the red photons that did and didn't pass through the object corresponds perfectly to an interference pattern between their yellow partners, neither of which passed through the object.  It is the latter that is observed at the detectors. [Update 8/29: now that I've done the simple calculation, I think this intuitive explanation is not so hot.  The phase shift imparted by the object "to the red photons" actually pertains to the entire red-yellow entangled pair that has come from NL1 even though it can be imparted by just "interacting" with the red beam, so it is not that the red photons interfere with the red photons from NL2, and the yellow with the yellow in the same way independently, so that the pattern could be observed on either color, with the statistical details perfectly correlated. Rather, without recombining the red photons with the beam, no interference could be observed between photons of a single color, be it red or yellow, because the "which-beam" information for each color is recorded in different beams of the other color.  The recombination of the red photons that have passed through the object with the undownconverted photons from the other output of the initial beamsplitter ensures that the red photons all end up in the same mode after crystal NL2 whether they came into the beam before the crystal or were produced in the crystal by downconversion, thereby ensuring that the red photons contain no record of which beam the yellow photons are in, and allowing the interference due to the phase shift imparted by the object to be observed on the yellow photons alone.]

As I mentioned, not having done the calculation, I don't think I fully understand what is happening.  [Update: Now that I have done a calculation of sorts, the questions raised in this paragraph are  answered in a further Update at the end of this post.  I now think that some of the recombinations of beams considered in this paragraph are not physically possible.]  In particular, I suspect that if the red beam that passes through the object were mixed with the downconverted beam on the lower arm of the interferometer after the downconversion, and then peeled off before detection, instead of having been mixed in before the downconversion and peeled off afterward, the interference pattern would not be observed, but I don't have clear argument why that should be.  [Update 8/29: the process is described ambiguously here.  If we could peel off the red photons that have passed through the object while leaving the ones that came from the downconversion at NL2, we would destroy the interference.  But we obviously can't do that; neither we nor our apparatus can tell these photons apart (and if we could, that would destroy interference anyway).  Peeling off *all* the red photons before detection actually would allow the interference to be seen, if we could have mixed back in the red photons first; the catch is that this mixing-back-in is probaby not physically possible.]  Anyone want to help out with an explanation?  I suspect one could show that this would be the same as peeling off the red photons from NL2 after the beamsplitter but before detection,  and only then recombining them with the red photons from the object, which would be the same as just throwing away the red photons from the object to begin with.  If one could image in this way, then that would allow signaling, so it must not work.  But I'd still prefer a more direct understanding via a comparison of the downconversion process with the red photons recombined before, versus after.  Similarly, I suspect that mixing in and then peeling off the red photons from the object before NL2 would not do the job, though I don't see a no-signaling argument in this case.  But it seems crucial, in order for the yellow photons to bear an imprint of interference between the red ones, that the red ones from the object be present during the downconversion process.

The news piece summarizing the article in Nature is much better than the one at New Scientist, in that it does explain that there are two pairs, and that the one member of one pair is passed through the object and recombined with something from the other pair.  But it does not make it clear that the recombination takes place before the second pair is created---indeed it strongly suggests the opposite:

According to the laws of quantum physics, if no one detects which path a photon took, the particle effectively has taken both routes, and a photon pair is created in each path at once, says Gabriela Barreto Lemos, a physicist at Austrian Academy of Sciences and a co-author on the latest paper.

In the first path, one photon in the pair passes through the object to be imaged, and the other does not. The photon that passed through the object is then recombined with its other ‘possible self’ — which travelled down the second path and not through the object — and is thrown away. The remaining photon from the second path is also reunited with itself from the first path and directed towards a camera, where it is used to build the image, despite having never interacted with the object.

Putting the quote from Barreta Lemos about a pair being created on each path before the description of the recombination suggests that both pair-creation events occur before the recombination, which is wrong. But the description in this article is much better than the New Scientist description---everything else about it seems correct, and it gets the crucial point that there are two pairs, one member of which passes through the object and is recombined with elements of the other pair at some point before detection, right even if it is misleading about exactly where the recombination point is.

[Update 8/28: clearly if we peel the red photons off before NL2, and then peel the red photons created by downconversion at NL2 off after NL2 but before the final beamsplitter and detectors, we don't get interference because the red photons peeled off at different times are in orthogonal modes, each associated with one of the two different beams of yellow photons to be combined at the final beamsplitter, so the interference is destroyed by the recording of "which-beam" information about the yellow photons, in the red photons. But does this mean if we recombine the red photons into the same mode, we restore interference? That must not be so, for it would allow signaling based on a decision to recombine or not in a region which could be arranged to be spacelike separated from the final beamsplitter and detectors.  But how do we see this more directly?  Having now done a highly idealized version of the calculation (based on notation like that in and around Eq. (1) of the paper) I see that if we could do this recombination, we would get interference.  But to do that we would need a nonphysical device, namely a one-way mirror, to do this final recombination.  If we wanted to do the other variant I discussed above, recombining the red photons that have passed the object with the red (and yellow) photons created at NL2 and then peeling all red photons off before the final detector, we would even need a dichroic one-way mirror (transparent to yellow, one-way for red), to recombine the red photons from the object with the beam coming from NL2.  So the only physical way to implement the process is to recombine the red photons that have passed through the object with light of the original wavelength in the lower arm of the interferometer before NL2; this just needs an ordinary dichroic mirror, which is a perfectly physical device.]

# Free will and retrocausality at Cambridge II: Conspiracy vs. Retrocausality; Signaling and Fine-Tuning

Expect (with moderate probability) substantial revisions to this post, hopefully including links to relevant talks from the Cambridge conference on retrocausality and free will in quantum theory, but for now I think it's best just to put this out there.

Conspiracy versus Retrocausality

One of the main things I hoped to straighten out for myself at the conference on retrocausality in Cambridge was whether the correlation between measurement settings and "hidden variables" involved in a retrocausal explanation of Bell-inequality-violating quantum correlations are necessarily "conspiratorial", as Bell himself seems to have thought.  The idea seems to be that correlations between measurement settings and hidden variables must be due to some "common cause" in the intersection of the backward light cones of the two.  That is, a kind of "conspiracy" coordinating the relevant hidden variables that can affect the meausrement outcome with all sorts of intricate processes that can affect which measurement is made, such as those affecting your "free" decision as to how to set a polarizer, or, in case you set up a mechanism to control the polarizer setting according to some apparatus reasonably viewed as random ("the Swiss national lottery machine" was the one envisioned by Bell), the functioning of this mechanism.  I left the conference convinced once again (after doubts on this score had been raised in my mind by some discussions at New Directions in the Philosophy of Physics 2013) that the retrocausal type of explanation Price has in mind is different from a conspiratorial one.

Deflationary accounts of causality: their impact on retrocausal explanation

Distinguishing "retrocausality" from "conspiratorial causality" is subtle, because it is not clear that causality makes sense as part of a fundamental physical theory.   (This is a point which, in this form, apparently goes back to Bertrand Russell early in this century.  It also reminds me of David Hume, although he was perhaps not limiting his "deflationary" account of causality to causality in physical theories.)  Causality might be a concept that makes sense at the fundamental level for some types of theory, e.g. a version ("interpretation") of quantum theory that takes measurement settings and outcomes as fundamental, taking an "instrumentalist" view of the quantum state as a means of calculating outcome probabilities giving settings, and not as itself real, without giving a further formal theoretical account of what is real.  But in general, a theory may give an account of logical implications between events, or more generally, correlations between them, without specifying which events cause, or exert some (perhaps probabilistic) causal influence on others.  The notion of causality may be something that is emergent, that appears from the perspective of beings like us, that are part of the world, and intervene in it, or model parts of it theoretically.  In our use of a theory to model parts of the world, we end up taking certain events as "exogenous".  Loosely speaking, they might be determined by us agents (using our "free will"), or by factors outside the model.  (And perhaps "determined" is the wrong word.)   If these "exogenous" events are correlated with other things in the model, we may speak of this correlation as causal influence.  This is a useful way of speaking, for example, if we control some of the exogenous variables:  roughly speaking, if we believe a model that describes correlations between these and other variables not taken as exogenous, then we say these variables are causally influenced by the variables we control that are correlated with them.  We find this sort of notion of causality valuable because it helps us decide how to influence those variables we can influence, in order to make it more likely that other variables, that we don't control directly, take values we want them to.  This view of causality, put forward for example in Judea Pearl's book "Causality", has been gaining acceptance over the last 10-15 years, but it has deeper roots.  Phil Dowe's talk at Cambridge was an especially clear exposition of this point of view on causality (emphasizing exogeneity of certain variables over the need for any strong notion of free will), and its relevance to retrocausality.

This makes the discussion of retrocausality more subtle because it raises the possibility that a retrocausal and a conspiratorial account of what's going on with a Bell experiment might describe the same correlations, between the Swiss National lottery machine, or whatever controls my whims in setting a polarizer, all the variables these things are influenced by, and the polarizer settings and outcomes in a Bell experiment, differing only in the causal relations they describe between these variables.  That might be true, if a retrocausalist decided to try to model the process by which the polarizer was set.  But the point of the retrocausal account seems to be that it is not necessary to model this to explain the correlations between measurement results.  The retrocausalist posits a lawlike relation of correlation between measurement settings and some of the hidden variables that are in the past light cone of both measurement outcomes.  As long as this retrocausal influence does not influence observable past events, but only the values of "hidden", although real, variables, there is nothing obviously more paradoxical about imagining this than about imagining----as we do all the time---that macroscopic variables that we exert some control over, such as measurement settings, are correlated with things in the future.   Indeed, as Huw Price has long (I have only recently realized for just how long) been pointing out, if we believe that the fundamental laws of physics are symmetric with respect to time-reversal, then it would be the absence of retrocausality, if we dismiss its possibility, and even if we accept its possibility to the limited extent needed to potentially explain Bell correlations, its relative scarcity, that needs explaining.  Part of the explanation, of course, is likely that causality, as mentioned above, is a notion that is useful for agents situated within the world, rather than one that applies to the "view from nowhere and nowhen" that some (e.g. Price, who I think coined the term "nowhen") think is, or should be,  taken by fundamental physical theories.  Therefore whatever asymmetries---- these could be somewhat local-in-spacetime even if extremely large-scale, or due to "spontaneous" (i.e. explicit, even if due to a small perturbation) symmetry-breaking --- are associated with our apparently symmetry-breaking experience of directionality of time may also be the explanation for why we introduce the causal arrows we do into our description, and therefore why we so rarely introduce retrocausal ones.  At the same time, such an explanation might well leave room for the limited retrocausality Price would like to introduce into our description, for the purpose of explaining Bell correlations, especially because such retrocausality does not allow backwards-in-time signaling.

Signaling (spacelike and backwards-timelike) and fine-tuning. Emergent no-signaling?

A theme that came up repeatedly at the conference was "fine-tuning"---that no-spacelike-signaling, and possibly also no-retrocausal-signaling, seem to require a kind of "fine-tuning" from a hidden variable model that uses them to explain quantum correlations.  Why, in Bohmian theory, if we have spacelike influence of variables we control on physically real (but not necessarily observable) variables, should things be arranged just so that we cannot use this influence to remotely control observable variables, i.e. signal?  Similarly one might ask why, if we have backwards-in-time influence of controllable variables on physically real variables, things are arranged just so that we cannot use this influence to remotely control observable variables at an earlier time?  I think --- and I think this possibility was raised at the conference --- that a possible explanation, suggested by the above discussion of causality, is that for macroscopic agents such as us, with usually-reliable memories, some degree of control over our environment and persistence over time, to arise, it may be necessary that the scope of such macroscopic "observable" influences be limited, in order that there be a coherent macroscopic story at all for us to tell---in order for us even be around to wonder about whether there could be such signalling or not.  (So the term "emergent no-signalling" in the section heading might be slightly misleading: signalling, causality, control, and limitations on signalling might all necessarily emerge together.) Such a story might end up involving thermodynamic arguments, about the sorts of structures that might emerge in a metastable equilibrium, or that might emerge in a dynamically stable state dependent on a temperature gradient, or something of the sort.  Indeed, the distribution of hidden variables (usually, positions and/or momenta) according to the squared modulus of the wavefunction, which is necessary to get agreement of Bohmian theory with quantum theory and also to prevent signaling (and which does seem like "fine-tuning" inasmuch as it requires a precise choice of probability distribution over initial conditions), has on various occasions been justified by arguments that it represents a kind of equilibrium that would be rapidly approached even if it did not initially obtain.  (I have no informed view at present on how good these arguments are, though I have at various times in the past read some of the relevant papers---Bohm himself, and Sheldon Goldstein, are the authors who come to mind.)

I should mention that at the conference the appeal of such statistical/thermodynamic  arguments for "emergent" no-signalling was questioned---I think by Matthew Leifer, who with Rob Spekkens has been one of the main proponents of the idea that no-signaling can appear like a kind of fine-tuning, and that it would be desirable to have a model which gave a satisfying explanation of it---on the grounds that one might expect "fluctuations" away from the equilibria, metastable structures, or steady states, but we don't observe small fluctuations away from no-signalling---the law seems to hold with certainty.  This is an important point, and although I suspect there are  adequate rejoinders, I don't see at the moment what these might be like.

# Free will and retrocausality in the quantum world, at Cambridge. I: Bell inequalities and retrocausality

I'm in Cambridge, where the conference on Free Will and Retrocausality in the Quantum World, organized (or rather, organised) by Huw Price and Matt Farr will begin in a few hours.  (My room at St. Catherine's is across from the chapel, and I'm being serenaded by a choir singing beautifully at a professional level of perfection and musicality---I saw them leaving the chapel yesterday and they looked, amazingly, to be mostly junior high school age.)  I'm hoping to understand more about how "retrocausality", in which effects occur before their causes, might help resolve some apparent problems with quantum theory, perhaps in ways that point to potentially deeper underlying theories such as a "quantum gravity". So, as much for my own use as anyone else's, I thought perhaps I should post about my current understanding of this possibility.

One of the main problems or puzzles with quantum theory that Huw and others (such as Matthew Leifer, who will be speaking) think retrocausality may be able to help with, is the existence of Bell-type inequality violations. At their simplest, these involve two spacelike-separated regions of spacetime, usually referred to as "Alice's laboratory" and "Bob's laboratory", at each of which different possible experiments can be done. The results of these experiments can be correlated, for example if they are done on a pair of particles, one of which has reached Alice's lab and the other Bob's, that have previously interacted, or were perhaps created simultaneously in the same event. Typically in actual experiments, these are a pair of photons created in a "downconversion" event in a nonlinear crystal.  In a "nonlinear"  optical process photon number is not conserved (so one can get a "nonlinearity" at the level of a Maxwell's equation where the intensity of the field is proportional to photon number; "nonlinearity" refers to the fact that the sum of two solutions is not required to be a solution).  In parametric downconversion, a photon is absorbed by the crystal which emits a pair of photons in its place, whose energy-momentum four-vectors add up to that of the absorbed photon (the process does conserve energy-momentum).   Conservation of angular momentum imposes correlations between the results of measurements made by "Alice" and "Bob" on the emitted photons. These are correlated even if the measurements are made sometime after the photons have separated far enough that the changes in the measurement apparatus that determine which component of polarization it measures (which we'll henceforth call the "polarization setting"), on one of the photons, are space-like separated from the measurement process on the other photon, so that effects of the polarization setting in Alice's laboratory, which one typically assumes can propagate only forward in time, i.e. in their forward light-cone, can't affect the setting or results in Bob's laboratory which is outside of this forward light-cone.  (And vice versa, interchanging Alice and Bob.)

Knowledge of how their pair of photons were prepared (via parametric downconversion and propagation to Alice and Bob's measurement sites) is encoded in a "quantum state" of the polarizations of the photon pair.  It gives us, for any pair of polarization settings that could be chosen by Alice and Bob, an ordinary classical joint probability distribution over the pair of random variables that are the outcomes of the given measurements.  We have different classical joint distributions, referring to different pairs of random variables, when different pairs of polarization settings are chosen.   The Bell "paradox" is that there is no way of introducing further random variables that are independent of these polarization settings, such that for each pair of polarization settings, and each assignment of values to the further random variables, Alice and Bob's measurement outcomes are independent of each other, but when the further random variables are averaged over, the experimentally observed correlations, for each pair of settings, are reproduced. In other words, the outcomes of the polarization measurements, and in particular the fact that they are correlated, can't be "explained" by variables uncorrelated with the settings. The nonexistence of such an explanation is implied by the violation of a type of inequality called a "Bell inequality". (It's equivalent to to such a violation, if "Bell inequality" is defined generally enough.)

How I stopped worrying and learned to love quantum correlations

One might have hoped to explain the correlations by having some physical quantities (sometimes referred to as "hidden variables") in the intersection of Alice and Bob's backward light-cone, whose effects, propagating forward in their light-cone to Alice and Bob's laboratories, interact their with the physical quantities describing the polarization settings to produce---whether deterministically or stochastically---the measurement outcomes at each sites, with their observed probabilities and correlations. The above "paradox" implies that this kind of "explanation" is not possible.

Some people, such as Tim Maudlin, seem to think that this implies that quantum theory is "nonlocal" in the sense of exhibiting some faster-than-light influence. I think this is wrong. If one wants to "explain" correlations by finding---or hypothesizing, as "hidden variables"---quantities conditional on which the probabilities of outcomes, for all possible measurement settings, factorize, then these cannot be independent of measurement settings. If one further requires that all such quantities must be localized in spacetime, and that their influence propagates (in some sense that I'm not too clear about at the moment, but that can probably be described in terms of differential equations---something like a conserved probability current might be involved) locally and forward in time, perhaps one gets into inconsistencies. But one can also just say that these correlations are a fact. We can have explanations of these sorts of fact---for example, for correlations in photon polarization measurements, the one alluded to above in terms of energy-momentum conservation and previous interaction or simultaneous creation---just not the sort of ultra-classical one some people wish for.

Retrocausality

It seems to me that what the retrocausality advocates bring to this issue is the possibility of something that is close to this type of classical explanation. It may allow for the removal of these types of correlation by conditioning on physical quantities. [Added July 31: this does not conflict with Bell's theorem, for the physical quantities are not required to be uncorrelated with measurement settings---indeed, being correlated with the measurement settings is to be expected if there is retrocausal influence from a measurement setting to physical quantities in the backwards light-cone of the measurement setting.] And unlike the Bohmian hidden variable theories, it hopes to avoid superluminal propagation of the influence of measurement settings to physical quantities, even unobservable ones.  It does this, however, by having the influence of measurement settings pursue a "zig-zag" path from Alice to Bob: in Alice's backward light-cone back to the region where Alice and Bob's backward light-cones intersect, then forward to Bob's laboratory. What advantages might this have over superluminal propagation? It probably satisfies some kind of spacetime continuity postulate, and seems more likely to be able to be Lorentz-invariant. (However, the relation between formal Lorentz invariance and lack of superluminal propagation is subtle, as Rafael Sorkin reminded me at breakfast today.)

# Probable signature of gravitational waves from early-universe inflation found in cosmic microwave background by BICEP2 collaboration.

Some quick links about the measurement, announced today, by the BICEP2 collaboration using a telescope at the South Pole equipped with transition edge sensors (TESs) read out with superconducting quantum interference devices (SQUIDs), of B-modes (or "curl") in the polarization of the cosmic microwave background (CMB) radiation, considered to be an imprint on the CMB of primordial graviational waves stirred up by the period of rapid expansion of the universe (probably from around 10-35--10-33 sec).  BICEP2 estimates the tensor-to-scalar ratio "r", an important parameter constraining models of inflation, to be 0.2 (+0.7 / -0.5).

Note that I'm not at all expert on any aspect of this!

Caltech press release.

Harvard-Smithsonian Center for Astrophysics press release.

Instrument paper: BICEP2 II: Experiment and three-year data set

BICEP/Keck homepage with the papers and other materials.

Good background blog post (semi-popular level) from Sean Carroll

Carroll's initial reaction.

Richard Easther on inflation, also anticipating the discover (also fairlybroadly accessible)

Very interesting reaction from a particle physicist at Résonaances.

Reaction from Liam McAllister guesting on Lubos Motl's blog.

Reaction from theoretical cosmology postdoc Sesh Nadathur.

NIST Quantum Sensors project homepage.

Besides a microwave telescope to collect and focus the relevant radiation, the experiment used transition-edge sensors (in which photons can trigger a quantum phase transition) read out by superconducting quantum interference devices (SQUIDs).  I don't know the details of how that works, but TE sensors have lots of applications (including in quantum cryptography), as do SQUIDs;  I'm looking forward to learning more about this one.

# Some ideas on food and entertainment for those attending SQUINT 2014 in Santa Fe

I'm missing SQUINT 2014 (bummer...) to give a talk at a workshop on Quantum Contextuality, Nonlocality, and the Foundations of Quantum Mechanics in Bad Honnef, Germany, followed by collaboration with Markus Mueller at Heidelberg, and a visit to Caslav Brukner's group and the IQOQI at Vienna.  Herewith some ideas for food and entertainment for SQUINTers in Santa Fe.

Cris Moore will of course provide good advice too.  For a high-endish foodie place, I like Ristra.  You can also eat in the bar there, more casual (woodtop tables instead of white tablecloths), a moderate amount of space (but won't fit an enormous group), some smaller plates.  Pretty reasonable prices (for the excellent quality).  Poblano relleno is one of the best vegetarian entrees I've had in a high-end restaurant---I think it is vegan.  Flash-fried calamari were also excellent... I've eaten here a lot with very few misses.  One of the maitres d' sings in a group I'm in, and we're working on tenor-baritone duets, so if Ed is there you can tell him Howard sent you but then you have to behave ;-).  The food should be good regardless.  If Jonathan is tending bar you can ask him for a flaming chartreuse after dinner... fun stuff and tasty too.  (I assume you're not driving.)  Wines by the glass are good, you should get good advice on pairing with food.

Next door to Ristra is Raaga... some of the best Indian food I've had in a restaurant, and reasonably priced for the quality.

I enjoyed a couple of lunches (fish tacos, grilled portobello sandwich, weird dessert creations...) at Restaurant Martin, was less thrilled by my one foray into dinner there.  Expensive for dinner, less so for lunch, a bit of a foodie vibe.

Fish and chips are excellent at Zia Café (best in town I think), so is the green chile pie--massive slice of a deep-dish quiche-like entity, sweet and hot at the same time.

I like the tapas at El Mesón, especially the fried eggplant, any fried seafood like oysters with salmorejo, roasted red peppers with goat cheese (more interesting than it sounds).  I've had better luck with their sherries (especially finos) better than their wines by the glass.  (I'd skip the Manchego with guava or whatever, as it's not that many slices and you can get cheese at a market.)  Tonight they will have a pretty solid jazz rhythm section, the Three Faces of Jazz, and there are often guests on various horn.  Straight-ahead standards and classic jazz, mostly bop to hard bop to cool jazz or whatever you want to call it.  "Funky Caribbean-infused jazz" with Ryan Finn on trombone on Sat. might be worth checking out too... I haven't heard him with this group but I've heard a few pretty solid solos from him with a big band.  Sounds fun.  The jazz is popular so you might want to make reservations (to eat in the bar/music space, there is also a restaurant area I've never eaten in) especially if you're more than a few people.

La Boca and Taverna La Boca are also fun for tapas, maybe less classically Spanish.  La Boca used to have half-price on a limited selection of tapas and \$1 off on sherry from 3-5 PM.  Not sure if they still do.

Il Piatto is relatively inexpensive Italian, pretty hearty, and they usually have some pretty good deals in fixed-price 3 course meals where you choose from the menu, or early bird specials and such.

Despite a kind of pretentious name Tanti Luci 221, at 221 Shelby, was really excellent the one time I tried it.  There's a bar menu served only in the bar area, where you can also order off the main menu.  They have a happy hour daily, where drinks are half price.  That makes them kinda reasonable.  The Manhattan I had was excellent, though maybe not all that traditional.

If you've got a car and want some down-home Salvadoran food, the Pupuseria y Restaurante Salvadoreño, in front of a motel on Cerillos, is excellent and cheap.

As far as entertainment, get a copy of the free Reporter (or look up their online calendar).  John Rangel and Chris Ishee are two of the best jazz pianists in town;  if either is playing, go.  Chris is also in Pollo Frito, a New Orleans funk outfit that's a lot of fun.  If they're playing at the original 2nd street brewery, it should be a fun time... decent pubby food and brews to eat while you listen.  Saxophonist Arlen Asher is one of the deans of the NM jazz scene, trumpeter and flugelhorn player Bobby Shew is also excellent, both quite straight-ahead.  Dave Anderson also recommended.  The one time I heard JQ Whitcomb on trumpet he was solid, but it's only been once.  I especially liked his compositions.  Faith Amour is a nice singer, last time I heard her was at Pranzo where the acoustics were pretty bad.  (Tiny's was better in that respect.)

For trad New Mexican (food that is) I especially like Tia Sophia's on Washington (I think), and The Shed for red chile enchiladas (and margaritas).

Gotta go.  It's Friday night, when all good grad students, faculty, and postdocs anywhere in the worlkd head for the nearest "Irish pub".

I had a look at Jacob Bekenstein's 1973 Physical Review D paper "Black holes and entropy" for the answer to my question about Susskind's presentation of the Bekenstein derivation of the formula stating that black hole entropy is proportional to horizon area.  An argument similar to the one in Susskind's talk appears in Section IV, except that massive particles are considered, rather than photons, and they can be assumed to be scalar so that the issue I raised, of entropy associated with polarization, is moot.  Bekenstein says:

we can be sure that the absolute minimum of information lost [as a particle falls into a black hole] is that contained in the answer to the question "does the particle exist or not?"  To start with, the answer [to this question] is known to be yes.  But after the particle falls in, one has no information whatever about the answer.  This is because from the point of view of this paper, one knows nothing about the physical conditions inside the black hole, and thus one cannot assess the likelihood of the particle continuing to exist or being destroyed.  One must, therefore, admit to the loss of one bit of information [...] at the very least."

Presumably for the particle to be destroyed, at least in a field-theoretic description, it must annihilate with some stuff that is already inside the black hole (or from the outside point of view, plastered against the horizon). This annihilation could, I guess, create some other particle. In fact it probably must, in order to conserve mass-energy.  My worry in the previous post about the entropy being due to the presence/absence of the particle inside the hole was that this would seem to need to be due to uncertainty about whether the particle fell into the hole in the first place, which did not seem to be part of the story Susskind was telling, and the associated worry that this would make the black hole mass uncertain, which also didn't seem to be a feature of the intended story although I wasn't sure. But the correct story seems to be that the particle definitely goes into the hole, and the uncertainty is about whether it subsequently annihilates with something else inside, in a process obeying all relevant conservation laws, rendering both of my worries inapplicable. I'd still like to see if Bekenstein wrote a version using photons, as Susskind's presentation does. And when I feel quite comfortable, I'll probably post a fairly full description of one (or more) versions of the argument. Prior to the Phys Rev D paper there was a 1972 Letter to Nuovo Cimento, which I plan to have a look at; perhaps it deals with photons. If you want to read Bekenstein's papers too, I suggest you have a look at his webpage.

# Question about Susskind's presentation of Bekenstein's black hole entropy derivation

I'm partway through viewing Leonard Susskind's excellent not-too-technical talk "Inside Black Holes" given at the Kavli Institute for Theoretical Physics at UC Santa Barbara on August 25.  Thanks to John Preskill,  @preskill, for recommending it.

I've decided to try using my blog as a discussion space about this talk, and ultimately perhaps about the "Harlow-Hayden conjecture" about how to avoid accepting the recent claim that black holes must have an information-destroying "firewall" near the horizon.  (I hope I've got that right.)  I'm using  Susskind's paper "Black hole complementarity and the Harlow-Hayden conjecture"  as my first source on the latter question.  It also seems to be a relatively nontechnical presentation (though much more technical than the talk so far)... that should be particularly accessible to quantum information theorists, although it seems to me he also does a good job of explaining the quantum information-theoretic concepts he uses to those not familiar with them.

But first things first.  I'm going to unembarassedly ask elementary questions about the talk and the paper until I understand.  First off, I've got a question about Susskind's "high-school level" presentation, in minutes 18-28 of the video, of Jacob Bekenstein's 1973 argument that in our quantum-mechanical world the entropy of a black hole is proportional to its area (i.e. the area of the horizon, the closed surface inside which nothing, not even light, can escape).   The formula, as given by Susskind, is

$S = (\frac{c^3}{4 \hbar G}) A$,

where $S$ is the entropy (in bits) of the black hole, and $A$ the area of its horizon.  (The constant here may have been tweaked by a small amounts, like $4 \pi$ or its inverse, to reflect considerations that Susskind alluded to but didn't describe, more subtle than those involved in Bekenstein's argument.)

Now, maybe the solution is just that given their wavelength of the same order of the hole, there is uncertainty about whether or not the photons actually get into the hole, and so the entropy of the black hole really is due to uncertainty about its total mass, and the mass M in the Bekenstein formula is just the expected value of mass?

I realize I could probably figure all this out by grabbing some papers, e.g. Bekenstein's original, or perhaps even by checking wikipedia, but I think there's some value in thinking out loud, and in having an actual interchange with people to clear up my confusion... one ends up understanding the concepts better, and remembering the solution.  So, if any physicists knowledgeable about black holes (or able and willing to intelligently speculate about them...) are reading this, straighten me out if you can, or at least let's discuss it and figure it out...

# Bohm on measurement in Bohmian quantum theory

Prompted, as described in the previous post, by Craig Callender's post on the uncertainty principle, I've gone back to David Bohm's original series of two papers "A suggested interpretation of the quantum theory in terms of "hidden" variables I" and "...II", published in Physical Review in 1952 (and reprinted in Wheeler and Zurek's classic collection "Quantum Theory and Measurement", Princeton University Press, 1983).  The Bohm papers and others appear to be downloadable here.

Question 1 of my previous post asked whether it is true that

"a "measurement of position" does not measure the pre-existing value of the variable called, in the theory, "position".  That is, if one considers a single trajectory in phase space (position and momentum, over time), entering an apparatus described as a "position measurement apparatus", that apparatus does not necessarily end up pointing to, approximately, the position of the particle when it entered the apparatus."

It is fairly clear from Bohm's papers that the answer is "Yes". In section 5 of the second paper, he writes

"in the measurement of an "observable," Q, we cannot obtain enough information to provide a complete specification of the state of an electron, because we cannot infer the precisely defined values of the particle momentum and position, which are, for example, needed if we wish to make precise predictions about the future behavior of the electron. [...] the measurement of an "observable" is not really a measurement of any physical property belonging to the observed system alone. Instead, the value of an "observable" measures only an incompletely predictable and controllable potentiality belonging just as much to the measuring apparatus as to the observed system itself."

Since the first sentence quoted says we cannot infer precise values of "momentum and position", it is possible to interpret it as referring to an uncertainty-principle-like tradeoff of precision in measurement of one versus the other, rather than a statement that it is not possible to measure either precisely, but I think that would be a misreading, as the rest of the quote, which clearly concerns any single observable, indicates. Later in the section, he unambiguously gives the answer "Yes" to a mutation of my Question 1 which substitutes momentum for position. Indeed, most of the section is concerned with using momentum measurement as an example of the general principle that the measurements described by standard quantum theory, when interpreted in his formalism, do not measure pre-existing properties of the measured system.

Here's a bit of one of two explicit examples he gives of momentum measurement:

"...consider a stationary state of an atom, of zero angular momentum. [...] the $\psi$-field for such a state is real, so that we obtain

$\mathbf{p} = \nabla S = 0.$

Thus, the particle is at rest. Nevertheless, we see from (14) and (15) that if the momentum "observable" is measured, a large value of this "observable" may be obtained if the $\psi$-field happens to have a large fourier coefficient, $a_\mathbf{p}$, for a high value of $\mathbf{p}$. The reason is that in the process of interaction with the measuring apparatus, the $\psi$-field is altered in such a way that it can give the electron particle a correspondingly large momentum, thus transferring some of the potential energy of interaction of the particle with its $\psi$-field into kinetic energy."

Note that the Bohmian theory involves writing the complex-valued wavefunction $\psi(\mathbf{x})$ as $R(\mathbf{x})e^{i S(\mathbf{x})}$, i.e. in terms of its (real) modulus $R$ and (real) phase $S$. Expressing the Schrödinger equation in terms of these variables is in fact probably what suggested the interpretation, since one gets something resembling classical equations of motion, but with a term that looks like a potential, but depends on $\psi$. Then one takes these classical-like equations of motion seriously, as governing the motions of actual particles that have definite positions and momenta. In order to stay in agreement with quantum theory concerning observed events such as the outcomes of measurements, m theory, one in addition keeps, from quantum theory, the assumption that the wavefunction $\psi$ evolves according to the Schrödinger equation. And one assumes that we don't know the particles' exact position but only that this is distributed with probability measure given (as quantum theory would predict for the outcome of a position measurement) by $R^2(\mathbf{x})$, and that the momentum is $\mathbf{p} = \nabla S$. That's why the real-valuedness of the wavefunction implies that momentum is zero: because the momentum, in Bohmian theory, is the gradient of the phase of the wavefunction.

For completeness we should reproduce Bohm's (15).

(15) $\psi = \sum_\mathbf{p} a_{\mathbf{p}} exp(i \mathbf{p}\cdot \mathbf{x} / \hbar).$

At least in the Wheeler and Zurek book, the equation has $p$ instead of $\mathbf{p}$ as the subscript on $\Sigma$, and $a_1$ instead of $a_\mathbf{p}$; I consider these typos, and have corrected them. (Bohm's reference to (14), which is essentially the same as (15) seems to me to be redundant.)

The upshot is that

"the actual particle momentum existing before the measurement took place is quite different from the numerical value obtained for the momentum "observable,"which, in the usual interpretation, is called the "momentum." "

It would be nice to have this worked out for a position measurement example, as well. The nicest thing, from my point of view, would be an example trajectory, for a definite initial position, under a position-measurement interaction, leading to a final position different from the initial one. I doubt this would be too hard, although it is generally considered to be the case that solving the Bohmian equations of motion is difficult in the technical sense of complexity theory. I don't recall just how difficult, but more difficult than solving the Schrödinger equation, which is sometimes taken as an argument against the Bohmian interpretation: why should nature do all that work, only to reproduce, because of the constraints mentioned above---distribution of $\mathbf{x}$ according to $R^2$, $\mathbf{p} = \nabla S$---observable consequences that can be more easily calculated using the Schrödinger equation?
I think I first heard of this complexity objection (which is of course something of a matter of taste in scientific theories, rather than a knockdown argument) from Daniel Gottesman, in a conversation at one of the Feynman Fests at the University of Maryland, although Antony Valentini (himself a Bohmian) has definitely stressed the ability of Bohmian mechanics to solve problems of high complexity, if one is allowed to violate the constraints that make it observationally indistinguishable from quantum theory. It is clear from rereading Bohm's 1952 papers that Bohm was excited about the physical possibility of going beyond these constraints, and thus beyond the limitations of standard quantum theory, if his theory was correct.

In fairness to Bohmianism, I should mention that in these papers Bohm suggests that the constraints that give standard quantum behavior may be an equilibrium, and in another paper he gives arguments in favor of this claim. Others have since taken up this line of argument and done more with it. I'm not familiar with the details. But the analogy with thermodynamics and statistical mechanics breaks down in at least one respect, that one can observe nonequilibrium phenomena, and processes of equilibration, with respect to standard thermodynamics, but nothing like this has so far been observed with respect to Bohmian quantum theory. (Of course that does not mean we shouldn't think harder, guided by Bohmian theory, about where such violations might be observed... I believe Valentini has suggested some possibilities in early-universe physics.)