Last week I saw an excellent talk by philosopher Wayne Myrvold.
The Reeh-Schlieder theorem says, roughly, that, in any reasonable quantum field theory, for any bounded region of spacetime R, any state can be approximated arbitrarily closely by operating on the vacuum state (or any state of bounded energy) with operators formed by smearing polynomials in the field operators with functions having support in R. This strikes many as counterintuitive, and Reinhard Werner has glossed the theorem as saying that “By acting on the vacuum with suitable operations in a terrestrial laboratory, an experimenter can create the Taj Mahal on (or even behind) the Moon!” This talk has two parts. First, I hope to convince listeners that the theorem is not counterintuitive, and that it follows immediately from facts that are already familiar fare to anyone who has digested the opening chapters of any standard introductory textbook of QFT. In the second, I will discuss what we can learn from the theorem about how relativistic causality is implemented in quantum field theories.
(Download MP4 video here.)
The topic was well-defined, and of reasonable scope. The theorem is easily and commonly misunderstood. And Wayne’s talk served to dissolve the confusion around it, by unpacking the theorem into a handful of pieces so that you could quickly see where the rub was. I would that all philosophy of physics were so well done.
Here are the key points as I saw them:
The vacuum state in QFTs, even non-interacting ones, is entangled over arbitrary distances (albeit by exponentially small amounts). You can think of this as every two space-like separated regions of spacetime sharing extremely diluted Bell pairs.
… [continue reading]
Over at PhysicsOverflow, Daniel Ranard asked a question that’s near and dear to my heart:
How deterministic are large open quantum systems (e.g. with humans)?
Consider some large system modeled as an open quantum system — say, a person in a room, where the walls of the room interact in a boring way with some environment. Begin with a pure initial state describing some comprehensible configuration. (Maybe the person is sitting down.) Generically, the system will be in a highly mixed state after some time. Both normal human experience and the study of decoherence suggest that this state will be a mixture of orthogonal pure states that describe classical-like configurations. Call these configurations branches.
How much does a pure state of the system branch over human time scales? There will soon be many (many) orthogonal branches with distinct microscopic details. But to what extent will probabilities be spread over macroscopically (and noticeably) different branches?
I answered the question over there as best I could. Below, I’ll reproduce my answer and indulge in slightly more detail and speculation.
This question is central to my research interests, in the sense that completing that research would necessarily let me give a precise, unambiguous answer. So I can only give an imprecise, hand-wavy one. I’ll write down the punchline, then work backwards.
The instantaneous rate of branching, as measured in entropy/time (e.g., bits/s), is given by the sum of all positive Lyapunov exponents for all non-thermalized degrees of freedom.
Most of the vagueness in this claim comes from defining/identifying degree of freedom that have thermalized, and dealing with cases of partial/incomplete thermalization; these problems exists classically.
The original question postulates that the macroscopic system starts in a quantum state corresponding to some comprehensible classical configuration, i.e., the system is initially in a quantum state whose Wigner function is localized around some classical point in phase space.… [continue reading]
The most profound discovery of science appears to be confirmed with essentially no wiggle room. The group led by Ronald Hanson at the Delft University of Technology in the Netherlands claim to have reported a loophole-free observation of Bell violations. Links:
I hope Matt Leifer is right and they give a Nobel Prize for this work.
EDIT Nov 12: Two other groups, who were clearly in a very close race, have just posted their loophole-free experiments: arXiv:1511.03189 and arXiv:1511.03190. (H/t Peter Morgan. Also, note the sequential numbers.) Delft’s group published as soon as they had sufficient statistics to reasonably exclude local realism, but the two runner-ups have collected gratifyingly larger samples, so their p-values are more like 1 in 10 million.… [continue reading]
[Other parts in this series: 1,2,3,4,5,6,7.]
In discussions of the many-worlds interpretation (MWI) and the process of wavefunction branching, folks sometimes ask whether the branching process conflicts with conservations laws like the conservation of energy.Here are some related questions from around the web, not addressing branching or MWI. None of them get answered particularly well. a There are actually two completely different objections that people sometimes make, which have to be addressed separately.
First possible objection: “If the universe splits into two branches, doesn’t the total amount of energy have to double?” This is the question Frank Wilczek appears to be addressing at the end of these notes.
I think this question can only be asked by someone who believes that many worlds is an interpretation that is just like Copenhagen (including, in particular, the idea that measurement events are different than normal unitary evolution) except that it simply declares that new worlds are created following measurements. But this is a misunderstanding of many worlds. MWI dispenses with collapse or any sort of departure from unitary evolution. The wavefunction just evolves along, maintaining its energy distributions, and energy doesn’t double when you mathematically identify a decomposition of the wavefunction into two orthogonal components.
Second possible objection: “If the universe starts out with some finite spread in energy, what happens if it then ‘branches’ into multiple worlds, some of which overlap with energy eigenstates outside that energy spread?” Or, another phrasing: “What happens if the basis in which the universe decoheres doesn’t commute with energy basis? Is it then possible to create energy, at least in some branches?” The answer is “no”, but it’s not obvious.… [continue reading]
I saw a neat talk at Perimeter a couple weeks ago on new integration techniques:
Speaker: Achim Kempf from University of Waterloo.
Title: “How to integrate by differentiating: new methods for QFTs and gravity”.
Abstract: I present a simple new all-purpose integration technique. It is quick to use, applies to functions as well as distributions and it is often easier than contour integration. (And it is not Feynman’s method). It also yields new quick ways to evaluate Fourier and Laplace transforms. The new methods express integration in terms of differentiation. Applied to QFT, the new methods can be used to express functional integration, i.e., path integrals, in terms of functional differentiation. This naturally yields the weak and strong coupling expansions as well as a host of other expansions that may be of use in quantum field theory, e.g., in the context of heat traces.
(Many talks hosted on PIRSA have a link to the mp4 file so you can directly download it. This talk does not, but you can right-click here and select “save as” to get the f4v file.This file format can be watched with VLC player. You can find it for any talk hosted by PIRSA by viewing the page source and searching the text for “.f4v”. There are many nice things about learning physics from videos, one of which is the ability to easily speed up the playback speed and skip around. In VLC player, playback speed can be incremented in 10% steps by pressing the left and right square brackets, ‘[‘ and ‘]’. a )
The technique is based on the familiar trick of extracting a functional derivate inside a path integral and using integration by parts.… [continue reading]
[Other parts in this series: 1,2,3,4,5,6,7.]
People often talk about “creating entanglement” or “creating a superposition” in the laboratory, and quite rightly think about superpositions and entanglement as resources for things like quantum-enhanced measurements and quantum computing.
However, it’s often not made explicit that a superposition is only defined relative to a particular preferred basis for a Hilbert space. A superposition is implicitly a superposition relative to the preferred basis . Schrödinger’s cat is a superposition relative to the preferred basis . Without a there being something special about these bases, the state is no more or less a superposition than and individually. Indeed, for a spin-1/2 system there is a mapping between bases for the Hilbert space and vector directions in real space (as well illustrated by the Bloch sphere); unless one specifies a preferred direction in real space to break rotational symmetry, there is no useful sense of putting that spin in a superposition.
Likewise, entanglement is only defined relative to a particular tensor decomposition of the Hilbert space into subsystems, . For any given (possibly mixed) state of , it’s always possible to write down an alternate decomposition relative to which the state has no entanglement.
So where do these preferred bases and subsystem structure come from? Why is it so useful to talk about these things as resources when their very existence seems to be dependent on our mathematical formalism? Generally it is because these preferred structures are determined by certain aspects of the dynamics out in the real world (as encoded in the Hamiltonian) that make certain physical operations possible and others completely infeasible.… [continue reading]
Folks have been asking about the new Nature Physics article by Pikovski et al., “Universal decoherence due to gravitational time dilation” . Here are some comments:
I think their calculation is probably correct for the model they are considering. One could imagine that they were placing their object in a superposition of two different locations in an electric (rather than gravitational field), and in this case we really would expect the internal degrees of freedom to evolve in two distinct ways. Any observer who was “part of the superposition” wouldn’t be able to tell locally whether their clock was ticking fast or slow, but it can be determined by bringing both clocks back together and comparing them.
It’s possible the center of mass (COM) gets shifted a bit, but you can avoid this complication by just assuming that the superposition separation is much bigger than the size of the object , and that the curvature of the gravitational field is very small compared to both.
Their model is a little weird, as hinted at by their observation that they get “Gaussian decoherence”, , rather than exponential, . The reason is that their evolution isn’t Markovian, as it is for any environment (like scattered or emitted photons) composed of small parts that interact for a bit of time and then leave. Rather, the COM is becoming more and more entangled with each of the internal degrees of freedom as time goes on.
Because they don’t emit any radiation, their “environment” (the internal DOF) is finite dimensional, and so you will eventually get recoherence. This isn’t a problem for Avagadro’s number of particles.
This only decoheres superpositions in the direction of the gravitational gradient, so it’s not particularly relevant for why things look classical above any given scale.
… [continue reading]
[This is a vague post intended to give some intuition about how particular toy models of decoherence fit in to the much hairier question of why the macroscopic world appears classical.]
A spatial superposition of a large object is a common model to explain the importance of decoherence in understanding the macroscopic classical world. If you take a rock and put it in a coherent superposition of two locations separated by a macroscopic distance, you find that the initial pure state of the rock is very, very, very quickly decohered into an incoherent mixture of the two positions by the combined effect of things like stray thermal photons, gas molecules, or even the cosmic microwave background.
Formally, the thing you are superposing is the center-of-mass (COM) variable of the rock. For simplicity one typically considers the internal state of the rock (i.e., all its degrees of freedom besides the COM) to be in a (possibly mixed) quantum state that is uncorrelated with the COM. This toy model then explains (with caveats) why the COM can be treated as a “classical variable”, but it doesn’t immediately explain why the rock as a whole can be considered classical. On might ask: what would that mean, anyways? Certainly, parts of the rock still have quantum aspects (e.g., its spectroscopic properties). For Schrödinger’s cat, how is the decoherence of its COM related the fact that the cat, considered holistically, is either dead or alive but not both?
Consider a macroscopic object with Avagadro’s number of particles N, which means it would be described classically in microscopic detail by 3N variables parameterizing configuration space in three dimensions. (Ignore spin.) We know at least two things immediately about the corresponding quantum system:
(1) Decoherence with the external environment prevents the system from exploring the entire Hilbert space associated with the 3N continuous degrees of freedom.… [continue reading]
I just posted my newest paper: “Decoherence from classically undetectable sources: A standard quantum limit for diffusion” (arXiv:1504.03250). [Edit: Now published as PRA 92, 010101(R) (2015).] The basic idea is to prove a standard quantum limit (SQL) that shows that some particles can be detected through the anomalous decoherence they induce even though they cannot be detected with any classical experiment. Hopefully, this is more evidence that people should think of big spatial superpositions as sensitive detectors, not just neat curiosities.
Here’s the abstract:
In the pursuit of speculative new particles, forces, and dimensions with vanishingly small influence on normal matter, understanding the ultimate physical limits of experimental sensitivity is essential. Here, I show that quantum decoherence offers a window into otherwise inaccessible realms. There is a standard quantum limit for diffusion that restricts some entanglement-generating phenomena, like soft collisions with new particle species, from having appreciable classical influence on normal matter. Such phenomena are classically undetectable but can be revealed by the anomalous decoherence they induce on non-classical superpositions with long-range coherence in phase space. This gives strong, novel motivation for the construction of matter interferometers and other experimental sources of large superpositions, which recently have seen rapid progress. Decoherence is always at least second order in the coupling strength, so such searches are best suited for soft, but not weak, interactions.
Here’s Figure 2:
Standard quantum limit for forces and momentum diffusion
. A test mass is initially placed in a minimal uncertainty wavepacket with a Wigner distribution
over phase space (top) that contains the bulk of its mass within a 2
-contour of a Gaussian distribution (dashed black line).
… [continue reading]
[Other parts in this series: 1,2,3,4,5,6,7.]
I am firmly of the view…that all the sciences are compatible and that detailed links can be, and are being, forged between them. But of course the links are subtle… a mathematical aspect of theory reduction that I regard as central, but which cannot be captured by the purely verbal arguments commonly employed in philosophical discussions of reduction. My contention here will be that many difficulties associated with reduction arise because they involve singular limits….What nonclassical phenomena emerge as h → 0? This sounds like nonsense, and indeed if the limit were not singular the answer would be: no such phenomena.
— Michael Berry
One of the great crimes against humanity occurs each year in introductory quantum mechanics courses when students are introduced to an limit, sometimes decorated with words involving “the correspondence principle”. The problem isn’t with the content per se, but with the suggestion that this somehow gives a satisfying answer to why quantum mechanics looks like classical mechanics on large scales.
Sometimes this limit takes the form of a path integral, where the transition probability for a particle to move from position to in a time is
where is the integral over all paths from to , and is the action for that path ( being the Lagrangian corresponding to the Hamiltonian ). As , the exponent containing the action spins wildly and averages to zero for all paths not in the immediate vicinity of the classical path that make the action stationary.
Other times this takes the form of Ehrenfest’s theorem, which shows that the expectation values of functions of position and momentum follow the classical equations of motion.… [continue reading]
In this post I want to lay out why I am a bit pessimistic about using quantum micromechanical resonators, usually of the optomechanical variety, for decoherence detection. I will need to rely on some simple ideas from 3-4 papers I have “in the pipeline” (read: partially written TeX files) that seek to make precise the sense in which decoherence detection allows us to detect classical undetectable phenomena, and to figure out exactly what sort of phenomena we should apply it to. So this post will sound vague without that supporting material. Hopefully it will still be useful, at least for the author.
The overarching idea is that decoherence detection is only particularly useful when the experimental probe can be placed in a superposition with respect to a probe’s natural pointer basis. Recall that the pointer basis is the basis in which the density matrix of the probe is normally restricted to be approximately diagonal by the interaction with the natural environment. Classically detectable phenomena are those which cause transitions within the pointer basis, i.e. driving the system from one pointer state to another. Classically undetectable phenomena are those which cause pure decoherence with respect to this basis, i.e. they add a decoherence factor to off-diagonal terms in this basis, but preserve on-diagonal terms.
The thing that makes this tricky to think about is that the pointer basis is overcomplete for most physically interesting situations, in particular for any continuous degree of freedom like the position of a molecule or a silicon nanoparticle. It’s impossible to perfectly localize a particle, and the part of the Hamiltonian that fights you on this, , causes a smearing effect that leads to the overcompleteness.… [continue reading]
Last month Scott Aaronson was kind enough to invite me out to MIT to give a seminar to the quantum information group. I presented a small uniqueness theorem which I think is an important intermediary result on the way to solving the set selection problem (or, equivalently, to obtaining an algorithm for breaking the wavefunction of the universe up into branches). I’m not sure when I’ll have a chance to write this up formally, so for now I’m just making the slides available here.
Scott’s a fantastic, thoughtful host, and I got a lot of great questions from the audience. Thanks to everyone there for having me.… [continue reading]
I had to brush up on my Hamilton-Jacobi mechanics to referee a paper. I’d like to share, from this Physics.StackExchange answer, Qmechanic’ clear catalog of the conceptually distinct functions all called “the action” in classical mechanics, taking care to specify their functional dependence:
At least three different quantities in physics are customary called an action and denoted with the letter .
The (off-shell) action
is a functional of the full position curve/path for all times in the interval . See also this question. (Here the words on-shell and off-shell refer to whether the equations of motion (eom) are satisfied or not.)
If the variational problem with well-posed boundary conditions, e.g. Dirichlet boundary conditions
has a unique extremal/classical path , it makes sense to define an on-shell action
which is a function of the boundary values. See e.g. MTW Section 21.1.
The Hamilton’s principal function in Hamilton-Jacobi equation is a function of the position coordinates integration constants , and time , see e.g. H. Goldstein, Classical Mechanics, chapter 10.
The total time derivative
is equal to the Lagrangian on-shell, as explained here. As a consequence, the Hamilton’s principal function can be interpreted as an action on-shell.
These sorts of distinctions are constantly swept under the rug in classical mechanics courses and textbooks (even good books like Goldstein). This leads to serious confusion on the part of the student and, more insidiously, it leads the student to think that this sort of confusion is normal. Ambiguity is baked into the notation! This is a special case of what I conjecture is a common phenomena in physics:
Original researcher thinks deeply, discovers a theory, and writes it down.
… [continue reading]
(This post is vague, and sheer speculation.)
Following a great conversation with Miles Stoudenmire here at PI, I went back and read a paper I forgot about: “Entanglement and the foundations of statistical mechanics” by Popescu et al.S. Popescu, A. Short, and A. Winter, “Entanglement and the foundations of statistical mechanics” Nature Physics 2, 754 – 758 (2006) [Free PDF]. a . This is one of those papers that has a great simple idea, where you’re not sure if it’s profound or trivial, and whether it’s well known or it’s novel. (They cite references 3-6 as “Significant results along similar lines”; let me know if you’ve read any of these and think they’re more useful.) Anyways, here’s some background on how I think about this.
If a pure quantum state is drawn at random (according to the Haar measure) from a -dimensional vector space , then the entanglement entropy
across a tensor decomposition into system and environment is highly likely to be almost the maximum
for any such choice of decomposition . More precisely, if we fix and let , then the fraction of the Haar volume of states that have entanglement entropy more than an exponentially small (in ) amount away from the maximum is suppressed exponentially (in ). This was known as Page’s conjectureD. Page, Average entropy of a subsystem. b , and was later provedS. Foong and S. Kanno, Proof of Page’s conjecture on the average entropy of a subsystem. c J. Sánchez-Ruiz, Simple proof of Page’s conjecture on the average entropy of a subsystem. d ; it is a straightforward consequence of the concentration of measure phenomenon.… [continue reading]
Lemos et al. have a relatively recent letterG. Lemos, V. Borish, G. Cole, S. Ramelow, R. Lapkiewicz, and A. Zeilinger, “Quantum imaging with undetected photons”, Nature 512, 409 (2014) [ arXiv:1401.4318 ]. a in Nature where they describe a method of imaging with undetected photons. (An experiment with the same essential quantum features was performed by Zou et al.X. Y. Zou, L. J. Wang, and L. Mandel, “Induced coherence and indistinguishability in optical interference”, Phys. Rev. Lett. 67, 318 (1991) [ PDF ]. b way back in 1991, but Lemos et al. have emphasized its implications for imaging.) The idea is conceptually related to decoherence detection, and I want to map one onto the other to flesh out the connection. Their figure 1 gives a schematic of the experiment, and is copied below.
Figure 1 from Lemos et al.: ''Schematic of the experiment. Laser light (green) splits at beam splitter BS1 into modes a and b. Beam a pumps nonlinear crystal NL1, where collinear down-conversion may produce a pair of photons of different wavelengths called signal (yellow) and idler (red). After passing through the object O, the idler reflects at dichroic mirror D2 to align with the idler produced in NL2, such that the final emerging idler f does not contain any information about which crystal produced the photon pair. Therefore, signals c and e combined at beam splitter BS2 interfere. Consequently, signal beams g and h reveal idler transmission properties of object O.''
The first two paragraphs of the letter contain all the meat, encrypted and condensed into an opaque nugget of the kind that Nature loves; it stands as a good example of the lamentable way many quantum experimental articles are written.… [continue reading]