John Preskill: My Lodestar of Awesomeness

I got back a couple days ago from John Preskill‘s 60th birthday symposium at Caltech.  To the general public, Preskill is probably best known for winning two bets against Stephen Hawking.  To readers of Shtetl-Optimized, he might be known for his leadership in quantum information science, his pioneering work in quantum error-correction, his beautiful lecture notes, or even his occasional comments here (though these days he has his own group blog and Twitter feed to keep him busy).  I know John as a friend, colleague, and mentor who’s done more for me than I can say.

The symposium was a blast—a chance to hear phenomenal talks, enjoy the California sun, and catch up with old friends like Dave Bacon (who stepped down as Pontiff before stepping down as Pontiff was cool).  The only bad part was that I inadvertently insulted John in my talk, by calling him my “lodestar of sanity.”  What I meant was that, for 13 years, I’ve known plenty of physicists who can be arbitrarily off-base when they talk about computer science and vice versa, but I’ve only ever known John to be on-base about either.  If you asked him a question involving, say, both Barrington’s Theorem and Majorana fermions, he’s one of the few people on earth who would know both, seem totally unfazed by your juxtaposing them, and probably have an answer that he’d carefully tailor to your level of knowledge and interest.  In a polyglot field like quantum information, that alone makes him invaluable.  But along with his penetrating insight comes enviable judgment and felicity of expression: unlike some of us (me), John always manages to tell the truth without offending his listeners.  If I were somehow entrusted with choosing a President of the United States, he’d be one of my first choices, certainly ahead of myself.

Anyway, it turned out that John didn’t like my use of the word “sane” to summarize the above: for him (understandably, in retrospect), it had connotations of being humorless and boring, two qualities I’ve never seen in him.  (Also, as I pointed out later, the amount of time John has spent helping me and patiently explaining stuff to me does weigh heavily against his sanity.)  So I hereby rename John my Lodestar of Awesomeness.

In case anyone cares, my talk was entitled “Hidden Variables as Fruitful Dead Ends”; the PowerPoint slides are here.  I spoke about a new preprint by Adam Bouland, Lynn Chua, George Lowther, and myself, on possibility and impossibility results for “ψ-epistemic theories” (a class of hidden-variable theories that was also the subject of the recent PBR Theorem, discussed previously on this blog).  My talk also included material from my old paper Quantum Computing and Hidden Variables.

The complete program is here.  A few highlights (feel free to mention others in the comments):

  • Patrick Hayden spoke about a beautiful result of himself and Alex May, on “where and when a qubit can be.”  After the talk, I commented that it’s lucky for the sake of Hayden and May’s induction proof that 3 happens to be the next integer after 2.  If you get that joke, then I think you’ll understand their result and vice versa.
  • Lenny Susskind—whose bestselling The Theoretical Minimum is on my to-read list—spoke about his views on the AMPS firewall argument.  As you know if you’ve been reading physics blogs, the firewall argument has been burning up (har, har) the world of quantum gravity for months, putting up for grabs aspects of black hole physics long considered settled (or not, depending on who you ask).  Lenny gave a typically-masterful summary, which for the first time enabled me to understand the role played in the AMPS argument by “the Zone” (a region near the black hole but outside its event horizon, in which the Hawking radiation behaves a little differently than it does when it’s further away).  I was particularly struck by Lenny’s comment that whether an observer falling into a black hole encounters a firewall might be “physics’ Axiom of Choice”: that is, we can only follow the logical consequences of theories we formulate outside black-hole event horizons, and maybe those theories simply don’t decide the firewall question one way or the other.  (Then again, maybe they do.)  Lenny also briefly mentioned a striking recent paper by Harlow and Hayden, which argues that the true resolution of the AMPS paradox might involve … wait for it … computational complexity, and specifically, the difficulty of solving QSZK (Quantum Statistical Zero Knowledge) problems in BQP.  And what’s a main piece evidence that QSZK⊄BQP?  Why, the collision lower bound, which I proved 12 years ago while a summer student at Caltech and an awestruck attendee of Preskill’s weekly group meetings.  Good thing no one told me back then that black holes were involved.
  • Charlie Bennett talked about things that I’ve never had the courage to give a talk about, like the Doomsday Argument and the Fermi Paradox.  But his disarming, avuncular manner made it all seem less crazy than it was.
  • Paul Ginsparg, founder of the arXiv, presented the results of a stylometric analysis of John Preskill’s and Alexei Kitaev’s research papers.  The main results were as follows: (1) John and Alexei are easily distinguishable from each other, due in part to the more latter’s “Russian” use of function words (“the,” “which,” “that,” etc.).   (2) Alexei, despite having lived in the US for more than a decade, is if anything becoming more “Russian” in his function word use over time. (3) Even more interestingly, John is also becoming more “Russian” in his function word use—a possible result of his long interaction with Alexei. (4) A joint paper by Kitaev and Preskill was indeed written by both of them.  (Update: While detained at the airport, Paul decided to post an online video of his talk.)

Speaking of which, the great Alexei Kitaev himself—the $3 million man—spoke about Berry curvature for many-body systems, but unfortunately I had to fly back early (y’know, 2-month-old baby) and missed his talk.  Maybe someone else can provide a summary.

Happy 60th birthday, John!

Two unrelated announcements.

1. Everyone who reads this blog should buy Sean Carroll’s two recent books: From Eternity to Here (about the arrow of time) and The Particle at the End of the Universe (about the Higgs boson and quantum field theory more generally).  They’re two of the best popular physics books I’ve ever read—in their honesty, humor, clarity, and total lack of pretense, they exemplify what every book in this genre should be but very few are.  If you need even more inducement, go watch Sean hit it out of the park on the Colbert Report (and then do it again).  I can’t watch those videos without seething with jealousy: given how many “OK”s and “y’know”s lard my every spoken utterance, I’ll probably never get invited to hawk a book on Colbert.  Which is a shame, because as it happens, my Quantum Computing Since Democritus book will finally be released in the US by Cambridge University Press on April 30th!  (It’s already available in the UK, but apparently needs to be shipped to the US by boat.)  And it’s loaded with new material, not contained in the online lecture notes.  And you can preorder it now.  And my hawking of Sean’s books is in no way whatsoever related to any hope that Sean might return the favor with my book.

2. Recent Turing Award winner Silvio Micali asks me to advertise the Second Cambridge Area Economics and Computation Day (CAEC’13), which will be held on Friday April 26 at MIT.  Anything for you, Silvio!  (At least for the next week or two.)

30 Responses to “John Preskill: My Lodestar of Awesomeness”

  1. Henning Dekant Says:

    Colbert totally should invite you to plug your book. Rumour has it that you can be pretty good in front of a camera (your TED talks are some of my favorites).

    And anyhow how are you ever going to pay Joy Christian if you can’t get some good book sales 😉

  2. Ajit R. Jadhav Says:

    0. Had a look at the book preview of Prof. Susskind’s book at Amazon.

    1. The following point caught the eye, simply because there was an exercise in a bold typeface following it.

    On page 2, the authors state the universe as a closed system:

    A system that is either the entire universe or is so isolated from everything else that it behaves as if nothing else exists is a closed system. [emphasis original]

    Thermodynamics usually classifies systems as: open, closed, and isolated. Open systems can exchange everything with surroundings: matter, heat, work. Closed systems cannot exchange mass (but may exchange work/energy). Isolated systems don’t exchange anything with any other objects/systems. The universe must be treated as an isolated system. (In fact, when you say universe, there already are no surroundings left to speak of.) I dwell on the isolated systems further in one of my comments at iMechanica, here.

    2. Coming back to the book, the table of contents, however, does suggest this to be a very valuable book, though I would have liked seeing a book like this deal with the action/extremum principles ahead of energy, so as to follow the historical order of the development… Of course, I haven’t read the main text of the book itself, and might have a look at it sometime later.


  3. John Preskill Says:

    If I had really been insulted I would have to be your Lodestar of Prickliness. Actually, I was hoping to be funny by pretending to be insulted, which must mean I will never be your Lodestar of Understated Humor.

    But wasn’t it a great conference! Getting all the people in one room who think I’m Awesome was Totally Awesome. Which means I’ll also never be your Lodestar of Modesty?

    Seriously (?), thanks for attending, for the Awesome Talk, and for all your very kind thoughts.

  4. Mateus Araújo Says:

    Research on hidden variables is something that breaks my heart. Every time a new paper on it appears on arXiv I feel a little bit more dead inside. I admit being guilty of doing research on it myself, but I swear I’m trying to stop.

    My point is, why do you consider it to be a fruitful dead end? It is certainly fruitful for generating interesting puzzles for us to solve, but philosophically speaking I’m not sure there has been any significant advance since Bell’s theorem…

  5. Łukasz Grabowski Says:

    The slides are unreadable in libreoffice. Any chance you’ll post a pdf?

  6. Chris Says:

    Your book is on sale in the US NOW!

  7. Daniel Freeman Says:

    Mateus (#4),

    There’s always the strange possibility that a nonlocal hidden variable theory (or perhaps even a particularly clever local one) will resonate with future physicists moreso than your Copenhagens or Many-Worlds of today.

    Even if, at times, the hidden variable theories can be at odds with Occam’s Razor, I think its instructive (as Scott’s paper shows) to see just how we can stretch and fold our understanding of quantum mechanics into an equivalent, if unfamiliar theory.

  8. Vadim Says:

    I’m thrilled that your book is finally coming out and have put it on pre-order. Of course I hate waiting a month for anything, so I’m hoping Leonard Susskind’s book keeps me busy until then, and Sean Carroll’s will be bought in short order if I make it through Susskind’s before yours ships. Thanks to all three of you for not forgetting us lay but scientifically curious people.

  9. Scott Says:

    Łukasz Grabowski #5:

      The slides are unreadable in libreoffice. Any chance you’ll post a pdf?

    Sorry, there really needs to be a better way to convert PowerPoint into some open format without totally destroying the animation sequences!

  10. Scott Says:

    Vadim #8: Thanks!!!

  11. Scott Says:

    Mateus Araújo #4:

      Research on hidden variables is something that breaks my heart. Every time a new paper on it appears on arXiv I feel a little bit more dead inside … My point is, why do you consider it to be a fruitful dead end? It is certainly fruitful for generating interesting puzzles for us to solve, but philosophically speaking I’m not sure there has been any significant advance since Bell’s theorem…

    Four responses:

    (1) I don’t, in fact, spend most of my time worrying about hidden variables, nor would I advise others to do so. The genesis of this latest paper was that I was asked to write a Nature Physics News & Views about the PBR Theorem. As part of my literature search for that article, I learned that there were actually still technical open problems about ψ-epistemic theories. So I thought, “why don’t I spend 5 minutes and try to solve them?” Then one thing led to another, two students got interested, George Lowther got interested via a post on MathOverflow … and a year later, here we are with a brand-new hidden-variable paper! 🙂

    (2) A lot of the time, interesting-enough puzzles is all I really ask for in a research topic! If the puzzles have some sort of philosophical “moral” to some people (even if not to me), then so much the better. There are plenty of research areas that can’t even deliver that, so when I find one that can, I’m reluctant to place too many further demands.

    (3) The argument I made in my talk is that the reason to study hidden-variable theories is not because we think they’re “true,” but simply because understanding the conditions under which hidden variables can or can’t work is a major part of understanding quantum mechanics itself. Let me put it this way: if I ask someone whether it’s possible to get a hidden-variable theory satisfying such-and-such assumptions, and the person—guided by “physical intuition”—expresses total confidence in an answer that turns out to be false (as has happened many times), I’d say the person then forfeits the right to dismiss all hidden-variable research as a waste of time. 🙂

    (4) Ten years ago, my interest in discrete-time hidden variable theories are what originally led me to prove the collision lower bound! For me, that was extremely powerful evidence that thinking about hidden variables can sometimes be fruitful.

  12. Mateus Araújo Says:

    Scott #9:
    Legend tells that it is possible to export to .odp from Microsoft Office, with acceptable results. Sorry I can’t test this.

  13. Mateus Araújo Says:

    Scott #11:

    I have no issue with your response (1), and (4) is more than enough to make me shut up, but I really don’t like reasons (2) and (3).

    (2) because I still think that we as researches should aspire to do something more than solving cool puzzles. Doesn’t need to be something deep, just please something that can have any kind of relevance!

    (3) do you really think that researching hidden variables can teach us something about quantum mechanics that we don’t already know? The current research I see in hidden variables is mostly filling details of details of details, without changing the overall theme that “yes hidden variables can work, as long as you give up some kind of natural indepedence assumption about how they should work”. This “independence assumption” can be Bell’s locality, no-conspiracy, non-contextuality, PBR’s preparation independence, Spekkens’ preparation noncontextuality, and surely other things that I can’t recall right now.

    When I see a new one the song “Alice Practice” always pops into my head, it goes like this: “*unintelligible*… Children shouldn’t play with dead things … *unintelligible* … Drop it, it’s dead *unintelligible*”

  14. Scott Says:

    Mateus #13: Well, let me tell you what I learned from the work on ψ-epistemic theories with Bouland, Chua, and Lowther.

    I learned that the quantum mechanics of a qutrit contains lots of interesting qualitative features that never show up in the quantum mechanics of a qubit—features that arise from CP2 having a more complicated geometry than CP1. Of course, the Kochen-Specker Theorem long provided one example of that, but there are many examples that have nothing to do with contextuality.

    I learned that some statements about QM are much easier to prove in complex Hilbert space than in real Hilbert space, for no other reason than that you have twice as many real dimensions to work with.

    I learned that the statement “projections onto orthogonal subspaces occur with probability 0” encodes, all by itself, a surprising amount of information about the Born rule.

    I learned all sorts of interesting tidbits about measure theory and differential geometry—e.g., the “fat Cantor set,” the Lebesgue decomposition theorem, the Radon-Nikodym theorem, the use of gauge-fixing and local coordinate systems … I never imagined any of that “fancy-schmancy stuff” arising in any problem that I actually wanted to solve, but now it has.

    Of course, I don’t know for sure that any of these things will prove useful to me in the future. But I certainly don’t know that they won’t!

  15. John Sidles Says:

    Mateus #4 and Daniel #7, it is natural to wonder (geometrically natural, to be specific) whether pulled-back quantum dynamics (pulled-back onto a truncated-rank product state-space, to be specific) is formally equivalent to a hidden variable theory in which the hidden variables are themselves non-local.

    This possibility is (arguably) unaesthetic if you prefer to think algebraically and conceive that the chief mystery of QM is informatic non-locality. Conversely, it is (arguably) a charming postulate for folks who prefer to think geometrically and conceive that the chief mystery of QM is thermodynamic extensivity.

    Hence my interest in Alexei Kitaev’s presentation (which I am still hoping that someone will report upon).

  16. Chris Says:

    Im going to order a copy of your book in the next couple of days (im in the U.K.). your comments above answered the one question i was going to ask prior to ordering (though i would have purchased it regardless) which was if there was any new or expanded material. Looking forward to reading it!

  17. Raoul Ohio Says:

    Re #4.
    The only thing I am sure about (in the topics of QM and Cosmology), is that we can not be sure we know the big picture, as opposed to some ‘local approximation’. But the picture comes slightly more into focus with every situation that is modeled and solved.

    Is time travel possible? Probably not. But, …, while in high school, I attempted to pass for a cool kid by buying some “Beatle Boots”. But they soon disappeared. Now, nearly half a century later, there is Sean Carroll wearing them on the “Colbert Report”.

  18. Pangloss Says:

    When are you going to post something on this year’s Abel prize winner? I’d like a CS perspective on Deligne’s work.

  19. Scott Says:

    Pangloss #18: Well, then you’ll have to get someone else to write it! Great though it is, I don’t really understand anything about Deligne’s work.

  20. David Poulin Says:

    Kitaev gave an outline of his current plan to classify topological phases. He didn’t explain in any details why he thought that this was the way to go, but he did explain the main ideas of the construction. Here is how I understood it. The idea is to associate a manifold to a given lattice Hamiltonian H, and to classify H by the topology of this manifold. The manifold is described by a chain complex equipped with connections. Let x,y,z… define points in space-time and let i,j,k… label a basis for the single site operators (they could for instance label the Pauli matrices). Then, we can compute ground state correlation functions like A(x,y,z,…)_ijk… the usual way. For instance, A(x,y)_11 could be the correlations between sigma_z at space-time locations x and y. This A will play the role of the connection: it is a tensor-valued field on spacetime. This choice is a natural extension of the more familiar Berry connection in the context of adiabatic evolution. So this ties in nicely with the idea that topological phases are classified by an equivalence relation defined by an adiabatic evolution through a gapped path. To define the chain complex, we need a boundary operator D. Recall that D is roughly a generalization of the gradient to manifolds, so it increases the rank of a tensor field by one. So the definition D A(x,y) = sum_z A(x,y,z) has the right mathematical structure, and this is the one he picks. He did not really explain why this particular choice is the right one, or if he did, I didn’t understand.

  21. eitan bachmat Says:

    One answer to comment 18 of Pangloss. Deligne proved many great results, but is probably best known for solving the Weil conjectures. Roughly speaking, these conjectures deal with the number of solutions of polynomial equations with integer coefficients in finite fields. They state that the number of solutions is the expected number with good control over the error term which is strongly related to the topology of the set of complex solutions. One major consequence of the Weil conjectures is good control over the growth of coefficients of
    some modular forms (cusp forms), again non trivial work of Deligne. These estimates were used by Lubotzky-Phillips and Sarnak to prove that their graph constructions (the LPS graphs) are Ramanujan and in particular expanders, yielding the first explicit (algorithmic) construction of these types of graphs which have proved to be so useful in TCS (See the survey paper of Hoory-Linial-Wigderson). More elementary
    constrcutions like the zigzag were given (much) later on.

  22. John Sidles Says:

    Mateus Araújo asks [#4] “Why do you consider it [hidden-variable theories] to be a fruitful dead end?”

    Mateus, this is an good question precisely because it has more than one good answer.

    To set the stage, please let me comment to Shtetl Optimized readers the (high-ranked) MathOverflow question “Where does a math person go to learn quantum mechanics?” to which the highest-rated answer is

    “I think it is difficult to learn quantum mechanics without first learning classical mechanics.”

    To this you yourself contributed the (perspicacious) comment:

    There’s a quick review of what I consider a good approach to classical mechanics in Terry Tao’s blog: “From Bose-Einstein condensates to the nonlinear Schrodinger equation (which begins with A quick review of classical mechanics)

    Rather like reading The Feynman Lectures on Physics, reading Terry Tao’s weblog is such an outstandingly good idea (for students especially) that it can also be an utterly terrible idea (for students). In this regard, a paradigmatically problematic passage is Terry’s starting assertion:

    Classical mechanics can be formulated in a number of essentially equivalent ways: Newtonian, Hamiltonian, and Lagrangian.

    which is followed by multiple occurrences of phrases like “a typical Hamiltonian”, and “a typical example of a symmetric Hamiltonian”, and “a typical Hamiltonian in this case”, and “a typical model”, and “a typical Hamiltonian in this setting” (the latter appears in three places).

    Here a distinction is falling-into-the-cracks — the cracks between math and physics, that is! — which is (arguably) crucial to fundamental quantum information theory, namely, the cases that Terry calls “typical” are (at least in the context of QIT) markedly atypical.

    Specifically, QIT is very largely concerned with Hamiltonian dynamical systems whose symplectic state-manifold (aka “quibits”) does not have the fiber-bundle topology that is physically associated to particle-potential systems. To appreciate that the set of Lagrangian dynamical systems is a restricted subset of Hamiltonian systems — and arguably not the most mathematically interesting or even the most physically important subset — just try to associate a (classical) Lagrangian to the (classical) Bloch equations!

    We thus appreciate that the 20th centory’s mathematical toolset for doing classical and quantum field theory — including staples of undergraduate education such as Hilbert’s state-space and Feynman’s path integrals — is strikingly ill-suited to QIT physics … to such a high degree, that too-early over-familiarity with early-and-mid 20th century quantum formalisms makes it difficult (for students especially) to grasp the ever-increasing power of more modern (and mathematically broader) geometric dynamical formalisms.

    Summary  Increasingly in the quantum 21st century, as in the -4th classical century, None But Geometers Enter Here“!Or as Robert Frost expressed it:

    When I was young my teachers were the old.
    I gave up fire for form till I was cold.
    I suffered like a metal being cast.
    I went to school to age to learn the past.

    Now I am old my teachers are the young.
    What can’t be molded must be cracked and sprung.
    I strain at lessons fit to start a suture.
    I go to school to youth to learn the future.

  23. Ben Standeven Says:

    @John Sidles:

    I don’t get it. You don’t need a Lagrangian to use the Hilbert space formalism, only a Hamiltonian.

  24. John Sidles Says:

    Ben Standeven #23, the geometric point is that the vital elements of Hamilton/Hilbert/Lindblad dynamical systems — that is, Hamiltonian potentials, closed non-degenerate symplectic forms, complex structures, and Lindblad measurement processes — all pullback naturally onto polynomial-dimension (Kählerian) submanifolds of Hilbert space.

    Metaphorically, the transition from Hilbert-state-space dynamics to Kähler-state-space dynamics is like moving from civilized Paris of the 1850s to the California Gold Fields. A great many civilized conveniences of Hilbert state-manifolds (exact superposition, exact relativistic invariance, and exact spectral theorems) become mere approximations on Kähler state-manifolds. Yet there are compensations … 21st century quantum systems engineers increasingly appreciate that exact superposition, relativistic invariance, and spectral purity, are in practice not required of our dynamical theories … it is entirely feasible — even advantageous! — for dynamical theorists to content themselves with local approximation as contrasted with global exactitude (as the transition from Euclidean to Riemannian geometry has previously shown us).

    Having experienced the excitement of California’s wild frontier in the 1850s, many young people never returned to Paris. Perhaps the same will prove true of the 21st century’s young quantum theorists, in regard to the exponentially increasing practical STEM attractions of pulled-back Kählerian dynamics!

  25. Daniel Tung Says:

    Hi Scott,
    One related and one off-topic question:
    1) I am interested in Susskind’s talk that you mentioned. Is there a chance that I can have his slides?

    2) On the issue of psi-epistemic vs psi-ontic.
    PBR’s paper showed that even non-orthogonal states must correspond to non-overlapping distributions on some ontic states, and this is what they claim that pure quantum states cannot be epistemic distributions over some ontic states.

    However, I think the correct implication of their proof is that pure states correspond to objective epistemic distributions over some ontic states – they still can represent our ignorance about the true ontic state, but in an objective way. Objective, in the sense that the (classical) preparation setup unambiguously determines the pure state.

    In fact this is nothing new, in classical thermodynamics a two canonical distributions almost do not overlap when N is very large, yet they represent our ignorance about the microstates.

  26. Scott Says:

    Daniel #25:

    (1) Unfortunately, Lenny’s talk was blackboard-only, and I don’t think it was recorded.

    (2) I either don’t understand or don’t agree with your interpretation of PBR. To avoid contradicting the PBR theorem (assuming their tensor-product axiom), it’s not enough to stipulate that the classical preparation setup unambiguously determines the pure state. Rather, you need that the ontic state unambiguously determines the pure state! And that’s a much stronger requirement, since it implies that the only epistemic theory you can get is a “trivial” one that essentially just restates quantum mechanics.

  27. Daniel Tung Says:

    Hi Scott,

    Yes, I agree that their result shows that ontic states unambiguously determines the pure state. But I wonder why this is trivial and just restates QM?
    It is like calling equilibrium thermodynamics a trivial statistical mechanics.

  28. Daniel Tung Says:

    BTW, those who say their result as showing that psi is ontic, sounds to me like saying the distributions in equilibrium thermodynamics are some ontic, physical fields.

  29. Scott Says:


      Yes, I agree that their result shows that ontic states unambiguously determines the pure state. But I wonder why this is trivial and just restates QM? It is like calling equilibrium thermodynamics a trivial statistical mechanics.

    OK, I think I finally understand your analogy to thermodynamics—and the answer is that what people had wanted to do with psi-epistemic theories was different. If the ontic state uniquely determined the pure state, then each ψ would correspond to a disjoint collection of ontic states. The ontic states for a given ψ might behave differently under measurement (e.g., each one could be deterministic), but they’d behave exactly like ψ as an ensemble. Call that a “thermodynamic hidden-variable theory.”

    The above can obviously be done, and for reasons that have nothing to do with the structure of quantum mechanics! Any time we have any random outcome, we can always reify whatever the source of the randomness is as a “hidden variable” (though of course, our “hidden variable” might fail to satisfy further natural properties, like locality).

    But we might ask whether, for quantum mechanics in particular, we can do something better / stronger / less obvious, and have different pure states correspond to overlapping collections of ontic states. And that turns out to be a mathematically interesting question.

    The context for the PBR theorem—which for me, played a absolutely crucial role in understanding its motivation—is that if you give up PBR’s tensor-product axiom, then you can get psi-epistemic theories where the same ontic state can correspond to multiple pure states (i.e., the “ontic distributions can overlap”). In 3 or more Hilbert space dimensions, the theories are ugly, but in 2 dimensions, there’s such a theory that’s simple and natural enough that you could almost believe it! See my paper with Bouland, Chua, and Lowther for more.

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