Let me try to give some reasons (there are many more) why they are useful: (1) absence of global phase, (2) description of subsystems, (3) definition of entropy. In any case, essentially every known theory in physics can be “compiled down” to where its predictions can be calculated to arbitrary accuracy by a computer toggling 0’s and 1’s—the only exceptions (not coincidentally) being the theories that aren’t yet fully defined or understood. So should we say, on that basis, that you don’t need any nontrivial math to do physics: no complex numbers, no linear algebra, no calculus, not even arithmetic? Alas, not if you actually want to understand what the theories say, which David Deutsch reminds us is more important than calculating with them… 🙂 I wonder what the Kolmogorov complexity size of that QM description would be, not including the seed?

Class of 2020 had finished.I learned a lot by teaching the class on Sunday since March and digesting complex ideas into drawings. Hope you enjoyed it as much as I did. All the recordings can be found here.This Occam-compatible postulate explains, naturally and even (arguably) very beautifully, why present-day experiments and simulations alike readily exhibit low-dimension spukhafte Fernwirkungen (like photon interference), but exhibit high-dimension spukhafte Fernwirkungen (like scalable quantum computation) only with very great difficulty such demonstrations perhaps being impossible even principle (as Kalai’s preprints argue). And I do have further speculations for the kind of micro-structure such a mechanism may have. However, in theory, you have to put it in terms of the micro-structure of the wavefunction.) I like the idea of a connection between quantum mechanics and consciousness. But if that connection exists it is not easy or obvious and nobody has suggested an experiment that shows its usefulness.

I applaud Scott for his intelligent discussions with both the people who overhype and the people who think its not worth studying.It’s true that the usual formulation of the uncertainty principle involves a peculiarity of the Schrödinger equation—namely, that position and momentum are conjugate observables—but I prefer the more abstract formulation, which applies to any pair of conjugate observables, in Hilbert spaces of any dimension (the finite case probably being the clearest). And in the latter case, yes, it’s just a logical consequence of the basic axioms of QM, the ones that talk about amplitudes. Niraj #27: Not sure if I understand the error. Had the OR/XOR distinction been relevant given the context, the mom could’ve added, “superposition doesn’t mean AND, and it doesn’t mean OR, and it doesn’t mean XOR either.” 🙂 People who do know a lot but think QC is not worth studying (Oded Goldreich might be in this category) QM needs Complex Numbers but they might have a more complex or a simpler way for working with them. We are accustomed to working with a+bi form or some other equivalent form but they might be using pulses of electricity or be more comfortable with some matrix notation….. Yes, entanglement is not a requirement of instantaneous action at a distance (IAD). IAD in QM (as in classical diffusion) comes about only because the Fourier theory itself has IAD built into it. And the Fourier theory comes in because measurements involve eigenstates.

That what quantum measurements show is a perfect randomness, is a hypothesis not a physically established fact. … Actually, it’s not even a hypothesis. It is just a conjecture because none has found a sensitive and accurate enough way to experimentally verify it. If you must refer to Popper’s falsifiability criterion (and I don’t care to), the matter has not yet experimentally come within the realm of falsifiability. I’m still trying to wrap my head around Scott’s point about preferred basis. I can’t work out what it means physically. I guess that means I will have to work on an actual understanding rather than just an intuitive one. gasarch #41: I can say from experience that Oded Goldreich does indeed know a lot, but the lot that he knows is not about QC (and he readily admits as much).Even if the global state of a system is given by a pure state, the state of a subsystem can in general not be described by a pure state. But the state of a subsystem can always be described by a density matrix, even if the global state of the system is itself given by a density matrix (or a pure state).

Richard Ernst’s density-matrix-centric text (with Geoffrey Bodenhausen and Alexander Wokaun) “Principles of nuclear magnetic resonance in one and two dimensions”. Ernst’ text surveys the comprehensive capacity of density matrices in simulating quantum dynamics in the (superficially) non-relativistic world of the Schrödinger equation and the Heisenberg picture (note: Ernst’ lively Nobel Biography supplies some needed real-world zest to his dry text). In turn, Ernst’s text prepares us to appreciate a third text: Scott regards (#71) “A fatal problem: namely, [post-quantum dynamical models are] utterly unable to explain why our universe only allows precisely the nonlocal behavior predicted by quantum mechanics (such as winning the CHSH game 85% of the time), and not even more nonlocal behavior than that. Occam’s blade is trembling in its sheath.” In case Born did say that: Born was not God—merely a Nobel laureate. This position (i.e., pre-emptying any possibility of there being any deeper mechanism) is not worthy of any serious attention—not in physics. May be (and just may be) Born shares a philosophic point with you, that’s all—at the most. That there is no mechanism (for what? I take it for randomness) is what Born might have said—and I am not sure if he really actually said that. Bohr could easily have said it, also Heisenberg. But Born? I am not sure.The von Neumann entropy must be computed from the density matrix (or rather from the eigenvalues e_i of the density matrix p as S(p) = Σ e_i ln(e_i)). Taking the probabilities from the Born rule instead (or the p_k from the definition of the density matrix) simply gives the wrong result. Of course, the aliens simulating our universe might be fine with that nonlocality, and you might be fine with it too! But what it does is to push the alleged pseudorandomness of quantum measurement outcomes to a level that’s disconnected from what we actually know about physics. Note, in particular, that it’s extremely important that none of us ever discover the pattern to the pseudorandomness, since if we did, we could break the whole structure of QM, communicate faster than light, etc. Personally, I’d say that it’s of limited interest to postulate a theoretical superstructure that has to be so intentionally sequestered from everything we know about the workings of the world, but YMMV. One thing that surprised me when I first took a Physics course is the importance Physicists give to intuition. fred #90: Yes, of course. But Bell’s Theorem tells us that such a simulation of our universe on a classical digital computer would necessarily be a nonlocal one (that is, the simulation would involve rapid signalling between memory cells corresponding to faraway events, violating the causal structure of the spacetime, even if we, living inside the universe, never actually experienced such signalling). It’s true that technically, the program would need a random number generator to make the final selection of a measurement outcome, and have it “really” be random (rather than pseudorandom). But I’ve never seen that as such a big deal—as a challenge to the Church-Turing Thesis or whatever—because even a deterministic program can easily output a list of probabilities, so that the only thing left for you to do would be to “spin the wheel.”