Download
1 / 27
270 likes | 370 Vues
Hidden Variables as Fruitful Dead Ends. Scott Aaronson (MIT). WWJPD?.
E N D
Hidden Variables as Fruitful Dead Ends Scott Aaronson (MIT)
WWJPD? Ever since I attended his group meetings as a 20-year-old summer student, John Preskill has been my unbreakable link between CS and physics—someone whose scientific judgments I’ve respected above all others’—my lodestar of sanity Goal of talk: By discussing hidden variables, show how little of his sanity I’ve learned
“God, Dice, Yadda Yadda” The “Einsteinian Impulse”: Quantum mechanics is a tool for calculating probabilities of measurement outcomes. It tells no clear story about what’s “really there” prior to measurement. Ergo, one should infer the existence of deeper laws, which tell the “real story” and from which the probability calculus can be derived (either exactly or as a limiting approximation) Don’t you need to be insane to still believe this in 2013??
The Sentient Quantum Computer So, what did it feel like to undergo a 210000-dimensional Fourier transform? It’s amazing how fast you forget If you believe that a sentient QC would need to have some definite experience—or distribution over possible experiences— hidden variables just might be for you
This Talk Tasting Menu of Hidden-Variable Theories No-Go Theorems: Bell, Kochen-Specker, and PBR New Results on -Epistemic Theories [ABCL’13] Computational Complexity and Hidden Variables Field Guide to Hidden-Variable Theories
A d-dimensional -Epistemic Theory is defined by: A set of “ontic states” (ontic = philosopher-speak for “real”) For each pure state |Hd, a probability measure over ontic states For each orthonormal basis B=(v1,…,vd) and i[d], a “response function” Ri,B:[0,1], satisfying Can trivially satisfy these axioms by setting =Hd, = the point measure concentrated on | itself, and Ri,B()=|vi||2 Gives a completely uninteresting restatement of quantum mechanics (called the “Beltrametti-Bugajski theory”) (Conservation of Probability) (Born Rule)
More Interesting Example: Kochen-Specker Theory Response functions Ri,B(): deterministically return basis vector closest to | Accounts beautifully for one qubit-epistemically! (One qutrit: Already a problem…) Observation: If |=0, then and can’t overlap Call the theory maximally nontrivial if (as above) and overlap whenever | and | are not orthogonal
Discrete Dynamical Theories Quantum state Quantum state Unitary matrix Probability distribution Probability distribution Stochastic matrix
Such a stochastic matrix S is trivial to find! “Product Dynamics” (a.k.a. “every Planck time is a whole new adventure!”) Some natural further requirements: “Indifference”: “Commutativity”: If UA,UB act only on A,B respectively, then Robustness to small perturbations in U and |
Bohmian Mechanics The “actual” particle positions x are a raft, floating passively on the (x,t) ocean My view: Bohm’s guiding equation only looks “inevitable” because he restricted attention to a weird Hilbert space… God only plays dice at the Big Bang! But then He smashes His dice, and lets x follow the ||2 distribution forever after Underappreciated Fact: In a finite-dimensional Hilbert space (like that of quantum gravity), we can’t possibly get Bohm’s kind of “determinism”
Schrödinger/Nagasawa Theory(based on iterative matrix scaling; originated in 1931) Set (i,j) entry of joint probabilities matrix to |uij|2, as a first guess Normalize the columns Normalize the rows Can prove this process converges for every U,|! Beautiful math involved: KL divergence, Max-Flow/Min-Cut Theorem…
Bell/CHSH No-Go Theorem Implication for dynamical theories:Impossible to satisfy both indifference and commutativity Implication for -epistemic theories: Can’t reproduce QM using =AliceJohn and “local” response functions
Kochen-Specker No-Go Theorem There exist unit vectors v1,…,v31R3 that can’t be colored red or blue so that in every orthonormal basis, exactly one vi is red Implication for dynamical theories: Can’t have dynamics in all bases that “mesh” with each other Implication for -epistemic theories: If theory is deterministic (Ri,B(){0,1}), then Ri,B() must depend on all vectors in B, not just on vi
PBR (Pusey-Barrett-Rudolph 2011)No-Go Theorem Suppose we assume =(“-epistemic theories must behave well under tensor product”) Then there’s a 2-qubit entangled measurement M, such that the only way to explain M’s behavior on the 4 states is using a “trivial” theory that doesn’t mix 0 and +. (Can be generalized to any pair of states, not just |0 and |+) Bell’s Theorem: Can’t “locally” simulate all separable measurements on a fixed entangled state PBR Theorem: Can’t “locally” simulate a fixed entangled measurement on all separable states (at least nontrivially so)
But suppose we drop PBR’s tensor assumption. Then: Theorem (A.-Bouland-Chua-Lowther ‘13):There’s a maximally-nontrivial -epistemic theory in any finite dimension d Albeit an extremely weird one!Solves the main open problem of Lewis et al. ‘12 Ideas of the construction: Cover Hd with -nets, for all =1/n Mix the states in pairs of small balls (B,B), where |,| both belong to some -net (“Mix” = make their ontic distributions overlap) To mix all non-orthogonal states, take a “convex combination” of countably many such theories
On the other hand, suppose we want our theory to be symmetric—meaning that and Theorem (ABCL’13):There’s no symmetric, maximally-nontrivial -epistemic theory in dimensions d3 To prove, easiest to start with “strongly symmetric” theories—special case where has the same form for every
“Speedo Region” Proof Sketch To generalize to the “merely” symmetric case (()=f(|||)), we use some measure theory and differential geometry, to show that the ’s can’t possibly “evade” | And strangely, our current proof works only for complex Hilbert spaces, not real Hilbert spaces Trying to adapt to the real case leads to a Kakeya-like problem Measuring | in the basis B={|1,|2,|3} must yield some outcome with nonzero probability—suppose |1 By sliding from |2 to |3, we can find a state | orthogonal to |1 such that | is nevertheless in the support of . Then applying B to | yields |1 with nonzero probability, contradicting the Born rule
Hidden Variables and Quantum Computing Some people believe scalable QC is fundamentally impossible I’ve never understood how such people could be right, unless Nature were describable by a “classical polynomial-time hidden variable theory” (some of the skeptics admit this, others don’t) Well-known problem: It’s incredibly hard to construct such a theory that doesn’t contradict QM on existing experiments! Sure/Shor separators
Needed: A “Sure/Shor separator” (A. 2004), between the many-particle quantum states we’re sure we can create and those that suffice for things like Shor’s algorithm PRINCIPLED LINE
Scalable Quantum Computing:“The Bell inequality violation of the 21st century” Admittedly, quantum computers seem to differ from Bell violation in being directly useful for something But in a recent advance, [A.-Arkhipov 2011] solved that problem! BosonSampling Recently demonstrated with 3-4 photons [Broome et al., Tillmann et al., Walmsley et al., Crespi et al.]
Ironically, dynamical hidden-variable theories could also increase the power of QC even further Yes, these theories reproduce standard QM at each individual time. But they also define a distribution over trajectories. And because of correlations, sampling a whole trajectory might be hard even for a quantum computer! • Concrete evidence comes from the Collision Problem: • Given a list of N numbers where every number appears twice, find any collision pair 13 10 4 1 8 7 12 9 11 5 6 4 2 13 10 3 2 7 9 11 5 1 6 12 3 8 Models graph isomorphism, breaking crypto hash functions Any quantum algorithm to solve the collision problem needs at least ~N1/3 steps [A.-Shi 2002] (and this is tight)
Two bitwise Fourier transforms How to solve the collision problem super-fast by sampling a trajectory[A. 2005] “Measurement” of 2nd register GOAL:When we inspect the hidden-variable trajectory, see both |i and |j with high probability
Hidden variable By sampling a trajectory, you can also do Grover search in ~N1/3 steps instead of ~N1/2 (!) Probability of observing the marked item after T iterations is ~T2/N N1/3 iterations of Grover’s quantum search algorithm
Conjectured World Map NP Satisfiability, Traveling Salesman, etc. DQPDynamical Quantum Polynomial Time Graph Isomorphism Approximate Shortest Vector Factoring BQPQuantumPolynomialTime PPolynomial Time
Upshot: If, at your death, your whole life flashed before you in an instant, then you could solve Graph Isomorphism in polynomial time • (Assuming you’d prepared beforehand by putting your brain in appropriate quantum states, and a dynamical hidden-variable theory satisfying certain reasonable axioms was true) But probably still not NP-complete problems! • DQP is basically the only example I know of a computational model that generalizes quantum computing, but only “slightly” • (Contrast with nonlinear quantum mechanics, postselection, closed timelike curves…)
Concluding Thought Hidden-variable theories are like mathematical sandcastles on the shores of QM Yes, they tend to topple over when pushed(by mathematical demands if they match QM’s predictions, or by experiments if they don’t) And yes, people who think they can live in one are almost certainly deluding themselves But it’s hard not to wonder: just how convincing a castle can one build, before the sand reasserts its sandiness? 80+ years after it was first asked, the answers to this question (both positive and negative) continue to offer surprises, making us wonder how well we really know sand and water…
Open Problems in Hiddenvariableology In the Schrödinger/Nagasawa theory, are the probabilities obtained by matrix scaling robust to small perturbations of U and |? Can we upper-bound the complexity of sampling hidden-variable histories? (Best upper bound I know is EXP) What’s the computational complexity of simulating Bohmian mechanics? Are there symmetric -epistemic theories in dimensions d3 that mix some ontic distributions (not necessarily all of them)? In -epistemic theories, what’s the largest possible amount of overlap between two ontic distributions and , in terms of |||?
