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Hidden Variables as Fruitful Dead Ends Scott Aaronson (MIT)

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WWJPD? Goal of talk: By discussing hidden variables, show how little of his sanity Ive learned Ever since I attended his group meetings as a 20-year-old summer student, John Preskill has been my unbreakable link between CS and physicssomeone whose scientific judgments Ive respected above all othersmy lodestar of sanity

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God, Dice, Yadda Yadda The Einsteinian Impulse: Quantum mechanics is a tool for calculating probabilities of measurement outcomes. It tells no clear story about whats 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) Dont you need to be insane to still believe this in 2013??

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The Sentient Quantum Computer So, what did it feel like to undergo a 2 10000 -dimensional Fourier transform? Its amazing how fast you forget If you believe that a sentient QC would need to have some definite experienceor distribution over possible experiences hidden variables just might be for you

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This Talk Tasting Menu of Hidden-Variable Theories No-Go Theorems: Bell, Kochen-Specker, and PBR New Results on -Epistemic Theories [ABCL13] Computational Complexity and Hidden Variables Same predictions as QMDifferent predictions Replace wavefunction -epistemic theories Lots of falsified ideas (Joy Christian, Stephen Wolfram…) Supplement wavefunction Bohmian mechanics, discrete dynamical theories Non-equilibrium Bohmian mechanics (Valentini) Field Guide to Hidden-Variable Theories

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A set of ontic states (ontic = philosopher-speak for real) For each pure state | H d, a probability measure over ontic states For each orthonormal basis B=(v 1,…,v d ) and i [d], a response function R i,B : [0,1], satisfying A d-dimensional -Epistemic Theory is defined by: (Conservation of Probability) (Born Rule) Can trivially satisfy these axioms by setting =H d, = the point measure concentrated on | itself, and R i,B ( )=| v i | | 2 Gives a completely uninteresting restatement of quantum mechanics (called the Beltrametti- Bugajski theory)

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Accounts beautifully for one qubit -epistemically! (One qutrit: Already a problem…) More Interesting Example: Kochen-Specker Theory Observation: If | =0, then and cant overlap Call the theory maximally nontrivial if (as above) and overlap whenever | and | are not orthogonal Response functions R i,B ( ): deterministically return basis vector closest to |

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Quantum state Probability distribution Unitary matrix Stochastic matrix Discrete Dynamical Theories

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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 U A,U B act only on A,B respectively, then Robustness to small perturbations in U and |

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Bohmian Mechanics Underappreciated Fact: In a finite-dimensional Hilbert space (like that of quantum gravity), we cant possibly get Bohms kind of determinism The actual particle positions x are a raft, floating passively on the (x,t) ocean God only plays dice at the Big Bang! But then He smashes His dice, and lets x follow the | | 2 distribution forever after My view: Bohms guiding equation only looks inevitable because he restricted attention to a weird Hilbert space…

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Schrödinger/Nagasawa Theory (based on iterative matrix scaling; originated in 1931) Normalize the columns Normalize the rows Set (i,j) entry of joint probabilities matrix to |u ij | 2, as a first guess Can prove this process converges for every U,| ! Beautiful math involved: KL divergence, Max-Flow/Min-Cut Theorem…

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Implication for dynamical theories: Impossible to satisfy both indifference and commutativity Implication for -epistemic theories: Cant reproduce QM using = Alice John and local response functions Bell/CHSH No-Go Theorem

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There exist unit vectors v 1,…,v 31 R 3 that cant be colored red or blue so that in every orthonormal basis, exactly one v i is red Kochen-Specker No-Go Theorem Implication for -epistemic theories: If theory is deterministic (R i,B ( ) {0,1}), then R i,B ( ) must depend on all vectors in B, not just on v i Implication for dynamical theories: Cant have dynamics in all bases that mesh with each other

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Suppose we assume = ( -epistemic theories must behave well under tensor product) Then theres a 2-qubit entangled measurement M, such that the only way to explain Ms behavior on the 4 states PBR (Pusey-Barrett-Rudolph 2011) No-Go Theorem is using a trivial theory that doesnt mix 0 and +. (Can be generalized to any pair of states, not just |0 and |+ ) Bells Theorem: Cant locally simulate all separable measurements on a fixed entangled state PBR Theorem: Cant locally simulate a fixed entangled measurement on all separable states (at least nontrivially so)

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But suppose we drop PBRs tensor assumption. Then: Theorem (A.-Bouland-Chua-Lowther 13): Theres a maximally- nontrivial -epistemic theory in any finite dimension d Cover H d 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 Albeit an extremely weird one! Solves the main open problem of Lewis et al. 12 Ideas of the construction:

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Theorem (ABCL13): Theres no symmetric, maximally- nontrivial -epistemic theory in dimensions d 3 To prove, easiest to start with strongly symmetric theoriesspecial case where has the same form for every On the other hand, suppose we want our theory to be symmetricmeaning that and

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Speedo Region Measuring | in the basis B={| 1,| 2,| 3 } must yield some outcome with nonzero probabilitysuppose | 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 Proof Sketch To generalize to the merely symmetric case ( ( )=f (| | |)), we use some measure theory and differential geometry, to show that the s cant 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

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Hidden Variables and Quantum Computing Well-known problem: Its incredibly hard to construct such a theory that doesnt contradict QM on existing experiments! Sure/Shor separators Some people believe scalable QC is fundamentally impossible Ive 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 dont)

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Needed: A Sure/Shor separator (A. 2004), between the many-particle quantum states were sure we can create and those that suffice for things like Shors algorithm PRINCIPLED LINE

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Scalable Quantum Computing: The Bell inequality violation of the 21 st century Admittedly, quantum computers seem to differ from Bell violation in being directly useful for something BosonSampling Recently demonstrated with 3-4 photons [Broome et al., Tillmann et al., Walmsley et al., Crespi et al.] But in a recent advance, [A.-Arkhipov 2011] solved that problem!

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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! Ironically, dynamical hidden-variable theories could also increase the power of QC even further Concrete evidence comes from the Collision Problem: Given a list of N numbers where every number appears twice, find any collision pair Any quantum algorithm to solve the collision problem needs at least ~N 1/3 steps [A.-Shi 2002] (and this is tight) 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

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How to solve the collision problem super-fast by sampling a trajectory [A. 2005] GOAL: When we inspect the hidden-variable trajectory, see both |i and |j with high probability Measurement of 2 nd register Two bitwise Fourier transforms

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By sampling a trajectory, you can also do Grover search in ~N 1/3 steps instead of ~N 1/2 (!) N 1/3 iterations of Grovers quantum search algorithm Probability of observing the marked item after T iterations is ~T 2 /N Hidden variable

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P Polynomial Time BQP Quantum Polynomial Time DQP Dynamical Quantum Polynomial Time NP Satisfiability, Traveling Salesman, etc. Factoring Graph Isomorphism Approximate Shortest Vector Conjectured World Map

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Upshot: If, at your death, your whole life flashed before you in an instant, then you could solve Graph Isomorphism in polynomial time (Assuming youd 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…)

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Hidden-variable theories are like mathematical sandcastles on the shores of QM Concluding Thought Yes, they tend to topple over when pushed (by mathematical demands if they match QMs predictions, or by experiments if they dont) 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… And yes, people who think they can live in one are almost certainly deluding themselves But its hard not to wonder: just how convincing a castle can one build, before the sand reasserts its sandiness?

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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) Whats the computational complexity of simulating Bohmian mechanics? Are there symmetric -epistemic theories in dimensions d 3 that mix some ontic distributions (not necessarily all of them)? In -epistemic theories, whats the largest possible amount of overlap between two ontic distributions and, in terms of | | |? Open Problems in Hiddenvariableology

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