Hidden Variables as Fruitful Dead Ends Scott Aaronson (MIT)
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Hidden Variables as Fruitful Dead Ends
Scott Aaronson (MIT)
WWJPD?
Goal of talk: By discussing hidden variables, show how little of his sanity I’ve 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 physics—someone whose scientific judgments I’ve respected above all others’—my lodestar of sanity
“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 TalkTasting 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
Same predictions as QM
Different 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
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
A d-dimensional -Epistemic Theory is defined by:
(Conservation of Probability)
(Born Rule)
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”)
Accounts beautifully for one qubit -epistemically!
(One qutrit: Already a problem…)
More Interesting Example: Kochen-Specker Theory
Observation: If |=0, then and can’t overlap
Call the theory maximally nontrivial if (as above) and overlap whenever | and | are not orthogonal
Response functions Ri,B(): deterministically return basis vector closest to |
1 11 1 1
1
N
N N NN N
u u
u u
2 2
1 11 1 1
2 21
N
N NNN N
s s
s s
Quantum state
Quantum state
Probability distribution
Probability distribution
Unitary matrix
Stochastic matrix
Discrete Dynamical Theories
Such a stochastic matrix S is trivial to find!2 2 2 2
1 1 1 1
2 2 2 2
N N N N
“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
Underappreciated Fact: In a finite-dimensional Hilbert space (like that of quantum gravity), we can’t possibly get Bohm’s 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: Bohm’s guiding equation only looks “inevitable” because he restricted
attention to a weird Hilbert space…
7 / 25 3/ 5 4 / 5 3/ 5
24 / 25 4 / 5 3/ 5 4 / 5
.360 .640
.640 .360
.360 .640
.078
.922
.130 .410
.230 .230
.019 .059
.461 .461
.013 .065
.347 .575
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 |uij|2, as a first guess
Can prove this process converges for every U,|! Beautiful math involved: KL divergence, Max-Flow/Min-Cut 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
Bell/CHSH 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 v i is red
Kochen-Specker No-Go Theorem
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
Implication for dynamical theories: Can’t have dynamics in all bases that “mesh” with each other
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
PBR (Pusey-Barrett-Rudolph 2011) No-Go Theorem
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
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
Albeit an extremely weird one!Solves the main open problem of Lewis et al. ‘12
Ideas of the construction:
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
On the other hand, suppose we want our theory to be symmetric—meaning that
and
“Speedo Region”
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
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
Hidden Variables and Quantum Computing
Well-known problem: It’s incredibly hard to construct such a theory that doesn’t contradict QM on existing experiments!
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)
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
BosonSamplingRecently 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!
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 ~N1/3 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
How to solve the collision problem super-fast by sampling a trajectory
[A. 2005]
1
1 1
2
N
i ii
i x i j xN
GOAL: When we inspect the hidden-variable trajectory, see both |i and |j with high probability
“Measurement” of 2nd register
Two bitwise Fourier transforms
1
2ii j x
By sampling a trajectory, you can also do Grover search in ~N1/3 steps instead of ~N1/2 (!)
N1/3 iterations
of Grover’s quantum
search algorithm
Probability of observing the marked item after T
iterations is ~T2/N
Hidden variable
PPolynomial
Time
BQPQuantum
PolynomialTime
DQPDynamical Quantum
Polynomial Time
NPSatisfiability, Traveling
Salesman, etc.
Factoring
Graph Isomorphism
Approximate Shortest Vector
Conjectured World Map
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…)
Hidden-variable theories are like mathematical sandcastles on the shores of QM
Concluding ThoughtYes, they tend to topple over when pushed(by mathematical demands if they match QM’s predictions, or by experiments if they don’t)
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 it’s hard not to wonder: just how convincing a castle can one build, before the sand reasserts its sandiness?
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 |||?
Open Problems in Hiddenvariableology
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