Hidden Variables as Fruitful Dead Ends Scott Aaronson (MIT)

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