Tsirelson’s problem and linear system games
William Slofstra
IQC, University of Waterloo
October 21st, 2016
includes joint work with Richard Cleve and Li Liu
Tsirelson’s problem and linear system games
William Slofstra
A speculative question
Conventional wisdom: Finite time / volume / energy / etc. =⇒
can always describe nature by finite-dimensional Hilbert spaces
But... many models in quantum mechanics and quantum field
theory require infinite-dimensional Hilbert spaces (e.g. CCR)
Could nature be “intrinsically” infinite-dimensional?
Answer: Probably not
But if it was... could we recognize that fact in an experiment?
(For instance, in a Bell-type experiment?)
Tsirelson’s problem and linear system games
William Slofstra
Non-local games (aka Bell-type experiments)
x
Referee
Alice
a
y
Bob
Referee
Win
Tsirelson’s problem and linear system games
Win/lose based on outputs a, b
and inputs x, y
Alice and Bob must cooperate
to win
b
Winning conditions known in
advance
Lose
Complication: players cannot
communicate while the game is
in progress
William Slofstra
Non-local games ct’d
x
Referee
Alice
a
y
Bob
Referee
Win
Tsirelson’s problem and linear system games
b
Lose
Suppose game is played many
times, with inputs drawn from
some public distribution π
To outside observer, Alice and
Bob’s strategy is described by:
P(a, b|x, y ) = the probability of
output (a, b) on input (x, y )
Correlation matrix: collection of
numbers {P(a, b|x, y )}
William Slofstra
Non-local games ct’d
x
Referee
Alice
a
Win
y
Bob
Referee
b
Lose
P(a, b|x, y ) = the probability of output (a, b) on
input (x, y )
With n possible questions and m possible answers,
correlation matrix {P(a, b|x, y )} is list of m2 n2
probabilities
Value of game: ωC = optimal winning probability using
correlations {P(a, b|x, y )} from fixed set of correlations C
Idea: if ωC1 < ωC2 then
• there is a correlation {P(a, b|x, y )} in C2 but not in C1 , and
• there is a (theoretical) Bell-type experiment to show this.
Tsirelson’s problem and linear system games
William Slofstra
Non-local games ct’d
Value of game ωC = optimal winning probability using strategies
{P(a, b|x, y )} from C
Idea: if ωC1 < ωC2 for some game then
• there is a correlation {P(a, b|x, y )} in C2 but not in C1 , and
• there is a (theoretical) Bell-type experiment to show this.
Bell’s theorem:
• Cc = classical correlation matrices of the form
P(a, b|x, y ) =
X
λi Ai (a|x)Bi (b|y ).
• Cq = quantum correlations
Then there are games with ωc < ωq .
Tsirelson’s problem and linear system games
William Slofstra
Quantum strategies
How do we describe a quantum strategy?
Use axioms of quantum mechanics:
• Physical system described by Hilbert space
• No communication ⇒ Alice and Bob each have their own
(finite dimensional) Hilbert spaces HA and HB
• Hilbert space for composite system is H = HA ⊗ HB
• Shared quantum state is a unit vector |ψi ∈ H
• Alice’s output on input x is modelled by measurement
operators {Max }a on HA
• Similarly Bob has measurement operators {Nby }b on HB
Quantum correlation: P(a, b|x, y ) = hψ| Max ⊗ Nby |ψi
Tsirelson’s problem and linear system games
William Slofstra
Quantum correlations
Compare: finite and infinite dimensional correlations
n
Cq = {P(a, b|x, y )} :P(a, b|x, y ) = hψ| Max ⊗ Nby |ψi where
|ψi ∈ HA ⊗ HB , where HA , HB fin dim’l
Max and Nby are projections on HA and HB
o
X
X y
Max = I and
Nb = I for all x, y
a
b
and
Cqs = same but HA and HB are possibly infinite-dimensional
Question: is ωq < ωqs for some game?
Answer: No!
Tsirelson’s problem and linear system games
William Slofstra
Quantum correlations ct’d
Cq : finite-dimensional correlations
Cqs : possibly-infinite-dimensional correlations
Question: is ωq < ωqs for some game?
Answer: No!
Let Cqa = Cq , so ωq = ωqa for any game
Then Cq ⊆ Cqs ⊆ Cqa , so ωqs = ωqa = ωq for any game.
Fortunately, this is not the end of the story
We’ve assumed that H = HA ⊗ HB ... maybe this is too restrictive
Tsirelson’s problem and linear system games
William Slofstra
Commuting-operator model
Another model for composite systems: commuting-operator model
In this model:
• Alice and Bob each have an algebra of observables A and B
• A and B act on the joint Hilbert space H
• A and B commute: if a ∈ A, b ∈ B, then ab = ba.
This model is used (for instance) in quantum field theory
Correlation set:
n
Cqc := {P(a, b|x, y )} : P(a, b|x, y ) = hψ| Max Nby |ψi ,
Max Nby = Nby Max
o
Hierarchy: Cq ⊆ Cqs ⊆ Cqa ⊆ Cqc
Tsirelson’s problem and linear system games
William Slofstra
Tsirelson’s problem
strong
Cq
⊆
Cqs
⊆
Cqa
⊆
Cqc
weak
Two models of QM: tensor product and commuting-operator
Tsirelson problems: is Ct , t ∈ {q, qs, qa} equal to Cqc
Fundamental questions:
1
What is the power of these models?
Strong Tsirelson: is Cq = Cqc ?
2
Is ωqa < ωqc for any game?
Since Cqa and Cqc are closed convex sets, equivalent to
Weak Tsirelson: is Cqa = Cqc ?
Tsirelson’s problem and linear system games
William Slofstra
What do we know?
strong
Cq
⊆
Cqs
⊆
Cqa
⊆
Cqc
weak
Theorem (Ozawa, JNPPSW, Fr)
Cqa = Cqc if and only if Connes’ embedding problem is true
Theorem (S)
Cqs 6= Cqc
Tsirelson’s problem and linear system games
William Slofstra
Other fundamental questions
1
Given a non-local game, can we compute the optimal value ωt
over strategies in Ct , t ∈ {qa, qc}?
2
Is Cq = Cqa ? In other words:
• Is Cq closed?
• Does every non-local game have an optimal finite-dimensional
strategy?
3
Given P ∈ Cq , is there a computable upper bound on the
dimension needed to realize P?
Tsirelson’s problem and linear system games
William Slofstra
What do we know?
Question: Given a non-local game, can we compute the optimal
value ωt over strategies in Ct , t ∈ {qa, qc}?
Brute force: search through strategies on HA = HB = Cn ,
converges to ωqa (from below)
Navascués, Pironio, Acı́n: Given a non-local game, there is a
hierarchy of SDPs which converge in value to ωqc (from above)
Problem: in both cases, no way to tell how close we are to the
correct answer
Theorem (S)
It is undecidable to tell if ωqc < 1
General cases of other questions completely open!
Tsirelson’s problem and linear system games
William Slofstra
Undecidability of quantum logic
Theorem (S)
It is undecidable to tell if ωqc < 1
We’re now talking about something practical (but mathematics,
not physics)
More generally: Can we tell if a statement like
Conditions on collection
some other condition
=⇒
of projections
on the collection
holds in all Hilbert spaces.
T. Fritz: Proof of above theorem implies quantum logic is
undecidable
Tsirelson’s problem and linear system games
William Slofstra
Two theorems
Theorem (S)
Cqs 6= Cqc
Theorem (S)
It is undecidable to tell if ωqc < 1
Theorems look very different...
But: proof follows from a single theorem in group theory
Connection with group theory comes from linear system games
Tsirelson’s problem and linear system games
William Slofstra
Linear system games
Start with m × n linear system Ax = b over Z2
Inputs:
• Alice receives 1 ≤ i ≤ m (an equation)
• Bob receives 1 ≤ j ≤ n (a variable)
Outputs:
• Alice outputs an assignment ak for all variables xk with
Aik 6= 0
• Bob outputs an assignment bj for xj
They win if:
• Aij = 0 (assignment irrelevant) or
• Aij 6= 0 and aj = bj (assignment consistent)
Tsirelson’s problem and linear system games
William Slofstra
Quantum solutions of Ax = b
Observables Xj such that
2
Xj2 = I for all j
Qn
Aij
= (−I )bi for all i
j=1 Xj
3
If Aij , Aik 6= 0, then Xj Xk = Xk Xj
1
(We’ve written linear equations multiplicatively)
Theorem (Cleve-Mittal,Cleve-Liu-S)
Let G be the game for linear system Ax = b. Then:
• G has a perfect strategy in Cqs if and only if Ax = b has a
finite-dimensional quantum solution
• G has a perfect strategy in Cqc if and only if Ax = b has a
quantum solution
Tsirelson’s problem and linear system games
William Slofstra
Group theory
Finitely presented group: algebraic structure generated by products
of symbols
X1 , . . . , Xn
and their inverses, satisfying some relations
Ri (X1 , . . . , Xn ), 1 ≤ i ≤ k
Important: the empty product e acts like 1, but
no scalars (i.e. no −1), no addition
Example:
ha, b : a2 = b 2 = e = (ab)3 i
Question: which relations determine interesting groups?
Tsirelson’s problem and linear system games
William Slofstra
Group theory ct’d
Is there a group generated by X1 , . . . , Xn satisfying the quantum
linear relations from before?
1 Xj2 = I for all j
Qn
Aij
2
= (−I )bi for all i
j=1 Xj
3
If Aij , Aik 6= 0, then Xj Xk = Xk Xj
Almost: replace −I with new symbol J
The solution group Γ of Ax = b is the group generated by
X1 , . . . , Xn , J such that
1 Xj2 = [Xj , J] = J 2 = e for all j
Qn
Aij
2
= J bi for all i
j=1 Xj
3 If Aij , Aik 6= 0, then [Xj , Xk ] = e
where [a, b] = aba−1 b −1 , e = group identity
Tsirelson’s problem and linear system games
William Slofstra
Group theory ct’d
The solution group Γ of Ax = b is the group generated by
X1 , . . . , Xn , J such that
2
Xj2 = [Xj , J] = J 2 = e for all j
Qn
Aij
= J bi for all i
j=1 Xj
3
If Aij , Aik 6= 0, then [Xj , Xk ] = e
1
where [a, b] = aba−1 b −1 , e = group identity
Theorem (Cleve-Mittal,Cleve-Liu-S)
Let G be the game for linear system Ax = b. Then:
• G has a perfect strategy in Cqs if and only if Γ has a
finite-dimensional representation with J 6= I
• G has a perfect strategy in Cqc if and only if J 6= e in Γ
Tsirelson’s problem and linear system games
William Slofstra
Groups and local compatibility
Suppose we can write down any group relations we want...
But: generators in the relation will be forced to commute!
Call this condition local compatibility
Local compatibility is (a priori) a very strong constraint
For instance, S3 is generated by a, b subject to the relations
a2 = b 2 = e, (ab)3 = e
If ab = ba, then (ab)3 = a3 b 3 = ab
So relations imply a = b, and S3 becomes Z2
Tsirelson’s problem and linear system games
William Slofstra
Group embedding theorem
Solution groups satisfy local compatibility
Nonetheless:
Solution groups are as complicated as general groups
Theorem (S)
Let G be any finitely-presented group, and suppose we are given J0
in the center of G such that J02 = e.
Then there is an injective homomorphism φ : G ,→ Γ, where Γ is
the solution group of a linear system Ax = b, with φ(J0 ) = J.
Tsirelson’s problem and linear system games
William Slofstra
How do we prove the embedding theorem?
Linear system Ax = b over Z2 equivalent to labelled hypergraph:
Edges are variables
Vertices are equations
v is adjacent to e if and only if Ave 6= 0
v is labelled by bi ∈ Z2
Given finitely-presented group G , we get Γ from a linear system
But what linear system?
Can answer this pictorially by writing down a hypergraph?
Tsirelson’s problem and linear system games
William Slofstra
The hypergraph by example
y
z
x
v
u
hx, y , z, u, v : xyxz = xuvu = e = x 2 = y 2 = · · · = v 2 i
Tsirelson’s problem and linear system games
William Slofstra
The end
hx, y , z, u, v : xyxz = xuvu = e = x 2 = y 2 = · · · = v 2 i
Thank-you!
Tsirelson’s problem and linear system games
William Slofstra