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Quantum entanglement, Tsirelson’s problem, and group theory

William Slofstra

University of Waterloo

April 4th, 2018

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Finite-dimensional models... of groups

G = x1, . . . , xn : r1, . . . , rm, a finitely presented group Finite-dimensional model of G: complex matrices X1, . . . , Xn such that ri(X1, . . . , Xn) = 1 for 1 ≤ i ≤ m a.k.a. a finite-dimensional representation G → GL(Cd)

Question: when can we recover G from its finite-dimensional models?

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Question: when can we recover G from its finite-dimensional models?

• G is linear: a subgroup of GL(Cd) for some d
• G is residually finite-dimensional (RFD): for every

w ∈ G \ {e}, there is a finite-dimensional representation φ with φ(w) = 1.

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Question: when can we recover G from its finite-dimensional models?

• G is linear: a subgroup of GL(Cd) for some d

G is residually finite-dimensional (RFD): for every w ∈ G \ {e}, there is a finite-dimensional representation φ with φ(w) = 1.

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Residually finite-dimensional groups

G is residually finite-dimensional (RFD): for every w ∈ G \ {e}, there is a finite-dimensional representation φ with φ(w) = 1. Are there non-RFD groups?

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Residually finite-dimensional groups

G is residually finite-dimensional (RFD): for every w ∈ G \ {e}, there is a finite-dimensional representation φ with φ(w) = 1. Are there non-RFD groups? Yes, lots...

• Baumslag-Solitar group: BS(2, 3) = x, y : xy2x−1 = y3
• Higman’s group:

a, b, c, d : aba−1 = b2, bcb−1 = c2, cdc−1 = d2, dad−1 = a2 (Higman’s group has no non-trivial finite-dimensional reps)

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Quantum mechanics (quantum probability)

... is a framework for working with physical systems Axioms of quantum mechanics

• Physical systems = Hilbert space H
• State of system = unit vector v ∈ H
• Measurement:

projections {Pa}a∈O on H with

a Pa = 1.

O = set of measurement outcomes Probability of measuring a is v∗Pav

• ...

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Surprising features of quantum mechanics

Uncertainty principle

Might not be able to measure two properties simultaneously Can measure {Ma} and {Nb} simultaneously only if MaNb = NbMa for all a, b

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Surprising features: contextuality

x7 x4 x1 x8 x5 x2 x9 x6 x3 1 1 1 −1 −1 −1

Problem: assign ±1 to x1, . . . , x9 so that

• 1. product across rows is 1
• 2. product across columns is −1

(this is a linear system over Z2 with 9 variables, 6 equations) Not possible by parity argument

Mermin-Peres magic square

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Surprising features: contextuality

x7 x4 x1 x8 x5 x2 x9 x6 x3 1 1 1 −1 −1 −1

Imagine a physical system, where state of system is set, and then we measure either a row or a column If magic square conditions are satisfied, then xi seems to depend on whether it is measured as part of a row or column

Mermin-Peres magic square

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Surprising features: contextuality

x7 x4 x1 x8 x5 x2 x9 x6 x3 1 1 1 −1 −1 −1

Quantum solution: there are unitaries X1, . . . , X9 such that

• 0. X 2

i = 1 for i = 1, . . . , 9

• 1. product across rows is 1
• 2. product across columns is −1
• 3. if Xi, Xj belong to the same

row or column, then XiXj = XjXi Interpretation: quantum mechanics is contextual!

Mermin-Peres magic square

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Quantum solutions of linear systems

Ax = b : m × n linear system over Z2 Quantum solution: Collection of unitaries X1, . . . , Xn ∈ U(H) such that

• 1. X 2

j = 1 for all j,

• 2. n

j=1 X Aij j

= (−1)bi for all i = 1, . . . , n,

• 3. XjXk = XkXj if Aij, Aik = 0 for some i.

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Quantum solutions of linear systems

Ax = b : m × n linear system over Z2 Solution group of Ax = b Γ(A, b) = x1, . . . , xn, J : x2

j = 1 = [xj, J] = J2 for all j

• j

xAij

j

= Jbi, i = 1, . . . , m [xj, xk] = 1 if Aij, Aik = 0, some i Group commutator: [a, b] = aba−1b−1

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Linear system non-local games

Mermin-Peres contextuality is hard to observe in an experiment m × n linear system Ax = b = ⇒ game with two separated players (Aravind, Cleve-Mittal)

A

equation index 1 ≤ i ≤ m satisfying assignment to variables in equation i

B

variable index 1 ≤ j ≤ n assignment to xj Inputs chosen at random Players win if Alice’s output is consistent with Bob’s output

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Non-local game

Linear system games are examples of non-local games:

A

x ∈ IA a ∈ OA

B

y ∈ IB b ∈ OB Win if V (a, b|x, y) = 1 Inputs chosen from IA × IB according to some distribution Winning condition: function V : OA × OB × IA × IB → {0, 1} Players know rules of game, want to cooperate to win Cannot communicate once game starts... may not be able to play perfectly

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Linear system non-local games: classical strategies

A

equation index 1 ≤ i ≤ m satisfying assignment to variables in equation i

B

variable index 1 ≤ j ≤ n assignment to xj Inputs chosen at random Players win if Alice’s

• utput is consistent

with Bob’s output

Classical strategies (deterministic or shared randomness): Optimal strategy achieved by a deterministic strategy Players can play perfectly if and only if Ax = b has a solution (Otherwise optimal strategy has success probability p < 1)

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Linear system non-local games: quantum strategies

A

equation index 1 ≤ i ≤ m satisfying assignment to variables in equation i

B

variable index 1 ≤ j ≤ n assignment to xj Inputs chosen at random Players win if Alice’s

• utput is consistent

with Bob’s output

Theorem (Cleve-Mittal)

The game associated to Ax = b has a perfect quantum strategy if and only if Ax = b has a finite-dimensional quantum solution. Mermin-Peres square: perfect quantum strategy Best classical strategy succeeds with probability 35/36 Bell test: success probability > 35/36 = ⇒ non-classicality

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Quantum strategies and quantum correlations

A

x ∈ IA a ∈ OA

B

y ∈ IB b ∈ OB Alice and Bob’s behaviour in a non-local game is described by a family of probability distributions {p(a, b|x, y)} ⊂ ROA×OB×IA×IB where p(a, b|x, y) = probability of output (a, b) on input (x, y) {p(a, b|x, y)} is called a correlation matrix

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Quantum strategies and quantum correlations

A B

Which correlations can arise in quantum mechanics?

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Quantum strategies and quantum correlations

HA ⊗ HB HB HB Which correlations can arise in quantum mechanics? Axiom for separated subsystems If HA and HB are Hilbert spaces of two separated systems, then joint sys- tem has Hilbert space HA ⊗ HB Important note: State of joint system does not have to be a product v ⊗ w We can have entangled states like e1 ⊗ e1 + e2 ⊗ e2

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Quantum correlations: formal definition

Quantum correlations: Alice and Bob generate output by measuring a shared quantum state

Definition

A correlation {p(a, b|x, y)} ∈ ROA×OB×IA×IB is quantum if there are:

• Hilbert spaces HA, HB,
• a state v ∈ HA ⊗ HB,
• measurements {Mx

a }a∈OA on HA for every x ∈ IA, and

• measurements {Ny

b }b∈OB on HB for every y ∈ IB,

such that p(a, b|x, y) = v∗Mx

a ⊗ Ny b v for all a, b, x, y.

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Sets of correlations

• Cc = Cc(OA, OB, IA, IB) := classical correlations
• Cq := quantum correlations with finite-dimensional Hilbert

spaces

• Cqs := Quantum correlations with any Hilbert spaces

All three sets are convex, Cc is closed Cq and Cqs capture behaviour of entangled states

What correlation set matches reality?

(what correlations can we generate in nature?)

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Bell tests rule out classical correlations

A

x ∈ IA a ∈ OA

B

y ∈ IB b ∈ OB Win if V (a, b|x, y) = 1 If Alice and Bob play game G using correlation p = {p(a, b|x, y)}, then success probability is ω(G; p) :=

• a,b,x,y

π(x, y)V (a, b|x, y)p(a, b|x, y). Classical success probability of game G is ωc(G) := maxp∈Cc ω(p) (maximum of linear functional over closed convex set)

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Bell tests rule out classical correlations II

Cq Cc ω(G, p) = ωc(G) p0 If ω(G; p0) > ωc(G), then p0 can’t be classical Non-classical correlations have been generated in experiments: Freedman and Clauser, 1972 Hensen et. al., TU Deflt, loop-hole free Bell tests, 2015

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Sets of correlations ct’d

• Cc = Cc(OA, OB, IA, IB) := classical correlations
• Cq := quantum correlations with finite-dimensional Hilbert

spaces

• Cqs := Quantum correlations with any Hilbert spaces

All three sets are convex, Cc is closed Cq and Cqs capture behaviour of entangled states

Which correlation set matches reality, and are there any other options?

Yes... Cqa := Cq = Cqs

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Commuting operator correlations

A B

Joint system

Recall: Tensor product approach to separated subsystems: Physical system = Hilbert space If HA and HB are Hilbert spaces of two separated systems, then joint system has Hilbert space HA ⊗ HB

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Commuting operator correlations

A B

Joint system

Different approach (used for instance in Haag-Kastler axioms): Commuting operator appraoch: Physical system = C ∗-algebra A, subsystem = subalgebra S Two subalgebras S1, S2 ⊆ A are separated if [S1, S2] = 0. (We can assume A = B(H), and states are still unit vectors in H)

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Commuting-operator correlations: formal definition

Definition

A correlation {p(a, b|x, y)} is commuting-operator if there is:

• a Hilbert space H and a state v ∈ H,
• measurements {Mx

a }a∈OA on H for every x ∈ IA, and

• measurements {Ny

b }b∈OB on H for every y ∈ IB,

such that

• Mx

a Ny b = Ny b Mx a for every a, b, x, y, and

• p(a, b|x, y) = v∗Mx

a Ny b v for all a, b, x, y.

(if H is finite-dimensional, then p ∈ Cq)

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Tsirelson’s problems

Cqc := commuting-operator correlations Cq ⊆ Cqs ⊆ Cqa ⊆ Cqc

strong weak

tensor product commuting operator All sets are convex... Cqc and Cqa are closed Which set describes reality? Are these sets even different? Tsirelson problems: is Ct, t ∈ {q, qs, qa} equal to Cqc? Strongest version is Cq

?

= Cqc: do ∞-dim’l commuting-operator correlations have finite-dimensional models?

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Recall: quantum solutions of linear systems

Ax = b : m × n linear system over Z2 Quantum solution: Collection of unitaries X1, . . . , Xn ∈ U(H) such that

• 1. X 2

j = 1 for all j,

• 2. n

j=1 X Aij j

= (−1)bi for all i = 1, . . . , n,

• 3. XjXk = XkXj if Aij, Aik = 0 for some i.

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Recall: quantum solutions of linear systems

Ax = b : m × n linear system over Z2 Solution group of Ax = b Γ(A, b) = x1, . . . , xn, J : x2

j = 1 = [xj, J] = J2 for all j

• j

xAij

j

= Jbi, i = 1, . . . , m [xj, xk] = 1 if Aij, Aik = 0, some i Group commutator: [a, b] = aba−1b−1

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Recall: Linear system non-local games

A

equation index 1 ≤ i ≤ m satisfying assignment to variables in equation i

B

variable index 1 ≤ j ≤ n assignment to xj Inputs chosen at random Players win if Alice’s

• utput is consistent

with Bob’s output

Theorem (Cleve-Mittal)

G = game associated to Ax = b. The following are equivalent:

• there is p ∈ Cq with ω(G; p) = 1,
• Ax = b has a finite-dimensional quantum solution, and
• J is non-trivial in a finite-dim’l representation of Γ(A, b)

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Theorem (Cleve-Mittal)

G = game associated to Ax = b. The following are equivalent:

• there is p ∈ Cq with ω(G; p) = 1,
• Ax = b has a finite-dimensional quantum solution, and
• J is non-trivial in a finite-dim’l representation of Γ(A, b).

Theorem (Cleve-Liu-S.)

G = game associated to Ax = b. The following are equivalent:

• there is p ∈ Cqc with ω(G; p) = 1,
• Ax = b has a quantum solution, and
• J is non-trivial in a representation of Γ(A, b).

Question: is there a group Γ(A, b) where J is non-trivial, but trivial

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Theorem (Cleve-Mittal)

G = game associated to Ax = b. The following are equivalent:

• there is p ∈ Cq with ω(G; p) = 1,
• Ax = b has a finite-dimensional quantum solution, and
• J is non-trivial in a finite-dim’l representation of Γ(A, b).

Theorem (Cleve-Liu-S.)

G = game associated to Ax = b. The following are equivalent:

• there is p ∈ Cqc with ω(G; p) = 1,
• Ax = b has a quantum solution, and
• J is non-trivial in a representation of Γ(A, b).

Question: is there a group Γ(A, b) where J is non-trivial, but trivial

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Theorem (Cleve-Mittal)

G = game associated to Ax = b. The following are equivalent:

• there is p ∈ Cq with ω(G; p) = 1,
• Ax = b has a finite-dimensional quantum solution, and
• J is non-trivial in a finite-dim’l representation of Γ(A, b).

Theorem (Cleve-Liu-S.)

G = game associated to Ax = b. The following are equivalent:

• there is p ∈ Cqc with ω(G; p) = 1,
• Ax = b has a quantum solution, and
• J is non-trivial in Γ(A, b).

Question: is there a group Γ(A, b) where J is non-trivial, but trivial

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Focus on the group theory

Theorem (Cleve-Mittal, Cleve-Liu-S.)

G = game associated to Ax = b. Then:

• there is p ∈ Cq with ω(G; p) = 1 if and only if J is non-trivial

in a finite-dim’l representation of Γ(A, b).

• there is p ∈ Cqc with ω(G; p) = 1 if and only if J is non-trivial

in Γ(A, b). Question: is there a group Γ(A, b) where J is non-trivial, but trivial in finite-dimensional representations? (In other words, can we find a non-residually-finite solution group?)

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

What groups are solution groups?

Γ(A, b) = x1, . . . , xn, J : x2

j = 1 = [xj, J] = J2 for all j

• j

xAij

j

= Jbi, i = 1, . . . , m [xj, xk] = 1 if Aij, Aik = 0, some i Imagine we can write down any group presentation we want, but generators in the relation will be forced to commute

Example (S3 = a, b : a2 = b2 = 1, (ab)3 = 1)

If ab = ba, then (ab)3 = a3b3 = ab So relations imply a = b, and S3 becomes Z2

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Group embedding theorem

Despite this seemingly strong restriction on our relations, solution groups are as complicated as general groups!

Theorem (S)

Any finitely-presented group G can be embedded in a solution group Γ(A, b). Given a central involution J0 of G, the embedding can be constructed to send J0 to J ∈ Γ(A, b). Gives us a way to translate group theory into non-local games

Corollary

There is a solution group Γ(A, b) where J = 1 but J is trivial in all finite-dim’l reps. As a result, Cq = Cqc.

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

How do we prove the embedding theorem

Can represent linear systems Ax = b / Z2 by hypergraphs

z y u v x x, y, z, u, v : xyxz = xuvu = e = x2 = y2 = · · · = v2

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Cq ⊆ Cqs ⊆ Cqa ⊆ Cqc

strong weak

tensor product commuting operator

• Refinement of this approach: Cqa = Cqs, i.e. Cq and Cqs are

not closed

• Size of parameters: |OA| = 8, |OB| = 2, but |IA|, |IB| ∼ 200.
• Dykema-Paulsen-Prakash: possible to get Cqa = Cqs with

|IA| = |IB| = 5, |OA| = |OB| = 2.

• Can we further reduce the number of inputs?

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

What else can we do with embedding theorem

Theorem (S)

Any finitely-presented group G can be embedded in a solution group Γ(A, b). Given a central involution J0 of G, the embedding can be constructed to send J0 to J ∈ Γ(A, b). Word problem for groups is undecidable, so:

• Undecidable to determine whether there is p ∈ Ct with

ω(G, p) = 1 for any t ∈ {q, qs, qa, qc}.

• T. Fritz: quantum logic is undecidable

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra

Cq ⊆ Cqs ⊆ Cqa ⊆ Cqc

strong weak

tensor product commuting operator Final question: what about the weak Tsirelson problem? This is equivalent to asking whether we can separate Cqc from Ct, t ∈ {q, qs, qa} with a Bell inequality

Theorem (Fritz, JNPPSW, Ozawa)

The weak Tsirelson problem is equivalent to the Connes embedding problem.

Quantum entanglement, Tsirelson’s problem, and group theory William Slofstra