1. Tsuyoshi Ito National Institute of Informatics (NII), Japan
Generalized Tsirelson Inequalities, Commuting Operator Provers, and Multi-Prover Interactive Proof Systems
Tsuyoshi Ito
National Institute of Informatics (NII), Japan
Joint work with Hirotada Kobayashi (NII), Daniel Preda (UC Berkeley), Xiaoming Sun (Tsinghua University), Andrew Yao (Tsinghua University)
arXiv: [quant-ph]
QIP 2008, Dec 17-21, 2007, New Delhi, India

2. Outline
Problem:
How to limit entanglement-assisted cheating in cooperative multi-player games (“non-local games”)
(≒ How to prove Tsirelson-type inequalities for quantum correlations)
Answer: Use commuting-operator players model
Specifically, we give the limit of cheating in:
Some specific n-player games (“n-player Magic Square game,” extending the most basic Tsirelson inequality)
A class of 3-player binary games with application in multi-prover interactive proofs

3. Magic Square game (3×3) [Aravind 2002] [Cleve, Høyer, Toner, Watrous 2004]
Constraints:
Products of 3 cells in each row/column = +1
Except: Product of 3 cells in column 3 = －1
(Clearly impossible)
3×3 matrix of ±1 values
A and B claim such a matrix exist. How to verify their claim?
Player A
Player B
Row/Column
Value of asked cell
Values of 3 cells
Cell
Referee
Game value = 17/18:
Referee knows players are telling a lie with probability ≧1/18

4. Magic Square game (3×3) [Aravind 2002] [Cleve, Høyer, Toner, Watrous 2004]
This verification breaks if players share prior entanglement
Player A
Player B
Row/Column
Value of asked cell
Values of 3 cells
Cell
Referee
Entangled game value = 1 (Entanglement-assisted perfect cheating; Pseudo-telepathy game)
How to prevent? → Ask 3 players [Sun, Yao, Preda, QIP 2007]

5. n-player version of Magic Square gameb1 … z1 z2 a2 y2 bn cn an
Constraints:
Products of n cells in each row/column = +1
Except: Product of n cells in column n = －1
(Clearly impossible)
What referee does:
Choose 1 row or 1 column
Ask n cells in chosen row/column to n players
Verify the constraint
n×n matrix of ±1 values
Referee A B i j ai bj … Z t zt
n players
Winning probability =
2n+E(a1b1…z1)+E(a2b2…z2)+…+E(anbn…zn) +E(a1bn…z2)+E(a2b1…z3)+…－E(anbn-1…z1) 4n
E(・): expected value

6. Unentangled case Game value = 1 2n 1－ a1 b1 b2 a2 n=2: ≦
4+E(a1b1)+E(a1b2)+E(a2b2)－E(a2b1) 8 ≦ 3 4
E(a1b1)+E(a1b2)+E(a2b2)－E(a2b1) ≦ 2
Clauser-Horne-Shimony-Holt (CHSH) inequality [1969]

7. Result: Entangled value of n-player Magic Square game
Game value = (<1) with entanglement
1+cos(1/2n) 2
E(a1b1…z1)+E(a2b2…z2)+…+E(anbn…zn) +E(a1bn…z2)+E(a2b1…z3)+…－E(anbn-1…z1)
≦ 2n cos 2n 1
Equivalently,
4+E(a1b1)+E(a1b2)+E(a2b2)－E(a2b1) 8 ≦
2+√2 4
E(a1b1)+E(a2b2)+…+E(anbn) +E(a1bn)+E(a2b1)+…－E(anbn-1)
≦ 2n cos 2n 1 a1 b1 b2 a2
n=2:
E(a1b1)+E(a1b2)+E(a2b2)－E(a2b1) ≦ 2√2
Tsirelson inequality for CHSH [Tsirelson 1980]
n=3:
[Sun, Yao, Preda, QIP 2007]
6+E(a1b1c1)+E(a2b2c2)+E(a3b3c3) +E(a1b3c2)+E(a2b1c3)－E(a3b2c1) 12 ≦
2+√3 4 a1 b1 c1 c2 a2 b2 b3 c3 a3
Special case is known [Wehner 2006]
Upper bound is achievable [Peres 1993]
E(a1b1c1)+E(a2b2c2)+E(a3b3c3) +E(a1b3c2)+E(a2b1c3)－E(a3b2c1)
≦ 3√3
The first example of 3(or more)-player Tsirelson inequality with maximum known

8. Entangled players model
Players prepare:
A state |Ψ〉 in HA⊗HB⊗HC
{±1}-valued observables Ai, Bj, Ck on HA, HB, HC
Probability Pr(ai,bj,ck)
= 〈Ψ| ⊗ ⊗ |Ψ〉
Expectation E(aibjck) = 〈Ψ| Ai⊗Bj⊗Ck |Ψ〉
Player A
Player B i j ai bj
Referee k ck
I+aiAi 2
I+bjBj 2
I+ckCk 2
Player C

9. Commuting-operator model [Tsirelson 1980, 1993]
Players prepare:
A state |Ψ〉 in H
{±1}-valued observables Ai, Bj, Ck on H, such that Ai, Bj, Ck pairwise commute for any i,j,k
Probability Pr(ai,bj,ck) = 〈Ψ| ・ ・ |Ψ〉
Expectation E(aibjck) = 〈Ψ|AiBjCk|Ψ〉
Player A
Player B i j ai bj
Referee k ck
I+aiAi 2
I+bjBj 2
I+ckCk 2
Player C
Note: Conceptual model, no physical realization

10. Comparing two models
Entangled model: Commutativity arises from tensor structure
Commuting-operator model: Explicitly require only commutativity
Not much is known:
Entanglement ⊆ Commuting-operator
Equivalent if 2 players and finite-dimensional state [Tsirelson 2006]
In general, … ???
[Navascues, Pironio, Acin 2007]:
SDP1 ≧ SDP2 ≧ SDP3 ≧ … ≧ SDPi ≧ … ≧ com ≧ ent

11. Proof idea
〈Ψ|A1B1C1|Ψ〉+〈Ψ|A2B2C2|Ψ〉+〈Ψ|A3B3C3|Ψ〉 +〈Ψ|A1B3C2|Ψ〉+〈Ψ|A2B1C3|Ψ〉－〈Ψ|A3B2C1|Ψ〉< by contradiction
〈Ψ|A1B1C1|Ψ〉=〈Ψ|A2B2C2|Ψ〉=〈Ψ|A3B3C3|Ψ〉 =〈Ψ|A1B3C2|Ψ〉=〈Ψ|A2B1C3|Ψ〉=－〈Ψ|A3B2C1|Ψ〉=1
Ai, Bj and Ck commute
Ai2=Bj2=Ck2=I a1 b1 c1 c2 a2 b2 b3 c3 a3
〈Ψ|(A1B1C1)(A3B2C1)(A2B2C2)(A2B1C3)× (A3B3C3) (A1B3C2)|Ψ〉=－1

12. Proof 〈Ψ|(A1B1C1)(A3B2C1)(A2B2C2)(A2B1C3)(A3B3C3) (A1B3C2)|Ψ〉=－1
cos-1〈Ψ|A1B1C1|Ψ〉=θ1, cos-1〈Ψ|A2B2C2|Ψ〉=θ2, cos-1〈Ψ|A3B3C3|Ψ〉=θ3, cos-1〈Ψ|A1B3C2|Ψ〉=θ4, cos-1〈Ψ|A2B1C3|Ψ〉=θ5, cos-1(－〈Ψ|A3B2C1|Ψ〉)=θ6
θ1+θ2+θ3+θ4+θ5+θ6≧π
Maximize ∑ cos θi → θi＝π/6 (∀i) a1 b1 c1 c2 a2 b2 b3 c3 a3
The same proof applies to general n

13. Limit of cheating in multi-prover interactive proofs [Kempe, Kobayashi, Matsumoto, Toner, Vidick 2007]
3-prover protocol where perfect cheating is impossible
2-prover protocol
Prover A
Prover B
Prover A
Prover B i j i j ai bj ai bj
Verifier
Verifier
i or j
ai or bj
Prover C

14. Ask 3 variables in one clause or Ask the same variables to all provers
Our approach
3-prover protocol where perfect cheating is impossible
Prover A
Prover B
3SAT
f(x1, …, xn) i j xi xj
Verifier
Ask 3 variables in one clause or Ask the same variables to all provers k xk
Prover C
This becomes a binary interactive proof system

15. NP-hardness of computing entangled value of games
Given 3-player game with entangled players, it is NP-hard to decide if game value =1 or <1-1/poly [Kempe, Kobayashi, Matsumoto, Toner, Vidick 2007]
Our result: Still NP-hard with binary games
(Proof method: KKMTV with modification to commuting-operator)
Implication: NP-hard to compute the bound in 3-party Tsirelson inequality with ±1-observables
Cf.
2 players, binary, whether value =1 or not ⇒ P by 2SAT [Cleve, Høyer, Toner, Watrous 2004]
2 players, binary, “XOR type game” ⇒ P by SDP [Tsirelson 1980]

16. Conclusion How to limit cheating by entangled players
Technique: commuting-operator model
Specific examples with known entangled value with 3 or more players
NP-hardness of computing the bound of 3-party Tsirelson inequality with ±1-valued observables

17. Open questions arXiv:0712.2163 [quant-ph]
Is tensor only useful for commutativity?
Deciding if entangled value =1 or <1－1/poly for 2-player game (not necessarily binary) → Still NP-hard! [Ito, Kobayashi, Matsumoto; in prep.]
Better bound of entangled value (like [Cleve, Gavinsky, Jain, 2007] in NP case)?
Thank you
arXiv: [quant-ph]