Title: Generalized Tsirelson Inequalities, Commuting Operator Provers, and MultiProver Interactive Proof Sy
1Generalized Tsirelson Inequalities,Commuting
Operator Provers,and Multi-Prover Interactive
Proof Systems
- Tsuyoshi Ito
- National Institute of Informatics (NII), Japan
Joint work withHirotada Kobayashi (NII),Daniel
Preda (UC Berkeley),Xiaoming Sun (Tsinghua
University),Andrew Yao (Tsinghua University)
arXiv0712.2163 quant-ph
QIP 2008, Dec 17-21, 2007, New Delhi, India
2Outline
Problem
How to limit entanglement-assisted cheatingin
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 gameswith application
in multi-prover interactive proofs
3Magic Square game (33)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)
33 matrix of 1 values
A and B claim such a matrix exist. How to verify
their claim?
Player A
Player B
Row/Column
Value ofasked cell
Values of3 cells
Cell
Referee
Game value 17/18
Referee knows players are telling a liewith
probability ?1/18
4Magic Square game (33)Aravind 2002 Cleve,
Høyer, Toner, Watrous 2004
This verification breaks if players share prior
entanglement
Player A
Player B
Row/Column
Value ofasked cell
Values of3 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
5n-player versionof Magic Square game
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
nn matrix of 1 values
Winning probability
E() expected value
6Unentangled case
n2
E(a1b1)E(a1b2)E(a2b2)-E(a2b1) ? 2
Clauser-Horne-Shimony-Holt (CHSH) inequality
1969
7Result Entangled valueof n-player Magic Square
game
Equivalently,
n2
E(a1b1)E(a1b2)E(a2b2)-E(a2b1) ? 2v2
Tsirelson inequality for CHSH Tsirelson 1980
n3
Sun, Yao, Preda, QIP 2007
- Special case is known Wehner 2006
- Upper bound is achievable Peres 1993
The first example of 3(or more)-player Tsirelson
inequality with maximum known
8Entangled players model
- Players prepare
- A state ?gt in HA?HB?HC
- 1-valued observables Ai, Bj, Ck on HA, HB,
HC - Probability Pr(ai,bj,ck)
- lt? ? ?
?gt - Expectation E(aibjck) lt? Ai?Bj?Ck ?gt
Player A
Player B
i
j
ai
bj
Referee
k
ck
Player C
9Commuting-operator modelTsirelson 1980, 1993
- Players prepare
- A state ?gt 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) lt?
?gt - Expectation E(aibjck) lt?AiBjCk?gt
Player A
Player B
i
j
ai
bj
Referee
k
ck
Player C
Note Conceptual model, no physical realization
10Comparing 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-dimensionalsta
te Tsirelson 2006 - In general, ???
Navascues, Pironio, Acin 2007 SDP1 ? SDP2 ?
SDP3 ? ? SDPi ? ? com ? ent
11Proof idea
lt?A1B1C1?gtlt?A2B2C2?gtlt?A3B3C3?gtlt?A1B3C2
?gtlt?A2B1C3?gt-lt?A3B2C1?gtlt 6 by
contradiction
lt?A1B1C1?gtlt?A2B2C2?gtlt?A3B3C3?gtlt?A1B3C2
?gtlt?A2B1C3?gt-lt?A3B2C1?gt1
- Ai, Bj and Ck commute
- Ai2Bj2Ck2I
lt?(A1B1C1)(A3B2C1)(A2B2C2)(A2B1C3)
(A3B3C3) (A1B3C2)?gt-1
12Proof
lt?(A1B1C1)(A3B2C1)(A2B2C2)(A2B1C3)(A3B3C3)
(A1B3C2)?gt-1
cos-1lt?A1B1C1?gt?1, cos-1lt?A2B2C2?gt?2,cos-1lt
?A3B3C3?gt?3, cos-1lt?A1B3C2?gt?4,cos-1lt?A2B1
C3?gt?5, cos-1(-lt?A3B2C1?gt)?6
?1?2?3?4?5?6?p
Maximize ? cos ?i ? ?ip/6 (?i)
The same proof applies to general n
13Limit of cheatingin multi-prover interactive
proofs Kempe, Kobayashi, Matsumoto, Toner,
Vidick 2007
3-prover protocolwhere 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
14Our approach
3-prover protocolwhere perfect cheating is
impossible
Prover A
Prover B
3SAT f(x1, , xn)
i
j
xi
xj
Verifier
k
xk
Ask 3 variables in one clauseorAsk the same
variables to all provers
Prover C
This becomes a binary interactive proof system
15NP-hardness of computingentangled value of games
- Given 3-player game with entangled players,it is
NP-hard to decide if game value 1 or
lt1-1/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
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
Cf.
16Conclusion
- How to limit cheating by entangled players
- Technique commuting-operator model
- Specific examples with known entangled valuewith
3 or more players - NP-hardness of computing the boundof 3-party
Tsirelson inequalitywith 1-valued observables
17Open questions
- Is tensor only useful for commutativity?
- Deciding if entangled value 1 or lt1-1/polyfor
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
arXiv0712.2163 quant-ph