Generalized Tsirelson Inequalities, Commuting Operator Provers, and MultiProver Interactive Proof Sy

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Generalized 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
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Outline
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

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Magic 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
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Magic 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
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n-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
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Unentangled case
n2
E(a1b1)E(a1b2)E(a2b2)-E(a2b1) ? 2
Clauser-Horne-Shimony-Holt (CHSH) inequality
1969
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Result 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
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Entangled 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
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Commuting-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
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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-dimensionalsta
  te Tsirelson 2006
• In general, ???

Navascues, Pironio, Acin 2007 SDP1 ? SDP2 ?
SDP3 ? ? SDPi ? ? com ? ent
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Proof 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
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Proof
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
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Limit 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
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Our 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
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NP-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.
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Conclusion

• 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

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Open 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