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This research explores the hardness of approximating entangled games, where players share quantum entanglement to improve their strategy. The study investigates the difficulty in computing the value of these games and the impact of entanglement on winning probabilities. It presents results showing the challenges in approximating the optimal value of entangled games, compared to classical and unique games. The proof outlines how distinguishing between different game values is NP-hard and discusses the limitations on approximating entangled games efficiently. The study sheds light on the quantum complexities and the power gained by entangled provers in game theory.
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QIP’08, Delhi Hardness of Approximating Entangled Games Thomas Vidick UC Berkeley Joint work with Julia Kempe, Hirotada Kobayashi, Keiji Matsumoto and Ben Toner
2-Prover Games • Verifier samples q,q’ in Q according to π • He sends q to A, q’ to B • He receives a,a’ as answers • Provers win iff V(a,a’|q,q’) = 1 Alice Bob q q’ Verifier Value of the game ω = Max. Winning Prob. (over all the prover’s strategies) Game G = (Q,A,π,V) of size |G| = poly(|Q|) a a’ accept reject
Alice Bob How hard is it to approximate ω? Input: game G = (Q,A,π,V) of size |G| = poly(|Q|) q’ Computing ω is NP-hard (Cook-Levin) Approximating ω is NP-hard (PCP Thm.) Approximating ω is NP-hard [Hastad’01] XOR-games • What is the situation in the quantum world? • Understand the power gained by entangled provers. How does it affect the value of a game? General classical games Alice Bob a a’ Unique games q Approximating ω is NP-hard [Khot’02] [CONJECTURE] Verifier accept reject
Entangled Games |ψ> The provers may share arbitrary entanglement q’ Alice Bob Value of the game ω* = Max. Winning Prob. (over all the prover’s strategies and shared |ψ>) a a’ • Provers can now produce nonlocal correlations: • - Bell inequalities 1962 • Hardness of approximating ω* is a major open question q Verifier accept reject
How hard is it to approximate ω*? (entangled provers) q’ Computing ω* is in P [CHTW04] (there is a semi- definite program) XOR-games What about general entangledgames? Alice Bob a a’ Unique games Can we also compute their value in P (is there an SDP?) q Approximating ω* is in P [KRT07] (there is an SDP) Verifier accept reject
Our Results Entangled games are hard to approximate • Theorem • There is an ε=1/poly(|Q|) such that distinguishing between ω*=1 and ω*≤1-ε is NP-hard for games with • Classical communication with three provers. • Quantum communication with two provers. • Corollary • Unless P=NP there are no SDPs of size poly(|Q|) which optimum approximates the value of entangled games (in contrast to XOR-games, unique games, single-prover quantum games). • The hierarchy of [PironioNavascuesAcin’07] cannot yield polynomial-sized SDPs (see also [DohertyLiangTonerWehner’08])
Outline of the proof • Do a reduction from the classical setting • Start with a classical 2-player game → NP-hard to distinguish between ω=1 and ω < 0.01 • Transform into a quantum game such that • ω=1 ω*=1 • ω<0.01 ω*<1-ε • (by construction) • (we show: ω*<1-εω<0.01) • Transform into a quantum game such that • Add tests that limit the prover’s use of entanglement • → They can do little better than using it as shared • randomness • → It is NP-hard to distinguish between ω*=1 and ω* < 1- ε
The modified game Alice Bob Alice Bob a a’ |a> |a’> q |q> q’ |q’> ω=1→ ω*=1 : Good provers from the classical game pass both tests. Verifier Verifier accept accept reject reject • With probability ½ do : • Classical Test : play classical game : send |q>, |q’> and check answers • Quantum Test : send |0>|q>|q’> + |1>|q’>|q> • After answer, do C-SWAP on last two registers, measure first in |±> basis, accept if |+> Quantum provers : |q> → |a>, |q’> → |a’> |0>|q>|q’> + |1>|q’>|q> → |0>|a>|a’> + |1>|a’>|a> → (|0>+|1>)|a>|a’> Classical provers : q → a, q’ → a’
Goal: ω*>1-εω>0.01 After you do the math… • The prover’s strategies can be described by orthogonal projectors for every q such that : • Classical test : Prob. that the verifier measures a,a’ as • answers to q,q’: • Swap-test : The W’s almost commute: → This is the key relation that we will use to round to a classical strategy
Special Case : ε=0 : W’s are diagonal in the same basis Set of projectors ↔ partition of space Norm of product ↔ dimension of the intersection We can easily simulate this classically by sampling using shared randomness → Classical strategy has the same success prob. as the quantum strategy
General case : ε = 1/poly • Are the W’s “almost diagonal” in a common basis? • If we can find Ŵ such that Ŵ≈ W and Ŵ can be simultaneously diagonalized, then we are done • “Do almost commutingmatrices nearly commute?” • Long-standing open question [Halmos’76] • Finally proven true for two hermitian matrices [Lin’97] • False for three or more hermitian matrices and the operator norm! • But… • - Our matrices have a special form (projectors) • - Our norm is not the operator norm • → It is an open question • We get around this by using a different rounding technique • → we lose a factor O(|Q|4 ε) in statistical distance
Hardness of computing the value of entangled games Computing ω* is in P 2-prover quantum games XOR-games 3-prover classical games Unique games Approximating ω* to within 1/poly is NP-hard Approximating ω* is in P
Open questions • Increase the gap! • A better gap could yield NEXP QMIP* • Prove that “almost commuting” projectors “nearly commute” • Find a better rounding procedure • Show some upper bounds • All known bounds depend on the amount of entanglement used