This review article is concerned with a recently uncovered connection between operator spaces, a noncommutative extension of Banach spaces, and quantum nonlocality, a striking phenomenon which underlies many of the applications of quantum mechanics to information theory, cryptography, and algorithms. Using the framework of nonlocal games, we relate measures of the nonlocality of quantum mechanics to certain norms in the Banach and operator space categories. We survey recent results that exploit this connection to derive large violations of Bell inequalities, study the complexity of the classical and quantum values of games and their relation to Grothendieck inequalities, and quantify the nonlocality of different classes of entangled states.
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A compactness argument, combined with a simple discretization, shows the existence of a finite d = d(ε, n), but does not give any estimate on the size of d.
A compactness argument together with a simple discretization of Ball(ℂd ⊗ ℂd) guarantees that one may always find a countable family of states that is guaranteed to enable strategies achieving values arbitrarily close to the optimum; here, we are interested in the existence of “natural” such families.
A formal definition of is beyond the scope of this survey; for our purposes, it will suffice to think of it as a C∗-algebra containing N copies of and with the universal property that for every family of unital ∗-homomorphisms , i = 1, …, N, there is a unital ∗-homomorphism which extends the πi.
All known proofs of Theorem 3.8 involve a reduction to Theorem 3.7.
Alternatively, random Gaussian vectors will achieve the same effect.
An observable is a Hermitian matrix that squares to identity.
As shown in Section III C, two-player quantum XOR games provide a bipartite scenario in which the maximally entangled state can be far from optimal, as demonstrated by the family of games Cn for which β∗(Cn) = 1 but .
General reductions relating the property of parallel repetition to that of direct sum are known; see, e.g., Ref. 37.
Here, one can verify, as was already the case for the entangled bias of classical XOR games, that restricting the supremum in (38) to Hermitian A, B does not affect its value.
L(N, K) is the same (operator) space as 𝒩ℂ(N, K)∗, introduced in Section IV B.
Note that here the o.s.s. is defined without explicitly specifying an embedding of in some . Although Ruan’s theorem assures that such an embedding must exist that leads to the sequence of norms (6), finding that embedding explicitly can be a difficult problem. In particular, for the simplest embedding is based on the universal C∗-algebra associated to the free group with n generators C∗(𝔽n).
Note that in case V(a, b, x, y) ∈ {0, 1} this corresponds to requiring that the players “win” each of the ℓ instances of the game played in parallel.
Of course Grothendieck gave a precise meaning to “reasonable”: it is required that the tensor norms satisfy simple compatibility conditions with the Banach space structure of the spaces to be combined.
Sometimes the function V is required to take values in {0, 1}. Our slightly more general definition can be interpreted as allowing for randomized predicates.
The players in an XOR game with quantum messages are always quantum, and it would be less natural to talk of the “classical” bias.
The presentation of JPn given above is in fact based on the construction by Regev, who in particular suggested the use of independent coefficients gi,x,a and gi,y,b where40 we used the same coefficients on both sides.
The proof given in Ref. 59 is highly non-constructive and only guarantees the existence of T.
The Schmidt rank of a bipartite pure state is the rank of its reduced density matrix on either subsystem. Its entanglement entropy is the Shannon entropy of the singular values of the reduced density matrix.
The state that can be extracted from the strategy constructed in the proof of Corollary 3.5 is rather complicated, with d3 coefficients chosen, up to normalization, as i.i.d. standard Gaussians. As will be shown in Section V B 1, this is to some extent necessary, as the natural generalization of maximally entangled states to multiple parties (Schmidt states, a class that encompasses the GHZ state) can only lead to bounded violations in multiplayer XOR games.
The term observable is used simultaneously for a quantity, such as spin or momentum, and the operator that measures the quantity.
This observation was already known to Tsirelson.
Unpublished work of the second author shows that the exponent of ε can be improved from 2 to 1, and that an exponential dependence on poly(ε−1) is necessary.
Using that C∗-algebras are operator spaces, one can verify that this definition for the minimal norm coincides with the one given in (11).
We will see in Sec. IV B that the lemma is false for general Bell functionals.
