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Register a new userEntangled games do not require much entanglement (withdrawn)
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Authors
Gus Gutoski
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We prove an explicit upper bound on the amount of entanglement required by any strategy in a two-player cooperative game with classical questions and quantum answers. Specifically, we show that every strategy for a game with n-bit questions and n-qubit answers can be implemented exactly by players who share an entangled state of no more than 5n qubits--a bound which is optimal to within a factor of 5/2. Previously, no upper bound at all was known on the amount of entanglement required even to approximate such a strategy. It follows that the problem of computing the value of these games is in NP, whereas previously this problem was not known to be computable.
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The behavior of games repeated in parallel, when played with quantumly entangled players, has received much attention in recent years. Quantum analogues of Razs classical parallel repetition theorem have been proved for many special classes of games. However, for general entangled games no parallel repetition theorem was known. We prove that the entangled value of a two-player game $G$ repeated $n$ times in parallel is at most $c_G n^{-1/4} log n$ for a constant $c_G$ depending on $G$, provided that the entangled value of $G$ is less than 1. In particular, this gives the first proof that the entangled value of a parallel repeated game must converge to 0 for all games whose entangled value is less than 1. Central to our proof is a combination of both classical and quantum correlated sampling.
We study the quantum moment problem: Given a conditional probability distribution together with some polynomial constraints, does there exist a quantum state rho and a collection of measurement operators such that (i) the probability of obtaining a particular outcome when a particular measurement is performed on rho is specified by the conditional probability distribution, and (ii) the measurement operators satisfy the constraints. For example, the constraints might specify that some measurement operators must commute. We show that if an instance of the quantum moment problem is unsatisfiable, then there exists a certificate of a particular form proving this. Our proof is based on a recent result in algebraic geometry, the noncommutative Positivstellensatz of Helton and McCullough [Trans. Amer. Math. Soc., 356(9):3721, 2004]. A special case of the quantum moment problem is to compute the value of one-round multi-prover games with entangled provers. Under the conjecture that the provers need only share states in finite-dimensional Hilbert spaces, we prove that a hierarchy of semidefinite programs similar to the one given by Navascues, Pironio and Acin [Phys. Rev. Lett., 98:010401, 2007] converges to the entangled value of the game. It follows that the class of languages recognized by a multi-prover interactive proof system where the provers share entanglement is recursive.
We investigate sharing of bipartite entanglement in a scenario where half of an entangled pair is possessed and projectively measured by one observer, called Alice, while the other half is subjected to measurements performed sequentially, independently, and unsharply, by multiple observers, called Bobs. We find that there is a limit on the number of observers in this entanglement distribution scenario. In particular, for a two-qubit maximally entangled initial shared state, no more than twelve Bobs can detect entanglement with a single Alice for arbitrary -- possibly unequal -- sharpness parameters of the measurements by the Bobs. Moreover, the number of Bobs remains unaltered for a finite range of near-maximal pure initial entanglement, a feature that also occurs in the case of equal sharpness parameters at the Bobs. Furthermore, we show that for non-maximally entangled shared pure states, the number of Bobs reduces with the amount of initial entanglement, providing a coarse-grained but operational measure of entanglement.
We show a parallel repetition theorem for the entangled value $omega^*(G)$ of any two-player one-round game $G$ where the questions $(x,y) in mathcal{X}timesmathcal{Y}$ to Alice and Bob are drawn from a product distribution on $mathcal{X}timesmathcal{Y}$. We show that for the $k$-fold product $G^k$ of the game $G$ (which represents the game $G$ played in parallel $k$ times independently), $ omega^*(G^k) =left(1-(1-omega^*(G))^3right)^{Omegaleft(frac{k}{log(|mathcal{A}| cdot |mathcal{B}|)}right)} $, where $mathcal{A}$ and $mathcal{B}$ represent the sets from which the answers of Alice and Bob are drawn.
We discuss the possibility of quantum interferences and entanglement of photons which exist at different intervals of time, i.e., one photon being recorded before the other has been created. The corresponding two-photon correlation function is shown to violate Bells inequalities.
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