Quantum information processing is at the crossroads of physics, mathematics and computer science. It is concerned with what we can and cannot do with quantum information that goes beyond the abilities of classical information processing devices. Communication complexity is an area of classical computer science that aims at quantifying the amount of communication necessary to solve distributed computational problems. Quantum communication complexity uses quantum mechanics to reduce the amount of communication that would be classically required.
Also, Aravind has generalized his own magic square idea @cite_14 to a two-player pseudo-telepathy game in which the players share @math Bell states, @math being an arbitrary odd number larger than 1.
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Bell inequality violations correspond to behavior of entangled quantum systems that cannot be simulated classically. We give two new two-player games with Bell inequality violations that are stronger, fully explicit, and arguably simpler than earlier work.The first game is based on the Hidden Matching problem of quantum communication complexity, introduced by Bar-Yossef, Jayram, and Kerenidis. This game can be won with probability 1 by a quantum strategy using a maximally entangled state with local dimension n (e.g., log n EPR-pairs), while we show that the winning probability of any classical strategy differs from 1 2 by at most O(log(n) sqrt(n)).The second game is based on the integrality gap for Unique Games by Khot and Vishnoi and the quantum rounding procedure of Kempe, Regev, and Toner. Here n-dimensional entanglement allows to win the game with probability 1 (log n)^2, while the best winning probability without entanglement is 1 n. This near-linear ratio ( Bell inequality violation'') is near-optimal, both in terms of the local dimension of the entangled state, and in terms of the number of possible outputs of the two players. We show that it is NP-hard to approximate, to within an additive constant, the maximum success probability of players sharing quantum entanglement in a two-player game with classical questions of logarithmic length and classical answers of constant length. As a corollary, the inclusion NEXP ⊆ MIP*, first shown by Ito and Vidick (FOCS'12) with three provers, holds with two provers only. The proof is based on a simpler, improved analysis of the low-degree test of Raz and Safra (STOC'97) against two entangled provers. In the generalized Russian cards problem, the three players Alice, Bob and Cath draw @math a , b and @math c cards, respectively, from a deck of @math a + b + c cards. Players only know their own cards and what the deck of cards is. Alice and Bob are then required to communicate their hand of cards to each other by way of public messages. For a natural number @math k , the communication is said to be @math k -safe if Cath does not learn whether or not Alice holds any given set of at most @math k cards that are not Cath's, a notion originally introduced as weak @math k -security by Swanson and Stinson. An elegant solution by Atkinson views the cards as points in a finite projective plane. We propose a general solution in the spirit of Atkinson's, although based on finite vector spaces rather than projective planes, and call it the geometric protocol'. Given arbitrary @math c , k > 0 , this protocol gives an informative and @math k -safe solution to the generalized Russian cards problem for infinitely many values of @math ( a , b , c ) with @math b = O ( a c ) . This improves on the collection of parameters for which solutions are known. In particular, it is the first solution which guarantees @math k -safety when Cath has more than one card. We introduce quantum XOR games, a model of two-player, one-round games that extends the model of XOR games by allowing the referee’s questions to the players to be quantum states. We give examples showing that quantum XOR games exhibit a wide range of behaviors that are known not to exist for standard XOR games, such as cases in which the use of entanglement leads to an arbitrarily large advantage over the use of no entanglement. By invoking two deep extensions of Grothendieck’s inequality, we present an efficient algorithm that gives a constant-factor approximation to the best performance that players can obtain in a given game, both in the case that they have no shared entanglement and that they share unlimited entanglement. As a byproduct of the algorithm, we prove some additional interesting properties of quantum XOR games, such as the fact that sharing a maximally entangled state of arbitrary dimension gives only a small advantage over having no entanglement at all. We present a protocol that transforms any quantum multi-prover interactive proof into a nonlocal game in which questions consist of logarithmic number of bits and answers of constant number of bits. As a corollary, it follows that the promise problem corresponding to the approximation of the nonlocal value to inverse polynomial accuracy is complete for QMIP*, and therefore NEXP-hard. This establishes that nonlocal games are provably harder than classical games without any complexity theory assumptions. Our result also indicates that gap amplification for nonlocal games may be impossible in general and provides a negative evidence for the feasibility of the gap amplification approach to the multi-prover variant of the quantum PCP conjecture. Isbell in 1959 was the first to find a weighted game without a minimum integer realization in which the affected players do not play a symmetric role in the game. His example has 12 players in a weighted decisive game, i.e. a weighted game for which a coalition wins iff its complement loses. The goal of this article is to provide a procedure for weighted games that allows finding out what is the minimum number of players needed to get a weighted game without a minimum integer weighted representation in which the affected players do not play a symmetric role in the game. We prove, by means of an algorithm, that the minimum number of voters required is nine. We show that any number of parties can coherently exchange any one pure quantum state for another, without communication, given prior shared entanglement. Two applications of this fact to the study of multi-prover quantum interactive proof systems are given. First, we prove that there exists a one-round two-prover quantum interactive proof system for which no finite amount of shared entanglement allows the provers to implement an optimal strategy. More specifically, for every fixed input string, there exists a sequence of strategies for the provers, with each strategy requiring more entanglement than the last, for which the probability for the provers to convince the verifier to accept approaches 1. It is not possible, however, for the provers to convince the verifier to accept with certainty with a finite amount of shared entanglement. The second application is a simple proof that multi-prover quantum interactive proofs can be transformed to have near-perfect completeness by the addition of one round of communication. Two players are independently placed on a commonly labelled network X. They cannot see each other but wish to meet in least expected time. We consider continuous and discrete versions, in which they may move at unit speed or between adjacent distinct nodes, respectively. There are two versions of the problem (asymmetric or symmetric), depending on whether or not we allow the players to use different strategies. After obtaining some optimality conditions for general networks, we specialize to the interval and circle networks. In the first setting, we extend the work of J. V. Howard; in the second we prove a conjecture concerning the optimal symmetric strategy. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 256–274, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002 nav.10011 Two players A and B are randomly placed on a line. The distribution of the distance between them is unknown except that the expected initial distance of the (two) players does not exceed some constant @math The players can move with maximal velocity 1 and would like to meet one another as soon as possible. Most of the paper deals with the asymmetric rendezvous in which each player can use a different trajectory. We find rendezvous trajectories which are efficient against all probability distributions in the above class. (It turns out that our trajectories do not depend on the value of @math ) We also obtain the minimax trajectory of player A if player B just waits for him. This trajectory oscillates with a geometrically increasing amplitude. It guarantees an expected meeting time not exceeding @math We show that, if player B also moves, then the expected meeting time can be reduced to @math The expected meeting time can be further reduced if the players use mixed strategies. We show that if player B rests, then the optimal strategy of player A is a mixture of geometric trajectories. It guarantees an expected meeting time not exceeding @math This value can be reduced even more (below @math ) if player B also moves according to a (correlated) mixed strategy. We also obtain a bound for the expected meeting time of the corresponding symmetric rendezvous problem. Suppose that @math players are placed randomly on the real line at consecutive integers, and faced in random directions. Each player has maximum speed one, cannot see the others, and doesn't know his relative position. What is the minimum time @math required to ensure that all the players can meet together at a single point, regardless of their initial placement? We prove that @math , @math , and @math is asymptotic to @math We also consider a variant of the problem which requires players who meet to stick together, and find in this case that three players require @math time units to ensure a meeting. This paper is thus a minimax version of the rendezvous search problem, which has hitherto been studied only in terms of minimizing the expected meeting time. Consider three players Alice, Bob and Cath who hold a, b and c cards, respectively, from a deck of d = a+b+c cards. The cards are all different and players only know their own cards. Suppose Alice and Bob wish to communicate their cards to each other without Cath learning whether Alice or Bob holds a specific card. Considering the cards as consecutive natural numbers 0, 1, . . . , we investigate general conditions for when Alice or Bob can safely announce the sum of the cards they hold modulo an appropriately chosen integer. We demonstrate that this holds whenever a, b > 2 and c = 1. Because Cath holds a single card, this also implies that Alice and Bob will learn the card deal from the other player’s announcement. This paper studies a cooperative card game called Hanabi from an algorithmic combinatorial game theory viewpoint. The aim of the game is to play cards from @math to @math in increasing order (this has to be done independently in @math different colors). Cards are drawn from a deck one by one. Drawn cards are either immediately played, discarded or stored for future use (overall each player can store up to @math cards). We introduce a simplified mathematical model of a single-player version of the game, and show several complexity results: the game is intractable in a general setting, but becomes tractable (and even linear) in some interesting restricted cases (i.e., for small values of @math and @math ). For an infinite family of @math -regular LPS Ramanujan graphs, we show that the diameter of these graphs is greater than or equal to @math , where @math is an odd prime number and @math is the number of vertices. On the other hand, for any @math -regular Ramanujan graph we show that the distance of only tiny fraction of all pairs of vertices is greater than @math . We also have some numerical experiments for LPS Ramanujan graphs and random Cayley graphs which suggest that the diameters are asymptotically @math and @math , respectively. These are consistent with Sarnak's expectation on the covering exponent of universal quantum gates and our conjecture for the optimal strong approximation for quadratic forms in 4 variables. Abstract : In the paper, demonstration is made of the validity of an iterative procedure suggested by George W. Brown for a two-person game. This method corresponds to each player choosing in turn the best pure strategy against the accumulated mixed strategy of his opponent up to then. For a constant ϵ, we prove a (N) lower bound on the (randomized) communication complexity of ϵ-Nash equilibrium in two-player N x N games. For n-player binary-action games we prove an exp(n) lower bound for the (randomized) communication complexity of (ϵ,ϵ)-weak approximate Nash equilibrium, which is a profile of mixed actions such that at least (1-ϵ)-fraction of the players are ϵ-best replying. Some real-world examples of simple games, like the procedure to amend the Canadian Constitution, are complete simple games with minimum. Using characteristic invariants for this class of games, we study different types of solution concepts. For an arbitrary number of players we get the nucleolus by means of a determinate compatible system of equations, characterize the maximality of the kernel and give a method to calculate semivalues. Several applications are found at the end of the paper. In this paper we show that sufficient multi-partite quantum entanglement helps in fair and unbiased election of a leader in a distributed network of processors with only linear classical communication complexity. We show that a total of @math distinct multi-partite maximally entanglement sets (ebits) are capable of supporting such a protocol in the presence of nodes that may lie and thus be biased. Here, @math is the number of nodes in the network. We also demonstrate the difficulty of performing unbiased and fair election of a leader with linear classical communication complexity in the absence of quantum entanglement even if all nodes have perfect random bit generators. We show that the presence of a sufficient number @math of biased agents leads to a non-zero limiting probability of biased election of the leader, whereas, the presence of a smaller number @math of biased agents matters little. We define two new related complexity classes motivated by the our leader election problem and discuss a few open questions. Recently Bogdanov and Viola (FOCS 2007) and Lovett (ECCC-07) constructed pseudorandom generators that fool degree k polynomials over F2 for an arbitrary constant k. We show that such generators can also be used to fool branching programs of width 2 and polynomial length that read k bits of inputs at a time. This model generalizes polynomials of degree k over F2 and includes some other interesting classes of functions, for instance k-DNF. The constructions of Bogdanov and Viola and Lovett consist of adding a constant number of independent copies of a generator that fools linear functions (an -biased set). It is natural to ask, in light of our first result, whether such generators can fool branching programs of width larger than 2. Our second result is a lower bound showing The problem of quantum dice rolling (DR)—a generalization of the problem of quantum coin flipping (CF) to more than two outcomes and parties—is studied in both its weak and strong variants. We prove by construction that quantum mechanics allows for (i) weak N-sided DR admitting arbitrarily small bias for any N and (ii) two-party strong N-sided DR saturating Kitaev's bound for any N. To derive (ii) we also prove by construction that quantum mechanics allows for (iii) strong imbalanced CF saturating Kitaev's bound for any degree of imbalance. Furthermore, as a corollary of (ii) we introduce a family of optimal 2m-party strong nm-sided DR protocols for any pair m and n. In the generalized Russian cards problem, Alice, Bob and Cath draw a, b and c cards, respectively, from a deck of size a+b+c. Alice and Bob must then communicate their entire hand to each other, without Cath learning the owner of a single card she does not hold. Unlike many traditional problems in cryptography, however, they are not allowed to encode or hide the messages they exchange from Cath. The problem is then to find methods through which they can achieve this. We propose a general four-step solution based on finite vector spaces, and call it the ''colouring protocol'', as it involves colourings of lines. Our main results show that the colouring protocol may be used to solve the generalized Russian cards problem in cases where a is a power of a prime, c=O(a^2) and b=O(c^2). This improves substantially on the set of parameters for which solutions are known to exist; in particular, it had not been shown previously that the problem could be solved in cases where the eavesdropper has more cards than one of the communicating players.