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| {{Orphan|date=January 2014}}
| | Hello, I'm Thad, a 20 year old from Gams, Switzerland.<br>My hobbies include (but are not limited to) Running, Rock collecting and watching Doctor Who.<br><br>Here is my homepage - http://www.hostgator1centcoupon.info/ - [http://room-valencia.com/?option=com_k2&view=itemlist&task=user&id=33086 room-valencia.com] - |
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| '''Quantum refereed game''' in quantum information processing is a class of [[Game theory|games]] in the [[general theory of quantum games]].<ref name=general_qg>{{cite journal|last=Gutoski|first=G|coauthors=Watrous J|title=Toward a general theory of quantum games|journal=Proceedings of the thirty-ninth annual ACM symposium on Theory of computing|year=2007|url=http://arxiv.org/pdf/quant-ph/0611234}}</ref> It is played between two players, Alice and Bob, and arbitrated by a referee. The referee outputs the pay-off for the players after interacting with them for a fixed number of rounds, while exchanging [[quantum state|quantum information]].
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| == Definition ==
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| An '''<math>n</math>-turn quantum referee''' performs <math>n</math> rounds of interaction with the player Alice and Bob. Each interaction involves receiving some quantum states from Alice and Bob, processing the quantum states together with the "left-over" state from the previous interaction, producing some output state, and sending part of the output state to the players. At the end of the <math>n</math> rounds, the referee processes the final state received from the players and decides the pay-off for Alice and Bob.
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| [[File:QuantumRefereedGames.svg|300px|framed|Quantum Refereed Games]]
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| Mathematically, an n-turn referee is a [[Multiple round quantum games#n-turn co-strategy|measuring co-strategy]] <math>\{R_a:a\in\Sigma\}</math> whose input spaces <math>\mathcal X_1,\cdots, \mathcal X_n</math> and output spaces <math>\mathcal Y_1,\cdots, \mathcal Y_n</math> are of the form
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| :<math>\mathcal X_k = \mathcal A_k\otimes \mathcal B_k</math> and <math> \mathcal Y_k = \mathcal C_k\otimes\mathcal D_k</math>
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| for [[complex numbers|complex]] [[inner product space|Euclidean spaces]] <math>\mathcal A_k,\mathcal B_k,\mathcal C_k</math> and <math>\mathcal D_k,\ 1\leq k\leq n</math>.
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| <math>\mathcal A_k,\mathcal B_k</math> represent the message sent by the referee to Alice and Bob during turn <math>k</math>, and <math>\mathcal C_k,\mathcal D_k</math> correspond to their responses. At the end of <math>n</math> turns, the referee produces an output <math>a\in\Sigma</math>
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| An '''<math>n</math>-turn quantum refereed game''' consists of an n-turn referee along with functions <math> V_A, V_B:\Sigma\mapsto\mathbb R</math> that maps each measurement output <math> a</math> to Alice's and Bob's pay-off.
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| Individual quantum refereed games may place specific restrictions on strategies Alice and Bob can choose from. For example, in [[Bell's theorem|nonlocal]] games <ref>{{cite journal|last=Cleve|first=R|coauthors=Hoyer P., Toner B., Watrous J.|title=Consequences and limits of nonlocal strategies|journal=Proceedings of the 19th Annual IEEE conference on Computational complexity|year=2004|pages=236–249}}</ref> and [[Quantum pseudo-telepathy|pseudo-telepathy]] games,<ref>{{cite journal|last=Brassard|first=G|coauthors=Broadbent A., Tapp A.|title=Quantum pseudo-telepathy|journal=Foundation of Physics|year=2005|pages=1877–1907}}</ref> Alice and Bob are allowed to share entanglement but are forbidden from communicating. In general, such restrictions may not apply in quantum refereed games.
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| == Zero sum quantum refereed game ==
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| Similar to classical [[zero sum game]], a '''zero sum quantum refereed game'''<ref name=general_qg /> refers a quantum refereed game with the additional constraint <math>V_A(a) + V_B(a) = 0</math>.
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| It is natural to assume Alice and Bob play independent [[Multiple round quantum games#n-turn strategy|strategies]] in a zero-sum quantum referred game, since it cannot simultaneously be to both player's advantage to communicate directly with one another or to initially share an [[quantum entanglement|entanglement state]] {reference}. In this case, Alice's and Bob's strategy can be represented by
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| :<math>A\in \mathcal S_n(\mathcal A_{1\cdots n}, \mathcal C_{1\cdots n})</math> and <math>B\in \mathcal S_n(\mathcal B_{1\cdots n}, \mathcal D_{1\cdots n})</math>
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| where <math>\mathcal S_n(\mathcal X_{1\cdots n},\mathcal Y_{1\cdots n})</math> is the set of all n-turn strategies having input space <math>\mathcal X_1,\cdots,\mathcal X_n</math> and output space <math>\mathcal Y_1,\cdots, \mathcal Y_n</math>.
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| The combined strategy is then <math>A\otimes B</math>.
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| === Min-Max Theorem ===
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| Define <math>V(a) = V_A(a) = -V_B(a)</math>, and <math>R = \sum_{a\in\Sigma} V(a) R_a</math>, then Alice's expected pay-off is <math>\sum_{a\in\Sigma}V(a)\langle A\otimes B,R_a\rangle = \langle A\otimes B,R\rangle</math>
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| The optimal strategy for Alice then lies in the min-max problem
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| :<math>\max_{A}\min_{B} \langle A\otimes B,R\rangle = \min_{B}\max_{A} \langle A\otimes B,R\rangle</math>.
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| The above equality holds because <math>A, B</math> are drawn from [[compact space|compact]] and [[convex set|convex]] sets <math>\mathcal S_n(\mathcal A_{1\cdots n}, \mathcal C_{1\cdots n})</math> and <math>\mathcal S_n(\mathcal B_{1\cdots n}, \mathcal D_{1\cdots n})</math>. It is called the '''[[Min-max theorem|Min-Max Theorem]]''' for zero-sum quantum games.
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| == Quantum Interactive Proof with Competing Provers ==
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| A quantum interactive proof with two competing provers is a generalization of the single prover [[Interactive Proof System#QIP|quantum interactive proof system]].<ref>{{cite journal|last=Kitaev|first=A|coauthors=Watrous J|title=Parallelization, amplification, and exponential time simulation of quantum interactive proof system|journal=Proceedings of the 32nd AMC Symposium on Theory of Computing|year=2000|pages=608–617}}</ref><ref>{{cite journal|last=Watrous|first=J|title=PSPACE has constant-round quantum interactive proof systems|journal=Theoretical Computer Science|year=2003|pages=575–588}}</ref> It can be modelled by zero-sum refereed games where Alice and Bob are the competing provers, and the referee is the verifier. The referee is assumed to be computationally bounded (polynomial size quantum circuit), whereas Alice and Bob can be computationally unrestricted. Alice, Bob and the referee receive a common string, and after fixed rounds of interactions (exchanging quantum information between the provers and the referee), the referee decides whether Alice wins or Bob wins.
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| === Classical RG ===
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| In the classical setting, RG can be viewed as the following problem. Alice, Bob, and the referee is given some statement. Alice is trying to convince the referee that the statement is true while Bob is trying to convince the referee that the statement is false. The referee, who has limited computing power, will look at the proofs provided by Alice and Bob, ask them questions, and at the end of the day decide which player is correct (wins). The goal is for the referee to find an algorithm such that if the statement is true, there is a way for Alice to win with probability greater than 3/4, and if the statement is false, there is a way for Bob to win with probability greater than 3/4.
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| In the language of complexity theory, a [[promise problem]] <math>L = (L_{\text{yes}}, L_{\text{no}})</math> has a classical refereed game (classical RG) if there exists a referee described by [[RP (complexity)|polynomial time randomized computation]], such that
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| :1. for each <math> x\in L_{\text{yes}}</math>, there is a strategy for Alice to win with probability ≥ 3/4, and
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| :2. for each <math>x\in L_{\text{no}}</math>, there is a strategy for Bob to win with probability ≥ 3/4.
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| It is known that RG = [[EXP]].<ref>{{cite journal|last=Koller|first=D|coauthors=Megiddo N|title=The complexity of two-person zero-sum games in extensive form|journal=Games and Economic Behavior|year=1992|pages=528–552}}</ref><ref>{{cite journal|last=Feige|first=U|coauthors=Kilian J|title=Making games short|journal=Proceedings of the Twenty-Ninth annual ACM Symbosium on Theory of Computing|year=1997|pages=506–516}}</ref>
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| === QRG ===
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| Quantum interactive proof systems with competing provers is a generalization of the classical RG where the referee is now restricted to polynomial-time generated [[quantum circuit]]s and may exchange quantum information with the players.<ref name=general_qg /> Therefore, QRG can be seen as the following problem. Alice, Bob and the referee is given some statement (it may involve a quantum state). Alice is trying to convince the referee the statement is true while Bob is trying to convince the referee the statement is false. The referee can ask the provers questions via quantum states, receive answers in quantum states, and analyse the received quantum states using a quantum computer. After communicating with Alice and Bob for <math>n</math> rounds, the referee decides whether the statement is true or false. If there is a way for the referee to make a correct decision with probability ≥ 3/4, the game is in QRG.
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| More formally, QRG denotes the complexity class for all promise problems having quantum refereed games defined as follows. Given a string <math>x</math>, a promise problem <math>L = (L_{\text{yes}}, L_{\text{no}})</math> is in QRG if there is a referee represented by a polynomial time generated quantum circuit such that
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| :1. if <math>x\in L_{\text{yes}}</math>, there exists a strategy for Alice to win with probability ≥ 3/4, and | |
| :2. if <math>x\in L_{\text{no}}</math>, there exists a strategy for Bob to win with probability ≥ 3/4.
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| It turns out that QRG = EXP — allowing the referee to use quantum circuit and send or receive quantum information does not give the referee any extra power. EXP ⊆ QRG follows from the fact that EXP = RG ⊆ QRG. proved QRG ⊆ EXP by a formulation of QRG using [[semidefinite programs]] (SDP).
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| ==== Semidefinite Program Formulation ====
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| For a quantum refereed game, at the end of all the interactions, the referee outputs one of the two possible outcomes <math>\{a,b\}</math> to indicate whether Alice wins <math>(a)</math> or Bob wins <math>(b)</math>.
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| Setting <math>V(a) = 1, V(b) = 0</math> results in a quantum refereed game whose value is the maximum winning probability for Alice.
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| Using the same notation as the zero sum quantum refereed game as above, the referee is represented by operators <math>\{R_a,R_b\}</math>, Alice may pick a strategy from <math>A\in\mathcal S_n(\mathcal A_{1\cdots n},\mathcal C_{1\cdots n})</math>, and Bob from <math>B\in\mathcal S_n(\mathcal B_{1\cdots n},\mathcal D_{1\cdots n})</math>. Define
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| :<math>\Omega_a(A) = \operatorname{Tr}_{\mathcal C_{1\cdots n}\otimes\mathcal A_{1\cdots n}}((A\otimes I_{\mathcal D_{1\cdots n}\otimes B_{1\cdots n}})R_a)</math> , and
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| :<math>\Omega_b(A) = \operatorname{Tr}_{\mathcal C_{1\cdots n}\otimes\mathcal A_{1\cdots n}}((A\otimes I_{\mathcal D_{1\cdots n}\otimes B_{1\cdots n}})R_b)</math>,
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| where <math>\operatorname{Tr}_{\mathcal X}(Z)</math> is the [[partial trace]] operator.
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| The referee outputs <math>a</math> with probability <math>\langle A\otimes B,R_a\rangle = \langle B,\Omega_a(A)\rangle</math>, and <math>b</math> with probability <math>\langle A\otimes B,R_b\rangle = \langle B,\Omega_b(A)\rangle</math>. <math>\{\Omega_a(A),\Omega_b(A)\}</math> can be considered as a co-strategy that merges Alice's strategy with the referee's.
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| For any given strategy <math>A</math> Alice chooses, the maximum winning probability for Bob is
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| :<math>\max_B\langle B,\Omega_b(A)\rangle</math>,
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| which, by the [[Multiple round quantum game#Maximum Output Probabilities|property]] of the [[Multiple round quantum game#Representation of strategy|strategy representation]], is equal to
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| :<math>\min\{p\geq 0:\Omega_b(A)\leq p Q,\ Q\in\text{co-}\mathcal S_n(\mathcal B_{1\cdots n},\mathcal D_{1\cdots n})\}</math>. | |
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| Therefore, to maximize Alice's winning probability, <math>p</math>, the maximum winning probability for Bob, needs to be minimized over all possible strategies. The goal is then to compute
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| :<math>
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| \begin{array}{rl}
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| \min & p \\
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| \text{subject to} & \Omega_b(A)\leq pQ, \\
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| & A\in \mathcal S_n(\mathcal A_{1\cdots n}, \mathcal C_{1\cdots n}),\\
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| & Q\in\text{co-}\mathcal S_n(\mathcal B_{1\cdots n},\mathcal D_{1\cdots n})
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| \end{array}
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| </math>.
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| This minimization problem can be expressed by the following SDP problem:<ref name=general_qg />
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| :<math>
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| \begin{array}{rll}
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| \min & \operatorname{Tr}(P_1) \\
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| \text{subject to} & \Omega_b(A_n)\leq Q, \\
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| &\operatorname{Tr}_{\mathcal C_k}(A_k) = A_{k-1}\otimes I_{\mathcal A_k} & (2\leq k\leq n),\\
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| &\operatorname{Tr}_{\mathcal C_1}(A_1) = I_{\mathcal A_1},\\
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| &Q_k = P_k\otimes I_{\mathcal D_k}&(1\leq k\leq n),\\ | |
| &\operatorname{Tr}_{\mathcal B_k}(P_k) = Q_{k-1}&(2\leq k\leq n),\\
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| &A_k\in\operatorname{Pos}(\mathcal C_{1\cdots k}\otimes A_{1\cdots k}) & (1\leq k\leq n),\\
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| &Q_k\in\operatorname{Pos}(\mathcal D_{1\cdots k}\otimes B_{1\cdots k}) & (1\leq k\leq n),\\
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| &P_k\in\operatorname{Pos}(\mathcal D_{1\cdots k}\otimes B_{1\cdots k}) & (1\leq k\leq n),\\
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| \end{array}
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| </math>.
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| The dimension of the input and output space of this SPD is exponential (from the tensor product states) in <math>n</math>, and the SDP has a size polynomial in the dimension of its input and output space. Since there are efficient algorithms that can solve SDP in polynomial-time,<ref>{{cite journal|last=KHACHIYAN|first=L|title=A polynomial time algorithm in linear programming|journal=Soviet Mathematics Kodlady 20|year=1979|pages=191–194}}</ref><ref>{{cite journal|last=Grotschel|first=M|coauthors=Lovasz L., Schrijver, A.|title=Geometric Algorithms and Combinatorial Optimization|journal=Springer-Verlag|year=1988}}</ref><ref>{{cite journal|last=Nesterov|first=Y|coauthors=Nemirovski A|title=Interior point polynomial algorithms in comvex programming|journal=SIAM Studies in Applied Mathematics 13|year=1994}}</ref> it follows that QRG ⊆ EXP.
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| == See also ==
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| *[[Min-max theorem]]
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| *[[Semidefinite programming]]
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| *[[QIP (complexity)]]
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| *[[Multiple round quantum games]]
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| == References ==
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| {{reflist}}
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| [[Category:Game theory]]
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