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Penny flip game Introduction Magic Square Game Noise in some quantum games. Piotr Gawron The Institute of Theoretical and Applied Informatics of the Polish Academy of Sciences 2008-11-16 Piotr Gawron Noise in some quantum games. 2008 Penny flip game Introduction Magic Square Game Table of contents Introduction Motivation Noise model Kraus operators One-qubit noise Multiqubit local channels Penny flip game The game Quantum extention Noise Magic Square Game The game Noise Success probability Piotr Gawron Noise in some quantum games. 2008 Penny flip game Introduction Magic Square Game Motivation Motivation The motivation for studies of noise in quantum games may be manifold: I how quantum game reduces to classical case? I how the odds change if the noise is present in quantum system implementing the game? I does the knowledge about the noise changes the players’ odds? Piotr Gawron Noise in some quantum games. 2008 Penny flip game Introduction Magic Square Game Noise model Noise model In the most general case quantum evolution is described by super operator Φ, which can be expressed in Kraus form: X Ek ρEk † , Φ(ρ) = (1) k where P k Ek † Ek = I. Piotr Gawron Noise in some quantum games. 2008 Introduction Penny flip game Magic Square Game Noise model In the literature, the following Kraus operators are considered to create typical noisy channels: nq pα pα pα o I depolarising channel: 1 − 3α 4 I, 4 σx , 4 σy , 4 σz , i h √ io nh 1√0 0 α I amplitude damping: , 0 1−α , 0 0 nh i h io 1√0 0 √0 I phase damping: , 0 1−α , 0 α Piotr Gawron Noise in some quantum games. 2008 Penny flip game Introduction Magic Square Game Noise model I I I √ 1 − αI, ασz , √ √ bit flip and 1 − αI, ασx , √ √ bit-phase flip 1 − αI, ασy . phase flip, √ Real parameter α ∈ [0, 1] represents here the amount of noise in the channel, σx , σy , σz are Pauli matrices. Piotr Gawron Noise in some quantum games. 2008 Penny flip game Introduction Magic Square Game Noise model Multiqubit local channels In order to apply noise operators to multiple qubits we form new set of operators acting on larger Hilbert space. We assume that we have set of n one-qubit Kraus operators {ek }. We construct new set of nN operators {Ek } that act on Hilbert space of N dimension P 2 . We† can write the action of the extended channel Φ(ρ) = k Ek ρEk in the following way: Φ(ρ) = n X ei1 ⊗ ei2 ⊗ . . . ⊗ eiN ρei†1 ⊗ ei†2 ⊗ . . . ⊗ ei†N . (2) i1 ,i2 ,...iN =1 When applying equation 2 to sets of the operators listed above we obtain one-parameter families of local noisy channels. Piotr Gawron Noise in some quantum games. 2008 Penny flip game Introduction Magic Square Game The game Penny flip game I There are two players Alice and Bob. I There is a penny hidden in a box. I The players can flip the penny. I At the beginning the penny is faced up, The game goes as follows: I I I I Alice choses to flip or not flip, now Bob choses to flip or not to flip, at last Alice plays, I the penny is revealed, I Alice wins if the penny is faced down, otherwise Bob wins. Piotr Gawron Noise in some quantum games. 2008 Magic Square Game Penny flip game Introduction The game N F NN -1 1 NF 1 -1 FN 1 -1 FF -1 1 Table: The game payoff matrix. N means not-flipping and F means flipping. One means that Alice won, minus one means that Bob won (it is zero-sum game). Piotr Gawron Noise in some quantum games. 2008 Introduction Penny flip game Magic Square Game Quantum extention Penny flip game Coin |ci facing heads up is denoted as U = |0i, facing tails up as D = |1i. Flip operator F = [ 01 10 ] is σx , not flipping operator N = [ 10 01 ] is identity I. Winning condition is described by the expectation value of σz operator hσz i|ci . We will restrict ourselves only to deterministic strategies. One can easily see that there is no winning strategy neither for Alice nor for Bob. Now we will cheat a little bit. We will exchange a qubit for the penny. The qubit can be in any normalised linear combination α|0i + β|1i. And what is even more important, it may be transformed by use of any unitary gate. Piotr Gawron Noise in some quantum games. 2008 Introduction Magic Square Game Penny flip game Quantum extention The cheater in this scheme is Alice. We assume that only she knows that the game is played with a qubit. So now Alice can rotate the qubit inany 1 √ direction. She may chose to apply the Hadamard gate H = 1 √ 2 2 1 √ 2 − √12 as her first and second movement. Bob has still two possibilities: two flip or not to flip. Lets see what happens in those two cases: 1. HIH|0i → |0i, 2. HNH|0i → |0i. It means that Bob’s actions have no influence whatsoever on the outcome of the game. More detailed analysis of this game including mixed (probabilistic) strategies may be found in [4]. Piotr Gawron Noise in some quantum games. 2008 Penny flip game Introduction Magic Square Game Noise We will assume that the qubit is influenced by the noise. To model this situation we will assume that the noise takes effect on the system, before Bob’s movement – Bob has to think a little bit making his decision, Alice is quick in her decision because she knows that they played with qubit rather than a penny. The case of depolarising channel was analysed in [1]. During each game we fix the noise type – it means that the qubit is coupled to the environment. We connect the noise ration α with time that Bob takes to make his decision. So now the scheme is as follows: ρf = HBεα (H|0ih0|H † )B † H † , (3) where B denotes Bob’s decision (I or N) and εα (·) CPTP map implementing noise parametrised by real parameter α. The expectation value of Alice win is given by: h$A i = h0|ρf |0i. (4) For Bob it is 1 − h$A i. Piotr Gawron Noise in some quantum games. 2008 Magic Square Game Penny flip game Introduction Noise channel depolarising amplitude damping phase damping bit flip phase flip bit-phase flip h$A i 1 − α/2 √ (√1 − α + 1)/2 ( 1 − α + 1)/2 1 1−α 1−α Table: Expectation value of Alice win in noisy penny flip game. From this simple case we learn two very important lessons: (i) when playing quantum games it is important to know and use quantum rules, (ii) the noise can influence the outcome of the game and even change the odds drastically. Piotr Gawron Noise in some quantum games. 2008 Introduction Penny flip game Magic Square Game The game Magic Square Game The magic square is a 3 × 3 matrix filled with numbers 0 or 1 so that the sum of entries in each row is even and the sum of entries in each column is odd. Although such a matrix cannot exist one can consider the following game. There are two players: Alice and Bob. Alice is given the number of the row, Bob is given the number of the column. Alice has to give the entries for a row and Bob has to give entries for a column so that the parity conditions are met. In addition, the intersection of the row and the column must agree. Alice and Bob can prepare a strategy but they are not allowed to communicate during the game. Piotr Gawron Noise in some quantum games. 2008 Penny flip game Introduction Magic Square Game The game There exists a (classical) strategy that leads to winning probability of 98 . If parties are allowed to share a quantum state they can achieve probability 1. In the quantum version of this game[3] Alice and Bob are allowed to share an entangled quantum state. The winning strategy is following. Alice and Bob share entangled state |Ψi = Piotr Gawron Noise in some quantum games. 1 (|0011i − |1100i − |0110i + |1001i). 2 (5) 2008 Introduction Penny flip game Magic Square Game The game Depending on the input (i.e. the specific row and column to be filled in) Alice and Bob apply unitary operators Ai ⊗ I and I ⊗ Bj , respectively, 0 0 1 −1 −1 1 1 i 1 −1 1 0i −i 1 −ii 11 −1 i 1 1 −1 1 1 0 A1 = √ 0 i 1 0 , A2 = , −i , A3 = 1 1 2 −ii 11 −1 2 11 −1 2 1 0 0i 1 −i −1 −1 −1 " # " # 1 0 0 1 1 1 i 1 i 1 −1 1 −1 1 −ii −i 0 0 1 −i 1 −1 1 i 1 −i , B2 = , B3 = √ B1 = 0 1 1 0 , 1 1 −i i 1 −i 1 i 2 −i 2 −1 2 0 1 −1 0 i 1 1 −i 1 −i where i and j denote the corresponding inputs. The final state is used to determine two bits of each answer. The remaining bits can be found by applying parity conditions. Piotr Gawron Noise in some quantum games. 2008 Introduction Magic Square Game Penny flip game Noise Adding the noise We analyse the influence of the noise on success probability when the noise operator is applied before the game gates [2]. The final state of this scheme is ρf = (Ai ⊗ Bj ) Φα (|ΨihΨ|) (A†i ⊗ Bj† ), where Φα is the superoperator realizing quantum channel parametrized by real number α. Probability Pi,j (α) is computed as the probability of measuring ρf in the state indicating success ! X Pi,j (α) = Tr ρf |ξi ihξi | , (6) i where |ξi i are the states that imply success. Piotr Gawron Noise in some quantum games. 2008 Introduction Penny flip game Magic Square Game Success probability We compute success probability Pi,j (α) for different inputs (i, j ∈ {1, 2, 3}) and different quantum channels. Our calculations show P that mean probability of success, P(α) = i,j∈{1,2,3} Pi,j (α), heavily depends on the noise level α. Piotr Gawron Noise in some quantum games. 2008 Magic Square Game Penny flip game Introduction Success probability 0.9 0.8 0.7 0.6 0.5 0 0.2 0.6 0.8 0.8 0.7 0.6 0 0.2 0.4 0.6 Error rate α Piotr Gawron Noise in some quantum games. depolarizing channel classical treshold 0.9 0.8 0.7 0.6 0.5 0 0.2 0.8 1 0.4 0.6 0.8 1 0.8 1 Error rate α 1 flipping channels classical treshold 0.9 0.5 1 Error rate α 1 Success probability P (α) 0.4 Success probability P (α) 1 amplitude damping channel classical treshold Success probability P (α) Success probability P (α) 1 phase damping channel classical treshold 0.9 0.8 0.7 0.6 0.5 0 0.2 0.4 0.6 Error rate α 2008 Penny flip game Introduction Magic Square Game Success probability Bibliography A. P. Flitney and D. Abbott. Quantum games with decoherence. Journal of Physics A: Mathematical and General, 38(2):449–459, 2005. P. Gawron, J. A. Miszczak, and J. Sladkowski. Noise effects in quantum magic squares game. International Journal of Quantum Information, 6(1 supp), 2008. N. David Mermin. Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett., 65(27):3373–3376, Dec 1990. E. W. Piotrowski and J. Sladkowski. An invitation to quantum game theory. Int. J. Theor. Phys., 42:1089 – 1099, 2003. Piotr Gawron Noise in some quantum games. 2008 Penny flip game Introduction Magic Square Game Success probability And now a question We would like to investigate the whole space of strategies. General equation for penny flip game is: ρf = A2 εγ (Bεβ (A1 εα (ρ)A†1 )B † )A†2 . Let α be one-parameter family of quantum channels. How to find a decomposition of this channel such that: εγ (εβ (εα (ρ))) = εx (ρ) what is the relation between the parameters: α + β + γ = x? Piotr Gawron Noise in some quantum games. 2008 Introduction Penny flip game Magic Square Game Success probability Thank you for your attention. Piotr Gawron Noise in some quantum games. 2008