Quantum Pseudo-Telepathy

Transcription

1 Quantum Pseudo-Telepathy Michail Lampis NTUA Quantum Pseudo-Telepathy p.1/24

2 Introduction In Multi-Party computations we are interested in measuring communication complexity. Communication by exchanging quantum bits exponential separation of complexity. Can quantum mechanics be used to completely eliminate communication in an MPC? Can quantum entanglement simulate an arbitrary communication channel? (EPR paradox) Quantum Pseudo-Telepathy p.2/24

3 Overview MPC problem demonstrates Quantum Pseudo-Telepathy It can be performed with 0 bits of communication by quantum players with probability 1. It can be performed with 0 bits of communication by classical players with probability < 1. Quantum Pseudo-Telepathy p.3/24

4 Setting Alice and Bob claim to have telepathic powers. Xavier and Yolande don t believe them Alice and Bob propose a game which will prove their case Alice and Bob must win every single round without communicating Quantum Pseudo-Telepathy p.4/24

5 Example of a game Xavier and Yolande write down a list of animals. Round Xavier Yolande 1 lion giraffe 2 tiger tiger 3 cat mouse At each round Alice and Bob say yes iff they were presented with the same animal. This game does not demonstrate pseudo-telepathy. Quantum Pseudo-Telepathy p.5/24

6 Formal definition of game A game is defined as G = {A, B, X, Y, P, W } A, B, X, Y are sets of possible answers and questions. P X Y is a promise on the questions. W X Y A B is the set of correct answers. Alice and Bob win when asked (x, y) and answer (a, b) if (x, y) / P or (x, y, a, b) W Quantum Pseudo-Telepathy p.6/24

7 Formal definition of pseudo-telepathy Deterministic strategy: functions f : X A, g : Y B Maximum success proportion ω(g) = max f,g {(x, y) P (x, y, f(x), g(y)) W } P Probabilistic strategy: distribution over deterministic strategies Probability of correct answer to (x, y) for strategy s P r s (win (x, y)) Quantum Pseudo-Telepathy p.7/24

8 Formal definition of pseudo-telepathy ( Maximum success probability ω(g) = max s min P r s(win (x, y)) (x,y) P Theorem: ω(g) ω(g) when questions are asked uniformly at random. P 1 When ω(g) P and we have a quantum winning strategy for G we say that G demonstrates Quantum Pseudo-Telepathy. Quantum Pseudo-Telepathy p.8/24

9 Impossible colouring game (i) Kochen-Specker Theorem There exists a finite set of vectors in R 3 which can not be {0, 1} coloured s.t. 1. No two orthogonal vectors are both coloured 1 2. For every three mutually orthogonal vectors at least one is coloured 1 Such a set has the Kochen-Specker property. Quantum Pseudo-Telepathy p.9/24

10 Impossible colouring game (ii) Let V be a set with the Kochen-Specker property. Alice is given a pair of orthogonal v 1, v 2 V or a triple of mutually orthogonal v 1, v 2, v 3 V Bob is given a vector v l V Promise: v l is one of the vectors given to Alice Question: Alice and Bob must {0, 1}-colour their vectors following the Kochen-Specker restrictions. They must give the same colour to v l Quantum Pseudo-Telepathy p.10/24

11 Impossible colouring game (iii) Quantum strategy: Alice and Bob share 4 entangled qbits in state i=0 i i Alice performs a measurement in base { v 1, v 2, v 3 } (if v 3 was not given she uses an arbitrary vector s.t. an orthonormal base is formed). She colours the vector she measured 1. Quantum Pseudo-Telepathy p.11/24

12 Impossible colouring game (iv) Bob performs a measurement in an orthonormal base which contains v l. If he measures v l he colours it 1. Because of entanglement Bob measures v l iff Alice measures v l. Therefore their colourings agree. A deterministic strategy that always wins would produce a proper coulouring of all vectors in V. Therefore ω(g) ω(g) < 1 Quantum Pseudo-Telepathy p.12/24

13 Magic Square game (i) X = Y = {1, 2, 3} A = {000, 011, 101, 110} B = {001, 010, 100, 111} P = X Y, Win: a[y] = b[x] Xavier asks Alice for row x of a magic square. Yolande asks Bob for column y. The answers must agree on their common bit. Quantum Pseudo-Telepathy p.13/24

14 Magic Square game (ii) A successful deterministic strategy would produce a magic square, i.e. a 3 3 square with rows of even parity and columns of odd parity. Hence ω(g) < 1 Quantum strategy: Alice and Bob share four qbits in state ( ). 1 2 They perform a computation depending on x, y and fill in the third bit to have appropriate parity. Quantum Pseudo-Telepathy p.14/24

15 Parity Games (i) Multiplayer games for n 3 players. Each player receives a bit-string x i {0, 1} l Promise: i x i = k2 l Victory: Each player outputs a single bit a i s.t. i a i = k(mod2) For n = 3, l = 1 we have the Mermin-GHZ game Quantum Pseudo-Telepathy p.15/24

16 Parity Games (ii) Quantum strategy: Each player has a bit of the state 1 2 ( 0 n + 1 n ). Each player performs computation e πix i/2 l 1 Each player applies Hadamard gate ( ) ( 0 1 ) Quantum Pseudo-Telepathy p.16/24

17 Parity Games (iii) Step 1 brings state to 1 2 ( 0 + ( 1) k 1 Step 2 brings state to 1 2 n 1 (a)=k(mod2) a Classical strategy for Mermin-GHZ game: a 0 + b 0 + c 0 0 a 0 + b 1 + c 1 1 a 1 + b 0 + c 1 1 a 1 + b 1 + c 0 1 Quantum Pseudo-Telepathy p.17/24

18 Deutsch-Jozsa Games (i) Alice and Bob are given bit strings x,y. They must answer with a=b iff x=y To prevent a x, b y x = y = 2 m, a = b = m. Promise: Either x = y or they differ in exactly 2 m 1 bits. Quantum strategy: Alice and Bob share 2m qbits in state 1 n j j j They each perform j ( 1) x j j Quantum Pseudo-Telepathy p.18/24

19 Deutsch-Jozsa Games (ii) They both apply Hadamard gates to their registers which gives a correct answer Classical solutions can t exist for sufficiently large m. Classical solutions exist for m = 1, 2 (easy) and m = 3. No classical solution for m = 4. Open problem for m > 4 (unknown boundary) Quantum Pseudo-Telepathy p.19/24

20 Matching Game (i) A perfect matching of size m is a partition of {0, 1,..., m 1} to m 2 sets of size 2. Alice receives an m-bit string x = x 0 x 1... x m 1, Bob receives a perfect matching M of size m. Alice must output a string a {0, 1} lg m and Bob a pair (α, β) M and a string b {0, 1} lg m s.t. x α x β = (α β) (a b) Quantum Pseudo-Telepathy p.20/24

21 Hidden Matching Problem Alice receives an m-bit string x = x 0 x 1... x m 1, Bob receives a perfect matching M of size m. Alice sends a message to Bob, and Bob outputs a triple (a, b, c) s.t. (a, b) M and c = x a x b. If Alice sends a message using classical bits, ω(lg m) bits are needed (Bar-Yossef, Jayram, Kerenidis) Quantum Pseudo-Telepathy p.21/24

22 Matching Game (ii) A classical winning strategy for the matching game could be reduced to a solution to the HMP using lg m bits (Alice send a to Bob). Therefore there is no classical winning strategy. Quantum strategy: Alice and Bob share lg m qbits each in state 1 m j j j Alice performs j ( 1) x j j Quantum Pseudo-Telepathy p.22/24

23 Matching Game (ii) Bob performs a partial measurement that brings the system to state 1 2 (( 1) x α αα + ( 1)x β ββ ) s.t. (α, β) M and outputs (α, β). Alice and Bob perform Hadamard transformations and full measurements and output the results a and b. Quantum Pseudo-Telepathy p.23/24

24 Further work How are pseudo-telepathy games related to each other? Categorize families of telepathy games. Investigate relation between complexity problems and pseudo-telepathy Investigate boundaries of pseudo-telepathy Quantum Pseudo-Telepathy p.24/24