Extended nonlocal games from quantum-classical games

Transcription

1 Extended nonlocl gmes from quntum-clssicl gmes Theory Seminr incent Russo niversity of Wterloo October 17, 2016

2 Outline Extended nonlocl gmes nd quntum-clssicl gmes Entngled vlues nd the dimension of entnglement Discussion Further work 2/34

3 Extended nonlocl gmes nd quntum-clssicl gmes 3/34

4 Extended nonlocl gmes An extended nonlocl gme (ENLG) is specified by: R x A R 0 R 1 y b B A probbility distribution π : X Y [0, 1] for lphbets X nd Y. A collection of mesurement opertors {P,b,x,y : A, b B, x X, y Y } Pos(R) where R is the spce corresponding to R nd A, B re lphbets. 4/34

5 Extended nonlocl gmes An (ENLG) is plyed in the following mnner: R x A R 0 R 1 y b B 1. Alice nd Bob present referee with register R. 2. Referee genertes (x, y) X Y ccording to π nd sends x to Alice nd y to Bob. Alice responds with nd Bob with b. 3. Referee mesures R w.r.t. mesurement {P,b,x,y, 1 P,b,x,y }. Outcome is either loss or win. 5/34

6 Entngled strtegies for ENLGs For n ENLG, n entngled strtegy consists of complex Eucliden spces R,, nd s well s Shred stte: D( R ), Mesurements: {A x } Pos(), {B y b } Pos(). 6/34

7 Entngled strtegies for ENLGs For n ENLG, n entngled strtegy consists of complex Eucliden spces R,, nd s well s Shred stte: D( R ), Mesurements: {A x } Pos(), {B y b } Pos(). Winning probbility for n entngled strtegy is given by: p = π(x, y) A x P,b,x,y B yb,. (x,y) X Y (,b) A B 6/34

8 Extended nonlocl gmes: Winning nd losing probbilities At the end of the protocol, the referee hs: 1. The stte t the end of the protocol: x,y,b D(R). 2. A mesurement the referee mkes on its prt of the stte ρ: P,b,x,y Pos(R). The respective winning nd losing probbilities re given by P,b,x,y, x,y,b nd 1 P,b,x,y, x,y,b. 7/34

9 Entngled vlues of ENLGs For ny ENLG denoted s H, the entngled vlue of H, denoted s ω (H), represents the supremum of the winning probbilities tken over ll entngled strtegies. We my lso write ωn (H) to denote the mximum winning probbility tken over ll entngled strtegies for which dim( ) N, so tht the entngled vlue of H is ω (H) = lim N ω N (H). 8/34

10 ENLGs nd steering ENLGs re ctully equivlent formultions of prticulr type of triprtite steering. Einstein, Podolsky, Rosen (1935): Cn Quntum-Mechnicl Description of Physicl Relity be Considered Complete? 9/34

11 ENLGs nd steering ENLGs re ctully equivlent formultions of prticulr type of triprtite steering. Biprtite steering ws initilly introduced by Schrödinger in 1936 in n ttempt to mke forml the spooky ction t distnce s discussed in the EPR pper Einstein, Podolsky, Rosen (1935): Cn Quntum-Mechnicl Description of Physicl Relity be Considered Complete? 9/34

12 Biprtite steering Alice nd Bob ech receive prt of quntum stte (sent by the referee). Their gol is to determine whether this stte is entngled. x A R 0 R 1 B Bob s mesurement device is trusted, wheres Alice s is not: Outcome of Alice s mesurements re only ±1 ( conclusive outcome) or 0 ( non-conclusive outcome). To demonstrte entnglement, Alice needs to steer Bob s stte by her choice of mesurement. 10/34

13 NLGs, ENLGs, nd steering Biprtite steering: A x R0 R 1 B 11/34

14 NLGs, ENLGs, nd steering Biprtite steering: A x R0 R 1 B Biprtite nonlocl gme: A x R0 R 1 y b B 11/34

15 NLGs, ENLGs, nd steering Biprtite steering: A x R0 R 1 Triprtite steering with two untrusted prties (ENLG): x R A R0 R 1 B y b Biprtite nonlocl gme: A x B R0 R 1 y b B 11/34

16 NLGs, ENLGs, nd steering Biprtite steering: A x R0 R 1 Triprtite steering with two untrusted prties (ENLG): x R A R0 R 1 B y b Biprtite nonlocl gme: A x R0 R 1 y b B Triprtite steering with one untrusted prty: R A x B R0 R 1 B 11/34

17 ENLG nd steering Triprtite steering; sme thing s before, only now we hve three prties where two members re untrusted nd one member is trusted. 12/34

18 ENLG nd steering Triprtite steering; sme thing s before, only now we hve three prties where two members re untrusted nd one member is trusted. In triprtite steering, Alice nd Bob re the untrusted prties, nd the referee is the trusted prty. 12/34

19 ENLG nd steering Triprtite steering; sme thing s before, only now we hve three prties where two members re untrusted nd one member is trusted. In triprtite steering, Alice nd Bob re the untrusted prties, nd the referee is the trusted prty. This isn t tlk on steering, but it s helpful to note tht proving something using ENLGs will lso sy something bout prticulr type of triprtite steering. 12/34

20 Quntum-clssicl gmes A quntum-clssicl gme (QCG) is coopertive gme plyed between Alice nd Bob ginst referee. X A S R 0 R 1 Y b B Specified by: A stte ρ D(X S Y) in registers (X, S, Y). Collection of mesurement opertors {Q,b : A, b B} Pos(S) for lphbets A nd B. 13/34

21 Quntum-clssicl gmes A (QCG) is plyed in the following mnner. X A S R 0 R 1 Y b B 1. Referee prepres (X, S, Y) in stte ρ nd sends X to Alice nd Y to Bob. 2. Alice responds with A nd Bob with b B. 3. Referee mesures S w.r.t. mesurement {Q,b, 1 Q,b }. The outcome of this mesurement results in 0 or 1, indicting loss or win. 14/34

22 Entngled strtegies for QCGs For QCG, n entngled strtegy consists of complex Eucliden spces nd s well s Shred stte: D( ), Mesurements: {A : A} Pos( X ), {B b : b B} Pos( Y). 15/34

23 Entngled strtegies for QCGs For QCG, n entngled strtegy consists of complex Eucliden spces nd s well s Shred stte: D( ), Mesurements: {A : A} Pos( X ), {B b : b B} Pos( Y). Winning probbility for entngled strtegy is given by: p = A Q,b B b, W ( ρ) W, (,b) A B where W is the unitry opertor tht corresponds to the nturl re-ordering of registers consistent with the tensor product opertors. 15/34

24 Entngled vlues for QCGs For ny QCG denoted s G, the entngled vlue of G, denoted s ω (G), represents the supremum of the winning probbilities tken over ll entngled strtegies. We my lso write ωn (G) to denote the mximum winning probbility tken over ll entngled strtegies for which dim( ) N, so tht the entngled vlue of G is ω (G) = lim N ω N (G). 16/34

25 Entngled vlues nd the dimension of entnglement 17/34

26 lues nd the dimension of shred entnglement Question: Does the dimensionlity of the stte tht Alice nd Bob shre determine how well Alice nd Bob perform? Regev, idick (2012): Quntum XOR gmes 18/34

27 lues nd the dimension of shred entnglement Question: Does the dimensionlity of the stte tht Alice nd Bob shre determine how well Alice nd Bob perform? Prtil nswer: In [Regev,idick (2012)], the uthors showed tht there exists specific clss of QCG such tht if the dimension of Alice nd Bob s quntum system, N, is finite then ωn (G) < 1, but ω (G) = 1. Regev, idick (2012): Quntum XOR gmes 18/34

28 lues nd the dimension of shred entnglement Question: Does the dimensionlity of the stte tht Alice nd Bob shre determine how well Alice nd Bob perform? Prtil nswer: In [Regev,idick (2012)], the uthors showed tht there exists specific clss of QCG such tht if the dimension of Alice nd Bob s quntum system, N, is finite then ωn (G) < 1, but ω (G) = 1. Wht bout ENLG? Regev, idick (2012): Quntum XOR gmes 18/34

29 Reltionship between ENLGs nd QCGs Min question: Does there lso exist n ENLG, H, such tht ω (H) = 1 nd ωn (H) < 1 when N is finite? 19/34

30 Reltionship between ENLGs nd QCGs Min question: Does there lso exist n ENLG, H, such tht ω (H) = 1 nd ωn (H) < 1 when N is finite? It is possible to construct n ENLG from ny QCG (not obvious). X A x R A S R0 R 1 R0 R 1 Y b y b B B 19/34

31 Reltionship between ENLGs nd QCGs Min question: Does there lso exist n ENLG, H, such tht ω (H) = 1 nd ωn (H) < 1 when N is finite? It is possible to construct n ENLG from ny QCG (not obvious). X A x R A S R0 R 1 R0 R 1 Y b y b B B From this construction, it turns out tht this property lso holds for ENLG, tht is, there does exist n ENLG such tht Alice nd Bob cn only win with certinty iff they shre n infinite-dimensionl stte. 19/34

32 Constructing ENLGs from QCGs Generl ide: Given strtegy for QCG, G, show tht it s possible to dpt this strtegy for n ENLG, H, nd vice-vers. Approch: Show tht for n rbitrry nd fixed strtegy for G, tht it s possible to dpt this strtegy for H. Show tht Alice nd Bob cnnot do ny better. 20/34

33 Key ide (for one direction of proof) Show tht for n rbitrry nd fixed strtegy for G, tht it s possible to dpt this strtegy for H. 21/34

34 Key ide (for one direction of proof) Show tht for n rbitrry nd fixed strtegy for G, tht it s possible to dpt this strtegy for H. Min restriction: In G, the referee is sending quntum registers, but in H, the referee is restricted to sending clssicl questions. X A x R A S R0 R 1 R0 R 1 Y b y b B B 21/34

35 Key ide (for one direction of proof) Show tht for n rbitrry nd fixed strtegy for G, tht it s possible to dpt this strtegy for H. Min restriction: In G, the referee is sending quntum registers, but in H, the referee is restricted to sending clssicl questions. X A x R A S R0 R 1 R0 R 1 Y b y b B B Key ide: se teleporttion to trnsmit X nd Y in gme H. 21/34

36 Attempt 1: Adpting vi teleporttion Protocol: 1. Alice nd Bob prepre ρ D( (X Y) ) in registers (, X, Y, ) such tht Alice/Ref nd Bob/Ref shre pirs of mximlly entngled sttes. 2. Referee desires to trnsmit sttes tht he cretes held in X nd Y to Alice nd Bob. To do so, he mesures (X, X ) nd (Y, Y ) in the Bell bsis to generte nd send (x, y) to Alice nd Bob. 3. Alice nd Bob complete the teleporttion protocol by pplying pproprite unitries to their system bsed on (x, y). 22/34

37 Attempt 1: Adpting vi teleporttion Protocol: 1. Alice nd Bob prepre ρ D( (X Y) ) in registers (, X, Y, ) such tht Alice/Ref nd Bob/Ref shre pirs of mximlly entngled sttes. 2. Referee desires to trnsmit sttes tht he cretes held in X nd Y to Alice nd Bob. To do so, he mesures (X, X ) nd (Y, Y ) in the Bell bsis to generte nd send (x, y) to Alice nd Bob. 3. Alice nd Bob complete the teleporttion protocol by pplying pproprite unitries to their system bsed on (x, y). Problem: The definition of n ENLG requires tht questions (x, y) re generted rndomly. 22/34

38 Attempt 2: Post-selected teleporttion protocol for H Problem: nill teleporttion is not enough (the questions (x, y) need to be generted independent of the stte of registers (X, Y)). 23/34

39 Attempt 2: Post-selected teleporttion protocol for H Problem: nill teleporttion is not enough (the questions (x, y) need to be generted independent of the stte of registers (X, Y)). Ide: Let (x, y) be selected t rndom, but then compre (x, y) to hypotheticl mesurement results tht would be obtined if the referee were to perform teleporttion. 23/34

40 Post-selected teleporttion protocol for H x A 1 x A 0 X X T x 1? = δ x,x1 R 0 S R 1 Y δ y,y1 B 0 Y y T y 1? = y B 1 b 24/34

41 Step 1: Post-selected teleporttion protocol for H Alice nd Bob prepre D( (X Y) ). A 0 X T B 0 Y T 25/34

42 Step 2: Post-selected teleporttion protocol for H Referee rndomly selects nd sends (x, y); keeps locl copy. Alice nd Bob respond with (, b). x A 1 x A 0 X T? = R 1 B 0 Y y T? = y B 1 b 26/34

43 Step 3: Post-selected teleporttion protocol for H Referee prepres ρ D(X S Y ). Performs teleporttion using (X, X ) nd (Y, Y ) resulting in outcomes (x 1, y 1 ). x A 1 x A 0 X X T x 1? = R 0 S R 1 Y B 0 Y y T y 1? = y B 1 b 27/34

44 Step 4: Post-selected teleporttion protocol for H 1. If x x 1 or y y 1 : teleporttion fils; Alice nd Bob win. 2. If x = x 1 nd y = y 1 : teleporttion succeeds; Referee mesures w.r.t. {P,b,x,y, 1 P,b,x,y }. x A 1 x A 0 X X T x 1? = δ x,x1 R 0 S R 1 Y δ y,y1 B 0 Y y T y 1? = y B 1 b 28/34

45 Discussion 29/34

46 Similr result for nonlocl gmes? Our result: There exists some ENLG such tht ω N (H) < 1 nd ω (H) = 1. Tht is, for certin ENLG, incresing mounts of entnglement yield higher winning probbilities. 30/34

47 Similr result for nonlocl gmes? Our result: There exists some ENLG such tht ω N (H) < 1 nd ω (H) = 1. Tht is, for certin ENLG, incresing mounts of entnglement yield higher winning probbilities. Known result: There exists some QCG such tht ω N (G) < 1 nd ω (G) = 1. Tht is, for certin QCG, incresing mounts of entnglement yield higher winning probbilities. 30/34

48 Similr result for nonlocl gmes? Our result: There exists some ENLG such tht ω N (H) < 1 nd ω (H) = 1. Tht is, for certin ENLG, incresing mounts of entnglement yield higher winning probbilities. Known result: There exists some QCG such tht ω N (G) < 1 nd ω (G) = 1. Tht is, for certin QCG, incresing mounts of entnglement yield higher winning probbilities. nknown: Does there exist some NLG, G, with similr properties, tht is ω N (G) < 1 nd ω (G) = 1? 30/34

49 ENLGs nd triprtite steering Recll, ENLG my be viewed equivlently s triprtite steering scenrio. 31/34

50 ENLGs nd triprtite steering Recll, ENLG my be viewed equivlently s triprtite steering scenrio. Proving results bout ENLGs gives us corresponding results bout triprtite steering. Wht does our result imply in the context of steering? 31/34

51 ENLGs nd triprtite steering Recll, ENLG my be viewed equivlently s triprtite steering scenrio. Proving results bout ENLGs gives us corresponding results bout triprtite steering. Wht does our result imply in the context of steering? Result: Our result implies the existence of triprtite steering inequlity tht is mximlly violted only by quntum stte with dimension pproching infinity. For ny finite-dimensionl stte, this steering inequlity cnnot be mximlly violted. 31/34

52 Further work 32/34

53 Open questions (Hrd problem): Does there exist ny nonlocl gme, G, such tht ω (G) = 1 nd ωn (G) < 1 for ll N? (More generl): Cn the study of extended nonlocl gmes revel nything further bout the properties of triprtite steering? 33/34

54 34/34 A relted question: Swpping rounds of communiction How powerful is the extended nonlocl gme model? Wht hppens when you substitute clssicl for quntum rounds of communiction or vice-vers? R0 R 1 A B R x y b R0 R 1 A B R x y E F R0 R 1 A B R C D b R0 R 1 A B R C D E F

55 Thnks Thnks for listening! This work is primrily bsed on: N. Johnston, R. Mittl,. R., J. Wtrous. Extended nonlocl gmes nd monogmy-of-entnglement gmes. Proc. R. Soc. A 472: , R., J. Wtrous. Extended nonlocl gmes from quntum-clssicl gmes. In preprtion. 34/34