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numerous books on this subject, e.g.[10]. In this paper we aim to show that there are quantum communication strategies outperform the classical ones in the game of bridge. Unfortunately, there is no best classical strategy. We have contacted Tommy Gullberg a World Life Master in duplicate bridge, author of many articles and books about the game, who told us: A profes- sional player plays with a strategy of his own preference since he finds it optimal. Thus if you go to a professional tournament you will encounter a wide range of different strategies, all with some strengths and some weaknesses, since there clearly is no such thing as an optimal classical bridge strategy”[11]. FIG. 1: Cards and Bidding for Bridge. An example of cards However, all the strategies must involve a ”slam- and bidding for the W and E partners. seeking convention”, which is a sub-strategy that play- ers use if they have very strong cards hope to win a big bonus. Although, quantum strategies are probably II. BRIDGE useful at many different situations in bridge, in this pa- per we only give an example for improving Roman Key- Bridge is one of the world’s most popular card games. card Blackwood (RKB)[12]. It is a very popular slam- It is a trick-taking game with the standard deck of 52 seeking strategy and often used in bridge games today. cards. The game is played by four players playing in If you play a game of bridge (in a tournament) it is very pairs with partners sitting at opposite sides of a table, likely that the RKB will be used, probably several times. and named as West (W )-East (E) and North (N)-South Therefore, the optimal classical strategy is likely to in- (S). The game consists of several deals each progress- clude it[11]. And even if it does not, other slam-seeking ing through two phases: the auction (called bidding), strategies like Gerber [13] differ from it only slightly and and playing the hand (called trick-taking) . The bidding the quantum protocol can be modified accordingly. To phase starts with the dealer and rotates around the table show that the problem at which the quantum protocols clockwise with each player making a bid for the contract. excel at is ubiquitous in bridge in the appendix we present The team that wins the auction undertakes to win a cer- how the strategy used for RKB can be modified to be tain number of tricks during the game. The second phase used in playing the hand. of the game is the standard trick-taking play. Here, the The following paragraph plunges deep into the nuances winners of the auction try to fulfill their contract (taking of bidding. A reader not interested in the game should as many tricks as the contract obliges them to) while the skip it and know only this: The task for the players is opposing team intends to prevent them from doing so. to decide if they have cards strong enough to try to get The team that succeeds with its task wins the round. In a bonus. To this end, one of them will ask questions both phases of Bridge, efficient communication is of vital (encoded in bids) and the other provide answers. In this importance for the partnership. This is even more true particular case there are two important questions but in the tournament version of the game - duplicate Bridge only one of them can be asked. Classical and quantum recognized by the International Olympic Committee as a ways of dealing with this problem are described in the sport. There, a larger group of partnerships play against next section. each other with prearranged hands to reduce the factor Consider the cards and bidding in a bridge scenario of chance to a minimum. What counts in this variation shown in figure 1. The bidding can be explained as fol- is earning more points with a given hand than the other lows, W looks at his cards. He has good cards in dia- partnerships had. This is done by winning the optimal monds ( ) and therefore makes the bid 1 , suggesting contract and flawless play. To find the optimal contract that the♦ partnership can win 1 more trick than♦ the min- both partners need to exchange information about their imum (which is always defined as 6 out of 13 possible). hands. This is done during the auction by bidding. Each E does not have particularly good cards in any suit and bid gives the other player some information but, as in therefore gives the answering bid of 1NT i.e., he suggests all auctions, the next bid has to be higher. This means 7 tricks without trump suit. By this bid, W understands that the more information is exchanged the higher the that E has no significant strength in and does not have contract will be and more difficult to make. In order to more than 4 cards in any suit (otherwise♦ he would have optimize the communication between the partners during suggested that suit). Therefore W suggests his second the auction phase various communication protocols have best suit, . E now knows that W prefers but can been devised. A short description of duplicate bridge is play with ♥ if necessary. E has more strength♦ in and given in the appendix. More can be found in any of the hence suggests♥ 3 i.e, 9 tricks with suit as trumps.♦ ♦ ♦ 3

The partnership has now settled the trump suit. Now they undertake a more difficult contract (by making more bids) in exchange for more information about the part- ner’s hand. W makes a so called cuebid of 4 , directly suggest 10 tricks but the cuebid implies that♣ W is aim- ing for 12 tricks. By choosing , W indicates that he has either no cards at all or one strong♣ card in the suit of . W is thus asking E with his bid if E has any key cards♣ (defined as four aces and trump King). E’s answer says that he indeed has the key cards needed. However W has no information about the last missing card of inter- est, the queen of trumps Q, Unfortunately, we have no standard tools to ask about♦ Q without exceeding the safety level of 5 . A typical♦ Bridge strategy is Roman ♦ FIG. 2: Quantum protocol of Bridge CCP. Alice (playing Key-card Blackwood 4NT (key-card refers to the king of East) holds two bits of information ai with i = 0, 1 (defined trumps and the four aces; NT stands for no trump)[12]. by her hand). Bob (playing West) chooses between b = 0 or If one partner wishes to ask the other whether he/she 1, which denotes in which bit of Alice he is interested. Alice has the queen of trumps in his hand he calls out a bid on chooses her measurement setting to be a = a0 ⊕ a1. Bob sets which meaning the players have agreed upon. The stan- his measurement according to his choice of b. After reading out her outcome A (which is encoded as a bit value), Alice dard answers for the partner are: For a bid 5 means sends, a one bit message to Bob, the value of m = A⊕a0. Bob that the partner has 0 or 3 key cards; similarly♣ 5 , 5 , ♦ ♥ computes his guess R for the value of the bit he wants to know and 5 means 1 or 4 key cards, 2 or 5 key cards without by adding the message to his local measurement outcome B ♠ the trump queen, and 2 or 5 key cards with the trump (also encoded as a a bit value), that is R = B ⊕ m. queen respectively. Clearly, there is no way to check for the trump queen without exceeding the level of 5 .Thus ♦ the classical bidding techniques are useless, and the part- a0 = 0 stands for 0 or 3 key cards, whereas a0 = 1 means ner W may be forced to guess whether the partner E has 1 or 4 of them; the bit a1 = 0 means that Alice has the trump queen or not. Q and a1 = 1 means that she does not have one. The assignments♦ of the particular values of the bits are arbi- trary. For simplicity we assume that for all the variables III. BRIDGE AS COMMUNICATION the value 0 corresponds to the most probable situation. COMPLEXITY PROBLEM If Alice has 2 or 5 key cards (the bit a0 is not defined) her answer is 5 or 5 just like in the standard Blackwood. ♥ ♠ The auction in Bridge can be regarded as a CCP. Let However, with the bits a0 and a1 well defined, it would HN , HE, HS and HW denote the hands of the players. be of an advantage for them to send a message to Bob There exists an optimal contract for W and E which is a allowing him to gain one of these values, or increase the function of all the hands f(HN ,HE,HS,HW ). The goal probability of guessing them. But, she can send exactly of the partners is to call the optimal contract, in other one bit of information to Bob without exceeding the crit- words: find the value of f, with as little communication ical value of 5 . That is, her response can be 5 or 5 . ♦ ♣ ♦ as possible. There are some additional constraints which Without knowing whether Bob is interested in a0 or a1 make the problem more challenging: (i) the amount of she can only, in the classical case, make a random choice, communication which the players are allowed to exchange or they could agree beforehand that in such a case she depends on the value of f and on the strategy of their sends, say a0. This highly limits their strategies. opponents, (ii) the function f may be hard to compute Clearly, their task is a certain CCP. This can be put even if all the hands are known, (iii) it is difficult to as follows: Alice has a random string of two bits a0 and compare the quantum and the classical strategies, since a1, while Bob makes his independent choice, which of the in most of the cases it is hard to find the optimal classical two bits he wants to learn, that is he fixes b = 0, 1, to strategies. learn ab (see Fig. 2).. But, Alice is allowed to send a Nevertheless, one can show that the bridge scenario in single bit message m to Bob (m = 0 in the form of 5 ♣ figure 1 is equivalent to a CCP, and prove that the quan- or m = 1as 5 ), while he is not allowed to send any ♦ tum strategies provide an advantage over classical ones. information. Let the bit b represent the type of information that Bob For such CCPs one is usually interested in the worst (playing West) is interested in (see Fig. 2). b = 0 cor- case success probability, or the average probability (the responds to Bob being interested in the key cards, and success is when Bob learns the correct value of a0 or a1). b = 1 to him being interested in Q. For Alice (playing One usually assumes uniformly distributed inputs. How- East) her bits will have the following♦ meaning: the bit ever, the nature of Bridge is such that this assumption 4 is not satisfied. Thus we must introduce the following Alice receives two bits a0 and a1. She chooses her figure of merit measurement setting according to the value of a = a0 ⊕ a1. The probability of a to be 0 is equal to p = p(a0 = I = p(a0 = i,a1 = j, b = k) 0)p(a1 = 0)+(1 p(a0 = 0))(1 p(a1 = 0)). Bob’s setting X × − − i,j,k is simply defined by his input bit b which can be 0 with

P (R = ab a0 = i,a1 = j, b = k,m) (1) probability q. After reading out her outcome A, Alice | prepares the message m = A a0 which she transmits to ⊕ where R is the value that Bob guesses for ab, after re- Bob. He then computes his guess value of the required ceiving m, and p(a ,a ,b) is the probability distribution bit, R, by adding the message to his outcome: R = B 0 1 ⊕ of a0,a1 and b. In simple words, I is the average proba- m. A simple calculation shows that R = ab as long as bility that Bob, after receiving the message m correctly A B = ab. ⊕ guesses the Alice’s bit he wants to know. Therefore, in our protocol Bob gets the correct value To find the maximal value of I achievable with classical of ab with probability I equal to resources, as in the case of any classical CCP, it suffices I = p(a,b)P (A B = ab a,b), (3) to check all deterministic encodings of a0 and a1 into one X ⊕ | a,b bit message m(a0,a1). The optimal deterministic strat- egy, that attains the maximum value of Ic corresponds to which can be put equivalently as Alice in each run sending a0 (or a1) as the message m. If Bob is interested in a (alternatively, a ), he gets its right 1 1 0 1 Q = + p(a,b)( 1)abE(a,b), (4) value. However, if he, in a given run, is interested in a1 2 2 X − a,b (alternatively, a0), his guess is the more probable value, which he infers from the known marginal distributions where the correlation function E(a,b) is given by P (A = p(a0) and p(a1). Thus one has B a,b) P (A = B a,b). The expression Q = | − ab6 | a,b p(a,b)( 1) E(a,b) is the left hand side of a bi- IC = max p(b =0)+ p(b = 1)max p(a1 = i b = 1) , P − { i { | } ased CHSH inequality considered in [15]. Its the maximal quantum value is given by [15] p(b = 1)+p(b = 0)max p(a0 = i b = 0) . i { | }} (2) Q = √2 q2 + (1 q)2 p2 + (1 p)2. (5) p − p −

In the game of duplicate bridge to win does not mean Thus, the maximal quantum value of IQ is to get more points than your opponents. It means to 1 get more points than other pairs playing with the same √ 2 2 2 2 IQ = 1+ 2 q + (1 q) p + (1 p)  . (6) cards. Because finding the optimal bid dramatically in- 2 p − p − creases the amount of points awarded (see ”Scoring and The exact probability distribution p(a0,a1,b) is diffi- match points” section of the appendix), doing so is the cult to estimate since it depends on the overall strategies necessary condition for winning. Therefore, the proba- of the partners and their opponents. We have asked the bility of guessing ab is the probability of winning and is bridge expert Tommy Gullberg for his estimates which referred to as such in the rest of the paper. are[11]: p(a0 = 0 b = 0) 0.5,p(a1 = 0 b = 1) 0.55 which implies p | 0.5. It≈ is also reasonable| to assume≈ q 0.75. With these≈ estimates the classical strategy (2) IV. QUANTUM BRIDGE gives≈ success probability 0.8875 while the quantum one (6) reaches 0.8953. As we have explained in the previous section, the task As we have explained before, the same quantum pro- that Alice and Bob have to perform in this situation re- tocol may be used in many different instances of bidding duces to communicating to the receiver one of the two and trick-taking. However, the probability distribution bits. The only problem is that it is the receiver that p(a0,a1,b) may differ from case to case. Therefore, in chooses what he is interested in and the communication the experimental section of the paper we have compared is restricted to a single bit. This kind of task is called a the performance of quantum and classical strategies for random access code and it is known that quantum infor- a wide range of parameters. Usually, p(a0 = 0 b = 0) | ≈ mation theory can deal with it more efficiently. p(a1 =0 b = 1) p and we use this approximation in all Let Alice and Bob make measurements on an entan- our figures.| ≈ gled state. Each of them can choose one of two observ- Let us stress that the game of Bridge as a whole is not a ables (Alice’s choice is denoted by a and Bob’s by b) with communication complexity problem. It only shares some binary outcomes A and B respectively. Consider a fol- properties of it, in certain clearly defined phases of the lowing protocol based on a 2 1 entanglement assisted game. For example its rules do not put any constraints on random access code discussed→ in Ref. [14]: the amount of communication between the partners but 5 only on the amount of communication about their cards. They can, for example ask about the other players strate- gies, discuss issues with the referee or simply ask the other partner to repeat his last bid if they did not hear it clearly. This means that, in the quantum strategy the players can make measurements on entangled pairs, and announce if they detected a particle, until both of them do. This information has nothing to do with their cards. Then they can carry on with the auction. This allows them to exploit the advantage of quantum states without having to worry about the efficiency of their detectors, which in standard communication complexity problems plays a crucial role. We also like to point out that in bridge offers no communication between partners that is not also available for the opposing team and hence eaves- FIG. 3: Experimental setup for Quantum Bridge CCP. UV dropping on the opposing team’s messages is pointless. light centered at wavelength of 390 nm are focused inside a Apart form the main advantage of having a larger prob- 2 mm thick BBO (β barium borate) nonlinear crystal, to pro- ability of playing the optimal contract, or better effi- duce photon pairs. Half wave plates (HWP) and two 1 mm ciency in the defense play, Alice and Bob also have a thick BBO crystals are used for compensation of longitudinal and transverse walk-offs. The emitted photons are coupled less obvious advantage. The rules of Bridge forbid the into 2 m single mode optical fibers (SMF) and passed through partners to use a secret strategy. Its detailed description a narrow-bandwidth interference filters (F) (∆λ = 1nm). Al- should be made available to the opponents before the ice ( or the E parter) uses (p : 1 − p) variable-ratio beam game. Moreover, the player making the bid after Alice splitter (VRBS) for her measurement basis choice with the (the sender) and before Bob (the receiver) can, before probabilities (p) and (1−p) (corresponding to her input value announcing his/her bid, ask the receiver what informa- a, see text) for the first and second base choice. Her measure- A1 A2 tion did Bob get and he must provide all the information ment observables and are realized by HWP oriented by φA1 and φA2 respectively. Bob uses (q : 1 − q) variable ratio that he has obtained. However, in the quantum case the beam splitter (VRBS) for his basis choice with the probabili- receiver can answer truthfully: I did not get any infor- ties (q) and (1−q) (corresponding to his input value b), for the mation yet because I haven’t measured my system yet and first and second basis. His observables B1 and B2 are realized

I cannot do it now because the choice of my measurement by HWP oriented by φB1 and φB2 respectively. The polariza- depends on your forthcoming bid. Therefore, using the tion measurements for Alice and Bob were performed using quantum strategy not only helps the partners to behave polarizing beam splitters (PBS) and single photon detectors more optimally, but also makes the game harder for the (D). opponents, as the overtly conveyed message carries no in- formation, until it is added to the result of the receiver. in [15]: Furthermore, in the quantum case the choice of what the receiver learns is delayed until the very last moment be- σx(q + (1 q)cos β)+ σz(1 q) sin β fore his bid, which gives him more knowledge about the A1 = − − , (q + (1 q)cos β)2 + (1 q)2 sin2 β opponents’ cards. This allows him to learn information q − − which is more relevant as he knows more about its con- σx(q (1 q)cos β) σz(1 q) sin β text. An advantage of this type is difficult to quantify A2 = − − − − , (q (1 q)cos β)2 + (1 q)2 sin2 β and delaying the choice of measurement requires high q − − − detection rates, therefore we do not dwell on this subject B1 = σx, anymore here. B2 = σx cos β + σz sin β, 1 ψ = ( 0 0 + 1 1 ) , (7) | i √2 | i| i | i| i V. EXPERIMENTAL REALIZATION where σz and σx are the standard Pauli operators, and 1 (q2 + (1 q)2)(p2 (1 p)2) Let’s describe our experiment, which demonstrates the cos β = − − − . (8) 2 q(1 q)(p2 + (1 p)2) quantum violation of the classical bound of (6) for various − − values of p and q. At the same time it is the experimental realization of a biased nonlocal game [15, 16], a quantum We have realized these quantum protocols by using po- CCP, and the quantum Bridge. The optimal state for larization entangled pairs of photons φ+ = ( HH + all these tasks is a two qubit maximally entangled state V V )/√2. In the experiment, UV| lighti centered| i at and the measurements of the parties are the ones given |wavelengthi of 390 nm was focused inside a 2 mm thick 6

1 and passed through a narrow-bandwidth interference fil- ters (F) (∆λ = 1 nm) to secure well defined spatial and

0.95 spectral emission modes (see Fig. 3)[17]. C(Q) Alice uses (p : 1 p) variable ratio beam splitter − 0.9 (VRBS) for her measurement basis choice with the prob- abilities (p) and (1 p) for the first and second basis − 0.85 choice. Her measurement observables A1 (corresponding to a = 0) and A2 (corresponding to a = 1) are real-

0.8 ized by HWP oriented by φA1 and φA2 respectively. Bob Success Probability I uses (q : 1 q) variable ratio beam splitter (VRBS) for − 0.75 his basis choice with the probabilities (q) and (1 q) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 − q for the first and second basis choice. His measurement observables B1 and B2 (corresponding to b = 0 and 1

FIG. 4: Experimental results for Quantum Bridge: (a) the respectively) are realized by HWP oriented by φB1 and IC q IQ q classical ( ) (dashed line), quantum ( ) (continuous φB2 respectively (see fig. 3) such as: line), and the experimental data points observed for the suc- cess probability I for p(a0 = 0|b = 0) = p(a1 = 0|b = 1) = (1 q)cos(π/2 β) p = 0.5. For q = 0.5, the obtained quantum success prob- tan φ 1 = − − , A q + (1 q) sin (π/2 β) ability is 0.848 ± 0.005 (the corresponding classical value is − − 0.75). For the value of q = 0.75, which we estimate to cor- (1 q) sin (π/2 β) tan φ 2 = − − − , respond to a common situation in Bridge, the quantum and A q (1 q)cos(π/2 β) classical values are 0.895 (the experimental quantum value is − − − 0.887 ± 0.005) and 0.875 respectively. φB1 = π/4,

φB2 = π/2 β. (9) − The polarization measurement was performed using polarizing beam splitters (PBS) and single-photon de- tectors (D) placed at the two output modes of the PBS. Our detectors are actively quenched Si-avalanche photo- diodes. All single detection events were registered using a VHDL programmed multichannel coincidence logic unit, with a time coincidence window of 1.7 ns. The measure- ment time for each setting was 200 seconds. We have tested the protocols for different probabilities p and q by changing the transmission coefficients of Al- ice’s and Bob’s VRBS. Fig. 4 shows the classical IC (q) (dashed line), quantum IQ(q) (continuous line), and ex- perimental data points observed for the success proba- bility versus q for the value of p = 0.5. Fig. 5 shows 3D plot for the classical ICS(p, q), quantum IQS (p, q), FIG. 5: Experimental results for quantum CCP protocol for and experimental data observed for the success proba- the whole spectrum of quantum strategies. The classical bility of CCP protocols versus p and q. Our results are IC (p, q) (blue plot), quantum IQ(p, q) (red plot), and exper- in very good agreement with the theoretical predictions imental data (green points) for the success probability I as and clearly demonstrate the advantage of the quantum functions of p and q. The classical value of IC (p, q) is calcu- strategy over the classical ones. lated from (2) under the assumption that p(a0 = 0|b = 0) = ′ ′2 ′ 2 p(a1 = 0|b =1)= p , which leads to p = p +(1−p ) . (Only half of IC (p, q) plot is shown to make the IQ(p, q) plot and the experimental data points visible). VI. CONCLUSION

In summary, we report an experimental realization of a BBO (β barium borate) nonlinear crystal. Photon pairs, quantum Bridge CCP protocol, which is the first demon- due to a degenerate emission of type-II spontaneous stration of a quantum CCP usable in a real-life scenario. parametric down-conversion, are collected in two spatial This was possible because we show that our quantum modes a and b. Half wave plates (HWP) and two 1 mm Bridge protocol corresponds to a biased nonlocal CHSH thick BBO crystals are used for compensation of longi- game, which in turn is equivalent to a 2 1 quantum tudinal and transversal walk-offs. The emitted photons random access code. Thus, our experiment→ is also a real- were coupled into 2 m single mode optical fibers (SMF) ization of an entanglement assisted random access code. 7

We also establish links between game theory, commu- APPENDIX: THE RULES OF DUPLICATE nication complexity, and quantum physics. Such quan- BRIDGE tum games and communication complexity protocols and their links can be generalized to n parties scenario. Con- Duplicate bridge is a metagame. The same bridge deal cerning Bridge, it is up to the World Bridge Federation to (i.e. the specific arrangement of the 52 cards into the decide whether to allow quantum resources and encoding four hands) is played at each table and scoring is based strategies in championships. A positive decision would on relative performance. This reduces the element of make this technique the first commonplace application chance while heightening the one of skill. Now we give of quantum communication complexity. A negative one the brief description of the game. More details can be would forbid quantum strategies and thus, would consti- found in e.g [10]. tute the first ever regulation of quantum resources sport. The results reported here will contribute to deeper un- derstanding of the possible impact of quantum resources Introduction on information and communication technologies. Duplicate bridge is one of the most popular card games, recognized by the International Olympic commit- tee as a sport (only two ”mind sports”: bridge and chess ACKNOWLEDGMENTS are recognized) and governed by the World Bridge Feder- ation for international competitions. It uses a standard The authors thank Johan Ahrens and Tommy Gullberg deck of 52 cards and is played by exactly four players. for discussions. This work was supported by the Swedish The players are usually named according to their seats Research Council (VR),the Linnaeus Center of Excel- directions as East, West, North and South. Among these lence ADOPT, TEAM programme of foundation for Pol- West and East form a partnership or a pair competing ish Science (FNP), NCN grant 2013/08/M/ST2/00626, against North and South. QUASAR (ERA-NET CHIST-ERA 7FP UE) and UK A tournament consists of several games. In each game EPSRC. the cards are distributed between the players. In rubber bridge a more casual version of the game this distribu- tion, called a deal, heavily influences the winning proba- bility. However, in duplicate bridge previously prepared deals are stored in bridge boards - simple four-way card [1] A. C. Yao, Proceedings of the 11th Annual ACM Sym- holders. They are used to enable each player’s hand to be posium on Theory of Computing, 209 (1979). passed intact to the next table that must play the same 56 [2] R. Cleve and H. Buhrman, Phys. Rev. A , 1201 (1997). deal, and final scores are calculated by comparing each [3] H. Buhrman, W. van Dam, P. Høyer, and A. Tapp, Phys.Rev. A 60, 2737 (1999). pair’s result with others who played the same hand. [4] H. Buhrman, R. Cleve, and W. van Dam, SIAM J. Com- Each game consists of two main phases: put. 30, 1829 (2001). [5] L. Hardy and W. van Dam, Phys.Rev. A 59, 2635 (1999). 1. The auction, also called bidding for a contract; ˇ ˙ [6] C Brukner, M. Zukowski, and A. Zeilinger, Phys. Rev. 2. Playing the hand, when the pair that won the auc- 89 Lett. , 197901, (2002). tion tries to take enough tricks to make the con- [7] E. Kushilevitz and N. Nisan, Communication complexity. (Cambridge University Press, England, 1997). tract. [8] C.ˇ Brukner, M. Zukowski,˙ J.-W. Pan, and A. Zeilinger, Phys. Rev. Lett. 92, 127901, (2004). [9] ”General conditions of contest”, Bidding in Bridge http://www.worldBridge.org/departments/rules/ GeneralConditionsOfContest2011.pdf. In order to understand how to bid, first one should [10] R. Klinger, P. Husband, and A. Kambites Basic Bridge, know the ranking of cards and suits. The deck of 52 London: Vixtor Gollancz, ISBN 0-575-05690-8 (1994). cards consists of 4 suits: spades , hearts , diamonds [11] T. Gullberg, private communication. ♠ ♥ ♦ [12] http://en.wikipedia.org/wiki/Blackwood convention and club . Spades are ranked as the highest and the next suit♣ is hearts ♠, together they are called the major [13] http://en.wikipedia.org/wiki/Gerber convention ♥ ˙ suits. They are followed by minor suits with diamonds [14] M. Paw lowski, and M. Zukowski, Phys. Rev. A 81, ♦ 042326, (2010). ranked higher than clubs . This ranking is important [15] T. Lawson, N. Linden, and S. Popescu, arXiv:1011.6245, in biding and scoring at the♣ end of the game. In a suit, (2010). cards are ranked from 2 being the lowest to ace being the [16] A. Winter, Nature 466, 1053 (2010). highest. et al. 74 [17] P. G. Kwiat Phys. Rev. Lett. , 4337-4341 (1995). The bidding phase starts with the dealer (one player [18] http://en.wikipedia.org/wiki/Bridge scoring is marked as dealer on each bridge board) and rotates 8 around the table clockwise with each player making a Scoring and match points call. A call is limited to a vocabulary of 38 words or phrases consisting of: If the declaring side won the number of tricks specified by contract (or more) they are awarded some points and a Bid which states a level and a denomination; a • their opponents get exactly the same amount of negative denomination can be any of the four suits or NT ones. If they won less the situation is reversed. (which stands for No Trump); given 7 levels of bid- If the declaring side made their contract, they get the ding and 5 denominations, there are 35 possible points for the following: bids; Odd tricks - the number of tricks specified by the • Double, only available when the last bid was made level of the contract. The amount of points de- • by an opponent; pends on the level of the contract, denomination and whether it was doubled or redoubled. The Redouble, only available after opponent’s double; points awarded for this are called contract points. • Overtricks - if the partnership won more tricks than Pass, when unwilling or unable to make one of the • • three preceding calls. contract specified they are counted as overtricks. They are worth less points than odd tricks. The This vocabulary is further limited by the requirement amount of points here also depends on denomi- that each bid has to be sufficient, i.e. it has to have either nation and whether the contract was doubled or a higher level than a previous bid or the same level and a redoubled but also on vulnerability. Whether or higher denomination. NT is higher than any of the suits. not the partnership is vulnerable is marked on the For example, if the last bid was 3 the next one can be bridge board. 3NT or 4 but not 3 . ♠ ♣ ♣ Slam bonus - if the partnership bids and wins 12 If three players in a row pass, the auction is over. A • or 13 tricks they are awarded a huge bonus which pair is said to have won the auction if the last bid was depends on their vulnerability. made by one of its players. This partnership is called the declaring side. The player on the declaring side who, dur- Double bonus - A bonus for making the contract ing the auction, first stated the denomination of the final • that has been doubled or redoubled. bid becomes ”declarer,” the declarer’s partner becomes Game bonus - A small bonus for making the con- the ”dummy,” and the opposing side become the ”de- • fenders.” The final bid also becomes the ”contract”. The tract is awarded. A much larger one is given if the declaring side promises to take, during the next phase, partnership scored more than 100 contract points. the number of tricks greater or equal to the level of the Even larger if they were vulnerable. contract plus 6 and the denomination becomes the trump If the declaring side failed to make their contract, their (in case of denomination NT, there is no trump). opponents get points for each undertrick. The amount of points is larger than for overtricks or even odd tricks and depends on vulnerability and whether the contract was Playing the hand doubled or redoubled. The tables with the exact point values can be found To begin play, the defender on the declarer’s left makes e.g. here [18]. An important implication of these scoring the opening lead by placing his selection face up on the rules is that finding the optimal contract is crucial for table. The dummy then spreads his hand on the table the game. This stems from the fact that overtricks are so that it is visible to every other player. The players worth less than odd tricks, game bonus is awarded only play clockwise around the table placing their cards on for contract points and slam bonus only applied if the the table, and each must ”follow suit” (that is, play a parties bid for a slam. For example, let us consider a card of the suit lead to the trick) if able. A player that case where the declaring side won 9 tricks with no trump cannot follow suit may either ”ruff” (play a trump) if while being vulnerable. If their contract was 2NT they there is a trump suit or ”sluff” (play a card of any other are awarded 70 contract points plus 30 for one overtrick suit). The player that plays either the highest trump or, plus 50 of game bonus, which gives them 150 total. If in a trick that contains no trumps, the highest card of the their contract was 3NT they get 100 contract points and suit led to the trick (1) wins the trick for its side and (2) a gem bonus of 500, which gives them 600 in total. If their proceeds to lead to the next trick. The declarer directs contract was 4NT they had one undertrick and they get the play of cards from the dummy in addition to playing 100 penalty points. It is therefore crucial for the declaring cards from his own hand. The play continues until all side neither to under- or overestimate their ability to win thirteen tricks are played. Then the score is calculated. tricks. 9

However, these points do not influence the final posi- tion of the partnership that scored them directly. Instead all the pairs that played the same deal are compared (the pairs that sat at North-South and East-West positions are compared separately). Then 2 match points (MPs) are awarded for each partnership that scored less points 1 with the same deal. 2 MP is awarded for each partner- ship that scored the same amount. Therefore, scoring 90 MPs when everyone else made only 70 is as good as scoring 2000.

Bidding strategies FIG. 6: Cards for defense play and signaling.

In duplicate bridge auction phase is much more impor- tant than playing the hand. Often, the second phase is the auction. Therefore they bear resemblance to commu- reduced to the declarer revealing his hand and claiming nication complexity problems. that he will take exactly the amount of tricks specified in the contract and the defenders agree ending the game. The auction phase is all about communication. The Strategies for playing the hand purpose of some early bids may be to exchange infor- mation rather than to set the final contract. For most The strategies for playing the hand are very different players, many calls (bids, doubles and redoubles, and for the declaring side and the defenders. This stems from sometimes even passes) are not made with the intention the fact that dummy’s cards are visible to everyone and that they become the final contract, but to describe the they are being played by the declarer. This leads to asym- strength and distribution of the player’s hand, so that metry. The declaring side is reduced to a single player the partnership can reach an informed conclusion on their and half of its cards are known to the opponents. The best contract, and/or to obstruct the opponents’ bidding. defenders, on the other hand, face a problem of coor- The set of agreements used by a partnership about the dinating their actions. Usually, they do not have many meaning of each call is referred to as a bidding system, opportunities to exchange information during the auction full details of which must be made available to the op- so they have to use the cards they play to send signals. ponents; ’secret’ systems are not allowed. An opponent One of the most common conventions is called Smith Pe- can ask the bidder’s partner to explain the meaning of ter. When one of the defenders plays the opening lead the call. the other plays a low spot card (2-5) if he would like his There are around 5.36 1028 different deals possible partner to play this suit once more and high spot card × which makes bidding strategies very complex, especially (6-10) if he wants to discourage his partner from playing that they have to take into the account possibility of the this suit. This strategy is called signalling. opponents interfering. To make it possible for humans to Again, the communication is limited because the play- play the game, they are divided into conventions - sub- ers aim at establishing a joint defence strategy as soon as strategies that are applicable only in certain situations, possible and their freedom of choosing the signals is con- e.g. Cappelletti convention is a strategy for interrupting strained by the cards they have. Therefore, the quantum opponents communication while sending the partner in- protocol described in the main text can be adapted also formation about the strongest suit; it can be used only to defence play. An example of this is presented in the after bid 1NT by the opponents by a player with moder- next part of the appendix. ately strong cards. Duplicate bridge is one of the most complex games Because making a contract for more than 100 contract and this short description presents only the very basics points or a slam leads to large amounts of bonus points of it. Interested reader should read any of the numerous there are multiple conventions designed to check if such books available on this subjects and try to play the game contracts are possible to make. One of the most popular himself/herself. is Roman Key Blackwood described in the main text. All the conventions, apart for the ones for interrupting the opponents, are designed to convey as much information APPENDIX: QUANTUM STRATEGIES FOR as possible while keeping the final bid as low as possible. DEFENCE PLAY This enables the players to stay at safe level in case they find out that their cards are not that good or to squeeze The quantum protocol described in the main text can in additional rounds of communication before concluding also be used for defense play and signaling. To see this 10 consider the cards given in figure 6. In this particular formation about the queen, we once again use the given game of Bridge, the W-E team is the defending team quantum protocol with the following variable definitions. and N is the dummy. In this given scenario, let assume If a0 = 0, encourages suit. If a0 = 1, discourages suit. If that W discards the A against S’s 4 contract. After a1 = 0, odd number of cards. If a1 = 1, even number of N’s response, East plays♥ the 10 to signal♦ a discouraging cards With q defined as the probability of W being inter- attitude towards the played suit♥ (see Fig 6). For W, it is ested in the attitude towards the suit we may define the interesting to know whether E possesses the Q or not. message m as: if m = 0, then play a small card, if m = 1, From the viewpoint of W, E can have 10 as♥ the highest then play a high card. Thus, the quantum protocol has card in the suit or East might have♥ e.g. Q and still once again improved the conditions of playing the best play 10 to signal a negative attitude. If♥ East has the card. Generally, whenever communication between the queen,♥ West should discard the 9 so that East can play partners is restricted to one single bit and there exists a the Q. If East does not have♥ the relevant card, there non-zero probability that the partner might be interested is a♥ good risk of letting the declarer win the 10 tricks in values of one of two bits of information, the protocol needed to fulfill the contract. In conclusion; West needs gives a quantum advantage. information about the Q. In order to extract the in- ♥