Quantum bidding in Bridge
S. Muhammad, A. Tavakoli, M. Kurant, M. Pawlowski, M. Zukowski, M. Bourennane
aa r X i v : . [ qu a n t - ph ] M a r . Quantum bidding in Bridge
Sadiq Muhammad, Armin Tavakoli, Maciej Kurant, MarcinPaw lowski,
3, 4
Marek ˙Zukowski, and Mohamed Bourennane Department of Physics, Stockholm University, S-10691, Stockholm, Sweden Department of Information Technology and Electrical Engineering, ETH Zurich, Switzerland Instytut Fizyki Teoretycznej i Astrofizyki, Uniwersytet Gda´nski, PL-80-952 Gda´nsk, Poland Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom (Dated: March 19, 2014)Quantum methods allow to reduce communication complexity of some computational tasks, withseveral separated partners, beyond classical constraints. Nevertheless, experimental demonstrationsof this fact are thus far limited to some abstract problems, far away from real-life tasks. We showhere, and demonstrate experimentally, that the power of reduction of communication complexity canbe harnessed to gain advantage in famous, immensely popular, card game - Bridge. The essence of awinning strategy in Bridge is efficient communication between the partners. The rules of the gameallow only specific form of communication, of a very low complexity (effectively one has a stronglimitations on number of exchanged bits). Surprisingly, our quantum technique is not violating theexisting rules of the game (as there is no increase in information flow). We show that our quantumBridge auction corresponds to a biased nonlocal Clauser-Horne-Shimony-Holt (CHSH) game, whichis equivalent to a 2 → PACS numbers: 03.65.Ud, 03.67.Mn, 42.50.Xa
I. INTRODUCTION
Quantum information science breaks the limitationsof conventional information transfer, cryptography andcomputation. Communication complexity problems(CCPs)[1] were shown to have quantum protocols, whichoutperform any classical counterparts. In CCPs, twotypes can be distinguished. The first type minimizesthe amount of information exchange necessary to solvea task with certainty [2–4]. The second type maximizesthe probability of successfully solving a task with a re-stricted amount of communication [4–6]. Such studiesaim, e.g., at a speed-up of a distributed computation byincreasing the communication efficiency, or at an opti-mization of VLSI circuits and data structures [7].Fundamentally, there exist connections between quan-tum CCPs, quantum games, and tests of the foundationsof quantum mechanics. It has been shown that for ev-ery CCP there is a corresponding quantum game andvice versa. Furthermore, it has also been proved that forevery Bell inequality and for a broad class of protocols,there always exists a multi-party CCP, for which the pro-tocol assisted by quantum states, which violate the Bellinequality, is more efficient than any classical protocol[6, 8]. However, in contrast with cryptography, the exist-ing demonstrations of quantum protocols reducing com-munication complexity are abstract problems, and haveno practical applications. Our aim is to apply quantumcorrelations in protocols related to a well known real-lifetask. The task that we are considering here is playing the card game of (Duplicate) Bridge.The essence of a successful game of Bridge is an effi-cient communication between the partners. Due to therules of the game, the form and the amount of infor-mation exchanged between the partners is severely re-stricted. We show that using the quantum resourcesthe players can increase their winning probability. Whatis important, our protocol does not require any changein the rules of the game (or perhaps the very existenceof it would force a change in the rules, so as to add amissing point: no quantum reduction of communicationcomplexity is allowed during the game). In order to useour scheme, the players need to share an entangled stateand locally measure its subsystems. Such procedure isnot against the rules of the World Bridge Federation [9],and it is not a method of transferring information (no‘Bell-telephone’ rule). Our aim is to show, that one canexploit the difference between the quantum and the clas-sical resources in CCPs, in which there are strict limitson the amount of communication, to gain advantage inBridge. So, whenever the rules governing a real-life sit-uation put a limit on the amount of communication, itshould also be specified whether quantum resources areallowed or not, since the quantum protocols, which out-perform the classical counterparts, are within the reachof the current state-of-the-art technology, as shown byour experiment. We present an experimental realizationof a quantum Bridge protocol, in which the quantum re-sources provide an advantage over the classical.
FIG. 1: Cards and Bidding for Bridge. An example of cardsand bidding for the W and E partners. II. BRIDGE
Bridge is one of the world’s most popular card games.It is a trick-taking game with the standard deck of 52cards. The game is played by four players playing inpairs with partners sitting at opposite sides of a table,and named as West ( W )-East ( E ) and North ( N )-South( S ). The game consists of several deals each progress-ing through two phases: the auction (called bidding),and playing the hand (called trick-taking) . The biddingphase starts with the dealer and rotates around the tableclockwise with each player making a bid for the contract.The team that wins the auction undertakes to win a cer-tain number of tricks during the game. The second phaseof the game is the standard trick-taking play. Here, thewinners of the auction try to fulfill their contract (takingas many tricks as the contract obliges them to) while theopposing team intends to prevent them from doing so.The team that succeeds with its task wins the round. Inboth phases of Bridge, efficient communication is of vitalimportance for the partnership. This is even more truein the tournament version of the game - duplicate Bridgerecognized by the International Olympic Committee as asport. There, a larger group of partnerships play againsteach other with prearranged hands to reduce the factorof chance to a minimum. What counts in this variationis earning more points with a given hand than the otherpartnerships had. This is done by winning the optimalcontract and flawless play. To find the optimal contractboth partners need to exchange information about theirhands. This is done during the auction by bidding. Eachbid gives the other player some information but, as inall auctions, the next bid has to be higher. This meansthat the more information is exchanged the higher thecontract will be and more difficult to make. In order tooptimize the communication between the partners duringthe auction phase various communication protocols havebeen devised. A short description of duplicate bridge isgiven in the appendix. More can be found in any of the numerous books on this subject, e.g.[10].In this paper we aim to show that there are quantumcommunication strategies outperform the classical onesin the game of bridge. Unfortunately, there is no bestclassical strategy. We have contacted Tommy Gullberga World Life Master in duplicate bridge, author of manyarticles and books about the game, who told us: A profes-sional player plays with a strategy of his own preferencesince he finds it optimal. Thus if you go to a professionaltournament you will encounter a wide range of differentstrategies, all with some strengths and some weaknesses,since there clearly is no such thing as an optimal classicalbridge strategy”[11].However, all the strategies must involve a ”slam-seeking convention”, which is a sub-strategy that play-ers use if they have very strong cards hope to win abig bonus. Although, quantum strategies are probablyuseful at many different situations in bridge, in this pa-per we only give an example for improving Roman Key-card Blackwood (RKB)[12]. It is a very popular slam-seeking strategy and often used in bridge games today.If you play a game of bridge (in a tournament) it is verylikely that the RKB will be used, probably several times.Therefore, the optimal classical strategy is likely to in-clude it[11]. And even if it does not, other slam-seekingstrategies like Gerber [13] differ from it only slightly andthe quantum protocol can be modified accordingly. Toshow that the problem at which the quantum protocolsexcel at is ubiquitous in bridge in the appendix we presenthow the strategy used for RKB can be modified to beused in playing the hand.The following paragraph plunges deep into the nuancesof bidding. A reader not interested in the game shouldskip it and know only this: The task for the players isto decide if they have cards strong enough to try to geta bonus. To this end, one of them will ask questions(encoded in bids) and the other provide answers. In thisparticular case there are two important questions butonly one of them can be asked. Classical and quantumways of dealing with this problem are described in thenext section.Consider the cards and bidding in a bridge scenarioshown in figure 1. The bidding can be explained as fol-lows, W looks at his cards. He has good cards in dia-monds ( ♦ ) and therefore makes the bid 1 ♦ , suggestingthat the partnership can win 1 more trick than the min-imum (which is always defined as 6 out of 13 possible).E does not have particularly good cards in any suit andtherefore gives the answering bid of 1 N T i.e., he suggests7 tricks without trump suit. By this bid, W understandsthat E has no significant strength in ♦ and does not havemore than 4 cards in any suit (otherwise he would havesuggested that suit). Therefore W suggests his secondbest suit, ♥ . E now knows that W prefers ♦ but canplay with ♥ if necessary. E has more strength in ♦ andhence suggests 3 ♦ i.e, 9 tricks with ♦ suit as trumps.The partnership has now settled the trump suit. Nowthey undertake a more difficult contract (by making morebids) in exchange for more information about the part-ner’s hand. W makes a so called cuebid of 4 ♣ , directlysuggest 10 tricks but the cuebid implies that W is aim-ing for 12 tricks. By choosing ♣ , W indicates that he haseither 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 saysthat he indeed has the key cards needed. However Whas no information about the last missing card of inter-est, the queen of trumps ♦ Q, Unfortunately, we have nostandard tools to ask about ♦ Q without exceeding thesafety level of 5 ♦ . A typical Bridge strategy is RomanKey-card Blackwood 4NT (key-card refers to the king oftrumps and the four aces; NT stands for no trump)[12].If one partner wishes to ask the other whether he/shehas the queen of trumps in his hand he calls out a bid onwhich meaning the players have agreed upon. The stan-dard answers for the partner are: For a bid 5 ♣ meansthat the partner has 0 or 3 key cards; similarly 5 ♦ , 5 ♥ ,and 5 ♠ means 1 or 4 key cards, 2 or 5 key cards withoutthe trump queen, and 2 or 5 key cards with the trumpqueen respectively. Clearly, there is no way to check forthe trump queen without exceeding the level of 5 ♦ .Thusthe classical bidding techniques are useless, and the part-ner W may be forced to guess whether the partner E hasthe trump queen or not. III. BRIDGE AS COMMUNICATIONCOMPLEXITY PROBLEM
The auction in Bridge can be regarded as a CCP. Let H N , H E , H S and H W denote the hands of the players.There exists an optimal contract for W and E which is afunction of all the hands f ( H N , H E , H S , H W ). The goalof the partners is to call the optimal contract, in otherwords: find the value of f , with as little communicationas possible. There are some additional constraints whichmake the problem more challenging: (i) the amount ofcommunication which the players are allowed to exchangedepends on the value of f and on the strategy of theiropponents, (ii) the function f may be hard to computeeven if all the hands are known, (iii) it is difficult tocompare the quantum and the classical strategies, sincein most of the cases it is hard to find the optimal classicalstrategies.Nevertheless, one can show that the bridge scenario infigure 1 is equivalent to a CCP, and prove that the quan-tum strategies provide an advantage over classical ones.Let the bit b represent the type of information that Bob(playing West) is interested in (see Fig. 2). b = 0 cor-responds to Bob being interested in the key cards, and b = 1 to him being interested in ♦ Q. For Alice (playingEast) her bits will have the following meaning: the bit
FIG. 2: Quantum protocol of Bridge CCP. Alice (playingEast) holds two bits of information a i with i = 0 , b = 0 or1, which denotes in which bit of Alice he is interested. Alicechooses her measurement setting to be a = a ⊕ a . Bob setshis measurement according to his choice of b . After readingout her outcome A (which is encoded as a bit value), Alicesends, a one bit message to Bob, the value of m = A ⊕ a . Bobcomputes his guess R for the value of the bit he wants to knowby adding the message to his local measurement outcome B (also encoded as a a bit value), that is R = B ⊕ m . a = 0 stands for 0 or 3 key cards, whereas a = 1 means1 or 4 of them; the bit a = 0 means that Alice has ♦ Q and a = 1 means that she does not have one. Theassignments of the particular values of the bits are arbi-trary. For simplicity we assume that for all the variablesthe value 0 corresponds to the most probable situation.If Alice has 2 or 5 key cards (the bit a is not defined) heranswer is 5 ♥ or 5 ♠ just like in the standard Blackwood.However, with the bits a and a well defined, it wouldbe of an advantage for them to send a message to Boballowing him to gain one of these values, or increase theprobability of guessing them. But, she can send exactlyone bit of information to Bob without exceeding the crit-ical value of 5 ♦ . That is, her response can be 5 ♣ or 5 ♦ .Without knowing whether Bob is interested in a or a she can only, in the classical case, make a random choice,or they could agree beforehand that in such a case shesends, say a . This highly limits their strategies.Clearly, their task is a certain CCP. This can be putas follows: Alice has a random string of two bits a and a , while Bob makes his independent choice, which of thetwo bits he wants to learn, that is he fixes b = 0 ,
1, tolearn a b (see Fig. 2).. But, Alice is allowed to send asingle bit message m to Bob ( m = 0 in the form of 5 ♣ or m = 1 as 5 ♦ ), while he is not allowed to send anyinformation.For such CCPs one is usually interested in the worstcase success probability, or the average probability (thesuccess is when Bob learns the correct value of a or a ).One usually assumes uniformly distributed inputs. How-ever, the nature of Bridge is such that this assumptionis not satisfied. Thus we must introduce the followingfigure of merit I = X i,j,k p ( a = i, a = j, b = k ) × P ( R = a b | a = i, a = j, b = k, m ) (1)where R is the value that Bob guesses for a b , after re-ceiving m , and p ( a , a , b ) is the probability distributionof a , a and b . In simple words, I is the average proba-bility that Bob, after receiving the message m correctlyguesses the Alice’s bit he wants to know.To find the maximal value of I achievable with classicalresources, as in the case of any classical CCP, it sufficesto check all deterministic encodings of a and a into onebit message m ( a , a ). The optimal deterministic strat-egy, that attains the maximum value of I c corresponds toAlice in each run sending a (or a ) as the message m . IfBob is interested in a (alternatively, a ), he gets its rightvalue. However, if he, in a given run, is interested in a (alternatively, a ), his guess is the more probable value,which he infers from the known marginal distributions p ( a ) and p ( a ). Thus one has I C = max { p ( b = 0) + p ( b = 1) max i { p ( a = i | b = 1) } ,p ( b = 1)+ p ( b = 0) max i { p ( a = i | b = 0) }} . (2)In the game of duplicate bridge to win does not meanto get more points than your opponents. It means toget more points than other pairs playing with the samecards. Because finding the optimal bid dramatically in-creases the amount of points awarded (see ”Scoring andmatch points” section of the appendix), doing so is thenecessary condition for winning. Therefore, the proba-bility of guessing a b is the probability of winning and isreferred to as such in the rest of the paper. IV. QUANTUM BRIDGE
As we have explained in the previous section, the taskthat Alice and Bob have to perform in this situation re-duces to communicating to the receiver one of the twobits. The only problem is that it is the receiver thatchooses what he is interested in and the communicationis restricted to a single bit. This kind of task is called arandom access code and it is known that quantum infor-mation theory can deal with it more efficiently.Let Alice and Bob make measurements on an entan-gled state. Each of them can choose one of two observ-ables (Alice’s choice is denoted by a and Bob’s by b ) withbinary outcomes A and B respectively. Consider a fol-lowing protocol based on a 2 → a and a . She chooses hermeasurement setting according to the value of a = a ⊕ a . The probability of a to be 0 is equal to p = p ( a =0) p ( a = 0)+(1 − p ( a = 0))(1 − p ( a = 0)). Bob’s settingis simply defined by his input bit b which can be 0 withprobability q . After reading out her outcome A , Aliceprepares the message m = A ⊕ a which she transmits toBob. He then computes his guess value of the requiredbit, R , by adding the message to his outcome: R = B ⊕ m . A simple calculation shows that R = a b as long as A ⊕ B = ab .Therefore, in our protocol Bob gets the correct valueof a b with probability I equal to I = X a,b p ( a, b ) P ( A ⊕ B = ab | a, b ) , (3)which can be put equivalently as Q = 12 + 12 X a,b p ( a, b )( − ab E ( a, b ) , (4)where the correlation function E ( a, b ) is given by P ( A = B | a, b ) − P ( A = B | a, b ). The expression Q = P a,b p ( a, b )( − ab E ( a, b ) is the left hand side of a bi-ased CHSH inequality considered in [15]. Its the maximalquantum value is given by [15] Q = √ p q + (1 − q ) p p + (1 − p ) . (5)Thus, the maximal quantum value of I Q is I Q = 12 (cid:16) √ p q + (1 − q ) p p + (1 − p ) (cid:17) . (6)The exact probability distribution p ( a , a , b ) is diffi-cult to estimate since it depends on the overall strategiesof the partners and their opponents. We have asked thebridge expert Tommy Gullberg for his estimates whichare[11]: p ( a = 0 | b = 0) ≈ . , p ( a = 0 | b = 1) ≈ . p ≈ .
5. It is also reasonable to assume q ≈ .
75. With these estimates the classical strategy (2)gives success probability 0.8875 while the quantum one(6) reaches 0.8953.As we have explained before, the same quantum pro-tocol may be used in many different instances of biddingand trick-taking. However, the probability distribution p ( a , a , b ) may differ from case to case. Therefore, inthe experimental section of the paper we have comparedthe performance of quantum and classical strategies fora wide range of parameters. Usually, p ( a = 0 | b = 0) ≈ p ( a = 0 | b = 1) ≈ p and we use this approximation in allour figures.Let us stress that the game of Bridge as a whole is not acommunication complexity problem. It only shares someproperties of it, in certain clearly defined phases of thegame. For example its rules do not put any constraints onthe amount of communication between the partners butonly 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 theother partner to repeat his last bid if they did not hearit clearly. This means that, in the quantum strategy theplayers can make measurements on entangled pairs, andannounce if they detected a particle, until both of themdo. This information has nothing to do with their cards.Then they can carry on with the auction. This allowsthem to exploit the advantage of quantum states withouthaving to worry about the efficiency of their detectors,which in standard communication complexity problemsplays a crucial role. We also like to point out that inbridge offers no communication between partners that isnot also available for the opposing team and hence eaves-dropping on the opposing team’s messages is pointless.Apart form the main advantage of having a larger prob-ability of playing the optimal contract, or better effi-ciency in the defense play, Alice and Bob also have aless obvious advantage. The rules of Bridge forbid thepartners to use a secret strategy. Its detailed descriptionshould be made available to the opponents before thegame. Moreover, the player making the bid after Alice(the sender) and before Bob (the receiver) can, beforeannouncing his/her bid, ask the receiver what informa-tion did Bob get and he must provide all the informationthat he has obtained. However, in the quantum case thereceiver can answer truthfully: I did not get any infor-mation yet because I haven’t measured my system yet andI cannot do it now because the choice of my measurementdepends on your forthcoming bid.
Therefore, using thequantum strategy not only helps the partners to behavemore optimally, but also makes the game harder for theopponents, as the overtly conveyed message carries no in-formation, until it is added to the result of the receiver.Furthermore, in the quantum case the choice of what thereceiver learns is delayed until the very last moment be-fore his bid, which gives him more knowledge about theopponents’ cards. This allows him to learn informationwhich is more relevant as he knows more about its con-text. An advantage of this type is difficult to quantifyand delaying the choice of measurement requires highdetection rates, therefore we do not dwell on this subjectanymore here.
V. EXPERIMENTAL REALIZATION
Let’s describe our experiment, which demonstrates thequantum violation of the classical bound of (6) for variousvalues of p and q . At the same time it is the experimentalrealization of a biased nonlocal game [15, 16], a quantumCCP, and the quantum Bridge. The optimal state forall these tasks is a two qubit maximally entangled stateand the measurements of the parties are the ones given FIG. 3: Experimental setup for Quantum Bridge CCP. UVlight centered at wavelength of 390 nm are focused inside a2 mm thick BBO ( β barium borate) nonlinear crystal, to pro-duce photon pairs. Half wave plates (HWP) and two 1 mmthick BBO crystals are used for compensation of longitudinaland transverse walk-offs. The emitted photons are coupledinto 2 m single mode optical fibers (SMF) and passed througha narrow-bandwidth interference filters (F) (∆ λ = 1 nm). Al-ice ( or the E parter) uses ( p : 1 − p ) variable-ratio beamsplitter (VRBS) for her measurement basis choice with theprobabilities ( p ) and (1 − p ) (corresponding to her input value a , see text) for the first and second base choice. Her measure-ment observables A and A are realized by HWP oriented by φ A and φ A respectively. Bob uses ( q : 1 − q ) variable ratiobeam splitter (VRBS) for his basis choice with the probabili-ties ( q ) and (1 − q ) (corresponding to his input value b ), for thefirst and second basis. His observables B and B are realizedby HWP oriented by φ B and φ B respectively. The polariza-tion measurements for Alice and Bob were performed usingpolarizing beam splitters (PBS) and single photon detectors(D). in [15]: A = σ x ( q + (1 − q ) cos β ) + σ z (1 − q ) sin β q ( q + (1 − q ) cos β ) + (1 − q ) sin β ,A = σ x ( q − (1 − q ) cos β ) − σ z (1 − q ) sin β q ( q − (1 − q ) cos β ) + (1 − q ) sin β ,B = σ x ,B = σ x cos β + σ z sin β, | ψ i = 1 √ | i| i + | i| i ) , (7)where σ z and σ x are the standard Pauli operators, andcos β = 12 ( q + (1 − q ) )( p − (1 − p ) ) q (1 − q )( p + (1 − p ) ) . (8)We have realized these quantum protocols by using po-larization entangled pairs of photons | φ + i = ( | HH i + | V V i ) / √
2. In the experiment, UV light centered atwavelength of 390 nm was focused inside a 2 mm thick q S u cce ss P r ob a b ilit y I C ( Q ) FIG. 4: Experimental results for Quantum Bridge: (a) theclassical I C ( q ) (dashed line), quantum I Q ( q ) (continuousline), and the experimental data points observed for the suc-cess probability I for p ( a = 0 | b = 0) = p ( a = 0 | b = 1) = p = 0 .
5. For q = 0 .
5, the obtained quantum success prob-ability is 0 . ± .
005 (the corresponding classical value is0 . q = 0 .
75, which we estimate to cor-respond to a common situation in Bridge, the quantum andclassical values are 0.895 (the experimental quantum value is0 . ± . .
875 respectively.FIG. 5: Experimental results for quantum CCP protocol forthe whole spectrum of quantum strategies. The classical I C ( p, q ) (blue plot), quantum I Q ( p, q ) (red plot), and exper-imental data (green points) for the success probability I asfunctions of p and q . The classical value of I C ( p, q ) is calcu-lated from (2) under the assumption that p ( a = 0 | b = 0) = p ( a = 0 | b = 1) = p ′ , which leads to p = p ′ + (1 − p ′ ) . (Onlyhalf of I C ( p, q ) plot is shown to make the I Q ( p, q ) plot andthe experimental data points visible). BBO ( β barium borate) nonlinear crystal. Photon pairs,due to a degenerate emission of type-II spontaneousparametric down-conversion, are collected in two spatialmodes a and b . Half wave plates (HWP) and two 1 mmthick BBO crystals are used for compensation of longi-tudinal and transversal walk-offs. The emitted photonswere coupled into 2 m single mode optical fibers (SMF) and passed through a narrow-bandwidth interference fil-ters (F) (∆ λ = 1 nm) to secure well defined spatial andspectral emission modes (see Fig. 3)[17].Alice uses ( p : 1 − p ) variable ratio beam splitter(VRBS) for her measurement basis choice with the prob-abilities ( p ) and (1 − p ) for the first and second basischoice. Her measurement observables A (correspondingto a = 0) and A (corresponding to a = 1) are real-ized by HWP oriented by φ A and φ A respectively. Bobuses ( q : 1 − q ) variable ratio beam splitter (VRBS) forhis basis choice with the probabilities ( q ) and (1 − q )for the first and second basis choice. His measurementobservables B and B (corresponding to b = 0 and 1respectively) are realized by HWP oriented by φ B and φ B respectively (see fig. 3) such as:tan φ A = (1 − q ) cos ( π/ − β ) q + (1 − q ) sin ( π/ − β ) , tan φ A = − (1 − q ) sin ( π/ − β ) q − (1 − q ) cos ( π/ − β ) ,φ B = π/ ,φ B = π/ − β. (9)The polarization measurement was performed usingpolarizing 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 aVHDL 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 I C ( q )(dashed line), quantum I Q ( 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 shows3 D plot for the classical I CS ( p, q ), quantum I QS ( p, q ),and experimental data observed for the success proba-bility of CCP protocols versus p and q . Our results arein very good agreement with the theoretical predictionsand clearly demonstrate the advantage of the quantumstrategy over the classical ones. VI. CONCLUSION
In summary, we report an experimental realization of aquantum Bridge CCP protocol, which is the first demon-stration of a quantum CCP usable in a real-life scenario.This was possible because we show that our quantumBridge protocol corresponds to a biased nonlocal CHSHgame, which in turn is equivalent to a 2 → n parties scenario. Con-cerning Bridge, it is up to the World Bridge Federation todecide whether to allow quantum resources and encodingstrategies in championships. A positive decision wouldmake this technique the first commonplace applicationof quantum communication complexity. A negative onewould forbid quantum strategies and thus, would consti-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 resourceson information and communication technologies. ACKNOWLEDGMENTS
The authors thank Johan Ahrens and Tommy Gullbergfor discussions. This work was supported by the SwedishResearch Council (VR),the Linnaeus Center of Excel-lence ADOPT, TEAM programme of foundation for Pol-ish Science (FNP), NCN grant 2013/08/M/ST2/00626,QUASAR (ERA-NET CHIST-ERA 7FP UE) and UKEPSRC. [1] A. C. Yao, Proceedings of the 11th Annual ACM Sym-posium on Theory of Computing, 209 (1979).[2] R. Cleve and H. Buhrman, Phys. Rev. A , 1201 (1997).[3] H. Buhrman, W. van Dam, P. Høyer, and A. Tapp,Phys.Rev. A , 2737 (1999).[4] H. Buhrman, R. Cleve, and W. van Dam, SIAM J. Com-put. , 1829 (2001).[5] L. Hardy and W. van Dam, Phys.Rev. A , 2635 (1999).[6] ˇC Brukner, M. ˙Zukowski, and A. Zeilinger, Phys. Rev.Lett. , 197901, (2002).[7] E. Kushilevitz and N. Nisan, Communication complexity .(Cambridge University Press, England, 1997).[8] ˇC. Brukner, M. ˙Zukowski, J.-W. Pan, and A. Zeilinger,Phys. Rev. Lett. Basic Bridge ,London: Vixtor Gollancz, ISBN 0-575-05690-8 (1994).[11] T. Gullberg, private communication.[12] http://en.wikipedia.org/wiki/Blackwood convention[13] http://en.wikipedia.org/wiki/Gerber convention[14] M. Paw lowski, and M. ˙Zukowski, Phys. Rev. A ,042326, (2010).[15] T. Lawson, N. Linden, and S. Popescu, arXiv:1011.6245,(2010).[16] A. Winter, Nature , 1053 (2010).[17] P. G. Kwiat et al. Phys. Rev. Lett. , 4337-4341 (1995).[18] http://en.wikipedia.org/wiki/Bridge scoring APPENDIX: THE RULES OF DUPLICATEBRIDGE
Duplicate bridge is a metagame. The same bridge deal(i.e. the specific arrangement of the 52 cards into thefour hands) is played at each table and scoring is basedon relative performance. This reduces the element ofchance while heightening the one of skill. Now we givethe brief description of the game. More details can befound in e.g [10].
Introduction
Duplicate bridge is one of the most popular cardgames, recognized by the International Olympic commit-tee as a sport (only two ”mind sports”: bridge and chessare recognized) and governed by the World Bridge Feder-ation for international competitions. It uses a standarddeck of 52 cards and is played by exactly four players.The players are usually named according to their seatsdirections as East, West, North and South. Among theseWest and East form a partnership or a pair competingagainst North and South.A tournament consists of several games. In each gamethe cards are distributed between the players. In rubberbridge a more casual version of the game this distribu-tion, called a deal, heavily influences the winning proba-bility. However, in duplicate bridge previously prepareddeals are stored in bridge boards - simple four-way cardholders. They are used to enable each player’s hand to bepassed intact to the next table that must play the samedeal, and final scores are calculated by comparing eachpair’s result with others who played the same hand.Each game consists of two main phases:1. The auction, also called bidding for a contract;2. Playing the hand, when the pair that won the auc-tion tries to take enough tricks to make the con-tract.
Bidding in Bridge
In order to understand how to bid, first one shouldknow the ranking of cards and suits. The deck of 52cards consists of 4 suits: spades ♠ , hearts ♥ , diamonds ♦ and club ♣ . Spades ♠ are ranked as the highest and thenext suit is hearts ♥ , together they are called the majorsuits. They are followed by minor suits with diamonds ♦ ranked higher than clubs ♣ . This ranking is importantin biding and scoring at the end of the game. In a suit,cards are ranked from 2 being the lowest to ace being thehighest.The bidding phase starts with the dealer (one playeris marked as dealer on each bridge board) and rotatesaround the table clockwise with each player making acall. A call is limited to a vocabulary of 38 words orphrases consisting of: • a Bid which states a level and a denomination; adenomination can be any of the four suits or NT(which stands for No Trump); given 7 levels of bid-ding and 5 denominations, there are 35 possiblebids; • Double, only available when the last bid was madeby an opponent; • Redouble, only available after opponent’s double; • Pass, when unwilling or unable to make one of thethree preceding calls.This vocabulary is further limited by the requirementthat each bid has to be sufficient, i.e. it has to have eithera higher level than a previous bid or the same level and ahigher denomination. NT is higher than any of the suits.For example, if the last bid was 3 ♠ the next one can be3NT or 4 ♣ but not 3 ♣ .If three players in a row pass, the auction is over. Apair is said to have won the auction if the last bid wasmade by one of its players. This partnership is called thedeclaring side. The player on the declaring side who, dur-ing the auction, first stated the denomination of the finalbid becomes ”declarer,” the declarer’s partner becomesthe ”dummy,” and the opposing side become the ”de-fenders.” The final bid also becomes the ”contract”. Thedeclaring side promises to take, during the next phase,the number of tricks greater or equal to the level of thecontract plus 6 and the denomination becomes the trump(in case of denomination NT, there is no trump). Playing the hand
To begin play, the defender on the declarer’s left makesthe opening lead by placing his selection face up on thetable. The dummy then spreads his hand on the tableso that it is visible to every other player. The playersplay clockwise around the table placing their cards onthe table, and each must ”follow suit” (that is, play acard of the suit lead to the trick) if able. A player thatcannot follow suit may either ”ruff” (play a trump) ifthere is a trump suit or ”sluff” (play a card of any othersuit). The player that plays either the highest trump or,in a trick that contains no trumps, the highest card of thesuit led to the trick (1) wins the trick for its side and (2)proceeds to lead to the next trick. The declarer directsthe play of cards from the dummy in addition to playingcards from his own hand. The play continues until allthirteen tricks are played. Then the score is calculated.
Scoring and match points
If the declaring side won the number of tricks specifiedby contract (or more) they are awarded some points andtheir opponents get exactly the same amount of negativeones. If they won less the situation is reversed.If the declaring side made their contract, they get thepoints for the following: • Odd tricks - the number of tricks specified by thelevel of the contract. The amount of points de-pends on the level of the contract, denominationand whether it was doubled or redoubled. Thepoints awarded for this are called contract points. • Overtricks - if the partnership won more tricks thancontract specified they are counted as overtricks.They are worth less points than odd tricks. Theamount of points here also depends on denomi-nation and whether the contract was doubled orredoubled but also on vulnerability. Whether ornot the partnership is vulnerable is marked on thebridge board. • Slam bonus - if the partnership bids and wins 12or 13 tricks they are awarded a huge bonus whichdepends on their vulnerability. • Double bonus - A bonus for making the contractthat has been doubled or redoubled. • Game bonus - A small bonus for making the con-tract is awarded. A much larger one is given if thepartnership scored more than 100 contract points.Even larger if they were vulnerable.If the declaring side failed to make their contract, theiropponents get points for each undertrick. The amount ofpoints is larger than for overtricks or even odd tricks anddepends on vulnerability and whether the contract wasdoubled or redoubled.The tables with the exact point values can be founde.g. here [18]. An important implication of these scoringrules is that finding the optimal contract is crucial forthe game. This stems from the fact that overtricks areworth less than odd tricks, game bonus is awarded onlyfor contract points and slam bonus only applied if theparties bid for a slam. For example, let us consider acase where the declaring side won 9 tricks with no trumpwhile being vulnerable. If their contract was 2NT theyare awarded 70 contract points plus 30 for one overtrickplus 50 of game bonus, which gives them 150 total. Iftheir contract was 3NT they get 100 contract points anda gem bonus of 500, which gives them 600 in total. If theircontract was 4NT they had one undertrick and they get100 penalty points. It is therefore crucial for the declaringside neither to under- or overestimate their ability to wintricks.However, these points do not influence the final posi-tion of the partnership that scored them directly. Insteadall the pairs that played the same deal are compared (thepairs that sat at North-South and East-West positionsare compared separately). Then 2 match points (MPs)are awarded for each partnership that scored less pointswith the same deal. MP is awarded for each partner-ship that scored the same amount. Therefore, scoring90 MPs when everyone else made only 70 is as good asscoring 2000.
Bidding strategies
In duplicate bridge auction phase is much more impor-tant than playing the hand. Often, the second phase isreduced to the declarer revealing his hand and claimingthat he will take exactly the amount of tricks specified inthe contract and the defenders agree ending the game.The auction phase is all about communication. Thepurpose of some early bids may be to exchange infor-mation rather than to set the final contract. For mostplayers, many calls (bids, doubles and redoubles, andsometimes even passes) are not made with the intentionthat they become the final contract, but to describe thestrength and distribution of the player’s hand, so thatthe partnership can reach an informed conclusion on theirbest contract, and/or to obstruct the opponents’ bidding.The set of agreements used by a partnership about themeaning of each call is referred to as a bidding system,full details of which must be made available to the op-ponents; ’secret’ systems are not allowed. An opponentcan ask the bidder’s partner to explain the meaning ofthe call.There are around 5 . × different deals possiblewhich makes bidding strategies very complex, especiallythat they have to take into the account possibility of theopponents interfering. To make it possible for humans toplay the game, they are divided into conventions - sub-strategies that are applicable only in certain situations,e.g. Cappelletti convention is a strategy for interruptingopponents communication while sending the partner in-formation about the strongest suit; it can be used onlyafter bid 1NT by the opponents by a player with moder-ately strong cards.Because making a contract for more than 100 contractpoints or a slam leads to large amounts of bonus pointsthere are multiple conventions designed to check if suchcontracts are possible to make. One of the most popularis Roman Key Blackwood described in the main text. Allthe conventions, apart for the ones for interrupting theopponents, are designed to convey as much informationas possible while keeping the final bid as low as possible.This enables the players to stay at safe level in case theyfind out that their cards are not that good or to squeezein additional rounds of communication before concluding FIG. 6: Cards for defense play and signaling. the auction. Therefore they bear resemblance to commu-nication complexity problems.
Strategies for playing the hand
The strategies for playing the hand are very differentfor the declaring side and the defenders. This stems fromthe fact that dummy’s cards are visible to everyone andthey are being played by the declarer. This leads to asym-metry. The declaring side is reduced to a single playerand half of its cards are known to the opponents. Thedefenders, on the other hand, face a problem of coor-dinating their actions. Usually, they do not have manyopportunities to exchange information during the auctionso they have to use the cards they play to send signals.One of the most common conventions is called Smith Pe-ter. When one of the defenders plays the opening leadthe other plays a low spot card (2-5) if he would like hispartner to play this suit once more and high spot card(6-10) if he wants to discourage his partner from playingthis suit. This strategy is called signalling.Again, the communication is limited because the play-ers aim at establishing a joint defence strategy as soon aspossible and their freedom of choosing the signals is con-strained by the cards they have. Therefore, the quantumprotocol described in the main text can be adapted alsoto defence play. An example of this is presented in thenext part of the appendix.Duplicate bridge is one of the most complex gamesand this short description presents only the very basicsof it. Interested reader should read any of the numerousbooks available on this subjects and try to play the gamehimself/herself.
APPENDIX: QUANTUM STRATEGIES FORDEFENCE PLAY
The quantum protocol described in the main text canalso be used for defense play and signaling. To see this0consider the cards given in figure 6. In this particulargame of Bridge, the W-E team is the defending teamand N is the dummy. In this given scenario, let assumethat W discards the ♥ A against S’s 4 ♦ contract. AfterN’s response, East plays the ♥
10 to signal a discouragingattitude towards the played suit (see Fig 6). For W, it isinteresting to know whether E possesses the ♥ Q or not.From the viewpoint of W, E can have ♥
10 as the highestcard in the suit or East might have e.g. ♥ Q and stillplay ♥
10 to signal a negative attitude. If East has thequeen, West should discard the ♥ ♥ Q . If East does not have the relevant card, thereis a good risk of letting the declarer win the 10 tricksneeded to fulfill the contract. In conclusion; West needsinformation about the ♥ Q . In order to extract the in- formation about the queen, we once again use the givenquantum protocol with the following variable definitions.If a = 0, encourages suit. If a = 1, discourages suit. If a = 0, odd number of cards. If a = 1, even number ofcards With q defined as the probability of W being inter-ested in the attitude towards the suit we may define themessage m as: if m = 0, then play a small card, if m = 1,then play a high card. Thus, the quantum protocol hasonce again improved the conditions of playing the bestcard. Generally, whenever communication between thepartners is restricted to one single bit and there exists anon-zero probability that the partner might be interestedin values of one of twotwo