Quantum Bidding in Bridge

Quantum methods allow us to reduce communication complexity of some computational tasks, with several separated partners, beyond classical constraints. Nevertheless, experimental demonstrations of this have thus far been limited to some abstract problems, far away from real-life tasks. We show here, and demonstrate experimentally, that the power of reduction of communication complexity can be harnessed to gain an advantage in a famous, immensely popular, card game—bridge. The essence of a winning strategy in bridge is efficient communication between the partners. The rules of the game allow only a specific form of communication, of very low complexity (effectively, one has strong limitations on the number of exchanged bits). Surprisingly, our quantum technique does not violate the existing rules of the game (as there is no increase in information flow). We show that our quantum bridge auction corresponds to a biased nonlocal Clauser-Horne-Shimony-Holt game, which is equivalent to a $2\to 1$ quantum random access code. Thus, our experiment is also a realization of such protocols. However, this correspondence is not complete, which enables the bridge players to have efficient strategies regardless of the quality of their detectors.

Popular Summary

Optimizing time and resources while successfully performing an information processing task is a fundamental challenge in communications and computation. Quantum information technologies break the limitations of conventional information transfer, cryptography, and computation. We consider the card game of duplicate bridge—which relies on efficient communication between partners—and show that players who employ quantum resources increase their probability of winning.

Communication complexity protocols are aimed at maximizing the probability of successfully solving a problem with a restricted amount of communication. Bridge is played with two competing pairs of players who must share information about their hands of cards. The rules of the game stipulate that the form and the amount of information exchanged between partners are severely restricted. Communication complexity protocols are therefore readily applicable to bridge. We present the experimental realization of the first quantum bridge protocol where the quantum resources provide an advantage over the classical resources, regardless of the cost of communication. Our technique relies on the partners sharing an entangled pair of photons; the players can exchange information to communicate their hands of cards, without violating any official rules of the game. Players who share entangled photons increase their probability of winning.

The World Bridge Federation has the authority to decide whether to allow quantum resources and encoding strategies in bridge championships (making this technique the first commonplace application of quantum communication complexity) or forbid quantum strategies (the first everyday regulation of quantum resources). Our work establishes links among game theory, communication complexity, and quantum physics, contributing to a deeper understanding of the advantages of quantum resources in information and communication technologies.

Figure 1

Quantum protocol of bridge CCP. Alice (playing East) holds two bits of information $a_i$ with $i=0$, 1 (defined by her hand). Bob (playing West) chooses between $b=0$ or 1, which denotes in which bit of Alice he is interested. Alice chooses her measurement setting to be $a=a_0\oplus a_1$. Bob sets his measurement according to his choice of $b$. After reading out her outcome $A$ (which is encoded as a bit value), Alice sends a one-bit message to Bob, the value of $m=A\oplus a_0$. Bob computes his guess $R$ for the value of the bit he wants to know by adding the message to his local measurement outcome $B$ (also encoded as a one-bit value), that is $R=B\oplus m$.

Figure 2

Experimental setup for quantum bridge CCP. UV light centered at a wavelength of 390 nm is focused inside a 2-mm-thick $\beta$ barium borate (BBO) nonlinear crystal to produce photon pairs. Half-wave plates (HWP) and two 1-mm-thick BBO crystals are used for compensation of longitudinal and transverse walk-offs. The emitted photons are coupled into 2-m SMF and passed through narrow-bandwidth interference filters (F) ($\mathrm{\Delta }\lambda =1\mathrm{nm}$). Alice (or the $E$ parter) uses a ($p$: $1-p$) variable-ratio beam splitter (VRBS) for her measurement basis choice, with the probabilities $(p)$ and ($1-p$) (corresponding to her input value $a$, see text) for the first and second base choices. Her measurement observables $A_1$ and $A_2$ are realized by HWP oriented by ${\varphi }_{A_1}$ and ${\varphi }_{A_2}$, respectively. Bob uses a ($q$: $1-q$) VRBS for his basis choice, with the probabilities $(q)$ and ($1-q$) (corresponding to his input value $b$) for the first and second bases. His observables $B_1$ and $B_2$ are realized by HWP oriented by ${\varphi }_{B_1}$ and ${\varphi }_{B_2}$, respectively. The polarization measurements for Alice and Bob were performed using polarizing beam splitters (PBS) and single-photon detectors (D).

Figure 3

Experimental results for quantum bridge: (a) the classical $I_C(q)$ (dashed line), quantum $I_Q(q)$ (continuous line), and the experimental data points observed for the success probability $I$ for $p(a_0=0|b=0)=p(a_1=0|b=1)=p=0.5$. For $q=0.5$, the obtained quantum success probability is $0.848\pm 0.005$ (the corresponding classical value is 0.75). For the value of $q=0.75$, which we estimate to correspond to a common situation in bridge, the quantum and classical values are 0.895 (the experimental quantum value is $0.887\pm 0.005$) and 0.875, respectively.

Figure 4

Experimental results for quantum CCP protocol for the whole spectrum of quantum strategies. We show the classical $I_C(p,q)$ (blue plot), quantum $I_Q(p,q)$ (red plot), and experimental data (green points) for the success probability $I$ as functions of $p$ and $q$. The classical value of $I_C(p,q)$ is calculated from Eq. (2) under the assumption that $p(a_0=0|b=0)=p(a_1=0|b=1)=p^{\prime }$, which leads to $p={p^{\prime }}^2+(1-p^{\prime }{)}^2$. [Only half of the $I_C(p,q)$ plot is shown to make the $I_Q(p,q)$ plot and the experimental data points visible].

Figure 5

Cards and bidding for bridge. An example of cards and bidding for the $W$ and $E$ partners.

Figure 6

Cards for defensive play and signaling.