Information sharing using entangled states and its applications to quantum card tricks

Abstract

Quantum superposition states, especially, quantum entangled states, are useful for several fields such as quantum computation, quantum cryptography, quantum game theory, and so on. In this paper, first, we propose methods to share entangled states without communication among players that may not know each other. Next, we introduce quantum cards and propose some quantum card tricks using this sharing method. For example, we propose tricks such that by sharing an entangled state between two spectators secretly, magicians make a spectator's card number and another spectator's card number the same.

Introduction

Quantum superposition states have many strange properties. For example, let

$$|0_x〉=1/\sqrt{2}(|0_z〉+|1_z〉\left)\text{,}\text{and}\right|1_x〉=1/\sqrt{2}\left(\right|0_z〉-|1_z〉)\text{.}$$

Here, let the states |0_z〉 and |1_z〉 be the states corresponding to the values 0 and 1 on z-axis. At that time, it is known that the states |0_x〉 and |1_x〉 mean the states corresponding to the values 0 and 1 on x-axis. We define the notation | ⋅〉 in Section 2, and these are called quantum bits or qubits as values corresponding to classical bits.

Now, Alice makes a state

$$\frac{1}{\sqrt{2}}\left(\right|0_z〉|0_x〉-|1_z〉|1_x〉)\text{.}$$

This state can also express as follows:

$$\frac{1}{\sqrt{2}}\left(\right|0_x〉|1_z〉+|1_x〉|0_z〉)\text{.}$$

Here, let us consider a game that Alice guesses by using the state whether Carol chooses 0 or 1. Namely, Alice predicts Carol's choice by the state. If Carol chooses 0, Alice measures on z-axis for the first qubit b₁ and measures on x-axis for the second qubit b₂. Then, b₁ ⊕ b₂ = 0. Otherwise, if Carol chooses 1, Alice measures on x-axis for the first qubit and measures on z-axis for the second qubit. Then, b₁ ⊕ b₂ = 1. Thus, Alice can make one state that possesses two values corresponding to Carol's will although Alice must know Carol's will before measurement.

This is a simple game, but represents a characteristic property of quantum states. The security of many quantum key distribution systems is also based on this property [1], [2], [8]. Distributors hide information against eavesdroppers by switching from one axis to another axis at random.

Moreover, Meyer proposed a quantum strategy for a coin flipping game, and showed that the quantum strategy has an advantage over the classical ones [12]. In addition, he also showed the importance of a relationship between quantum game theory and quantum algorithms. After that, other types of quantum strategies have been also proposed. For example, Eisert et al. proposed a quantum strategy with entangled states for a famous two-player game called the Prisoner's Dilemma [7] (also see Refs. [4], [5], [6], [10]). Marinatto et al. also proposed a quantum strategy with entangled states for another famous two-player game called the Battle of the Sexes [11]. For these games, they showed quantum Nash equilibriums different from the classical ones by using quantum states. As a summary of this field, see, e.g., Ref. [9].

In this paper, first, we propose methods to share entangled states without communication among players that do not share them. This means that players sharing entangled states may not know each other. Next, we propose some quantum card tricks based on this method. By defining quantum cards, magicians show some card tricks to spectators. For example, we propose tricks such that by sharing an entangled state between two spectators secretly, magicians make a spectator's card number and another spectator's card number the same. Namely, by showing some quantum tricks, we propose methods manipulating as if a player's choice were led by another player's will and methods sharing/guessing a player's choice among some other players without knowing the choice. In Ref. [14], we have proposed the similar quantum tricks, i.e., quantum coin and card tricks. The card tricks in that paper use cards whose numbers are written in both sides. On the other hand, in this paper, we use cards whose numbers are written in one side. In addition, we also focus on tricks using remote communications among players (magicians and spectators).

Our results seem to be related to game theory, especially, quantum pseudo-telepathy (see, e.g., Ref. [3] and references therein). Now, let us consider two players, Alice and Bob. Assume that they cannot communicate with each other, but that they share entangled states beforehand. Pseudo-telepathy game is as follows. Let x be Alice's input and y be Bob's input. Moreover, let w be a winning condition. Then, the players make a strategy f(x, y, g(x, y)), where g is a strategy computing the players' outputs. The game is whether they can find outputs corresponding to w = f(x, y, g(x, y)). In order to compute g, the players use the entangled states in the quantum strategy. In addition, in quantum pseudo-telepathy game, every player has the same ability generally. On the other hand, although our strategies also use entangled states, there exist two types of players, magicians and spectators. Magicians know all the information of strategies, but spectators know only a part of them. Moreover, magicians can also manipulate spectators' information (i.e., quantum states) in our games. Consequently, our tricks are to make strategies that lead to outputs corresponding magicians' will although quantum pseudo-telepathy game is to find outputs corresponding to a winning condition.

The remainder of this paper has the following organization. In Section 2, we define notations and basic operations used in this paper. In Section 3, we propose methods sharing entangled states among players without communication. In Section 4, first, we define quantum cards. Then, we show some magician's techniques, and propose several quantum card tricks. Finally, in Section 5, we provide some concluding remarks.