Quantum mechanics has been incorporated into many fields called information. The most famous results are a quantum factoring algorithm by Shor [1] and a quantum database search algorithm by Grover [2]. Moreover, the studies on quantum information have succeeded in such as quantum computation, quantum circuit, quantum cryptography, quantum communication complexity, and so on. Recently, game theory based on quantum mechanics, quantum game theory, has been also proposed and it has been shown that quantum game theory is more powerful than classical one.

Game theory is one of the most famous decision making methods and has been used in many situations both theoretically and practically. There had existed the basic concept with respect to these games since early times but in corporation with Morgenstern, von Neumann [3] firstly constructed the theory systematically. However, the main principle of this theory was based on classical physics although he was familiar with quantum mechanics.

In 1998, for a coin flipping game, Meyer [4] proposed a quantum strategy for the first time and showed that the quantum strategy has an advantage over classical ones. This game is called PQ Penny Flip.

PQ Penny Flip: The starship Enterprise is facing some immanent—and apparently inescapable—calamity when Q appears on the bridge and offers to help, provided Captain Picard can beat him at penny flipping: Picard is to place a penny head up in a box, whereupon they will take turns (Q, then Picard, then Q) flipping the penny (or not), without being able to see it. Q wins if the penny is head up when they open the box.

This game is a two-player zero-sum game and the probability that each player wins is at most 1/2 with classical strategies. Meyer showed a quantum strategy with which Q can always win by using a superposition of quantum states effectively. In this game, Picard is constrained to play classically. The quantum strategy is then executed in the following way. Let and represent the head and tail of the penny, respectively. First, Picard prepares and Q applies a Walsh-Hadamard operation defined in the next section to the state:

Next, Picard decides classically whether he flips it or not. However, the state does not change even if Picard flipped it. Finally, Q applies to the state and always obtains. This means that Q always wins.

Moreover, he also showed the importance of a relationship between quantum game theory and quantum algorithms.

Later, other types of quantum strategies have been also proposed. In their strategies, all the players can use quantum operations. For example, Eisert et al. [5] proposed a quantum strategy with entanglement for a famous two-player game called the Prisoner’s Dilemma (also see Du et al. [6,7], Eisert and Wilkens [8], and Iqbal and Toor [9]). In their strategy, entanglement plays an important role. For another famous two-player game called the Battle of the Sexes, Marinatto et al. [10] also proposed a quantum strategy with entanglement. For these games, they showed quantum Nash equilibriums different from classical ones. Furthermore, there are many results being related to games such as the Monty Hall problem by D’Ariano et al. [11], Flitney and Abbott [12], and Li et al. [13], Parrondo’s game by Flitney et al. [14], games in economics by Piotrowski and Sładkowski [15–17], Newcomb’s paradox by Piotrowski and Sładkowski [18], and so on.

In this paper, we study Newcomb’s problem. Newcomb’s problem is a thought experiment between two players, Alice and Bob. Alice is a common human being. On the other hand, Bob may be a wizard having an ability to predict the future, or not. Bob can predict Alice’s will if he is a wizard. Then, the problem is as follows:

Newcomb’s problem: Bob prepares two boxes, Box₁ and Box₂, and Alice can select either Box₂ or both boxes. Box₁ contains $1. Box₂ contains $1,000 only if Alice selects only Box₂; otherwise Box₂ is empty($0). Which is better for Alice?

No one knows the answer except Bob. Namely, there exists no best classical strategy. Therefore, this problem is called Newcomb’s paradox. We show some quantum strategies for Newcomb’s paradox by using entanglement.

It is thought that entanglement is essential as the main power of quantum information and many results mentioned above also have used entanglement effectively. First, we show some basic quantum strategies with entanglement. In the other related studies mentioned above, each player operates only each assigned qubit although the states are entangled. On the other hand, our proposed strategies operate not only one qubit but also states between two qubits. Consequently, we show that our quantum strategies with entanglement are more powerful than classical ones.

Finally, we show some quantum strategies for Newcomb’s paradox. Piotrowski and Sładkowski showed a quantum solution for Newcomb’s paradox by using Meyer’s strategy [18]. Newcomb’s paradox is whether a player can predict another player’s will. We also study this problem by applying our strategies. Then, we obtain positive results. That is, in some case, a player can predict another player’s will.

The remainder of this paper has the following organization. In Section 2, first, we define notations and basic operations used in this paper. Moreover, as the tools of our quantum strategies, we show two fundamental lemmas with relation to entanglement. In Section 3, we denote two types of two-player zero-sum games. We then show that in these games, each player cannot win with certainty with classical strategies but one side player can win with certainty with quantum ones. In Section 4, we study Newcomb’s paradox. We modify this problem and show some quantum strategies to it by using the results in Section 3. Finally, in Section 5, we provide some concluding remarks.

In this section, first, we define some notations used in this paper. Let be a bitwise exclusive-or operator, i.e.,. Let be the negation of a bit for, i.e.,. Moreover, let be the inner product of and, where and for ().

Next, we define some basic operations. Let and, where is Dirac notation and is the transposed matrix of a matrix. Let be the identity matrix. This operation means no operation. A Walsh-Hadamard operation is

(and). Note that. This operation is used when we make a superposition of states. As an operation used when they flip a coin classically, players use an operation,

(and). We also define a phase-shift operation by

(and), where.

Moreover, we define an operation between two qubits. Here, we denote an -qubit state by, where is a tensor product. Then, let be a Controlled Not gate,

(, where the first bit is the controlled bit and the second bit is the target bit). We denote the operation by when the -th bit is the controlled bit and the -th bit is the target bit.

Finally, we show two basic results operating entangled states. These results can be used as tools of making a specific state in order that one side player always wins games by using our quantum strategies mentioned in the following sections.

Lemma 2.1 Let

be a -qubit entangled state, where () and. Then,

when we apply the Walsh-Hadamard operation to all the qubits of.

Proof. When we apply to all the qubits of,

where. Then, the statement of this lemma is then satisfied.

This lemma means that a player can obtain a state satisfying if he makes the state and that he can obtain a state satisfying if he makes the state.

Next, we show a result used as a player’s quantum strategy when another player flips coins classically.

Lemma 2.2 Let be the -qubit entangled state of Lemma 2.1, and let

Now, let the operation be applied to some qubits of, i.e.,

where. Note that if has been executed. Then,

Proof. When we apply to all the qubits of,

where the last expression is obtained by noting that except for the states corresponding to and, the states vanish. The statement of this lemma is then satisfied.

In this section, we study Newcomb’s paradox (Free Will problem) and show some quantum strategies for this problem. A quantum solution of this problem is shown by using Meyer’s quantum strategy by Piotrowski and Sładkowski [18]. We study this problem by using our results in previous section. A problem is as follows:

Newcomb’s problem: Let Bob have the ability to predict Alice’s will. Now, Bob prepares two boxes, Box₁ and Box₂, and Alice can select either Box₂ or both boxes. Box₁ contains $1. Box₂ contains $1,000 only if Alice selects only Box₂; otherwise Box₂ is empty($0). Which is better for Alice?

The payoff matrix is shown in Figure 8. The focus of this problem is whether Bob can really predict Alice’s will, or whether Bob can control Alice’s will. Obviously, Alice’s strategy is selecting both boxes if Bob cannot predict Alice’s will. Now, we modify this problem as simplified problems. In addition, we observe only strategies on Box₂.

Newcomb1: First, Alice decides whether she selects either Box₂ or not, but Bob cannot know her selection. Box₂ contains $1,000 only if Alice selects Box₂; otherwise Box₂ is empty($0). Can Bob let Alice select her first will even if Alice changes her will after the first selection?

Theorem 4.1 For Newcomb1, there exists a quantum strategy that can be positively solved.

Proof. Alice selects a state if she selects Box₂; otherwise she selects a state. Note that Bob does not know the state. To this state, Bob applies H, Alice applies X if she changes her will, and Bob applies H. Fi

nally, the state is if her first selection is Box₂; otherwise it is. Then, Bob can let Alice select her first will.

This strategy uses Meyer’s strategy. Moreover, we can also show other proofs by using Theorem 3.3 or Theorem 3.4. For Newcomb1, Bob does not permit Alice’s change. Next, by using -Coin Flip(G), we modify this problem to problems such that Bob permits Alice’s change.

Newcomb2: Alice prepares coins and puts them into a box in the state of (H,D,…, D), where (). Next, Bob flips some coins(or not), Alice flips coins() under keeping the value of secret from Bob, and Bob flips some coins(or not), where let Alice do not change her will if is even; otherwise let she change her will. Can Bob know whether is even or odd?

Theorem 4.2 For Newcomb2, there exist a quantum strategy that can be positively solved.

Proof. This result is immediately obtained by Theorem 3.4.

We can regard this problem as Newcomb’s problem when the value of is her will, i.e., she selects Box₂ when is even; otherwise she selects both boxes. Thus, Bob can predict Alice’s will. We can also modify Newcomb’s problem by using -Coin Flip and Extended -Coin Flip.

Next, we denote that we can also modify Alice’s will in Newcomb2 to -player’s will.

-Newcomb2: First, players, , decide whether they selects or not. They prepare coins and puts them into a box in the state of (H,D,…, D), where (), and let player () deal with the -th coin. Now, Bob flips some coins(or not). Next, each () flips his/her coin(or not), where let the number of -player’s flips be (), and let they do not change their will if is even; otherwise let they change their will. Finally, Bob flips some coins(or not). Can Bob know whether is even or odd?

The quantum strategy for Bob is same as Newcomb2.

Finally, we study the Chicken game. The payoff matrix is shown in Figure 9. In the Chicken game, the Nash equilibrium points are (Swerve, Drive straight) and (Drive straight, Swerve). By classical strategies, the probability that each player selects either Swerve or Drive straight is 1/2. This means that either (Swerve, Swerve) or (Drive straight, Drive straight) may be selected. On the other hand, by our quantum strategies, either (Swerve, Drive straight) and (Drive straight, Swerve) can be selected with certainty because Bob can predict Alice’s will.

In this paper, we proposed quantum strategies with entanglement for -coin flipping games. These games are multi-qubit variations of the quantum strategy by Meyer [4]. One player is constrained to play classically but the other player can use a quantum strategy. We then showed that by using a technique in quantum communication complexity theory, our quantum strategies have an advantage over classical ones. For a player using the quantum strategy, the entanglement is used in order to obtain the information of the enemy and the player can always win the games. Moreover, by rewriting Newcomb’s paradox including multi-player’s will, we also showed that we can use our results as its quantum strategies and that a player can have an ability to predict another player’s will.

Can Bob always win our games even if both players use quantum strategies? This answer is “No”. Meyer also showed that Q cannot always win if Picard can also use a quantum strategy [4]. In the same reason, Bob cannot always win our games if Alice can also use a quantum strategy. Therefore, it is an interesting question whether Bob can always win the games in the case when both players are constrained to only execute some restricted quantum operations.