ORIGINAL PAPER

A scheme to cancel out superiority of quantum strategies in coin-tossinggame

H-F Ren1, Q-L Wang1*, R-M Lian1 and S-X Hou2

1Department of Physics and Electronics, XinZhou Teachers University, XinZhou 034000, People’s Republic of China

2Department of Light Industry, Hebei Polytechnic University, Langfang 065001, People’s Republic of China

Received: 13 July 2013 / Accepted: 01 October 2013 / Published online: 12 November 2013

Abstract: Recently, it has been reported that quantum strategies are more successful than classical ones in coin-tossing

game and the player adopted quantum-tossing operation could control the game entirely. In this paper, we propose a new

scheme such that quantum game returns to be a unbiased two-player zero-sum one, in which the player, who originally uses

classical strategies, makes an appropriate quantum measurement on quantum game.

Keywords: Quantum game; Quantum coin; Quantum measurement

PACS Nos.: 03.67.-a; 02.50.Le; 03.65.-w

1. Introduction

Classical game theory has been applied successfully to

economic and industrial decision models in order to resolve

and determine the best possible strategy [1–7]. Advantage

of quantum strategies for two-player zero-sum game has

been demonstrated [2, 3] that is, if one person adopts a

quantum strategy, he always wins the game. Recently, the

protocol has been discussed a lot which has led a new sight

into nature of information and it opens a new range of

potential applications [8–16]. Wang et al. [5] have exten-

ded this case into a roulette with N states. Chen et al. [7]

have studied noisy quantum games and have shown that

handicapped player with classical means can delay his

action for a sufficiently long time and then quantum ver-

sion reverts to classical zero-sum game under decoherence.

Li [13] has investigated the impact of quantum measure-

ments for a special game of three states, named Monty Hall

problem and concluded that quantum measurements can

make this biased game tending to be unbiased. In this

paper, we attempt to make an appropriate measurement on

a quantum coin-tossing system and study the impact of

quantum measurements on quantum coins.

2. Monty hall problem

Goldenberg et al. [2, 6] have discussed quantum strategies

of quantum measurement of Monty Hall problem. For a

classical case, there are one particle and three boxes.

Firstly, Alice places the particle into one of the boxes and

does not change its position, while Bob does not know

which box the particle is in. Then, Bob chooses a box and

he would win the game if particle is in this box, otherwise

he loses game. Obviously, Bob has a probability 13

to win

the game and he thinks that game is biased. So he may

argue after he chooses a box (eg. 0) and Alice should reveal

an empty box, for example, box 2 from the other ones (1

and 2). Then they do the same game for unrevealed boxes 0

and 1. However, it is preponderant for Bob to change his

original action to choose box 1 and the probability winning

the game is 23

and sticking to 0, probability of him is 13. So

this game is biased for Alice now. However, after Alice

reveals an empty box 2 she could make the game to be

unbiased for both of them by quantum strategy [7].

For quantum version, there are one quantum particle and

three boxes 0, 1 and 2. After Alice places quantum particle

into one of the boxes where states of the boxes can be

described by quantum states, |0i, |1i and |2i. Here, suppose

that Alice prepares a superposition state:

jwip ¼1ffiffiffi

3p ðj0i þ j1i þ j2iÞ: ð1Þ

*Corresponding author, E-mail: leky.w@mail.nankai.edu.cn

Indian J Phys (March 2014) 88(3):271–274

DOI 10.1007/s12648-013-0404-3

� 2013 IACS

Based on that state, Bob has a probability 13

to win the

game. After he chooses a box (eg. 0), Alice reveals an

empty one from 1 and 2 (eg. 2). Then his winning proba-

bility will be 23

if he chooses box 1. In classical Monty Hall

problem, once Alice places the particle into one box she

could not change its position. Corresponding to this, once

Alice prepares the above superposition states in quantum

version she could not evolve that state.

Now, suppose that Alice is allowed to make quantum

measurement after she reveals an empty box 2. State of the

particle can be shown by density matrix

qp ¼1

3j0ih0j þ 2

3j1ih1j: ð2Þ

The box that particle is in, is known to Alice, so she can

determine how to measure boxes 0 and 1. Alice can adopt

following orthogonal bases

j/i0 ¼ cos a0j0i þ sin a0j1i; ð3Þ

j/i?0 ¼ � sin a0j0i þ cos a0j1i; ð4Þ

and

j/i1 ¼ cos a1j0i þ sin a1j1i; ð5Þ

j/i?1 ¼ � sin a1j0i þ cos a1j1i; ð6Þ

where 0� a0; a1� p2.

The pure strategy of Bob is to choose |0i or |1i, that is,

Nq ¼ h0jqj0i; Vq ¼ h1jqj1i: ð7Þ

Normally, Bob’s strategy is classical mixed ones,

SBðgÞ ¼ gN þ ð1� gÞV ; ð8Þ

where g ¼ 12. And the probability that Bob wins is as

following,

PB ¼ SBðgÞq0P ¼ Nq0Pþ Vq0

P¼ 1

2: ð9Þ

From above discussion, it can be found that Monty Hall

problem can tend to be unbiased for both players, Alice and

Bob, by quantum strategy of quantum measurement [6].

Based on this, we study effect of quantum strategy for

quantum game of one coin.

3. How to cancel out the superiority of quantum game

These two players, Alice and Bob, make a coin tossing game

[2]. Firstly, Alice places a coin into a box and original state of

the coin is known by both Alice and Bob. Then Bob and Alice

toss the box by classical strategy in turn. At last, after Bob

tosses the coin secondly, box is opened. If the state of the coin

is head up, Bob wins game and otherwise, he loses. Bob has a

chance of 12

to win the game for classical game. However,

once Bob adopts two steps of quantum tossing instead of

classical ones he can always win the game. That is to say,

Bob can entirely control this game by quantum strategies and

this implies that quantum strategies are more successful than

classical ones.

It is obvious that original unbiased game changes to be

biased if one of the players uses quantum strategies. The

other one, Alice, must be unwilling. However, in this

quantum game if she makes an quantum measurement on

the system at appropriate moment she could find that game

returns to be unbiased. It is well known that a classical coin

placed in a box has two possible states, head up and tail up,

which can be described by following matrixes

j1i � jheadi ¼ 1

0

� �

ð10Þ

j0i � jtaili ¼ 0

1

� �

; ð11Þ

and the corresponding density matrixes for quantum coin

are q ¼ 1 0

0 0

� �

for j1i; q ¼ 0 0

0 1

� �

for |0i.

Now Alice and Bob come to play the quantum coin

flipping game. Here, Alice still adopts classical strategies

while Bob uses quantum ones. Suppose original state of

quantum coin placed in the box by Alice is |1i or |0i,density matrix is

q1 ¼ j1ih1j ¼1 0

0 0

� �

ð12Þ

or

q1 ¼ j0ih0j ¼0 0

0 1

� �

ð13Þ

Then, Bob acts on the coin by quantum tossing, unitary

transformation S1 or S2. The density matrix becomes

q2 ¼ Siq1Syi ¼ G2 ¼

1

2

1 1

1 1

� �

; ð14Þ

where, i = 1 for q1 ¼1 0

0 0

� �

; i ¼ 2 for q1 ¼0 0

0 1

� �

and

S1 ¼1ffiffiffi

2p 1 1

1 �1

� �

; S2 ¼1ffiffiffi

2p 1 1

�1 1

� �

ð15Þ

Because Alice’s classical tossing can not change the

density matrix, q3 = q2 = G2, Bob can obtain the state he

wants by quantum tossing again. Here, density matrix of

the state would be q4 ¼ Sþ1 G2S1 ¼1 0

0 0

� �

if Bob uses

quantum tossing and unitary transformation S1?; density

matrix would be q4 ¼ Sþ2 G2S2 ¼0 0

0 1

� �

if Bob uses S2?.

272 H-F Ren et al.

Altogether, if the state of the coin is known Bob, he

could transform density matrix of coin into a special matrix

G2 by quantum tossing, which could not be transformed by

classical strategy adopted by Alice. Thus, Bob can control

the game by an appropriate quantum strategy. It is shown

that quantum strategies are more successful than classical

ones for quantum coin game [2]. Now, Alice finds that

Bob’s tossing is quantum strategy instead of classical one

and the game turns to be biased, she could make a quantum

measurement on the system after initial density matrix is

transformed into G2 by quantum strategy adopted by Bob.

The density matrix q2 = G2 can be described by

q ¼ G2 ¼1

2ðj1ih1j þ j1ih0j þ j0ih1j þ j0ih0jÞ; ð16Þ

At this time, Alice could make a quantum measurement on

quantum system and the measurement operator could be

M0 or M1, here

M0 ¼ j0ih0j;M1 ¼ j1ih1j: ð17Þ

From [13], the state of such a quantum system would

therefore be described by density operator

q0

P ¼X

MmqMþm ¼1

2ðj0ih0j þ j1ih1jÞ: ð18Þ

For the state j0i and j1i orthogonal bases of von Neumann

measurement that Alice adopts are

j/i0 ¼ cos a0j0i þ sin a0j1i ð19Þ

j/i?0 ¼ � sin a0j0i þ cos a0j1i ð20Þ

and

j/i1 ¼ cos a1j0i þ sin a1j1i ð21Þ

j/i?1 ¼ � sin a1j0i þ cos a1j1i ð22Þ

respectively, where 0� a0; a1� p2. Then density matrix

becomes

q0

P ¼1

2ðcos2 a0j/i0h/j þ sin2 a0j/i?0 h/jÞ

þ 1

2ðcos2 a1j/i1h/j þ sin2 a1j/i?1 h/jÞ ð23Þ

That is, G2 is transformed into a classical superposition

of the states, j0i or j1i. Afterwards, Bob adopts quantum

strategy. However, either of the unitary transformations

(U2 ¼ Sy1 or S

y1) could not change qP

0, that is,

U2q0

PUy2 ¼

1ffiffiffi

2p 1 0

0 1

� �

¼ 1

2ðj0ih0j þ j1ih1jÞ ¼ q

0

P: ð24Þ

Bob could not control the game by quantum strategy any

longer and the game becomes a classical game from here

on out. Then, after Alice adopts classical tossing for qP

0the

box is opened. She may choose j0i or j1i. Her strategy is

NP ¼ h0jqj0i; ð25Þ

or

VP ¼ h1jqj1i: ð26Þ

Suppose her choice is a classical superposition strategy

SBðgÞ ¼ gN þ ð1� gÞV : ð27Þ

According to previous appointment, Alice wins if the state

is j0i, otherwise, she loses. The probability that Alice wins

would be

PB ¼ SBðgÞq0

P ð28Þ¼ Nq

0Pþ Vq

0P

ð29Þ

Based on orthogonal bases of von Neumann measurement

Eqs. (17)–(20), we can derive that

Nq0P¼ 1

2gðcos4 a0 þ sin4 a0Þ þ

1

2g� 2 cos2 a1 sin2 a1

ð30Þ

and

Vq0P¼ 1

2ð1� gÞ � 2 sin2 a0 cos2 a0 þ

1

2ð1� gÞðsin4 a1

þ cos4 a1Þ:ð31Þ

Then,

PB ¼ SBðgÞq0

P ð32Þ

¼ 1

2½gðcos4 a0 þ sin4 a0Þ þ 2ð1� gÞ sin2 a0 cos2 a0� ð33Þ

þ 1

2½ð1� gÞðsin4 a1 þ cos4 a1 þ 2g cos2 a1 sin2 a1Þ: ð34Þ

The concrete orthogonal bases may be

j/i0 ¼ j/i1 ¼1ffiffiffi

2p ðj0i þ j1iÞ; ð35Þ

j/i?0 ¼ j/i?1 ¼

1ffiffiffi

2p ð�j0i þ j1iÞ: ð36Þ

That is to say,

cos a0 ¼ sin a0 ¼1ffiffiffi

2p ð37Þ

cos a1 ¼ sin a1 ¼1ffiffiffi

2p : ð38Þ

Then Eq. (33) becomes

PB ¼ SBðgÞq0P¼ 1

2ð39Þ

Here, we find that probability that Alice wins return to

be 12

and has nothing to do with classical probability g.

Quantum strategies in coin-tossing game 273

4. Conclusions

In conclusion, for classical coin tossing game, probability

that either of the two players wins the game is 12

and neither

of them could control the game. But once one of the

players, eg. Bob, adopts the quantum strategies instead of

the classical ones, he could always obtain the state of the

system that he wants. Namely, Bob’s quantum strategies

are more successful than Alice’s classical ones. However,

after the other player, Alice, finds the strategy that Bob

adopted in first step, she could choose an appropriate

quantum measurement on the box in this moment and the

biased game would be unbiased again for each other. Bob

could not control the game by his own quantum strategies

any more. That is to say, Alice’s quantum measurement

makes Bob’s quantum strategies lose his superiority.

Acknowledgement This work was supported by the Natural Sci-

ence Foundation for Young Scientists of Shanxi Province, China

(Grant No. 2010021003-5); and Xinzhou Teachers University

Foundation.

References

[1] R D Luce and H Raiffa Games and Decisions (New York: Dover

Publications) (1989)

[2] L Goldenberg, L Vaidman and S Wiesner Phys. Rev. Lett. 823356 (1999)

[3] D A Meyer Phys. Rev. Lett. 82 1052 (1999)

[4] J Eisert, M Wilkens and M Lewenstein Phys. Rev. Lett 83 3077

(1999)

[5] X-B Wang, L C Kwek and C H Oh Phys. Lett. A 278 44 (2000)

[6] Ch-F Li et al. Phys. Lett. A 280 257 (2001)

[7] J-L Chen, L C Kwek and C H Oh Phys. Rev. A 65 052320 (2002)

[8] E W Piotrowski, J. Sładkowski Int. J. Theor. Phys. 42 1089

(2003)

[9] E Ahmed et al. Int. J. Theor. Phys. 45 880 (2006)

[10] H-F Ren and Q-L Wang Int. J. Theor. Phys. 47 1828 (2008)

[11] T D Cohen Indian. J. Phys. 83 681 (2009)

[12] H Prakash Indian. J. Phys. 84 1021 (2010)

[13] Q-L Wang et al. Indian. J. Phys. 85 471 (2011)

[14] M A Nielsen and I L Chuang Quantum Computation and

Quantum Information (New York: Cambridge University Press)

(2010)

[15] Z Yang, C Hu and Q Meng Indian. J. Phys. 86 977 (2012)

[16] Y P Zhang, W Qiao and J Sun Indian. J. Phys. 87 271 (2013)

274 H-F Ren et al.