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Fluctuation and Noise Letters
Vol. 2, No. 4 (2002) L257–L262
c World Scientific Publishing Company
QUANTUM PARRONDO GAMES: BIASED AND UNBIASED
DAVID A. MEYER
Department of Mathematics, University of California/San Diego
La Jolla, CA 92093-0112
[email protected]
HEATHER BLUMER
Institute for Physical Sciences
Los Alamos, NM 87544
[email protected]
Received 30 May 2002
Revised 18 October 2002
Accepted 21 October 2002
Although the dissipative character of thermal diffusion, which is modelled by Parrondo
games, might seem necessary to achieve the “paradoxical” behaviour corresponding to
ratcheting, quantum Parrondo games with similar properties can be constructed. There
are subtle differences, however, which manifest themselves in distinct asymptotic behaviours for biased and unbiased games.
Keywords: Quantum ratchet; quantum lattice gas automaton.
1.
Classical Parrondo Games
Parrondo invented the coin flipping game described as having “paradoxical” properties to illustrate the working of thermal ratchets [1]. Quite recently, ratcheting
has also been observed for quantum particles [2]. This raises a natural question:
Are there quantum Parrondo games? ([3, 4]).
A game of chance is a losing game if the expected return over many plays is
negative. Parrondo’s surprising observation is that alternating plays of two losing
games can be a winning game [1, 5]. Let game A be a flip of a negatively biased
coin (with Pr(win) = 12 − ) and let game B consist of two biased coins — a
1
− ) that is flipped when the gambler’s
negatively biased one (with Pr(win) = 10
capital is a multiple of 3 and a positively biased one (with Pr(win) = 34 − ) that is
flipped otherwise. With sufficiently small but positive , these biases make A and B
losing games, but if the games are alternated in the sequence AABB repeatedly, the
combination is a winning game. Figure 1 shows the expected winnings as a function
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D. A. Meyer & H. Blumer
<x>
2
AABB
1
20
40
-1
60
80
100
t
A
B
-2
Fig. 1. The expected payoffs for games B, A and AABB as a function of number of plays t.
Although A and B are losing games, the combination AABB is a winning game.
of number of plays when the player’s capital starts at 0 and increases (decreases)
by 1 after a win (loss). Other sequences of As and Bs show similar behaviour, with
varying average rate of increase; even random sequences of A and B increase the
gambler’s capital, on average [1, 5].
This capital corresponds to a discretization of the position of a Brownian particle
diffusing in a one-dimensional potential. Game A models a potential with constant
negative gradient and game B models a tilted sawtooth potential; alternating the
games mimics a “flashing ratchet” which drives the particle uphill. A quantum
Parrondo game, therefore, should be a discrete model for a particle evolving unitarily
in appropriate potentials [4].a
2.
Quantum Parrondo Games
The first quantum game studied involved flipping a “quantum coin” [7]; the following
generalization provides a framework within which to search for a Parrondo effect. A
quantum coin is realized by, for example, the spin of a spin- 12 particle. Its state is a
2
2
superposition
1 0 a|↓ + b|↑ with |a| + |b| = 1, where |↓ and |↑ are the eigenstates of
σz = 0 −1 . An unbiased coin flip is represented by the unitary transition matrix
1 i √
means that the amplitude for the coin to remain in the same σz
i 1 / 2, which
√
√
eigenstate is 1/ 2, while it has amplitude i/ 2 to change eigenstates.b Letting
|x denote capital, and identifying |↓ and |↑ with ‘lose’ and ‘win’, respectively, a
play of the game is the unitary transformation defined by linear extension from the
a An alternative ‘quantization’ of Parrondo games has been suggested by Flitney, Ng and Abbott
[6].
b This may be compared with the stochastic transition matrix for an uncorrelated, unbiased classical
coin: 11 11 /2, where the rows and columns are labelled by ‘head’ and ‘tail’, which means that the
probability to get the same result again is 1/2 and the probability of it changing is also 1/2.
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Quantum Parrondo Games: Biased and Unbiased
L259
<x>
4
BAAAA
2
20
40
60
80
100
t
A
B
-2
-4
Fig. 2. The expected payoffs for quantum games A, B and BAAAA as a function of number of
plays t. Although A and B are losing games, the alternating sequence BAAAA is a winning game.
For short times the expectation values change quadratically.
following mapping of the basis vectors:
|x, ↓ →
|x, ↑ →
1 √ |x − 1, ↓ + i|x + 1, ↑ ,
2
1 √ i|x − 1, ↓ + |x + 1, ↑ .
2
This rule, in fact, is a discretization of the evolution of a Dirac particle in
1 + 1 dimensions [8] for which potentials are implemented by x-dependent phase
multiplication [9]. Thus the quantum versions of the A and B games incorporate
the unbiased transition matrix, multiplied
by a phase e−iV (x) where VA (x) = αx
1
has a constant gradient and VB (x) = β 1− 2 (x mod 3) +VA (x) is a tilted sawtooth
potential.
Figure 2 shows the results of 100 play simulations of these quantum games with
α = 2π/5000 and β = π/3. The evolution of the expected winnings is decreasing
for games A and B, and increasing for the repeating sequence BAAAA, indicating
that there is a quantum Parrondo phenomenon. Other, but not all, sequences of As
and Bs also display a Parrondo effect for this choice of parameters. The sequence
BAAAA shows one of the largest among repeating sequences of length no more
than 5. (Random sequences of As and Bs, of course, are no longer purely quantum
mechanical.) In each case, the expectations change approximately quadratically
with the number of plays, so the quantum Parrondo phenomenon appears to be even
more pronounced than the classical one. The fact that these games are equivalent
to discretizations of the relativistic Dirac equation, however, implies that these
quadratic trends cannot persist for infinitely many plays [4]. The results of longer
simulations, shown in Fig. 3, confirm this deduction — the expectation values are
periodic in time. For random stopping times, however, both the A and B games are
losing games and the BAAAA quantum game is a winning game. In this sense these
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D. A. Meyer & H. Blumer
<x>
100
BAAAA
1000
-100
2000
3000
4000
5000
t
B
-200
-300
-400
-500
A
Fig. 3. The expected payoffs for quantum games A, B and BAAAA as a function of number
of plays t. Although the curves are periodic, for random times (or on average), A and B have
negative expected payoffs while BAAAA has a positive expected payoff.
quantum games display a Parrondo phenomenon. Furthermore, the periodicity is
inversely proportional to the magnitude of the gradient of VA , so by accepting an
arbitrarily small bias the gambler can maintain a positive expected return for an
arbitrarily large number of plays of the alternating game.
3.
Unbiased Games
These results, explained in detail in our previous paper on Parrondo games [4], do
not obviously predict the outcome for an unbiased quantum Parrondo game, e.g.,
one constructed from A and B with α = 0. In this case the repeating sequence
BAAAA is a winning game for arbitrarily long times, as shown in Fig. 4. After an
initial quadratic growth, the expected winnings now increase only linearly with the
number of plays, so there is no violation of relativity. And without a biased potential VA (x), there is no bound quantum state leading to (approximately) periodic
behaviour [10–15].
Thus there are quantum Parrondo games for which the difference between the
losing and winning games grows quadratically for some finite time (faster than the
linear growth in classical Parrondo games). In order that the expected winnings
grow arbitrarily large, however, the A and B quantum games must be unbiased. In
this case the quantum ratchet effect continues for all times, but is only linear with
time — as it is classically.
Acknowledgements
We thank Derek Abbott, Jeff Rabin and Ruth Williams for useful discussions. This
work has been partially supported by the National Security Agency (NSA) and
Advanced Research and Development Activity (ARDA) under Army Research Office
(ARO) grant number DAAD19-01-1-0520 and by the Air Force Office of Scientific
Research (AFOSR) under grant number F49620-01-1-0494.
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Quantum Parrondo Games: Biased and Unbiased
L261
<x>
BAAAA
60
50
40
30
20
10
A
200
400
600
800
B 1000
t
Fig. 4. The expected payoffs for quantum games A, B and BAAAA as a function of number of
plays t, in the case where A and B are unbiased. Now the expected winnings for BAAAA increase
approximately linearly for large times.
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