Implementing Parrondo's paradox with two coin quantum walks
IImplementing Parrondo’s paradox with two coin quantum walks
Jishnu Rajendran and Colin Benjamin ∗ School of Physical Sciences, National Institute of Science Education & Research, HBNI, Jatni-752050, India
Parrondo’s paradox is ubiquitous in games, ratchets and random walks.The apparent paradox,devised by J. M. R. Parrondo, that two losing games A and B can produce an winning outcome hasbeen adapted in many physical and biological systems to explain their working. However, proposalson demonstrating Parrondo’s paradox using quantum walks failed for large number of steps. In thiswork, we show that instead of a single coin if we consider a two coin initial state which may or maynot be entangled, we can observe a genuine Parrondo’s paradox with quantum walks. Further wefocus on reasons for this and pin down the asymmetry in initial two-coin state or asymmetry in shiftoperator, either of which are necessary for observing a genuine Parrondo’s paradox. We extend ourwork to a 3-coin initial state too with similar results. The implications of our work for observingquantum ratchet like behavior using quantum walks is also discussed. I. INTRODUCTION
Parrondo’s paradox consists of a sequence of games, individually each of which are losing games but provide awinning outcome when played in a deterministic or random order. It has been shown that Parrondo’s games haveimportant applications in many physical and biological systems[1, 2]. Quantum version of Parrondo’s games wereintroduced in Refs.[3-6]. Quantum version of the classical random walk on other hand was introduced in 1993 inRef.[7] and is developed and studied extensively throughout the years[8]. In Refs.[3, 6, 9] Parrondo’s games areexplored using 1-D discrete time quantum walk(DTQW). When a game is played, the net expectation of position ofthe walker defines a win or a loss. It has been already shown that quantum walk version of Parrondo’s paradox doesnot exist in the asymptotic limits [3, 4]. The need for studying Parrondo’s games via quantum walks is necessitated bythe search for applications in building better algorithms[10] and to explain physical process like quantum ratchets[11].
II. MOTIVATION
Our motivation in this work is to implement a genuine Parrondo’s paradox via quantum walks. We show that whileprevious attempts at implementing Parrondo’s paradox with quantum walks failed in the asymptotic limits[3, 4] ourmethod using two coin initial states gives a genuine Parrondo’s paradox even in the asymptotic limits.Parrondo’s game as originally introduced in Refs.[12, 13] is a gambling game. A player plays against a bank witha choice of two games A and B , whose outcomes are determined by the toss of biased coins. Each of these gamesis losing when played in isolation but when played alternately or in some other deterministic or random sequence(such as ABB . . . , ABAB . . . , etc.) can become a winning game. Owing to this counter-intuitive nature, Parrondo’sgames are also referred to as Parrondo’s paradox. The apparent paradox that two losing games A and B can producea winning outcome when played in an alternating sequence was originally devised by Juan M. R. Parrondo as apedagogical illustration of the Brownian ratchet[13]. Parrondo’s games have important applications in many physicaland biological systems, e.g., in control theory the random/deterministic combination of two unstable systems canproduce a overall stable system[1].The 1-D discrete time quantum walk(DTQW) implementation of Parrondo’s paradox is as follows: Considertwo games A and B played alternately in time. Game A and B are represented by different quantum operators U ( α A , β A , γ A ) and U ( α B , β B , γ B )[14, 15], U ( α, β, γ ) = (cid:18) e iα cos β − e − iγ sin βe iγ sin β e − iα cos β (cid:19) . (1)The initial state of the quantum walker is | Ψ (cid:105) = √ | (cid:105) ⊗ ( | (cid:105) − i | (cid:105) ) , where first ket refers to the position space andsecond ket refers to the single coin space which is initially in a superposition of heads and tails. The shift in the ∗ [email protected] a r X i v : . [ qu a n t - ph ] J a n position space, say from | n (cid:105) to | n − (cid:105) or | n + 1 (cid:105) , is defined by a unitary operator called shift operator( S ) defined as, S = ∞ (cid:88) n = −∞ | n + 1 (cid:105)(cid:104) n | ⊗ | (cid:105)(cid:104) | + ∞ (cid:88) n = −∞ | n − (cid:105)(cid:104) n | ⊗ | (cid:105)(cid:104) | . (2)Games A and B are played alternately in different time steps, i.e., game A is played on time steps t = nq and game B is played on time steps t (cid:54) = nq , where q is the period and n is an integer. The evolution operator can be written as: U = (cid:26) S · U ( α A , β A , γ A ) if t = nq, n ∈ Z S · U ( α B , β B , γ B ) if t (cid:54) = nq, n ∈ Z (3)and the final state after N steps is given by | Ψ N (cid:105) = U N | Ψ (cid:105) . For q = 3, it means we play games with the time sequence ABBABB . . . . As denoted in Fig. 1, after N steps, if the probability P R of the walker to be found to the right of theorigin, is greater than the probability P L to be found to the left of the origin, i.e., P R − P L >
0, we consider the player towin. Similarly, if P R − P L <
0, the player losses. If P R − P L = 0, it means the player neither loses nor wins, it’s a draw.By making use of the above scheme, Parrondo’s games using 1-D DTQW are formulated. The game is constructedwith two losing games A and B having two different biased coin operators U A ( α A , β A , γ A ) and U B ( α B , β B , γ B ), if weset α A = − , β A = 45 , γ A = 0 , α B = 0 , β B = 88 , γ B = 0, U SA = U S ( − , , U SB = U S (0 , , −
16) as in Fig. 2(a).We form a game with sequences
ABBB . . . . This results in winning at the beginning but in the asymptotic limitthe player will lose as in Fig. 2(b), one can check for different sequences like
ABAB . . . ABBABB . . . etc. and in allcases in the asymptotic limits we lose. Hence Parrondo’s paradox does not exist in case of 1-D DTQW. This factwas noted in Refs. [3, 4] also. In particular, Ref. [3] shows with many different sequences like ABAB.., AABAAB..,etc, at large steps there is no Parrondo’s paradox. Hence our motivation to find circumstances for the existence of agenuine Parrondo’s paradox in quantum walks. It is possible to show the convergence of quantum walk as obtainedin Fig. 2(a) analytically. The analytical form of convergence mentioned in Theroem 1 of Ref.[16] for a single coinquantum walk can be used for calculating the asymptotic limit. E = − (1 − (cid:112) − | e iα cos β | ) λ (4)where, λ = ie iα cos β ( − e iγ ) sin β + ie − iα cos β ( − e − iγ ) sin β | e iα cos β | where E is the convergence value of the quantum walk. The convergence for the single coin quantum walk whenthe coin operates AAAA . . . and
BBBB . . . ( A = U ( − , , B = U (0 , , − − . − . A = Identity (I) and B = N OT (X) operators are used then the classicalwalk we have P R − P L = 0 for AAAA . . . and
BBBB . . . as well as
ABAB . . . . This is in conformity with results of aclassical random walk which has a gaussian distribution with mean as well as median equal to zero.
III. PARRONDO’S PARADOX USING TWO COIN INITIAL STATE
As in the previous section, the elements of our two coin quantum walk are the walker, coins, evolution operatorsfor both the coins, walker and a set of observables. The walker is a quantum system with its position denoted as | position (cid:105) residing in a Hilbert space of infinite but countable dimension H P . The basis states | i (cid:105) P which span H P ,and any superposition of the form (cid:80) i α i | i (cid:105) p which are subject to (cid:80) i | α i | = 1, are valid states for the walker [17].The walker is usually initialized at the ‘origin’, i.e., | position (cid:105) = | (cid:105) P . The two coin initial state is a quantum systemin a 4-D Hilbert space H EC . We denote the two coin initial state as | coin (cid:105) , which may or may not be entangled- | coin (cid:105) = cos (cid:18) θ (cid:19) | (cid:105) + i sin (cid:18) θ (cid:19) | (cid:105) . (5)The initial state of the quantum walker resides in the Hilbert space H T = H P ⊗ H EC and has the form: | ψ (cid:105) = | position (cid:105) ⊗ | coin (cid:105) (6) FIG. 1. Pictorial illustration of the conditions for win or loss for QWs on a line. (a) (b)
FIG. 2. a) P R − P L of the walker after t steps, with initial state | Ψ (cid:105) = √ | (cid:105)⊗ ( | (cid:105)− i | (cid:105) ), and coin operator A = U S ( − , , B = U S (0 , , −
16) (green line). b) P R − P L of the walker with games played in sequence ABBBABBB . . . (i.e., q = 4), A = U S ( − , , B = U S (0 , , −
16) (1600 steps), herein initially you win( steps < which using Eq. 5, gives | ψ (cid:105) = | (cid:105) ⊗ (cid:0) cos (cid:0) θ (cid:1) | (cid:105) + i sin (cid:0) θ (cid:1) | (cid:105) (cid:1) . Evolution operators used are unitary as beforeand since the coin is a bipartite system, the coin is defined as the tensor product of two single-qubit coin operators: C EC = U α k ,β k ,γ k ⊗ U α l ,β l ,γ l , where k , l can be any of the Game A and B . The evolution operator is fully separable,thus any entanglement in the coins is due to the initial states used. The conditional shift operator S EC allows thewalker to move either forward or backward, depending on the state of the coins. The operator S EC = (cid:88) i | i + 1 (cid:105) pp (cid:104) i | ⊗ | (cid:105) cc (cid:104) | + (cid:88) i | i (cid:105) pp (cid:104) i | ⊗ | (cid:105) cc (cid:104) | + (cid:88) i | i (cid:105) pp (cid:104) i | ⊗ | (cid:105) cc (cid:104) | + (cid:88) i | i − (cid:105) pp (cid:104) i | ⊗ | (cid:105) cc (cid:104) | (7)incorporates the stochastic behavior of the random walk with a two coin initial state. It is only when the coin isin the | (cid:105) or | (cid:105) state that the walker moves either forward or backward else the walker does not move. The fullevolution operator has the structure U T = S EC . ( I p ⊗ C EC ) and one can mathematically represent a two coin quantumwalk after N steps as | ψ (cid:105) N = ( U T ) N | ψ (cid:105) , where | ψ (cid:105) denotes the initial state of the walker and the coins. As definedbefore, winning and losing in context of Parrondo’s game, after N time steps if the probability P R of the walker tobe found to the right of the origin is greater than the probability to be found left of the origin, i.e., P R − P L > P R − P L < P R − P L = 0 it implies a draw.In order to obtain a genuine Parrondo’s paradox the two games A and B are now played on the two coin space asfollows: U A ⊗ U B is operated on the two coins and in the next step U B ⊗ U A is played on the two coins. Thus, forthe first coin we have the series ABAB . . . while on the second coin we have
BABA . . . . The coin operators can as (a) (b)(c) (d)
FIG. 3. a) Parrondo walk is evident even at large number of steps for partially entangled coin states ( θ = π/
4) when
ABAB . . . is played on first coin &
BABA . . . on second coin. b) However, when
AAAA . . . is played on first &
BBBB . . . on secondcoin, one gets a losing outcome. In c) we show similar to a partially entangled state a non-entangled state ( θ = 0) also givesa Parrondo’s paradox for large number of steps when ABAB . . . is played on first coin &
BABA . . . on second coin and finallyin d) we show that P R − P L is negative at large steps when AAA . . . and
BBB . . . are played on the two coins. before be defined as- X = A ⊗ B = C EC = U ( − , , ⊗ U (0 , , − Y = B ⊗ A = C (cid:48) EC = U (0 , , − ⊗ U ( − , , XY XY . . . and the plot for P R − P L as shown in Fig. 3(a)is obtained. It is evident that the sequence XY XY . . . provides a winning outcome for two losing games at largenumber of steps. The fact that individually the sequence
AAA . . . on first coin and
BBB . . . on second coin give alosing outcome can be seen from P R − P L plot in Fig. 3(b). IV. DISCUSSION
From Fig. 3 one can convincingly conclude that to obtain a genuine Parrondo’s paradox via quantum walks oneneeds a non-entangled or a partially entangled two coin state. When a single coin was considered (as in Fig. 2) theoutcome of Parrondo’s games did not give rise to the paradox for quantum walk with large number of steps. In orderto obtain a Parrondo’s paradox, what is needed is a two-coin state. Finally, what are the plausible reasons for thesuccess of the two coin initial state as compared to the single coin state? We can start by identifying the reasonswhich do not lead to Parrondo’s paradox. First, entanglement has no or marginal role. Maximally entangled coinslead to a draw as the probability distribution is perfectly symmetric as noted before in Ref.[17], on the other handnon-entangled or partially entangled coins lead to a Parrondo’s paradox. Further, in Fig. 4, we plot the amount ofentanglement present in a quantum system, i.e., the concurrence[18]. The concurrence is zero for a separable state
FIG. 4. Plot of Concurrence(Green), P R − P L (red, solid) for ABAB.. on first coin and BABA...on second coin, and finally P R − P L (red, dashed) for AAAA.. on first coin and BBBB...on second coin. Note that Parrondo’s paradox is observed for0 < θ < π/ π/ < θ < π with the definition as in Fig. 1. In the region π/ < θ < π/ P R − P L for ABAB.. on first coin and BABA...on secondcoin for state | (cid:105) , b) Plot of P R − P L for ABAB.. on first coin and BABA...on second coin for state | (cid:105) , c) Plot of P R − P L for AAAA.. on first coin and BBBB...on second coin for state | (cid:105) and finally d) Plot of P R − P L for AAAA.. on first coin andBBBB...on second coin for state | (cid:105) . For both | (cid:105) as well as | (cid:105) state there is no Parrondo’s paradox. For state | (cid:105) thereis a role reversal and thus our definition for Parrondo’s paradox as used in Fig. 1 is also reversed. FIG. 6. Parrondo’s paradox with a shift operator with one wait state (Eq. 8). a) Plot of P R − P L for ABAB.. on first coinand BABA...on second coin for state | (cid:105) , b) Plot of P R − P L for AAAA.. on first coin and BBBB.. on second coin for state | > , c) Plot of P R − P L for ABAB.. on first coin and BABA.. on second coin for state | (cid:105) and finally d) Plot of P R − P L for AAAA.. on first coin and BBBB.. on second coin for state | (cid:105) . For both | (cid:105) as well as | (cid:105) state there is now Parrondo’sparadox with shift operator as defined in Eq. 8. For state | (cid:105) there is a role reversal and thus our definition for Parrondo’sparadox as used in Fig. 1 is also reversed. and one for a maximally entangled state. Fig.4 shows the concurrence for our arbitrary two coin state as a functionof θ . One sees that Parrondo’s paradox is observed for 0 < θ < π/ π/ < θ < π with the definition as inFig. 1. In the region π/ < θ < π/ | (cid:105) and | (cid:105) in Fig. 5. Both do not lead to the Parrondo’s paradox. This may give the impression thatonly when one has initial 2-coin state | (cid:105) or | (cid:105) composed of orthogonal coin states do we see a Parrondo’s paradox.However, it’s not the complete picture. The shift operator plays a non-trivial role. If we change the shift operator,see Eq. 7 from two wait states to just a single wait state as in Eq. 8 then a different picture emerges. S EC = (cid:88) i | i +1 (cid:105) pp (cid:104) i |⊗| (cid:105) cc (cid:104) | + (cid:88) i | i +1 (cid:105) pp (cid:104) i |⊗| (cid:105) cc (cid:104) | + (cid:88) i | i (cid:105) pp (cid:104) i |⊗| (cid:105) cc (cid:104) | + (cid:88) i | i − (cid:105) pp (cid:104) i |⊗| (cid:105) cc (cid:104) | (8)In Fig. 6, we plot P R − P L for around 800 time steps for both initial states- | (cid:105) as well as | (cid:105) with the new shiftoperator defined with a single wait state as in Eq. 8. In this case for both | (cid:105) and | (cid:105) states we see Parrondo’sparadox. To conclude the most plausible reason for observing the Parrondo’s paradox is both due to some asymmetrywhich comes into play in a two coin state and is not possible to include in the single coin state. The asymmetry maybe in the initial quantum state or in the shift operator.Finally, what are the implications for more than two coin initial state? To test this we consider two different 3 coininitial states: | (cid:105) and | (cid:105) . Similar to the 2 coin case discussed earlier, the coin is defined as the tensor product ofthree single-qubit coin operators: C EC = U α k ,β k ,γ k ⊗ U α l ,β l ,γ l ⊗ U α m ,β m ,γ m , where k , l and m can be any of the games ABAB... ON COIN 1BABA... ON COIN 2ABAB... ON COIN 3INITIAL STATE: |010> ABAB ... ON COIN 1BABA... ON COIN 2 INITIAL STATE: |11>ABAB... ON COIN 1BABA... ON COIN 2ABAB... ON COIN 3 INITIAL STATE: |000> AAAA... ON COIN 1BBBB... ON COIN 2 AAAA... ON COIN 3INITIAL STATE: |000> P R P L P R P L aacc dd steps P R P L steps AAAA... ON COIN 1BBBB... ON COIN 2AAAA... ON COIN 3INITIAL STATE: |010> bb P R P L stepssteps FIG. 7. Parrondo’s paradox with an initial 3-coin state. a) Plot of P R − P L for ABAB.. on first and third coins and BABA...onsecond coin for state | (cid:105) , b) Plot of P R − P L for AAAA.. on first and third coins and BBBB...on second coin for state | (cid:105) ,c) Plot of P R − P L for ABAB.. on first coin and BABA...on second coin for state | (cid:105) and finally d) Plot of P R − P L forAAAA.. on first and third coins and BBBB...on second coin for state | (cid:105) . For | (cid:105) state there is no Parrondo’s paradox.For state | (cid:105) we see a Parrondo’s paradox however there is a role reversal and thus our definition for Parrondo’s paradox asused in Fig. 1 is also reversed. A and B . The conditional shift operator S EC allows the walker to move either forward or backward, depending onthe state of the coins and is defined as- S EC = (cid:88) i | i + 2 (cid:105) pp (cid:104) i | ⊗ | (cid:105) cc (cid:104) | + (cid:88) i | i + 1 (cid:105) pp (cid:104) i | ⊗ | (cid:105) cc (cid:104) | + (cid:88) i | i (cid:105) pp (cid:104) i | ⊗ | (cid:105) cc (cid:104) | + (cid:88) i | i (cid:105) pp (cid:104) i | ⊗ | (cid:105) cc (cid:104) | + (cid:88) i | i (cid:105) pp (cid:104) i | ⊗ | (cid:105) cc (cid:104) | + (cid:88) i | i (cid:105) pp (cid:104) i | ⊗ | (cid:105) cc (cid:104) | + (cid:88) i | i − (cid:105) pp (cid:104) i | ⊗ | (cid:105) cc (cid:104) | + (cid:88) i | i − (cid:105) pp (cid:104) i | ⊗ | (cid:105) cc (cid:104) | (9)incorporates the stochastic behavior of the random walk with a three coin initial state. When the coin is in the | (cid:105) or | (cid:105) state that the walker moves two steps at once either forward or backward, while when the coin is in state | (cid:105) or | (cid:105) the walker moves one step either forward or backward and for the rest of the cases the walker remainsfixed. The full evolution operator similar for the two coin case has the structure U T = S EC . ( I p ⊗ C EC ) and the threecoin quantum walk after N steps is written as | ψ (cid:105) N = ( U T ) N | ψ (cid:105) , where | ψ (cid:105) denotes the initial state of the walkerand the coins. In order to obtain a genuine Parrondo’s paradox the two games A and B are now played on the threecoin space as follows: U A ⊗ U B ⊗ U A is operated on the three coins and in the next step U B ⊗ U A ⊗ U B is playedon the three coins. Thus, for the first and third coins we have the series ABAB . . . while on second coin we have
BABA . . . . The coin operators can as before be defined as- X = A ⊗ B ⊗ A = C EC = U ( − , , ⊗ U (0 , , − ⊗ U ( − , , Y = B ⊗ A ⊗ B = C (cid:48) EC = U (0 , , − ⊗ U ( − , , ⊗ U (0 , , − XY XY . . . and the plot for P R − P L as shown in Fig. 7(a) isobtained. It is evident that the sequence XY XY . . . provides a winning outcome for two losing games even at largenumber of steps. The fact that individually the sequence
AAA . . . on first and third coins while
BBB . . . on secondcoin gives a losing outcome can be seen from P R − P L plot in Fig. 7(b). In Fig. 7(c) and (d) we plot the median P R − P L for the initial 3-coin state | (cid:105) , we confirm the absence of any Parrondo’s paradox for this initial state,confirming the trend seen for 2-coin initial states with symmetric shift operator.To conclude, this section the initial state has a great bearing on having the Parrondo’s paradox in a quantum walkor not. In both the two coin and three coin state when coins are orthogonal we see the paradox and for the casewhen they are not paradox disappears. Of course the aforesaid is subject to the qualification that the shift operatorwhich controls the position of the coin state has an important bearing. Asymmetry in either the initial two coin state,e.g., | (cid:105) or | (cid:105) or an asymmetric shift operator (in case the initial state is | (cid:105) or | (cid:105) ) is necessary for obtaining aParrondo’s paradox with quantum walks. V. CONCLUSION
Our goal in this work was to show evidence of a genuine Parrondo’s paradox using quantum walks and we show thisusing a two coin state. We also considered entanglement between the two coins and showed that maximally entangledstates do not show any paradox while non-entangled as well as partially entangled states do show the paradox. Wealso tried to understand the reasons behind this paradox. The most plausible reason behind observing the paradoxwith two or higher coin initial states is the introduction of asymmetry either in the initial coin state or in the shiftoperator with one or two wait states in addition to left or right shifts. Our work can be considered as a demonstrationof a quantum ratchet too, implying particle transport against an applied bias in presence of noise or perturbations. Inour case the noise parameter can be considered to reduce entanglement, thus looking at Fig. 4, from zero asymmetryin probability distribution, i.e., non-directed transport, when there is maximal entanglement to finite asymmetry inprobability distribution, i.e., directed transport when there is no entanglement, is a clear marker of quantum ratchetlike behavior of our system. The quantum ratchet analogies in Parrondo’s paradox with quantum walks were alsonoticed in Ref. [6], however without any entanglement. New quantum walks are of great interest to the communityas their investigation may lead to new quantum algorithms, which are of great interest to the quantum computationcommunity at present. [1] A. Allison and D. Abbott, Control systems with stochastic feedback, Chaos, Vol. , No. 3, pp. 715–724, (2001)[2] J. M. R. Parrondo and L. Dinis, Brownian motion and gambling: from ratchets to paradoxical games, ContemporaryPhysics, Vol , pp. 147–157, (2004)[3] A. P. Flitney, Quantum Parrondo’s games using quantum walks. arXiv preprint arXiv: (2012).[4] M. Li, Y. S. Zhang, G.-C. Guo, Qunatum Parrondo’s games constructed by quantum random walk, Fluct. Noise Lett. , (2013).[5] C. M. Chandrashekar, S. Banerjee, Parrondos game using a discrete-time quantum walk, Physics Lett. A
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VII. COMPETING INTERESTS
We have no competing interests.
VIII. AUTHOR CONTRIBUTIONS
C.B. conceived the proposal, J.R. did the calculations on the advice of C.B., J.R. and C.B. wrote the paper, J.R.and C.B analyzed the results.
IX. FUNDING
SCIENCE & ENGINEERING RESEARCH BOARD, NEW DELHI, GOVT. OF INDIA funded this research underGRANT NO. EMR/2015/001836.
X. RESEARCH ETHICS
This study did not require any prior ethical assesment.
XI. ANIMAL ETHICS
This study did not require any prior ethical assessment
XII. PERMISSION TO CARRY OUT FIELDWORK
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