We analyze a randomly perturbed quantum version of the baker's
transformation, a prototype of an area-conserving chaotic map. By numerically
simulating the perturbed evolution, we estimate the information needed to
follow a perturbed Hilbert-space vector in time. We find that the Landauer
erasure cost associated with this information grows very rapidly and becomes
much larger than the maximum statistical entropy given by the logarithm of the
dimension of Hilbert space. The quantum baker's map thus displays a
hypersensitivity to perturbations that is analogous to behavior found earlier
in the classical case. This hypersensitivity characterizes ``quantum chaos'' in
a way that is directly relevant to statistical physics.Comment: 8 pages, LATEX, 3 Postscript figures appended as uuencoded fil

A. Peres

C. M. Caves

Carlton M. Caves

G. J. Chaitin

I. C. Percival

J. Ford

K. R. W. Jones

M. Saraceno

N. L. Balazs

R. Landauer

R. Schack

Rüdiger Schack

S. Weigert

V. I. Arnold

W. H. Zurek

W. K. Wootters

English

arXiv

Schack, R.

Caves, C. M.

Royal Holloway Research Online

Hypersensitivity to perturbations in the quantum baker's map

Crossref

Physical Review Letters

Hypersensitivity to perturbations in the quantum baker’s map

arXiv:chao-dyn/9303002v1  6 Mar 1993Hypersensitivity to Perturbations in the QuantumBaker’s Map∗Ru¨diger Schack(a)and Carlton M. Caves(a,b)(a)Department of Physics and Astronomy, University of New MexicoAlbuquerque, NM 87131(b)Santa Fe Institute, 1660 Old Pecos Trail, Suite A, Santa Fe, NM 87501January 14, 1993AbstractWe analyze a randomly perturbed quantum version of the baker’s transforma-tion, a prototype of an area-conserving chaotic map. By numerically simulatingthe perturbed evolution, we estimate the information needed to follow a perturbedHilbert-space vector in time. We find that the Landauer erasure cost associated withthis information grows very rapidly and becomes much larger than the maximumstatistical entropy given by the logarithm of the dimension of Hilbert space. Thequantum baker’s map thus displays a hypersensitivity to perturbations that is analo-gous to behavior found earlier in the classical case. This hypersensitivity characterizes“quantum chaos” in a way that is directly relevant to statistical physics.Great progress has been made in studying manifestations of chaos in quantum systems [1],yet there still is controversy as to whether quantum chaos exists at all [2, 3]. A chief reasonfor this is that the most important characteristic of classical chaotic systems—exponentialdivergence of trajectories starting at arbitrarily close initial points in phase space—is absentfrom quantum systems simply because the existence of a quantum scale makes meaninglessthe concept of two arbitrarily close points in phase space.Any attempt to find exponential divergence of trajectories of Hilbert-space vectorsfounders immediately, because the linear Schro¨dinger equation, with its unitary evolution,preserves Hilbert-space inner products. Yet the unitary linear evolution of the Schro¨dingerequation must be irrelevant to the issue of quantum chaos [2], since any Hamiltonianclassical chaotic system can be described by an analogous area-conserving linear Liouville∗Submitted to Physical Review Letters1equation. Two probability distributions in classical phase space—we call these phase-spacepatterns—that are initially close together (in terms of an overlap integral) stay close to-gether forever, because the Liouville equation is area conserving. This leads to the questionwe address in the present article: Are there manifestations of Hamiltonian classical chaosin the Liouville equation that are also present in quantum mechanics?In an earlier paper [4] we analyzed a prototype of an area-conserving chaotic map,the baker’s transformation [5], in the Liouville representation; i. e., we focused on phase-space patterns instead of on single trajectories. Our analysis was guided by the question[6] of how available work decreases with time when the baker’s map is subjected to area-conserving random perturbations. An area-conserving abstract mapping corresponds toan energy-conserving phase-space system, so we identify two negative contributions tofree energy. The conventional one is ordinary entropy, which measures how incompleteknowledge about a system reduces our ability to extract work. The other contributionarises from Landauer’s principle [7, 8] that there is an unavoidable energy cost of kBT ln 2connected with the erasure of one bit of information. It follows from Landauer’s principlethat the information, quantified by algorithmic information [9], needed to give a completedescription of a system state also reduces the amount of available work and thus shouldmake a further negative contribution to free energy [10, 11].In our earlier paper [4] we compared two strategies for preserving the ability to extractwork from a system. The first strategy is to keep track of the perturbed phase-space patternin fine-grained detail, in an attempt to preserve the work inherent in the initial condition.The second strategy, which we call coarse graining, is to average over the perturbation andto put up with the resulting ordinary entropy increase. We found, for the perturbed baker’smap, that the information needed to implement the first strategy is overwhelmingly largerthan the entropy increase of the second strategy. This means that the free-energy costof tracking the perturbed pattern in fine-grained detail is enormous and far greater thanthe cost of the entropy increase that results from coarse graining. We conjecture that thishypersensitivity to perturbations is a general feature of perturbed classical chaotic systems,and we regard it as the desired manifestation of classical chaos in the Liouville equation.In the present paper we compare the two strategies for preserving work in the case of aquantum system, a quantum version of the baker’s map [12]. Using numerical simulation,we find essentially the same behavior as in the classical case, as was suggested using heuristicarguments in Ref. [6]. The quantum baker’s map displays hypersensitivity to perturbationsand thus can be said to exhibit quantum chaos.The concept of algorithmic information has been used before to investigate quantumchaos [2, 13]. If one defines a chaotic system as one where the algorithmic informationneeded to predict a single (unperturbed) trajectory grows linearly with time (or numberof steps) [14], then there is classical chaos, but no quantum chaos [2]. Our approach, byfocusing on patterns in phase space instead of trajectories, uses a framework where classicaland quantum mechanics can be treated on analogous footings. Moreover, since Landauer’sprinciple gives information an explicit physical meaning by connecting it to available work,our characterization in terms of hypersensitivity to perturbations is directly relevant tostatistical physics.The classical baker’s transformation maps the unit square 0 ≤ q, p ≤ 1 onto itself2according tof : (q, p) 7−→ (2q − [2q], (p+ [2q])/2) , (1)where the square brackets denote the integer part. There is no unique way to quantize aclassical map. Here we adopt a quantized baker’s map introduced by Balazs and Voros[12] and put in more symmetrical form by Saraceno [15]. Position and momentum spaceare discretized, placing the lattice sites at half-integer values qj = pj = (j + 1/2)/2Nfor j = 0, . . . , 2N − 1. The dimension 2N of Hilbert space is assumed to be even. Forconsistency of units, let the quantum scale on phase space be 2pih¯ = 1/2N . Position andmomentum basis kets are denoted by |qj〉 and |pj〉. A transformation between these twobases is performed by the operator G2N , defined by the matrix elements(G2N )jk = 〈pj|qk〉 =√2pih¯ e−ipjqk/h¯ . (2)The quantum baker’s map is now defined by the matrixB = G−12N(GN 00 GN), (3)where, as throughout this article, matrix elements and vector coordinates are given relativeto the position basis.The perturbation operator we use is constructed to resemble the type of perturbationused in our previous work [4] on the classical baker’s map. We partition phase space intoan even number 2Nc = 2N/wc of congruent perturbation cells, where 2Nc and wc ≥ 2 areintegral divisors of 2N . In the following we use perturbation cells that are vertical stripesextending over the entire p range. Then each perturbation cell contains wc q-eigenstates.A perturbation operator that perturbs each perturbation cell independently has the formof a matrix with zero elements everywhere except for 2Nc square blocks of size wc on thediagonal. We choose these square blocks so that they correspond to a shift in the p directionin a wc-dimensional subspace. Let the momentum shift in the nth perturbation cell (n =0, . . . , 2Nc − 1) be αin, where the real number α is the magnitude of the momentum shiftand in ∈ {−1, 1}. The symmetry condition i2Nc−n−1 = in (n = 0, . . . , Nc − 1) avoids rapidoscillations and thus ensures similarity to the classical case. The perturbation operatorUα;i0,...,iNc−1 is defined by(Uα;i0,...,iNc−1)jk = δjkei2piφk , (4)where φ0 = 0, φk = φk−1 + αin(k) (k = 1, . . . , 2N − 1) with n(k) = [k/wc]. The parameterα characterizes the “strength” of the perturbation, whereas wc/2N = 1/2Nc is the area ofthe perturbation cells.A perturbed time step consists of first applying the unperturbed time-evolution operatorB, followed by a perturbation operator Uα;i0,...,iNc−1 with i0, . . . , iNc−1 ∈ {−1, 1} chosen atrandom, α being fixed. We thus allow for a different perturbation at each step, in contrastto Ref. [16] where a particular perturbed evolution operator was applied repeatedly. Aftern time steps, the number of different perturbation sequences—or histories—is 2nNc.Our specialization to vertically striped perturbation cells involves no restriction relativeto our work on the classical baker’s map. There we allowed for (2mwc/2N)×2−m rectangular3perturbation cells, which are the image of vertical stripes underm applications of the baker’smap, 2m ≤ 2N/wc = 2Nc. Likewise, the perturbation operator for “rectangular” quantum-mechanical perturbation cells is U ′ = BmUB−m where U is given by Eq. (4). Using U withinitial state |ψ0〉 is equivalent to using U ′ with initial state Bm|ψ0〉. The freedom to choosem—we use m = 0 for vertical stripes—is the same as the classical freedom to choose theinitial position of the “decimal point” in the symbolic representation of the baker’s map.As a preliminary step, we show that perturbed evolution leads after several steps toan ensemble of vectors that is similar to an ensemble of vectors distributed randomly onHilbert space. A useful criterion for determining the randomness of an ensemble of vectors|ψ〉 is based on the moments of the quantity [17]W (|ψ〉) = −2N−1∑i=0|〈qi|ψ〉|2 log(|〈qi|ψ〉|2) , (5)which we call W -entropy to distinguish it from ordinary entropy (log denotes the binarylogarithm, as throughout this paper). For random vectors in 2N = 16-dimensional Hilbertspace, the mean and standard deviation of the W -entropy are given by W = 3.434± 0.178bits, a result obtained on a computer by calculating W (|ψ〉) for a large number of vectors|ψ〉 chosen at random from an ensemble distributed uniformly over Hilbert space [17].(This mean value agrees with the exact formula for the mean value [18], W = [Ψ(2N +1) − Ψ(2)]/ ln 2, where Ψ is the digamma function.) We compare the result for randomvectors with the first two moments of the W -entropy for an ensemble of vectors createdby the perturbed baker’s map in a 16-dimensional Hilbert space. Choosing 2Nc = 2 andα = 0.025 = 0.4/2N , we create an ensemble of approximately 20 000 perturbed vectorsby applying different randomly chosen histories for n = 15 perturbed time steps to theinitial vector |ψ0〉 = |p5〉. We find for the first two moments of the W -entropy the valuesW = 3.428± 0.184 bits, very close to the moments for a random sample.As a second check of randomness, we calculate the distribution of Hilbert-space an-gles θ = cos−1(|〈ψ′|ψ〉|) between vectors |ψ〉 and |ψ′〉 that have evolved under the sameperturbed quantum baker’s map applied to the same initial state as in the previous ex-ample. We compute the Hilbert-space angle between each pair of vectors in each of threeensembles of approximately 16 000 vectors created by applying different randomly chosenperturbation histories for 15, 23, and 31 steps. In addition, we compute the Hilbert-spaceangle between each pair of the 210 vectors after 10 steps. The resulting distributions ofHilbert-space angles are displayed in Fig. 1. After 31 steps the closest pair of vectors is27.1◦ apart, a striking demonstration of the “size” of 16-dimensional Hilbert space, i. e., ofhow many widely separated vectors Hilbert space can accommodate even for a relativelysmall dimension. For comparison, Fig. 1 also shows the distribution f(θ) = 30(sin θ)29 cos θof Hilbert-space angles for a set of random vectors. These results show clearly how theensemble is randomized by the perturbed quantum baker’s map.We proceed now to compare the two strategies for extracting work outlined above—coarse graining versus following the evolved vector in fine-grained detail. We estimatethe conditional algorithmic information ∆I needed—given background information—tospecify a typical perturbed vector after n steps and compare it to the increase in ordinaryentropy ∆H that results from averaging over the perturbation. Our first example uses, as4random vectorsn=10n=15n=23n=31pi / 2pi / 400123Hilbert-space angle (rad)Probability densityFigure 1: Distribution of Hilbert-space angles for vectors evolving under the perturbedquantum baker’s map, shown for different numbers of perturbed time steps n. For com-parison, the distribution for random vectors is also shown. The dimension of Hilbert spaceis 2N = 16, the number of perturbation cells is 2Nc = 2, the perturbation strength isα = 0.025 = 0.4/2N , and the initial vector is |ψ0〉 = |p5〉.before, a 2N = 16-dimensional Hilbert space, partitioned into 2Nc = 2 vertically stripedperturbation cells. We choose a fixed perturbation amplitude α = 0.025 = 0.4/2N andan initial pure state |ψ0〉 = |p5〉, i. e., a momentum eigenstate, which corresponds to ahorizontal stripe in the unit square. This perturbation can be described completely bygiving one bit per step, to specify which of the two possible perturbation operators Uα;1and Uα;−1 is applied. If the logarithmic term [11] that keeps track of the number of stepsn is neglected, this sets an upper bound on the information ∆I. This upper bound isrealized only if two different histories of perturbed time steps always lead to two differentvectors at some level of resolution on Hilbert space. We choose a resolution that regardstwo vectors as different if their Hilbert-space angle exceeds δθ = pi/50 = 3.6◦ (|〈ψ′|ψ〉|smaller than 0.998). By comparing numerically all possible histories, we find that, through15 perturbed time steps, all trajectories lead to distinguishable vectors. Figure 2 showsthe resulting linear increase in the information ∆I.Figure 2 also shows the ordinary entropy increase ∆H , obtained by determining theentropy of the density matrix that results from averaging over all possible histories. Itcan be seen that ∆I is always larger than ∆H . Indeed, ∆H saturates at the value∆Hmax = log 2N = 4 bits, the logarithm of the dimension of Hilbert space, whereas ∆I isonly limited by ∆Imax ≃ −2(2N − 1) log(δθ/2) ≃ 150 bits, which is the logarithm of thenumber of different vectors Hilbert space can accommodate [6, 19]. Whereas ∆Hmax growslogarithmically with the dimension 2N of Hilbert space, the maximum information ∆Imaxgrows linearly with 2N and is enormous for macroscopic systems.As in the classical case [4], the information ∆I grows more dramatically when the num-50510150 5 10 15∆H∆Imax∆Hnumber of steps nbitsFigure 2: Conditional algorithmic information ∆I needed to track a vector evolving underthe perturbed quantum baker’s map, compared to the increase in ordinary entropy ∆Hthat results from averaging over the perturbation. The parameters are the same as inFig. 1.0510150 1 2 3 4∆H∆Imax∆Hbitsnumber of steps nFigure 3: ∆I and ∆H as in Fig. 2, but with parameters 2N = 64, 2Nc = 16, α = 0.05 ≃3/2N , and |ψ0〉 = B−5|p1〉.6ber of perturbation cells is large. Figure 3 displays results for a 64-dimensional Hilbert spacewith 16 vertically striped perturbation cells, each containing four position eigenstates. Theperturbation strength is α = 0.05 ≃ 3/2N , and the initial state is |ψ0〉 = B−5|p1〉, a statewhose image under B has negligible support outside the leftmost perturbation cell. Thismeans that, in order to describe the perturbed state after the first time step, in the pertur-bation operator Uα;i0,...,i7 only the sign i0 referring to the leftmost perturbation cell must bespecified. Since B2|ψ0〉, B3|ψ0〉, and B4|ψ0〉 extend over 2, 4, and 8 perturbation cells, weexpect the number of bits needed to specify the perturbed state to grow as∑n−1j=0 2j = 2n−1until the state extends over all perturbation cells. This behavior is verified in Fig. 3 usingthe same method as for Fig. 2.Given these results and those of our previous paper [4], we have demonstrated similarhypersensitivity to perturbation in a classically chaotic system and its quantum analogue.In both cases the large information needed to track the perturbed evolution is due to thelarge number of possible ways to perturb a state [6, 20]. In the classical domain, chaos opensup the large space of possibilities—phase-space patterns with structure on finer and finerscales. Quantum mechanics operates inherently in an enormous space of possibilities—thepure states on Hilbert space. Hypersensitivity to perturbations means that more work canbe extracted by coarse graining than by following the perturbed evolution in fine-graineddetail. Our results provide a motivation for coarse graining and thus an explanation of thesecond law of thermodynamics.RS acknowledges the support of a fellowship from the Deutsche Forschungsgemeinschaft.References[1] See, e. g., F. Haake, Quantum Signatures of Chaos (Springer, New York, 1991); Quan-tum Chaos, edited by H. A. Cerdeira, R. Ramaswamy, M. C. Gutzwiller, and G. Casati(World Scientific, Singapore, 1991); Quantum Chaos, Quantum Measurement, editedby P. Cvitanovic´, I. Percival, and A. Wirzba (Kluwer, Dordrecht, 1992).[2] J. Ford, G. Mantica, and G. H. Ristow, Physica D 50, 493 (1991).[3] A. Peres, in Quantum Chaos, Quantum Measurement, edited by P. Cvitanovic´, I.Percival, and A. Wirzba (Kluwer, Dordrecht, 1992), p. 249.[4] R. Schack and C. M. Caves, Phys. Rev. Lett. 69, 3413 (1992).[5] V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (Benjamin, NewYork, 1968).[6] C. M. Caves, in Physical Origins of Time Asymmetry, edited by J. J. Halliwell,J. Pe´rez-Mercader, and W. H. Zurek (Cambridge University Press, Cambridge, Eng-land, 1993).[7] R. Landauer, IBM J. Res. Develop. 5, 183 (1961).[8] R. Landauer, Nature 355, 779 (1988).7[9] G. J. Chaitin, Information, Randomness, and Incompleteness (World Scientific, Sin-gapore, 1987).[10] W. H. Zurek, Nature 341, 119 (1989).[11] W. H. Zurek, Phys. Rev. A 40, 4731 (1989).[12] N. L. Balazs and A. Voros, Ann. Phys. 190, 1 (1989).[13] S. Weigert, Phys. Rev. A 43, 6597 (1991).[14] J. Ford, Phys. Today 36(4), 40 (1983).[15] M. Saraceno, Ann. Phys. 199, 37 (1990).[16] A. Peres, in Quantum Chaos, edited by H. A. Cerdeira, R. Ramaswamy, M. C.Gutzwiller, and G. Casati (World Scientific, Singapore, 1991), p. 73.[17] W. K. Wootters, Foundations of Physics 20, 1365 (1990).[18] K. R. W. Jones, J. Phys. A 23, L1247 (1990).[19] I. C. Percival, in Quantum Chaos, Quantum Measurement, edited by P. Cvitanovic´, I.Percival, and A. Wirzba (Kluwer, Dordrecht, 1992), p. 199.[20] C. M. Caves, “Information and entropy”, submitted to Phys. Rev. E.8

Hypersensitivity to Perturbations in the Quantum Baker's Map

Hypersensitivity to Perturbations in the Quantum Baker's Map

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