Quantized baker map

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Leonardo Ermann and Marcos Saraceno (2012), Scholarpedia, 7(12):9860. doi:10.4249/scholarpedia.9860 revision #129387 [link to/cite this article]
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Curator: Leonardo Ermann

Quantized baker map

The baker's transformation is one of the simplest models where all features of chaos are present. It was devised by E. Hopf in 1937 in the context of ergodic theory [1]. It is defined geometrically, just like the horseshoe map and the cat map. Unlike other simple chaotic models coming from Hamiltonian systems as the Chirikov Standard map and Kicked Harper model all its classical structures can be described analytically. The quantum version of the map is used to explore the relationship between classical and quantum mechanics in the semiclassical limit, when the dynamics is chaotic (i.e. quantum chaos). The relative simplicity of the classical description - provided by symbolic dynamics - and the elegant and flexible quantization scheme devised by Balazs and Voros allow very detailed analytical and/or numerical studies of central open questions in the semiclassical description of chaotic systems, like the commutativity of quantization with time propagation and the comparison of classical and quantum invariant structures [2],[3],[4]. This scheme is sufficiently general to be applied to most piecewise linear maps like the tri-baker and the non-dissipative horseshoe map. When implemented on tensor product Hilbert spaces (in particular on a system of qubits) it provides a simple implementable model of unitary propagation in the circuit model of quantum computation. It has also been used to study the emergence of fractal Weyl laws in the quantum theory of open systems.


The Classical Map

Figure 1: The geometrical action of the baker's map on a basket of bread in the unit square phase space.

The phase space coordinates \(p,q\in[0,1)\times[0,1)\) are transformed as \begin{eqnarray} p^\prime&=&\frac{p+[2q]}{2}\\ q^\prime&=&2q-[2q] \end{eqnarray} where the bracket indicates the integer part. The action of the map is that of the baker compressing the dough (in \(p\)) and stretching it (in \(q\)). The additional action of cutting and putting on top is what brings in the mixing character - and the only non-linear feature - of the transformation. The map is area-preserving, uniformly hyperbolic with Lyapunov exponent \(\lambda=\log{2}\) and with the coordinate axes providing the stable (\(p=\)cst) and unstable (\(q=\)cst) foliations. The definition of the baker transformation can be found in [5].

The classical map can be described geometrically as in Figure 1. In all figures the unit square has coordinates \((q,p)\), where \(q\) and \(p\) are the horizontal and vertical axis respectively.

Symbolic dynamics

The baker map can be easily represented as a Bernoulli shift in a doubly infinite sequence of binaries. If \(p,q\in[0,1)\times[0,1)\) are represented as binaries \[q=\sum_{i=1}^\infty \varepsilon_{i-1} \left(\frac{1}{2}\right)^{i}, \ \ \ p=\sum_{i=1}^\infty \varepsilon_{-i} \left(\frac{1}{2}\right)^{i}\] then the action of the map upon symbols is \[ (p\bullet q)=\ldots\varepsilon_{-2}\varepsilon_{-1}\bullet\varepsilon_{0}\varepsilon_{1}\varepsilon_{2}\ldots \xrightarrow{\mathcal{B}} (p^\prime\bullet q^\prime)=\ldots\varepsilon_{-2}\varepsilon_{-1}\varepsilon_{0}\bullet\varepsilon_{1}\varepsilon_{2}\ldots \]

Periodic orbits

Figure 2: Prime periodic orbits of the baker map. Left panel shows periodic orbits with \(L\leq5\). Right panel corresponds to \(L=6\).

Invariant structures are especially simple to describe in the symbolic picture. Periodic orbits are obviously obtained by infinitely repeating a finite binary pattern \(\nu\) of \(L\) bits \((\ldots\nu\nu\nu\ldots)\). The coordinates \(p\) and \(q\) of the periodic trajectories can be obtained explicitly as follows:

  • An integer number \(n_\nu\) can be associated to a given pattern \(\nu=\varepsilon_0\ldots\varepsilon_{L-1}\) as \(n_\nu=\sum_{i=0}^{L-1}\varepsilon_{i}2^{L-1-i}\) (with \(0\le n_\nu< 2^L\)).

The same can be done for \(n_{\bar{\nu}}\) where \(\bar{\nu}\) is constructed reversing the order of the bits of \(\nu\).

  • The coordinates of the initial periodic point are obtained as \(q=n_\nu/(2^L-1)\) and \(p=n_{\bar{\nu}}/(2^L-1)\).
  • The other points on the trajectory are obtained by cyclical shifts of the bits (i.e. map iterations).

The periodic orbits excluding repetitions (i.e. prime periodic orbits) up to period \(L=6\) and their corresponding patterns \(\nu\) are shown in Figure 2.

Symbolic picture is also useful for describing other structures like homoclinic and heteroclinic manifolds. Description of orbits in the stable and unstable manifold of the periodic orbit \(\nu\) can be represented as \((\)anything \(\nu\nu\nu\ldots)\) and \((\ldots\nu\nu\nu\) anything\()\) respectively. A trajectory is homoclinic to a periodic trajectory if it comes arbitrarily close to it in the distant past and in the far future. Its symbolic dynamics is given by the periodic sequence \(\nu\) broken by a finite nonperiodic \(h\) as \((\ldots\nu\nu\nu h\nu\nu\nu\ldots)\). The same can be done for the heteroclinic case where past and future of the trajectory converge asymptotically to different periodic trajectories as \((\ldots\nu\nu\nu h\nu^\prime\nu^\prime\nu^\prime\ldots)\). Periodic orbits, homoclinic and heteroclinic trajectories all have rational coordinates, while generic chaotic trajectories have irrational coordinates.


The baker map has two basic symmetries

  • Time reversal: An anticanonical symmetry related to the interchange of \(p\) and \(q\) and the reversal \(t\rightarrow -t\). It means geometrically that each trajectory will have a partner reflected on the main diagonal of the square.
  • Parity: Phase space symmetry given by the interchange \(p\rightarrow1-p\) and \(q\rightarrow1-q\).

Periodic orbits can be self-symmetric or appear as distinct symmetric pairs related by parity or time reversal.

Generating function

Figure 3: Allowed regions of the generating function (in blue) for the first three iterates of the map.

Like any symplectic transformation the baker's map can be specified in terms of a generating function which, because of the alignment of the stable and unstable manifolds with the coordinate axes, must be of the mixed type \[ W_{\epsilon_0}(p_1,q_0).=2p_1 q_0-\epsilon_0 p_1 -\epsilon_0 q_0 \] \begin{eqnarray} \frac {\partial W}{\partial p_1}&=&q_1 =2q_0-\epsilon_0\\ \frac {\partial W}{\partial q_0}&=&p_0 =2p_1-\epsilon_0 \end{eqnarray} \(\epsilon_0\) is the leading bit of \(q_0\) (and of \(p_1\)). \(W\) has two allowed components \(W_0, W_1\) and a classically forbidden region.

The great advantage of the map is that this generating function is very simple to iterate \(T\) times leading to the explicit form \[ W^T(p_{_T},q_0).=2^Tp_{_T} q_0-\nu p_T -\nu^\dagger q_0 \] where now \(\nu=\varepsilon_0\ldots\varepsilon_{T-1}\) is the leading string of length \(L=T\) of \(q_0\) and \(\nu^\dagger=\varepsilon_{T-1}\ldots\varepsilon_0\) is the leading string of \(p_T\). The allowed region now has \(2^T\) disconnected pieces of size \(2^{-T}\times2^{-T}\) as shown in Figure 3.

Quantization procedure

There is no unique prescription for the quantization of a finite canonical transformation. Basic requirements are that the resulting map be unitary (if the dynamics is closed), that the symmetries be preserved, and that appropriate boundary conditions are established. Then semiclassical arguments lead to the form \(\exp {\rm i}W/\hbar\) where \(W\) is the generating function of the transformation in appropriate coordinates.

The Balazs-Voros-Saraceno quantization follows this general route [2], [3] and [4]. Quasi periodic boundary conditions both in coordinate and momentum are imposed thus turning the phase space into a torus. \[ \psi(q+1)=e^{2\pi i \chi_q}\psi(q), \ \ \ \ \ \ \tilde{\psi}(p+1)=e^{-2\pi i \chi_p}\tilde{\psi}(p) \] where \(\psi(q)\) and \(\tilde{\psi}(p)\) are the wavefunctions in position and momentum spaces respectively. As a consequence the value of \(\hbar\) can only take discrete values according to \(\hbar=1/(2\pi N)\) where \(N\) is the integer dimension of the Hilbert space. Moreover position and momenta become discrete as \[ \vert q_n\rangle=\left\vert\frac{n+\chi_p}{N}\right\rangle, \ \ \ \ \ \ \vert p_m\rangle=\left\vert\frac{m+\chi_q}{N}\right\rangle \] with \(n,m=0,1,\ldots,N-1\). The transformation kernel is given by the unitary discretized Fourier transform \[ \langle p_m\vert q_n\rangle=\frac{1}{\sqrt{N}}e^{-2\pi i (m+\chi_q)(n+\chi_p)/N}\equiv\left(F_{N}^{\chi_q,\chi_p}\right) \] Two Floquet parameters \(\chi_q,\chi_p\) characterize the quantization procedure. In the semiclassical limit the torus - of unit area - is spanned by \(N\to\infty\) quantum states.

The quantization of the map then proceeds directly (for \(N\)= even), discretizing the generating function in the mixed representation, setting forbidden matrix elements to zero and Fourier transforming back to coordinate representation to obtain the quantum map \[ \boxed{B=\left(F_{N}^{\chi_q,\chi_p}\right)^{-1}\left(\begin{array}{cc} F_{N/2}^{\chi_q,\chi_p}& 0 \\ 0&F_{N/2}^{\chi_q,\chi_p}\end{array} \right)} \] The only choice of Floquet angles for which the map preserves time reversal and parity is given by antiperiodic boundary conditions \(\chi_q=\chi_p=1/2\), and this choice has been retained in almost all applications with the following simplification in notation \(G_N\equiv F_{N}^{1/2,1/2}\). Time reversal is implemented by \(G_N\) followed by complex conjugation and leads to the antiunitary symmetry \(G_N B G_N^{\dagger}=\left(B^{-1}\right)^*\). The parity operator is \(R\equiv-G^2_N\) and commutes with the map.

The map is thus constructed from two non-commuting Fourier matrices and in spite of its simplicity it cannot be diagonalized exactly. Unlike the extensively studied cat maps the eigenvalues are non-degenerate and follow quite well the random matrix Bohigas-Giannoni-Schmit conjecture. Some anomalies occur when \(N\) is proportional to a high power of 2 [6], [7].

Moreover the simplicity of the iterated generating function allows also the direct quantization of the iterates of the map, resulting in the following matrices \begin{eqnarray} W^{(0)} \to B^{(0)}&=& G^{-1}_N G_N ~~~~~~~~~~~~~\text{(identity)}\\ \tag{1} W^{(1)} \to B^{(1)}&=& G^{-1}_N\left( \begin{array}{cc} G_{N/2}&0 \\ 0& G_{N/2} \end{array}\right)\\ W^{(2)} \to B^{(2)}&=& G^{-1}_N\left( \begin{array}{cccc} G_{N/4}&0&0&0 \\ 0&0&G_{N/4}&0 \\ 0&G_{N/4}&0&0 \\ 0&0&0&G_{N/4} \end{array}\right)\\ \cdots &.& \cdots \end{eqnarray} The Fourier blocks in the mixed matrix elements precisely mimic the allowed regions of the mixed generating function in Figure 3 (note that p-axis in the matrix representation is reversed compared to the one in phase-space plots). Provided \(2^T\) divides \(N\) this procedure provides valid quantizations of the iterates of the baker's map respecting unitarity, time reversal and parity symmetries and the discrete nature of coordinate and momenta required for the compact phase space. Because they retain these essential quantum features, but use the classical generating function for their construction they were called semiquantum propagators.

Quantum propagation

Figure 4: Mixed propagators for \(B^1,B^2,B^3\). Top row \(N=128\), bottom row \(N=190\). The phase and amplitude of matrix elements are coded as hue (color) and brightness with zero coded as black (from [8]).

One of the central problems in the semiclassical description of chaotic systems is to understand the differences between classical and quantum evolution and how the accumulation of errors arising from the stationary phase calculation of path integrals spoil the long time description of quantum dynamics in terms of classical elements. For the baker dynamics this question can be given very precise answers testable analytically or numerically. The quantization of the one step map provides the short time propagator which is then iterated exactly by simple matrix multiplication and compared with the direct quantization of the propagated classical map as given by \(B^{(T)}\). One is then probing the non-commutativity of time propagation and quantization,which is an essential feature of generic systems. The main features of the differences can be seen in Figure 4 for the mixed propagators for \(T=1,2,3\). The matrix arrays are plotted with hue and brightness coding the phase and amplitude of each matrix element. The close correspondence with the allowed and forbidden regions in Figure 3 is apparent, with the main difference being the presence of real and decaying amplitudes in the forbidden regions and anomalies along the discontinuities of the map. For longer times the correspondence remains close up to times \(T\approx\log_2 N\), where the allowed regions shrink to pixel size and all correspondence with the classical propagator is lost.

A more detailed analysis of the semiclassical description of the map can be found in [9], [10], [11], [12], [13], [8] and [14].

Spectral properties

Figure 5: Smoothed density of quasienergies calculated exactly (full line) and semiclassically (dotted line). The full spectrum involves \(N=1024\) quasienergies with average spacing \(2\pi/N\) smoothed with a width \(2\pi/5\) (from [9]).

The spectral features of the unitary map consist of quasienergies on the unit circle and the corresponding eigenfunctions \[ B\vert\psi_i\rangle=\mathrm{e}^{\mathrm{i}\theta_i}\vert\psi_i\rangle \] The semiclassical link between classical invariant properties and the quantum spectrum arises from the Gutzwiller trace formula, which provides a semiclassical representation of the density of states as a sum over periodic orbits of the classical system. In our case it takes the form \[ \rho^k_{sc}(\epsilon)=\frac{1}{2\pi}\sum^{\infty}_{T=-\infty}{\rm Tr}(B^T)_{sc} \mathrm{e}^{-\mathrm{i} \epsilon T}\mathrm{e}^{-\frac{1}{2}(T/k)^2} \] where \({\rm Tr}(B^T)_{sc}\) are the traces of the propagator, evaluated semiclassically in terms of periodic orbits as \[ {\rm Tr}(B^T)_{sc}= \sum_{\nu=0}^{2^T-1}\frac{ 2^{T/2}}{2^T-1}\mathrm{e}^{2\pi{\rm i}NS_\nu} \]

Here \(k\) is a smoothing parameter which on the time side of the sum limits the amount of traces involved, while on the quasienergy side it smoothes the spectrum with a Gaussian width \(2\pi/k\). In Figure 5 we show the excellent agreement between the exact and semiclassical densities obtained for a very smoothed spectrum. The broad oscillations in the density are very well explained by the actions and stabilities of a small number of periodic orbits. However, any attempt to increase the resolution up to the point where individual quasienergies are resolved (\(k\approx N\)) will face the fact - inescapable for chaotic maps - that the number of periodic orbits that contribute to the computation of the traces increase with \(T\) as \(\mathrm{e}^{\lambda T}\) with \(\lambda=\log{2}\) in line with the topological entropy of the map. Thus the calculations become rapidly unmanageable, quite independently of issues of accuracy. The problem can be partially alleviated by resummation techniques involving the spectral determinant.

Figure 6: Eigenstates of the quantum baker map in the Husimi representation for \(N=130\).

The relationship of the eigenfunctions with classical structures is best revealed in the Husimi representation, which displays a wave function in phase space. It is defined as \(\vert\langle q,p \vert\psi_i\rangle\vert^2\) where \(\vert q,p \rangle\) is a coherent state centered at \((q,p)\). Figure 6 shows three eigenstates in Husimi representation: on the left with a clear scar of an orbit pair of period three, on the center an irregular eigenfunction, and on the right a state with an enhancement on the homoclinic neighbourhood of the fixed point. The eigenstates of the map have been described in a periodic orbit state basis in [15]

Other quantization schemes

Figure 7: Circuit representation of the semiquantum propagator. Notice the close correspondence with Figure 3. The allowed regions of the generating function now are replaced by Fourier blocks of the appropriate dimensions. Thin lines are qubits.

Although quite natural, the BVS quantization is not the only possible one, nor is there any compelling reason to be preferred to others (an alternative quantization of the baker transformation can be found in [16]). The inverse map is also symplectic, has a generating function, which can be discretized and yields a different unitary matrix, which has all the same classical features, but differs quantum mechanically. Another possibility stems from the fact that \((B^{(1)})^T\neq B^{(T)}\), so that for example the product \(B^{(2)}{B^{(1)}}^\dagger\) is also a one step map that is not the same as \(B^{(1)}\). Of course many other combinations yield a multiplicity of possibilities. To analyze the common features of these different quantizations it is convenient to resort to the language of quantum circuits that arise in the fields of quantum information and quantum computation.

An even dimensional Hilbert space can be considered as the tensor product of a qubit of dimension \(2\) and an \(N/2\)-dimensional Hilbert space. Likewise spaces of dimension \(M2^T\) where the semiquantum propagators are defined are tensor products of \(T\) qubits with a space of dimension \(M\). With this in mind we obtain a very simple circuit representation of the mixed propagators in equation (1) as shown in Figure 7.

By combining these different maps and using some elementary circuit algebra we arrive at the conclusion that all maps in the coordinate representation can be factored as a product of an "essential baker map" common to all the quantizations and an almost diagonal "diffraction kernel" where all the differences reside [7]. The essential baker has a simple matrix propagator but in general its spectral properties are as intractable as the original one. There is one exception when the Hilbert space is a space of qubits of dimension \(2^T\). In that case the essential baker can be exactly diagonalized by a Hadamard transformation, and constitutes the only instance where the full spectral properties are analytically known [6], [17], [18]. When the baker eigenfunctions are expanded in the Hadamard basis very interesting multifractal structure are uncovered [19], [20].

Other directions

The quantum baker map was simulated using wave-optical methods [21] and applying linear optics [22]. Other simulation was proposed in a 3-qubit NMR quantum computer in [23], where the quantum Fourier transform was successfully performed [24]. The efficient realization of the map in terms of quantum gates was shown in [25].

The initial usefulness of the model as a probe for quantum chaos in closed systems (e.g. hypersensitivity to perturbations in [26]) has been significantly extended in several new directions. The main one is the incorporation of techniques of quantum information and issues of decoherence and nonunitary evolution. The fact that the Fourier blocks used in the construction of the map when \(N=2^L\) can be easily factorized into one and two qubit unitary gates allow a very natural representation of the operation of the map in terms of efficient quantum circuits [7]. The generation of entanglement between different components of the circuit has been studied [27], as well as the rate of entropy production when the map is coupled to a diffusive environment. The map has also been used as a model for a complex environment in interaction with simple qubit systems and also with a quantum walker [28].

The quantum baker map was also used in the study of classical and quantum transport properties in a chains of coupled maps [29], [30], [31], [32] and [33].

The other direction where the map and its generalizations has been useful is in the study of open quantum maps as models of chaotic scattering [34], [17], [18], [35] and [36]. The advantage here is that openings that coincide with Markov partitions for the map are easily engineered and lead to tractable numerical and analytical tests for the distribution of resonances in the semiclassical limit. Some analytical results concerning the distribution of resonances according to a fractal Weyl law have been obtained using a version of the map where the Fourier blocks are replaced by Walsh-Hadamard matrices.


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Classical Baker map

Quantum Fourier transform

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