---
title: Quantum Open Baker's Maps
url: https://www.emergentmind.com/topics/quantum-open-baker-s-maps
type: topic
---

# Quantum Open Baker's Maps

Quantum open baker’s maps are finite-dimensional non-unitary quantizations of classical open baker dynamics on the torus. They function as analytically tractable models of open quantum chaos and of scattering resonances: the quantum propagator has spectrum in the unit disk, eigenvalues of large modulus are interpreted as long-lived resonances, and the dominant semiclassical structures are determined by the classical trapped set, typically a product Cantor set of dimension \(\delta=\frac{\log|\mathcal A|}{\log M}\). The subject has developed along three closely connected lines: precise operator constructions for open baker dynamics, spectral-gap results based on fractal uncertainty principles, and fractal Weyl laws controlling the number of eigenvalues in annuli; later work has extended the framework to continuous openings, decohered projections, and anisotropic higher-dimensional models [1608.02238] [2202.10591] [2602.20392].

## 1. Canonical construction of the open quantum baker map

A standard formulation starts from a triple
\[
(M,\mathcal A,\chi), \qquad M\in\mathbb N,\quad \mathcal A\subset\{0,\dots,M-1\},\quad \chi\in C_0^\infty((0,1);[0,1]).
\]
Here \(M\) is the base of the baker map, \(\mathcal A\) is the alphabet of allowed branches, and \(\chi\) is a smooth cutoff used to localize in the allowed region. In one common normalization one assumes \(N=KM\) and works on \(\ell_N^2=\ell^2(\mathbb Z_N)\), while in the fractal uncertainty principle setting one often takes \(N=M^k\) [2202.10591] [1608.02238].

The quantum open baker’s map is
\[
B_N=\mathcal F_N^* \begin{pmatrix} \chi_{N/M}\mathcal F_{N/M}\chi_{N/M} & & \\ & \ddots & \\ & & \chi_{N/M}\mathcal F_{N/M}\chi_{N/M} \end{pmatrix} I_{\mathcal A,M},
\]
where \(\mathcal F_N\) is the unitary discrete Fourier transform on \(\ell_N^2\), \(\chi_{N/M}\) is the discretized cutoff, and \(I_{\mathcal A,M}\) is the diagonal projector selecting the branches indexed by \(\mathcal A\). Equivalently,
\[
B_N=\sum_{a\in\mathcal A}\mathcal F_N^*\Pi_a^*\chi_{N/M}\mathcal F_{N/M}\chi_{N/M}\Pi_a.
\]
This operator is the discrete quantum analogue of the open classical baker map on \(\mathbb T^2\) [2202.10591].

The underlying classical map is
\[
\varkappa_{M,\mathcal A}:(y,\eta)\mapsto (x,\xi)=\Big(My-a,\frac{\eta+a}{M}\Big),
\]
defined on the strips
\[
\left(\frac aM,\frac{a+1}{M}\right)\times(0,1), \qquad a\in\mathcal A.
\]
Thus the openness is implemented by allowing only the branches indexed by \(\mathcal A\), with the cutoff \(\chi\) smoothing the corresponding phase-space localization [1608.02238].

This finite-dimensional construction is the canonical model for open quantum baker dynamics. It is non-unitary by design, and \((2\pi N)^{-1}\) plays the role of the semiclassical parameter \(h\) [2202.10591].

## 2. Trapped sets, resonances, and semiclassical organization

The central classical invariant is the trapped set. For the open baker map it is described in terms of the limiting Cantor set
\[
\mathcal C_\infty=\bigcap_k \bigcup_{j\in\mathcal C_k} \Big[\frac j{M^k},\frac{j+1}{M^k}\Big],
\]
where
\[
\mathcal C_k = \Big\{ \sum_{j=0}^{k-1} a_j M^j \;:\; a_j\in\mathcal A \Big\}\subset\mathbb Z_N.
\]
The forward and backward trapped sets are
\[
\Gamma_+=\{x\in(0,1),\ \xi\in\mathcal C_\infty\}, \qquad \Gamma_-=\{\xi\in(0,1),\ x\in\mathcal C_\infty\},
\]
and the full trapped set is \(\Gamma=\Gamma_+\cap\Gamma_-\). Its fractal dimension is
\[
\delta=\frac{\log|\mathcal A|}{\log M}\in(0,1).
\]
This \(\delta\) is the basic exponent in the fractal Weyl law [1608.02238].

The quantum spectrum lies in the unit disk since \(\|B_N\|\le 1\). The eigenvalues are interpreted as model resonances, with the correspondence
\[
\lambda=e^{-i\omega\log M}=M^{-i\omega},
\]
so that a strip \(\{\Im \omega \ge -\nu\}\) corresponds to an annulus \(\{|\lambda|\ge M^{-\nu}\}\). The associated counting function is
\[
\mathcal N_N(\nu)=\left|\operatorname{Spec}(B_N)\cap\{|\lambda|\ge M^{-\nu}\}\right|,
\]
with multiplicities. These are the eigenvalues not too small in modulus, i.e. the long-lived part of the non-unitary spectrum [2202.10591].

A recurrent semiclassical heuristic is that long-lived quantum states should localize near the trapped set, so the number of such states should be proportional to the number of semiclassical states that can fit near a set of dimension \(\delta\), namely \(\sim N^\delta\). Later rigorous results show that this heuristic captures the correct leading exponent for broad classes of annuli, but not the full structure of spectral gaps and counting exponents [2202.10591] [2602.20392].

## 3. Spectral gaps and the fractal uncertainty principle

A major development was the introduction of a discrete fractal uncertainty principle for open quantum baker’s maps. Dyatlov–Jin study the operator
\[
r_k=\|\mathbf 1_{\mathcal C_k}\mathcal F_N\mathbf 1_{\mathcal C_k}\|_{\ell^2_N\to\ell^2_N}
\]
and define the fractal uncertainty exponent
\[
\beta(M,\mathcal A) = -\lim_{k\to\infty}\frac{\log r_k}{k\log M}.
\]
The key estimate is
\[
\|\mathbf 1_{\mathcal C_k}\mathcal F_N\mathbf 1_{\mathcal C_k}\|_{\ell^2_N\to\ell^2_N} \le C_\varepsilon N^{-\beta+\varepsilon},
\]
and it leads to an essential spectral gap
\[
\limsup_{N\to\infty}\max\{|\lambda|:\lambda\in\Sp(B_N)\}\le M^{-\beta},
\]
for some
\[
\beta>\max\Big(0,\frac12-\delta\Big).
\]
This strictly improves the standard pressure bound for all \(\delta\in(0,1)\), including the regime \(\delta>\tfrac12\) where the pressure bound is trivial [1608.02238].

The same work also connects spectral improvement to additive combinatorics. Using additive energy, one obtains a lower bound of the form
\[
\beta\ge \frac34\Big(\frac12-\delta\Big)+\frac{\gamma}{8},
\]
provided the additive energy of the discrete Cantor sets satisfies the stated decay condition. This shows that the spectral gap is not determined by \(\delta\) alone: arithmetic structure matters [1608.02238].

This point became sharper in later work. "Improved fractal Weyl bounds matching improved spectral gaps for hyperbolic surfaces and open quantum maps" [2602.20392] proves an improved fractal Weyl bound for quantum open baker’s maps that matches both the improved FUP gap \(\beta\) and the additive-energy gap \(\beta_E\). The resulting exponent
\[
m(\nu,\delta)=\min \left(4(\nu-\beta),\,4(\nu-\beta_E),\,2\Bigl(\nu-\bigl(\tfrac{1}{2}-\delta\bigr)\Bigr),\,\delta\right)
\]
shows explicitly that improved resonance-free annuli and improved resonance counting are governed by the same fractal-analytic mechanisms [2602.20392].

The FUP framework has also been extended to genuinely anisotropic settings. For the 2D anisotropic quantum open baker’s map, with base \(\mathbf M=(M_1,M_2)\), \(M_1>M_2\ge 2\), and trapped set a Bedford–McMullen carpet, one has an anisotropic discrete FUP
\[
\|\mathds{1}_{\mathcal Y_k}\mathcal F_{\mathbf N}\mathds{1}_{\mathcal X_k}\|_{\ell^2_{\mathbf N}\to \ell^2_{\mathbf N}} \le C N_2^{-\beta}
\]
under non-full-row or non-full-column hypotheses, and consequently an essential spectral gap
\[
\limsup_{N\to\infty}\max\{|\lambda|:\lambda\in \operatorname{Sp}(B_{\mathbf N})\} \le M_2^{-\beta}.
\]
This extends the 1D discrete FUP strategy to self-affine trapped sets with anisotropic scaling [2606.23160].

## 4. Fractal Weyl laws and eigenvalue counting in annuli

The counting problem asks how many eigenvalues lie in the annulus \(\{|\lambda|\ge M^{-\nu}\}\). Dyatlov–Jin proved that for every \(\nu>0\) and \(\varepsilon>0\),
\[
\mathcal N_k(\nu)=\mathcal O\big(N^{m(\delta,\nu)+\varepsilon}\big), \qquad m(\delta,\nu)=\min(2\nu+2\delta-1,\ \delta).
\]
For \(\nu\ge \frac{1-\delta}{2}\), this gives the standard fractal Weyl exponent \(\delta\); for smaller \(\nu\), the exponent decreases linearly and vanishes at the pressure gap threshold \(\nu=\frac12-\delta\) [1608.02238].

A sharper result was obtained in "Weyl Laws for Open Quantum Maps" [2202.10591]. For each fixed \(\nu>0\),
\[
\mathcal N_N(\nu)=\mathcal O(N^\delta), \qquad N=KM\to\infty.
\]
This removes the previous \(\epsilon\)-loss and improves the earlier \(\mathcal O(N^{\delta+\epsilon})\) bound to the sharp exponent \(\delta\). If the cutoff has Gevrey regularity,
\[
\chi\in \mathcal G^s_{\mathrm c((0,1))}, \qquad s>1,
\]
then for all \(\nu\ge 1\) and all sufficiently large \(N=KM\),
\[
\mathcal N_N(\nu)\le C\,N^\delta\,\nu^{(1-\delta)s}.
\]
The Gevrey case therefore yields explicit dependence on the annulus depth \(\nu\) [2202.10591].

The proof strategy has become structurally standard. It begins with nonstationary phase estimates and one-step propagation of singularities, which imply that \(B_N\) pushes mass toward the next Cantor-like strip while \(B_N^*\) propagates in reverse. Iterating this propagation yields an approximate inverse
\[
I=Z(\lambda)(B_N-\lambda)+\mathcal R(\lambda)+A,
\]
where \(A\) is a finite-rank localizer onto a neighborhood of the trapped set. The rank estimate for \(A\) is precisely where \(\delta\) enters. Choosing propagation time \(\ell\sim \log_M N\) makes the rank scale like \(N^\delta\), and Jensen’s formula applied to a determinant built from the parametrix converts this rank control into eigenvalue counting [2202.10591].

The 2026 refinement modifies the determinant stage. Instead of the older determinant, it uses
\[
F(\lambda):=\det\!\bigl(I-B(\lambda)^4\bigr),
\]
which allows sharper trace-class estimates and brings the improved FUP and additive-energy exponents directly into the counting argument. A common misconception is that trapped-set dimension alone fixes the optimal counting exponent in every annulus; the improved results show that the fine fractal arithmetic encoded in \(\beta\) and \(\beta_E\) can further sharpen the bound near the spectral gap [2602.20392].

## 5. Continuous openings, tribaker maps, and periodic-orbit organization

Not all open baker models use a fully absorbing opening. In the continuously open quantum tribaker map, the opening is described by a reflectivity function
\[
F_R(q,p)\in[R,1],
\]
so trajectories are partially reflected rather than completely removed. The quantum map is
\[
\widetilde U = G_N^{-1} P\, G_{N/3} P,
\]
and long-lived resonances are those with \(|z_j|\) close to \(1\). The corresponding classical object is a continuous repeller obtained from finite-time forward and backward trapped intensity distributions rather than a strict escape/no-escape set [1711.03852].

Two reflectivity profiles were studied: a Fermi-Dirac-like step smoothing and a sinusoidal reflectivity. The central semiclassical conclusion is that the shortest periodic orbits belonging to the classical repeller of the fully open map remain robust in a perturbative regime and continue to support the long-lived resonances. Scar functions built from such short periodic orbits form an efficient nonorthogonal basis, and the overlap between the exact quantum continuous-repeller distribution and the semiclassical one satisfies
\[
O>0.99
\]
in all tested cases. For step-like reflectivity, the number of scar functions needed is significantly reduced, similarly to the completely open situation; for sinusoidal reflectivity, the reduction is less pronounced and the spectral behavior deviates more strongly from the discontinuous case [1711.03852].

The same work also emphasizes that continuous openings alter spectral scaling qualitatively. The strong oscillations typical of discontinuous openings in the scaling of the number of long-lived resonances are almost absent, and the authors suggest this may indicate a different Weyl-law regime. Thus the fully open repeller remains the organizing semiclassical structure in the perturbative regime, but continuous reflectivity changes the detailed resonance statistics [1711.03852].

## 6. Adjacent open-system formulations and operator-theoretic viewpoints

A closely related construction is the quantum Bernoulli map, defined as a projection of the quantum baker map with instant decoherence in the position basis after each step:
\[
\rho_{\tau+1}=U_B^Q\rho_\tau=\left[T\rho_\tau T^\dagger\right]_{\rm diagonal}.
\]
This produces a dissipative, non-unitary effective map on density matrices. The paper constructs quantum decaying states represented by density matrices, derives decay laws
\[
(U_B^Q)^\tau B_\alpha \approx 2^{-\alpha\tau} B_\alpha,
\]
and shows that the evolving quantum Bernoulli polynomials develop a quasi-fractal structure down to the resolution scale
\[
\Delta q \sim \frac1N \sim \hbar.
\]
The construction is explicitly framed as conceptually close to the broader class of open quantum baker maps, since the openness comes from repeated decoherence and irreversible coarse-graining rather than from a spatial hole [1110.5122].

Another operator-theoretic viewpoint comes from truncations of the closed quantum baker propagator. In the OTOC formulation with projector observables, the projected evolution
\[
\tilde{U}^t = P(0)\,U^t\,P(0)
\]
is a subunitary truncated matrix, and
\[
f(t)=\sum_{i=1}^{J}\mu_i(t)\bigl(1-\mu_i(t)\bigr)
\]
is determined by the squared singular values \(\mu_i(t)\) of the truncated propagator. This is exactly the structure familiar from open quantum baker maps: a projector turns unitary evolution into a truncated non-unitary operator whose singular values encode contraction, escape, and scrambling [1810.12029].

A more semiclassical but technically adjacent representation appears in the transfer-matrix approach to baker traces. There the baker transfer matrix \(M\) is non-unitary, while the reflected baker transfer matrix \(M'\) is exactly unitary; in the circuit representation, the nonunitarity of \(M\) is restricted to a single one-qubit gate
\[
\tilde H=\frac{1}{\sqrt2} \begin{pmatrix} 1&1\\ 1&x \end{pmatrix}.
\]
Although this is not itself an open baker-map construction, it isolates a controlled nonunitary core inside a baker-map circuit and provides a useful comparison point for open and truncated models [1004.1626].

Taken together, these adjacent formulations clarify that “openness” in baker systems can be realized in several mathematically distinct ways: by deleting branches in phase space, by continuous reflectivity, by repeated decoherence, or by projector-induced truncation. The shared themes are nonunitarity, resonance-like decay, and localization on dynamically defined fractal or symbolic structures.

Source: https://www.emergentmind.com/topics/quantum-open-baker-s-maps