Introduction

Several recent papers1,2,3,4,5,6,7 have considered nonlinear generalizations of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation8,9 for qudits. The superoperators resulting from these evolutions each take the form of a positive trace-preserving (PTP) channel10,11 \(X \mapsto \phi (X)/\textrm{tr}[\phi (X)]\), with X a density matrix and \(\phi \) a positive map. The positivity of this normalized PTP channel follows from the positivity of \(\phi \) and \(\textrm{tr}[\phi (X)] > 0\). It’s trace preservation property is actually a trace fixing one, but these are physically equivalent when applied to normalized initial states. Kowalski and Rembieliński1, and also Rembieliński and Caban3, considered cases with linear \(\phi \) and \(\textrm{tr}[\phi (X)] \ne 1\), extending Gisin’s 1981 model12 to mixed states. We call these channels nonlinear in normalization only (NINO) to emphasize that the nonlinearity in this case serves only to conserve trace. We might think of NINO channels as being “mildly” nonlinear. In particular, they satisfy a convex quasilinearity property13, preventing superluminal signaling14,15,16,17,18. The main difference between linear completely positive trace-preserving (CPTP) and NINO channels are that the generators of linear CPTP evolution are negative definite, leading to strictly nonexpansive dynamics, whereas NINO channels support non-CP19,20,21,22 and entropy decreasing1 processes that amplify the Bloch vector1,6. Hence we can interpret the NINO master equation as extending the GKSL equation to non-Hermitian Hamiltonians. Fernengel and Drossel2 studied cases where \(\phi \) is nonlinear and \(\textrm{tr}[\phi (X)]=1\), a family of state-dependent CPTP channels obtained by adding state-dependence to a Hamiltonian and set of Lindblad jump operators. This is a stronger form of nonlinearity, supporting rich dynamical phenomena such as such Hopf bifurcations and strange attractors usually associated with classical nonlinear systems2. State-dependent CPTP channels also support Bloch-ball torsion. Torsion can be created from the product of an SO(3) rotation generator \(J_\mu \) with the projection of the Bloch vector along the twist axis. Abrams and Lloyd23 and Childs and Young24 investigated state discrimination with z-axis torsion. Kłobus et al. 7 observed Feigenbaum’s universal period doubling in a mean field model simulating torsion. Torsion also arises in a qubit friendly extension25 of a rigorous duality between nonlinear mean field theory and the BBGKY hierarchy for n interacting bosons in the \(n \rightarrow \infty \) limit26,27,28,29,30. Many of these nonlinear models come from mean field theory.

Methods

In this paper we investigate qubit PTP channels with both nonlinear \(\phi \) and \(\textrm{tr}[\phi (X)] \ne 1\) that support generalized Lorenz attractors. The first version, which we call Lor63, implements Lorenz’s 1963 model31

$$\begin{aligned} \frac{dx}{dt} \, = \, & {} \sigma (y-x), \end{aligned}$$
(1)
$$\begin{aligned} \frac{dy}{dt}\,=\, & {} \rho x - y - g x z, \end{aligned}$$
(2)
$$\begin{aligned} \frac{dz}{dt} \, = \, & {} - \beta z + g x y, \end{aligned}$$
(3)

where \(\textbf{r} = (x,y,z) = \textrm{tr}(X {\varvec{\sigma }})\) is the Bloch vector. However here we increase the nonlinearity by a factor of \(g \gg 1\) to shrink the attractor sufficiently as to contain it within the Bloch sphere. The master equation for the Lor63 qubit in the Pauli basis is

$$\begin{aligned} \frac{dX}{dt} = \frac{\sigma ^a }{2} \bigg ( \frac{dr^a}{dt} \bigg ), \ \ \frac{dr^a}{dt} = \textrm{tr} \bigg ( \! \frac{dX}{dt} \sigma ^a \!\bigg ) \! = G^{a b}(\textbf{r}) \, r^b = ( L + g x J_x)^{a b} r^b \!, \end{aligned}$$
(4)

where \(a, b \in \{ 1,2,3 \}\) and

$$\begin{aligned} L = \begin{pmatrix} -\sigma &{} \sigma &{} 0 \\ \rho &{} -1 &{} 0 \\ 0 &{} 0 &{} -\beta \\ \end{pmatrix} \! = L_{+} + L_{-}, \ \ L_{+} \! = \! \begin{pmatrix} -\sigma &{} \frac{\rho + \sigma }{2} &{} 0 \\ \frac{\rho + \sigma }{2} &{} -1 &{} 0 \\ 0 &{} 0 &{} -\beta \\ \end{pmatrix} \! = \big ( { \frac{\rho + \sigma }{2}} \big ) \, \lambda _1 - D, \end{aligned}$$
(5)
$$\begin{aligned} L_{-} \! \, =\, \! \big ( \frac{\rho - \sigma }{2} \big ) \, J_z, \ \ \lambda _1 \! = \! \begin{pmatrix} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{pmatrix} \!, \ \ D \! = \! \begin{pmatrix} \sigma &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} \beta \\ \end{pmatrix} \!, \ \ J_{x} \! = \! \begin{pmatrix} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} -1 \\ 0 &{} 1 &{} 0 \\ \end{pmatrix} \!, \ \ J_{z} \! = \! \begin{pmatrix} 0 &{} -1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{pmatrix} \!. \ \ \ \ \ \end{aligned}$$
(6)

\(X \in {\mathbb C^{2 \times 2}}\) is a Hermitian positive-semidefinite matrix with unit trace. Model parameters \(\rho \), \(\sigma \), \(\beta \), g are given in Table 1. The nonlinear generator \(G^{ab}(\textbf{r})\) is a \(3 \times 3\) real matrix that depends on the Bloch vector \(\textbf{r}\). We decompose it into a linear (\(\textbf{r}\) independent) operator L plus x-axis torsion. The \(J\)'s are SO(3) generators: \( (J_a)_{bc} = - \varepsilon _{abc}\) with \(\varepsilon \) the Levi-Civita symbol. L is decomposed into symmetric and antisymmetric parts implementing a non-Hermitian Hamiltonian iL. \(\lambda _1\) is a Gell-Mann matrix. Note that \(\lambda _1\) has a positive eigenvalue corresponding to an amplifying and entropy decreasing non-CP process1. Techniques for constructing Gell-Mann matrices and other symmetric generators from jump operators are given in6.

Table 1 Lor63 model parameters.
Full size table

Result and discussion

The Lor63 qubit is simulated in Fig. 1. The blue points indicate random initial conditions. Trajectories rapidly approach one of the two disc-shaped sets (pink or cyan) and bounce back and forth between them in an unpredictable manner, mirroring the aperiodic reversals of the Malkus waterwheel lying in its Fourier representation32.

Figure 1
figure 1

Bloch ball dynamics of the Lor63 qubit. The faint yellow wireframe shows the Bloch sphere. Equator states \(\left| \pm \right\rangle = 2^{-1/2} (\left| 0 \right\rangle \pm \left| 1 \right\rangle )\) and \(\left| \pm i \right\rangle = 2^{-1/2} (\left| 0 \right\rangle \pm i \left| 1 \right\rangle )\) are also indicated with black dots and cyan lines. The model parameters used in the simulation are given in Table 1.

Full size image
Figure 2
figure 2

Bloch ball dynamics of the GP butterfly qubit. Blue dots indicate random initial conditions.

Full size image

A Lorenz-like attractor can also be created from the z-axis torsion coming from the Gross-Pitaevskii (GP) equation24,33,34, leading to an aesthetic attracting set shown in Fig. 2. We call this channel the GP butterfly. The GP butterfly qubit has an especially simple master equation:

$$\begin{aligned} \frac{dr^a}{dt} = \textrm{tr} \bigg ( \! \frac{dX}{dt} \sigma ^a \!\bigg ) \! = G^{a b}(\textbf{r}) \, r^b = ( m \lambda _4 + g z J_z)^{a b} r^b \!, \end{aligned}$$
(7)

where \(m=10\), \(g=40\), and

$$\begin{aligned} \lambda _4 = \begin{pmatrix} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 \\ \end{pmatrix} \end{aligned}$$
(8)

is another Gell–Mann matrix. The symmetric generator \(\lambda _4\) can be implemented with Lindblad jump operators6.

Conclusion

In conclusion, we have proposed nonlinear PTP channels for the generation of Lorenz-like attractors in the Bloch ball. Despite its early prominence the Lorenz system defied rigorous analysis until rather recently when, in 2002, Tucker35 established the existence of a strange attractor. Classical electrical circuits have been used to implement the Lorenz attractor and other chaotic and hyperchaotic attractors36,37,38,39, which might find cryptographic application40,41. It is tempting to speculate that chaotic attractors will find application in quantum technology as well. However it is important to recognize the very large nonlinear coupling strengths required, making experimental realization especially challenging.