Stability of quantum chaos against weak non-unitarity

TL;DR

The paper investigates how fixed, weakly non-unitary quantum dynamics, implemented as $T=\boldsymbol\zeta U$ with Haar-random $U$, influence chaotic signatures and information scrambling. By examining radial eigenvalue statistics, azimuthal spectral correlations via the spectral form factor, and purification dynamics, it shows that purification is exponentially slow in system size due to a ring-like eigenvalue distribution with a sharp outer edge and an exponentially small radial gap, while azimuthal level repulsion and a ramp in the averaged SFF persist up to a timescale $t_*$ that is itself exponential in $N$. Yamamoto's theorem links late-time singular values to eigenvalues, underpinning the connection between spectral properties and purification dynamics. The results reveal how chaotic spectral statistics survive in a non-unitary setting and offer insights into tensor-network contraction complexity and monitored dynamics, including nontrivial limit-noncommutativity between weak non-unitarity and system size.

Abstract

We study the quantum dynamics generated by the repeated action of a non-unitary evolution operator on a system of qubits. Breaking unitarity can lead to the purification of mixed initial states, which corresponds to the loss of sensitivity to initial conditions, and hence the absence of a key signature of dynamical chaos. However, the scrambling of quantum information can delay purification to times that are exponential in system size. Here we study purification in systems whose evolution operators are fixed in time, where all aspects of the dynamics are in principle encoded in spectral properties of the evolution operator for a single time step. The operators that we study consist of global Haar random unitary operators and non-unitary single-qubit operations. We show that exponentially slow purification arises from a distribution of eigenvalues in the complex plane that forms a ring with sharp edges at large radii, with the eigenvalue density exponentially large near these edges. We argue that the sharp edges of the eigenvalue distribution arise from level attraction along the radial direction in the complex plane. By calculating the spectral form factor we also show that there is level repulsion around the azimuthal direction, even close to the outer edge of the ring of eigenvalues. Our results connect this spectral signature of quantum chaos to the sensitivity of the system to its initial conditions.

Figures (11)

• Figure 1: Calculations of the averages (a,b) $\langle\text{Tr}[(\tilde{T}(t)\tilde{T}^\dagger(t))^2]\rangle$ and (c,d) $\langle\text{Tr}[\tilde{T}(t)\tilde{T}^\dagger(t)]^2\rangle$ using Eq. \ref{['eq:Weingarten']}. Here we show the dominant pairs of permutations, which have $\sigma=\tau$. In each diagram, $\circ$ and $\bullet$ represent basis states $i_m,i_m^*$ and $j_m,j_m^*$ (corresponding to row and column indices, respectively, of $U$ and $U^*$) in Eq. (\ref{['eq:Weingarten']}), and each horizontal bond corresponds to $\zeta$. Solid and dashed vertical lines represent the permutations $\sigma$ and $\tau$, with red and yellow indicating the identity and two-replica swap, respectively. Domain walls, indicated by thick arrows, appear when the permutations change across a layer of $\zeta$, leading to higher moments $\text{Tr}[\zeta^{2n}]$ with $n>1$. (a,b) Diagrams with one (a) and three (b) domain walls. After summing over the allowed locations of domain walls, the overall contributions from these classes of diagrams are $t\text{Wg}(1^2)^{t-1}\text{Tr}[\zeta^2]^{2t-2}\text{Tr}[\zeta^4]$ and $\frac{t^3}{6}\text{Wg}(1^2)^{t-1}\text{Tr}[\zeta^2]^{2t-6}\text{Tr}[\zeta^4]^3$, respectively. (c,d) Diagrams with zero (c) and two (d) domain walls. The overall contributions from these classes of diagrams are $\text{Wg}(1^2)^{t-1}\text{Tr}[\zeta^2]^{2t}$ and $\frac{t^2}{2}\text{Wg}(1^2)^{t-1}\text{Tr}[\zeta^2]^{2t-4}\text{Tr}[\zeta^4]^2$, respectively.
• Figure 2: Schematics of next-to-leading pairs of permutations, with $\sigma=\tau$, involving non-equal-time pairings in the calculation of (a) $\langle\text{Tr}[(T^t(T^\dagger)^t)^2]\rangle$ and (b) $\langle\text{Tr}[T^t(T^\dagger)^t]\rangle$. Factors of $\text{Tr}[\zeta^4]$ arise from the configurations, indicated by thick arrows, where adjacent basis states ($\circ$ and $\bullet$ linked by a bond) are connected by distinct sets of equal- (black) or non-equal-time (colors other than black) pairings. In (a) the equal-time pairings over time intervals $t_1,t_2,t_3$ and $t_4$ are as shown in Fig. \ref{['fig:Random']}(b) (although here the diagram is 'unfolded'), while non-equal-time pairings generate three factors of $\text{Tr}[\zeta^4]$. The overall contribution from this class of diagrams, generated by summing over $t_1,\ldots,t_4$, is $\frac{t^5}{30}\text{Wg}(1^{2t-2})\text{Tr}[\zeta^2]^{2t-6}\text{Tr}[\zeta^4]^3$. (b) Non-equal-time pairings generate two factors of $\text{Tr}[\zeta^4]$. The overall contribution from this class of diagrams is $\frac{t^4}{24}\text{Wg}(1^{t-1})\text{Tr}[\zeta^2]^{t-4}\text{Tr}[\zeta^4]^2$.
• Figure 3: Purification of maximally mixed initial state under the model $T(t)=T^t$ with time-translation invariance. (a) Ensemble-averaged second Rényi entropy versus $t$ at several $N$ and $h=0.05$. (b) Ensemble-averaged second Rényi entropy versus $Dt/t_*^2$ at several $h$ and $N=12$. In both panels, the black curves underneath represents early-time prediction of Eq. (\ref{['eq:EarlyEntropy']}). The dashed lines indicate the time scales $t_*$ where we anticipate deviations from Eq. (\ref{['eq:EarlyEntropy']}) due to time-translation invariance and, in panel (a), their equal separation on a logarithmic scale reflects the fact that $t_*$ is exponential in $N$.
• Figure 4: (a) Ratio of averages $\braket{\Delta_\sigma(t)}/\braket{\Delta_\rho}$, with $\Delta_{\sigma}\equiv t^{-1}[\sigma_0(t)-\sigma_1(t)]$ and $\Delta_{\rho}=\rho_0-\rho_1$, as a function of $t/t_*$ for various system sizes $N=8,9,10,11,12$ (increasing opacity) and values of $h$. As $t$ increases and exceeds $t_*$, the singular value gap converges to the eigenvalue gap, as required by Yamamoto's theorem. (b) The eigenspectral gap of Eq. (\ref{['eq:1']}) is exponentially small in $N$. The dashed lines show $\langle\Delta_\rho\rangle = A t_*^{-1}[\ln t_*]^{-1/2}$ [Eq. (\ref{['eq:Gap']})] with the coefficient $A \approx 0.4$.
• Figure 5: Left: eigenvalue distribution of $T=\zeta U$ within a ring whose outer and inner radii are respectively $e^{\pm\rho_N}=[\cosh^N(2h)]^{\pm1/2}$ (dashed). The unit circle is shown solid. Right: schematics of radial attraction between eigenvalues in non-unitary Dyson Brownian motion.