We study unitary evolution of bipartite entanglement in a circuit with nearest-neighbor random gates. Deriving a compact nonunitary description of purity dynamics on qudits we find a sudden transition in the purity relaxation rate the origin of which is in the underlying boundary localized eigenmodes---the skin effect. We provide the full solution of the problem, being one of the simplest iterations of two-site matrices, namely, that each is a sum of only two projectors. This leads to rich dynamics influenced by the Jordan normal form of the kernel and, most importantly, a spectrum that is completely discontinuous in the thermodynamic limit. It provides a simple example of how a seemingly innocuous many-body unitary evolution can harbor interesting mathematical effects: an effective nonsymmetric Toeplitz transfer matrix description causes a phantom relaxation, such that the correct relaxation rate is not given by the matrix spectrum, but rather by its pseudospectrum.
We study unitary evolution of bipartite entanglement in a circuit with nearest-neighbor random gates. Deriving a compact nonunitary description of purity dynamics on qudits we find a sudden transition in the purity relaxation rate the origin of which is in the underlying boundary localized eigenmodes---the skin effect. We provide the full solution of the problem, being one of the simplest iterations of two-site matrices, namely, that each is a sum of only two projectors. This leads to rich dynamics influenced by the Jordan normal form of the kernel and, most importantly, a spectrum that is completely discontinuous in the thermodynamic limit. It provides a simple example of how a seemingly innocuous many-body unitary evolution can harbor interesting mathematical effects: an effective nonsymmetric Toeplitz transfer matrix description causes a phantom relaxation, such that the correct relaxation rate is not given by the matrix spectrum, but rather by its pseudospectrum.
Phantom relaxation is relaxation with a rate that is not given by a finite spectral gap. Studying the average purity dynamics in a staircase random Haar circuit and the spectral decomposition of a nonsymmetric matrix describing the underlying Markovian evolution, we explain how that can arise out of an ordinary-looking spectrum. Crucial are alternating expansion coefficients that diverge in the thermodynamic limit due to the non-Hermitian skin effect in the matrix describing the average purity dynamics under an overall unitary evolution. The mysterious phantom relaxation emerges out of localized generalized eigenvectors describing the Jordan normal form kernel, and, independently, also out of interesting trigonometric sums due to localized true eigenvectors. All this shows that when dealing with non-Hermitian matrices it can happen that the spectrum is not the relevant object; rather, it is the pseudospectrum or, equivalently, a delicate cancellation enabled by localized eigenvectors.
We study purity decay---a measure of bipartite entanglement---in a chain of n qubits under the action of various geometries of nearest-neighbor random two-site unitary gates. We use a Markov chain description of average purity evolution, using further reduction to obtain a transfer matrix of only polynomial dimension in n. In most circuits, an exception being the brick-wall configuration, purity decays to its asymptotic value in two stages: the initial thermodynamically relevant decay persisting up to extensive times is $\ensuremath{\sim}{\ensuremath{\lambda}}_{\mathrm{eff}}^{t}$, with ${\ensuremath{\lambda}}_{\mathrm{eff}}$ not necessarily being in the spectrum of the transfer matrix, while the ultimate asymptotic decay is given by the second largest eigenvalue ${\ensuremath{\lambda}}_{2}$ of the transfer matrix. The effective rate ${\ensuremath{\lambda}}_{\mathrm{eff}}$ depends on the location of bipartition boundaries as well as on the geometry of applied gates.
We study random quantum circuits and their rate of producing bipartite entanglement, specifically with respect to the choice of 2-qubit gates and the order (protocol) in which these are applied.The problem is mapped to a Markovian process, and we prove that there are large spectral equivalence classes-different configurations have the same spectrum.Optimal gates and the protocol that generate entanglement with the fastest theoretically possible rate are identified.Relaxation towards the asymptotic thermal entanglement proceeds via a series of phase transitions in the local relaxation rate, which is a consequence of non-Hermiticity.In particular, non-Hermiticity can cause the rate to be either faster or, even more interestingly, slower than predicted by the matrix eigenvalue gap.This result is caused by expansion coefficients that grow exponentially with system size, resulting in a "phantom" eigenvalue and is due to nonorthogonality of non-Hermitian eigenvectors.We numerically demonstrate that the phenomenon also occurs in random circuits with nonoptimal generic gates, random U(4) gates, and also without spatial or temporal randomness, suggesting that it could be of wide importance in other non-Hermitian settings, including correlations.
We study random quantum circuits and their rate of producing bipartite entanglement, specifically with respect to the choice of 2-qubit gates and the order (protocol) in which these are applied. The problem is mapped to a Markovian process, and we prove that there are large spectral equivalence classes—different configurations have the same spectrum. Optimal gates and the protocol that generate entanglement with the fastest theoretically possible rate are identified. Relaxation towards the asymptotic thermal entanglement proceeds via a series of phase transitions in the local relaxation rate, which is a consequence of non-Hermiticity. In particular, non-Hermiticity can cause the rate to be either faster or, even more interestingly, slower than predicted by the matrix eigenvalue gap. This result is caused by expansion coefficients that grow exponentially with system size, resulting in a “phantom” eigenvalue and is due to nonorthogonality of non-Hermitian eigenvectors. We numerically demonstrate that the phenomenon also occurs in random circuits with nonoptimal generic gates, random U(4) gates, and also without spatial or temporal randomness, suggesting that it could be of wide importance in other non-Hermitian settings, including correlations.
Abstract We want to understand how relaxation process from an initial non-generic state proceeds towards a long-time typical state reached under unitary quantum evolution. One would expect that after some initial correlation time relaxation will be a simple exponential decay with constant decay rate. We show that this is not necessarily the case. Studying various Floquet systems with fixed two-qubit gates, and focusing on purity and out-of-time-ordered correlation functions, we find that in many situations relaxation proceeds in two phases of exponential decay having different relaxation rates. Namely, in the thermodynamic limit the relaxation rate exhibits a change at a critical time proportional to system’s size. The initial thermodynamically relevant rate can be slower or faster than the asymptotic one, demonstrating that the recently discovered phantom relaxation, in which the decay is slower than predicted by a nonzero transfer matrix gap, is not limited to only random circuits.
The non-Hermitian skin effect (NHSE), characterized by a macroscopic accumulation of eigenstates at the edge of a system with open boundaries, is often ascribed to a non-trivial point-gap topology of the Bloch Hamiltonian. We revisit this connection and separate the question of the NHSE as a spectral reconstruction effect in clean systems from the question of stable topological protection. For a Hatano-Nelson ladder, where point-gap winding and non-normality can be varied independently, we demonstrate that, in a clean translationally invariant multiband setting, the NHSE can occur without point-gap winding and, conversely, that point-gap winding can persist without the NHSE. These results establish that in the clean case the connection between point-gap winding and the NHSE only holds in the scalar one-band case but, in general, not in the multiband case. Even more importantly, the eigenspectrum of non-normal operators is generically highly sensitive to boundary conditions and perturbations, and therefore does not constitute a stable object encoding topological information. Instead, topological properties are reflected in the splitting of the singular-value spectrum for finite systems and, in the semi-infinite limit, correspond to boundary-localized kernel modes implied by the index of the corresponding Toeplitz operator.