Rank $n$ swapping algebra for $operatorname{PGL}_n$ Fock-Goncharov $mathcal{X}$ moduli space

The {em rank $n$ swapping algebra} is a Poisson algebra defined on the set of ordered pairs of points of the circle using linking numbers, whose geometric model is given by a certain subspace of $(mathbb{K}^n times mathbb{K}^{n*})^r/operatorname{GL}(n,mathbb{K})$. For any ideal triangulation of $D_k$---a disk with $k$ points on its boundary, using determinants, we find an injective Poisson algebra homomorphism from the fraction algebra generated by the Fock--Goncharov coordinates for $mathcal{X}_{operatorname{PGL}_n,D_k}$ to the rank $n$ swapping multifraction algebra for $r=kcdot(n-1)$ with respect to the (Atiyah--Bott--)Goldman Poisson bracket and the swapping bracket. This is the building block of the general surface case. Two such injective Poisson algebra homomorphisms related to two ideal triangulations $mathcal{T}$ and $mathcal{T}$ are compatible with each other under the flips.

Swapping algebra, Virasoro algebra and discrete integrable system

We induce a Poisson algebra ${cdot,cdot}_{mathcal{C}_{n,N}}$ on the configuration space $mathcal{C}_{n,N}$ of $N$ twisted polygons in $mathbb{RP}^{n-1}$ from the swapping algebra cite{L12}, which is found coincide with Faddeev-Takhtajan-Volkov algebra for $n=2$. There is another Poisson algebra ${cdot,cdot}_{S2}$ on $mathcal{C}_{2,N}$ induced from the first Adler-Gelfand-Dickey Poissson algebra by Miura transformation. By observing that these two Poisson algebras are asymptotically related to the dual to the Virasoro algebra, finally, we prove that ${cdot,cdot}_{mathcal{C}_{2,N}}$ and ${cdot,cdot}_{S2}$ are Schouten commute.

Rank $n$ swapping algebra for the $operatorname{PSL}(n, mathbb{R})$ Hitchin component

F. Labourie [arXiv:1212.5015] characterized the Hitchin components for $operatorname{PSL}(n, mathbb{R})$ for any $n>1$ by using the swapping algebra, where the swapping algebra should be understood as a ring equipped with a Poisson bracket. We introduce the rank $n$ swapping algebra, which is the quotient of the swapping algebra by the $(n+1)times(n+1)$ determinant relations. The main results are the well-definedness of the rank $n$ swapping algebra and the cross-ratio in its fraction algebra. As a consequence, we use the sub fraction algebra of the rank $n$ swapping algebra generated by these cross-ratios to characterize the $operatorname{PSL}(n, mathbb{R})$ Hitchin component for a fixed $n>1$. We also show the relation between the rank $2$ swapping algebra and the cluster $mathcal{X}_{operatorname{PGL}(2,mathbb{R}),D_k}$-space.

Fisher information under decoherence in Bloch representation

The dynamics of two variants of quantum Fisher information under decoherence are investigated from a geometrical point of view. We first derive the explicit formulas of these two quantities for a single qubit in terms of the Bloch vector. Moreover, we obtain analytical results for them under three different decoherence channels, which are expressed as affine transformation matrices. Using the hierarchy equation method, we numerically study the dynamics of both the two information in a dissipative model and compare the numerical results with the analytical ones obtained by applying the rotating-wave approximation. We further express the two information quantities in terms of the Bloch vector for a qudit, by expanding the density matrix and Hermitian operators in a common set of generators of the Lie algebra $mathfrak{su}(d)$. By calculating the dynamical quantum Fisher information, we find that the collisional dephasing significantly diminishes the precision of phase parameter with the Ramsey interferometry.

Entanglement Dynamics of Two Qubits in a Common Bath

We derive a set of hierarchical equations for qubits interacting with a Lorentz-broadened cavity mode at zero temperature, without using the rotating-wave, Born, and Markovian approximations. We use this exact method to reexamine the entanglement dynamics of two qubits interacting with a common bath, which was previously solved only under the rotating-wave and single-excitation approximations. With the exact hierarchy equation method used here, we observe significant differences in the resulting physics, compared to the previous results with various approximations. Double excitations due to counter-rotating-wave terms are also found to have remarkable effects on the dynamics of entanglement.

Reduced fidelity approach for quantum phase transitions in spin-1/2 dimerized Heisenberg chains

We use reduced fidelity approach to characterize quantum phase transitions in the one-dimensional spin-1/2 dimerized Heisenberg chain in the antiferromagnetic case. The reduced fidelity susceptibilities between two nearest-neighboring spin pairs are considered. We find that they are directly related to the square of the second derivative of the ground-state energy. This enables us to conclude that the former might be a more effective indicator of the second-order quantum phase transitions than the latter. Two further exemplifications are given to confirm the conclusion is available for a broad class of systems with SU(2) and translation symmetries. Moreover, a general connection between reduced fidelity susceptibility and quantum phase transitions is illustrated.

Operator fidelity susceptibility, decoherence and quantum criticality

The extension of the notion of quantum fidelity from the state-space level to the operator one can be used to study environment-induced decoherence. state-dependent operator fidelity sucepti- bility (OFS), the leading order term for slightly different operator parameters, is shown to have a nontrivial behavior when the environment is at critical points. Two different contributions to OFS are identified which have distinct physical origins and temporal dependence. Exact results for the finite-temperature decoherence caused by a bath described by the Ising model in transverse field are obtained.

Operator fidelity susceptibility: an indicator of quantum criticality

We introduce the operator fidelity and propose to use its susceptibility for characterizing the sensitivity of quantum systems to perturbations. Two typical models are addressed: one is the transverse Ising model exhibiting a quantum phase transition, and the other is the one dimensional Heisenberg spin chain with next-nearest-neighbor interactions, which has the degeneracy. It is revealed that the operator fidelity susceptibility is a good indicator of quantum criticality regardless of the system degeneracy.

Disentanglement in a quantum critical environment

We study the dynamical process of disentanglement of two qubits and two qutrits coupled to an Ising spin chain in a transverse field, which exhibits a quantum phase transition. We use the concurrence and negativity to quantify entanglement of two qubits and two qutrits, respectively. Explicit connections between the concurrence (negativity) and the decoherence factors are given for two initial states, the pure maximally entangled state and the mixed Werner state. We find that the concurrence and negativity decay exponentially with fourth power of time in the vicinity of critical point of the environmental system.