We apply the fidelity metric approach to analyze two recently introduced models that exhibit a quantum phase transition to a topologically ordered phase. These quantum models have a known connection to classical statistical mechanical models; we exploit this mapping to obtain the scaling of the fidelity metric tensor near criticality. The topological phase transitions manifest themselves in divergences of the fidelity metric across the phase boundaries. These results provide evidence that the fidelity approach is a valuable tool to investigate novel phases lacking a clear characterization in terms of local order parameters.
The notion of fidelity susceptibility, introduced within the context of quantum metric tensor, has been an important quantity to characterize the criticality near quantum phase transitions. We demonstrate that for topological phase transitions in Dirac models, provided the momentum space is treated as the manifold of the quantum metric, the fidelity susceptibility coincides with the curvature function whose integration gives the topological invariant. Thus the quantum criticality of the curvature function near a topological phase transition also describes the criticality of the fidelity suscep…
We present a numerical study of a quantum phase transition from a spin-polarized to a topologically ordered phase in a system of spin-1/2 particles on a torus. We demonstrate that this non-symmetry-breaking topological quantum phase transition (TOQPT) is of second order. The transition is analyzed via the ground state energy and fidelity, block entanglement, Wilson loops, and the recently proposed topological entropy. Only the topological entropy distinguishes the TOQPT from a standard QPT, and remarkably, does so already for small system sizes. Thus the topological entropy serves as a proper …
We perform the fidelity analysis for Boltzmann-Gibbs-like states in order to investigate whether the topological order of 1D fermionic systems at zero temperature is maintained at finite temperatures. We use quantum walk protocols that are known to simulate topological phases and the respective quantum phase transitions for chiral symmetric Hamiltonians. Using the standard approaches of the fidelity analysis and the study of edge states, we conclude that no thermal-like phase transitions occur as temperature increases, i.e., the topological behaviour is washed out gradually. We also show that …
We study the reduced fidelity between local states of lattice systems exhibiting topological order. By exploiting mappings to spin models with classical order, we are able to analytically extract the scaling behavior of the reduced fidelity at the corresponding quantum phase transitions out of the topologically ordered phases. Our results suggest that the reduced fidelity, albeit being a local measure, generically serves as a faithful marker of a topological quantum phase transition.
Quantum phase transitions that take place between two distinct topological phases remain an unexplored area for the applicability of the fidelity approach. Here, we apply this method to spin systems in two and three dimensions and show that the fidelity susceptibility can be used to determine the boundary between different topological phases particular to these models, while at the same time offering information about the critical exponent of the correlation length. The success of this approach relies on its independence on local order parameters or breaking symmetry mechanisms, with which non…
In this article, we study the high order term of the fidelity of the Heisenberg chain with next-nearest-neighbor interaction and analyze its connection with quantum phase transition of Beresinskii-Kosterlitz-Thouless type happened in the system. We calculate the fidelity susceptibility of the system and find that although the phase transition point can't be well characterized by the fidelity susceptibility, it can be effectively picked out by the higher order of the ground-state fidelity for finite-size systems.
The fidelity susceptibility and entanglement entropy in a system of two-leg $XXZ$ spin ladder with rung coupling is investigated by using exact diagonalization of the system. The effects of rung coupling on fidelity susceptibility, entanglement entropy and quantum phase transition are analyzed. It is found that the quantum phase transition between two different $XY$ phases can be well characterized by the fidelity susceptibility. Though the quantum phase transition from $XY$ phase to rung singlet phase can be hardly detected by fidelity susceptibility, it can be predicted by the first derivati…
The global-state fidelity cannot characterize those quantum phase transitions (QPTs) induced by continuous level crossing due to its collapse around each crossing point. In this paper, we take the isotropic Lipkin-Meshkov-Glick (LMG) model and the antiferromagnetic one-dimensional Heisenberg model as examples to show that the partial-state fidelity can signal such level-crossing QPTs. Extending to the thermodynamic limit we introduce the partial-state fidelity susceptibility and study its scaling behavior. The maximum of the partial-state fidelity susceptibility goes like $N$ for the LMG model…
We propose a new approach to quantum phase transitions in terms of the density-functional fidelity, which measures the similarity between density distributions of two ground states in parameter space. The key feature of the approach, as we will show, is that the density-functional fidelity can be measured easily in experiments. Both the validity and versatility of the approach are checked by the Lipkin-Meshkov-Glick model and the one-dimensional Hubbard model.
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…
We study the behaviour of the fidelity and the Uhlmann connection in two-dimensional systems of free fermions that exhibit non-trivial topological behavior. In particular, we use the fidelity and a quantity closely related to the Uhlmann factor in order to detect phase transitions at zero and finite temperature for topological insulators and superconductors. We show that at zero temperature both quantities predict quantum phase transitions: a sudden drop of fidelity indicates an abrupt change of the spectrum of the state, while the behavior of the Uhlmann connection signals equally rapid chang…
It is proved that a qubit encoded in excited states of a V-type quantum system cannot be perfectly transferred to the state of the cavity field mode using a single rectangular laser pulse. This obstacle can be overcome by using a two-stage protocol, in which the fidelity of a state-mapping operation can be increased to nearly one.
The fidelity of quantum cloning is very often limited by the accompanying unwanted transitions. We show how the fidelity can be improved by using a coherent field to cycle away the unwanted transitions. We demonstrate this explicitly in the context of the model of Simon {\it et al.} [J. Mod. Opt. {\bf 47}, 233 (2000) ; Phys. Rev. Lett. {\bf 84}, 2993 (2000)]. We also investigate the effects of the number of atoms on the quality of quantum cloning. We show that the universality of the scheme can be maintained by choosing the cycling field according to the input state of the qubit.
Driving a certain transition without including undesired transitions is an ubiquitous problem in quantum control and the implementation of quantum information processing. This problem gets the more challenging the weaker the desired transition couples to the control field, and the denser the system's spectrum is. With the explicit example of a trapped ion we show how temporally shaped driving helps to increase the fidelity of a gate operation beyond the regular spectral selection of resonantly driven transitions. We chose the explicit example of side-band transitions, since those couple more w…
Fast and high-fidelity transfer of edge states via dynamical control of topological phases and effects of dissipation
Hokkaido University（北海道大学）；National Institute of Technology, Asahikawa College（国立长春冈技术短期大学）；RIKEN（理化学研究所）；Ritsumeikan University（立命馆大学）；Osaka University（大阪大学）；The University of Tokyo（东京大学）
We propose the generation of entangled qubits by utilizing the properties of edge states appearing at one end of a periodically driven (Floquet) superconducting qubit chain. Such qubits are naturally protected by the system's topology and their manipulation is possible through adiabatic control of the system parameters. By utilizing a Y-junction geometry, we then develop a protocol to perform high-fidelity transfer of entangled qubits from one end to another end of a qubit chain. Our quantum state transfer protocol is found to be robust against disorder and imperfection in the system parameter…
We study the quantum fidelity approach to characterize thermal phase transitions. Specifically, we focus on the mixed-state fidelity induced by a perturbation in temperature. We consider the behavior of fidelity in two types of second-order thermal phase transitions (based on the type of non-analiticity of free energy), and we find that usual fidelity criteria for identifying critical points is more applicable to the case of $λ$ transitions (divergent second derivatives of free energy). Our study also reveals limitations of the fidelity approach: sensitivity to high temperature thermal fluctua…
The operator fidelity is a measure of the information-theoretic distinguishability between perturbed and unperturbed evolutions. The response of this measure to the perturbation may be formulated in terms of the operator fidelity susceptibility (OFS), a quantity which has been used to investigate the parameter spaces of quantum systems in order to discriminate their regular and chaotic regimes. In this work we numerically study the OFS for a pair of non-linearly coupled two-dimensional harmonic oscillators, a model which is equivalent to that of a hydrogen atom in a uniform external magnetic f…
The fidelity and entanglement entropy in an antiferromagnetic-ferromagnetic alternating Heisenberg chain are investigated by using the method of density-matrix renormalization-group. The effect of anisotropy on fidelity and entanglement entropy are investigated. The relations between fidelity, entanglement entropy and quantum phase transition are analyzed. It is found that the quantum phase transition point can be well characterized by both the ground-state entropy and fidelity for large system.
Fidelity approach to quantum phase transitions uses the overlap between ground states of the system to gain some information about its quantum phases. Such an overlap is called fidelity. We illustrate how this approach works in the one dimensional quantum Ising model in the transverse field. Several closed-form analytical expressions for fidelity are discussed. An example of what insights fidelity provides into the dynamics of quantum phase transitions is carefully described. The role of fidelity in central spin systems is pointed out.
The Uhlmann connection is a mixed state generalisation of the Berry connection. The latter has a very important role in the study of topological phases at zero temperature. Closely related, the quantum fidelity is an information theoretical quantity which is a measure of distinguishability of quantum states. Moreover, it has been extensively used in the analysis of quantum phase transitions. In this work, we study topological phase transitions in 1D and 2D topological superconductors with long-range hopping and pairing amplitudes, using the fidelity and the quantity $Δ$ closely related to the …
The fidelity and entropy in an easy-axis antiferromagnetic Heisenberg spin-1 chain are studied numerically. By using the method of density-matrix renormalization group, the effects of anisotropy on fidelity and entanglement entropy are investigated. Their relations with quantum phase transition are analyzed. It is found that the quantum phase transition from the Haldane spin liquid to Néel spin solid can be well characterized by the fidelity. The phase transition can be hardly detected by the entropy but it can be successfully detected by the first deviation of the entropy.
We derive a general formula of the reduced fidelity susceptibility when the reduced density matrix is $2\times2$ block-diagonal. By using this result and the continuous unitary transformations, we study finite-size scaling of the reduced fidelity susceptibility in the Lipkin-Meshkov-Glick Model. It is found that it can be used to characterize quantum phase transitions, implying that we can extract information of quantum phase transitions only from the fidelity of a subsystem, which is of practical meaning in experiments.
For an extended Harper model, the fidelity for two lowest band edge states corresponding to different model parameters, the fidelity susceptibility and the von Neumann entropy of the lowest band edge states, and the spectrum-averaged von Neumann entropy are studied numerically, respectively. The fidelity is near one when parameters are in the same phase or same phase boundary; otherwise it is close to zero. There are drastic changes in fidelity when one parameter is at phase boundaries. For fidelity susceptibility the finite scaling analysis performed, the critical exponents $α$, $β$, and $ν$ …
Fidelity Strange Correlators for Average Symmetry-Protected Topological Phases
The Pennsylvania State University（宾夕法尼亚州立大学）；University of Colorado, Boulder（科罗拉多大学博尔德分校）；Fudan University（复旦大学）；Collaborative Innovation Center of Advanced Microstructures（先进微结构协同创新中心）
We study the quantum phase transitions in the two-dimensional spin-orbit models in terms of fidelity susceptibility and reduced fidelity susceptibility. An order-to-order phase transition is identified by fidelity susceptibility in the two-dimensional Heisenberg XXZ model with Dzyaloshinsky-Moriya interaction on a square lattice. The finite size scaling of fidelity susceptibility shows a power-law divergence at criticality, which indicates the quantum phase transition is of second order. Two distinct types of quantum phase transitions are witnessed by fidelity susceptibility in Kitaev-Heisenbe…
Motivated by recent development in quantum fidelity and fidelity susceptibility, we study relations among Lie algebra, fidelity susceptibility and quantum phase transition for a two-state system and the Lipkin-Meshkov-Glick model. We get the fidelity susceptibility for SU(2) and SU(1,1) algebraic structure models. From this relation, the validity of the fidelity susceptibility to signal for the quantum phase transition is also verified in these two systems. At the same time, we obtain the geometric phase in these two systems in the process of calculating the fidelity susceptibility. In additio…
We present a detailed study of the fidelity, the entanglement entropy, and the entanglement spectrum, for a dimerized chain of spinless fermions---a simplified Su-Schrieffer-Heeger (SSH) model---with open boundary conditions which is a well-known example for a model supporting a symmetry protected topological (SPT) phase. In the non-interacting case the Hamiltonian matrix is tridiagonal and the eigenvalues and -vectors can be given explicitly as a function of a single parameter which is known analytically for odd chain lengths and can be determined numerically in the even length case. From a s…
The fidelity approach to the Gaussian transitions in spin-one XXZ spin chains with three different values of Ising-like anisotropy lambda is analyzed by means of the density matrix renormalization group (DMRG) technique for systems of large sizes. We find that, despite the success in the cases of lambda=2.59 and 1, the fidelity susceptibility fails to detect the Gaussian transition for lambda=0.5. Thus our results demonstrate the limitation of the fidelity susceptibility in characterizing quantum phase transitions, which was proposed recently in general frameworks.
We present some aspects of the fidelity approach to phase transitions based on lower and upper bounds on the fidelity susceptibility that are expressed in terms of thermodynamic quantities. Both commutative and non commutative cases are considered. In the commutative case, in addition, a relation between the fidelity and the nonequilibrium work done on the system in a process from an equilibrium initial state to an equilibrium final state has been obtained by using the Jarzynski equality.
We review briefly the quantum fidelity approach to quantum phase transitions in a pedagogical manner. We try to relate all established but scattered results on the leading term of the fidelity into a systematic theoretical framework, which might provide an alternative paradigm for understanding quantum critical phenomena. The definition of the fidelity and the scaling behavior of its leading term, as well as their explicit applications to the one-dimensional transverse-field Ising model and the Lipkin-Meshkov-Glick model, are introduced at the graduate-student level. In addition, we survey als…
Topological insulators and topological superconductors display various topological phases that are characterized by different Chern numbers or by gapless edge states. In this work we show that various quantum information methods such as the von Neumann entropy, entanglement spectrum, fidelity, and fidelity spectrum may be used to detect and distinguish topological phases and their transitions. As an example we consider a two-dimensional $p$-wave superconductor, with Rashba spin-orbit coupling and a Zeeman term. The nature of the phases and their changes are clarified by the eigenvectors of the…
When a system undergoes a quantum phase transition, the ground-state wave-function shows a change of nature, which can be monitored using the fidelity concept. We introduce two Quantum Monte Carlo schemes that allow the computation of fidelity and its susceptibility for large interacting many-body systems. These methods are illustrated on a two-dimensional Heisenberg model, where fidelity estimators show marked behaviours at two successive quantum phase transitions. We also develop a scaling theory which relates the divergence of the fidelity susceptibility to the critical exponent of the corr…
We evaluate the Uhlmann fidelity between two one-mode displaced thermal states as the maximal probability transition between appropriate purifications of the given states. The optimal purifications defining the fidelity are proved to be two-mode displaced Gaussian states.
Fidelity zeros and Lee-Yang theory of quantum phase transitions
College of Physics, Nanjing University of Aeronautics and Astronautics（南京航空航天大学物理学院）；Key Laboratory of Aerospace Information Materials and Physics (NUAA), MIIT（工业和信息化部航空航天信息材料与物理重点实验室）
We study the energy level crossings of the states and thermal fidelity for a two-qubit system in the presence of a transverse and inhomogeneous magnetic field. It is shown clearly the effects of the anisotropic factor of the magnetic field through the contour figures of energy level crossing in two subspaces, the isotropy subspace and anisotropy subspace. We calculate the quantum fidelity between the ground state and the state of the system at temperature $T$, and the results show the strong effect of the anisotropic factor again. In addition, by making use of the transition of Yangian generat…
We have applied the real space quantum renormalization group approach to study the topological quantum phase transition in the one-dimensional chain of a spinless p-wave superconductor. We investigate the behavior of local compressibility and ground-state fidelity of the Kitaev chain. We show that the topological phase transition is signaled by the maximum of local compressibility at the quantum critical point tuned by the chemical potential. Moreover, a sudden drop of the ground-state fidelity and the divergence of fidelity susceptibility at the topological quantum critical point have been us…
We analyze the scaling behavior of the fidelity, and the corresponding susceptibility, emerging in finite-size many-body systems whenever a given control parameter $λ$ is varied across a quantum phase transition. For this purpose we consider a finite-size scaling (FSS) framework. Our working hypothesis is based on a scaling assumption of the fidelity in terms of the FSS variables associated to $λ$ and to its variation $δλ$. This framework entails the FSS predictions for continuous transitions, and meanwhile enables to extend them to first-order transitions, where the FSS becomes qualitatively …
We present a method for achieving high fidelity state preparation and measurement (SPAM) using trapped ion hyperfine qubits with nuclear spins higher than $I = 1/2$. The ground states of these higher nuclear spin isotopes do not afford a simple frequency-selective state preparation scheme. We circumvent this limitation by stroboscopically driving strong and weak transitions, blending fast optical pumping using dipole transitions and narrow microwave or optical quadrupole transitions. We demonstrate this method with the $I=3/2$ isotope $^{137}\mbox{Ba}^+$ to achieve a SPAM infidelity of $\left(…
Mean fidelity amplitude and parametric energy--energy correlations are calculated exactly for a regular system, which is subject to a chaotic random perturbation. It turns out that in this particular case under the average both quantities are identical. The result is compared with the susceptibility of chaotic systems against random perturbations. Regular systems are more susceptible against random perturbations than chaotic ones.
We study the fidelity susceptibility of two SU(2)-invariant reduced density matrices. Due to the commuting property of these matrices, analytical results for reduced fidelity susceptibility are obtained and can be applied to study quantum phase transitions in SU(2)-invariant systems. As an example, we analyze the quantum criticality of the spin-one bilinear-biquadratic model via the fidelity approach.
We study the pinning quantum phase transition in a Tonks-Girardeau gas, both in equilibrium and out-of-equilibrium, using the ground state fidelity and the Loschmidt echo as diagnostic tools. The ground state fidelity (GSF) will have a dramatic decrease when the atomic density approaches the commensurate density of one particle per lattice well. This decrease is a signature of the pinning transition from the Tonks to the Mott insulating phase. We study the applicability of the fidelity for diagnosing the pinning transition in experimentally realistic scenarios. Our results are in excellent agr…
We describe a scheme that enables a strong Jaynes-Cummings coupling between a topological qubit and a superconducting flux qubit. The coupling strength is dependent on the phase difference between two superconductors on a topological insulator and may be expediently controlled by a phase controller. With this coherent coupling and single-qubit rotations arbitrary unitary operations on the two-qubit hybrid system of topological and flux qubits can be performed. Numerical simulations show that quantum state transfer and entanglement distributing between the topological and superconducting flux q…
A general relation between quantum phase transitions and the second derivative of the fidelity (or the "fidelity susceptibility") is proposed. The validity and the limitation of the fidelity susceptibility in characterizing quantum phase transitions is thus established. Moreover, based on the bosonization method, general formulas of the fidelity and the fidelity susceptibility are obtained for a class of one-dimensional gapless systems known as the Tomonaga-Luttinger liquid. Applying these formulas to the one-dimensional spin-1/2 $XXZ$ model, we find that quantum phase transitions, even of the…
We compute the fidelity between the ground states of general quadratic fermionic hamiltonians and analyze its connections with quantum phase transitions. Each of these systems is characterized by a $L\times L$ real matrix whose polar decomposition, into a non-negative $Λ$ and a unitary $T$, contains all the relevant ground state (GS) information. The boundaries between different regions in the GS phase diagram are given by the points of, possibly asymptotic, singularity of $Λ$. This latter in turn implies a critical drop of the fidelity function. We present general results as well as their exe…
We introduce a partial state fidelity approach to quantum phase transitions. We consider a superconducting lattice with a magnetic impurity inserted at its centre, and look at the fidelity between partial (either one-site or two-site) quantum states. In the vicinity of the point of the quantum phase transition, we observe a sudden drop of the fidelity between two one-site partial states corresponding to the impurity location and its close vicinity. In the case of two-site states, the fidelity reveals the transition point as long as one of the two electron sites is located at the impurity, whil…
Quantum fidelity between two density matrices, $F(ρ_1,ρ_2)$ is usually defined as the trace of the operator ${\cal F}=\sqrt{\sqrt{ρ_1} ρ_2 \sqrt{ρ_1}}$. We study the logarithmic spectrum of this operator, which we denote by {\it fidelity spectrum}, in the cases of the $XX$ spin chain in a magnetic field, a magnetic impurity inserted in a conventional superconductor and a bulk superconductor at finite temperature. When the density matrices are equal, $ρ_1=ρ_2$, the fidelity spectrum reduces to the entanglement spectrum. We find that the fidelity spectrum can be a useful tool in giving a detaile…
We characterize excited state quantum phase transitions in the two dimensional limit of the vibron model with the quantum fidelity susceptibility, comparing the obtained results with the information provided by the participation ratio. As an application, we perform fits using a four-body algebraic Hamiltonian to bending vibrational data for several molecular species and, using the optimized eigenvalues and eigenstates, we locate the eigenstate closest to the barrier to linearity and determine the linear or bent character of the different overtones.
In this paper we analyze the ground state phase diagram of a class of fermionic Hamiltonians by looking at the fidelity of ground states corresponding to slightly different Hamiltonian parameters. The Hamiltonians under investigation can be considered as the variable range generalization of the fermionic Hamiltonian obtained by the Jordan-Wigner transformation of the XY spin-chain in a transverse magnetic field. Under periodic boundary conditions, the matrices of the problem become circulant and the models are exactly solvable. Their free-ends counterparts are instead analyzed numerically. In …
In this paper, we explore the differences between classical logarithmic fidelity and quantum fidelity. The classical logarithmic fidelity is found to be always extensive while the quantum one manifests distinct size dependence in different phases. Illustrated by the anisotropic Lipkin-Meshkov-Glick model, we found numerically and analytically that the logarithmic fidelity scales like N in the symmetry-broken phase and scales like N^0 in the polarized phase. The singular behavior around the critical point is also investigated.
We study finite-temperature Dynamical Quantum Phase Transitions (DQPTs) by means of the fidelity and the interferometric Loschmidt Echo (LE) induced metrics. We analyse the associated dynamical susceptibilities (Riemannian metrics), and derive analytic expressions for the case of two-band Hamiltonians. At zero temperature the two quantities are identical, nevertheless, at finite temperatures they behave very differently. Using the fidelity LE, the zero temperature DQPTs are gradually washed away with temperature, while the interferometric counterpart exhibits finite-temperature Phase Transitio…
We study the dynamics and readout of topological qubits encoded by zero-energy Majorana bound states in a topological superconductor. We take into account bosonic modes due to the electromagnetic environment which couple the Majorana manifold to above-gap continuum quasi-particles. This coupling causes the degenerate ground state of the topological superconductor to be dressed in a polaron-like manner by quasi-particle states and bosons, and the system to become gapless. Topological protection and hence full coherence is only maintained if the qubit is operated and read out within the low-ener…
Transition Probability (fidelity) for pairs of density operators can be defined as "functor" in the hierarchy of "all" quantum systems and also within any quantum system. The introduction of "amplitudes" for density operators allows for a more intuitive treatment of these quantities, also pointing to a natural parallel transport. The latter is governed by a remarkable gauge theory with strong relations to the Riemann-Bures metric.
