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.
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…
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 t…
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…
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 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…
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.
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 …
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…
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 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 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 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…
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…
A unified fidelity-based framework for quantum speed limits in open quantum systems
Instituto de Física Enrique Gaviola (IFEG-UNC-CONICET), Universidad Nacional de Córdoba（恩里克·加维亚多物理研究所，科尔多瓦国立大学）；Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación（科尔多瓦国立大学数学、天文学、物理与计算学院）；Instituto de Física La Plata (IFLP, CONICET-UNLP)（拉普拉塔物理研究所，国家科学研究委员会，拉普拉塔国立大学）；Universidad Nacional de La Plata. Facultad de Ciencias Exactas（拉普拉塔国立大学理学院）；Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)（国家科学研究和技术理事会）
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 $ν$ …
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.
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 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…
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…
QmDFT for Polycyclic Aromatics: Balancing Embedding Ground-State Fidelity and Experimental Gap Estimation
Centre for Development of Advanced Computing (C-DAC)（高级计算发展中心）；Shah and Anchor Kutchhi Engineering College（沙阿和安卡奇希工程学院）；SRM University - AP（SRM大学-安得拉邦校区）
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.
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 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 …
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 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 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(…
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 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 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 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.
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 …
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.
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…
Here, we show that, although quantum fidelity can truly identify two quantum phase transitions of a one-dimensional spin-1/2 quantum Ising model with competing nearest and next-nearest neighbour interactions in a transverse magnetic field, it may not be a suitable approach for analyzing its ground-state phase diagram.
We study fidelity susceptibility in one-dimensional asymmetric Hubbard model, and show that the fidelity susceptibility can be used to identify the universality class of the quantum phase transitions in this model. The critical exponents are found to be 0 and 2 for cases of half-filling and away from half-filling respectively.
In the present work, we investigate the intrinsic relation between quantum fidelity susceptibility (QFS) and the dynamical structure factor. We give a concise proof of the QFS beyond the perturbation theory. With the QFS in the Lehmann representation, we point out that the QFS actually the negative-two-power moment of dynamical structure factor, and illuminate the inherent relation between physical quantities in the linear response theory. Moreover, we discuss the generalized fidelity susceptibility (GFS) of any quantum relevant operator, that may not be coupled to the driving parameter, prese…
Finite-size resource scaling for learning quantum phase transitions with fidelity-based support vector machines
Dipartimento Interateneo di Fisica, Università di Bari（巴里大学跨校物理系）；INFN, Sezione di Bari（意大利国家核物理研究所巴里分部）；International Centre for Theory of Quantum Technologies, University of Gdańsk（格但斯克大学量子技术理论国际中心）；Dipartimento Interateneo di Fisica, Politecnico di Bari（巴里理工大学跨校物理系）
Matrix product states, a key ingredient of numerical algorithms widely employed in the simulation of quantum spin chains, provide an intriguing tool for quantum phase transition engineering. At critical values of the control parameters on which their constituent matrices depend, singularities in the expectation values of certain observables can appear, in spite of the analyticity of the ground state energy. For this class of generalized quantum phase transitions we test the validity of the recently introduced fidelity approach, where the overlap modulus of ground states corresponding to slight…
We study the critical properties of the Lipkin-Meshkov-Glick Model in terms of the fidelity susceptibility. By using the Holstein-Primakoff transformation, we obtain explicitly the critical exponent of the fidelity susceptibility around the second-order quantum phase transition point. Our results provide a rare analytical case for the fidelity susceptibility in describing the universality class in quantum critical behavior. The different critical exponents in two phases are non-trivial results, indicating the fidelity susceptibility is not always extensive.
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…
We study the fidelity susceptibility in an antiferromagnetic spin-1 XXZ chain numerically. By using the density-matrix renormalization group method, the effects of the alternating single-site anisotropy $D$ on fidelity susceptibility are investigated. Its relation with the quantum phase transition is analyzed. It is found that the quantum phase transition from the Haldane spin liquid to periodic Néel spin solid can be well characterized by the fidelity. Finite size scaling of fidelity susceptibility shows a power-law divergence at criticality, which indicates the quantum phase transition is of…
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.
We study the fidelity approach to quantum phase transitions (QPTs) and apply it to general thermal phase transitions (PTs). We analyze two particular cases: the Stoner-Hubbard itinerant electron model of magnetism and the BCS theory of superconductivity. In both cases we show that the sudden drop of the mixed state fidelity marks the line of the phase transition. We conduct a detailed analysis of the general case of systems given by mutually commuting Hamiltonians, where the non-analyticity of the fidelity is directly related to the non-analyticity of the relevant response functions (susceptib…
We put forth a hitherto unexplored control strategy that enables finite-speed, high-fidelity transport of a quantum wavepacket through a low-temperature dissipative medium. The control consists in confining the wavepacket within a shallow anharmonic trap (tweezer), whose nonuniform velocity is steered so as to maximize the transfer fidelity between two locations. A relevant scenario is a quantum impurity moving through an ultracold gas. Unlike shortcuts to adiabaticity, our approach can simultaneously cope with wavepacket leakage via non-adiabatic and phonon-mediated processes, provided both a…
The fidelity of two pure states (also known as transition probability) is a symmetric function of two operators, and well-founded operationally as an event probability in a certain preparation-test pair. Motivated by the idea that the fidelity is the continuous quantum extension of the combinatorial equality function, we enquire whether there exists a symmetric operational way of obtaining the fidelity. It is shown that this is impossible. Finally, we discuss the optimal universal approximation by a quantum operation.
