Quantum Jensen-Shannon Divergence Research Articles

Should the multi-layer transportation network structure be reduced?

ABSTRACT The increasing diversity of transportation modes and the rapid expansion of transportation networks present significant challenges for modeling multi-layer comprehensive transportation networks. It is crucial to determine whether aggregating certain layers is a viable option for balancing complexity reduction and information preservation. This decision defines the layered structures and informs subsequent analyses of these networks. Two-dimensional factors, namely topological structures and transportation attributes, are considered to enhance understanding of the similarities among network layers. The relative entropy and the Gini index are employed as metrics to assess information gain or loss resulting from layer aggregation or segregation, guiding decisions on network reduction. Furthermore, an integrated similarity measure, based on the quantum Jensen-Shannon divergence and the Gower distance, is utilized to identify the optimal aggregation sequences. Two real-world transportation networks serve as case studies. Results demonstrate that these transportation networks are more effectively maintained with layer-separated structures, preserving maximum information.

Steered quantum coherence and entropic uncertainty relation in the cluster Ising model

Abstract In this paper, we investigate the cluster Ising model (CIM) via steered quantum coherence (SQC) and entropic uncertainty relation (EUR). We present the behavior of SQC quantified by the L1 norm, relative entropy and quantum Jensen–Shannon divergence. We also demonstrate the properties of EUR in the CIM. In addition, we provide a comparative analysis of these measures and present detailed numerical results.

Entropic uncertainty relations and quantum coherence in the two-dimensional XXZ spin model with Dzyaloshinskii–Moriya interaction

Quantum renormalization group (QRG) is a tractable method for studying the criticalities of one-dimensional (1D) and two-dimensional (2D) many-body systems. By employing the QRG method, we first derive the effective Hamiltonian and QRG equations of a 2D XXZ model with Dzyaloshinskii–Moriya (DM) interaction analytically. The linear-entropy-based uncertainty, the quantum discord (QD), and the multipartite quantum coherence based on the square root of the quantum Jensen–Shannon divergence of the 2D XXZ model are then studied as the indicators of quantum phase transitions (QPTs). The nonanalytic and scaling behaviors of the uncertainty, QD and quantum coherence are also analyzed through numerical calculations. Moreover, we investigate the effect of the easy-axis anisotropy parameter and DM interaction on the QPT. We find that the uncertainty, QD, and quantum coherence can all be utilized to detect QPTs. Our findings could shed new light on the observable of the QPT of the many-body system with the uncertainty and quantum coherence, and enrich the application of QRG method to Heisenberg spin models.

Quantum teleportation and dynamics of quantum coherence and metrological non-classical correlations for open two-qubit systems

We investigate the dynamics of non-classical correlations and quantum coherence in open quantum systems by employing metrics like local quantum Fisher information, local quantum uncertainty, and quantum Jensen-Shannon divergence. Our focus here is on a system of two qubits in two distinct physical situations: the first one when the two qubits are coupled to a cavity field whether the system is closed or open, while the second consists of two qubits immersed in dephasing reservoirs. Our study places significant emphasis on how the evolution of these quantum criterion is influenced by the initial state’s purity (whether pure or mixed) and the nature of the environment (whether Markovian or non-Markovian). We observe that a decrease in the initial state’s purity corresponds to a reduction in both quantum correlations and quantum coherence, whereas higher purity enhances these quantumness. Furthermore, we establish a quantum teleportation strategy based on the two different physical scenarios. In this approach, the resulting state of the two qubits functions as a quantum channel integrated into a quantum teleportation protocol. We also analyze how the purity of the initial state and the Markovian or non-Markovian regimes impact the quantum teleportation process.

Quantum coherence and its distribution in a two-dimensional Heisenberg XY model

We investigate quantum phase transitions in the two-dimensional Heisenberg XY system. A basis-independent measure that is square root of quantum Jensen–Shannon divergence is used to identify quantum phase transitions. This method is suitable for discussing the multipartite coherence in the quantum spin system. By analyzing the local coherence and collective coherence in the model, we find that both quantities can effectively detect the critical point and they will exhibit a trade-off relation. When close to the quantum critical point, the scaling behavior are obtained for both measures. We also studied the distribution of quantum coherence and it was found that the reduced tripartite system is polygamous and the five-site block state does not obey additivity relation.

Some applications of the Hermite–Hadamard inequality for log‐convex functions in quantum divergences

One of the beautiful and very simple inequalities for a convex function is the Hermite–Hadamard inequality. The concept of ‐convexity is a strong variant of convexity. In this paper, by the Hermite–Hadamard inequality, we introduce two‐parametric Tsallis quantum relative entropy, two‐parametric Tsallis–Lin quantum relative entropy, and two‐parametric quantum Jensen–Shannon divergence in quantum information theory. Then some properties of quantum Tsallis–Jensen–Shannon divergence for two density matrices are investigated by this inequality.

Quantum metrics based upon classical Jensen–Shannon divergence

Jensen–Shannon divergence is an important distinguishability measure between probability distributions that finds interesting applications within the context of Information Theory. In particular, this classical divergence belongs to a remarkable class of divergences known as Csiszár or f-divergences. In this paper we analyze the problem of obtaining a distance measure between two quantum states starting from the classical Jensen–Shannon divergence between two probability distributions. Considering the Jensen–Shannon divergence as a Csiszár divergence, we first focus on the problem of distinguishability between two pure quantum states. We find a quantum version of the classical Jensen–Shannon divergence that differs from the previously introduced Quantum Jensen–Shannon Divergence. The two quantum versions of Jensen–Shannon divergence have different interpretations within the framework of Quantum Information Theory. Whereas the former quantum version of Jensen–Shannon divergence can be interpreted as the Holevo bound, the alternative quantum version obtained in this work equals the accessible information. Furthermore, we obtain a monoparametric family of metrics between two quantum pure states. Finally, we extend this family of metrics to the case of mixed quantum states by means of the concept of purification.

Universal order-parameter and quantum phase transition for two-dimensional q-state quantum Potts model

We investigate quantum phase transitions for q-state quantum Potts models (q = 2,3,4) on a square lattice and for the Ising model on a honeycomb lattice by using the infinite projected entangled-pair state algorithm with a simplified updating scheme. We extend the universal order parameter to a two-dimensional lattice system, which allows us to explore quantum phase transitions with symmetry-broken order for any translation-invariant quantum lattice system of the symmetry group G. The universal order parameter is zero in the symmetric phase, and it ranges from zero to unity in the symmetry-broken phase. The ground-state fidelity per lattice site is computed, and a pinch point is identified on the fidelity surface near the critical point. The results offer another example highlighting the connection between (i) critical points for a quantum many-body system undergoing a quantum phase-transition and (ii) pinch points on a fidelity surface. In addition, we discuss three quantum coherence measures: the quantum Jensen–Shannon divergence, the relative entropy of coherence, and the l1 norm of coherence, which are singular at the critical point, thereby identifying quantum phase transitions.

Entropic Bounds on Information Backflow.

In the dynamics of open quantum systems, the backflow of information to the reduced system under study has been suggested as the actual physical mechanism inducing memory and thus leading to non-Markovian quantum dynamics. To this aim, the trace-distance revivals between distinct evolved system states have been shown to be subordinated to the establishment of system-environment correlations or changes in the environmental state. We show that this interpretation can be substantiated also for a class of entropic quantifiers. We exploit a suitably regularized version of Umegaki's quantum relative entropy, known as telescopic relative entropy, that is tightly connected to the quantum Jensen-Shannon divergence. In particular, we derive general upper bounds on the telescopic relative entropy revivals conditioned and determined by the formation of correlations and changes in the environment. We illustrate our findings by means of examples, considering in particular the Jaynes-Cummings model and a phase covariant dynamics.

Quantum coherence and spin nematic to nematic quantum phase transitions in biquadratic spin-1 and spin-2 XY chains with rhombic single-ion anisotropy

We investigate quantum phase transitions and quantum coherence in infinite biquadratic spin-1 and -2 XY chains with rhombic single-ion anisotropy. All considered coherence measures such as the ${l}_{1}$ norm of coherence, the relative entropy of coherence, and the quantum Jensen-Shannon divergence, and the quantum mutual information show consistently that singular behaviors occur for the spin-1 system, which enables to identify quantum phase transitions. For the spin-2 system, the relative entropy of coherence and the quantum mutual information properly detect no singular behavior in the whole system parameter range, while the ${l}_{1}$ norm of coherence and the quantum Jensen-Shannon divergence show a conflicting singular behavior of their first-order derivatives. Examining local magnetic moments and spin quadrupole moments lead to the explicit identification of novel orderings of spin quadrupole moments with zero magnetic moments in the whole parameter space. We find the three uniaxial spin nematic quadrupole phases for the spin-1 system and the two biaxial spin nematic phases for the spin-2 system. For the spin-2 system, the two orthogonal biaxial spin nematic states are connected adiabatically without an explicit phase transition, which can be called quantum crossover. The quantum crossover region is estimated by using the quantum fidelity. Whereas for the spin-1 system, the two discontinuous quantum phase transitions occur between three distinct uniaxial spin nematic phases. We discuss the quantum coherence measures and the quantum mutual information in connection with the quantum phase transitions including the quantum crossover.

The metric property of the quantum Jensen-Shannon divergence

In this short note, we prove that the square root of the quantum Jensen-Shannon divergence is a true metric on the cone of positive matrices, and hence in particular on the quantum state space.

Metrics induced by Jensen-Shannon and related divergences on positive definite matrices

We study metric properties of symmetric divergences on Hermitian positive definite matrices. In particular, we prove that the square root of these divergences is a distance metric. As a corollary we obtain a proof of the metric property for Quantum Jensen-Shannon-(Tsallis) divergences (parameterized by α∈[0,2]). When specialized to α=1, we obtain as a corollary a proof of the metric property of the Quantum Jensen-Shannon divergence that was conjectured by Lamberti et al. (2008) [13], and recently also proved by Virosztek (2019) [28]. A more intricate argument also establishes metric properties of Jensen-Rényi divergences (for α∈(0,1)); this argument develops a technique that may be of independent interest.

Quantum divergences with p-power means

We study several properties of a family of quantum divergences with p-power means. We also show that the Hellinger distance with log Euclidean mean and the quantum Jensen-Shannon divergence can be rediscovered from our class of quantum distances via the limit processes.

Influence of Stark-shift on quantum coherence and non-classical correlations for two two-level atoms interacting with a single-mode cavity field

An exact analytic solution for two two-level atoms coupled with a multi-photon single-mode electromagnetic cavity field in the presence of the Stark shift is derived. We assume that the field is initially prepared in a coherent state and the two atoms are initially prepared in excited state. Considering the atomic level shifts generated by the Stark shift effect, the dynamical behavior of both quantum coherence (QC) measured using a quantum Jensen–Shannon divergence and of quantum correlations captured by quantum discord (QD) are investigated. It is shown that the intensity-dependent Stark-shift in the cavity and the number of coherent state photons plays a key role in enhancing or destroying both QC and QD during the process of intrinsic decoherence. We remarked that increasing the Stark-shift parameters, the frequencies of the transition for the mode of the cavity field, and photons number destroy both the amount of QC and QD and affected their periodicity. More importantly, QC and QD exhibit similar behavior and both show a revival phenomenon. We believe that the present work shows that the quantum information protocols based on physical resources in optical systems could be controlled by adjusting the Stark-shift parameters.

A variant of Horn's problem and the derivative principle

Identifying the spectrum of the sum of two given Hermitian matrices with fixed eigenvalues is the famous Horn's problem. In this note, we investigate a variant of Horn's problem, i.e., we identify the probability density function (abbr. pdf) of the diagonals of the sum of two random Hermitian matrices with given spectra. We then use it to re-derive the pdf of the eigenvalues of the sum of two random Hermitian matrices with given eigenvalues via the derivative principle, a powerful tool used to get the exact probability distribution by reducing to the corresponding distribution of diagonal entries. We can also recover Jean-Bernard Zuber's recent results. Moreover, as an illustration, we derive the analytical expressions of eigenvalues of the sum of two random Hermitian matrices from GUE(n) or Wishart ensemble by the derivative principle, respectively. We also investigate the statistics of exponential of random matrices and connect them with Golden-Thompson inequality, and partially answer a question proposed by Forrester. Some potential applications in quantum information theory, such as uniform average quantum Jensen-Shannon divergence and average coherence of uniform mixture of two orbits, are discussed.

Influence of the quantum-jump-based feedback on the Rydberg atoms system

We investigated the influence of the quantum-jump-based feedback on the behaviors of the three-level Rydberg system by making use of relative entropy coherence and the square root of quantum Jensen–Shannon divergence. It is shown that quantum relative entropy coherence and quantum Jensen–Shannon divergence are enhanced by particular quantum-jump-based feedbacks and increase with the enhancement of the detuning δ. We also explored the behavior of the entropic uncertainty relationship, and lower uncertainty bounds are obtained with quantum-jump-based feedbacks. The effect of different detunings δ on the entropic uncertainty relationship is also discussed, and it is found that the uncertainty bound decreases when the detuning δ increases.

Quantum coherence and quantum Fisher information in the XXZ system

Abstract We investigate the quantum phase transition of the XXZ model via quantum coherence and quantum Fisher information. The quantum phase transition of the nearest, the next-nearest, and the third-nearest particles is demonstrated by quantum Fisher information and quantum coherence measures, i.e., relative entropy, l1 norm, quantum Jensen-Shannon divergence, respectively. Both analytical and numerical results are presented in detail. This work demonstrates that quantum coherence and quantum Fisher information can be used as new quantum measures for characterizing the quantum phase transition of the XXZ spin chain systems.

Can a Quantum Walk Tell Which Is Which?A Study of Quantum Walk-Based Graph Similarity.

We consider the problem of measuring the similarity between two graphs using continuous-time quantum walks and comparing their time-evolution by means of the quantum Jensen-Shannon divergence. Contrary to previous works that focused solely on undirected graphs, here we consider the case of both directed and undirected graphs. We also consider the use of alternative Hamiltonians as well as the possibility of integrating additional node-level topological information into the proposed framework. We set up a graph classification task and we provide empirical evidence that: (1) our similarity measure can effectively incorporate the edge directionality information, leading to a significant improvement in classification accuracy; (2) the choice of the quantum walk Hamiltonian does not have a significant effect on the classification accuracy; (3) the addition of node-level topological information improves the classification accuracy in some but not all cases. We also theoretically prove that under certain constraints, the proposed similarity measure is positive definite and thus a valid kernel measure. Finally, we describe a fully quantum procedure to compute the kernel.

Topological quantum phase transitions in the 2-D Kitaev honeycomb model

We study the topological quantum phase transition in the 2-D Kitaev honeycomb model by making use of the square root of the quantum Jensen–Shannon divergence and find that the square root of the quantum Jensen–Shannon divergence exhibits singular behaviors at the critical point of quantum phase transition. The scaling behaviors of the square root of the quantum Jensen–Shannon divergence are also examined from the first-order derivatives, and we demonstrate that the square root of the quantum Jensen–Shannon divergence obeys universal finite-size scaling laws. Furthermore, we explore the performance of quantum discord and the relative entropy coherence of the system. It is shown that quantum discord and relative entropy coherence display similar critical behaviors, and the square root of the quantum Jensen–Shannon divergence and quantum discord can serve as good indicators for quantum phase transitions.

Dynamics of entanglement and state-space trajectories followed by a system of four-qubit in the presence of random telegraph noise: common environment (CE) versus independent environments (IEs)

The paper investigates the dynamics of entanglement and explores some geometrical characteristics of the trajectories in state space, in four-qubit Greenberger-Horne-Zeilinger (GHZ) - and W-type states, coupled to common and independent classical random telegraph noise (RTN) sources. It is shown from numerical simulations that: (i) the dynamics of entanglement depends drastically not only on the input configuration of the qubits and the presence or absence of memory effects, but also on whether the qubits are coupled to the RTN in a CE or IEs; (ii) a considerable amount of entanglement can be indefinitely trapped when the qubits are embedded in a CE; (iii) the CE configuration preserves better the entanglement initially shared between the qubits than the IEs configuration, however, for W-type states, there is a period of time and/or certain values of the purity for which, the opposite can be found. Thanks to the results obtained in our earlier works on three-qubit models, we are able to conclude that entanglement becomes more robustly protected from decay when the number of qubits of the system increases. Finally, we find that the trajectories in state space of the system quantified by the quantum Jensen Shannon divergence (QJSD) between the time-evolved states of the qubits and some reference states may be curvilinear or chaotic.