Quantum Jensen-Shannon Divergence Research Articles

Quantum coherence of the Heisenberg spin models with Dzyaloshinsky-Moriya interactions

We study quantum coherence in a spin chain with both symmetric exchange and antisymmetric Dzyaloshinsky-Moriya couplings. Quantum coherence is quantified using the recently introduced quantum Jensen-Shannon divergence, which has the property that it is easily calculable and has several desirable mathematical properties. We calculate exactly the coherence for arbitrary number of spins at zero temperature in various limiting cases. The σzσz interaction tunes the amount of coherence in the system, and the antisymmetric coupling changes the nature of the coherence. We also investigate the effect of non-zero temperature by looking at a two-spin system and find similar behavior, with temperature dampening the coherence. The characteristic behavior of coherence resembles that of entanglement and is opposite to that of discord. The distribution of the coherence on the spins is investigated and found that it arises entirely due to the correlations between the spins.

Quantum kernels for unattributed graphs using discrete-time quantum walks

In this paper, we develop a new family of graph kernels where the graph structure is probed by means of a discrete-time quantum walk. Given a pair of graphs, we let a quantum walk evolve on each graph and compute a density matrix with each walk. With the density matrices for the pair of graphs to hand, the kernel between the graphs is defined as the negative exponential of the quantum Jensen–Shannon divergence between their density matrices. In order to cope with large graph structures, we propose to construct a sparser version of the original graphs using the simplification method introduced in Qiu and Hancock (2007). To this end, we compute the minimum spanning tree over the commute time matrix of a graph. This spanning tree representation minimizes the number of edges of the original graph while preserving most of its structural information. The kernel between two graphs is then computed on their respective minimum spanning trees. We evaluate the performance of the proposed kernels on several standard graph datasets and we demonstrate their effectiveness and efficiency.

Distribution of Quantum Coherence in Multipartite Systems.

The distribution of coherence in multipartite systems is examined. We use a new coherence measure with entropic nature and metric properties, based on the quantum Jensen-Shannon divergence. The metric property allows for the coherence to be decomposed into various contributions, which arise from local and intrinsic coherences. We find that there are trade-off relations between the various contributions of coherence, as a function of parameters of the quantum state. In bipartite systems the coherence resides on individual sites or is distributed among the sites, which contribute in a complementary way. In more complex systems, the characteristics of the coherence can display more subtle changes with respect to the parameters of the quantum state. In the case of the XXZ Heisenberg model, the coherence changes from a monogamous to a polygamous nature. This allows us to define the shareability of coherence, leading to monogamy relations for coherence.

Measuring graph similarity through continuous-time quantum walks and the quantum Jensen-Shannon divergence.

In this paper we propose a quantum algorithm to measure the similarity between a pair of unattributed graphs. We design an experiment where the two graphs are merged by establishing a complete set of connections between their nodes and the resulting structure is probed through the evolution of continuous-time quantum walks. In order to analyze the behavior of the walks without causing wave function collapse, we base our analysis on the recently introduced quantum Jensen-Shannon divergence. In particular, we show that the divergence between the evolution of two suitably initialized quantum walks over this structure is maximum when the original pair of graphs is isomorphic. We also prove that under special conditions the divergence is minimum when the sets of eigenvalues of the Hamiltonians associated with the two original graphs have an empty intersection.

Quantum skew divergence

In this paper, we study the quantum generalisation of the skew divergence, which is a dissimilarity measure between distributions introduced by Lee in the context of natural language processing. We provide an in-depth study of the quantum skew divergence, including its relation to other state distinguishability measures. Finally, we present a number of important applications: new continuity inequalities for the quantum Jensen-Shannon divergence and the Holevo information, and a new and short proof of Bravyi's Small Incremental Mixing conjecture.

A quantum Jensen–Shannon graph kernel for unattributed graphs

In this paper, we use the quantum Jensen–Shannon divergence as a means of measuring the information theoretic dissimilarity of graphs and thus develop a novel graph kernel. In quantum mechanics, the quantum Jensen–Shannon divergence can be used to measure the dissimilarity of quantum systems specified in terms of their density matrices. We commence by computing the density matrix associated with a continuous-time quantum walk over each graph being compared. In particular, we adopt the closed form solution of the density matrix introduced in Rossi et al. (2013) [27,28] to reduce the computational complexity and to avoid the cumbersome task of simulating the quantum walk evolution explicitly. Next, we compare the mixed states represented by the density matrices using the quantum Jensen–Shannon divergence. With the quantum states for a pair of graphs described by their density matrices to hand, the quantum graph kernel between the pair of graphs is defined using the quantum Jensen–Shannon divergence between the graph density matrices. We evaluate the performance of our kernel on several standard graph datasets from both bioinformatics and computer vision. The experimental results demonstrate the effectiveness of the proposed quantum graph kernel.

Quantum coupled mutation finder: predicting functionally or structurally important sites in proteins using quantum Jensen-Shannon divergence and CUDA programming

BackgroundThe identification of functionally or structurally important non-conserved residue sites in protein MSAs is an important challenge for understanding the structural basis and molecular mechanism of protein functions. Despite the rich literature on compensatory mutations as well as sequence conservation analysis for the detection of those important residues, previous methods often rely on classical information-theoretic measures. However, these measures usually do not take into account dis/similarities of amino acids which are likely to be crucial for those residues. In this study, we present a new method, the Quantum Coupled Mutation Finder (QCMF) that incorporates significant dis/similar amino acid pair signals in the prediction of functionally or structurally important sites.ResultsThe result of this study is twofold. First, using the essential sites of two human proteins, namely epidermal growth factor receptor (EGFR) and glucokinase (GCK), we tested the QCMF-method. The QCMF includes two metrics based on quantum Jensen-Shannon divergence to measure both sequence conservation and compensatory mutations. We found that the QCMF reaches an improved performance in identifying essential sites from MSAs of both proteins with a significantly higher Matthews correlation coefficient (MCC) value in comparison to previous methods. Second, using a data set of 153 proteins, we made a pairwise comparison between QCMF and three conventional methods. This comparison study strongly suggests that QCMF complements the conventional methods for the identification of correlated mutations in MSAs.ConclusionsQCMF utilizes the notion of entanglement, which is a major resource of quantum information, to model significant dissimilar and similar amino acid pair signals in the detection of functionally or structurally important sites. Our results suggest that on the one hand QCMF significantly outperforms the previous method, which mainly focuses on dissimilar amino acid signals, to detect essential sites in proteins. On the other hand, it is complementary to the existing methods for the identification of correlated mutations. The method of QCMF is computationally intensive. To ensure a feasible computation time of the QCMF’s algorithm, we leveraged Compute Unified Device Architecture (CUDA).The QCMF server is freely accessible at http://qcmf.informatik.uni-goettingen.de/.

Characterizing graph symmetries through quantum Jensen-Shannon divergence

In this paper we investigate the connection between quantum walks and graph symmetries. We begin by designing an experiment that allows us to analyze the behavior of the quantum walks on the graph without causing the wave function collapse. To achieve this, we base our analysis on the recently introduced quantum Jensen-Shannon divergence. In particular, we show that the quantum Jensen-Shannon divergence between the evolution of two quantum walks with suitably defined initial states is maximum when the graph presents symmetries. Hence, we assign to each pair of nodes of the graph a value of the divergence, and we average over all pairs of nodes to characterize the degree of symmetry possessed by a graph.

Distance between quantum states in the presence of initial qubit-environment correlations: A comparative study

The time evolution of the trace distance between two states of an open quantum system may increase due to initial system-environment correlations, thus exhibiting a breakdown of distance contractivity of the reduced dynamics. We analyze how the time evolution of the distance depends on the chosen distance measure. Here we elucidate the behavior of the trace distance, the Hilbert- Schmidt distance, the Bures distance, the Hellinger distance and the quantum Jensen-Shannon divergence for two system-environment setups, namely a qubit bi-linearly coupled to an infinite and a finite size environment with the latter composed of harmonic oscillators.

Properties of classical and quantum Jensen-Shannon divergence

Jensen-Shannon divergence (JD) is a symmetrized and smoothed version of the most important divergence measure of information theory, Kullback divergence. As opposed to Kullback divergence it determines in a very direct way a metric; indeed, it is the square of a metric. We consider a family of divergence measures (JD_alpha for alpha>0), the Jensen divergences of order alpha, which generalize JD as JD_1=JD. Using a result of Schoenberg, we prove that JD_alpha is the square of a metric for alpha lies in the interval (0,2], and that the resulting metric space of probability distributions can be isometrically embedded in a real Hilbert space. Quantum Jensen-Shannon divergence (QJD) is a symmetrized and smoothed version of quantum relative entropy and can be extended to a family of quantum Jensen divergences of order alpha (QJD_alpha). We strengthen results by Lamberti et al. by proving that for qubits and pure states, QJD_alpha^1/2 is a metric space which can be isometrically embedded in a real Hilbert space when alpha lies in the interval (0,2]. In analogy with Burbea and Rao's generalization of JD, we also define general QJD by associating a Jensen-type quantity to any weighted family of states. Appropriate interpretations of quantities introduced are discussed and bounds are derived in terms of the total variation and trace distance.

JENSEN–SHANNON DIVERGENCE AS A MEASURE OF THE DEGREE OF ENTANGLEMENT

The notion of distance in Hilbert space is relevant in many scenarios. In particular, "distances" between quantum states play a central role in quantum information theory. An appropriate measure of distance is the quantum Jensen Shannon divergence (QJSD) between quantum states. Here we study this distance as a geometrical measure of entanglement and apply it to different families of states.

Metric character of the quantum Jensen-Shannon divergence

In a recent paper, the generalization of the Jensen-Shannon divergence in the context of quantum theory has been studied [Majtey et al., Phys. Rev. A 72, 052310 (2005)]. This distance between quantum states has shown to verify several of the properties required for a good distinguishability measure. Here we investigate the metric character of this distance. More precisely we show, formally for pure states and by means of a numerical procedure for mixed states, that its square root verifies the triangle inequality.