This paper builds a graph Laplacian from the ground state mutual information matrix of the isotropic antiferromagnetic Heisenberg (XXZ) spin chain and uses it as a case study in what this class of construction does and does not establish. A first-pass analysis reproduces several textbook-looking signatures, including a statistically significant nonzero finite-size intercept in the lowest nonzero eigenvalue that persists and grows under periodic boundaries, naively suggestive of an entanglement-induced mass scale. This reading does not survive scrutiny: the intercept's magnitude varies by an order of magnitude under different adjacency normalizations and can change sign under an AICc-preferred finite-size model with logarithmic corrections. Repeating the construction with the classical, non-entanglement, connected spin correlator reproduces the same intercept phenomenon more cleanly; toggling only quantum coherence on and off via discord leaves it almost unchanged; and restricting to an unambiguous entanglement-only measure, two-body concurrence, collapses the intercept to a value statistically indistinguishable from zero. Across every construction tested, the intercept tracks whether the underlying matrix has long-range support, not whether it is quantum in origin.
This paper builds a graph Laplacian from the ground state mutual information matrix of the isotropic antiferromagnetic Heisenberg (XXZ) spin chain and uses it as a case study in what this class of construction does and does not establish. A first-pass analysis reproduces several textbook-looking signatures, including a statistically significant nonzero finite-size intercept in the lowest nonzero eigenvalue that persists and grows under periodic boundaries, naively suggestive of an entanglement-induced mass scale. This reading does not survive scrutiny: the intercept's magnitude varies by an order of magnitude under different adjacency normalizations and can change sign under an AICc-preferred finite-size model with logarithmic corrections. Repeating the construction with the classical, non-entanglement, connected spin correlator reproduces the same intercept phenomenon more cleanly; toggling only quantum coherence on and off via discord leaves it almost unchanged; and restricting to an unambiguous entanglement-only measure, two-body concurrence, collapses the intercept to a value statistically indistinguishable from zero. Across every construction tested, the intercept tracks whether the underlying matrix has long-range support, not whether it is quantum in origin.
The quantum discord, which quantifies the amount of quantum correlations present between parts of a system, is investigated for antiferromagnetic spin systems. The discord for a pair of spins in the many-spin ground state is related to the diagonal and off-diagonal spin-spin correlation functions and the local magnetization. For isotropic and translationally invariant states, the discord is shown to be a function of the diagonal correlation function only. Thus, near a thermal/quantum critical point, the discord for a pair of spins shows long-range behavior, analogously of the correlation function. The discord exhibits a kink singularity as a function of the anisotropy parameter for the the ground state of the Heisenberg model, for both nearest-neighbor spins as well as for well-separated spins. The preferred measurement basis for the minimum conditional entropy, which determines the discord, changes discontinuously across the critical point. The conditional entropy distribution over all possible the measurement basis is investigated. For the isotropic model, the distribution is just a delta-function, whereas it has a twin-peak structure for anisotropic model. It is shown that the average value and the mean-square fluctuation of the conditional entropy also show a signature of the critical-point behavior.
This work is at the interface of graph theory and quantum mechanics. Quantum correlations epitomize the usefulness of quantum mechanics. Quantum discord is an interesting facet of bipartite quantum correlations. Earlier, it was shown that every combinatorial graph corresponds to quantum states whose characteristics are reflected in the structure of the underlined graph. A number of combinatorial relations between quantum discord and simple graphs were studied. To extend the scope of these studies, we need to generalize the earlier concepts applicable to simple graphs to weighted graphs, corresponding to a diverse class of quantum states. To this effect, we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and call them graph Laplacian quantum states. We find the graph theoretic conditions for zero and nonzero quantum discord for these states. We apply these results on some important pure two qubit states, as well as a number of mixed quantum states, such as the Werner, Isotropic, and [Formula: see text]-states. We also consider graph Laplacian states corresponding to simple graphs as a special case.
We explore quantum and classical correlations along with coherence in the ground states of spin-1 Heisenberg chains, namely the one-dimensional XXZ model and the one-dimensional bilinear biquadratic model, with the techniques of density matrix renormalization group theory. Exploiting the tools of quantum information theory, that is, by studying quantum discord, quantum mutual information, and three recently introduced coherence measures in the reduced density matrix of two nearest neighbor spins in the bulk, we investigate the quantum phase transitions and special symmetry points in these models. We point out the relative strengths and weaknesses of correlation and coherence measures as figures of merit to witness the quantum phase transitions and symmetry points in the considered spin-1 Heisenberg chains. In particular, we demonstrate that, as none of the studied measures can detect the infinite-order Kosterlitz-Thouless transition in the XXZ model, they appear to be able to signal the existence of the same type of transition in the biliear biquadratic model. However, we argue that what is actually detected by the measures here is the SU(3) symmetry point of the model rather than the infinite-order quantum phase transition. Moreover, we show in the XXZ model that examining even single site coherence can be sufficient to spotlight the second-order phase transition and the SU(2) symmetry point.
We examine the behavior of quantum correlations of spin pairs in a finite anisotropic XY spin chain immersed in a transverse magnetic field, through the analysis of the quantum discord and the conventional and quadratic one-way information deficits. We first provide a brief review of these measures, showing that the last ones can be obtained as particular cases of a generalized information deficit based on general entropic forms. All of these measures coincide with an entanglement entropy in the case of pure states, but can be non-zero in separable mixed states, vanishing just for classically correlated states. It is then shown that their behavior in the exact ground state of the chain exhibits similar features, deviating significantly from that of the pair entanglement below the critical field. In contrast with entanglement, they reach full range in this region, becoming independent of the pair separation and coupling range in the immediate vicinity of the factorizing field. It is also shown, however, that significant differences between the quantum discord and the information deficits arise in the local minimizing measurement that defines them. Both analytical and numerical results are provided.
The quantum discord, which quantifies the amount of quantum correlation present between parts of a system, is investigated for spin systems. The discord for a pair of spins in the many-spin ground state, computed by minimizing the conditional entropy of the two spins, is related to the diagonal and off-diagonal spin–spin correlation functions and the local magnetization. We show that for isotropic and translationally invariant states, the discord depends only on the diagonal spin correlation function. Thus, near a thermal/quantum critical point, the pairwise discord exhibits long-range behavior, analogously to the correlation function. The discord exhibits a kink singularity as a function of the anisotropy parameter for the ground state of the anisotropic Heisenberg model, for both nearest-neighbor spins as well as for well-separated spins. The preferred measurement basis for the minimum conditional entropy, which determines the discord, changes discontinuously across the critical point. We also investigate, from exact numerical diagonalization, the distribution of conditional entropies as the measurement basis is varied. For the isotropic model, the conditional entropy distribution is a δ-function, whereas it has a twin-peak structure for the anisotropic model. It is shown that the average value and the mean-square fluctuation of the conditional entropy also show a signature of the critical-point behavior.
We examine the behavior of quantum correlations of spin pairs in a finite anisotropic XY spin chain immersed in a transverse magnetic field, through the analysis of the quantum discord and the conventional and quadratic one way-information deficits. We first provide a brief review of these measures, showing that the last ones can be obtained as particular cases of a generalized information deficit based on general entropic forms. All these measures coincide with an entanglement entropy in the case of pure states, but can be non-zero in separable mixed states, vanishing just for classically correlated states. It is then shown that their behavior in the exact ground state of the chain exhibits similar features, deviating significantly from that of the pair entanglement below the critical field. In contrast with entanglement, they reach full range in this region, becoming independent of the pair separation and coupling range in the immediate vicinity of the factorizing field. It is also shown, however, that significant differences between the quantum discord and the information deficits arise in the local minimizing measurement that defines them. Both analytical and numerical results are provided.