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Register a new userNote on von Neumann and Renyi entropies of a Graph
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We conjecture that all connected graphs of order $n$ have von Neumann entropy at least as great as the star $K_{1,n-1}$ and prove this for almost all graphs of order $n$. We show that connected graphs of order $n$ have Renyi 2-entropy at least as great as $K_{1,n-1}$ and for $alpha>1$, $K_n$ maximizes Renyi $alpha$-entropy over graphs of order $n$. We show that adding an edge to a graph can lower its von Neumann entropy.
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An alternative method is presented for extracting the von Neumann entropy $-operatorname{Tr} (rho ln rho)$ from $operatorname{Tr} (rho^n)$ for integer $n$ in a quantum system with density matrix $rho$. Instead of relying on direct analytic continuation in $n$, the method uses a generating function $-operatorname{Tr} { rho ln [(1-z rho) / (1-z)] }$ of an auxiliary complex variable $z$. The generating function has a Taylor series that is absolutely convergent within $|z|<1$, and may be analytically continued in $z$ to $z = -infty$ where it gives the von Neumann entropy. As an example, we use the method to calculate analytically the CFT entanglement entropy of two intervals in the small cross ratio limit, reproducing a result that Calabrese et al. obtained by direct analytic continuation in $n$. Further examples are provided by numerical calculations of the entanglement entropy of two intervals for general cross ratios, and of one interval at finite temperature and finite interval length.
The R{e}nyi and von Neumann entropies of the thermal state in the generalized uncertainty principle (GUP)-corrected single harmonic oscillator system are explicitly computed within the first order of the GUP parameter $alpha$. While the von Neumann entropy with $alpha = 0$ exhibits a monotonically increasing behavior in external temperature, the nonzero GUP parameter makes the decreasing behavior of the von Neumann entropy at the large temperature region. As a result, the von Neumann entropy is maximized at the finite temperature if $alpha eq 0$. The R{e}nyi entropy $S_{gamma}$ with nonzero $alpha$ also exhibits similar behavior at the large temperature region. In this region the R{e}nyi entropy exhibit decreasing behavior with increasing the temperature. The decreasing rate becomes larger when the order of the R{e}nyi entropy $gamma$ is smaller.
We present a method to measure the von Neumann entanglement entropy of ground states of quantum many-body systems which does not require access to the system wave function. The technique is based on a direct thermodynamic study of entanglement Hamiltonians, whose functional form is available from field theoretical insights. The method is applicable to classical simulations such as quantum Monte Carlo methods, and to experiments that allow for thermodynamic measurements such as the density of states, accessible via quantum quenches. We benchmark our technique on critical quantum spin chains, and apply it to several two-dimensional quantum magnets, where we are able to unambiguously determine the onset of area law in the entanglement entropy, the number of Goldstone bosons, and to check a recent conjecture on geometric entanglement contribution at critical points described by strongly coupled field theories.
The R{e}nyi and von Neumann entropies of various bipartite Gaussian states are derived analytically. We also discuss on the tripartite purification for the bipartite states when some particular conditions hold. The generalization to non-Gaussian states is briefly discussed.
A unital ring is called clean (resp. strongly clean) if every element can be written as the sum of an invertible element and an idempotent (resp. an invertible element and an idempotent that commutes). T.Y. Lam proposed a question: which von Neumann algebras are clean as rings? In this paper, we characterize strongly clean von Neumann algebras and prove that all finite von Neumann algebras and all separable infinite factors are clean.
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