We study Rényi entropy of locally excited states with considering the thermal and boundary effects respectively in two dimensional conformal field theories (CFTs). Firstly, we consider locally excited states obtained by acting primary operators on a thermal state in low temperature limit. The Rényi entropy is summation of contribution from thermal effect and local excitation. Secondly, we mainly study the Rényi entropy of locally excited states in 2D CFT with a boundary. We show that the time evolution of Rényi entropy is affected by the boundary, but does not depend on the boundary condition. Moreover, we show that the maximal value of Rényi entropy always coincides with the log of quantum dimension of the primary operator. In terms of quasi-particle interpretation, the boundary behaves as an infinite potential barrier which reflects any energy moving towards it.We study Rényi entropy of locally excited states with considering the thermal and boundary effects respectively in two dimensional conformal field theories (CFTs). Firstly, we consider locally excited states obtained by acting primary operators on a thermal state in low temperature limit. The Rényi entropy is summation of contribution from thermal effect and local excitation. Secondly, we mainly study the Rényi entropy of locally excited states in 2D CFT with a boundary. We show that the time evolution of Rényi entropy is affected by the boundary, but does not depend on the boundary condition. Moreover, we show that the maximal value of Rényi entropy always coincides with the log of quantum dimension of the primary operator. In terms of quasi-particle interpretation, the boundary behaves as an infinite potential barrier which reflects any energy moving towards it.

arXiv:1501.00757Journal of High Energy PhysicsApr 1, 201552 citations

Quantum Tsallis Entropy with Localization Characteristics

Qi Han, Shuai Wang, Lijie Gou +1 more

International Journal of Theoretical PhysicsAug 1, 20251 citation

Renyi Entropy and the Uncertainty Relations

Iwo Białynicki‐Birula

AIP conference proceedingsJan 1, 200743 citations

Numerical evaluation of Rényi entropy with path integrals

Miha Srdinšek

Quantum phase transitions and quantum entanglement can be studied by quantum Rényi entropy, a generalization of Von Neumann entropy. However, Rényi entropy calculations have been plagued by their computational complexity. In this thesis we present a new numerical technique based on thermodynamic integration via regularized integration paths. The optimal path is constructed by minimizing the variance of the Rényi entropy estimator leading to an efficient evaluation of its quantity for complex quantum systems of distinguishable particles with unprecedented entanglement and size. We implement the method in path integral Monte Carlo (PIMC) and path integral molecular dynamics (PIMD) schemes. We demonstrate in the case of the one-dimensional Ising model in transverse magnetic field, that the method outperforms some of the existing state-of-the-art numerical methods. We then study the complex of formic acid dimer and discover that its two shared protons are entangled even above room temperature. By the means of hybrid PIMC-PIMD scheme we then fully resolve the phase diagram of the discrete ϕ4 model, and show that the peak of the Rényi entropy at finite temperature exactly coincides with the quantum critical point (QCP) and the freezing of the field in the imaginary time, interpreted as the local moments formation. We then use the Rényi entropy to determine the phase diagram of high-pressure water ice phases VIII, VII and X. The quantum Rényi entropy of protons sharply peaks at the pressure of 43 GPa, which is a signature of a QCP. The peak precisely coincides with the freezing of protons in the imaginary time, where phase X is lost, and the formation of a macroscopic polarization at lower temperature, characteristic of phase VIII. In this way we are able to scan the full phase diagram of high pressure water ice by looking at the changes in Rényi entropy behavior as a function of pressure and temperature.Quantum phase transitions and quantum entanglement can be studied by quantum Rényi entropy, a generalization of Von Neumann entropy. However, Rényi entropy calculations have been plagued by their computational complexity. In this thesis we present a new numerical technique based on thermodynamic integration via regularized integration paths. The optimal path is constructed by minimizing the variance of the Rényi entropy estimator leading to an efficient evaluation of its quantity for complex quantum systems of distinguishable particles with unprecedented entanglement and size. We implement the method in path integral Monte Carlo (PIMC) and path integral molecular dynamics (PIMD) schemes. We demonstrate in the case of the one-dimensional Ising model in transverse magnetic field, that the method outperforms some of the existing state-of-the-art numerical methods. We then study the complex of formic acid dimer and discover that its two shared protons are entangled even above room temperature. By the means of hybrid PIMC-PIMD scheme we then fully resolve the phase diagram of the discrete ϕ4 model, and show that the peak of the Rényi entropy at finite temperature exactly coincides with the quantum critical point (QCP) and the freezing of the field in the imaginary time, interpreted as the local moments formation. We then use the Rényi entropy to determine the phase diagram of high-pressure water ice phases VIII, VII and X. The quantum Rényi entropy of protons sharply peaks at the pressure of 43 GPa, which is a signature of a QCP. The peak precisely coincides with the freezing of protons in the imaginary time, where phase X is lost, and the formation of a macroscopic polarization at lower temperature, characteristic of phase VIII. In this way we are able to scan the full phase diagram of high pressure water ice by looking at the changes in Rényi entropy behavior as a function of pressure and temperature.

Sep 26, 20230 citations

Entanglement entropy for descendent local operators in 2D CFTs

Bin Chen, Wu-zhong Guo, Song He +1 more

We mainly study the Rényi entropy and entanglement entropy of the states locally excited by the descendent operators in two dimensional conformal field theories (CFTs). In rational CFTs, we prove that the increase of entanglement entropy and Rényi entropy for a class of descendent operators, which are generated by ${\mathrm{\mathcal{L}}}^{\left(-\right)}{\overline{\mathrm{\mathcal{L}}}}^{\left(-\right)}$ onto the primary operator, always coincide with the logarithmic of quantum dimension of the corresponding primary operator. That means the Rényi entropy and entanglement entropy for these descendent operators are the same as the ones of their corresponding primary operator. For 2D rational CFTs with a boundary, we confirm that the Rényi entropy always coincides with the logarithmic of quantum dimension of the primary operator during some periods of the evolution. Furthermore, we consider more general descendent operators generated by $\sum {d}_{\left\{{n}_i\right\}\left\{{n}_j\right\}}\left({\displaystyle {\prod}_i{L}_{-}{{}_n}_{{}_i}{\displaystyle {\prod}_j{{\overline{L}}_{-n}}_{{}_j}}}\right)$ on the primary operator. For these operators, the entanglement entropy and Rényi entropy get additional corrections, as the mixing of holomorphic and anti-holomorphic Virasoro generators enhance the entanglement. Finally, we employ perturbative CFT techniques to evaluate the Rényi entropy of the excited operators in deformed CFT. The Rényi and entanglement entropies are increased, and get contributions not only from local excited operators but also from global deformation of the theory.We mainly study the Rényi entropy and entanglement entropy of the states locally excited by the descendent operators in two dimensional conformal field theories (CFTs). In rational CFTs, we prove that the increase of entanglement entropy and Rényi entropy for a class of descendent operators, which are generated by $$ {\mathrm{\mathcal{L}}}^{\left(-\right)}{\overline{\mathrm{\mathcal{L}}}}^{\left(-\right)} $$ onto the primary operator, always coincide with the logarithmic of quantum dimension of the corresponding primary operator. That means the Rényi entropy and entanglement entropy for these descendent operators are the same as the ones of their corresponding primary operator. For 2D rational CFTs with a boundary, we confirm that the Rényi entropy always coincides with the logarithmic of quantum dimension of the primary operator during some periods of the evolution. Furthermore, we consider more general descendent operators generated by $$ {\displaystyle \sum {d}_{\left\{{n}_i\right\}\left\{{n}_j\right\}}\left({\displaystyle {\prod}_i{L}_{-}{{}_n}_{{}_i}{\displaystyle {\prod}_j{{\overline{L}}_{-n}}_{{}_j}}}\right)} $$ on the primary operator. For these operators, the entanglement entropy and Rényi entropy get additional corrections, as the mixing of holomorphic and anti-holomorphic Virasoro generators enhance the entanglement. Finally, we employ perturbative CFT techniques to evaluate the Rényi entropy of the excited operators in deformed CFT. The Rényi and entanglement entropies are increased, and get contributions not only from local excited operators but also from global deformation of the theory.

arXiv:1507.01157Journal of High Energy PhysicsOct 1, 201559 citations

Quantum Entropy Regularization

Richard N. Silver

Springer eBooksJan 1, 19992 citations

{\mathrm{\mathcal{L}}}^{\left(-\right)}{\overline{\mathrm{\mathcal{L}}}}^{\left(-\right)}

onto the primary operator, always coincide with the logarithmic of quantum dimension of the corresponding primary operator. That means the Rényi entropy and entanglement entropy for these descendent operators are the same as the ones of their corresponding primary operator. For 2D rational CFTs with a boundary, we confirm that the Rényi entropy always coincides with the logarithmic of quantum dimension of the primary operator during some periods of the evolution. Furthermore, we consider more general descendent operators generated by

\sum {d}_{\left\{{n}_i\right\}\left\{{n}_j\right\}}\left({\displaystyle {\prod}_i{L}_{-}{{}_n}_{{}_i}{\displaystyle {\prod}_j{{\overline{L}}_{-n}}_{{}_j}}}\right)

on the primary operator. For these operators, the entanglement entropy and Rényi entropy get additional corrections, as the mixing of holomorphic and anti-holomorphic Virasoro generators enhance the entanglement. Finally, we employ perturbative CFT techniques to evaluate the Rényi entropy of the excited operators in deformed CFT. The Rényi and entanglement entropies are increased, and get contributions not only from local excited operators but also from global deformation of the theory.