Abstract An alternative method is presented for extracting the von Neumann entropy − Tr(ρ ln ρ) from Tr(ρn) for integer n in a quantum system with density matrix ρ. Instead of relying on direct analytic continuation in n, the method uses a generating function − Tr{ρ ln[(1 − zρ)/(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 = −∞ 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.
Abstract An alternative method is presented for extracting the von Neumann entropy − Tr(ρ ln ρ) from Tr(ρn) for integer n in a quantum system with density matrix ρ. Instead of relying on direct analytic continuation in n, the method uses a generating function − Tr{ρ ln[(1 − zρ)/(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 = −∞ 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.
Extracting operationally meaningful quantities such as von Neumann entropy and mutual information is central to characterizing many-body quantum systems, yet experiments and numerics often provide direct access only to integer Rényi entropies. Whereas conventional extrapolation relies on a prescribed fitting ansatz, in PRL 137, 100202 we introduced an alternative approach based on stabilized analytic continuation (SAC), which avoids such an ansatz and is inherently robust to noise. Here, we further develop this framework in the noiseless setting and benchmark its performance across several nontrivial many-body problems. We first show that, besides the intrinsic ambiguity of reconstructing the von Neumann entropy from finitely many Rényi samples, analytic continuation can also fail because of genuine nonanalyticities in the Rényi function arising from zeros of Trρz. We derive universal zero-free domains for Trρz and systematically sharpen these bounds using additional spectral information about ρ. We then benchmark the performance of SAC in three settings: (i) extracting topological entanglement entropy from DMRG Rényi data for the toric code and Kagome Heisenberg models; (ii) reconstructing the mutual information between disjoint intervals in a (1+1)d compact-boson CFT; and (iii) recovering finite-time ballistic growth of the von Neumann entropy from integer Rényi entropies that cross over toward subballistic growth in diffusive quantum dynamics. In the latter setting, analytic continuation is expected to fail in the long time limit. We illustrate this mechanism with a physically motivated two-sector model, where the asymptotic separation between von Neumann and higher-Rényi growth is accompanied by a zero of Trρz pinching the real axis at z=1, thereby producing a first-order Rényi phase transition.
Abstract Analytically continuing the von Neumann entropy from Rényi entropies is a challenging task in quantum field theory. While the n -th Rényi entropy can be computed using the replica method in the path integral representation of quantum field theory, the analytic continuation can only be achieved for some simple systems on a case-by-case basis. In this work, we propose a general framework to tackle this problem using classical and quantum neural networks with supervised learning. We begin by studying several examples with known von Neumann entropy, where the input data is generated by representing TrρAn Tr ρ A n with a generating function. We adopt KerasTuner to determine the optimal network architecture and hyperparameters with limited data. In addition, we frame a similar problem in terms of quantum machine learning models, where the expressivity of the quantum models for the entanglement entropy as a partial Fourier series is established. Our proposed methods can accurately predict the von Neumann and Rényi entropies numerically, highlighting the potential of deep learning techniques for solving problems in quantum information theory.
Von Neumann defined the informational entropy of a density matrix ρ by the expression S(ρ)=−tr(ρ ln ρ). [J. Von Neumann, Die Mathematischen Grundlagen der Quantummechanik (Springer-Verlag, Berlin, 1932)]. Here, starting from the definitions of Shannon entropy and of Renyi entropy of random variables, and using some (reasonable) rules of inference, two expressions are obtained for the quantum entropy of a given square (deterministic) matrix, irrespective of any probabilistic framework and/or any randomization technique. These definitions apply directly to linear operators on Hilbert spaces even when they are not of positive trace class, and they contain Von Neumann entropy as a special case. These new measures of uncertainty could provide new approaches to some problems related to structure complexity.
Using arguments built on ergodicity, we derive an analytical expression for the Renyi entanglement entropies which, we conjecture, applies to the finite-energy density eigenstates of chaotic many-body Hamiltonians. The expression is a universal function of the density of states and is valid even when the subsystem is a finite fraction of the total system-a regime in which the reduced density matrix is not thermal. We find that in the thermodynamic limit, only the von Neumann entropy density is independent of the subsystem to the total system ratio V_{A}/V, while the Renyi entropy densities depend nonlinearly on V_{A}/V. Surprisingly, Renyi entropies S_{n} for n>1 are convex functions of the subsystem size, with a volume law coefficient that depends on V_{A}/V, and exceeds that of a thermal mixed state at the same energy density. We provide two different arguments to support our results: the first one relies on a many-body version of Berry's formula for chaotic quantum-mechanical systems, and is closely related to the eigenstate thermalization hypothesis. The second argument relies on the assumption that for a fixed energy in a subsystem, all states in its complement allowed by the energy conservation are equally likely. We perform an exact diagonalization study on quantum spin-chain Hamiltonians to test our analytical predictions.
Pseudo entropy extends entanglement entropy from a single quantum state to a pair of nonorthogonal states and is generally complex. Taking advantage of Cauchy-Riemann equations, Kramers-Kronig relations, and analytic continuation in state parameters, we show how and to what extent real and imaginary parts of pseudo entropy can be derived from ordinary entanglement entropy. For families with holomorphic coefficients in finite-dimensional Hilbert spaces, the reduced transition matrix equals the ordinary reduced density matrix formula evaluated at complex parameters. Then a convergent Taylor series gives the real and imaginary parts of pseudo entropy from even and odd derivatives of entanglement entropy at the real midpoint. The matrix identity also gives formulae for excess pseudo entropy and Renyi entropies, and interpolation formulae for families with polynomial coefficients. We apply these results to boundary-state quenches and thermal states in conformal field theory, and to fermionic and bosonic Gaussian states and quenches. In conformal field theory, Kramers-Kronig relations give the first moment of the imaginary part in terms of the central charge and one-point functions for boundary states.
We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy S A = −Tr ρ A logρ A corresponding to the reduced density matrix ρ A of a subsystem A . For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c , we re-derive the result of Holzhey et al when A is a finite interval of length in an infinite system, and extend it to many other cases: finite systems, finite temperatures, and when A consists of an arbitrary number of disjoint intervals. For such a system away from its critical point, when the correlation length ξ is large but finite, we show that , where is the number of boundary points of A . These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.