Reduced density matrix and entanglement entropy of permutationally invariant quantum many body systems - Chap. 9

Mario Salerno, Vladislav Popkov

Vladislav Popkov, Mario Salerno

Mario Salerno, Vladislav Popkov

Spectral properties of the reduced density matrix (RDM) of permutational invariant quantum many-body systems are investigated. The RDM block diagonalization which accounts for all symmetries of the Hamiltonian is achieved. The analytical expression of the RDM spectrum is provided for arbitrary parameters and rigorously proved in the thermodynamical limit. The existence of several sum rules and recurrence relations among RDM eigenvalues is also demonstrated and the distribution function of RDM eigenvalues (including degeneracies) characterized. In particular, we prove that the distribution function approaches a two-dimensional Gaussian in the limit of large subsystem sizes n>>1. As a physical application we discuss the von Neumann entropy (VNE) of a block of size n for a system of hard-core bosons on a complete graph, as a function of n and of the temperature T. The occurrence of a crossover of VNE from purely logarithmic behavior at T=0 to a purely linear behavior in n for T≥Tc, is demonstrated.

優希 宮﨑

It is becoming clear that the concepts originating from quantum information theory1are useful for the analysis of quantum many-body physics. For example, topological ordered phases which appear in quantum Hall effect can not be explained by conventional Ginzburg-Landau theory [1], because we cannot define any order parameter and no symmetry breaking occurs. However, it has been shown that it is possible to characterize quantum phases with high generality beyond the framework of the Ginzburg-Landau theory by considering entanglement entropy, which is one of quantum information stored in the quantum state itself (density matrix) [2, 3]. We can mathematically express a quantum state by a density matrix. Given the density matrix of a system, we can, in principle, obtain quantum information about the system, including its entanglement entropy, when the system is divided into two parts. However, in the field of solid state physics, experimentally obtaining the density matrix of a system is challenging. In artificial quantum systems including cold atomic gases in an optical lattice [4], trapped-ion systems [5], etc., the reconstruction of a density matrix of multi-qubit systems has been successfully achieved by preparing the identical state repeatedly and measuring the various microscopic observables, taking advantage of the cleanness and high controllability of such systems. This process is known as quantum state tomography (QST). However, scaling up QST is hard due to the exponentially increasing cost with respect to the system size. Moreover, especially in the experiments of cold atomic gases in an optical lattice, the local rotation of quantization axes is hard within currently-available optical techniques. In this work, we focus on the permutationally invariant part of the density matrix (PIDM), and investigate the role of PIDM in quantum many-body systems. PIDM is defined as the average over all possible site permutations of the density matrix. In the context of QST, it is recognized that PIDM requires only a polynomial number of measurements, in contrast to the exponential cost associated with reconstructing a full density matrix [6]. We examine how this experimentally tractable quantity reflects the property of the ground state of the system across a quantum phase transition. In this doctoral thesis, we develop a calculation method of second-order Rényi entropy of PIDM using density-matrix renormalization-group algorithm and tensor network. Applying this procedure, we compute the second-order Rényi entropy of PIDM for the ground state of the transverse-field Ising and XXZ chains. As a result, we find that the Rényi entropy of PIDM quantifies the uncertainness raised in discarding information regarding special structure by symmetrization, in other words, it quantifies the special information of the original quantum state. Moreover, we obtain the insight that the difference of the scaling law of the Rényi entropy of PIDM reflects the difference of quantum phases[7].

Pasquale Calabrese

Andrea Coser

In this thesis we study entanglement in one-dimensional critical quantum many-body systems and in particular we will focus on disconnected regions. Given a system in a pure state, to quantify the amount of entanglement between a multicomponent subsystem and the rest of the full system we can use as an entanglement measure the renowned entanglement entropy. However, if we are interested in the entanglement shared among the disconnected regions, the entanglement entropy fails to be a good quantifier. The reason is that the state of the subsystem is in general mixed once the rest is traced out, and the entanglement entropy is a good measure of entanglement only for pure states. A good measure of entanglement in mixed states is the logarithmic negativity, which is the quantitative version of Peres' criterion of separability. The main advantage of the negativity with respect to other entanglement measures is the simplicity of its definition in terms of the density matrix describing the quantum state. Since the definition does not require any variational calculus, it is much more easily computable than any other entanglement measure, and therefore we can obtain some results also in complicated settings such as many-body systems. We will be mostly interested in the configuration where the full system is divided into two parts, A and B. If we do not have access to the degrees of freedom in B, we can describe subsystem A through its reduced density matrix, where the degrees of freedom in B have been traced out. The residual subsystem in A will in general be left in a mixed state. Subsystem A is then divided again into two parts, A_1 and A_2, and we will be interested in the entanglement shared by these two components. Knowing the full density matrix, the logarithmic negativity can be easily computed from the eigenvalues of its partial transpose with respect to the degrees of freedom living in one of the two subsystems A_1 or A_2. However, the full density matrix of a many-body state is in general unaccessible, even numerically, since the size of the matrix grows exponentially with the size of the system. If we concentrate on critical systems whose low-energy physics can be described by a quantum field theory, we can resort to its powerful tools to compute the entanglement properties. In particular the main tool is the replica trick. The entanglement entropy is obtained from the moments of the reduced density matrix, while the negativity can be computed from the moments of the partial transpose. However, even the computation of the integer-order moments is not an easy task, and exact analytical results are known only for the simplest quantum field theories. Hence, throughout the thesis we will usually consider conformal field theories, which have an enhanced set of symmetries and therefore allow for some exact computations. In the Introduction, we will try to frame this work by briefly reviewing some of the main topics in the literature of quantum many-body systems and quantum field theories where entanglement plays a crucial role. We will also describe what are the main features that are requested to a `good' measure of entanglement, and we will review some of the main measures that have been considered so far in the context of quantum information. We will define the entanglement negativity and stress its main advantages, as well as its drawbacks. After reviewing some basic facts of conformal field theories, we will give some technical details on the computation of entanglement entropy and logarithmic negativity in general quantum field theories, which will be needed in the rest of the thesis. In Chap. 2 we will start the study of the entanglement of several disjoint disconnected regions by considering the entanglement that the union of these regions share with the rest of the system. We will compute the integer-order Rényi entropies of the subsystem from which the entanglement entropy can be obtained through the replica limit. Unfortunately, the exact analytic continuation to real order is still out of reach and therefore the entanglement entropy cannot be computed exactly. However numerical extrapolations allow to obtain some accurate estimates. In the remaining chapters we will focus on the entanglement shared between two non complementary disjoint regions, through the computations of the entanglement negativity in different settings. In Chap. 3, we will consider a global quantum quench starting from a conformal boundary state and evolving through a conformal evolution. A general formula for the mutual information and the logarithmic negativity for two adjacent and disjoint intervals is given in the spacetime limit. In Chap. 4 we focus on the XY chain and we recover a formula for the partial transpose as a sum of four fermionic auxiliary Gaussian density matrices, by generalizing some previous results for the reduced density matrix of spin systems and for the partial transpose of pure fermionc Gaussian states. Even if the computation of the negativity is still out of reach, we can obtain formulas for the integer-order moments of the partial transpose in terms of the correlation matrices relative to the two components and to the region in between them. In Chap. 5 we will study the entanglement negativity for a free Dirac fermion field. Starting from the lattice results, we will obtain a path integral representation of the partial transpose which can be easily generalized to the continuum. With this representation of the partial transpose, we can construct all its integer-order moments and compute them for the simple conformal field theory of the free fermion. Again, the computation of the negativity could not be accomplished due to technical difficulties in the analytic continuation to real order. The same computation has been extended also to the modular invariant Dirac fermion and the Ising model, and the obtained formulas coincide with the ones already present in the literature. This analysis draws an interesting connection between some terms appearing in the formulas for the moments of the partial transpose (as well as of the R\'enyi entropies) found for the lattice models and the ones found for the corresponding theories describing their scaling limit. Whenever possible, all the analytical results will be checked against numerical calculations performed on simple free chain models. In Chaps. 2 and 3 we will consider bosonic Gaussian states, specifically the harmonic chain in its ground state and out of equilibrium, while in Chaps. 4 and 5 we will consider fermionic systems, specifically the XX and Ising spin chains, and the tight-binding model. Finally, we will draw some conclusions and discuss some open problems.