置換不変密度行列が保持する量子情報量を用いた量子多体現象の解析
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].
•Jan 1, 2024•0 citations