We study entanglement in a system of three coupled quantum harmonic oscillators. Specifically, we use the Schmidt decomposition to analyze how the entanglement is distributed among the three subsystems. The Schmidt decomposition is a powerful mathematical tool for characterizing bipartite entanglement in composite quantum systems. It allows to write a multipartite quantum state as a sum of product states between the subsystems, with coefficients known as Schmidt coefficients. We apply this decomposition to the general quantum state of three coupled oscillators and study how the Schmidt coefficients evolve as the interaction strengths between the oscillators are varied. This provides insight into how entanglement is shared between the different bipartitions of the overall three-particle system. Our results advance the fundamental understanding of multipartite entanglement in networked quantum systems. They also have implications for quantum information processing using multiple entangled nodes.
We study entanglement in a system of three coupled quantum harmonic oscillators. Specifically, we use the Schmidt decomposition to analyze how the entanglement is distributed among the three subsystems. The Schmidt decomposition is a powerful mathematical tool for characterizing bipartite entanglement in composite quantum systems. It allows to write a multipartite quantum state as a sum of product states between the subsystems, with coefficients known as Schmidt coefficients. We apply this decomposition to the general quantum state of three coupled oscillators and study how the Schmidt coefficients evolve as the interaction strengths between the oscillators are varied. This provides insight into how entanglement is shared between the different bipartitions of the overall three-particle system. Our results advance the fundamental understanding of multipartite entanglement in networked quantum systems. They also have implications for quantum information processing using multiple entangled nodes.
This study presents an analytical investigation of entanglement quantification in three-qubit pure states through the Acín canonical representation, which serves as a generalization of the two-qubit Schmidt decomposition. Driven by the intricacies of multipartite entanglement and the shortcomings of current measures, we employ a global concurrence measure derived from the generalized concurrence of various bipartitions within the system. The characteristics of this measure are explored both analytically and numerically across a range of SLOCC entanglement classes, such as product, biseparable, GHZ-type, and W-type states. When compared to the three-tangle and tripartite negativity, our findings indicate that this measure yields complementary insights into multipartite correlations. The outcomes underscore the significance of this measure in elucidating the structure and distribution of entanglement within three-qubit quantum systems.
Abstract In this paper we study the entanglement in symmetric N-quDit systems. In particular we use generalizations to U(D) of spin U(2) coherent states (CSs) and their projections on definite parity c ∈ Z 2 D − 1 (multicomponent Schrödinger cat) states and we analyse their reduced density matrices when tracing out M < N quDits. The eigenvalues (or Schmidt coefficients) of these reduced density matrices are completely characterized, allowing to prove a theorem for the decomposition of a N-quDit Schrödinger cat state with a given parity c into a sum over all possible parities of tensor products of Schrödinger cat states of N − M and M particles. Diverse asymptotic properties of the Schmidt eigenvalues are studied and, in particular, for the (rescaled) double thermodynamic limit ( N , M → ∞ , M / N fixed), we reproduce and generalize to quDits known results for photon loss of parity adapted CSs of the harmonic oscillator, thus providing an unified Schmidt decomposition for both multi-quDits and (multi-mode) photons. These results allow to determine the entanglement properties of these states and also their decoherence properties under quDit loss, where we demonstrate the robustness of these states.
Entanglement plays a crucial role in quantum physics and is the key resource in quantum information processing. In entanglement theory, Schmidt decomposition is a powerful tool to analyze the fundamental properties and structure of quantum entanglement. This work introduces a hybrid quantum-classical algorithm for Schmidt decomposition of bipartite pure states on near-term quantum devices. First, we show that the Schmidt decomposition task could be accomplished by maximizing a cost function utilizing bi-local quantum neural networks. Based on this, we propose a variational quantum algorithm for Schmidt decomposition (named VQASD) of which the cost function evaluation notably requires only one estimate of expectation with no extra copies of the input state. In this sense, VQASD outperforms existent approaches in resource cost and hardware efficiency. Second, by further exploring VQASD, we introduce a variational quantum algorithm to estimate the logarithm negativity, which can be applied to efficiently quantify entanglement of bipartite pure states. Third, we experimentally implement our algorithm on Quantum Leaf using the IoP CAS superconducting quantum processor. Both experimental implementations and numerical simulations exhibit the validity and practicality of our methods for analyzing and quantifying entanglement on near-term quantum devices.
Efficient methods to access the entanglement of a quantum many-body state, where the complexity generally scales exponentially with the system size N, have long been a concern. Here we propose the Schmidt tensor network state (Schmidt TNS) that efficiently represents the Schmidt decomposition of finite- and even infinite-size quantum states with nontrivial bipartition boundary. The key idea is to represent the Schmidt coefficients (i.e., entanglement spectrum) and transformations in the decomposition to tensor networks (TNs) with linearly scaled complexity versus N. Specifically, the transformations are written as the TNs formed by local unitary tensors, and the Schmidt coefficients are encoded in a positive-definite matrix product state (MPS). Translational invariance can be imposed on the TNs and MPS for the infinite-size cases. The validity of Schmidt TNS is demonstrated by simulating the ground state of the quasi-one-dimensional spin model with geometrical frustration. Our results show that the MPS encoding the Schmidt coefficients is weakly entangled even when the entanglement entropy of the decomposed state is strong. This justifies the efficiency of using MPS to encode the Schmidt coefficients, and promises an exponential speedup on the full-state sampling tasks.
Schmidt decomposition is a widely employed tool of quantum theory which plays a key role for distinguishable particles in scenarios such as entanglement characterization, theory of measurement and state purification. Yet, its formulation for identical particles remains controversial, jeopardizing its application to analyze general many-body quantum systems. Here we prove, using a newly developed approach, a universal Schmidt decomposition which allows faithful quantification of the physical entanglement due to the identity of particles. We find that it is affected by single-particle measurement localization and state overlap. We study paradigmatic two-particle systems where identical qubits and qutrits are located in the same place or in separated places. For the case of two qutrits in the same place, we show that their entanglement behavior, whose physical interpretation is given, differs from that obtained before by different methods. Our results are generalizable to multiparticle systems and open the way for further developments in quantum information processing exploiting particle identity as a resource.
In this study, local unitary (LU) properties of two- and three-qubit quantum systems are studied. Specifically, the Schmidt decomposition approach to LU classification for two qubits is re-examined using a more general approach, i.e. higher order singular value decomposition (HOSVD). Later, HOSVD is used to classify three-qubit pure states by LU operations. We found that due to HOSVD, it is possible to characterize the entanglement classes of three qubits by the eigenvalues distribution of its one-particle reduced density matrices. This finding generalized the similar classification results in the literature and it is hoped that this work will fill in the gap of LU classification from bipartite to multipartite quantum states by using HOSVD.