Abstract The problem of quantifying entanglement in multiparticle quantum systems can be addressed using techniques from the invariant theory of Lie groups. We briefly review this theory, and then develop these techniques for application to entanglement of multiple qubits.
Abstract Entanglement is a key ingredient for quantum technologies and a fundamental signature of quantumness in a broad range of phenomena encompassing many-body physics, thermodynamics, cosmology and life sciences. For arbitrary multiparticle systems, entanglement quantification typically involves nontrivial optimisation problems, and it may require demanding tomographical techniques. Here, we develop an experimentally feasible approach to the evaluation of geometric measures of multiparticle entanglement. Our framework provides analytical results for particular classes of mixed states of N qubits, and computable lower bounds to global, partial, or genuine multiparticle entanglement of any general state. For global and partial entanglement, useful bounds are obtained with minimum effort, requiring local measurements in just three settings for any N. For genuine entanglement, a number of measurements scaling linearly with N are required. We demonstrate the power of our approach to estimate and quantify different types of multiparticle entanglement in a variety of N-qubit states useful for quantum information processing and recently engineered in laboratories with quantum optics and trapped ion setups.
This paper introduces the Entanglement Dimension, a novel group invariant derived from the application of quantum information theory concepts to algebraic structures. Traditional group invariants, such as order, nilpotency class, and solvability length, offer crucial insights but often fail to distinguish between non-isomorphic groups sharing many common properties. By mapping group elements or their representations to quantum states and subsequently quantifying the entanglement present in superpositions formed from these states, we propose a new measure that captures nuanced aspects of group structure. We rigorously define the Entanglement Dimension and demonstrate its computation for various finite groups, including cyclic, dihedral, and symmetric groups. Our results indicate that this invariant can differentiate between groups previously indistinguishable by classical methods, providing a finer characterization of their internal symmetries and interactions. The Entanglement Dimension offers a fresh perspective on group theory, bridging algebraic structures with quantum mechanical principles and opening new avenues for research in both fields, particularly in areas like quantum computing and fundamental physics.
This dissertation is a study of several aspects of discrete time quantum control. After reviewing the necessary background from quantum mechanics and control theory we construct a simple model and solve a control problem on it as a proof of concept for the discrete control formulation developed herein. We apply optimal control to solve a state preparation problem on a physically motivated time discretized single qubit system with simplified dynamics to enable the Bellman equation to be solved both numerically and approximately analytically. However for more than one qubit classical theories of control become problematic, a notable difficulty being the exponential increase in the size of the state space with the number of particles. To study workarounds for this problem we quantify how much information a quantum measurement can extract from the system and then use the results to show that a naive classical technique for controlling a multiparticle system performs poorly when the system has "quantum" properties. We then consider the problem of protecting the entanglement of a bipartite quantum system using discrete open loop control with a technique that seems to have no classical analog. The states of the bipartite system, called X-states, have a special structure which is generalized to states of systems with more than two qubits. We develop the properties of these generalized X-states which are a class of mixed states with interesting multipartite entanglement properties
This dissertation is a study of several aspects of discrete time quantum control. After reviewing the necessary background from quantum mechanics and control theory we construct a simple model and solve a control problem on it as a proof of concept for the discrete control formulation developed herein. We apply optimal control to solve a state preparation problem on a physically motivated time discretized single qubit system with simplified dynamics to enable the Bellman equation to be solved both numerically and approximately analytically. However for more than one qubit classical theories of control become problematic, a notable difficulty being the exponential increase in the size of the state space with the number of particles. To study workarounds for this problem we quantify how much information a quantum measurement can extract from the system and then use the results to show that a naive classical technique for controlling a multiparticle system performs poorly when the system has quantum properties. We then consider the problem of protecting the entanglement of a bipartite quantum system using discrete open loop control with a technique that seems to have no classical analog. The states of the bipartite system, called X-states, have a special structure which is generalized to states of systems with more than two qubits. We develop the properties of these generalized X-states which are a class of mixed states with interesting multipartite entanglement properties