Reveals Hierarchical Approximations to Geometric Measure of Entanglement in Multiparticle Systems

Researchers are continually seeking more effective ways to quantify entanglement in complex quantum systems. Lisa T. Weinbrenner, Albert Rico, and Kenneth Goodenough, from the Naturwissenschaftlich-Technische Fakultät at Universität Siegen, alongside Xiao-Dong Yu and Otfried Gühne, have developed a novel method for determining the maximal overlap of a multiparticle state with product states, offering a new approach to the geometric measure of entanglement. This work introduces three hierarchical approximations which demonstrably converge to the true value, representing a significant advance as it allows for more efficient calculation of entanglement and has broader implications for optimisation problems in physics and multilinear algebra. Furthermore, the findings provide tools for identifying weakly entangled states and designing robust separability tests, ultimately illuminating the inherent challenges in determining whether a quantum state is entangled or not.

This research addresses a fundamental problem in quantum physics where understanding entanglement, a key resource for technologies like quantum computing and cryptography, requires quantifying the distance between a quantum state and the set of separable states.

The team achieved this by considering multiple copies of the pure state and formulating a series of increasingly accurate approximations, offering a powerful new tool for analysing multipartite entanglement. The study reveals that the geometric measure of entanglement, which quantifies entanglement as the geometric distance to separable states, can be computed using these hierarchical approximations.
Researchers leveraged concepts from multilinear algebra, specifically the injective tensor norm, to develop this approach, which is applicable to systems with any number of particles and dimensions. Experiments show that the hierarchies provide both upper and lower bounds on the entanglement measure.

Quantifying multipartite entanglement via multi-copy overlap and symmetric projection is a challenging task

Scientists are developing novel methods to quantify entanglement in multiparticle quantum systems, addressing a computationally difficult problem central to quantum information theory. The research introduces a technique for determining the maximal overlap of a pure multiparticle state with product states by examining multiple copies of the original state.

This approach yields three hierarchical approximations, each demonstrably converging towards the true value, offering increasingly precise results with greater computational effort. The study pioneers a multi-copy approach, rewriting the maximal overlap problem, defined as Λ2(ψ) = max |abc⟩|⟨abc|ψ⟩|2 for a three-partite state |ψ⟩ , within a two-copy Hilbert space.

Researchers then express this as Λ2(ψ) = max |abc⟩|⟨abc|⊗2|ψ⟩⊗2|, enabling manipulation using projection operators. Crucially, the team harnessed the symmetry of the quantum state to project onto the symmetric subspace of two particles, simplifying the calculation without altering the final result. Experiments employ the symmetric projection Π2 = (11 + V )/2, where V represents the SWAP operator, acting on the two copies of the state.

This projection acts trivially on the individual particle states but nontrivially on the entangled state, allowing for a more efficient computation of the maximal overlap. The work demonstrates that these hierarchical approximations are complete, meaning they converge to the actual solution as the level of approximation increases.

Numerical illustrations confirm the convergence behaviour for three- and five-qubit states, validating the method’s efficacy. Furthermore, the study extends the applicability of these hierarchies to construct entanglement witnesses for weakly entangled bipartite states and to design robust separability tests for mixed multiparticle states.

This innovative methodology also sheds light on the inherent complexity of determining separability, advancing understanding of fundamental limits in quantum information processing. The research builds upon earlier work concerning hierarchies for separability and optimization problems, offering a generalization without requiring symmetry assumptions, and establishing a complete hierarchy for the geometric measure of entanglement.

Convergence of hierarchical approximations to maximal overlap and entanglement bounds offers promising results

Scientists have developed a novel method for determining the maximal overlap of a pure multiparticle quantum state with product states, utilising multiple copies of the initial state. This research introduces three hierarchical approximations to this problem, all mathematically proven to converge towards the actual value.

The team measured the geometric measure of entanglement, a key indicator of quantum correlation, and demonstrated its computation through these new hierarchies. Results demonstrate that the approach allows for tackling optimizations over stochastic local transformations and designing strong separability tests for complex multiparticle states.

Experiments revealed that the maximal overlap, denoted as Λ2(ψ), can be calculated by considering the limit of increasingly accurate approximations. Specifically, the research establishes two-sided bounds for Λ2(ψ) expressed as dk∥|Fk⟩∥2/k ≤Λ2(ψ) ≤∥|Fk⟩∥2/k, where dk are coefficients converging to one as the number of copies, k, increases.

The team showed that applying symmetric projectors, denoted as Πk, to k copies of the quantum state leads to increasingly tighter upper bounds on Λ2. Measurements confirm that these bounds converge to the true value of the injective tensor norm, a crucial parameter in multilinear algebra. Data shows the hierarchies are complete, meaning the approximations improve with each iteration, approaching the actual solution as k tends to infinity.

The hierarchies rely on a multi-copy approach, where operators act on multiple instances of the target state, refining the bounds with each added copy. Numerical illustrations for three- and five-qubit states demonstrate the convergence behaviour of these hierarchies. Furthermore, the study reveals connections to entanglement witnesses, distillability checks, and characterization of entanglement in mixed states.

The breakthrough delivers a new understanding of the complexity of separability tests, building upon earlier work by Helfgott and Friedland. Scientists recorded that the formulation extends these previous approaches without requiring symmetry assumptions, offering a complete and robust solution. This work advances the ability to quantify entanglement, with potential applications in quantum metrology, cryptography, and communication.

Hierarchical approximations quantify multiparticle entanglement and separability in many-body systems

Scientists have developed a novel method for determining the maximal overlap of a multiparticle quantum state with product states, utilising multiple copies of the original state. This research introduces three hierarchical approximations, each demonstrably converging towards the true value of this overlap.

The approach allows for calculation of the geometric measure of entanglement and offers potential applications in optimisation problems involving stochastic local transformations, the creation of entanglement witnesses for weakly entangled systems, and the design of robust separability tests for mixed states. The findings advance understanding of the complexity inherent in determining whether a quantum state is separable, meaning it can be described as a simple product of independent particle states.

Researchers demonstrated the convergence behaviour of these hierarchies for specific three- and five-qubit states, and adapted the results for constructing entanglement witnesses and assessing distillability. Notably, one of the hierarchies shares connections with existing product tests for entanglement, leading to a refined version applicable to multiple copies of the state.

This work extends previous mathematical investigations concerning the injective norm of tensors, removing the need for symmetry assumptions and providing a complete formulation. Acknowledging limitations, the authors note that the initial estimate used in their derivation was not entirely optimal, potentially neglecting some product structure.

Future research could focus on refining this estimate to improve the accuracy and efficiency of the hierarchical approximations. The team also suggests exploring the application of these hierarchies to a wider range of quantum systems and investigating their potential for developing more powerful tools for quantum information processing and fundamental tests of quantum mechanics.