[Paper Review] The geometric measure of entanglement of pure states with nonnegative amplitudes and the spectral theory of nonnegative tensors

Shenglong Hu, Liqun Qi+1 more|arXiv (Cornell University)|Mar 16, 2012

Tensor decomposition and applications21 references17 citations

TL;DR

This paper establishes a direct link between the geometric measure of entanglement for pure quantum states with nonnegative amplitudes and the spectral theory of nonnegative tensors, showing that the geometric measure equals the Z-spectral radius of the underlying symmetric tensor. It introduces an analytical elimination method for qubit and qutrit states and a randomized convergent algorithm for qudit states, enabling efficient computation via tensor eigenvalue theory.

ABSTRACT

The geometric measure of entanglement for a symmetric pure state with nonnegative amplitudes has attracted much attention. On the other hand, the spectral theory of nonnegative tensors (hypermatrices) has been developed rapidly. In this paper, we show how the spectral theory of nonnegative tensors can be applied to the study of the geometric measure of entanglement for a pure state with nonnegative amplitudes. Especially, an elimination method for computing the geometric measure of entanglement for symmetric pure multipartite qubit or qutrit states with nonnegative amplitudes is given. For symmetric pure multipartite qudit states with nonnegative amplitudes, a numerical algorithm with randomization is presented and proven to be convergent. We show that for the geometric measure of entanglement for pure states with nonnegative amplitudes, the nonsymmetric ones can be converted to the symmetric ones.

Motivation & Objective

• To bridge the geometric measure of entanglement in quantum states with nonnegative amplitudes and the spectral theory of nonnegative tensors.
• To develop analytical and numerical methods for computing the geometric measure of entanglement in symmetric and nonsymmetric pure states with nonnegative amplitudes.
• To prove that the geometric measure of such states corresponds to the Z-spectral radius of the associated nonnegative tensor.
• To establish a convergent randomized algorithm for computing the geometric measure in symmetric multipartite qudit states with nonnegative amplitudes.
• To demonstrate that nonsymmetric states with nonnegative amplitudes can be transformed into symmetric ones for computational purposes.

Proposed method

• Define the geometric measure of entanglement as the minimal distance to a separable state, equivalent to the largest singular value of the state's amplitude tensor.
• Establish that for symmetric pure states with nonnegative amplitudes, the geometric measure equals the Z-spectral radius of the corresponding nonnegative symmetric tensor.
• Propose a variable elimination method based on polynomial system solving to analytically compute the geometric measure for symmetric qubit and qutrit states.
• Develop a randomized numerical algorithm using the shifted higher-order power method, initialized from the positive orthant intersected with the unit sphere.
• Prove convergence of the algorithm with positive probability, dependent on the spectral gap between the largest and second-largest nonnegative Z-eigenvalues.
• Show that nonsymmetric states with nonnegative amplitudes can be symmetrized via a transformation, allowing application of symmetric methods.

Experimental results

Research questions

• RQ1 How can the spectral theory of nonnegative tensors be used to compute the geometric measure of entanglement for pure quantum states with nonnegative amplitudes?
• RQ2 What is the precise relationship between the Z-spectral radius of a nonnegative tensor and the geometric measure of entanglement of its corresponding symmetric quantum state?
• RQ3 Can an analytical method be derived for computing the geometric measure of symmetric multipartite qutrit states with nonnegative amplitudes?
• RQ4 What conditions ensure the convergence of a randomized numerical algorithm for computing the geometric measure in symmetric qudit states?
• RQ5 To what extent can nonsymmetric pure states with nonnegative amplitudes be reduced to symmetric cases for entanglement measure computation?

Key findings

• The geometric measure of entanglement for a symmetric pure state with nonnegative amplitudes is equal to the Z-spectral radius of the corresponding nonnegative symmetric tensor.
• An analytical elimination method is provided for symmetric multipartite qubit and qutrit states, offering a new derivation for the qutrit case.
• For symmetric multipartite qudit states with nonnegative amplitudes, a randomized numerical algorithm is proposed and proven to converge with positive probability.
• The convergence probability of the algorithm is determined by the spectral gap between the largest and second-largest nonnegative Z-eigenvalues of the underlying tensor.
• Nonsymmetric pure states with nonnegative amplitudes can be transformed into symmetric ones, enabling the use of symmetric computation methods.
• The largest singular value of the amplitude tensor corresponds to the geometric measure of entanglement for nonsymmetric states with nonnegative amplitudes.