[Paper Review] Geometric measure of entanglement of symmetric d-qubits is polynomial-time computable

Shmuel Friedland, Li Wang|arXiv (Cornell University)|Aug 3, 2016

Tensor decomposition and applications20 references3 citations

TL;DR

This paper establishes that the geometric measure of entanglement for symmetric d-qubits is polynomial-time computable by deriving a closed-form formula for the spectral norm of symmetric d-mode tensors in two variables. The method reduces the problem to finding roots of a univariate polynomial of degree at most (d−1)²+1, enabling efficient computation via standard root-finding algorithms, with a polynomial-time approximation for exceptional cases.

ABSTRACT

We give a simple formula for finding the spectral norm of d-mode symmetric tensor in two variables over the complex or real numbers in terms of the complex or real roots of a corresponding polynomial in one complex variable. This result implies that the geometric measure of entanglement of symmetric d-qubits is polynomial-time computable. We discuss a generalization to d-mode symmetric tensor in more than two variables.

Motivation & Objective

• To resolve the computational complexity of the geometric measure of entanglement for symmetric d-qubits, which is central in quantum information theory.
• To develop an analytic method for computing the spectral norm of symmetric d-qubits over complex and real fields.
• To show that the spectral norm computation reduces to solving a univariate polynomial equation, enabling polynomial-time algorithms.
• To extend the framework to real symmetric tensors and provide approximation for exceptional cases.
• To conjecture polynomial-time computability for symmetric d-qudits with n ≥ 3 levels.

Proposed method

• Derives a formula for the spectral norm of a symmetric d-qubit as the maximum of a function involving roots of a univariate polynomial of degree at most (d−1)²+1.
• Uses the spectral norm characterization via Banach's theorem, relating it to critical points of homogeneous polynomials on the unit sphere.
• Applies degree theory to bound the number of fixed and anti-fixed points of polynomial maps in complex and real spaces.
• Reduces the problem to computing complex or real roots of a derived polynomial zv(z) − u(z), depending on the tensor's symmetry and field.
• Employs standard numerical root-finding algorithms with polynomial-time precision guarantees for univariate polynomials.
• Provides a polynomial-time approximation algorithm for the exceptional family of symmetric d-qubits where the main formula fails.

Experimental results

Research questions

• RQ1 Can the geometric measure of entanglement of symmetric d-qubits be computed in polynomial time?
• RQ2 What is the relationship between the spectral norm of a symmetric d-qubit and the roots of a univariate polynomial?
• RQ3 How does the spectral norm computation differ between real and complex symmetric d-qubits?
• RQ4 What is the structure of critical points of the polynomial map associated with symmetric tensors?
• RQ5 Is the spectral norm of symmetric d-qudits with n ≥ 3 levels also polynomial-time computable?

Key findings

• The spectral norm of a symmetric d-qubit is computable in polynomial time by reducing the problem to finding roots of a univariate polynomial of degree at most (d−1)²+1.
• For real symmetric d-qubits, the real spectral norm depends only on the real roots of a polynomial of degree at most d+1.
• The geometric measure of entanglement of symmetric d-qubits is polynomial-time computable due to efficient root-finding algorithms for univariate polynomials.
• In Example A.15, the spectral norm was computed as 0.3953, matching the known value √(5/32).
• In Example A.16, the spectral norm was computed as 0.4125, consistent with the theoretical value √(28/243).
• The method provides a polynomial-time approximation for the exceptional family of symmetric d-qubits where the main formula does not apply.