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Calculating the Projective Norm of higher-order tensors using a gradient descent algorithm

Aaditya Rudra, Maria Anastasia Jivulescu

TL;DR

The paper addresses the NP-hard problem of computing the projective tensor norm and proposes a gradient-descent algorithm that converges to the global minimum, yielding the nuclear rank decomposition for higher-order tensors. It builds a CPD-based framework with adaptive and symmetric variants, including extensions to density matrices, to estimate the projective norm for both pure and mixed quantum states. The approach is validated against analytic results for low-order tensors and benchmarked on higher-order cases, demonstrating accurate entanglement detection and scalable computation, with open-source code provided. This work offers a practical, theoretically grounded tool for quantifying entanglement via the projective norm in large Hilbert spaces.

Abstract

Projective Norms are a class of tensor norms that map on the input and output spaces. These norms are useful for providing a measure of entanglement. Calculating the projective norms is an NP-hard problem, which creates challenges in computing due to the complexity of the exponentially growing parameter space for higher-order tensors. We develop a novel gradient descent algorithm to estimate the projective norm of higher-order tensors. The algorithm guarantees convergence to a minimum nuclear rank decomposition of our given tensor. We extend our algorithm to symmetric tensors and density matrices. We demonstrate the performance of our algorithm by computing the nuclear rank and the projective norm for both pure and mixed states and provide numerical evidence for the same.

Calculating the Projective Norm of higher-order tensors using a gradient descent algorithm

TL;DR

The paper addresses the NP-hard problem of computing the projective tensor norm and proposes a gradient-descent algorithm that converges to the global minimum, yielding the nuclear rank decomposition for higher-order tensors. It builds a CPD-based framework with adaptive and symmetric variants, including extensions to density matrices, to estimate the projective norm for both pure and mixed quantum states. The approach is validated against analytic results for low-order tensors and benchmarked on higher-order cases, demonstrating accurate entanglement detection and scalable computation, with open-source code provided. This work offers a practical, theoretically grounded tool for quantifying entanglement via the projective norm in large Hilbert spaces.

Abstract

Projective Norms are a class of tensor norms that map on the input and output spaces. These norms are useful for providing a measure of entanglement. Calculating the projective norms is an NP-hard problem, which creates challenges in computing due to the complexity of the exponentially growing parameter space for higher-order tensors. We develop a novel gradient descent algorithm to estimate the projective norm of higher-order tensors. The algorithm guarantees convergence to a minimum nuclear rank decomposition of our given tensor. We extend our algorithm to symmetric tensors and density matrices. We demonstrate the performance of our algorithm by computing the nuclear rank and the projective norm for both pure and mixed states and provide numerical evidence for the same.

Paper Structure

This paper contains 13 sections, 4 theorems, 29 equations, 19 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. The projective norm and its application in quantum information theory
  3. Algorithm
  4. Alternating Method
  5. Canonical Polyadic Decomposition
  6. Adaptive Rank Canonical Polyadic Decomposition
  7. Nuclear Rank Canonical Polyadic Decomposition
  8. Symmetric Nuclear Rank Canonical Polyadic Decomposition
  9. Nuclear Rank Canonical Polyadic Density Matrix Decomposition
  10. Computation and Results
  11. Projective Norm Computations for Tensors
  12. Projective Norm Computations for Density Matrices
  13. Conclusions

Key Result

Proposition 3

A multipartite pure state $|\psi\rangle \in H_1 \otimes \cdots \otimes H_k$, $\|\psi\|_2=1$, is separable if and only if

Figures (19)

  • Figure 1: Projective Norm and Nuclear rank of the Bell state $|\phi^+\rangle$ in $\mathbb C$
  • Figure 2: Projective Norm and Nuclear rank of a 2 qubit separable state in $\mathbb C$
  • Figure 3: Projective Norm and Nuclear rank of the 3 qubit GHZ state in $\mathbb C$
  • Figure 4: Projective Norm and Nuclear rank of the 3 qubit W state in $\mathbb R$
  • Figure 5: Projective Norm and Nuclear rank of the 3 qubit W state in $\mathbb C$
  • ...and 14 more figures

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Proposition 3
  • Theorem 4
  • Proposition 5
  • Corollary 6
  • Definition 7