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Tensors, entanglement, separability, and their complexity

Shmuel Friedland

TL;DR

The paper unifies quantum entanglement quantification with tensor-norm analysis by linking the geometric measure of entanglement to the spectral norm and an energy-like quantity to the nuclear norm. It provides sharp, norm-based characterizations of separability for density tensors in Hermitian, bi-symmetric, symmetric, and bi-skew-symmetric settings, and demonstrates polynomial-time computability for these norms in fixed dimension, while establishing NP-hardness in the general case. The work also shows that the separable states form a semi-algebraic set and that density-tensor separability aligns with strong separability in several tensor subspaces, enabling algebraic and geometric analysis. Overall, it offers a rigorous framework connecting entanglement measures to computable tensor norms and outlines a clear complexity frontier between tractable and intractable instances, with implications for quantum information and tensor optimization.

Abstract

One of the most challenging problems in quantum physics is to quantify the entanglement of $d$-partite states and their separability. We show here that these problems are best addressed using tensors. The geometric measure of entanglement of a pure state is one of most natural ways to quantify the entanglement, which is simply related to the spectral norm of a tensor state. On the other hand, the logarithm of the nuclear norm of the state and density tensors can be considered as its ``energy''. We first show that the most geometric measure entangled $d$-partite state has the minimum spectral norm and maximum nuclear norm. Second, we introduce the notion of Hermitian and density tensors, and the subspace of bi-symmetric Hermitian tensors, which correspond to Bosons. We show that separable density tensors, and strongly separable bi-symmetric density tensors are characterized by the value (equal to one) of their corresponding nuclear norms. In general, these characterizations are NP-hard to verify. Third, we show that the above quantities are computed in polynomial time when we restrict our attentions to Bosons: symmetric $d$-qubits, or more generally to symmetric $d$-qunits in $C^n$, and the corresponding bi-symmetric Hermtian density tensors, for a fixed value of $n$.

Tensors, entanglement, separability, and their complexity

TL;DR

The paper unifies quantum entanglement quantification with tensor-norm analysis by linking the geometric measure of entanglement to the spectral norm and an energy-like quantity to the nuclear norm. It provides sharp, norm-based characterizations of separability for density tensors in Hermitian, bi-symmetric, symmetric, and bi-skew-symmetric settings, and demonstrates polynomial-time computability for these norms in fixed dimension, while establishing NP-hardness in the general case. The work also shows that the separable states form a semi-algebraic set and that density-tensor separability aligns with strong separability in several tensor subspaces, enabling algebraic and geometric analysis. Overall, it offers a rigorous framework connecting entanglement measures to computable tensor norms and outlines a clear complexity frontier between tractable and intractable instances, with implications for quantum information and tensor optimization.

Abstract

One of the most challenging problems in quantum physics is to quantify the entanglement of -partite states and their separability. We show here that these problems are best addressed using tensors. The geometric measure of entanglement of a pure state is one of most natural ways to quantify the entanglement, which is simply related to the spectral norm of a tensor state. On the other hand, the logarithm of the nuclear norm of the state and density tensors can be considered as its ``energy''. We first show that the most geometric measure entangled -partite state has the minimum spectral norm and maximum nuclear norm. Second, we introduce the notion of Hermitian and density tensors, and the subspace of bi-symmetric Hermitian tensors, which correspond to Bosons. We show that separable density tensors, and strongly separable bi-symmetric density tensors are characterized by the value (equal to one) of their corresponding nuclear norms. In general, these characterizations are NP-hard to verify. Third, we show that the above quantities are computed in polynomial time when we restrict our attentions to Bosons: symmetric -qubits, or more generally to symmetric -qunits in , and the corresponding bi-symmetric Hermtian density tensors, for a fixed value of .

Paper Structure

This paper contains 28 sections, 45 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Related works
  4. A short summary of the paper
  5. Tensors
  6. Basic notions
  7. Symmetric tensors
  8. Fermions
  9. Nuclear rank
  10. Tensor formulations of entanglement
  11. Two measurements of entanglement
  12. Some known results on $\alpha_{\infty,\mathbf{n},\mathbb{F}}$
  13. The most geometric entangled 4-qubit
  14. Geometric measure of entanglement of Bosons
  15. Geometric measure of entanglement of Fermions
  16. ...and 13 more sections

Key Result

Theorem 2.1

Let $d\geqslant 2$ and $2\mathbf{1}_d\leqslant \mathbf{n}\in\mathbb{N}^d$. Assume that $0\ne\mathcal{X}\in\mathbb{F}^{\mathbf{n}}$. Then

Theorems & Definitions (89)

  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 79 more