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Monogamy of entanglement between cones

Guillaume Aubrun, Alexander Müller-Hermes, Martin Plávala

TL;DR

The paper generalizes monogamy of entanglement to the setting of general convex cones by analyzing the minimal and maximal tensor products $\mathsf{C}_A \otimes_{\min} \mathsf{C}_B$ and $\mathsf{C}_A \otimes_{\max} \mathsf{C}_B$ through an extendibility hierarchy. It proves that $\mathsf{C}_A \otimes_{\min} \mathsf{C}_B$ equals the infinite-extendibility cone $\mathrm{Ext}_{\infty}(\mathsf{C}_A,\mathsf{C}_B,\phi)$ for any interior $\phi$ of $\mathsf{C}_B^*$, and characterizes when the hierarchy stops finitely: this occurs for all $\mathsf{C}_A$ precisely when the base $K_\phi$ is affinely equivalent to a product of at most $k$ simplices. The authors connect this finite stoppage to a polytope-theoretic property and establish a new product-of-simplices characterization, leveraging a de Finetti-type representation. The results unify and extend quantum monogamy insights to a broad convex-cone framework and reveal deep connections between tensor-product structure and polytope geometry, with notable implications for entanglement-breaking maps and separability criteria in both classical and quantum contexts.

Abstract

A separable quantum state shared between parties $A$ and $B$ can be symmetrically extended to a quantum state shared between party $A$ and parties $B_1,\ldots ,B_k$ for every $k\in\mathbf{N}$. Quantum states that are not separable, i.e., entangled, do not have this property. This phenomenon is known as "monogamy of entanglement". We show that monogamy is not only a feature of quantum theory, but that it characterizes the minimal tensor product of general pairs of convex cones $\mathsf{C}_A$ and $\mathsf{C}_B$: The elements of the minimal tensor product $\mathsf{C}_A\otimes_{\min} \mathsf{C}_B$ are precisely the tensors that can be symmetrically extended to elements in the maximal tensor product $\mathsf{C}_A\otimes_{\max} \mathsf{C}^{\otimes_{\max} k}_B$ for every $k\in\mathbf{N}$. Equivalently, the minimal tensor product of two cones is the intersection of the nested sets of $k$-extendible tensors. It is a natural question when the minimal tensor product $\mathsf{C}_A\otimes_{\min} \mathsf{C}_B$ coincides with the set of $k$-extendible tensors for some finite $k$. We show that this is universally the case for every cone $\mathsf{C}_A$ if and only if $\mathsf{C}_B$ is a polyhedral cone with a base given by a product of simplices. Our proof makes use of a new characterization of products of simplices up to affine equivalence that we believe is of independent interest.

Monogamy of entanglement between cones

TL;DR

The paper generalizes monogamy of entanglement to the setting of general convex cones by analyzing the minimal and maximal tensor products and through an extendibility hierarchy. It proves that equals the infinite-extendibility cone for any interior of , and characterizes when the hierarchy stops finitely: this occurs for all precisely when the base is affinely equivalent to a product of at most simplices. The authors connect this finite stoppage to a polytope-theoretic property and establish a new product-of-simplices characterization, leveraging a de Finetti-type representation. The results unify and extend quantum monogamy insights to a broad convex-cone framework and reveal deep connections between tensor-product structure and polytope geometry, with notable implications for entanglement-breaking maps and separability criteria in both classical and quantum contexts.

Abstract

A separable quantum state shared between parties and can be symmetrically extended to a quantum state shared between party and parties for every . Quantum states that are not separable, i.e., entangled, do not have this property. This phenomenon is known as "monogamy of entanglement". We show that monogamy is not only a feature of quantum theory, but that it characterizes the minimal tensor product of general pairs of convex cones and : The elements of the minimal tensor product are precisely the tensors that can be symmetrically extended to elements in the maximal tensor product for every . Equivalently, the minimal tensor product of two cones is the intersection of the nested sets of -extendible tensors. It is a natural question when the minimal tensor product coincides with the set of -extendible tensors for some finite . We show that this is universally the case for every cone if and only if is a polyhedral cone with a base given by a product of simplices. Our proof makes use of a new characterization of products of simplices up to affine equivalence that we believe is of independent interest.
Paper Structure (9 sections, 13 theorems, 55 equations, 2 figures)

This paper contains 9 sections, 13 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic properties of the extendibility hierarchy and examples
  3. The extendibility hierarchy
  4. The dual of the extendibility hierarchy
  5. Examples
  6. Proof of Theorem \ref{['thm:Main1']}
  7. Proof of Theorem \ref{['thm:Main2']}
  8. Proof of Theorem \ref{['theorem:polysimplices']}
  9. Proof of Proposition \ref{['prop:quantum']}

Key Result

Theorem 1

Consider proper cones $\mathsf{C}_A\subseteq V_A$ and $\mathsf{C}_B\subseteq V_B$ and a linear form $\phi$ in the interior of $\mathsf{C}_B^*$. Then, we have where, for an integer $k \geqslant 1$, the $k$th reduction map$\gamma_k^\phi : V_B^{\otimes k} \to V_B$ is defined as

Figures (2)

  • Figure 1: The planar case of Theorem \ref{['theorem:polysimplices']}: if a polygon is neither a triangle nor a parallelogram, it has disjoint edges whose affine hulls intersect.
  • Figure 2: The cone $\mathsf{C}_B$ over a square

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Proposition 5: Proved in Appendix \ref{['app:quantum']}
  • Corollary 6
  • Lemma 7
  • proof
  • Proposition 8
  • ...and 15 more