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The joint numerical range of three $4\times 4$ matrices

Piotr Pikul, Ilya Spitkovsky, Konrad Szymański, Stephan Weis, Karol Życzkowski

TL;DR

This work analyzes the joint numerical range $W(A_1,A_2,A_3)$ for three Hermitian $4x4$ matrices, revealing a rich boundary structure beyond the generic smooth case. It provides a complete 15-type classification of non-elliptic (rank-3) boundary faces, with explicit $4x4$ examples for each class, and proves that the intersection of three distinct one-dimensional faces yields a corner point in dimension four. By introducing the separable joint numerical range via a two-qubit tensor-product framework, the paper compares its boundary to that of $W$ and develops computational SDP methods to study separable faces under PPT constraints. The findings illuminate the intricate geometry of quantum-state-induced convex bodies and have implications for entanglement geometry and classical-quantum separability in bipartite systems.

Abstract

We analyze the joint numerical range $W$ of three complex hermitian matrices of order four. In the generic case this $3D$ convex set has a smooth boundary. We analyze non-generic structures and investigate non-elliptic faces in the boundary $\partial W$. Fifteen possible classes regarding the numbers of non-elliptic faces are identified and an explicit example is presented for each class. Secondly, it is shown that a nonempty intersection of three mutually distinct one-dimensional faces is a corner point. Thirdly, introducing a tensor product structure into $\mathbb C^4=\mathbb C^2\otimes\mathbb C^2$, one defines the separable joint numerical range -- a subset of $W$ useful in studies of quantum entanglement. The boundary of the separable joint numerical range is compared with that of $W$.

The joint numerical range of three $4\times 4$ matrices

TL;DR

This work analyzes the joint numerical range for three Hermitian matrices, revealing a rich boundary structure beyond the generic smooth case. It provides a complete 15-type classification of non-elliptic (rank-3) boundary faces, with explicit examples for each class, and proves that the intersection of three distinct one-dimensional faces yields a corner point in dimension four. By introducing the separable joint numerical range via a two-qubit tensor-product framework, the paper compares its boundary to that of and develops computational SDP methods to study separable faces under PPT constraints. The findings illuminate the intricate geometry of quantum-state-induced convex bodies and have implications for entanglement geometry and classical-quantum separability in bipartite systems.

Abstract

We analyze the joint numerical range of three complex hermitian matrices of order four. In the generic case this convex set has a smooth boundary. We analyze non-generic structures and investigate non-elliptic faces in the boundary . Fifteen possible classes regarding the numbers of non-elliptic faces are identified and an explicit example is presented for each class. Secondly, it is shown that a nonempty intersection of three mutually distinct one-dimensional faces is a corner point. Thirdly, introducing a tensor product structure into , one defines the separable joint numerical range -- a subset of useful in studies of quantum entanglement. The boundary of the separable joint numerical range is compared with that of .

Paper Structure

This paper contains 7 sections, 22 theorems, 86 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Algebra and geometry of the joint numerical range
  3. Numbers of non-elliptic faces
  4. Examples
  5. One-dimensional faces and corner points
  6. On elliptic faces
  7. Separable joint numerical range

Key Result

Theorem 2.1

There is a one-to-one correspondence between access sequences of orthogonal projections $P_0\geq P_1\geq\dots\geq P_\ell$ with respect to $A_1,\dots,A_k$ and access sequences of faces $F_0 \supset F_1 \supset \dots \supset F_\ell$ with respect to $W$. For every $i=0,\ldots,\ell$ we have $F_i=w(\math

Figures (2)

  • Figure 1: Examples of numerical ranges of a $3\times 3$ matrix $B$ in the complex plane exemplifying the shapes of type 0 (oval), 1 (loaf) and 2 (droplet). Black asterisks mark the (in some cases, duplicate) eigenvalues of $B$. Small black dots mark the nonexposed faces of the numerical ranges --- endpoints of segments smoothly connected to the rest of the boundary. The missing numerical range of type 3 is a triangle, which is the convex hull of the three eigenvalues of a normal matrix, $BB^*=B^*B$.
  • Figure 2: (a) Separable numerical range (red) for randomly sampled matrices \ref{['eq:randommat']}; according to Remark \ref{['rem:numerics']}, the boundary contains a ruled surface (yellow region) -- the fact that it is a family of segments is exemplified by showing a family of them. (b) Joint and separable numerical range (transparent yellow and solid red, respectively), illustrating the content of Lemma \ref{['lem:facesnr']}: faces of the joint numerical range (the two circles at the top and bottom) contain points of the separable numerical range (black dots).

Theorems & Definitions (72)

  • Definition 1.1: Non-elliptic faces
  • Theorem 2.1
  • Example 2.2
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • Corollary 2.7
  • proof
  • ...and 62 more