Non-Subhomogeneity of Minimal Operator Systems over Positive Semidefinite and Lorentz Cones

Technical Summary: Non-Subhomogeneity of Minimal Operator Systems over Positive Semidefinite and Lorentz Cones

Problem Statement
The paper investigates the structural properties of minimal operator systems associated with specific convex cones, specifically the cone of positive semidefinite (PSD) matrices $\text{Mat}_k(\mathbb{C})_+$ for $k \geq 2$ and the Lorentz cones $L_m$ for $m \geq 4$. The central question is whether these minimal operator systems are subhomogeneous.

An operator system is defined as $d$-subhomogeneous if it admits a unital complete order embedding into $\text{Mat}_d(A)$ for some commutative $C^*$-algebra $A$. In geometric terms, this corresponds to the existence of a finite-dimensional realization where the matrix size $d$ remains fixed regardless of the level of the system. The paper aims to determine if such a fixed $d$ exists for the specified cones.

Methodology
The proof strategy relies on duality and the construction of specific positive maps. The author employs the following logical reductions and constructions:

1. Reduction via Retracts: Using Lemma 2.1, the author establishes that if a cone $P$ is a unital positive retract of a cone $Q$, then the non-subhomogeneity of the minimal system over $P$ implies the non-subhomogeneity of the minimal system over $Q$.

   • Lemma 2.2 demonstrates that $\text{Mat}_2(\mathbb{C})_+$ is a retract of $\text{Mat}_k(\mathbb{C})_+$ for all $k \geq 2$, and $L_4$ is a retract of $L_m$ for all $m \geq 4$.
   • Consequently, the problem reduces to proving that the minimal system over $\text{Mat}_2(\mathbb{C})_+$ (which is order-isomorphic to $L_4$) is not $d$-subhomogeneous for any $d$.

2. Duality and Extreme Rays: The author utilizes the Jamiołkowski–Choi correspondence, which identifies positive maps $\Phi: \text{Mat}_2(\mathbb{C}) \to \text{Mat}_s(\mathbb{C})$ with block-positive matrices.

   • Lemma 3.1 establishes that if there exists an extreme positive map $\Phi$ from $\text{Mat}_2(\mathbb{C})$ to $\text{Mat}_s(\mathbb{C})$ (where $s > d$) such that $\Phi(I_2)$ is invertible, then the minimal system is not $d$-subhomogeneous. This is because such a map cannot be decomposed into a sum of compressions of maps into $\text{Mat}_d(\mathbb{C})$ due to rank constraints.

3. Explicit Construction: To satisfy the conditions of Lemma 3.1, the author constructs a specific family of maps using Woronowicz's symmetric-power construction (Section 4).

   • They define a map $\Phi$ on the symmetric tensor power $H = \text{Sym}_n(E)$ where $E = \mathbb{C}^2$ and $n = 2r+1$.
   • A key component is an alternating diagonal operator $\sigma$ acting on the symmetric power of dimension $n-1$.
   • The map is defined via an inverse of a specific Hermitian operator $\rho$ constructed from $\sigma$ and the embedding of symmetric powers.
   • The author proves that for every $r \geq 1$, this construction yields a unital, irreducible, extreme positive map $\Phi_r: \text{Mat}_2(\mathbb{C}) \to \text{Mat}_{2r+2}(\mathbb{C})$ where the output dimension $s = 2r+2$ can be arbitrarily large.

Key Contributions and Results

• Main Theorem (Theorem 2.3): For every $k \geq 2$ and $m \geq 4$, the minimal operator systems over $\text{Mat}_k(\mathbb{C})_+$ and $L_m$ are not $d$-subhomogeneous for any $d \in \mathbb{N}$.
• Construction of Extreme Maps: The paper explicitly constructs extreme positive maps from $\text{Mat}_2(\mathbb{C})$ to $\text{Mat}_s(\mathbb{C})$ of arbitrarily large output dimension $s$, where the value at the identity is invertible. This construction utilizes an alternating diagonal operator within the symmetric power framework.
• Entanglement Interpretation: The result is equivalently stated in terms of quantum information: For fixed $k \geq 2$ and $d$, there exist $s > d$ and an entangled positive operator $X \geq 0$ on $\mathbb{C}^k \otimes \mathbb{C}^s$ such that every compression of the second factor to $\mathbb{C}^d$ results in a separable operator.
• Sharpness of Thresholds: Remark 5.1 notes that the threshold for Lorentz cones is sharp. $L_2$ is simplicial (1-subhomogeneous), and $L_3$ admits a $2 \times 2$ realization (2-subhomogeneous). The non-subhomogeneity property strictly begins at $L_4$.

Significance and Scope
The paper addresses a specific question raised in prior literature [4] regarding the subhomogeneity of minimal operator systems over PSD cones. By proving that these systems lack a fixed finite-dimensional realization, the work clarifies the limitations of matrix-convex descriptions for these specific cones.

The author explicitly states in the AI Declaration that the results were produced almost entirely by an artificial intelligence system (ChatGPT 5.6 Sol), with the human author acting as a facilitator for posing the question, suggesting approaches, evaluating output, and editing the text. Consequently, the author does not claim ownership and will not submit the paper to a mathematical journal, inviting comments and corrections instead. The work serves as a rigorous verification of a mathematical result generated by AI, demonstrating the capability of such systems to construct complex proofs involving operator theory, tensor products, and convex geometry, provided they are subject to human verification and refinement.