On Non-Existence of Stabilizer Absolutely Maximally Entangled States in Even Local Dimensions

1. Problem Statement

The paper addresses the existence of Absolutely Maximally Entangled (AME) states within the class of stabilizer states (specifically graph states) for specific system configurations.

• AME States: Pure multipartite quantum states where every balanced reduction (tracing out half the parties) results in a maximally mixed state. They are crucial for quantum error correction, secret sharing, and holographic duality (AdS/CFT).
• The Conflict: While AME states are desirable, their existence is not guaranteed for all numbers of parties ($N$) and local dimensions ($d$). A specific open question has been whether stabilizer AME states exist for $N=4k$ parties with even local dimensions $d$.
• Specific Context: Recent work by Cha (arXiv:2603.13442) proved the non-existence of a stabilizer AME(4,6) state (4 parties, dimension 6) by decomposing dimension 6 into qubit $\otimes$ qutrit structures. The authors of this paper aim to generalize this result to an infinite family of states using a different, graph-theoretical approach.

2. Methodology

The authors employ a graph-theoretical analysis of stabilizer states rather than the algebraic decomposition used in previous works.

• Formalism:
  • They define graph states $|G\rangle$ via an adjacency matrix $\Gamma$ over local dimension $d$.
  • The stabilizer group $S$ is generated by operators $g_i = X_i \prod_{j} Z_j^{\Gamma_{ij}}$.
  • Lemma 1: They establish a necessary and sufficient condition for a graph state to be AME: For every non-identity stabilizer element $S \in S \setminus \{I\}$, the partial trace over $\lceil N/2 \rceil$ parties must vanish ($\text{Tr}_{\lceil N/2 \rceil}(S) = 0$).
• Proof Strategy:
  • The authors attempt to prove the non-existence of such states by contradiction. They assume an AME graph state exists for $N=4k$ and even $d$.
  • They construct a specific subset of stabilizer operators $B$ involving products of generators.
  • They formulate a system of linear equations over the finite ring $\mathbb{Z}_d$ (Eq. 14) to determine if there exists a stabilizer operator that acts as the Identity ($I$) on exactly $N/2$ qudits.
  • Key Insight: If such an operator exists, the partial trace is non-zero, violating the AME condition (Lemma 1).
  • Determinant Analysis: They analyze the determinant ($\Delta_j$) of the matrix associated with the system of equations. Using Fact 1, they argue that if the determinant is not a unit in $\mathbb{Z}_d$ (i.e., not coprime to $d$), the system has a non-trivial solution (a kernel), implying the existence of the "bad" stabilizer operator.
  • Summation Argument: They consider the alternating sum of these determinants (Eq. 15). By expanding via Laplace expansion and utilizing the symmetry of the adjacency matrix ($\Gamma_{a,b} = \Gamma_{b,a}$), they show that the terms cancel out in pairs, resulting in a total sum of zero (Eq. 18).
  • Parity Contradiction: This summation leads to a contradiction regarding the parity (odd/even nature) of the determinants in $\mathbb{Z}_d$ (where $d$ is even). Specifically, if all determinants were units (odd), the parity of the sums in Eq. 19 would be inconsistent. Therefore, at least one determinant must be non-invertible (not a unit), guaranteeing a non-trivial solution.

3. Key Contributions

• Generalized Non-Existence Theorem: The paper proves that no stabilizer AME states exist for any system with $N = 4k$ parties (where $k \in \mathbb{N}^+$) and even local dimension $d$.
• Graph-Theoretical Approach: Unlike the decomposition method used for AME(4,6), this proof relies directly on the structural properties of the graph state's adjacency matrix and linear algebra over finite rings. This provides an independent verification of the AME(4,6) case.
• Extension to Composite Dimensions: The result is extended to a broader corollary: No stabilizer AME states exist for $N=4k$ and any local dimension $d \equiv 2 \pmod 4$. This covers the $d=6$ case and generalizes it to $d=2, 6, 10, 14, \dots$.

4. Results

• Theorem 1: There are no AME $N$-partite graph states for $N=4k$ and even $d$.
• Corollary 1: Combining their result with Cha's work, they conclude that for $N=4k$ and $d \equiv 2 \pmod 4$, no stabilizer AME states exist at all (not just graph states, as all qubit stabilizer states are locally equivalent to graph states, and the decomposition logic extends to composite dimensions).
• Specific Case Resolution: The paper independently confirms the non-existence of the AME(4,6) stabilizer state, clarifying its characterization within the stabilizer formalism.

5. Significance and Implications

• Constraints on Quantum Architectures: The result imposes strict limits on the construction of highly entangled states using Clifford circuits (which generate stabilizer states). It implies that for specific configurations ($4k$ parties, even dimensions), one cannot rely on stabilizer codes to achieve absolute maximal entanglement.
• Distinction between Stabilizer and Non-Stabilizer: The findings highlight a fundamental boundary between stabilizer states and "magic" states (non-stabilizer states). Since stabilizer states are insufficient for these specific AME configurations, achieving such entanglement requires non-Clifford resources, which are essential for universal quantum computation.
• Future Directions: The authors note that it remains an open problem whether these exclusion results extend to other numbers of parties (e.g., $N \neq 4k$) or other local dimensions. Understanding these obstructions is vital for mapping the landscape of multipartite entanglement and designing quantum error-correcting codes.

In summary, the paper provides a rigorous, graph-theoretical proof that a large class of potential AME states (specifically those with $4k$ parties and even local dimensions) cannot be realized as stabilizer states, thereby narrowing the search space for such states and deepening the understanding of the limitations of Clifford-based quantum systems.