Stabilizer AME States: Entanglement & Codes

• The paper identifies stabilizer AME states as multipartite pure states satisfying maximal mixing conditions through Abelian subgroup constraints, thereby underpinning robust quantum error correction.
• They are constructed via graph-state criteria and classical MDS code methods, enabling explicit extraction of stabilizer codes and practical protocols like secret sharing.
• No-go theorems and unique stabilizer support patterns delineate the limits of the stabilizer formalism, highlighting cases where non-stabilizer AME states may emerge.

Searching arXiv for the cited papers and closely related AME/stabilizer sources to ground the article in current literature. arxiv_search(query="Absolutely maximally entangled stabilizer graph states quantum error correction", max_results=10) Stabilizer absolutely maximally entangled states are multipartite pure states that simultaneously satisfy the AME condition—every reduction to at most half of the parties is maximally mixed—and the stabilizer condition of being the unique common $+1$-eigenstate of an Abelian subgroup of a generalized Pauli group. In the AME literature they occupy a central position because the same objects admit equivalent or closely parallel descriptions in terms of graph states, pure quantum error-correcting codes, quantum MDS behavior, perfect tensors, and secret-sharing resources. At the same time, recent results show that the stabilizer formalism is not universal: in composite local dimensions, and especially in the four-party six-dimensional case, stabilizer AME states are provably absent even though non-stabilizer AME states may still exist (Helwig, 2013, Raissi et al., 2017, Cha, 13 Mar 2026).

1. Definition and stabilizer setting

For $n$ parties of local dimension $d$, an AME state is a pure state

$|\Psi\rangle \in (\mathbb C^d)^{\otimes n}$

such that for every subset $S\subseteq[n]$ with $|S|\le \lfloor n/2\rfloor$,

$$\rho_S=\operatorname{Tr}_{S^c}(|\Psi\rangle\langle\Psi|)=\frac{I}{d^{|S|}}.$$

Equivalently, maximal entanglement is present across every bipartition; for $n=4$, this reduces to the requirement that every two-party marginal is completely mixed (Cha, 13 Mar 2026). In the coding-theoretic language, AME states are pure one-dimensional quantum codes whose low-weight reductions are fixed by maximal-mixing conditions (Huber et al., 2017).

In the prime-dimensional stabilizer formalism, the single-site generalized Pauli operators are

$$X=\sum_{k=0}^{d-1}|k+1\rangle\langle k|,\qquad Z=\sum_{k=0}^{d-1}\omega^k |k\rangle\langle k|,\qquad \omega=e^{2\pi i/d},$$

and a stabilizer state $|\Phi\rangle$ is specified by an Abelian subgroup $n$0 such that

$n$1

If the subgroup fixes a subspace rather than a unique vector, that subspace is a stabilizer code (Mazurek et al., 2019). A stabilizer AME state is therefore an AME state whose entanglement structure is encoded by commuting Pauli constraints.

Graph states provide the most common normal form. For prime local dimension $n$2, a graph with weighted adjacency matrix $n$3 yields stabilizer generators

$n$4

so every qudit graph state is a stabilizer state, and every stabilizer state is local-Clifford equivalent to a graph state (Helwig, 2013). This places stabilizer AME states inside a finite-field and symplectic framework in which bipartite entanglement can be read directly from graph data.

2. Graph-state criteria and code-theoretic correspondences

For graph states, the AME property can be tested by a linear-independence condition on the adjacency matrix. If $n$5 with $n$6, then a graph state is AME iff the truncated row vectors

$n$7

are linearly independent in $n$8 for every such $n$9 (Helwig, 2013). In equivalent terms, every balanced cut has full-rank cross-adjacency block. This criterion is the graph-state version of the statement that all balanced bipartitions are maximally entangled.

A major construction route starts from classical MDS codes. For minimal-support AME states, the support strings form a classical MDS code, and conversely a suitable MDS code yields an AME state by uniform superposition over codewords. In the linear case, with generator matrix $d$0, the state takes the form

$d$1

and the stabilizer can be read directly from the generator and parity-check matrices: $d$2-type stabilizers come from rows of $d$3, $d$4-type stabilizers from rows of $d$5 (Raissi et al., 2017). This is the most explicit route from classical coding theory to stabilizer AME states.

The correspondence can be pushed further. For every stabilizer $d$6-qudit AME state with prime local dimension, one obtains not just one code but an entire family: for each

$d$7

there is a stabilizer code encoding $d$8 logical qudits into $d$9 physical qudits. The extraction is constructive: after bringing a graph-state stabilizer into a suitable standard form by Gauss-Jordan elimination, column permutations, row swaps, and generator multiplication, truncating $|\Psi\rangle \in (\mathbb C^d)^{\otimes n}$0 designated inputs leaves $|\Psi\rangle \in (\mathbb C^d)^{\otimes n}$1 commuting stabilizer generators and $|\Psi\rangle \in (\mathbb C^d)^{\otimes n}$2 logical Pauli operators on the remaining $|\Psi\rangle \in (\mathbb C^d)^{\otimes n}$3 sites (Mazurek et al., 2019). The six-qubit AME state realizes this explicitly as $|\Psi\rangle \in (\mathbb C^d)^{\otimes n}$4, $|\Psi\rangle \in (\mathbb C^d)^{\otimes n}$5, and $|\Psi\rangle \in (\mathbb C^d)^{\otimes n}$6 codes.

3. Stabilizer support structure and operational consequences

Stabilizer AME states have a rigid support pattern that is not visible from the AME definition alone. For an $|\Psi\rangle \in (\mathbb C^d)^{\otimes n}$7-qubit AME stabilizer state, if $|\Psi\rangle \in (\mathbb C^d)^{\otimes n}$8 is any subset of size

$|\Psi\rangle \in (\mathbb C^d)^{\otimes n}$9

then the subgroup of stabilizers supported only on $S\subseteq[n]$0 is nontrivial; more sharply, when $S\subseteq[n]$1 is odd there is exactly one nonidentity stabilizer supported on $S\subseteq[n]$2, and when $S\subseteq[n]$3 is even there are exactly three. In the qudit extension, the theorem becomes an existence statement: any choice of $S\subseteq[n]$4 parties supports at least one nontrivial stabilizer (Sudevan et al., 2024). This support theorem is a distinct structural signature of the stabilizer subclass.

That support property has an immediate operational interpretation in majority-agreed key distribution. AME maximal mixing implies that observables supported on at most $S\subseteq[n]$5 parties have vanishing expectation, so a usable perfect correlation must involve at least $S\subseteq[n]$6 parties. Stabilizer structure then upgrades necessity to necessity-and-sufficiency: for any two communicating parties, the cooperation of exactly

$S\subseteq[n]$7

other parties is sufficient, because a stabilizer supported on the resulting $S\subseteq[n]$8-party set always exists (Sudevan et al., 2024). In this sense, AME enforces a majority threshold, while the stabilizer subgroup makes that threshold algebraically usable.

The same algebraic control underlies secret sharing and tensor-network applications. Even-party AME graph states directly realize threshold and ramp quantum secret-sharing schemes in graph-state language, with authorized sets decoding by unitaries derived from graph labels (Helwig, 2013). In a different direction, AME-derived stabilizer codes can be concatenated by Bell-type projections, so entanglement swapping with an AME state becomes encoding into the associated code; this mechanism was used to analyze tensor-network and holographic-code constructions built from the six-qubit AME tensor (Mazurek et al., 2019).

4. Existence results and no-go theorems

The qubit landscape is highly constrained. AME states exist for qubits only at $S\subseteq[n]$9; there is no AME$|S|\le \lfloor n/2\rfloor$0, no AME$|S|\le \lfloor n/2\rfloor$1, and no qubit AME state for $|S|\le \lfloor n/2\rfloor$2 (Huber et al., 2017). Since stabilizer AME states are a subclass, these absences are already stabilizer no-go results. The four-qubit obstruction is especially fundamental: recent work rederives it by invariant theory, coding-theoretic arguments, Pauli commutation relations, shadow inequalities, and Lovász-theta methods, showing that a pure four-qubit AME state—and therefore any four-qubit stabilizer AME state—cannot exist (Huber et al., 26 Jun 2025).

Minimal-support existence has an exact coding characterization. An $|S|\le \lfloor n/2\rfloor$3 state of minimal support exists iff there is an MDS code over $|S|\le \lfloor n/2\rfloor$4 of length $|S|\le \lfloor n/2\rfloor$5 and minimum distance

$|S|\le \lfloor n/2\rfloor$6

For prime-power $|S|\le \lfloor n/2\rfloor$7, this yields the standard sufficient family $|S|\le \lfloor n/2\rfloor$8 of minimal support, but the pattern fails at the first non-prime-power case: for $|S|\le \lfloor n/2\rfloor$9, there is no $$\rho_S=\operatorname{Tr}_{S^c}(|\Psi\rangle\langle\Psi|)=\frac{I}{d^{|S|}}.$$0 state of minimal support and $$\rho_S=\operatorname{Tr}_{S^c}(|\Psi\rangle\langle\Psi|)=\frac{I}{d^{|S|}}.$$1 (Bernal, 2018). This does not by itself rule out non-minimal-support or non-stabilizer AME$$\rho_S=\operatorname{Tr}_{S^c}(|\Psi\rangle\langle\Psi|)=\frac{I}{d^{|S|}}.$$2, but it already shows that the usual finite-field template breaks down at $$\rho_S=\operatorname{Tr}_{S^c}(|\Psi\rangle\langle\Psi|)=\frac{I}{d^{|S|}}.$$3.

General nonexistence tools come from quantum weight enumerators and shadow inequalities. Viewing AME states as pure one-dimensional codes, one can compute the unitary and shadow enumerators from the prescribed subsystem purities and test positivity constraints. This excludes many $$\rho_S=\operatorname{Tr}_{S^c}(|\Psi\rangle\langle\Psi|)=\frac{I}{d^{|S|}}.$$4 pairs beyond Scott’s bound; for example, the shadow enumerator rules out additional AME cases in dimensions $$\rho_S=\operatorname{Tr}_{S^c}(|\Psi\rangle\langle\Psi|)=\frac{I}{d^{|S|}}.$$5, and those exclusions automatically apply to the stabilizer subclass (Huber et al., 2017).

The sharpest recent stabilizer no-go in composite dimension is Cha’s theorem: $$\rho_S=\operatorname{Tr}_{S^c}(|\Psi\rangle\langle\Psi|)=\frac{I}{d^{|S|}}.$$6 Its proof uses the factorization theorem that every stabilizer state on $$\rho_S=\operatorname{Tr}_{S^c}(|\Psi\rangle\langle\Psi|)=\frac{I}{d^{|S|}}.$$7 six-dimensional systems is locally unitarily equivalent to a tensor product of a qubit stabilizer state and a qutrit stabilizer state,

$$\rho_S=\operatorname{Tr}_{S^c}(|\Psi\rangle\langle\Psi|)=\frac{I}{d^{|S|}}.$$8

A hypothetical stabilizer $$\rho_S=\operatorname{Tr}_{S^c}(|\Psi\rangle\langle\Psi|)=\frac{I}{d^{|S|}}.$$9 would therefore imply an $n=4$0, which is impossible. More generally, if a stabilizer $n=4$1 exists and $n=4$2 is decomposed into prime-power factors, then an $n=4$3 exists for every nonempty subset $n=4$4; for $n=4$5, this yields the corollary that there is no stabilizer $n=4$6 for any $n=4$7 (Cha, 13 Mar 2026).

An independent graph-state obstruction strengthens the picture in even local dimension. No AME graph state exists for

$n=4$8

Combined with Cha’s theorem, this implies that there are no $n=4$9-partite stabilizer AME states of local dimension $$X=\sum_{k=0}^{d-1}|k+1\rangle\langle k|,\qquad Z=\sum_{k=0}^{d-1}\omega^k |k\rangle\langle k|,\qquad \omega=e^{2\pi i/d},$$0 when $$X=\sum_{k=0}^{d-1}|k+1\rangle\langle k|,\qquad Z=\sum_{k=0}^{d-1}\omega^k |k\rangle\langle k|,\qquad \omega=e^{2\pi i/d},$$1 (Wójcik et al., 18 Mar 2026). Thus the failure of stabilizer constructions at $$X=\sum_{k=0}^{d-1}|k+1\rangle\langle k|,\qquad Z=\sum_{k=0}^{d-1}\omega^k |k\rangle\langle k|,\qquad \omega=e^{2\pi i/d},$$2 is part of an infinite family rather than an isolated anomaly.

5. Composite and mixed local dimensions, and the non-stabilizer boundary

A central conceptual distinction in current AME theory is between stabilizer and non-stabilizer states. The six-dimensional four-party case is exemplary: the no-go theorems rule out stabilizer and graph-state realizations, but they do not rule out AME$$X=\sum_{k=0}^{d-1}|k+1\rangle\langle k|,\qquad Z=\sum_{k=0}^{d-1}\omega^k |k\rangle\langle k|,\qquad \omega=e^{2\pi i/d},$$3 itself. The stabilizer obstruction should therefore be interpreted as a limitation of the stabilizer formalism rather than a complete classification of all AME states at that parameter point (Cha, 13 Mar 2026).

This boundary is now explicit at the circuit level. Explicit quantum circuits have been constructed for non-stabilizer AME$$X=\sum_{k=0}^{d-1}|k+1\rangle\langle k|,\qquad Z=\sum_{k=0}^{d-1}\omega^k |k\rangle\langle k|,\qquad \omega=e^{2\pi i/d},$$4, AME$$X=\sum_{k=0}^{d-1}|k+1\rangle\langle k|,\qquad Z=\sum_{k=0}^{d-1}\omega^k |k\rangle\langle k|,\qquad \omega=e^{2\pi i/d},$$5, and AME$$X=\sum_{k=0}^{d-1}|k+1\rangle\langle k|,\qquad Z=\sum_{k=0}^{d-1}\omega^k |k\rangle\langle k|,\qquad \omega=e^{2\pi i/d},$$6 states that neither have a graph-state representation nor are locally equivalent to such states. The circuits isolate the source of non-stabilizerness in a non-Clifford diagonal layer $$X=\sum_{k=0}^{d-1}|k+1\rangle\langle k|,\qquad Z=\sum_{k=0}^{d-1}\omega^k |k\rangle\langle k|,\qquad \omega=e^{2\pi i/d},$$7; the remaining ingredients are generalized Fourier and controlled gates of the same general type used in graph-state preparation (Casas et al., 7 Apr 2025). For four parties, this places stabilizer AME states inside the broader class of multi-unitary constructions, but not exhaustively so.

The notion of stabilizer AME states also generalizes beyond uniform local dimension. In a mixed-dimensional Hilbert space

$$X=\sum_{k=0}^{d-1}|k+1\rangle\langle k|,\qquad Z=\sum_{k=0}^{d-1}\omega^k |k\rangle\langle k|,\qquad \omega=e^{2\pi i/d},$$8

one introduces dimensional weight $$X=\sum_{k=0}^{d-1}|k+1\rangle\langle k|,\qquad Z=\sum_{k=0}^{d-1}\omega^k |k\rangle\langle k|,\qquad \omega=e^{2\pi i/d},$$9, a mixed-dimensional Singleton bound, and an AME condition indexed by subsystem product dimension rather than subsystem cardinality: $|\Phi\rangle$0 In that setting, a pure state is AME iff its span is a $|\Phi\rangle$1 code (Ball et al., 20 Oct 2025). The stabilizer formalism is broadened as well: the stabilizer group need only be an Abelian subgroup of unitary operators, often permutation-phase operators, and explicit mixed-dimensional AME examples exist in spaces such as $|\Phi\rangle$2.

6. Analytical tools and broader structural perspective

Much of the modern analysis of stabilizer AME states proceeds through code invariants rather than direct state construction. The Shor–Laflamme enumerators $|\Phi\rangle$3 and $|\Phi\rangle$4, Rains’ unitary enumerators $|\Phi\rangle$5, and the shadow enumerator $|\Phi\rangle$6 translate subsystem purities and low-weight error constraints into polynomial relations. The quantum MacWilliams identity

$|\Phi\rangle$7

and the shadow transform

$|\Phi\rangle$8

give necessary feasibility conditions for putative AME or stabilizer AME parameters (Huber et al., 2017). For stabilizer codes, these enumerators acquire a direct counting meaning: $|\Phi\rangle$9 counts stabilizer elements of weight $n$00, and $n$01 counts normalizer elements of weight $n$02.

A more recent perspective emphasizes stabilizer extremality. Stabilizer states uniquely saturate uncertainty principles involving the support size of the characteristic function or, in odd prime dimension, the Wigner function. They are also extremal for convex, local-unitary-invariant information measures within the class of states sharing the same stabilizer group (Bu, 2024). This suggests that when the mean state $n$03 is a stabilizer AME state, its subsystem entropy vector is maximal inside that stabilizer-group fiber, although AME itself is still defined by the sharper condition of exact maximal mixing on all reductions up to half the parties.

Taken together, these results place stabilizer AME states at a mathematically rigid intersection of multipartite entanglement, Pauli-module structure, and optimal coding behavior. In prime or prime-power settings they are often the canonical constructions; in composite and mixed settings they remain a powerful but non-exhaustive subclass. Current theory therefore treats stabilizer AME states both as a source of explicit AME families and as a precisely delimited frontier beyond which genuinely non-stabilizer multipartite entanglement must appear.