Holant-Indistinguishability

• Holant-Indistinguishability is defined as the property where two sets of signatures produce identical Holant values on every admissible grid, reflecting a structural refinement of holographic equivalence.
• Recent results establish that for real-valued signatures, indistinguishability is equivalent to orthogonal equivalence, thereby affirming a key conjecture in the field.
• Vanishing signatures and orbit-closure phenomena play critical roles as obstructions to a full converse of the Holant theorem, highlighting complex invariant theory challenges.

Searching arXiv for recent and foundational papers on Holant-indistinguishability and related converse results. Holant-indistinguishability is the property that two sets of signatures parameterize exactly the same Holant values on every admissible signature grid, so that no Holant instance can distinguish between them. Within the Holant framework, this notion is a structural refinement of holographic equivalence: Valiant’s Holant theorem gives a forward implication from a change of basis to indistinguishability, while recent converse results show that, for real-valued signatures, indistinguishability is exactly orthogonal equivalence, and that in the general linear setting the essential obstruction is vanishing signatures together with orbit-closure phenomena in invariant theory (Young, 2024, Cai et al., 13 Sep 2025).

1. Formal definition within the Holant framework

Given a set $\mathcal{F}$ of signatures on domain $[q]$, an $\mathcal{F}$-grid $\Omega$ is a graph whose vertices are labeled by signatures from $\mathcal{F}$. Its Holant value is

$\holant_\Omega(\mathcal{F}) := \sum_{\sigma: E \to [q]} \prod_{v \in V} F_v(\sigma|_{\delta(v)}),$

where $F_v$ is the signature at vertex $v$ and $\sigma$ assigns domain values to edges. If $\mathcal{F}$ and $[q]$0 are sets of real-valued signatures with the same domain and a matching bijection between corresponding signatures of equal arity, then they are Holant-indistinguishable when, for every $[q]$1-grid $[q]$2,

$[q]$3

where $[q]$4 is obtained by replacing each occurrence of a signature in $[q]$5 by its matched signature in $[q]$6. In words, no Holant problem can distinguish $[q]$7 from $[q]$8 as parameters (Young, 2024).

The bipartite analogue is Bi-Holant-indistinguishability, where one compares bipartite or bigraded gadget systems. In that setting, the same principle applies: equality of all Bi-Holant evaluations defines indistinguishability (Cai et al., 13 Sep 2025).

A central background fact is Valiant’s Holant theorem: if two signature sets are related by a holographic transformation, then they are Holant-indistinguishable. Thus indistinguishability is always implied by an invertible change of basis, and the converse asks whether every indistinguishability phenomenon comes from such a transformation (Cai et al., 13 Sep 2025).

2. Real-valued converse: indistinguishability equals orthogonal equivalence

For real-valued signatures, the converse is affirmative in an orthogonal form. Let $[q]$9 and let $\mathcal{F}$0 be an $\mathcal{F}$1-ary tensor with vector flattening $\mathcal{F}$2. The transformed signature is defined by

$\mathcal{F}$3

and for a set $\mathcal{F}$4 one writes $\mathcal{F}$5. Two signature sets are ortho-equivalent if there exists $\mathcal{F}$6 such that $\mathcal{F}$7 (Young, 2024).

The main theorem of "The Converse of the Real Orthogonal Holant Theorem" states that for sets of real-valued signatures the following are equivalent:

$\mathcal{F}$8

Accordingly, two sets of real-valued signatures are Holant-indistinguishable if and only if they are related by a real orthogonal transformation. The paper presents this as resolving a partially open conjecture of Xia (2010) (Young, 2024).

The proof strategy emphasizes that the full converse for general linear transformations does not hold, but that the orthogonal real case remains highly general. Its key technical tool is intertwiner duality: the set of tensors invariant under the stabilizer group of $\mathcal{F}$9 matches exactly those that can be built as quantum $\Omega$0-gadgets. The proof proceeds by induction on domain size and separates the domain using signature matrices’ diagonals and restriction gadgets (Young, 2024).

A broader interpretation stated in the paper is that, in real-valued settings, Holant evaluations capture all information up to orthogonal transformation, so there are no combinatorial invariants in Holant beyond those induced by $\Omega$1 linear-algebraic actions (Young, 2024).

3. Failure of the full converse and the role of vanishing signatures

The most general converse to the Holant theorem is false. A decisive obstruction is the existence of vanishing signatures: signature sets that are indistinguishable from the zero signature because every admissible Holant value is zero (Cai et al., 2012, Cai et al., 13 Sep 2025).

In the Boolean symmetric setting, "A Complete Dichotomy Rises from the Capture of Vanishing Signatures" defines a set $\Omega$2 to be vanishing if

$\Omega$3

The paper gives a complete characterization of symmetric vanishing signatures via the classes $\Omega$4 and $\Omega$5 and states: a set of symmetric signatures $\Omega$6 is vanishing if and only if $\Omega$7 or $\Omega$8. It also records an explicit counterexample to a general converse: Holant problems with $\Omega$9 and with $\mathcal{F}$0 always yield the same Holant value on any grid, even though this indistinguishability is not explained by a holographic transformation alone (Cai et al., 2012).

The later paper "Vanishing Signatures, Orbit Closure, and the Converse of the Holant Theorem" isolates vanishing signatures as the only true obstacle to a converse. It proves two near-converses. First, finite signature sets are Holant-indistinguishable if and only if their $\mathcal{F}$1-orbit closures intersect: $\mathcal{F}$2 Second, if the sets are quantum-nonvanishing and Holant-indistinguishable, then there exists a holographic transformation $\mathcal{F}$3 such that $\mathcal{F}$4. The same paper states that finite $\mathcal{F}$5 is Bi-Holant-vanishing if and only if

$\mathcal{F}$6

This turns vanishing into a geometric condition on orbit closure (Cai et al., 13 Sep 2025).

The notion of quantum-nonvanishing is itself structural. The paper defines