The Holant theorem is a powerful tool for studying the computational complexity of counting problems in the Holant framework. Due to the great expressiveness of the Holant framework, a converse to the Holant theorem would itself be a very powerful counting indistinguishability theorem. The most general converse does not hold, but we prove the following, still highly general, version: if any two sets of real-valued signatures are Holant-indistinguishable, then they are equivalent up to an orthogonal transformation. This resolves a partially open conjecture of Xia (2010). Consequences of this theorem include the well-known result that homomorphism counts from all graphs determine a graph up to isomorphism, the classical sufficient condition for simultaneous orthogonal similarity of sets of real matrices, and a combinatorial characterization of simultaneosly orthogonally decomposable (odeco) sets of symmetric tensors.

翻译：Holant定理是研究Holant框架下计数问题计算复杂性的有力工具。由于Holant框架具有极强的表达能力，Holant定理的逆定理本身将成为一个非常强大的计数不可区分性定理。最一般的逆定理并不成立，但我们证明了以下仍具有高度一般性的版本：若任意两组实值签名是Holant不可区分的，则它们在一个正交变换下等价。这解决了夏（2010）提出的部分开放猜想。该定理的推论包括：所有图的同态计数可确定图同构（这一众所周知的结果）、实矩阵集合同时正交相似的经典充分条件，以及对称张量集合同时正交可分解（odeco）的组合刻画。