Multivariate Lee–Yang Property

• The multivariate Lee–Yang property is defined by generating polynomials of tensors that remain zero-free within specified domains, extending the classical Lee–Yang theorem to multivariate settings.
• It exhibits robust closure properties under tensor contractions and semigroup operations, which facilitate unique eigenvalue structures and efficient quantum state preparation.
• Applications include statistical physics, quantum Hamiltonians, and Fourier quasicrystals, where it unifies results on zero-freeness, spectral analysis, and computational tractability.

The multivariate Lee–Yang property is a central concept at the intersection of statistical physics, tensor network theory, quantum many-body systems, and harmonic analysis. It generalizes the celebrated Lee–Yang theorem on the zeros of partition functions in statistical mechanics to multivariate polynomials and tensors, providing a rigorous framework for describing zero-free regions, uniqueness theorems, structure under quantum operations, and the spectral theory of physically relevant Hamiltonians and Fourier quasicrystals. In the multivariate setting, the property characterizes those multilinear forms (or, equivalently, tensors) whose generating polynomials have zeros confined entirely outside a prescribed domain, typically a polydisk or its complement in $\mathbb C^n$. This zero-freeness imposes strong algebraic, analytic, and physical constraints, leading to closure properties, uniqueness results, efficient algorithmic consequences for quantum state preparation, and links to eigenvalue distributions in both quantum and quasicrystalline systems (Wong et al., 3 Feb 2026, Alon et al., 2023).

1. Formal Definition and Algebraic Structure

Let $\psi\in(\mathbb C^2)^{\otimes n}$ denote an $n$-qubit tensor, indexed by bitstrings $x\in\{0,1\}^n$. Define its generating multilinear polynomial by

$$f_\psi(z_1,\ldots,z_n) = \sum_{x\in\{0,1\}^n} \psi_x \prod_{i:x_i=1} z_i .$$

Fixing a radius $r > 0$, let $\mathbb D_r = \{z\in\mathbb C : |z|<r\}$ and define the polydisk $$\mathbb D_r^n = \mathbb D_r \times \cdots \times \mathbb D_r$$. The tensor $\psi$ is termed a Lee–Yang tensor of radius $r$ (notation: $\psi \in LY_n(r)$) if

$$f_\psi(z_1, \ldots, z_n) \neq 0 \quad \forall (z_1,\ldots,z_n) \in \mathbb D_r^n.$$

This definition extends naturally to families with variable radii $r=(r_1,\ldots,r_n)$. The union $$LY(r) = \bigcup_{n\ge0} LY_n(\underbrace{r,\ldots,r}_{n})$$ forms the totality of Lee–Yang tensors with every variable at least radius $r$ (Wong et al., 3 Feb 2026).

The more general notion of a multivariate Lee–Yang polynomial—central to harmonic analysis and quasicrystal theory—is given as follows. For the polynomial ring $\mathbb{C}[z]_{\leq d}$ on $n$ variables $z_1,\ldots,z_n$ of (possibly non-uniform) multi-degree $d = (d_1,\ldots,d_n)$, a polynomial $p(z_1,\ldots,z_n)$ is Lee–Yang (of multi-degree $d$) if

$$p(z_1,\ldots,z_n) \neq 0 \qquad\text{whenever}\qquad |z_i|<1\ \forall i$$

and

$$p(z_1,\ldots,z_n) \neq 0 \qquad\text{whenever}\qquad |z_i|>1\ \forall i.$$

By Möbius invariance and Hurwitz’s theorem, this is equivalent to requiring that for any vector $\ell\in\mathbb R_+^n$, the one-variable exponential polynomial $f(x) = p(e^{i\ell_1 x}, \ldots, e^{i\ell_n x})$ has only real zeros (Alon et al., 2023).

2. Tensor-Network Closure Properties and Operational Structure

Lee–Yang tensors possess remarkable closure properties under natural tensor network operations:

• Tensor contraction: If $\psi\in LY_n(r)$, contracting tensor indices $i$ and $j$ yields $$\phi \in LY_{n-2}(r_1,\ldots,\widehat r_i,\ldots,\widehat r_j,\ldots,r_n)$$ whenever $r_i r_j > 1$. For $r_i r_j = 1$, either $\phi \equiv 0$ or $\phi$ remains Lee–Yang with the same radius structure. This structural lemma is central to recursion and renormalization arguments.
• Semigroup closure: The invertible elements of $LY(r)$ form a semigroup under composition; if $A,B\in LY(r)$, then $AB\in LY(r)$.
• Behavior under quantum operations:
  • Postselected Pauli measurements in the $X$ or $Y$ basis at $r \ge 1$ either annihilate the state or yield another Lee–Yang tensor of lower rank.
  • Single-qubit Pauli channels of the form $$\mathcal E(\rho)=p_0\rho + p_1 X\rho X + p_2 Y\rho Y + p_3 Z\rho Z$$ are Lee–Yang ($LY(1)$ in Choi form) whenever $\min\{p_0, p_3\} \ge \max\{p_1, p_2\}$ (Wong et al., 3 Feb 2026).

These closure results generalize the original techniques of Asano, Ruelle, and Suzuki–Fisher, and are foundational for the recursive and combinatorial analysis of quantum states and partition functions.

3. The Threshold Phenomenon at Radius $r=1$

The value $r=1$ signals a sharp threshold in the analytic, algebraic, and computational properties of Lee–Yang tensors and associated operators.

• Uniqueness of Eigenvectors: Any Hermitian operator $H\in LY_n(r)$ for $r>1$ admits a unique principal eigenvector (the eigenvector with largest magnitude eigenvalue). This is a nontrivial quantum analog of the Perron–Frobenius theorem and ensures the non-degeneracy of certain ground states.
• Quasi-polynomial State Preparation: States in $LY_n(r)$ for fixed $r>1$ can be prepared with quasipolynomial overhead: each $X$-basis amplitude can be approximated to relative error $\epsilon$ in classical time and on quantum circuits of size $n^{O(\log (n/\epsilon))}$. This exploits Barvinok’s polynomial-interpolation method and Grover–Rudolph-type superposition preparation (Wong et al., 3 Feb 2026).
• Implications: For $r>1$, efficient preparation and classical approximation of quantum states are possible. At the limiting case $r=1$, zero-freeness is no longer strict, and structural uniqueness may fail. This delineates a frontier in Hamiltonian complexity and the algorithmic tractability of many-body quantum states.

4. Applications in Statistical Physics, Quantum Hamiltonians, and Quasicrystal Theory

In statistical physics, the property underlies extensions of the classical Lee–Yang circle theorem, notably for quantum spin systems:

• Zero-freeness for Partition Functions: The multivariate Lee–Yang property guarantees that partition functions (viewed as polynomials in auxiliary complex parameters) are zero-free in appropriate domains for ferromagnetic models such as the Ising model with transverse field and Heisenberg-type Hamiltonians, unifying previously disparate zero-free region results.
• EPR-like Hamiltonians: For two-local Hamiltonians $H_s$ built from projectors onto deformed EPR states $|\phi_s\rangle = |00\rangle + s|11\rangle$, the Lee–Yang radius of the ground state on any graph of $n$ vertices is at least $r = 1/\sqrt s$, and the spectral gap $\Delta$ is at least $1-s^2$. This supports conjectures regarding polynomial-time quantum adiabatic algorithms for the ground energy of the Heisenberg antiferromagnet (quantum Max-Cut) on bipartite graphs when $s=1-O(1/n)$ (Wong et al., 3 Feb 2026).
• Fourier Quasicrystals (FQ): Recent work shows that one-dimensional Fourier quasicrystals arise precisely as supports of exponential polynomials associated with Lee–Yang polynomials. Necessary and sufficient conditions for generating non-periodic, unit-coefficient, uniformly discrete FQs are that the polynomial $p$ is irreducible and its gradient does not vanish at torus zeros. Extremal choices of $p$ interpolate between Poissonian and Circular Unitary Ensemble (CUE) gap distributions (Alon et al., 2023).

5. Genericity, Examples, and Spectral Statistics

The set of Lee–Yang polynomials with desirable spectral and combinatorial properties forms a semi-algebraic, open dense subset of the relevant polynomial spaces:

• Genericity: For $n \geq 2$, almost every Lee–Yang polynomial yields an FQ with non-periodic, unit weights, and uniformly discrete support; such polynomials are stable under suitable perturbations.
• Spectral regimes: Poisson gap statistics arise from maximal tensor-product Lee–Yang polynomials $p(z) = \prod_{j=1}^n (1-z_j)$, while CUE statistics are realized by determinants of the form $p_U(z) = \det(I - \operatorname{diag}(z) U)$ for $U\in U(n)$. This reveals a rich interpolation between integrable and chaotic behavior and unifies ad hoc constructions for spectrum statistics in quantum graphs and FQs (Alon et al., 2023).

6. Significance and Broader Implications

The multivariate Lee–Yang property establishes powerful analytical boundaries for the zero sets of physical partition functions, ground-state wavefunctions, and quasicrystalline spectra. It offers a unified language and toolkit, making possible deep results across several domains:

• It underpins sign-structure results (e.g., Griffiths’ inequalities), semigroup behaviors under stochastic and quantum operations, and efficient preparation protocols for special quantum states.
• The property recovers and extends the classical Lee–Yang circle theorem, its quantum analogs (e.g., Suzuki–Fisher), and provides generic methods for constructing FQs with prescribed statistical properties.
• In Hamiltonian complexity, it identifies a phase boundary ($r=1$) at which qualitative changes in computational hardness, uniqueness, and preparation emerge, guiding both classical and quantum algorithm development.

A plausible implication is that further study of Lee–Yang tensors, polynomials, and their operator-theoretic closure structures will underpin advances in quantum algorithms, statistical physics, and harmonic analysis. The universality seen in the application to FQs and quantum ground states suggests a central organizing role for the Lee–Yang property in future research at the quantum-classical interface (Wong et al., 3 Feb 2026, Alon et al., 2023).