Bi-Symmetric Hermitian Tensors

• Bi-Symmetric Hermitian tensors are higher-order structures generalizing Hermitian matrices with conjugate and block symmetry, ensuring real-valued polynomial forms.
• Their structured decompositions, including CPS and orthogonal factorizations, enable efficient rank analysis and facilitate convex relaxation in optimization problems.
• They are crucial in quantum information, differential geometry, and polynomial optimization, modeling density operators, curvature tensors, and separability tests.

A bi-symmetric Hermitian tensor is a higher-order generalization of a Hermitian matrix that maintains two essential types of symmetry: Hermitian symmetry (conjugate symmetry between two groups of tensor indices) and full or partial symmetry within each group of indices. These tensors form the principal algebraic structures underpinning diverse modern applications in quantum information theory, complex polynomial optimization, spectral analysis, and differential geometry.

1. Formal Definition and Symmetry Structure

A complex tensor of even order $2d$ and dimension $n$, denoted $$A = (a_{i_1,\ldots,i_d,\;\bar{j}_1,\ldots,\bar{j}_d}) \in \mathbb{C}^{n^{2d}}$$, is termed bi-symmetric Hermitian (also known as a conjugate partial-symmetric (CPS) tensor) if it satisfies the following symmetry properties:

• Hermitian (Conjugate) Symmetry:

$$a_{i_1,\ldots,i_d,\;\bar{j}_1,\ldots,\bar{j}_d} = \overline{a_{j_1,\ldots,j_d,\;\bar{i}_1,\ldots,\bar{i}_d}}$$

for all multi-indices.

• Partial (Block) Symmetry:

The entries are invariant under any permutation within the first $d$ indices and within the last $d$ indices, i.e.,

$$a_{i_1,\ldots,i_d,\;\bar{j}_1,\ldots,\bar{j}_d} = a_{i_{\pi(1)},\ldots,i_{\pi(d)},\;\bar{j}_{\tau(1)},\ldots,\bar{j}_{\tau(d)}}$$

for all permutations $\pi, \tau$ in the symmetric group $\mathfrak{S}_d$.

When $d=1$, these reduce to standard Hermitian matrices. This structure ensures that the associated homogeneous complex polynomial (with $d$ variables and their conjugates symmetrized in the two groups) is real-valued if and only if the tensor coefficients satisfy the above conjugate symmetry constraint (Jiang et al., 2015). This intertwining of symmetry and Hermitian structure is crucial in separating the bi-symmetric Hermitian tensors from more general Hermitian or symmetric tensors (Fu et al., 2018, Huang et al., 2021).

2. Decomposition and Rank Structure

Rank-One and Orthogonal Decompositions

Bi-symmetric Hermitian tensors admit unique and highly structured decompositions:

• CPS Decomposition: Any CPS (bi-symmetric Hermitian) tensor $\mathcal{A}$ can be written as a real linear combination of symmetric rank-one terms:

$$\mathcal{A} = \sum_{i=1}^r \lambda_i\, a_i^{\otimes d} \otimes \overline{a_i}^{\otimes d}, \quad \lambda_i \in \mathbb{R},\; a_i \in \mathbb{C}^n$$

(Fu et al., 2018, Huang et al., 2021).

• Orthogonal Matrix Decomposition: For fourth-order CPS tensors, one may unfold $\mathcal{A}$ into a Hermitian matrix $M(\mathcal{A})$ (by merging paired indices). The spectral decomposition of $M(\mathcal{A})$ yields

$$M(\mathcal{A}) = \sum_{i=1}^r \lambda_i\, e_i e_i^*,$$

with the $e_i$ “folded” into symmetric matrices $E_i$, satisfying $E_i=E_i^T$, leading to

$$\mathcal{A} = \sum_{i=1}^r \lambda_i\, E_i \otimes \overline{E_i}.$$

This establishes connections to matrix analysis (Cholesky, spectral decompositions) and provides explicit bounds on various tensor ranks (Huang et al., 2021).

Ranks

Three primary notions of rank are relevant:

Rank notion	Definition	Inequality
CP-rank	Minimal number of arbitrary rank-1 terms	$\leq$ PS-rank
PS-rank	Minimal number, preserving partial symmetry	$=$ CPS-rank
CPS-rank	Minimal number in CPS decomposition

Generally, $$\text{rank}(\mathcal{A}) \leq \text{PS-rank}(\mathcal{A}) = \text{CPS-rank}(\mathcal{A})$$, with strict inequality possible. This invalidates a direct analogue of Comon's conjecture in the CPS setting (Fu et al., 2018).

3. Spectral Theory and Positive (Semi-)Definiteness

Eigenvalue Theory

Multiple eigenvalue notions extend the familiar Hermitian matrix theory:

• C-eigenvalue: For $F$ a conjugate partial-symmetric tensor and $x \in \mathbb{C}^n$ with $x^H x = 1$, $\lambda \in \mathbb{R}$ is a C-eigenvalue if

$$F(\bullet, x, \ldots, x, x, \ldots, x) = \lambda\, x$$

(Jiang et al., 2015, Chen et al., 17 Aug 2025).

• $\hat{H}$-eigenvalue (Editor’s term): $\lambda \in \mathbb{R}$, with $x \neq 0$, such that

$$\sum_{i_2, \dots, i_m, j_1, \dots, j_m} a_{i\,i_2,\ldots,i_m,\;\bar{j}_1,\ldots,\bar{j}_m}\, x_{i_2} \dots x_{i_m}\, \overline{x_{j_1}\dots x_{j_m}} = \lambda\, \overline{x_i}\, |x_i|^{2m-2}$$

(Chen et al., 17 Aug 2025).

All such eigenvalues are always real for bi-symmetric Hermitian tensors (Jiang et al., 2015, Chen et al., 17 Aug 2025). Positive definiteness of the tensor is equivalent to strict positivity of all $\hat{H}$-eigenvalues, mirroring the classical spectral theorem for Hermitian matrices.

Inclusion Sets

Extensions of Geršgorin’s circle theorem localize eigenvalues via explicit sums over off-diagonal entries, yielding nested sets $K_{\mathrm{ger}}(A)$, $K_{\mathrm{llk}}(A)$, $K_{\mathrm{ll}}(A)$ such that

$$\sigma_\mathrm{H}(A) \subseteq K_{\mathrm{ll}}(A) \subseteq K_{\mathrm{llk}}(A) \subseteq K_{\mathrm{ger}}(A)$$

(Chen et al., 17 Aug 2025).

Positive Definiteness Criteria

Several easily checkable criteria are established:

• Diagonal Dominance: If all diagonal entries dominate the sum of magnitudes of off-diagonal terms, positive definiteness/semi-definiteness follows (Chen et al., 17 Aug 2025).
• LL and LLK Tensors: These refinements provide sharper criteria by comparing various off-diagonal blocks.

For real tensors, there is a hierarchy among notions of PSD: general quartic (or polynomial) PSD, matrix PSD, and general PSD defined by positivity over the cone of symmetric or Hermitian matrices (Huang et al., 2021).

4. Applications in Quantum Information and Geometry

Bi-symmetric Hermitian tensors serve as natural generalizations of density matrices for multipartite quantum states:

• Quantum Mixed States: The tensor encodes the density operator for $d$-partite or Bosonic systems. The bi-symmetry captures physically required exchange symmetry for Bosons (Ni, 2019, Friedland, 25 Sep 2025).
• Entanglement and Separability: Separable bi-symmetric density tensors correspond precisely to those with b-nuclear norm equal to $1$, with the dual b-spectral norm serving as a geometric measure of entanglement. Strong separability is characterized by representation as a convex combination of pure symmetric tensors $x^{\otimes d} \otimes \overline{x}^{\otimes d}$ (Friedland, 25 Sep 2025).
• Computational Complexity: While separability and norm evaluation for general density tensors are NP-hard, restriction to the bi-symmetric (Bosonic) case allows polynomial-time algorithms when the local dimension $n$ is fixed and $d$ varies (Friedland, 25 Sep 2025).
• Curvature in Complex Differential Geometry: The Chern and Riemannian curvature tensors on Hermitian manifolds can be encoded as 4th order bi-symmetric Hermitian tensors. Positivity of holomorphic sectional curvature corresponds directly to positive definiteness of the associated tensor (Yang et al., 2016, Chen et al., 17 Aug 2025).

5. Optimization, Approximation, and Algorithmics

Best Rank-One Approximation and Banach’s Theorem

The maximal value of the bi-symmetric Hermitian form on the unit sphere coincides with the maximal absolute eigenvalue (“singular value”) of the tensor: $$\max_{\|x\|=1} |A(x^{\otimes d},\overline{x}^{\otimes d})| = \max_{X \in T(\mathrm{CPS})} |\langle M_{T}(A), X \rangle|, \quad \mathrm{rank}(X)=1, \;\mathrm{tr}(X)=1$$ (Fu et al., 2018). This is a tensor-level analogue of classical results for matrices and Banach’s symmetrization theorem (Jiang et al., 2015).

Matricization and Convex Relaxation

Matricization, or unfolding of a CPS tensor into a Hermitian matrix through a suitable pairing of indices, allows the transfer of tensor optimization problems into convex (SDP) frameworks. When the unfolded matrix is rank-one, so is the original tensor. Convex relaxation techniques such as dropping rank constraints or adding nuclear norm penalties are empirically demonstrated to be effective, especially when the tensor possesses bi-symmetric Hermitian structure (Fu et al., 2018).

Separability Certification

For Hermitian tensors (including the bi-symmetric case), separability can be decided via truncated moment problems and solved with a hierarchy of SDP relaxations. Flat truncation conditions guarantee the ability to extract positive semidefinite rank-one atoms, thus certifying separability (Dressler et al., 2020). For low psd-rank tensors, additional flattening and tensor decomposition steps enable effective separability detection and the explicit computation of psd decompositions.

6. Triangular and Structural Decompositions

For third-order bi-symmetric Hermitian tensors, a generalization of Cholesky decomposition is possible. Any positive semidefinite third-order Hermitian tensor admits a triangular decomposition: there exists a lower triangular sub-symmetric tensor $L$ such that the original tensor is its “cubic power,” with explicit slice-wise construction formulas. This structure allows reduction of spectral analysis and eigenvalue computation to analysis of the triangular factor (Qi et al., 23 Dec 2024).

7. Geometric and Algebraic Spectral Properties

The characteristic polynomial and eigenstructure of bi-symmetric Hermitian tensors can be approached using algebraic geometry. For generic symmetric tensors (and a fortiori for bi-symmetric Hermitian tensors, which correspond to a finite symmetry group action rather than a continuous one), the spectrum is a strong invariant, and the fiber of the map assigning characteristic polynomials to tensors is typically finite (Galuppi et al., 2023).

This finiteness has significant consequences for identifiability and reconstruction from spectral data, ensuring, for instance, that the spectral decomposition is unique or highly structured in the bi-symmetric Hermitian case. In quantum many-body settings, Banach’s theorem allows restriction to symmetric product states for maximizing or minimizing invariant functionals (Friedland, 25 Sep 2025).

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The bi-symmetric Hermitian tensor formalism thus provides a versatile and robust toolkit for encoding, analyzing, and optimizing complex interactions that preserve both Hermitian and block-symmetric structure, with broad applications across quantum physics, optimization, geometry, and computational mathematics (Jiang et al., 2015, Fu et al., 2018, Ni, 2019, Nie et al., 2019, Dressler et al., 2020, Huang et al., 2021, Qi et al., 23 Dec 2024, Chen et al., 17 Aug 2025, Friedland, 25 Sep 2025).