Rank-Three Tensor Q

• Rank-three tensor Q is a third-order multidimensional array defined by its minimal decomposition into rank-one tensors.
• Research on rank-three tensor Q examines decomposition uniqueness, complementarity conditions, and numerical methods like Jacobian analysis.
• Applications span representation theory, quantum entanglement, and efficient low-rank approximations in signal processing and scientific computing.

A rank-three tensor Q refers to a third-order multidimensional array or multilinear operator, which appears in a wide range of mathematical, physical, data-theoretic, and computational contexts. The paper of its structural, algebraic, and analytical properties encompasses notions of rank, decomposition theory, complementarity classes, representation theory, and model-theoretic frameworks in applied mathematics, mathematical physics, signal processing, and quantum information. The following sections cover the foundational definitions, generic rank problem, complementarity and Q-tensor theory, algebraic and representation-theoretic aspects, physical and statistical modeling, and connections to computational applications.

1. Structural and Algebraic Definition of Rank-Three Tensor Q

A rank-three tensor is an element $Q \in V_1 \otimes V_2 \otimes V_3$ for finite-dimensional real or complex vector spaces $V_1, V_2, V_3$ (or, more generally, over division rings including $\mathbb{H}$). In a basis, $Q$ has coordinates $Q_{ijk}$, $i=1,\ldots,N_1$, $j=1,\ldots,N_2$, $k=1,\ldots,N_3$. The tensor rank, denoted $\mathrm{rank}(Q)$, is the smallest integer $R$ such that

$$Q = \sum_{r=1}^R a^{(r)} \otimes b^{(r)} \otimes c^{(r)},$$

with $a^{(r)} \in V_1$, $b^{(r)} \in V_2$, $c^{(r)} \in V_3$.

For symmetric tensors, $Q$ is invariant under any permutation of indices and can be written as

$$Q = \sum_{r=1}^R a^{(r)} \otimes a^{(r)} \otimes a^{(r)}.$$

The notion of (symmetric) tensor rank varies according to imposed symmetries.

Multiple distinct notions appear:

• Generic/typical rank: The rank occurring on a Zariski open dense subset of all tensors of given shape.
• Symmetric rank: Minimal $R$ in a symmetric decomposition.
• Border rank: Minimal $r$ such that $Q$ is a limit of tensors of rank $r$.
• Q-tensor property: In context of complementarity problems, $Q$ designates a tensor satisfying solubility conditions (see §3).

2. Generic and Typical Ranks—Jacobian Approach and Decomposition Uniqueness

The generic (also called typical) rank is central to understanding tensor factor analysis and identifiability:

• For a general third-order tensor $Q$, the canonical decomposition (CAND/PARAFAC) is determined by parametrizing the set of possible rank-one decompositions and analyzing the associated parameter-to-tensor map. Given parameters (the $a^{(r)}$, $b^{(r)}$, $c^{(r)}$ vectors), a mapping $y : \mathbb{R}^M \to \mathbb{R}^{N_1 N_2 N_3}$ is defined, and the Jacobian $J$ of $y$ with respect to all parameters is computed.
• The generic rank $R$ is the minimal $R$ such that the Jacobian achieves maximal possible rank $M = R(N_1 + N_2 + N_3 - 2)$ (for unconstrained case). If $J$ reaches this rank, then almost every tensor of that shape has rank $R$.
• For tensors with imposed structure (e.g., symmetry, symmetric matrix slices, or double-centering), the parameter counts and Jacobians are suitably modified; for symmetric $Q$, $M=R N$, and the maximal rank is restricted by the dimension of the symmetric tensor space, $\binom{N + 2}{3}$ for $N \times N \times N$.

The numerical Jacobian procedure is algorithmically general and can reproduce classical algebraic results as well as certify generic ranks in regimes inaccessible to closed-form analysis. It identifies R for which the parameter-to-tensor map is locally surjective, also yielding fiber dimension $F = M - D$, whose zeroing ($F=0$) signals essential uniqueness of the decomposition up to expected indeterminacies (0802.2371).

3. Q-Tensor Theory and Tensor Complementarity Problems

The term "Q-tensor" has a specific and technically crucial role in complementarity theory:

• For $m$th order $n$-dimensional real tensor $Q$, the tensor complementarity problem (TCP) is

$$\text{Find } x \in \mathbb{R}^n:~x \geq 0,~q + Q x^{m-1} \geq 0,~x^\top(q + Q x^{m-1})=0,$$

with $$(Q x^{m-1})_i = \sum_{j_2,\ldots,j_m} Q_{i j_2\cdots j_m} x_{j_2}\cdots x_{j_m}$$.

• $Q$ is called a Q-tensor iff this problem is solvable for every $q \in \mathbb{R}^n$.
• Subclasses of Q-tensors include P-tensors (strong positivity: for all nonzero $x$, some $x_i(Q x^{m-1})_i > 0$), strictly semi-positive tensors, R- and $R_0$-tensors defined by nonexistence of nontrivial solutions in special TCPs.

Key theorems:

• For nonnegative $Q$, $Q$ is a Q-tensor if and only if all principal diagonal entries are positive, and in this case the Q, R, and strictly semi-positive notions coincide.
• For rank-one symmetric tensors, Q-tensor, S-tensor, positivity, and $R_0$-tensor properties are equivalent; this is a tensor generalization of results on rank-one Q-matrices (Sharma et al., 2023).
• For symmetric $Q$ of rank two and dimension two, Q-tensor implies $R_0$-tensor, but for rank three and higher, the implications are not clarified (open problem).

4. Representation Theory, Symmetry, and Algebraic Constructions

Rank-three tensors appear prominently in the representation theory of groups, Lie algebras, and quantum groups:

• In Temperley-Lieb algebra representations ($TL_N(Q)$), a tensor Q of rank $r$ arises via projection onto a subspace with specific orthogonality and unitarity criteria. For a rank-three projection, parameter Q must satisfy trace and unitarity relations; in many cases, allowed values are discrete, governed by inequalities such as $4r > n^2$ (Bytsko, 2015).
• In symmetry breaking and grand unified models, antisymmetric rank-three tensors Q act as scalar fields whose vacuum expectation values break symmetry. The paper (Adler, 2015) derives branching rules for SU(n) broken by such tensors, identifies representation content under subgroups, clarifies $U(1)$ generator mismatches, and gives explicit mass spectra for the physical scalar content post-breaking.
• In the theory of symmetric traceless and antisymmetric tensor models with tetrahedral interactions, rank-three tensors Q admit combinatorial $1/N$ expansions structurally analogous to those appearing in the Sachdev-Ye-Kitaev (SYK) model, with dominance of melonic diagrams (Benedetti et al., 2017, Bonzom, 2019).

5. Computational Aspects, Decomposition Algorithms, and Low-Rank Approximation

The computational identification and exploitation of rank-three tensor structure is fundamental across disciplines:

• In algorithmic tensor rank determination and decomposition, recent results provide efficient (polynomial in $n$, $d$) algorithms for constant-rank (e.g., rank-three) tensors expressed via depth-3 arithmetic circuits, with rigorous guarantees on uniqueness (under "semantic rank") and error-correction; in the rank-three setting, the decomposition $Q = c_1 \ell_1^d + c_2 \ell_2^d + c_3 \ell_3^d$ can be recovered via black-box access using cluster-preserving coordinate subspaces (Peleg et al., 2022).
• Low-rank approximation of symmetric third-order tensors uses primal-dual moment/matrix optimization frameworks, with global optima certified via dual certificate checking. For orthogonally decomposable tensors, the best rank-$r$ approximation is computed exactly; the error of "quantified quasi-optimal" approximations is tightly bounded in terms of a control parameter $\sigma$ (Hu et al., 2023).
• Quantized tensor decompositions (QTT, transposed QTT, Tucker-QTT), when applied to the coefficients of solutions to PDEs with isolated singularities, give tensor rank bounds for three-way tensors that grow only polylogarithmically in the error tolerance, enabling efficient representations for computational physics and scientific computing (Marcati et al., 2019).
• Advanced third-order tensor singular value decompositions (T-SVD, TQt-SVD) and their respective notions of tubal rank, T-rank, and TQt-rank provide structural tools for signal processing, video/image modeling, and quantum channel/entanglement analysis, with associated invariance properties under specified transformations (Qi et al., 2021, Miao et al., 2021).

6. Entanglement, Complexity, and Theoretical Implications

In quantum information and the complexity theory of tensors:

• The tensor rank of a pure state (represented as a three-way tensor) is an integer invariant that quantifies multipartite entanglement. Well-known states such as GHZ (rank 2) and W (rank 3, but border rank 2) illustrate this; rigorous uniqueness and maximal rank criteria (e.g., via Kruskal's theorem) govern the correspondence between structural decompositions and operational entanglement (Bruzda et al., 2019).
• Tensor rank is not, in general, multiplicative under tensor product: there exist tensors Q with $\mathrm{rank}(Q\otimes Q) < \mathrm{rank}(Q)^2$; this "compression" has important implications in algebraic complexity theory, especially in connection with fast matrix multiplication and border/asymptotic rank concepts. The nonmultiplicativity does not arise for 2-tensors (matrices), highlighting a strict difference between higher-order and matrix settings (Christandl et al., 2017).

7. Summary Table of Contexts and Properties

Property or Application	Key Results or Insights	Reference
Generic/typical rank	Jacobian-based numerical methods; structure-dependent; matches classical algebraic results	(0802.2371)
Complementarity problem (Q-tensors)	Q-tensor iff all principal diagonal entries positive (nonnegative case); equivalence with semi-positive/R-tensor	(Song et al., 2014, Sharma et al., 2023)
Symmetry breaking in SU(n)	Explicit subrepresentation content; calculable scalar masses and U(1) mismatch understood	(Adler, 2015)
Temperley-Lieb algebra representation	Existence of unitary projection-based representations; Q values tightly constrained for rank-three	(Bytsko, 2015)
Multi-linear rank and bilinear maps	Plural typical ranks linked to nonsingular bilinear maps and determinantal properties	(Sumi et al., 2015)
Tensor rank and entanglement	Rank as measure of entanglement; non-uniqueness and border rank subtleties	(Bruzda et al., 2019)
Low-rank approximation/certification	Primal-dual certification methods; exact for orthogonally decomposable tensors	(Hu et al., 2023)
Noncommutative algebras (quaternions)	Optimal decomposition in $2\times 2\times 2$ case over $\mathbb{H}$ is 3; upper bound is tight	(Liang et al., 2020)

8. Concluding Perspectives

Rank-three tensor Q serves as a unifying object across mathematics, theoretical physics, and applied computation. Its paper reveals deep links between multilinear algebra, representation theory, optimization, operator theory, and computational complexity. The interdependencies among notions of rank, symmetry constraints, complementarity solvability, and algebraic or analytic structure remain a vibrant area of ongoing research, with implications ranging from unique decomposability and identifiability in statistics to the emergence of universal behavior and symmetry breaking in quantum field theory.