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arXiv:2609.04341

Schatten norms and determinants of linear combinations of matrix tensor powers via virtual representations

Martin Áron Juhász, Mihály Weiner

math.RTquant-phSubmitted Sep 3, 2026• Announced Sep 7, 2026
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Abstract

Let $$X_n=\sum_{i=1}^s t_i A_i^{\otimes n},$$ where $A_1,\ldots,A_s\in M_d(\mathbb C)$ and $t_1,\ldots,t_s\in\mathbb C$ are fixed, while $n$ grows. Direct computation of determinants or Schatten norms of $X_n$ is exponential in $n$. For a single tensor power these quantities are elementary, and even the determinant of a generic two-term combination admits a reduction to polynomially many scalar factors; however, no analogous elementary reduction is available for three or more terms. We give an exact representation-theoretic method which, for fixed $d$ and $s$, computes $\|X_n\|_p$, $0<p<\infty$, and determinants in polynomial time in $n$. Schur--Weyl duality yields a simultaneous block decomposition, while Jacobi--Trudi identities in the Grothendieck ring replace Schur modules by signed combinations of tensor products of symmetric powers. For $d=3$, each irreducible contribution reduces to the difference of two explicitly computable symmetric-power terms, leading to an open-source implementation. In a single-thread CPU benchmark, a genuine three-term $3\times3$ trace-norm problem with $n=18$ is evaluated in about $47$ seconds, whereas just storing the unreduced matrix would require approximately $2.4\times10^{18}$ bytes. Direct and reduced computations agree to relative error below $3.4\times10^{-15}$ throughout their common range $n\leq9$.

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Let Xn=∑i=1stiAi⊗n,X_n=\sum_{i=1}^s t_i A_i^{\otimes n},Xn​=∑i=1s​ti​Ai⊗n​, where A1,…,As∈Md(C)A_1,\ldots,A_s\in M_d(\mathbb C)A1​,…,As​∈Md​(C) and t1,…,ts∈Ct_1,\ldots,t_s\in\mathbb Ct1​,…,ts​∈C are fixed, while nnn grows. Direct computation of determinants or Schatten norms of XnX_nXn​ is exponential in nnn. For a single tensor power these quantities are elementary, and even the determinant of a generic two-term combination admits a reduction to polynomially many scalar factors; however, no analogous elementary reduction is available for three or more terms.
We give an exact representation-theoretic method which, for fixed ddd and sss, computes ∥Xn∥p\|X_n\|_p∥Xn​∥p​, 0<p<∞0<p<\infty0<p<∞, and determinants in polynomial time in nnn. Schur--Weyl duality yields a simultaneous block decomposition, while Jacobi--Trudi identities in the Grothendieck ring replace Schur modules by signed combinations of tensor products of symmetric powers.
For d=3d=3d=3, each irreducible contribution reduces to the difference of two explicitly computable symmetric-power terms, leading to an open-source implementation. In a single-thread CPU benchmark, a genuine three-term 3×33\times33×3 trace-norm problem with n=18n=18n=18 is evaluated in about 474747 seconds, whereas just storing the unreduced matrix would require approximately 2.4×10182.4\times10^{18}2.4×1018 bytes. Direct and reduced computations agree to relative error below 3.4×10−153.4\times10^{-15}3.4×10−15 throughout their common range n≤9n\leq9n≤9.
Trace norms are fundamental to quantum information theory, yet in many-body systems their evaluation remains a major computational bottleneck, as it generally requires diagonalizing exponentially large operators. Here, we overcome this bottleneck by introducing a controlled tensor-network algorithm for estimating the trace norm of matrix product operators without full diagonalization. The key idea is to combine Zolotarev's rational approximation to the sign function with a variational formulation solved using a density-matrix-renormalization-group-like algorithm. The resulting approximation is systematically improvable, with its accuracy controlled by the rational approximation parameters and the spectral weight near zero. Beyond the reach of exact diagonalization, we demonstrate controlled trace-norm calculations for entanglement negativity, quantum fidelity and quantum Fisher information, achieving substantially improved accuracy over polynomial-based Lanczos approaches. Our results establish trace-norm-based quantities as practical tensor-network observables, opening a route toward tensor-network studies of quantum information in mixed states.
The tensor rank decomposition is a useful tool for the geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able to estimate how many tensor rank decompositions can approximate a given tensor. More precisely, finding an approximate symmetric tensor rank decomposition of a symmetric tensor QQQ with an error allowance ΔΔΔ is to find vectors ϕiϕ^iϕi satisfying ∥Q−∑i=1Rϕi⊗ϕi⋯⊗ϕi∥2≤Δ\|Q-\sum_{i=1}^R ϕ^i\otimes ϕ^i\cdots \otimes ϕ^i\|^2 \leq Δ∥Q−∑i=1R​ϕi⊗ϕi⋯⊗ϕi∥2≤Δ. The volume of all possible such ϕiϕ^iϕi is an interesting quantity which measures the amount of possible decompositions for a tensor QQQ within an allowance. While it would be difficult to evaluate this quantity for each QQQ, we find an explicit formula for a similar quantity by integrating over all QQQ of unit norm. The expression as a function of ΔΔΔ is given by the product of a hypergeometric function and a power function. We also extend the formula to generic decompositions of non-symmetric tensors. The derivation depends on the existence (convergence) of the partition function of a matrix model which appeared in the context of the CTM.
We give an explicit treatment of the third tensor power of the Clifford representation on nnn qutrits, for n≥2n\geq2n≥2. This representation has five pairwise inequivalent irreducible constituents, with multiplicities 1,1,1,1,21,1,1,1,21,1,1,1,2. The decomposition reduces the Clifford twirl of a tensor cube to four real parameters. Two of them are spectral moments of the input. The other two are cubic invariants expressed as sums of cubes of the Weyl characteristic function and the discrete Wigner function. On trace-one Hermitian matrices, the invariant polynomials of degree at most three are spanned by the constant function and four algebraically independent functions. These functions do not separate Clifford orbits. For pure inputs a single parameter suffices. We recover its range and the characterization of its maximum by stabilizer states. We bound the stabilizer fidelity, determine full separability across the three copies in the symmetric family, and recover the criterion for a complex projective 333-design. We classify the jointly invariant quantum Boolean functions. For stabilizer testing and for testing unknown observables we give explicit bounds on copies and queries. We also give character formulas for the finite projective Clifford groups and record the complete character table for two qutrits.
We show that the spectral norm of a d d -mode real or complex symmetric tensor in n n variables can be computed by finding the fixed points of the corresponding polynomial map. For a generic complex symmetric tensor the number of fixed points is finite, and we give upper and lower bounds for the number of fixed points. For n = 2 n=2 we show that these fixed points are the roots of a corresponding univariate polynomial of degree at most ( d − 1 ) 2 + 1 (d-1)^2+1 , except certain cases, which are completely analyzed. In particular, for n = 2 n=2 the spectral norm of d d -symmetric tensor is polynomially computable in d d with a given relative precision. For a fixed n > 2 n>2 we show that the spectral norm of a d d -mode symmetric tensor is polynomially computable in d d with a given relative precision with respect to the Hilbert-Schmidt norm of the tensor. These results show that the geometric measure of entanglement of d d -mode symmetric qunits on C n \mathbb {C}^n are polynomially computable for a fixed n n .
It is well known that tensor decompositions show separations, that is, constraints on local terms (such as positivity) may entail an arbitrarily high cost in their representation. Here, we show that many of these separations disappear in the approximate case. Specifically, for every approximation error ɛ and norm, we define the approximate rank as the minimum rank of an element in the ɛ-ball with respect to that norm. For positive semidefinite matrices, we show that the separations between rank, purification rank, and separable rank disappear for a large class of Schatten p-norms. For non-negative tensors, we show that the separations between rank, positive semidefinite rank, and non-negative rank disappear for all ℓp-norms with p > 1. For the trace norm (p = 1), we obtain upper bounds that depend on the ambient dimension. We also provide a deterministic algorithm to obtain the approximate decomposition attaining our bounds. Our main tool is an approximate version of the Carathéodory theorem. Our results imply that many separations are not robust under small perturbations of the tensor, with implications in quantum many-body systems and communication complexity.
In multiparameter quantum estimation, incompatible measurements can impose trade-offs among estimation errors. Understanding the precision attainable with a finite number of jointly measured copies is therefore important, but direct optimization becomes costly as the tensor product space grows exponentially. For independent and identically distributed (i.i.d.) copies, the quantum statistical model is invariant under permutations of the copies. We exploit this symmetry through Schur--Weyl duality to remove permutation multiplicity spaces while using their dimensions as weights. This gives a reduced statistical model equivalent to the original under arbitrary collective measurements. On this reduced model, we derive an exact conic formulation of the tight Cramér--Rao bound and a semidefinite program for its Nagaoka--Hayashi (NH) relaxation. For fixed single-copy dimension, the reduced operator representation grows polynomially rather than exponentially with the number of copies. To approximate the tight bound, we use an exact quadratic identity to correct finite-ray dual solutions and obtain rigorous upper and lower bounds. For fixed copy number and parameter dimension, sufficiently fine meshes admit feasible primal points, and dual solutions with bounded objective gaps have uniformly bounded compact-search radii. These results yield quadratic convergence in the projective covering radius. Numerical examples illustrate the quadratic scaling of the mesh correction term and the computational reach of the reduced NH formulation.