Sessions

We present a new explicit formula for the determinant that contains superexponentially fewer terms than the usual Leibniz formula. As an immediate corollary of our formula, we show that the tensor rank of the $n \times n$ determinant tensor is no larger than the $n$-th Bell number, which is much smaller than the previously best known upper bounds when $n \geq 4$. Over fields of non-zero characteristic we obtain even tighter upper bounds, and we also slightly improve the known lower bounds. In particular, we show that the $4 \times 4$ determinant over $\mathbb{F}_2$ has tensor rank exactly equal to $12$. Our results also improve upon the best known upper bound for the Waring rank of the determinant when $n \geq 17$, and lead to a new family of axis-aligned polytopes that tile $\mathbb{R}^n$.

A pure state of an $n$-level quantum system can be regarded as a norm-one vector $\textbf{x} \in \mathbb{C}^n$ defined up to a global phase factor ($\textbf{x} \sim \textup{e}^{\textup{i}\varphi}\textbf{x}$). In my talk I will show that, for any $n \times n$ distance matrix $(D_{ij})$ and any $p \geq 2$, the map $d_p(\mathbf{x},\mathbf{y}) := (\sum_{i<j} D_{i j}^p |x_i y_j - x_j y_i|^2 )^{1/p} $ is a bona fide distance, in particular satisfying the triangle inequality. It is a far-reaching generalization of the standard distance between pure quantum states given by the wedge product $\|\textbf{x} \wedge \textbf{y}\| = \sqrt{\sum_{i<j} |x_i y_j - x_j y_i|^2}$. I will also discuss how this result carries over to the $n \rightarrow +\infty$ case, offering a robust way to lift the metric structure from some underlying metric space $(X,D)$ to the suitably defined `space of wave functions', and — time permitted — touch upon the mixed states case. The talk builds upon papers [1,2].
[1] TM, R. Bistroń, Distances between pure quantum states induced by a distance matrix, arXiv:2509.14727 [math-ph]
[2] R. Bistroń, M. Eckstein, S. Friedland, TM, K. Życzkowski, A new class of distances on complex projective spaces, Linear Algebra Appl. 721, 577--611, (2024)

One of the most challenging problems in quantum physics is to quantify the entanglement of $d$-partite states and their separability. In general, these quantities are NP-hard to compute. We show that some quantities are computed in polynomial time when we restrict our attention to Bosons: symmetric $d$-qubits, or more generally to symmetric $d$-qunits (symmetric $d$-tensors) in $\mathbb{C}^n$, and the corresponding bi-symmetric Hermitian density tensors, for a fixed value of $n$.
Reference: S. Friedland, Tensors, entanglement, separability, and their complexity, arXiv:2509.21639, 2025.

We develop new techniques for proving smoothed analysis bounds for algorithmic problems related to tensor decompositions, and their applications. Smoothed analysis is a powerful paradigm for proving robust, polynomial time guarantees of algorithms on worst-case computationally intractable problems. While polynomial time smoothed analysis guarantees are desirable, they have been challenging to obtain as they entail lower bounds on the least singular value of random matrices with limited randomness. The matrices we consider have entries that are given by polynomials of a few underlying base random variables.
We present two new general techniques for proving least singular value bounds of higher-order lifts of random matrices. This allows us to get simpler proofs of existing smoothed analysis results, and obtain new smoothed analysis guarantees in previously open settings.

Ranks are ubiquitous invariants of tensors, each qualitatively measuring a structure of tensors. Thus, relations among different structures of tensors are reflected by relations among their corresponding ranks. In this talk, we discuss the relation between geometric rank and subrank, where the former quantifies the shape of the kernel of tensors and the latter measures the computational complexity of tensors. We will show that the geometric rank is bounded by some function in the subrank. Moreover, we will prove that this function is a quadratic polynomial for order three tensors. If time permits, we will also discuss potential extensions of this quadratic polynomial bound to higher-order tensors.

In this talk we introduce degree $k$ t-Hermitian forms, obtained via a synthesis of the tensor transformation law and the t-product of third order tensors. Degree k $t$-Hermitian forms uniquely correspond to $t$-Hermitian partially symmetric tensors, which arise as canonical equivalence class representatives within the $*$-algebra of Hermitian tensors equipped with the $t$-product (or more generally for higher order tensors, the $t$-Einstein product); these tensors decompose into collections of degree $k$ Hermitian forms under the discrete Fourier transformation. This lift of degree $k$ Hermitian forms admits a universal characterization, preserving properties such as positivity. Furthermore, this framework provides a natural setting for studying structured interactions amongst degree $k$ Hermitian forms, and, under certain conditions, extends the classical spectral decomposition of Hermitian tensors and their associated forms.