arXiv subjects

Isaac Dobes

Publications and source records attributed to Isaac Dobes.

6 recordsLinked to original sources

arXiv2026

Higher Degree $t$-Hermitian Forms and Positivity-Preserving Contractions

In this article we introduce higher-degree $t$-Hermitian forms, a tubal analogue of ordinary Hermitian forms of arbitrary degree. Through a synthesis of multilinear matrix multiplication and the $t$-product on third-order tensors, we show that $t$-Hermitian forms are in bijection with odd order tubal tensors satisfying certain symmetry conditions, which we call $t$-conjugate partial symmetry. After applying the Fast Fourier Transform along the tubal mode of the corresponding tubal tensor, $t$-Hermitian forms decompose into a family of classical Hermitian forms. This decomposition enables us to characterize positivity of $t$-Hermitian forms in terms of the spectra of the conjugate partially symmetric Fourier slices of its corresponding tubal tensor, yielding a tubal analogue of the spectral theorem for classical higher degree Hermitian forms. Then, we study classical Hermitian forms induced by contractions along the tubal mode of $t$-conjugate partially symmetric tensors, characterizing when such contractions preserve positivity and deriving quantitative lower bounds for the positivity margin of the resulting classical Hermitian forms.

math.SP↗

arXiv2025

New Identity for Cayley's First Hyperdeterminant with Applications to Symmetric Tensors and Entanglement

In this article, a new formula for computing Cayley's first hyperdeterminant in terms of the Levi-Civita symbol is given. It is then shown that this formula can be used to compute the hyperdeterminant of symmetric tensors in polynomial time with respect to their order (assuming fixed side length). Applications to quantifying the entanglement of states of bosonic quantum systems are then discussed. Additionally, in order to obtain the fast calculation of the hyperdeterminant on symmetric tensors, generalized elimination and duplication matrices are defined and their explicit formulas are derived.

quant-ph↗

arXiv2025

Cayley's First Hyperdeterminant is an Entanglement Measure

Previously, it was shown that both the concurrence and $n$-tangle on $2n$-qubit pure quantum states can be expressed in terms of Cayley's first hyperdeterminant \cite{dobes2024qubits}, indicating that Cayley's first hyperdeterminant, denoted $\mathrm{hdet}$, captures some aspects of a state's $2n$-way entanglement. In this paper, we rigorously prove that on both pure and mixed states, $|\mathrm{hdet}|^{2/d}$ is identically zero on separable states, is an LU invariant, and is non-increasing on average under LOCC, thus demonstrating that $|\mathrm{hdet}|^{d/2}$ is a physically meaningful and legitimate entanglement measure. Moreover, we discuss a few key examples to illustrate the particular type of entanglement Cayley's first hyperdeterminant is detecting: genuine full $d$-level GHZ-type entanglement across all $2n$ parties. Combined, this establishes Cayley's first hyperdeterminant (or $|\mathrm{hdet}|^{2/d}$ to be precise), as a physically significant generalization of the $G$-concurrence and the $n$-tangle to $2n$-qudit states.

quant-ph↗

arXiv2025

Local Unitary Equivalence of Tripartite Quantum States In Terms of Trace Identities

In this paper we present a modified version of the proof given Jing-Yang-Zhao's paper "Local Unitary Equivalence of Quantum States and Simultaneous Orthogonal Equivalence," which established the correspondence between local unitary (LU) equivalence and simultaneous orthogonal equivalence of bipartite quantum states. Our modified proof utilizes a hypermatrix algebra framework, and with this framework we are able to generalize this correspondence to tripartite quantum states. Finally, we apply a generalization of Specht's criterion proved in Futorny-Horn-Sergeichuk' paper "Specht's Criterion for Systems of Linear Mappings" to \textit{essentially} reduce the problem of local unitary equivalence of tripartite quantum states to checking trace identities and a few other LU invariants. We also note that all of these results can be extended to arbitrary multipartite quantum states, however there are some practical limitations.

quant-ph↗

arXiv2024

Classifying Density Matrices of 2 and 3 Qubit States Up To LU Equivalence

In this paper we present a modified version of the proof given Jing-Yang-Zhao's paper titled "Local Unitary Equivalence of Quantum States and Simultaneous Orthogonal Equivalence," which established the correspondance between local unitary equivalence and simultaneous orthogonal equivalence of $2$-qubits. Our modified proof utilizes a hypermatrix algebra framework, and through this framework we are able to generalize this correspondence to $3$-qubits. Finally, we apply a generalization of Specht's criterion (first proved in "Specht's Criterion for Systems of Linear Mappings" by V. Futorney, R. A. Horn, and V. V. Sergeichuk) to reduce the problem of local unitary equivalence of $3$-qubits to checking trace identities and a few other easy-to-check properties. We also note that all of these results can be extended to $2$ and $3$ qudits if we relax the notion of LU equivalence to quasi-LU equivalence, as defined in the aforementioned paper by Jing et. al.

quant-ph↗

arXiv2023

Qubits as Hypermatrices and Entanglement

In this paper, we represent $n$-qubits as hypermatrices and consider various applications to quantum entanglement. In particular, we use the higher-order singular value decomposition of hypermatrices to prove that the $\pi$-transpose is an LU invariant. Additionally, through our construction we show that the matrix representation of the combinatorial hyperdeterminant of $2n$-qubits can be expressed as a product of the second Pauli matrix, allowing us to derive a formula for the combinatorial hyperdeterminant of $2n$-qubits in terms of the $n$-tangle.

quant-ph↗