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Shmuel Friedland

Publications and source records attributed to Shmuel Friedland.

At least 19 recordsLinked to original sources
arXiv2026

Complete invariants for simultaneous similarity

Always dealing with an arbitrary field we consider the variety $(k^{n\times n})^{p}$ under the action of $GL_{n}$ by simultaneous similarity. We define discrete and continuous invariants which completely determine the orbits. The discrete invariants induce a disjoint decomposition of the variety into finitely many locally closed $GL_{n}$-stable subsets and for each of these we construct finitely many invariant morphisms to $k$ separating the orbits. The complicated action of $GL_{n}$ by similarity is reduced to left multiplication of a product of $GL_{l_{i}}$'s on a product of $k^{l_{i}\times m_{i}}$'s. An analogous result holds for the left-right action of $GL_{m}\times GL_{n}$ on $(k^{m\times n })^{p}$ and more generally for all varieties of finite dimensional modules over some finitely generated algebra.

math.RT↗
arXiv2026

A generalization of Frenkel's formula

We generalize Frenkel's integral formula for traces of operators to operators. The resulting formula holds for bounded self-adjoint positive operators and $p$-Schatten class of compact positive operators.

math.FA↗
arXiv2025

Barrier relaxations of the classical and quantum optimal transport problems

In the last fifteen years a significant progress was achieved by considering an entropic relaxation of the classical multi-partite optimal transport problem (MPOTP). The entropic relaxation gives rise to the rescaling problem of a given tensor. This rescaling can be achieved fast with the Sinkhorn type algorithms. Recently, it was shown that a similar approach works for the quantum MPOTP. However, the analog of the rescaling Sinkhorn algorithm is much more complicated than in the classical MPOTP. In this paper we show that the interior point method (IPM) for the primary and dual problems of classical and quantum MPOTP problems can be considered as barrier relaxations of the optimal transport problems (OTP). It is well known that the dual of the OTP are advantageous as it has much less variables than the primary problem. The IPM for the dual problem of the classical MPOTP are not as fast as the Sinkhorn type algorithm. However, IPM method for the dual of the quantum MPOTP seems to work quite efficiently.

math.OC↗
arXiv2025

Tensors, entanglement, separability, and their complexity

One of the most challenging problems in quantum physics is to quantify the entanglement of $d$-partite states and their separability. We show here that these problems are best addressed using tensors. The geometric measure of entanglement of a pure state is one of most natural ways to quantify the entanglement, which is simply related to the spectral norm of a tensor state. On the other hand, the logarithm of the nuclear norm of the state and density tensors can be considered as its ``energy''. We first show that the most geometric measure entangled $d$-partite state has the minimum spectral norm and maximum nuclear norm. Second, we introduce the notion of Hermitian and density tensors, and the subspace of bi-symmetric Hermitian tensors, which correspond to Bosons. We show that separable density tensors, and strongly separable bi-symmetric density tensors are characterized by the value (equal to one) of their corresponding nuclear norms. In general, these characterizations are NP-hard to verify. Third, we show that the above quantities are computed in polynomial time when we restrict our attentions to Bosons: symmetric $d$-qubits, or more generally to symmetric $d$-qunits in $C^n$, and the corresponding bi-symmetric Hermtian density tensors, for a fixed value of $n$.

quant-ph↗
arXiv2025

Complexity of Geometric programming in the Turing model and application to nonnegative tensors

We consider a version of geometric programming problem consisting in minimizing a function given by the maximum of finitely many log-Laplace transforms of discrete nonnegative measures on a Euclidean space. Under a coerciveness assumption, we show that an $\varepsilon$-minimizer can be computed in a time that is polynomial in the input size and in $|\log\varepsilon|$. This is obtained by establishing bit-size estimates on approximate minimizers and by applying the ellipsoid method. We also derive polynomial iteration complexity bounds for the interior-point method applied to the same class of problems. We deduce that the spectral radius of a partially symmetric, weakly irreducible nonnegative tensor can be approximated within an $\varepsilon$-error in polynomial time. For strongly irreducible tensors, we show in addition that the logarithm of the positive eigenvector is polynomial time approximable. Our results also yield that the the maximum of a nonnegative homogeneous $d$-form in the $\ell_d$ unit ball can be approximated in polynomial time. In particular, the spectral radius of uniform weighted hypergraphs and some known upper bounds for the clique number of uniform hypergraphs are polynomial time computable. In contrast, we provide an example showing that the Phase I approach needs exponentially many bits to solve the feasibility problem in geometric programming.

math.OC↗
arXiv2024

On semidefinite programming characterizations of the numerical radius and its dual norm

We state and give self contained proofs of semidefinite programming characterizations of the numerical radius and its dual norm for matrices. We show that the computation of the numerical radius and its dual norm within $\varepsilon$ precision are polynomially time computable in the data and $|\log \varepsilon |$ using either the ellipsoid method or the short step, primal interior point method. We apply our results to give a simple formula for the spectral and nuclear norm of $2\times n\times m$ real tensor in terms of the numerical radius and its dual norm.

math.NA↗
arXiv2023

A new class of distances on complex projective spaces

The complex projective space $\mathbb{P}(\mathbb{C}^n)$ can be interpreted as the space of all quantum pure states of size $n$. A distance on this space, interesting from the perspective of quantum physics, can be induced from a classical distance defined on the $n$-point probability simplex by the `earth mover problem'. We show that this construction leads to a quantity satisfying the triangle inequality, which yields a true distance on complex projective space belonging to the family of quantum $2$-Wasserstein distances.

math-ph↗
arXiv2023

Interior point method in tensor optimal transport

We study a tensor optimal transport (TOT) problem for $d\ge 2$ discrete measures. This is a linear programming problem on $d$-tensors. We introduces an interior point method (ipm) for $d$-TOT with a corresponding barrier function. Using a "short-step" ipm following central path within $\varepsilon$ precision we estimate the number of iterations.

math.OC↗
arXiv2022

Haagerup bound for quaternionic Grothendieck inequality

We present here several versions of the Grothendieck inequality over the skew field of quaternions: The first one is the standard Grothendieck inequality for rectangular matrices, and two additional inequalities for self-adjoint matrices, as introduced by the first and the last authors in a recent paper. We give several results on ``conic Grothendieck inequality'': as Nesterov $π/2$-Theorem, which corresponds to the cones of positive semidefinite matrices; the Goemans--Williamson inequality, which corresponds to the cones of weighted Laplacians; the diagonally dominant matrices. The most challenging technical part of this paper is the proof of the analog of Haagerup result that the inverse of the hypergeometric function $x {}_2F_1(\frac{1}{2}, \frac{1}{2}; 3; x^2)$ has first positive Taylor coefficient and all other Taylor coefficients are nonpositive.

math.FA↗
arXiv2022

Quantum Optimal Transport

We analyze a quantum version of the Monge--Kantorovich optimal transport problem. The quantum transport cost related to a Hermitian cost matrix $C$ is minimized over the set of all bipartite coupling states $ρ^{AB}$ with fixed reduced density matrices $ρ^A$ and $ρ^B$ of size $m$ and $n$. The minimum quantum optimal transport cost $\rT^Q_{C}(ρ^A,ρ^B)$ can be efficiently computed using semidefinite programming. In the case $m=n$ the cost $\rT^Q_{C}$ gives a semidistance if and only if $C$ is positive semidefinite and vanishes exactly on the subspace of symmetric matrices. Furthermore, if $C$ satisfies the above conditions, then $\sqrt{\rT^Q_{C}}$ induces a quantum analogue of the Wasserstein-2 distance. Taking the quantum cost matrix $C^Q$ to be the projector on the antisymmetric subspace, we provide a semi-analytic expression for $\rT^Q_{C^Q}$ for any pair of single-qubit states and show that its square root yields a transport distance on the Bloch ball. Numerical simulations suggest that this property holds also in higher dimensions. Assuming that the cost matrix suffers decoherence and that the density matrices become diagonal, we study the quantum-to-classical transition of the Earth mover's distance, propose a continuous family of interpolating distances, and demonstrate that the quantum transport is cheaper than the classical one. Furthermore, we introduce a related quantity -- the SWAP-fidelity -- and compare its properties with the standard Uhlmann--Jozsa fidelity. We also discuss the quantum optimal transport for general $d$-partite systems.

quant-ph↗
arXiv2022

Tensor rank and entanglement of pure quantum states

The rank of a tensor is analyzed in context of quantum entanglement. A pure quantum state $\bf v$ of a composite system consisting of $d$ subsystems with $n$ levels each is viewed as a vector in the $d$-fold tensor product of $n$-dimensional Hilbert space and can be identified with a tensor with $d$ indices, each running from $1$ to $n$. We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state ${\bf v}$ is called {\sl entangled}, if it {\sl cannot} be written in the product form, ${\bf v} \ne {\bf v}_1 \otimes {\bf v}_2 \otimes \cdots \otimes {\bf v}_d$, what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed..

quant-ph↗
arXiv2021

Quantum Monge-Kantorovich problem and transport distance between density matrices

A quantum version of the Monge--Kantorovich optimal transport problem is analyzed. The transport cost is minimized over the set of all bipartite coupling states $ρ^{AB}$, such that both of its reduced density matrices $ρ^A$ and $ρ^B$ of dimension $N$ are fixed. We show that, selecting the quantum cost matrix to be proportional to the projector on the antisymmetric subspace, the minimal transport cost leads to a semidistance between $ρ^A$ and $ρ^B$, which is bounded from below by the rescaled Bures distance and from above by the root infidelity. In the single qubit case we provide a semi-analytic expression for the optimal transport cost between any two states and prove that its square root satisfies the triangle inequality and yields an analogue of the Wasserstein distance of order two on the set of density matrices. We introduce an associated measure of proximity of quantum states, called SWAP-fidelity, and discuss its properties and applications in quantum machine learning.

quant-ph↗
arXiv2021

Tensor optimal transport, distance between sets of measures and tensor scaling

We study the optimal transport problem for $d>2$ discrete measures. This is a linear programming problem on $d$-tensors. It gives a way to compute a "distance" between two sets of discrete measures. We introduce an entropic regularization term, which gives rise to a scaling of tensors. We give a variation of the celebrated Sinkhorn scaling algorithm. We show that this algorithm can be viewed as a partial minimization algorithm of a strictly convex function. Under appropriate conditions the rate of convergence is geometric and we estimate the rate. Our results are generalizations of known results for the classical case of two discrete measures.

cs.CV↗
arXiv2021

Infimum of a matrix norm of A induced by an absolute vector norm

We characterize the infimum of a matrix norm of a square matrix A induced by an absolute norm, over the fields of real and complex numbers. Usually this infimum is greater than the spectral radius of A. If A is sign equivalent to a nonnegative matrix B then this infimum is the spectral radius of B.

math.FA↗
arXiv2021

Upper bounds for the spectral norm of symmetric tensors

The maximum of the absolute value of a real homogeneous polynomial of degree $d\ge 3$ on the unit sphere corresponds to the spectral norm of the induced real $d$-symmetric tensor $\mathcal{S}$. We give two sequences of upper bounds on the spectral norm of $\mathcal{S}$, which are stated in terms of certain roots of the Hilbert-Schmidt norms of corresponding iterates. We show that these sequences are converging to a limit, which is the minimal value of these upper bounds. Some generalizations to iterates of homogeneous polynomial maps are discussed.

math.FA↗
arXiv2021

Graph isomorphism and Gaussian boson sampling

We introduce a connection between a near-term quantum computing device, specifically a Gaussian boson sampler, and the graph isomorphism problem. We propose a scheme where graphs are encoded into quantum states of light, whose properties are then probed with photon-number-resolving detectors. We prove that the probabilities of different photon-detection events in this setup can be combined to give a complete set of graph invariants. Two graphs are isomorphic if and only if their detection probabilities are equivalent. We present additional ways that the measurement probabilities can be combined or coarse-grained to make experimental tests more amenable. We benchmark these methods with numerical simulations on the Titan supercomputer for several graph families: pairs of isospectral nonisomorphic graphs, isospectral regular graphs, and strongly regular graphs.

quant-ph↗