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arXiv:2406.08509

Noncommutative Bohnenblust--Hille Inequality for qudit systems

Joseph Slote, Alexander L'vovich Vol'berg, Haonan Zhang

Published Jun 3, 2024• 0 citations
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Abstract

Previous noncommutative Bohnenblust--Hille (BH) inequalities addressed operator decompositions in the tensor-product space $M_2(\mathbb{C})^{\otimes n}$; \emph{i.e.,} for systems of qubits \cite{HCP22,VZ23}. Here we prove noncommutative BH inequalities for operators decomposed in tensor-product spaces of arbitrary local dimension, \emph{i.e.,} $M_K(\mathbb{C})^{\otimes n}$ for any $K\geq2$ or on systems of $K$-level qudits. We treat operator decompositions in both the Gell-Mann and Heisenberg--Weyl basis, reducing to the recently-proved commutative hypercube BH \cite{DMP} and cyclic group BH \cite{SVZ} inequalities respectively. As an application we discuss learning qudit quantum observables.

DOI
10.48550/arxiv.2406.08509

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Previous noncommutative Bohnenblust--Hille (BH) inequalities addressed operator decompositions in the tensor-product space M2(C)⊗nM_2(\mathbb{C})^{\otimes n}M2​(C)⊗n; \emph{i.e.,} for systems of qubits \cite{HCP22,VZ23}. Here we prove noncommutative BH inequalities for operators decomposed in tensor-product spaces of arbitrary local dimension, \emph{i.e.,} MK(C)⊗nM_K(\mathbb{C})^{\otimes n}MK​(C)⊗n for any K≥2K\geq2K≥2 or on systems of KKK-level qudits. We treat operator decompositions in both the Gell-Mann and Heisenberg--Weyl basis, reducing to the recently-proved commutative hypercube BH \cite{DMP} and cyclic group BH \cite{SVZ} inequalities respectively. As an application we discuss learning qudit quantum observables.
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We prove here several tightness results for such quantum inequalities as 1) the comparison of operator norm and product norm of ddd-local hamiltonians, 2) Bohnenblust--Hille inequality for ddd-local hamiltonians, 3) for quantum Fourier entropy-influence conjecture. We give a deterministic prove of the sharpness of Bohnenblust--Hille exponent. We also discuss the quantum Aaronson--Ambainis conjecture in a special case of anti-commuting Pauli strings.
Abstract Chapter 17 deals with the description of high-dimensional quantum systems and entanglement between them. We start with the generalized Bloch decomposition and present several useful options for the choice of operator bases, including the generalized Gell-Mann basis, the polarization-operator basis, and the Weyl-operator basis. We apply these bases for the example of isotropic two-qudit states and for the detection of bound entanglement for systems of two qutrits. We then turn to the problem of detecting entanglement in high-dimensional systems. For this purpose we review mutually unbiased bases and employ them for entanglement detection. We then discuss entanglement detection using entropic uncertainty relations, before we show how to obtain lower bounds on pure-state fidelities, on the entanglement of formation, and on the Schmidt number from measurement in pairs of mutually unbiased bases
Recent efforts in Analysis of Boolean Functions aim to extend core results to new spaces, including to the slice ([n]k)\binom{[n]}{k}(k[n]​), the hypergrid [K]n[K]^n[K]n, and noncommutative spaces (matrix algebras). We present here a new way to relate functions on the hypergrid (or products of cyclic groups) to their harmonic extensions over the polytorus. We show the supremum of a function fff over products of the cyclic group {exp⁡(2πik/K)}k=1K\{\exp(2πi k/K)\}_{k=1}^K{exp(2πik/K)}k=1K​ controls the supremum of fff over the entire polytorus ({z∈C:∣z∣=1}n)(\{z\in\mathbf{C}:|z|=1\}^n)({z∈C:∣z∣=1}n), with multiplicative constant CCC depending on KKK and deg(f)\text{deg}(f)deg(f) only. This Remez-type inequality appears to be the first such estimate that is dimension-free (i.e., CCC does not depend on nnn). This dimension-free Remez-type inequality removes the main technical barrier to giving O(log⁡n)\mathcal{O}(\log n)O(logn) sample complexity, polytime algorithms for learning low-degree polynomials on the hypergrid and low-degree observables on level-KKK qudit systems. In particular, our dimension-free Remez inequality implies new Bohnenblust--Hille-type estimates which are central to the learning algorithms and appear unobtainable via standard techniques. Thus we extend to new spaces a recent line of work \cite{EI22, CHP, VZ22} that gave similarly efficient methods for learning low-degree polynomials on the hypercube and observables on qubits. An additional product of these efforts is a new class of distributions over which arbitrary quantum observables are well-approximated by their low-degree truncations -- a phenomenon that greatly extends the reach of low-degree learning in quantum science \cite{CHP}.