Stronger Welch Bounds and Optimal Approximate $k$-Designs

TL;DR

This paper develops strengthened bounds for quantum state distributions, providing a way to measure how well finite sets approximate ideal designs, and identifies optimal approximate 3-designs like SICs and MUBs.

Contribution

It introduces new sharp inequalities for approximate quantum designs, characterizes minimal error at fixed size, and computes the spectrum of key operators for the first time.

Findings

SICs and MUBs saturate the bounds for approximate 3-designs.

Derived bounds are sharp even below the size of exact k-designs.

Numerical evidence suggests no complete MUB set exists in dimension 6.

Abstract

A fundamental question asks how uniformly finite sets of pure quantum states can be distributed in a Hilbert space. The Welch bounds address this question, and are saturated by $k$-designs, i.e. sets of states reproducing the $k$-th Haar moments. However, these bounds quickly become uninformative when the number of states is below that required for an exact $k$-design. We derive strengthened Welch-type inequalities that remain sharp in this regime by exploiting rank constraints from partial transposition and spectral properties of the partially transposed Haar moment operator. We prove that the deviation from the Welch bound captures the average-case approximation error, hence characterizing a natural notion of minimum achievable error at fixed cardinality. For $k=3$, we prove that SICs and complete MUB sets saturate our bounds, making them optimal approximate 3-designs of their cardinality. This leads to a natural variational criterion to rule out the existence of a complete set MUBs, which we use to obtain numerical evidence against such set in dimension $6$. As a key technical ingredient, we compute the complete spectrum of the partially transposed symmetric-subspace projector, including multiplicities and eigenvectors, which may find applications beyond the present work.