Stronger Welch Bounds and Optimal Approximate $k$-Designs

TL;DR

This work develops stronger Welch-type bounds for finite ensembles of pure quantum states by exploiting rank constraints from partial transposition and the spectral structure of the partially transposed Haar moment operator. It introduces a natural average-case design error, showing it is controlled by the excess $k$-frame potential above the Haar value, and proves that the deviation from the Welch bound captures the typical approximation error to a $k$-design. For $k=3$, the authors show SICs and complete MUBs saturate these sharpened bounds, establishing them as optimal approximate $3$-designs at their cardinalities and enabling a variational criterion against the existence of a complete MUB set in dimension $6$. The key technical achievement is the complete spectrum of the partially transposed symmetric-subspace projector, which may have further applications in representation theory and quantum information.

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 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.

Figures (2)

• Figure 1: Block decomposition of $\mathcal{F}_{2,2}$ for $k=4,\ k'=2$. The $k'$-design constraints force agreement with Haar on all blocks with $r+r'\ge k-k'=2$, leaving only the red blocks as free variables.
• Figure 2: Lower bounds and heuristic minima for the off-diagonal overlap moment ($\sum_{i \not =j} \bigl|\braket{\psi_i}{\psi_j}\bigr|^{2k}= N^{2}\mathcal{E}_k(\chi)-N$). Left:$d=3$, $k=3,4$: Welch bound, strengthened bound (Thm. \ref{['Th:StrongerWB']}), and heuristic minima versus $N$. Black points/dotted lines mark the design threshold $N_{\min}(k,d)$, where the curves coincide. For $k=3$, the heuristic values nearly saturate the strengthened bound. Right:$k=3$ in the $2$-design regime $N=d(d+1)$: Welch bound, sharpened bound (Thm. \ref{['Th:VS-WB']}), and heuristic minima versus $d$. The Welch bound is trivial from $d=4$, while the sharpened bound remains non-trivial. The minimum is consistent with $0$ for $d\le 5$ and $d=7$, but shows a clear positive gap (about $20\%$) at $d=6$, adding evidence against a complete set of MUBs in $d=6$. The code used to generate the plots is available at Code.

Theorems & Definitions (18)

• Theorem 1: Welch bounds WelchBound
• Definition 1: Complex projective $k$-design
• Theorem 2: Welch saturation and designs WelchBoundGeom(2012)klappenecker2005Roy2007
• Theorem 3: Spectrum of $\rho_{n,m}$
• Theorem 4
• proof
• Theorem 5: Stronger Welch bounds
• proof
• Theorem 6: Stronger Welch bound for designs
• Theorem 7: Full spectrum of $\rho_{n,m}$