Maximal ℓ₂-Norm Coherence

• Maximal ℓ₂-norm coherence is defined as the maximum ℓ₂-norm measure of nonorthogonality in structured sets such as frames, matrices, quantum states, and lattices.
• Key theoretical results include the Welch bound in frame theory, sampling thresholds for orthonormal matrices, and explicit bounds in quantum coherence measures.
• Practical applications span quantum resource theories, compressed sensing, and lattice constructions where minimizing coherence improves recovery, stability, and performance.

Maximal ℓ₂-norm coherence is a central concept in quantifying the degree of nonorthogonality, interference, or “spread” in structured sets of vectors, matrices, lattices, or quantum states. It appears across several mathematical and physical domains, including frame theory, matrix analysis, quantum information theory, and lattice geometry. Common to all contexts is the use of the ℓ₂-norm—either vector or Hilbert–Schmidt/Frobenius norm—to measure maximal overlap or off-diagonal magnitude, often seeking to minimize this quantity under structural constraints. This article surveys definitions, fundamental results, operational significance, and key constructions of maximal ℓ₂-norm coherence, with specific reference to the literature cited in (Jing et al., 2020, Ipsen et al., 2012, Datta et al., 2018, Fukshansky et al., 2020, Li et al., 8 Nov 2025), and (Yang et al., 2021).

1. Definitions and Fundamental Forms

The precise definition of maximal ℓ₂-norm coherence is context-dependent:

• Frame and Matrix Setting: For an $m \times n$ real or complex matrix $Q$ with orthonormal columns ($Q^T Q = I_n$), the maximal ℓ₂-norm coherence (“coherence parameter”; often denoted $\mu$) is

$\mu(Q) = \max_{1 \leq i \leq m} \|e_i^T Q\|_2^2,$

where $e_i^T Q$ is the $i$th row, and $\|\,\cdot\,\|_2$ is the vector 2-norm (Ipsen et al., 2012). For a frame $F = \{f_i\}_{i=1}^N$ of unit vectors in $\mathbb{R}^d$ or $\mathbb{C}^d$,

$$\mu(F) = \max_{i \neq j} |\langle f_i, f_j \rangle| \,.$$

• Quantum State Coherence: Given a density operator $\rho$ on a finite-dimensional Hilbert space and a fixed orthonormal basis $\{|i\rangle\}$, the Hilbert–Schmidt (matrix) ℓ₂-norm “distance-to-diagonal” coherence is

$$C_{2,2}(\rho) = \|\rho - \Delta(\rho)\|_2 = \left( \sum_{i \neq j} |\rho_{ij}|^2 \right)^{1/2}$$

with $\Delta(\rho)$ the diagonal pinching of $\rho$ (Jing et al., 2020, Li et al., 8 Nov 2025).

• Lattice Context: For a full-rank lattice $L \subset \mathbb{R}^d$, the maximal ℓ₂-norm coherence among minimal vectors $S'(L)$ (one from each $\pm v$ pair) is

$$\mu_{\max}(L) = \max_{x \neq y \in S'(L)} \frac{|\langle x, y \rangle|}{\|x\| \|y\|} \,.$$

If all $x \in S'(L)$ are normalized to unit length, this reduces to the standard frame coherence (Fukshansky et al., 2020).

2. Key Theoretical Results and Bounds

Frame and ETF Theory

A central lower bound is provided by the Welch bound for any $N$ unit vectors in $d$ dimensions:

$\mu \geq \sqrt{\frac{N - d}{d(N-1)}} \,,$

with equality attained if and only if the frame is an equiangular tight frame (ETF) (Datta et al., 2018). For the special $(d+1, d)$ simplex ETF, $\mu = 1/d$.

When ETFs of a given size do not exist, constructions aim to produce unit-norm tight frames (UNTFs) with $\mu$ as close as possible to the Welch bound, often using combinatorial block designs to append optimally chosen vectors to an initial ETF (Datta et al., 2018). Constructed UNTFs typically achieve

$$\mu \leq \max\left\{ \frac{1}{\sqrt{d}}, \frac{3}{d-1} \right\}$$

for large $d$, the leading term being $1/\sqrt{d}$.

Sampling from Orthonormal Matrices

For an $m \times n$ orthonormal matrix $Q$, the coherence $\mu$ satisfies

$\frac{n}{m} \leq \mu \leq 1\,,$

with the lower bound achieved by “most uniform” distributions (e.g., scaled Hadamard submatrices), the upper bound by maximally “spiky” cases (Ipsen et al., 2012). The key operational result relates $\mu$ to the number of sampled rows $c$ needed for well-conditioned submatrices:

$c \gtrsim m\,\mu\,\ln n \,.$

Smaller $\mu$ translates to fewer samples needed to guarantee, with high probability, both full rank and reasonable condition number.

Quantum Information and Hilbert–Schmidt ℓ₂-Norm

For quantum states, the Hilbert–Schmidt “distance-to-diagonal” coherence $C_{2,2}(\rho)$ is bounded as

$$C_{2,2}(\rho)^2 = \operatorname{Tr}(\rho^2) - \sum_i \rho_{ii}^2 \leq 1 - \frac{1}{n}$$

with equality for the maximally coherent state $$|\psi_n\rangle = \frac{1}{\sqrt{n}} \sum_{i = 1}^n |i\rangle$$ (Jing et al., 2020, Li et al., 8 Nov 2025). The “maximal” $l_2$-norm coherence—allowing basis optimization—is

$$C_{l_2}^{\max}(\rho) = \sqrt{\operatorname{Tr}(\rho^2) - \frac{1}{n}} \,,$$

thereby directly relating maximal coherence to quantum purity.

Lattice Constructions

For cyclotomic lattices $\Lambda_n \subset \mathbb{R}^{\varphi(n)}$, the maximal coherence among minimal vectors is characterized by

$$\mu_{\max}(\Lambda_n) = \begin{cases} 0, & n \ \text{a power of }2, \ 1/(p - 1), & p = \text{smallest odd prime dividing } n\,, \end{cases}$$

which can be made arbitrarily small as $n$ gains small odd prime divisors (Fukshansky et al., 2020). In contrast, irreducible root lattices $A_d$, $D_d$, and $E_6$–$E_8$ have $\mu_{\max}=1/2$ universally.

3. Operational and Structural Significance

Quantum Foundations

Maximal $l_2$-norm coherence quantifies the highest possible wave-like interference (off-diagonal magnitude) achievable by a quantum state under optimal basis transformation. The difference between $C_{l_2}^{\max}(\rho)$ and $C_{l_2}(\rho)$ defines a measure of path-predictability or “particle” property, and together with normalized linear entropy, forms a tradeoff (wave–particle–mixedness triality):

$$\frac{d}{d-1}\,C_{l_2}(\rho)^2 + M_l(\rho) + P(\rho)^2 = 1\,,$$

where $M_l(\rho)$ is the mixedness and $P(\rho)$ the single-path predictability (Li et al., 8 Nov 2025).

Random Matrix Sampling and Numerical Algorithms

In randomized linear algebra, maximal ℓ₂-norm coherence governs how many random samples are required for stability and accuracy in applications like preconditioned least squares and approximate matrix factorizations. Numerical experiments confirm that the $O(m\mu \ln n)$ threshold is sharp for achieving well-conditioned submatrices with high probability (Ipsen et al., 2012). Coherence also guides the design and assessment of fast matrix-generation algorithms for prescribed leverage scores and row norms.

Combinatorial and Geometric Constructions

Block designs enable the construction of large UNTFs with low maximal coherence by controlling intersection properties of appended vectors, approaching optimal lower bounds when ETFs at the desired size are unobtainable (Datta et al., 2018). In lattice geometry, the existence of lattices with minimal coherence is intimately tied to number-theoretic properties of their generating fields.

4. Constraints and Limitations

Monotonicity and Coherence Axioms

In quantum resource theory, not all natural ℓ₂-based quantities yield valid coherence measures. Theorem 1 of (Jing et al., 2020) establishes that no measure $C_v$ defined via a unitarily-invariant matrix norm (e.g., Hilbert–Schmidt, i.e., $q = p = 2$) satisfies monotonicity under incoherent operations. Explicit counterexamples are provided in dimension 4 or 6. Only $\ell_{1,p}$-norms ($q=1$, $1 \leq p \leq 2$) yield valid measures.

Extreme Values and Achievability

For matrices, the extreme values $\frac{n}{m} \leq \mu \leq 1$ are only achieved for highly structured cases (e.g., Hadamard matrices or canonical basis vectors). In quantum theory, the maximal achievable $C_{2,2}$ for a given system size is $\sqrt{1 - 1/n}$, uniquely realized by the maximally coherent superposition. In lattice theory, the minimal possible coherence depends on the arithmetic of dimension; cyclotomic lattices provide sequences with $\mu_{\max}$ vanishing as $n$ grows via small odd factors, while in other structured families (root lattices) a strict lower bound persists.

5. Applications and Interpretive Remarks

Signal Processing, Sparse Recovery, and Coding

Low maximal ℓ₂-norm coherence is advantageous for frames and lattices used in compressed sensing, sparse dictionary learning, and error-correcting codes. Small coherence ensures near-orthogonality, supporting unique recovery and resilience to erasures. Frames or point sets constructed via block designs or cyclotomic lattices can realize minimal coherence values at high redundancy (Fukshansky et al., 2020, Datta et al., 2018).

Quantum Information and Channel Resource Theories

In quantum dynamics, maximal $\ell_2$-norm coherence quantifies both fundamental limitations (through resource monotonicity) and potential for purity enhancement by noisy channels. The maximal-increasing-static-resource (MISR) measure (Yang et al., 2021) uses the increase in purity (above the maximally mixed baseline) to quantify dynamical coherence-generating power, with analytic formulas derived for amplitude-damping channels.

Randomized Numerical Algorithms

Matrix coherence directly informs the efficiency of randomized algorithms for linear algebra, including sampling-based preconditioners and matrix sketching. Lower coherence translates to drastically reduced sample complexity and improved numerical stability (Ipsen et al., 2012).

6. Related Measures and Ongoing Developments

Two prominent related measures are:

• Average Coherence: The mean pairwise inner product (or corresponding lattice cosine) among all minimal vectors, which captures the aggregate level of nonorthogonality and, in conjunction with the orthogonality defect, may be optimized for simultaneous packing and signal processing objectives (Fukshansky et al., 2020).
• Leverage-score–based Refinements: In matrix sampling, replacing $\mu$ by suitable aggregates of leverage scores allows for sharper bounds in highly nonuniform settings.

Remaining challenges include the explicit construction of low-coherence frames and lattices in specified dimensions, the search for tight upper and lower bounds in combinatorially constrained settings, and the detailed analysis of coherence in random matrix and random state ensembles.

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In summary, maximal ℓ₂-norm coherence serves as a fundamental quantitative invariant for near-orthogonality in frames, matrices, lattices, and quantum states, governing both structural limitations and algorithmic performance across a spectrum of mathematical, physical, and computational problems.