The Surprisingly Small Zoo of Natural Norms and Metrics
When you first encounter the theory of norms and metrics, you might despair: there are so many. Every textbook introduces a new one. Matrix norms alone fill entire chapters. Surely any structure this promiscuous can’t have much to say.
But here’s the thing—once you commit to what “natural” means in your setting, the sea of possibilities collapses dramatically. The right question isn’t “which norm should I use?” but rather “what structure do I want my norm to respect?” Symmetries, composition rules, monotonicity under coarse-graining, operational interpretability—pick your desiderata and the answer often becomes nearly unique.
This post surveys the main characterization theorems that constrain the zoo. Think of it as a field guide: given your symmetry requirements, here’s what you’re allowed to have.
Vector Space Norms: When Does a Norm “Really Come From Geometry”?
Inner Product Structure Forces Itself on You
If you want angles, orthogonality, and Pythagoras—i.e., if you want your norm to arise from an inner product—there’s a remarkably clean characterization:
Jordan–von Neumann Theorem: A norm $|\cdot|$ comes from an inner product if and only if it satisfies the parallelogram law:
\[\|x+y\|^2 + \|x-y\|^2 = 2\|x\|^2 + 2\|y\|^2\]When this holds, the inner product is recovered by polarization.
Physics punchline: If you assume “Euclidean” structure (rotational invariance + Pythagoras-type additivity of squared lengths), you’re essentially forced into $L^2$ / Hilbert space. The parallelogram law is the Pythagorean theorem in disguise.
Finite Dimensions: Don’t Overthink It
On $\mathbb{R}^n$, all norms are equivalent—they induce the same topology. So “naturalness” isn’t about convergence or existence of limits; it’s about symmetry and operational meaning. The topology doesn’t care, but the physics does.
Matrix Norms: Unitary Invariance Classifies (Almost) Everything
For physics applications, you almost always want basis-independence: the norm shouldn’t change under a change of orthonormal basis. This is the key constraint.
The Classification Theorem
Let $A \in \mathbb{C}^{n \times n}$ with singular values $\sigma(A) = (\sigma_1, \ldots, \sigma_n)$.
Theorem (von Neumann / Ky Fan / Schatten): A norm $|\cdot|$ is unitarily invariant (UIN),
\[\|UAV\| = \|A\| \quad \forall \text{ unitary } U, V\]if and only if there exists a symmetric gauge norm $g$ on $\mathbb{R}^n$ such that
\[\|A\| = g(\sigma(A))\]Translation: choosing a unitarily invariant norm is equivalent to choosing a permutation- and sign-invariant norm on singular values. The whole infinite-dimensional space of possible matrix norms collapses to something parameterized by functions on vectors of singular values.
The Canonical Families
Within unitarily invariant norms, the most principled ones come from $\ell_p$ gauges on singular values:
| Name | Definition | $p$ | When to Use |
|---|---|---|---|
| Operator/Spectral | $|A|_\infty = \sigma_1$ | $\infty$ | Worst-case amplification, dynamics |
| Frobenius/Hilbert-Schmidt | $|A|_2 = \sqrt{\mathrm{Tr}(A^*A)}$ | $2$ | RMS error, Fourier-friendly |
| Trace/Nuclear | $|A|_1 = \sum_i \sigma_i$ | $1$ | Total mass, low-rank optimization |
Other named UINs (Ky Fan $k$-norms, etc.) are also symmetric gauges, but the Schatten $p$-norms are the backbone.
Submultiplicativity: Compatibility with Composition
If you want your norm to play nicely with composition of linear maps—crucial for dynamics, where you care about $|A^n|$—you want submultiplicativity: $|AB| \leq |A| |B|$.
The canonical way to get this is via an induced operator norm:
\[\|A\|_{v \to v} = \sup_{x \neq 0} \frac{\|Ax\|_v}{\|x\|_v}\]For $\ell_p$ vector norms, this gives the standard induced operator norms. The cases $p = 1, 2, \infty$ have especially clean interpretations (max column sum, spectral radius, max row sum respectively for $p = 1, 2, \infty$… well, $p=2$ is the spectral norm).
Measures and Symmetry: Haar Is the Archetype
If matrix norms are constrained by unitary invariance, measures are constrained by group invariance. The poster child is Haar measure.
Haar Measure Theorem: On any locally compact topological group $G$, there exists a nonzero left-invariant measure, and it is unique up to scale.
Group structure + invariance requirement → essential uniqueness. This is the template.
Riemannian Metrics on Lie Groups: Algebraic Control
If you want a metric compatible with a symmetry group acting transitively (the typical situation in physics), then metrics correspond to algebraic data.
On a Lie group $G$:
- Left-invariant Riemannian metrics ↔ inner products on the Lie algebra $\mathfrak{g}$
If you demand bi-invariance (left and right), existence becomes restrictive. For compact semisimple groups it exists and is closely tied to the Killing form.
On homogeneous spaces $G/H$, $G$-invariant metrics correspond to $H$-invariant inner products on the tangent space at the identity coset.
Physics punchline: If your system has a big symmetry group, “natural” metrics are exactly those invariant under that group—often a finite-parameter family, or unique up to scale.
Information Geometry: Monotonicity Picks Out Canonical Metrics
This is where things get really satisfying for physics applications.
Classical: Fisher Is Essentially Unique
If you want a Riemannian metric on probability distributions that is invariant under sufficient statistics / coarse-graining (Markov morphisms), then:
Čencov’s Theorem: The Fisher information metric is (essentially) the unique monotone Riemannian metric, up to overall scale.
This is remarkable. Among all possible ways to measure distances between probability distributions, requiring that “distinguishability shouldn’t increase under coarse-graining” leaves you with essentially one choice.
Quantum: A Classified Family
For density matrices, if you want a Riemannian metric monotone under CPTP maps (quantum channels), uniqueness fails—but there’s a complete classification:
Petz Classification: Monotone quantum Riemannian metrics correspond to operator monotone functions.
Special cases include:
- Bures / Quantum Fisher: Minimal monotone metric; tied to state distinguishability
- Kubo-Mori / Bogoliubov: Linked to linear response / relative entropy Hessian
Requiring “distinguishability should not increase under noisy processing” basically forces you into the Fisher/Bures family.
Selection Principles: A Cheat Sheet
Here’s my attempt at a taxonomy. Given your desiderata, here’s what you get:
| Principle | What It Picks Out |
|---|---|
| Basis/coordinate independence | Unitarily invariant norms → symmetric gauges on singular values |
| Composition/dynamics | Induced operator norms; spectral norm canonical in Hilbert space |
| Energy/Parseval structure | $L^2$, Hilbert-Schmidt/Frobenius; inner-product norms via parallelogram law |
| Operational distinguishability | Fisher (classical, unique), Bures/quantum Fisher (quantum) |
| Convex optimization / low-rank | Trace norm as convex proxy for rank; $\ell_1$ for sparsity |
| Worst-case vs average-case | Operator norm (worst), trace norm (total), Frobenius (RMS) |
The Real Message
The zoo isn’t that big once you know what you’re looking for.
If you want unitary invariance + inner-product geometry → Frobenius / Hilbert-Schmidt
If you want unitary invariance + operational distinguishability → trace norm (discrimination bounds) and Bures/quantum Fisher (infinitesimal distinguishability)
If you want dynamics control → operator norm (spectral norm), induced norms
If you want monotonicity under coarse-graining → Fisher (classical) / Petz family (quantum)
The point isn’t to memorize norms—it’s to recognize that asking for natural structure constrains you heavily. This is philosophically satisfying: the mathematical universe isn’t arbitrary. Impose reasonable requirements and you land on a small number of distinguished objects.
(As usual in physics, “reasonable” is doing a lot of work in that sentence. But that’s fine. Figuring out what’s reasonable for your problem is the problem.)