Special Types of Metric Spaces  

Types > s.a. Inner Product; norm; Ultrametric.
* Polish space: A complete separable metric space; > s.a. causal curves.
* Length space (path metric space): A metric space with distance d(x, y) equal to the lower bound on the length of curves between x and y (because of the triangle inequality, it suffices to ask that there exist a curve γ such that d(x, y) = length(γ)).
@ General references: Uspenskij T&A(04) [Urysohn universal metric space].
@ Between physical theories: Calmet & Calmet MPLA(11) [and information theory]; Anselmi & Buttazzo PRD(11)-a1105 [as a measure of Lorentz-symmetry violation].
@ Lorentzian length spaces: Kunzinger & Sämann AGAG-a1711; Grant et al AGAG(19)-a1804 [and spacetime inextendibility]; > s.a. manifolds with metrics; causality conditions.
@ Other special types: Mascioni DM(05) [finite spaces with random distances]; Akhvlediani et al T&A(10) [Hausdorff and Gromov distances]; Ben Rekeb et al T&A(13) [non-expansive homeomorphisms]; Guido et al JFA(17)-a1512 [between von Neumann algebras]; Bhatia et al LMP(19)-a1901 [positive-definite matrices].

Examples > s.a. spectral geometry; yang-mills gauge theory.
* On Rn: One can define d(x, y):= supi |xiyi|, or dp(x, y):= [∑i \(|x^i - y^i|^p\)]1/p; The case p = 2 is the Euclidean d.
* On a vector space: The space X can be given a norm compatible with d iff d(ax, ay) = |a| d(x, y).
* For locally finite subsets of Rn: d(S, S'):= min{2−1/2, inf D(S, S')}, where D(S, S') is the half-line defined by D(S, S'):= {a > 0 | SB1/aS' + Ba & S'B1/aS + Ba} [@ Gouéré mp/02].
* For complex functions:

d(f, g):= dx |f(x) − g(x)|2 F(x) ,   for some positive real function F .

* For probability distributions / measures: Examples are the Fisher metric and the Wasserstein metric; Another possibility is

d(P, P'):= arccos(i=1N Pi1/2 P'i1/2) .

* For spectra: The log-spectral distance (symmetric; Wikipedia page) and the Itakura-Saito distance (non-symmetric; Wikipedia page).
* For paths in a metric space (X, d): Given two paths σ and τ: I → X,

d*(σ, τ):= supt ∈ I d(σ(t), τ(t)) ,

or, for I = [0, ∞), D(σ, τ):= ∑n=1 2n [Fn(σ, τ) / (1+Fn)], where Fn(σ, τ):= sup0 ≤ tn d[σ(t), τ(t)].
* For knots / links: The smallest number of crossings needed to go from one to the other.
* For unlabelled posets: (a) One possibility is to call d(P, Q) the minimal number of relationships that must be changed in P to get a poset isomorphic to Q; (b) Another possibility is to use subposets.
@ On discrete / finite spaces: Iochum et al JGP(01) [from non-commutative geometry]; > s.a. graphs.
@ Probability measures: [Fisher metric]; Raviculé et al PRA(97); Casas et al qp/04 [vs Hilbert-space states]; Abe et al JSP(07) [l1 distance]; Budzyński et al CQG(08)-a0712 [and gravitational-wave data analysis]; Costa et al a1210 [geometrical approach]; > s.a. Wikipedia page on Statistical Distances.
@ Other: Nabutovsky CMP(96) [triangulations of a compact manifold; D ≥ 4]; Crooks PRL(07) [equilibrium states, thermodynamic length]; Schuhmacher & Xia AAP(08) [point-process distributions]; Kar & Rajeev PRD(12)-a1207 [on spacetime, non-Riemannian metric from a scalar quantum field theory]; Nekvinda & Zindulka Ord(12) [monotone]; Eldering & Vankerschaver DG&A(14)-a1401 [on the space of parametrized curves modulo rigid transformations].

For Quantum States > s.a. coherent states; types of metrics; Propagator; riemannian geometry.
* Bures metric: Introduced by Uhlmann; It generalizes the Fubini-Study metric to mixed states.
@ General references: Wootters PhD(80), PRD(81); Braunstein & Caves PRL(94); Raviculé et al PRA(97); Dodonov et al PS(99)qp/98 ["energy-sensitive"]; Rieffel DocM(99)m.OA; Ozawa PLA(00)qp [re Hilbert-Schmidt]; Trifonov & Donev qp/00-wd; in Giovannetti et al PRA(03)qp/02; Lee et al PRL(03)qp; Majtey et al EPJD(05)qp/04 [and distinguishability]; Arbatsky qp/05 [quantum angle]; Li et al a1512 [modulus fidelity, for many-body systems]; Shivam et al IJQI(18)-a1609 [based on quantum relative entropy].
@ Various types of distances: Brody & Hughston JGP(01) [Fubini-Study d]; Lamberti et al IJQI(09)-a0807-conf, Osán & Lamberti PRA(13)-a1303 [based on entropy and purification]; Cohen PRA(09)-a0906 [statistical distance]; Anshu et al IEEE(16)-a1404v3 [trace distance]; Wang et al a2007 [physical distance]; Girolami & Anzà a2012 [weighted distances, between many-body states].
@ Between density matrices: Życzkowski & Słomczyński JPA(98)qp/97 [Monge]; Petz & Sudar in(99)qp/01 [Fisher d]; Slater JMP(06);
> s.a. mixed states.
@ Bures metric: Twamley JPA(96) [thermal squeezed states]; Slater PLA(98)qp/97; Dittmann JPA(99) [explicit formulae].
@ Between classical and quantum states: Klauder qp/03; Abernethy & Klauder FP(05)qp/04.

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