Special Types of Metric Spaces| 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 |xi
− yi|,
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
| S ∩ B1/a⊂ S'
+ Ba & S' ∩
B1/a ⊂ S
+ 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∞
2−n
[Fn(σ, τ)
/ (1+Fn)], where
Fn(σ, τ):=
sup0 ≤ t ≤ n
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.
main page
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send feedback and suggestions to bombelli at olemiss.edu – modified 11 dec 2020