Special
Types of Metric Spaces| Special
Types of Metric Spaces |
Types > s.a. Inner Product; norm; Ultrametric.
* Polish space: A complete
separable metric space.
* 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 
such
that d(x, y)
= length()).
@ References: Uspenskij T&A(04) [Urysohn universal metric space].
Examples > s.a. yang-mills
gauge theory.
* On Rn:
One can define d(x, y):=
supi |xi–yi|,
or dp(x, y):=
[
i |xi – 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}
[@ Gouere 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: One is the Fisher metric; Another possibility is
d(P, P'):= arccos(i=1N Pi1/2 P'i1/2)
.
* For paths in a metric
space (X, d): Given two paths 
and 
:
I → X,
d*(, ):=
supt in I d((t), (t))
,
or, for I = [0,
), D(,):= n=1infty
2–n [Fn(,)/(1+Fn)],
where Fn(, ):=
sup_{0 
t n} d[(t), (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]; Budzynski
et al CQG(08)-a0712 [and gravitational wave data analysis].
@ Other: Nabutovsky CMP(96)
[triangulations of a compact manifold; D 
4];
Crooks PRL(07)
[equilibrium states, thermodynamic length].
For Quantum States > s.a. mixed
states; types
of metrics; Propagator; riemannian geometry.
* Bures metric: Introduced by Uhlmann; 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;
Brody & Hughston JGP(01)
[Fubini-Study d];
in Giovannetti
et
al PRA(03)qp/02;
Lee et al PRL(03)qp;
Majtey et al qp/04 [and
distinguishability]; Arbatsky qp/05 [quantum
angle]; Lamberti et al a0807-in
[based on entropy and purification].
@ Between density matrices: Zyczkowski & Slomczynski JPA(98)qp/97 [Monge];
Petz & Sudar in(99)qp/01 [Fisher d];
Slater JMP(06).
@ 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.
Other Types > see distance on and between manifolds
with metrics.
@ References: Mascioni DM(05)
[finite spaces with random distances].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
3 jul 2008