Noncommutative Mathematics for Quantum Systems

Index 179

Hopf algebra, 32

joint distribution

of a quantum stochastic process, 30

Kac type compact quantum group, 48
Kochen-Specker theorem, 21

L ́evy process
classical, 24,
on a dual semigroup, 101
on a Hopf *-algebra, 33
on an involutive bialgebra, 32
L ́evy-Khintchine formula, 26

marginal distribution
of a quantum stochastic process, 30
measurable dynamical system, 157
mixed state, 6
monotone
independence, 92, 101
L ́evy process, 101
product, 92,
monotone independence, 60, 62

observable, 5
operator process, 30
orderkconvergence, 168

Poisson generating functional, 40
positive
completely, 125
element, 124
map, 124
probability space
*-algebraic, 4, 29
algebraic, 87
classical, 4
quantum, 4, 29
pure state, 6

quadratic generating functional, 40
quantum

probability space, 4, 29

random variable, 5, 29

stochastic process, 29

quantum cancellation property, 43

quantum group

algebraic, 45

compact, 44

Kac type, 48

random variable

classical, 5

quantum, 5, 29

reciprocal Cauchy-Stieltjes

transform, 56

Schoenberg correspondence, 35, 103

Sch ̈urmann triple, 36

state, 124

Haar, 44

mixed, 6

pure, 6

tracial, 124

vector, 6

stationarity of increments, 32, 102

Stieltjes transform, 55

stochastic

independence, 23, 89

process

classical, 24

quantum, 29

subordinator, 25

surjective Schu ̈rmann triple, 37

Sweedler notation, 31

tensor

independence, 31, 90, 101

L ́evy process, 101

time-reversal, 103

topological dynamical system

classical, 123

quantum/noncommutative, 124