Quantum systems allow active state spaces - Quantum information

96 Pages·2013·2.32 MB·English

Quantum systems allow active state spaces
Chris Heunen
University of Oxford
July 2, 2013
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Relationship between classical and quantum
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Relationship between classical and quantum
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Relationship between classical and quantum
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Part I
Algebras of observables
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C*-algebras
-algebra of operators that is closed
⇤
AW*-algebras
abstract/algebraic version of W*-algebra
von Neumann algebras / W*-algebras
-algebra of operators that is weakly closed
⇤
Jordan algebras
JC/JW-algebras: real version of above
Algebras of observables
Observables are primitive, states are derived
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Jordan algebras
JC/JW-algebras: real version of above
Algebras of observables
Observables are primitive, states are derived
C*-algebras
-algebra of operators that is closed
⇤
AW*-algebras
abstract/algebraic version of W*-algebra
von Neumann algebras / W*-algebras
-algebra of operators that is weakly closed
⇤
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Algebras of observables
Observables are primitive, states are derived
C*-algebras
-algebra of operators that is closed
⇤
AW*-algebras
abstract/algebraic version of W*-algebra
von Neumann algebras / W*-algebras
-algebra of operators that is weakly closed
⇤
Jordan algebras
JC/JW-algebras: real version of above
4/34
Algebras of observables
Observables are primitive, states are derived
C*-algebras
-algebra of operators that is closed
⇤
AW*-algebras
abstract/algebraic version of W*-algebra
von Neumann algebras / W*-algebras
-algebra of operators that is weakly closed
⇤
Jordan algebras
JC/JW-algebras: real version of above
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Theorem: Everycommutativeoperatoralgebra
I
is of this form.
I Can recover states (as maps C(X) C): “spectrum”
!
Constructions on states transfer to observables:
X +Y C(X) C(Y)
7! ⌦
X Y C(X) C(Y
⇥ 7! �
Equivalence of categories: states determine everything
Classical mechanics
I If X is a state space,
then C(X) = f : X C is an operator algebra.
{ ! }
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