Taken from Condition Expectation in Quantum Probabilty by Denes Petz.
In quantum probability there are a number of fundamental questions that ask how faithfully can one quantise classical probability. Suppose that is a (classical) probability space and a sub--algebra. The conditional expectation of some integrable function (with respect to some -space) relative to is the orthogonal projection onto the closed subspace :
, .
Suppose now that is a quantum probability space and that is some C*-subalgebra of . Can we always define a conditional expectation with respect to ? The answer turns out to be not always, although this paper gives sufficient conditions for the existence of such a projection. Briefly, things work the other way around. Distinguished states give rise to quantum conditional expectations — and these conditional expectations define a subalgebra. We can’t necessarily start with a subalgebra and find the state which gives rise to it — Theorem 2 gives necessary conditions in which this approach does work.
Let be a probability space and a sub--algebra of . In 1954 Moy observed that the property
( are bounded)
plays a crucial role in the characterisation of the conditional expectation operator
.
More or less at the same time Nakamura, Turumaru and Umegaki created the algebraic form of the conditional expectation (i.e. with respect to quantum probability). Here we deal with states on a von Neumann algebra . A state is called tracial if for all . When is finite dimensional (and so it is a direct sum of some matrix algebras), then it always possesses a tracial state . In this case every state has the form with a unique , called the density of . Due to the lack of unbounded operators the finite dimensional case is rather easily computable.
Let be a fixed state on and a linear mapping such that
- if .
- .
- for all .
- for every .
Property 3. implies that is weak continuous (I can’t seem to show this) and hence is weak closed (as the kernel of the continuous map ). It also follows easily that is an algebra () and the range of
This just implies that is a projection — it’s nice to know we don’t have to assume this. I can prove that easily. Suppose now that . We can assume that is positive because everything in sight is linear. By assumption there exists a such that . By linearity, must be positive also. Now, using property 3.
.
As is faithful this implies that so that as required.
Such an is called a –preserving conditional expectation onto .
Theorem 1
Let be a tracial state on and von Neuman subalgebra of . Then there exists a unique conditional expectation preserving and mapping onto .
Proof : The reference proffered is Umegaki although I can’t find the proof of this theorem in it
defines an inner product on (the Hilbert-Schmidt inner product). To avoid the completion process assume that is finite dimensional and so it is a Hilbert space. is a closed subspace of and we consider the the orthogonal projection . The properties 1. to 4. are verified straightforwardly.
is defined in terms of the conditional expectation associated to . What Petz does here is prove that .
The case of tracial reminds very much of the commutative situation. From this point of view the following example may be surprising. If
and , then for a state on the conditional expectation preserving , exists if and only if . The Takesaki theorem clarifies the existence of the conditional expectation by means of the modular group of the state.
Theorem 2 (Takesaki)
Let be a von Neumann subalgebra of and a state on . Then the -preserving conditional expectation from to exists if and only if is stable under the modular group of . If the conditional expectation exists then it is unique.
Modular Groups
It would probably be wise to find out about these objects before I say any more. This is taken from here and has zero rigour so we are just getting a flavour and then we’ll move on.
Suppose that is a von Neumann algebra acting on a Hilbert space , and is a separating, cyclic vector of of norm . We write for the state on , so that is constructed from via the GNS construction. We can define an antilinear operator on with domain by setting
for all ,
and similarly we can define an antilinear operator on with domain by setting
for all .
These operators are closable, and we denote their closures by and . They have polar decompositions
,
.
where is an antilinear isometry called the modular conjugation and is a positive self adjoint operator called the modular operator. The main result of Tomita–Takesaki theory states that
, the commutant of , and
There is a 1-parameter family of modular automorphisms of associated to the state , defined by
.
Proof of Theorem 2
See Takesaki (no access)
If we try to apply the argument (Umegaki’s to Theorem 1) which leads to success for a tracial state then we arrive at troubles with the positivity condition 1. Accardi and Cecchini have defined a mapping , which is completely positive (need to look at this — I have no idea why this might jeapardise condition 4.), always exists and therefore possesses only a weaker form of condition 4. , called -conditional expectation, does not map onto . In fact, they did not consider postulates like 1. to 4. but extended to von Neumann algebras the construction of measure-theoretic conditional expectation. If is a bounded -measurable function, then is a functional on , and its restriction determines an -measurable function such that
for all .
So in this construction is on the duality .
The following result is standard in Tomita-Takesaki theory
Lemma 3
If is a state on and , then there exists a function
so that
- is bounded, continuous and is analytic on the interior of its domain.
- for all .
- for all .
Theorem 4
Let be von Neumann algebras and a state on . If () is the modular group of () then the formula
determines a completely positive mapping . For
for every $latex a\in A$
holds if and only if for every .
I better stop at this point and go and find out more about completely positive maps. The modular group of I imagine is a departure from the classical case.
Leave a comment
Comments feed for this article