Understand projection

Til: Since I understand mathematical morphology which is a complete lattice, and understand the logic of all complete lattices by association, since projection is a complete lattice, i already understand the logic of projections.

1.2 The “Logic” of Projections
As stressed by von Neumann, the {0,1}
{0,1}
-valued observables may be regarded as encoding propositions about—or, to use his phrasing, properties of—the state of the system. It is not difficult to show that a self-adjoint operator P
P
with spectrum contained in the two-point set {0,1}
{
0
,
1
}
must be a projection; i.e., P2=P
P
2
=
P
. Such operators are in one-to-one correspondence with the closed subspaces of H
H
. Indeed, if P
P
is a projection, its range is closed, and any closed subspace is the range of a unique projection. If u
u
is any unit vector, then ⟨Pu,u⟩=||Pu||2

P
u
,
u

=
|
|
P
u
|
|
2
is the expected value of the corresponding observable in the state represented by u
u
. Since this observable is {0,1}
{
0
,
1
}
-valued, we can interpret this expected value as the probability that a measurement of the observable will produce the “affirmative” answer 1. In particular, the affirmative answer will have probability 1 if and only if Pu = u; that is, u
u
lies in the range of P
P
. Von Neumann concludes that

… the relation between the properties of a physical system on the one hand, and the projections on the other, makes possible a sort of logical calculus with these. However, in contrast to the concepts of ordinary logic, this system is extended by the concept of “simultaneous decidability” which is characteristic for quantum mechanics. (1932: 253)

Let’s examine this “logical calculus” of projections. Ordered by set-inclusion, the closed subspaces of H
H
form a complete lattice, in which the meet (greatest lower bound) of a set of subspaces is their intersection, while their join (least upper bound) is the closed span of their union. Since a typical closed subspace has infinitely many complementary closed subspaces, this lattice is not distributive; however, it is orthocomplemented by the mapping

M→M⊥={v∈H∣∀u∈M(⟨v,u⟩=0)}.
M

M

=
{
v

H


u

M
(

v
,
u

=
0
)
}
.
In view of the above-mentioned one-one correspondence between closed subspaces and projections, we may impose upon the set L(H)
L
(
H
)
the structure of a complete orthocomplemented lattice, defining P≤Q
P

Q
, where ran(P)⊆ran(Q)
ran
(
P
)

ran
(
Q
)
and P′=1−P
P

=
1

P
(so that ran(P′)=ran(P)⊥)
ran
(
P

)
=
ran
(
P
)

)
. It is straightforward that P≤Q
P

Q
just in case PQ=QP=P
P
Q
=
Q
P
=
P
. More generally, if PQ = QP, then PQ=P∧Q
P
Q
=
P

Q
, the meet of P
P
and Q
Q
in L(H)
L
(
H
)
; also in this case their join is given by P∨Q=P+Q−PQ
P

Q
=
P
+
Q

P
Q
.

1.1 Lemma:
Let P
P
and Q
Q
be projection operators on the Hilbert space H
H
. The following are equivalent:

PQ=QP
P
Q
=
Q
P
The sublattice of L(H)
L
(
H
)
generated by P,Q,P′
P
,
Q
,
P

and Q′
Q

is Boolean
P,Q
P
,
Q
lie in a common Boolean sub-ortholattice of L(H)
L
(
H
)
.
Adhering to the idea that commuting observables—in particular, projections—are simultaneously measurable, we conclude that the members of a Boolean sub-ortholattice of L(H)
L(H)
are simultaneously testable. This suggests that we can maintain a classical logical interpretation of the meet, join and orthocomplement as applied to commuting projections.

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