In this section the formalism of orthodox quantum logic (OQL) is presented. Usu- ally, this formalism is introduced by making an analogy with the way classical logic derives from the state space formalism of classical mechanics (Isham, 1995). Propo- sitions are then identified with sets of states, and logical connectives are introduced by suggesting manipulations of these sets. The downside of this approach for the present discussion is that it starts from the idea that projections can be seen as propositions, which is the very idea that is being questioned here. Instead, I pro- vide a short derivation of OQL from the formalism of quantum probability. The upshot of this approach is that OQL necessarily presents itself to us as part of the quantum formalism, in that there are no real philosophical choices involved in the adoption of OQL. Rather, the philosophical work lies in providing an understanding of the role of this structure. Minimally, it is just a mathematical oddity that has no conceptual role to play, and maximally it points us towards a reinvestigation of what kind of reasoning is correct in a world in which quantum mechanics is true. I take it that the truth lies somewhere in between these two extremes. But before one can investigate where precisely, one has to know what OQL is.
To understand how one can derive quantum logic from quantum probability it is useful to consider a classical analogy. In the classical case the probability function is a map P :F → [0,1], whereF is some σ-algebra. The structural properties of
P are reflections of the the structure of the σ-algebra F. Specifically, as a set of subsets,F has the natural partial order
∆1≤∆2 iff ∆1 ⊂∆2. (8.1)
This partial order also has another possible characterization.
Proposition 8.1. For a measurable space (Ω,F), for every ∆1,∆2, ∆1 ≤ ∆2 if
and only if P(∆1)≤P(∆2) for every probability measure P on(Ω,F).
Proof. Suppose ∆1≤∆2. Then for every probability measure Pit holds that
P(∆1) =P(∆1∩∆2)≤P(∆1∩∆2) +P(∆c1∩∆2) =P(∆2). (8.2)
Conversely, suppose ∆1 is not a subset of ∆2. Then ∆3 := ∆1∩∆c2 is not empty.
Now choose P such that P(∆3) = 1. Then P(∆1) = 1 6≤ 0 = P(∆2), which is a
contradiction.
The partial order onF may thus also be understood as a structure imposed on it by the set of all probability functions. In an analogous way a relation may be introduced onL(H) that turns it into a partial ordered set.
Orthodox quantum logic
8.2
Theorem 8.1. The relation
P1 ≤P2 iff ∀P:P(P1)≤P(P2) (8.3)
is a partial order on L(H).
Proof. From the properties of probability functions it follows directly that this re- lation is a preorder (i.e., reflexive and transitive). With a little more effort it can be seen that it is also antisymmetric. What has to be shown for this is that if
P(P1) = P(P2) for all P, then P1 = P2. The proof relies on the following three
identities.
P PP ψ =PP ψ, P PP⊥ψ =0 andkP ψk=kψk q
Tr(P Pψ)
for every projection operatorP and every vector ψ, where
P⊥:=1−P (8.4)
denotes the orthocomplement ofP, andPP ψis the projection onto the line spanned
by P ψ, i.e.,
PP ψφ:=
hP ψ, φi
hP ψ, P ψiP ψ. (8.5)
Now suppose for two projectionsP1, P2 thatP(P1) =P(P2) for allP, then
kP2ψ−P1ψk=k(P2−P1)P1ψ+ (P2−P1)P1⊥ψk ≤k(P2−P1)P1ψk+k(P2−P1)P1⊥ψk=kP2⊥P1ψk+kP2P1⊥ψk =kP1ψk q Tr(P2⊥PP1ψ) +kP ⊥ 1 ψk q Tr(P2PP⊥ 1 ψ) =kP1ψk q Tr(P1⊥PP1ψ) +kP ⊥ 1 ψk q Tr(P1PP⊥ 1 ψ) = 0,
where in the last line it was used that P 7→ Tr(P PP1ψ) and P 7→ Tr(P PP⊥
1 ψ)
are probability functions for all P1 and ψ and that P(P1) = P(P2) if and only if
P(P1⊥) =P(P2⊥).
This theorem establishes that quantum probability functions introduce a partial order on L(H) analogous to the way classical probability functions introduce a partial order on F. The partial order on F is the one associated with the usual lattice structure: the join of two sets corresponds to the union, and their meet with the intersection. Note that, given a partially ordered set, the meet and join of two elements (if they exist) are unique. Thus union and intersection are operations that derive from the partial order of set-inclusion. Similarly, the meet and join in the orthodox quantum logic of Birkhoff and von Neumann (1936) (see also (von
Neumann, 1932,§III.5)) derive from the partial order (8.3) on L(H). To see this it is required to show that indeed every pair of projections has a meet and a join with respect to the partial order (8.3). This, in turn, is best shown by proving that the partial order is equivalent to the usual one.
Proposition 8.2. Let (L(H),≤) be the poset with ≤given by (8.3), then
P1 ≤P2 iff P1H ⊂P2H iff P1P2 =P2P1 =P1. (8.6)
Proof. Start with the first left-to-right implication. Suppose P1 ≤P2 and letψ ∈
P1H. By making use of the fact that P 7→ Tr(P Pψ) is a probability function, it
follows that kψ−P2ψk2=kP2⊥ψk2 =kψk2Tr(P ⊥ 2 Pψ) =kψk2(1−Tr(P 2Pψ))≤ kψk2(1−Tr(P1Pψ)) = 0. (8.7) ThusP2ψ=ψand ψ∈P2H.
Now consider the second left-to-right implication. Because P2 is idempotent, it
follows directly fromP1H ⊂P2HthatP2P1 =P1. The second equation is obtained
by taking adjoints:
P1P2 = ((P1P2)∗)∗ = (P2∗P1∗)
∗
= (P2P1)∗ =P1∗ =P1. (8.8)
The last step closes the loop. Suppose P1P2 = P2P1 = P1, and define P3 :=
P2−P1. ThenP3 is a projection operator:
P3∗ = (P2−P1)∗=P2∗−P
∗
1 =P2−P1 =P3,
P32= (P2−P1)(P2−P1) =P2−P1P2−P2P1+P1=P2−P1=P3.
(8.9) It also follows directly that P3 and P1 are orthogonal. Finally then, for every
probability functionP, one has
P(P1)≤P(P1) +P(P3) =P(P1+P3) =P(P2). (8.10)
Proposition 8.2 shows that the partial order onL(H) can be identified with the convenient partial order of set inclusion on the set of subsets ofH of the formPH
forP ∈L(H), i.e., there is an isomorphism (L(H),≤)'(L0(H),⊂) with
L0(H) :={PH |P ∈L(H)}. (8.11)
The upshot is that meets and joins of projection operators can be understood as operations on sets provided one restricts attention to the closed linear subspaces.
Orthodox quantum logic
8.2
SoP1∧P2 identifies with the biggest subspacePHthat satisfiesPH ⊂P1H ∩P2H,
and P1∨P2 with the smallest subspacePHthat satisfiesP1H ∪P2H ⊂PH. What
is needed now is a way to characterize sets of the form PH that helps to see that these meets and joins indeed exist. This characterization is given by Theorem A.2 which states that a subset ofHis of the formPHfor some projectionP if and only if it is a closed linear subspace.
Showing the existence of meets and joins for closed linear subspaces is relatively straightforward. For any two closed linear subspaces K1,K2, the intersection is
again a closed linear subspace. This is also the largest linear subspace that is a subspace of both K1 and K2. Thus
K1∧ K2=K1∩ K2. (8.12)
Similarly, for any pair of closed linear subspacesK1,K2 one can take the linear span
of all the vectors in K1∪ K2. Although this gives a linear subspace, in general it
will not be closed. This problem is solved by taking the closure of the set.5 Thus
K1∨ K2= span(K1∪ K2). (8.13)
Since L(H) as a partially ordered set is isomorphic to the set of closed linear subspaces with set inclusion, lattice operations on L(H) can be completely under- stood in terms of the lattice operations on closed linear subspaces. Specifically, if PKdenotes the projection such that PKH=K, then
P1∨P2 =PP1H∨P2H, P1∧P2 =PP1H∧P2H, (8.14)
where the join and meet for closed linear subspaces are given by (8.13) and (8.12). Although this shows the existence of joins and meets of projection operators, it does not give a very insightful formulation of these operations. In some cases, a clearer formulation can be given purely in terms of projection operators, as seen in the following example.
Example 8.1. SupposeP1 andP2 are two compatible projection operators. Then
P1∧P2 =P1P2,
P1∨P2 =P1+P2−P1P2.
(8.15)
4
Thus far I have shown that quantum probability gives rise to a partial order on L(H), which in turn introduces a join and meet that turn it into a lattice. In
5IfKis a linear subspace, then its closureKconsists of all the limits of all converging sequences
the classical case the lattice structure onF gives rise to a unique complement. On (L(H),≤), however, complements are not unique. For example, in C2 every pair
of unequal one-dimensional projection operators are complements of each other. This is because any two vectorsψ1, ψ2 that aren’t on the same line, together span
the whole Hilbert space (Pψ1∨Pψ2 =1). This non-uniqueness already implies that
(L(H),≤) is not Boolean. But despite this non-uniqueness, there is still a particular complement singled out by quantum probability.
Theorem 8.2. For every P ∈ L(H) the orthocomplement P⊥ is the unique com- plement that satisfies
P(P) +P(P⊥) = 1 (8.16) for all quantum probability functionsP.
Proof. It is easy to check that the orthocomplement is a complement and satisfies (8.16), so the task is to prove uniqueness. SupposeP ∈L(H) and Pc is a comple- ment that satisfies (8.16), then it has to be shown thatPc=P⊥. Ifψ∈PH, then
kPcψk2 =hPcψ, Pcψi=hψ, Pcψi = Tr(PψPc) =Pψ(Pc) = 1−Pψ(P) = 0. (8.17) Ifψ∈P⊥H, then kψ−Pcψk2 =k(1−Pc)ψk2=h(1−Pc)ψ,(1−Pc)ψi=hψ,(1−Pc)ψi = Tr(Pψ(1−Pc)) = Tr(Pψ)−Tr(PψPc) = 1−(1−Pψ(P)) = 0. (8.18) Thus for arbitraryψ∈ H:
Pcψ=PcP ψ+PcP⊥ψ= 0 +P⊥ψ. (8.19)
This theorem provides the final step to conclude that the formalism of quantum probability alone has all the ingredients to turnL(H) in a complemented lattice. To conclude this section, let me state explicitly what is meant by orthodox quantum logic in the remainder of this dissertation.
Definition 8.2. The orthodox quantum logic for a system described by a Hilbert space H is the set L(H) of projection operators understood as a complemented lattice (L(H),≤,∧,∨,⊥) with the lattice structure given by (8.3) and (8.4).
The interpretation of orthodox quantum logic