Projection Lattices

The projections on a Hilbert space form a lattice through their correspondence with closed subspaces. This gives a precise language for inclusion and combination of sharp measurement events. The order structure must be distinguished from arbitrary operator multiplication and from sequential measurement probabilities.

Order means containment of certain states

For orthogonal projections $P,Q$, write $P\leq Q$ if $\operatorname{ran}P\subseteq\operatorname{ran}Q$. Equivalently,

$$PQ=QP=P.$$

A state in which $P$ is certainly true then makes $Q$ certainly true. This is a structural implication, not a claim that every probability ordering for one particular state implies $P\leq Q$.

The meet $P\wedge Q$ projects onto the intersection of ranges. The join $P\vee Q$ projects onto their closed linear span. Closure matters in infinite dimensions. The orthocomplement is $P^\perp=I-P$.

Compatible and incompatible events

When $P$ and $Q$ commute,

$$P\wedge Q=PQ,\qquad P\vee Q=P+Q-PQ.$$

For noncommuting projections, $PQ$ is usually neither self-adjoint nor idempotent and cannot be the lattice meet.

For example, let $P=\lvert0\rangle\langle0\rvert$ and $Q=\lvert+\rangle\langle+\rvert$. Their ranges are distinct lines, so $P\wedge Q=0$. Nevertheless $PQ\ne0$. If a system initially in $\lvert0\rangle$ is tested first for $P$ and then for $Q$, the probability of two yes outcomes is $1/2$.

There is no contradiction: lattice conjunction asks for a vector belonging to both ranges, whereas the sequential experiment includes state change. For ideal Lüders measurements, its probability is

$$p(P\text{ then }Q)=\operatorname{Tr}(\rho PQP).$$

The effect $PQP$ records that sequence and need not be a projection.

Probability measures on the lattice

For mutually orthogonal projections $\{P_i\}$, their join is their sum, and quantum probabilities add:

$$p\!\left(\sum_iP_i\right)=\sum_i p(P_i).$$

The finite or countable additivity required depends on the setting. Gleason’s theorem studies how suitable probability assignments on these events recover density operators under explicit dimension and regularity assumptions.

The lattice contains no Hamiltonian or state-update rule by itself. Those require additional data. Its role is the structure of sharp propositions, not a complete account of quantum dynamics.

Exercises

For $P\leq Q$, prove that $Q-P$ is a projection orthogonal to $P$.

Solution

It is self-adjoint and $(Q-P)^2=Q-QP-PQ+P=Q-P$. Moreover $P(Q-P)=PQ-P=0$. This realizes the decomposition of $Q$ into the part already contained in $P$ and the remaining orthogonal part, anticipating orthomodularity.

References

• Birkhoff, Garrett, and John von Neumann, “The logic of quantum mechanics,” Annals of Mathematics 37, 823–843 (1936). Original paper.
• Gleason, Andrew M., “Measures on the closed subspaces of a Hilbert space,” Journal of Mathematics and Mechanics 6, 885–893 (1957). Paper.