It is shown that for any quantum logic $L$ one can find a concrete logic $K$ and a surjective homomorphism $f$ from $K$ onto $L$ such that $f$ maps the centre of $K$ onto the centre of $L$ . Moreover, one can ensure that each finite set of compatible elements in $L$ is the image of a compatible subset of $K$ . This result is "best possible" - let a logic $L$ be the homomorphic image of a concrete logic under a homomorphism such that, if $F$ is a finite subset of the pre-image of a compatible subset of $L$ , then $F$ is compatible. Then $L$ must be concrete. In the second part one considers embeddings into concrete logics. It is shown that any concrete logic can be embedded into a concrete logic with preassigned centre and an abundance of two-valued measures. Finally, one proves that an arbitrary logic can be mapped into a concrete logic by a centrally additive mapping which preserves the ordering and complementation.It is shown that for any quantum logic $L$ one can find a concrete logic $K$ and a surjective homomorphism $f$ from $K$ onto $L$ such that $f$ maps the centre of $K$ onto the centre of $L$. Moreover, one can ensure that each finite set of compatible elements in $L$ is the image of a compatible subset of $K$. This result is "best possible" - let a logic $L$ be the homomorphic image of a concrete logic under a homomorphism such that, if $F$ is a finite subset of the pre-image of a compatible subset of $L$, then $F$ is compatible. Then $L$ must be concrete. In the second part one considers embeddings into concrete logics. It is shown that any concrete logic can be embedded into a concrete logic with preassigned centre and an abundance of two-valued measures. Finally, one proves that an arbitrary logic can be mapped into a concrete logic by a centrally additive mapping which preserves the ordering and complementation.

Applications of MathematicsJan 1, 19858 citations

Preserver problems for the logics associated to Hilbert spaces and related Grassmannians

Mark Pankov

We consider the standard quantum logic ${\mathcal L}(H)$ associated to a complex Hilbert space $H$ , i.e. the lattice of closed subspaces of $H$ together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic ${\mathcal L}(H)$ ) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of ${\mathcal L}(H)$ and the associated Grassmannians which preserve the orthogonality or compatibility relation. The first result in this direction was classical Wigner's theorem.We consider the standard quantum logic ${\mathcal L}(H)$ associated to a complex Hilbert space $H$, i.e. the lattice of closed subspaces of $H$ together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic ${\mathcal L}(H)$) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of ${\mathcal L}(H)$ and the associated Grassmannians which preserve the orthogonality or compatibility relation. The first result in this direction was classical Wigner's theorem.

arXiv:1702.03157Feb 10, 20170 citations

On the logic of quantum logic

C. Piron

Journal of Philosophical LogicJan 1, 197724 citations

Coupling of quantum logics

Sylvia Pulmannová

International Journal of Theoretical PhysicsSep 1, 198316 citations

On joint distribution in quantum logics. I. Compatible observables

Anatolij Dvurečenskij

The notion of a joint distribution in $\sigma$ -finite measures of observables of a quantum logic defined on some system of $\sigma$ -independent Boolean sub- $\sigma$ -algebras of a Boolean $\sigma$ -algebra is studied. In the present first part of the paper the author studies a joint distribution of compatible observables. It is shown that it may exists, although a joint obsevable of compatible observables need not exist.The notion of a joint distribution in $\sigma$-finite measures of observables of a quantum logic defined on some system of $\sigma$-independent Boolean sub-$\sigma$-algebras of a Boolean $\sigma$-algebra is studied. In the present first part of the paper the author studies a joint distribution of compatible observables. It is shown that it may exists, although a joint obsevable of compatible observables need not exist.

Applications of MathematicsJan 1, 19872 citations

It is shown that for any quantum logic

L

one can find a concrete logic

K

and a surjective homomorphism

f

from

K

onto

L

such that

f

maps the centre of

K

onto the centre of

L

. Moreover, one can ensure that each finite set of compatible elements in

L

is the image of a compatible subset of

K

. This result is "best possible" - let a logic

L

be the homomorphic image of a concrete logic under a homomorphism such that, if

F

is a finite subset of the pre-image of a compatible subset of

L

, then

F

is compatible. Then

L

must be concrete. In the second part one considers embeddings into concrete logics. It is shown that any concrete logic can be embedded into a concrete logic with preassigned centre and an abundance of two-valued measures. Finally, one proves that an arbitrary logic can be mapped into a concrete logic by a centrally additive mapping which preserves the ordering and complementation.

We consider the standard quantum logic

{\mathcal L}(H)

associated to a complex Hilbert space

H

, i.e. the lattice of closed subspaces of

H

together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic

{\mathcal L}(H)

) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of

{\mathcal L}(H)

and the associated Grassmannians which preserve the orthogonality or compatibility relation. The first result in this direction was classical Wigner's theorem.

The notion of a joint distribution in

\sigma

-finite measures of observables of a quantum logic defined on some system of

\sigma

-independent Boolean sub-

\sigma

-algebras of a Boolean

\sigma

-algebra is studied. In the present first part of the paper the author studies a joint distribution of compatible observables. It is shown that it may exists, although a joint obsevable of compatible observables need not exist.