Research Article | Open Access
Fedor Herbut, "State-Dependent Implication and Equivalence in Quantum Logic", Advances in Mathematical Physics, vol. 2012, Article ID 385341, 23 pages, 2012. https://doi.org/10.1155/2012/385341
State-Dependent Implication and Equivalence in Quantum Logic
Ideal occurrence of an event (projector) leads to the known change of a state (density operator) into (the Lüders state). It is shown that two events and give the same Lüders state if and only if the equivalence relation is valid. This relation determines equivalence classes. The set of them and each class, are studied in detail. It is proved that the range projector of the Lüders state can be evaluated as , where denotes the greatest lower bound, and is the null projector of . State-dependent implication extends absolute implication (which, in turn, determines the entire structure of quantum logic). and are investigated in a closely related way to mutual benefit. Inherent in the preorder is the state-dependent equivalence , defining equivalence classes in a given Boolean subalgebra. The quotient set, in which the classes are the elements, has itself a partially ordered structure, and so has each class. In a complete Boolean subalgebra, both structures are complete lattices. Physical meanings are discussed.
The basic object of this study is the concept of a quantum-mechanical state (density operator) and its change when an event (projector) with positive probability occurs (the result is obtained) in ideal measurement. The terms “state” and “density operator” as well as “event,” and “projector” will be used interchangeably.
As it is well known, ideal measurement is the simplest special case of measurement of the first kind (synonyms: predictive, repeatable, and nondemolition measurement). It causes a change of state according to the Lüders selective (or definite-result) formula See [1–3]. We call the Lüders state, and a state-determining projector (it is all meant with respect to the initially given state ).
Usually one treats the special case of a pure-state . Then, as easily seen, (1.1) takes the simple form which is sometimes called the von Neumann-Lüders projection (it is actually a normalized projection). Von Neumann treated the even more special case when the event is elementary (an atom) . The change of state then is
The Lüders state was postulated by Lüders. It was derived by several authors including the present one [5, 6] (and the first derivation was repeated in different context in ; see also references in these articles).
For a different approach, see references [8–10]. In Khrennikovʼs terminology, one deals with the postulates of Lüders and von Neumann (and he carefully examines their effect on some foundational issues).
As it was mentioned, in my view, ideal measurement is the simplest kind of measurement, and of which kind a measurement is depends on the interaction of object and measuring instrument (an elaboration of this will be presented in a followup). Both Lüders and von Neumann treat ideal measurement, but the former allows degenerate eigenvalues of the measured observable, whereas the latter is confined to the special case of complete observables, that is, to ones with all eigenvalues nondegenerate.
Incidentally, I have shown in my mentioned article  (see Section 3 there) that the Lüders change of state can be obtained by measuring a suitable complete observable as its refinement (in the way of von Neumann), but the choice of the latter depends on the state in which the measurement is performed. Khrennikov has rediscovered essentially the same result independently in .
Change of state (1.1) is made use of in the concept of state-dependent (SD) implication , where and are events, and is a given state [11–13]. Absolute (or state independent) implication in quantum logic (the complete lattice of all projectors in the Hilbert space of the given quantum system) is such that if an event occurs in a state , that is, if , then also the implied event occurs; and this is so for every state.
SD implication , which is a generalization if the absolute implication (cf. Corollary 2.3 below), is, on the other hand, based on the following reasoning. If an event occurs in an ideal way in a given state and, after the occurrence, another event ipso facto also occurs, then one says that implies state-dependently and one writes .
When one takes into account that ideal occurrence gives rise to (1.1), one obtains the equivalence relation Evidently, the SD implication relation (1.4) is restricted to nonzero-probability state-dependently implying events .
SD implication is defined on a given Boolean subalgebra of (to achieve transitivity) . SD implication on is a preorder [11, 14]. As every preorder, being reflexive and transitive but not antisymmetric, SD implication defines an equivalence relation on , SD equivalence , as follows: SD equivalence breaks up into equivalence classes, the elements of the SD quotient set . By this all zero-probability projectors are taken to form one class. (This will be seen to be correct, cf. Remark 2.2.) The SD equivalence class in to which a given projector belongs will be denoted by .
The SD implication concept is further elaborated in the next section. But, unlike in [11, 12], this time the powerful structure of Boolean algebras is not made use of. Only poset (partially ordered set) theory is utilized, and poset structure is introduced in the SD quotient set .
2. SD Implication Further Elaborated
Simple algebraic relations characterizing SD implication and SD equivalence are now derived.
Proposition 2.1. If , then if and only if is valid.
Proof. On account of commutation under the trace, idempotency of projectors, and commutation in , the mentioned former definition (1.4) of state-dependent implication boils down to , or equivalently to . Since is a projector, one can further equivalently write , and (no other positive operator then zero gives trace zero). Finally the Lüders lemma (page 326 in , cf. also Remark 4.3) gives .
Remark 2.2. Relation (1.4) as the definition of SD implication cannot be extended to projectors for which , but the equivalent definition for can. Hence, we extend formally SD implication to these projectors. If an event is such that , then it (formally) state-dependently implies every other event in . This statement cannot be falsified because cannot occur in . Hence it is formally true though trivial (more on this in Lemma 4.7).
One should note the equality of sets due to (cf. for analogy proof of Proposition 2.1).
Corollary 2.3. SD implication generalizes and extends absolute implication, that is, the latter is a special case of the former, and the former can be valid for pairs of events for which the latter is not valid.
Proof. If , then relation (1.4) is obviously valid, and ensues.
SD implication , being reflexive, transitive but not antisymmetric, is a partial order . As for every partial order, one can take the SD (equivalence) quotient , the elements of which are the SD equivalence classes in . Next, one can induce the corresponding partial-order relation in the quotient set and thus obtain a proper binary relation for quantum logic.
We proceed by acquiring a simple insight on which the poset structure of is based.
Proposition 2.4. Let and be distinct SD equivalence classes, that is, elements of , such that and so that . Let further and be arbitrary elements in the same respective classes. Then also . In other words, if one projector from one class implies state dependently a projector from another class, then every projector from the former class implies state dependently every projector from the latter one.
Proof. The very definition of the classes implies and . In view of the assumed relation , and due to the validity of transitivity of SD implication in , follows.
On ground of Proposition 2.4 we can define a poset structure in .
Definition 2.5. One has if for all and for all the relation is valid.
Lemma 2.6. SD implication thus defined in the SD quotient set is a partial order relation, that is, it has the properties of reflexivity, transitivity, and antisymmetry.
Proof. Reflexivity. If is an arbitrary SD equivalence class in , and is its arbitrary element, one has and , so that, according to Definition 2.5, also and are valid.
Transitivity. If and , and are arbitrary elements from the respective classes, then, according to Proposition 2.4, the relations and hold true. Transitivity of SD implication in then implies the same for the SD quotient set: from ensues .
Antisymmetry. Let and be valid. Then Definition 2.5 implies and , having as its consequence, that is, one obtains .
Remark 2.7. Having in mind that can be arbitrarily chosen in , we may call the SD quotient set SD quantum logic paralleling the set of all projectors, quantum logic . Since the term “quantum logic” has also another meaning as a branch of mathematical physics, one might use the term “SD quantum logic” also to designate a part of quantum logic (meant as the mentioned branch), the part treated in [11–13], and in this paper (cf. also the last passage in Section 11).
Proposition 2.8. Two projectors are state-dependently equivalent if and only if is satisfied.
Proof. Relation (1.5) and Proposition 2.1 imply and . Since all projectors in commute, necessity of the claimed condition obviously follows. As to sufficiency, the claimed condition and idempotency obviously imply and , which, in view of Proposition 2.1 and relation (1.5), is tantamount to .
Remark 2.9. The SD quotient set is complemented because orthocomplementation in takes each SD equivalence class onto another, and none is self-complemented. Finite Boolean subalgebras thus consist of an even number of elements.
Evidently, SD equivalence can be extended to the entire quantum logic . In what immediately follows we study SD equivalence in the entire , and we write the equivalence classes as (without index). Subsequently, beginning with Section 8, we shall return to restricting our investigation to to obtain additional insight in SD quantum logic.
3. Lüders States
We now express SD implication in terms of a (known but, perhaps, not well known) projector relation.
Lemma 3.1. Denoting by the range projector of a given density operator , the general claim is valid that an event has probability in the state if and only if .
Proof. The proof is given for the reader's convenience in Appendix A.
Corollary 3.2. Denoting the range projector of the Lüders state (cf. (1.1)) by , and calling it Lüders projector, relation (1.4) and Lemma 3.1 entail the following necessary and sufficient condition for SD implication in terms of it:
Evidently, the projector in Corollary 3.2 is somehow inherent in . We want to find out how, and proceed by evaluating the Lüders projector . One needs, not , the initial range projector (of the initially given state ), but its ortho-complementary projector, the null-space projector, which one denotes by and call the initial null projector. Besides, one shall need more insight into the properties of that appears in (1.1). As stated, one calls such a projector a state determining one.
Lemma 3.3. Let and be an arbitrary projector and an arbitrary density operator, respectively. Then the following equivalence relation is valid
This general equivalence is known, but perhaps not well known. For the reader's convenience it is proved in Appendix B.
Definition 3.5. On account of Corollary 3.4, we can define a new projector as follows: We call the redundancy projector (for reasons that will become clear below).
Proposition 3.6. The redundancy projector of a state-determining projector in (1.1) satisfies both and , where , being the identity operator, is the Lüders null projector and is the initial null projector (that of ).
Proof. Definition 3.5 makes orthogonal to , hence it must be a subprojector of the null projector , , because , where is the identity operator.
Let , where is the range of , that is, the null space of the Lüders state (cf. (1.1)). Further, let be a spectral decomposition of the initially given density operator into eigen-ray-projectors with all eigenvalues positive. This is always possible. Then, in view of (1.1), implying
Since is any state vector in , and the vectors span the range of the initial state , the last relations mean that the image subspace is orthogonal to , that is, that maps the null space into the null space : Further, being a subprojector of , one has . Finally, since is also a sub-projector of the idempotent operator (cf. Definition 3.5), it follows that Therefore, transitivity of the subset relation implies , or equivalently as claimed.
Theorem 3.7. The Lüders projector can be evaluated in terms of a state determining projector (cf. (1.1)) and the initial null projector as follows: where is the greatest lower bound (glb) or largest common subprojector of and .
Proof. Let us introduce or equivalently
Since is a sub-projector of the initial null projector , one has implying
Hence, also is a determining projector for (cf. (1.1)), and all that has been proved for the determining projector in (1.1) is valid also for . In particular, , (definition of ).
The definition of implies . On the other hand, it is seen from (3.10) that . Hence, due to transitivity of absolute implication, . Since also Proposition 3.6 is valid for and its redundancy projector , one has , and follows.
On can see from (3.10) that is orthogonal to : , and so is its subprojector : . The projector orthogonality amounts to , where the suffix denotes taking the ortho-complementary projector. Comparing this with (3.12), we conclude that because only zero is the common subprojector of a projector and its ortho-complementary projector.
Returning to the definition of , we see that we are led to the equality , which, due to the definition (3.10) of , completes the proof.
Theorem 3.7 immediately implies the following conclusions.
Corollary 3.8. The redundancy projector of any state-determining projector is given by .
Corollary 3.9. A projector is a Lüders projector if and only if its redundancy projector is zero, and if and only if the only common subprojector of and is zero:
Corollary 3.10. If , or at least , then .
Proof. The first commutation is equivalent, as well known, to reduction of in each eigen-subspace of . Since the null-space of is its eigen-subspace corresponding to the eigenvalue zero, reduces in it, hence it commutes with that projects onto the latter. Then the glb equals the product as claimed.
Remark 3.12. If the initial density operator is nonsingular, that is, when , then the Lüders projector is the only state determining projector as obvious from Theorem 3.7.
4. Generalized Lüders Lemma
The SD equivalence relation implies that both and are state determining projectors for the same Lüders state (cf. (1.1)). We want to clarify if the converse implication is also valid.
Theorem 4.1. Generalized Lüders lemma. If are two projectors and is a density operator, then implies .
Corollary 4.2. If , then .
Proof. One can write . Then Theorem 4.1 implies the claim.
Remark 4.3. The claim of Corollary 4.2 was proved by Lüders  (page 326) in the following more general form: If is a positive operator, and , where the dagger denotes adjoining (of the linear operator ), then is valid. This auxiliary result is usually called "the Lüders lemma."
Lemma 4.4. The equivalence relation gives, in general, finer equivalence classes in then the equivalence relation .
Proof. The proof is given by a simple example. We take the two-dimensional spin- space and we define: , , . Then , though . Thus, though is not valid. The latter, finer, equivalence relation breaks up the coarser class , so that the two distinct finer classes and are within it.
Remark 4.5. Theorem 4.1 can be put as the claim that and are equivalent, that is, they both define the same equivalence relation in .
Lemma 4.6. Relation is valid if and only if and act equally on each element of the range of . In particular, SD equivalence is equivalent to .
Proof. The general element of the range is of the form , and SD equivalence in Proposition 2.8 amounts to . This bears out the first claim of the lemma. Since the last relation is obviously equivalent to , also the second claim is shown to hold true.
Lemma 4.7. One has , or equivalently , if and only if . The last relation means that is a subprojector of the initial null projector , that is, that is valid. Thus, the SD equivalence class consists of all subprojectors of . The projectors and are the minimal and the maximal elements in the class, respectively, (cf. Remark 2.2).
Proof. Sufficiency. Let the subprojector condition be valid. Then .
Necessity. Let hold true. Then, according to Lemma 4.6, also is true. Finally, as claimed.
Lemma 4.8. One has , or equivalently if and only if . The last relation means that is a subprojector of , that is, that is valid. Thus, the class consists of all projectors of which is a subprojector. The projectors and the identity operator are the minimal and the maximal elements of the class, respectively.
Proof. Follows immediately from Lemma 4.6.
Lemma 4.9. The equivalence implies , where is defined as the common glb of and .
Proposition 4.10. Let , and let by definition , (cf. Lemma 4.9). Then is equivalent to the simultaneous validity of the following two sub-projector relations
Proof. Sufficiency. Let the two subprojector relations be valid. Then one can write and . SD equivalence then immediately follows.
Necessity. and Lemma 4.9 imply , and one has
Proposition 4.10 has introduced the entities and in the general case. Incidentally, there is a general lemma that says: Two arbitrary projectors and commute if and only if and are orthogonal (since we actually do not need this claim, let its proof be left to the reader).
5. Complete-Lattice Structure of the SD Equivalence Classes in
We assume that an arbitrary density operator is given and subject to scrutiny the SD equivalence classes. On account of Remark 4.5 we know that each such class contains precisely all projectors that are state determining for one and the same Lüders state . We begin by paying attention to the Lüders projectors in the classes.
Proposition 5.1. (A) Each SD equivalence class contains a Lüders projector , and it can be evaluated from any element of the class according to Theorem 3.7.
(B) Each Lüders projector is the unique minimal element in its class : implies .
(C) The following bijection maps the SD quotient set onto the set of all Lüders projectors in , which we denote by :
Proof. (A) We assume that is any state-dependent equivalence class, and that is any of its elements. We take as defined by the relation in Theorem 3.7. Since is a sub-projector of , it follows that we have , and then, according to the relation in Theorem 3.7, . Thus, each class does contain at least one Lüders projector.
(B) That is immediately seen from Corollary 3.4. This makes the unique minimal element in (there cannot be two distinct minimal elements, because they would have to be subprojectors of each other, and is a partial order, the anti-symmetric property of which implies the equality).
(C) is now obvious.
It is now useful to be reminded of a general property of the complete lattice .
Lemma 5.2. Let be any nonzero projector. The set of all lower bounds is a complete lattice.
Though this a known and simple property of complete lattices, we prove it for the reader's convenience in Appendix C.
Theorem 5.3. An arbitrary SD equivalence class , as a poset subset of , is isomorphic to the complete lattice . The isomorphism acts as follows:
Proof. Let . Then, on account of the assumed orthogonalities , one has Thus, one has , symbolically if and only if , symbolically .
We have the immediate consequence.
Corollary 5.4. Each SD equivalence class is a complete lattice with the Lüders projector as the minimal element and as the maximal element.
The set of all Lüders projectors , as any subset of , is a poset. It has a lattice structure and it is easily connected with the ortho-complemented SD quotient set.
Theorem 5.5. (A) The poset has the structure of a complete glb-semilattice, the operation of glb (greatest lower bound) being the same as in the entire quantum logic . Its minimal element is the zero projector.
(B) There is a simple bijection mapping onto the SD quotient set that associates with each Lüders projector its SD equivalence class, and the inverse of which singles out the minimal element in the SD equivalence class:
The term "glb-semilattice" is consistent with the terminology of Birkhoff, who used the synonymous term "meet-semilattice" (cf. the definition on page 22 in ).
Proof of Theorem 5.5. Let be any subset of , and let be the greatest lower bound of the projectors in the subset taken in . We show now that it is a Lüders projector. Corollary 3.9 implies that a projector is a Lüders projector if and only if only the zero projector is a common lower bound of it and .
We take as a common sub-projector of and of . On account of being a lower bound (a sub-projector) of each in the subset, is a common sub-projector of each and of . Since each is a Lüders projector, must be zero.
6. Physical Meaning of Complete-Lattice Structure
The physical meaning of an event (projector) consists in the fact that it can occur in a given state (density operator) , when , or “not occur” when . In the latter case the opposite event (ortho-complementary event) occurs. (There is a third possibility, can de indeterminate, that is, . Then it is made to occur or not occur in an ensemble of measurements giving relative frequencies.)
Let be an arbitrary set of projectors in . One may wonder if the set can, as a whole, occur? The answer is of course, that it can. This is the case when all the events in it occur. One may further wonder if this occurrence of the set of events can be equivalently replaced by occurrence of a single event. Affirmative answer is given by the following claim.
Proposition 6.1. The system of relations is equivalent to the single relation .
Proof. Necessity. On account of Lemma 3.1, one can rewrite the system of relations in Proposition 6.1 equivalently as follows:
Being a common lower bound of the set , must satisfy . This is equivalent, due to Lemma 3.1, to .
Sufficiency. Assuming , or equivalently , and taking into account that , transitivity implies . This is equivalent to the system of relations in Proposition 6.1.
Let us turn to the other extreme case. Nonoccurrence of the set of events also amounts to that of a single event.
Proposition 6.2. Non-occurrence of a set of events is equivalent to non-occurrence of the single event , where denotes the least upper bound.
Proof. The non-occurrence in question is equivalent to the occurrence of the set of opposite events. This, in turn, is, according to Proposition 6.1, equivalent to the occurrence of . On account of the corresponding de Morgan rule, this projector can be rewritten as . Occurrence of this event is equivalent to non-occurrence of the opposite event .
Propositions 6.1 and 6.2 have made it clear that the physical meaning of the greatest lower bound and the least upper bound in a complete lattice is occurrence and non-occurrence of a set of events, respectively.
7. Physical Meaning of the Complete-Lattice Structure of the SD Equivalence Classes
We have seen that every SD equivalence class contains precisely all those events, that, as projectors, are the state-determining ones for one and the same Lüders state. These events have the same probability in the initially given state , and, more importantly, they give rise to the same change of state in ideal measurement. In this sense, they are indistinguishable, and the SD equivalence class that they constitute, as a whole, correctly replaces its individual elements.
According to Proposition 6.1, the minimal element in the class, which is its glb, and which is the Lüders projector , determines via its occurrence the occurrence of the whole class. Its maximal element , via its non-occurrence determines that of the entire class. Thus, the two extreme elements in the class make possible the occurrence or non-occurrence of the class as a set of indistinguishable events (as far as probability and ideal measurement go).
One wonders if the elements of the class that are intermediate between the extreme ones do have any physical meaning. They are distinguished via the redundancy projectors (cf. Theorem 5.3). They have a subtle physical meaning of redundancy. We elaborate this point.
(A) The Lüders projector of any class has no redundancy as far as occurrence goes in the sense that they occur (have probability one) in the corresponding Lüders state and there exists no other event that occurs in and that implies the event .
(B) All other events that are state-dependent ly equivalent to do have redundancy, and a kind of its measure is uniquely expressed by the redundancy projector because (cf. Definition 3.5). They occur in , but there exists another event that also occurs in this state and it implies (it is, of course, the Lüders state).
(C) As to the events that are implied by but belong to another SD-class, one must distinguish two distinct sets of Lüders entities, besides that were in focus so far, also another set connected to the SD-class to which belongs.
For such events the redundancy projector satisfies formally the same last relation , but this time with .
Next we return to restricting to one of its Boolean subalgebras.
8. SD Equivalence Classes in and Their Minimal Elements
After having restricted to an arbitrary Boolean subalgebra , “shrinking” further into is mathematically necessary to upgrade the preorder into an order relation (cf. Lemma 2.6). In other words, the SD implication is actually valid in the quotient set. Physically, the events are now the classes, and the question arises how do they “occur” or “not occur.”
The subalgebra can be a finite lattice, if it is closed to taking the greatest lower bound (glb) and the least upper bound (lub) of at most finite subsets; it can be a -lattice if it contains the glb and lub of its subsets with at most countably infinite number of elements, and, finally, it can be a complete lattice if the glb and lub of any subset is in it. We will call this the lattice type of .
In restricting to each SD equivalence class is part of the corresponding SD equivalence class in : The importance of the intersection relation (8.1) is in the fact that if two projectors are in , then they belong to one and the same SD equivalence class if and only if they are equivalent in . For the latter, we have, besides , additional necessary and sufficient conditions like in Lemma 4.6 and in Proposition 4.10.
It was seen in Corollary 5.4 that each SD equivalence class in is a complete lattice. Hence, it follows from (8.1) that each SD equivalence class in is also a lattice of the same lattice type as that of .
To acquire a feeling for the SD equivalence classes defined by (8.1), we consider two extreme cases. Firstly, we assume that is nonsingular. Then and . According to Remark 3.12, each projector in equals the minimal element in its SD equivalence class. Hence all SD equivalence classes in have only one element. Relation (8.1) then tells us that also the SD equivalence classes all consist of one element and if .
Secondly, let us take a singular density operator, but let consist of only two elements. They must, of course, be and . Then, only and (cf. Lemmas 4.7 and 4.8) have nonempty intersections with . The former intersection consists of only the maximal element in the class, whereas the latter intersection contains only the minimal element of the corresponding class. We proceed by investigating the general case of in detail.
If not stated otherwise, we assume that is a complete lattice in what follows. We introduce two key entities for further study.
Definition 8.1. Let be the greatest subprojector of that is in : . We call the Boolean initial null projector. Further, let be the least among the projectors that are greater than and belong to : . We call the Boolean initial range projector.
Note that, since , , and are complete lattices (cf. Lemma 5.2 and its proof in Appendix C), so are the above intersections.
Proposition 8.2. The Boolean null projector and the Boolean range projector are dual entities, that is, they are the ortho-complementary projectors of each other: .
Proof. The SD equivalence classes and are dual to each other. Hence, so are the maximal element in the former and the minimal element in the latter.
Remark 8.3. Evidently, , or equivalently if and only if , or equivalently .
Proposition 8.4. One has and . The zero projector and are the least and the greatest elements, respectively, in : , whereas and the identity operator are the least and the greatest elements, respectively, in : .
Proof. The projector is by Definition 8.1 a subprojector of . Hence, , or . The projector is, by definition, the greatest subprojector of that belongs to . Hence it is the greatest element in . Further, as we know (cf. lemma 2 and lemma 3), one has , or equivalently , if and only if . Hence, is the minimal element of . The rest is obvious.
Definition 8.5. We call a projector a core element in if the relation is satisfied. The set of all core elements in we call the core set and denote it by .
Proposition 8.6. The core set is . It is a complete lattice.
Proof. The definition means that each core element is orthogonal to , or equivalently that it is a subprojector of the ortho-complementary projector of (cf. Proposition 8.2). The poset is the intersection of two complete lattices, hence it is itself a complete lattice
(B) Every core element is the least element of its class : . Hence, each class contains a unique core element.
Proof. (A) Let be any state-dependent equivalence class in , and let be any of its elements. The rhs of (8.2) is state-dependently equivalent to because is equivalent to . In other words, the lhs belongs to . We prove that it is a core element: . Hence, according to Definition 8.5, it is a core element, and therefore, each class does contain at least one core element.
(B) Let be a core element of . Then, according to Theorem 8.7, . Multiplication gives , that is, as claimed. This makes the unique least element in the class to which it belongs.
The core elements in the SD equivalence classes in play the same role as the Lüders range projectors in the SD equivalence classes in (cf. Corollary 5.4).
9. Further Investigation of the SD Equivalence Classes in
Some SD equivalence classes in do and others do not contain an SD equivalence class , that is, they may be vacant in this respect. We now derive some necessary conditions for nonvacant classes .
Proposition 9.1. Each SD equivalence class is vacant unless is an upper bound for the minimal element, the Lüders range projector , in the class.
Proof. The core element , if it exists in a class , must be an upper bound of the minimal element in the class. Since, further, (cf. Proposition 8.6), transitivity establishes the claim.
Corollary 9.2. In the special case when , or equivalently, , SD equivalence classes in can appear only in those SD equivalence classes in in which , and then not necessarily in all (take, e.g., a non-singular and the minimal Boolean subalgebra).
We now investigate the properties of the intermediate elements of the SD equivalence classes in analogously as one did in .
Theorem 9.3. (A) Each element of each state-dependent equivalence class can (uniquely) be written as follows:
Conversely, for each core element , each projector determines an element that is state-dependently equivalent to . In this way one has a one-to-one map of onto any class .
(B) The map associating with each the image , where is the core element of any given SD equivalence class in , is an isomorphism of the poset onto the poset .
Proof. (A) Theorem 8.7 has established that each SD equivalence class contains a unique core element and that the latter is the least element of the class. The projector is obviously in since one can write . That is a subprojector of is seen as follows. We take into account that , which implies , and (cf. Proposition 8.4). Thus, .
Conversely, we assume that One has (cf. Definition 8.5 and Proposition 8.4), and, taking the ortho-complement of this inequality, ensues. Hence, transitivity and (9.3) give , implying . This orthogonality makes it possible to add to .
Finally, as we know from Proposition 8.4, . Hence, as claimed. The map at issue is clearly a one-to-one map because it can be inverted beginning with any element of the image set .
(B) To prove isomorphism, let , . Let us, further, multiply out the product One obtains if one utilizes the orthogonalities stemming from (cf. Definition 8.5), and . Obviously, if and only if , then as claimed.
Corollary 9.4. Each SD equivalence class has, as a poset, the same structure as the poset . In particular, in the former, its core element is the least and is the greatest element.
Corollary 9.5. Each SD equivalence class has precisely one element, the core element , if and only if . This is the case, for example, when is non-singular, because then , and . But may be non-zero, and .
Corollary 9.6. Theorem 9.3 establishes an isomorphism between any two SD equivalence classes in (in an obvious way).
The reader who remembers well the results in the decomposition of entire quantum logic into SD equivalence classes is rightly perplexed. He wonders how did it happen that restriction to a Boolean subalgebra made all the SD equivalence classes mutually isomorphic, whereas the larger SD equivalence classes (cf. (1.1)) are not isomorphic due to the fact that the Lüders null projectors are, in general, different for different classes.
To throw additional light on this puzzling fact, we shortly repeat the relevant parts of the above arguments. The (minimal) core elements in the classes are obtained from an arbitrary element in the class in a completely parallel way as in (cf. Theorem 8.7 above and Theorem 3.7). As a consequence, the core elements satisfy parallelly to the Lüders range projectors for which is valid (cf. Definition 8.5 above and Corollary 3.9).
Also Theorem 9.3 above could have been put in the form of a map of (see below) onto , to display further parallelism with , where the counterpart is . So far the classes in and in go parallelly.
What makes a difference is the fact that in a Boolean subalgebra any two projectors commute. Hence, glb takes the form of multiplication. If the glb is zero, as it is in our case (cf. Definition 8.5), we have orthogonality, leading to and, taking the ortho-complement of this inequality, to . Since in a Boolean subalgebra the last inequality means , we have as a consequence.
Thus, , the counterpart of , has disappeared from the scene (whereas in does remain in general).
Remark 9.7. If , then, taking the ortho-complementary projectors, we obtain implying . In other words, in the classes considered, disappears from the maximal element in the class, which is now . Thus, the considered classes are mutually isomorphic just like the SD equivalence classes in that they may contain.
10. The Core Decomposition
Theorem 10.1. The core set has the structure of a complete lattice with the greatest lower bound operation being the same as in , but the least upper bound operation being, in general, different. The least element in is the zero projector , and the greatest element is .
Proof. Let be any subset of , and let be the glb of the projectors in taken in . We show that it is a core element.
Arguing ab contrario, we assume that there exists a non-zero projector in that is a common subprojector of and of . But then, on account of being a lower bound (a subprojector) of each and of , it cannot be non-zero (cf. Definition 8.5), and follows. It is obvious that zero is the least element in .
The projector is a core element (cf. Definition 8.5) since it is orthogonal to as its ortho-complementary projector. That it is the largest element in is evident from Proposition 8.6.
It is an elementary fact that every complete glb-semilattice with a greatest element is a complete lattice because the lub of any subset is the glb of the subset of all upper bounds of the given subset (for the reader's convenience the latter claim is proved in Appendix ).
The lub in need not be the same as the lub of the same subset in , in general, because, in the latter case, it equals the glb of the subset of all upper bounds of the given subset in . This is the glb of a larger set, hence it is, in general, less than the mentioned glb in .
Unlike in , the core set has isomorphic complete lattices.
Theorem 10.2. For each , the poset , which we call the core class determined by , with the absolute implication defined in , is isomorphic to ; hence, it is a complete lattice like .
One should note that is the set of all projector sums of and an element of . A sum of two projectors is a projector if the terms are orthogonal as well known. It is the case here because (cf. Theorem 9.3 and Proposition 8.4), (cf. Proposition 8.6), and and are orthogonal to each other (cf. Proposition 8.2).
Definition 10.3. We call the (nonoverlapping) decomposition of the core decomposition of the given Boolean subalgebra of into core classes.
We can picture as a matrix of elements such that the rows are the SD equivalence classes and the columns are the core classes. Then, the rows are enumerated by the core elements (elements of ), and the columns by the elements of .
One should pay attention to the fact that the same notation is used for summation (+ and ) in two different senses: as a summation of projectors like in the core class , and as a union of non-overlapping subsets as in Definition 10.3.
Remark 10.4. In view of Remark 9.7, we see that if we restrict ourselves to (a subset of ), that is, if we consider only those SD equivalence classes in that may contain a SD equivalence class in (that are nonvacant in this respect), then we have a decomposition of parallelling the core decomposition of . But in , when no is chosen, there is no motivation for this.
11. Back to the SD Quotient Set
There is more insight to be gained about the poset structure of the SD quotient set .
A remarkable property of is the following fact  (Lemma 1 there). A SD equivalence class implies state-dependently another such class if and only if there is an event in the former and an event in the latter so that , that is, the former event implies the latter one in the absolute sense. Since this result is of great importance for the aims of this study, we re derive it (with new insight) to make the present text self-contained.
Proposition 11.1. If , then . Conversely, if , then and such that .
Proof. The relation can be rewritten as (cf. Corollary 2.3), then the claimed relation immediately follows (cf. Definition 2.5). If , then (cf. Propositions 2.4 and 2.1). Hence, , that is, , and, taking and one has .
Remark 11.2. In the partially ordered set one can replace by because Proposition 11.1 allows one to define via that the respective CD equivalence classes contain.
For the sake of generality, let us assume now that the Boolean subalgebra is of any lattice type (finite, - or complete, cf. the second passage in section 8).
Theorem 11.3. (A) The map associating with each projector the SD equivalence class to which it belongs is a poset homomorphism of the Boolean subalgebra onto the Boolean algebra .
(B) is of the same lattice type as .
Proof. (A) In view of Proposition 11.1, it is obvious that the map taking onto by associating with each element the SD equivalence class to which it belongs is a homomorphism of a poset with absolute implication "" onto the poset with the same implication "." The map evidently preserves the ortho-complementation operation. We prove now that it preserves also the glb and lub operations.
Let be an arbitrary subset of of a finite number of elements if is a finite lattice, of at most a countably infinite number of elements if is a -lattice, and a completely arbitrary subset if is a complete lattice.
The SD equivalence class is a lower bound of the set of classes because so is for the set of projectors . If is an arbitrary lower bound of , then such that (cf. Proposition 11.1 and its proof). Let . Since , one has .
On the other hand, because also is a lattice of the specified lattice type (cf. (8.1)). Hence, on account of the last inequality, , and is seen to be the glb of . Thus the map in question preserves the glb operation. Symmetrically (or dually) one can prove preservation of the lub operation.
To prove that the quotient set is an ortho-complemented lattice, let again be an arbitrary subset of (of cardinality corresponding to the lattice type of ). Since the first Morgan rule is valid in , and the surjection is such that , one obtains the validity of the first Morgan rule in . Symmetrically one proves that also the second Morgan rule is true.
To show that is a Boolean algebra, we have to prove that the glb and the lub operations are mutually distributive. It is, actually, a straightforward consequence of the preservation of these operations in the map : Since is a Boolean subalgebra, it is distributive, and one further has The converse distributivity of lub with respect to glb is proved analogously.
(B) We prove now that is a lattice of the same lattice type as . Let be an arbitrary subset of the former (of cardinality corresponding to the lattice type of ). Then , being the glb of , makes (according to the preceding passage) the glb of . Symmetrically one deals with the lub operation.
The results of this and the next section, in conjunction with all the results of the four articles on the topic ([11–13], and this paper) make it, hopefully, desirable to accept the proposed term “SD quantum logic’’ in its two meanings (cf. Remark 2.7).
12. Physical Meaning of the Quotient Set
We have seen that SD implication extends absolute implication in a given Boolean subalgebra of quantum logic , but there is a price to be paid: SD implication is a preorder. Inherent in it is an equivalence relation that makes equivalent events indistinguishable. These are the events that have same probability in the given state , and change this state in the same way when ideally measured.
When one makes the transition to the SD quotient set (by a homomorphism of a Boolean subalgebra to a Boolean algebra of the same kind, cf. Theorem 11.3), then one can even forget about the SD implication, because can be characterized by the equivalence relation in and absolute implication (cf. Remark 11.2).
This throws new light on SD implication. The SD quotient set, that is, the SD quantum logic (cf. Remark 2.7), can be viewed, instead of as extending absolute implication in , as a contraction, a shrinking of sets of indistinguishable events with respect to into, effectively, single events (the corresponding SD equivalence classes in ).
This is where, when is a complete lattice, the extreme elements of the SD equivalence classes acquire physical meaning. As it has been shown in Propositions 6.1 and 6.2 (cf. also Corollary 9.4), occurrence of a SD equivalence class amounts to the occurrence of its minimal element, the core element , and its non-occurrence to that of its maximal element .
If for a set of SD equivalence classes, that is, elements of the SD quantum logic , the question of occurrence or non-occurrence appears, then the physical answer is given by the corresponding glb and lub operation in the core set (cf. Definition 8.5 and Proposition 8.6).
13. An Illustration
Let the state space be a separable, that is, a countably infinite dimensional, complex Hilbert space , and let be a given complete orthonormal basis in . We consider an observable defined by its spectral form and a state vector The last two sums are over all even integers, and all in them are non-zero.
We define the Boolean subalgebra as the set of all spectral projectors of the observable . Let be an arbitrary set of integers including the vacant set, and let be the unique decomposition of into the subset of all even integers that it contains and the subset of all odd integers in it. Let us denote by the power set, that is, the set of all subsets (the vacant set included) of the set of all integers.
It is easily seen that can be used as the index set of the spectral projectors, that is, that , . and are isomorphic posets with respect to the map . In absolute implication "" defined in , and in the set-theoretical inclusion relation "" are the partial orders. Evidently, and are complete lattices.
The SD equivalence classes are defined by the SD equivalence relation Evidently, if and only if . Hence. the SD equivalence classes are the sets with as the core elements.
To find the greatest elements in the SD equivalence classes, we note that , where the summation goes over all odd integers, is the greatest spectral projector that takes into zero. It is the Boolean null projector . The greatest elements in the SD equivalence classes are . The Boolean range projector is , where the summation goes over all even integers.
In particular, the projectors and are the extreme elements of when , and and are the extreme elements of if (cf. Proposition 8.4).
Incidentally, the initial range projector is , and the initial null projector is . In a given SD equivalence class in the Lüders range projector is .
The core set is . The core classes are Thus an arbitrary element in is uniquely decomposed as , where , and . If one pictures as a matrix of elements (projectors) so that the SD equivalence classes are the rows and the core classes are the columns, then shows in which row, and displays in which column the element is.
One could replace the above density operator by a mixed state, and make the observable incomplete with arbitrary multiplicities of the eigenvalues, but still with a purely discrete spectrum. Such an observable would be the general one that is measured exactly. But the additional intricacies would then diminish the transparency of the illustration.
A. Proof of Lemma 3.1
Claiming the general validity of the equivalence relation where is the range projector of .
Let be an eigen-basis of spanning its range, that is, let and , where is the positive spectrum of . Then is equivalent to