The Quantum Logic of Direct-Sum Decompositions

The Quantum Logic of Direct-Sum Decompositions

David Ellerman
University of California Riverside

Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the ”quantum logic” of subspaces of a general vector space–which is then specialized to the closed subspaces of a Hilbert space. But there is a ”dual” progression. The notion of a partition (or quotient set or equivalence relation) is dual (in a category-theoretic sense) to the notion of a subset. Hence the Boolean logic of subsets has a dual logic of partitions. Then the dual progression is from that logic of partitions to the quantum logic of direct-sum decompositions (i.e., the vector space version of a set partition) of a general vector space–which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the logic to express measurement by any self-adjoint operators rather than just the projection operators associated with subspaces. In this introductory paper, the focus is on the quantum logic of direct-sum decompositions of a finite-dimensional vector space (including such a Hilbert space). The primary special case examined is finite vector spaces over where the pedagogical model of quantum mechanics over sets (QM/Sets) is formulated. In the Appendix, the combinatorics of direct-sum decompositions of finite vector spaces over is analyzed with computations for the case of QM/Sets where .

1 Introduction

This paper is an introduction to quantum logic based on direct-sum decompositions rather than on subspaces of vector spaces. This allows the logic to express measurement by any self-adjoint operators rather than just the projection operators corresponding to subspaces. A direct-sum decomposition (DSD) of a vector space over a base field is a set of (nonzero) subspaces that are disjoint (i.e., their pair-wise intersections are the zero space ) and that span the space so that each vector has a unique expression with each (with only a finite number of ’s nonzero). For introductory purposes, it is best to assume is finite dimensional (although many of the proofs are more general).

Each self-adjoint operator, and in general diagonalizable operator, has eigenspaces that form a direct-sum decomposition of the vector space but the notion of a direct-sum decomposition makes sense over arbitrary vector spaces independently of an operator. For instance, in the pedagogical model of ”quantum mechanics over sets” or QM/Sets ([12]; [14]),the vector space is so the only operators (always assumed diagonalizable) are projection operators . But given a set of basis vectors for , any real-valued ”random variable” or function determines a DSD of (where is the power-set and is the image or ”spectrum” of ”eigenvalues” of the numerical attribute ). Thus the concept of a direct-sum decomposition of a vector space allows one to capture many of the relevant properties of such a real-valued ”observable” even though it does not take values in the base field (which is only in QM/Sets). It is only as the base field is increased up to the complex numbers that all real-valued observables can be ”internalized” as self-adjoint operators.

The genesis of the usual quantum logic of projection operators or subspaces can be seen as starting with the Boolean logic of subsets in the Boolean lattice of subsets of a universe set and taking the vector space version of a ”subset”–which is a subspace. That yields the lattice of subspaces of an arbitrary vector space–which can then be specialized to the lattice of (closed) subspaces of a Hilbert space for the strictly quantum mechanical case. In category theory, the notion of a subobject, such as a subset or subspace, has a dual notion of a quotient object. Thus the dual concept to a subset is the concept of a quotient set, equivalence relation, or partition on a set . That gives rise to the idea of the logic of partitions ([11]; [13]) instead of the Boolean logic of subsets. The origin and topic of this paper is the vector space version of a partition, namely a direct-sum decomposition (NB: not a quotient space). The analogue of a partial Boolean algebra [19] is then a ”partial partition algebra” of DSDs on an arbitrary vector space (our topic here)–which can then be specialized to a Hilbert space for the strictly quantum mechanical interpretation or specialized to a vector space over for pedagogical purposes.

Figure 1: Progressions from sets to vector spaces starting with dual concepts of subset and partition.

There is a natural partial order (”refinement” as with partitions on sets) on the DSDs of a vector space and there is a minimum element , the indiscrete DSD (nicknamed the ”blob”) which consists of the whole space . A DSD is ”atomic” in the partial order if there is no DSD between it and the minimum DSD , and the atomic DSDs are the binary ones consisting of just two subspaces. Each atomic DSD determines a pair of projection operators, and the indiscrete DSD also determines a pair of projection operators, namely the zero operator and the identity operator . Conversely, each projection operator on an arbitrary vector space (other than the identity or zero operator) determines an atomic DSD consisting of the image of and the image of , while the identity and zero operators determine the indiscrete DSD. In that sense, the quantum logic of DSDs extends the quantum logic of projection operators associated with atomic DSDs. In the quantum logic of (closed) subspaces, only measurement of projection operators (associated with atomic DSDs and the blob) can be represented, so the quantum logic of direct-sum decompositions allows the representation of the measurement of any self-adjoint operators.

2 The partial partition algebra of direct-sum decompositions

2.1 Compatibility of DSDs

Let be a finite dimensional vector space over a field . A direct sum decomposition (DSD) of is a set of subspaces such that (the zero space) for and which span the space: written . Let be the set of DSDs of .

In the algebra of partitions on a set, the operations of join, meet, and implication are always defined, but in the context of ”vector space partitions,” i.e., DSDs, we need to define a notion of compatibility. If we were dealing with operators (and their associated DSDs of eigenspaces), then compatibility would be defined by commutativity. But we are dealing with DSDs directly with no assumption that they are the eigenspace DSDs of operators.

Given two DSDs and , their proto-join is the set of non-zero subspaces . If the two DSDs and were defined as the eigenspace DSDs of two operators, then the space spanned by the proto-join would be the space spanned by the simultaneous eigenvectors of the two operators, and that space would be the kernel of the commutator of the two operators. If the two operators commuted, then their commutator would be the zero operator whose kernel is the whole space so the proto-join would span the whole space. Hence the natural definition of compatibility without any mention of operators is:

and are compatible, written , if the proto-join spans the whole space .

The indiscrete DSD (the ”blob”) is compatible with all DSDs, i.e., for any .

2.2 The join of compatible DSDs

When two DSDs are compatible, the proto-join is the join:

Join of DSDs when .

The binary relation of compatibility on DSDs is reflexive and symmetric. The indiscrete DSD acts as the identity for the join: for any DSD .

In a set of mutually compatible DSDs, we need to show that the join operation preserves compatibility. If , it is trivial that and , but for a third DSD with and , does ?

Lemma 1

Let the DSDs and be compatible so that is a DSD and thus any has a unique expression where . Let so that . If , then .

Proof: Let so that . Then . If , then itself and are two different expressions for of vectors in a direct sum, so .

Theorem 2

Given three DSDs, , , and that are mutually compatible, i.e., , , and , then .

Proof: We need to prove is compatible with , i.e., that . Consider any nonzero where since , such that .

Now since , for each nonzero , such that . But since , by the Lemma, only is nonzero, so .

Symmetrically, since , for each , such that . But since , by the Lemma, only is nonzero, so .

Now since is a DSD, there is a unique expression for each nonzero where . Hence by uniqueness: . But since and and , we have . Thus

. Since was arbitrary,


2.3 The meet of two DSDs

For any two DSDs and , the meet is the DSD whose subspaces are direct sums of subspaces from and the direct sum of subspaces from and are minimal subspaces in that regard. That is, is the meet if there is a set partition on and a set partition such that:

and that holds for no more refined partitions on the index sets. If and , then it is trivial that .

As in the movie of the same name,

Figure 2: ”The blob absorbs everything it meets”: .

2.4 The refinement partial order on DSDs

The partial order on the DSDs of is defined as for set partitions but with subspaces replacing subsets:

refines , written , if for every , such that .

If holds, then each so and as well as as expected.

Proposition 1

Where it exists, the join is the least upper bound of and .

Proof: If , i.e., and have a common upper bound , then and . Given a nonzero , it can be expressed uniquely as where and as where . But since is a DSD, there is a unique expression so where where and so and thus . Hence is the least upper bound of and .

Corollary 1

If and have a common upper bound, i.e., , then .

Two DSDs and need not have a common upper bound so is not a join-semilattice.

Lemma 3

Given a DSD , let and both be direct sums of some ’s. If is nonzero, then it is also a direct sum of some ’s.

Proof: Consider a nonzero so there is a unique expression where and a unique expression where . Since is a DSD, there is also a unique expression so for each nonzero . . Thus for any such , is a common direct summand to and , so . Thus every nonzero element is in a direct sum of ’s for and thus is the direct sum of that are common direct summands of and .

Proposition 2

The meet is the greatest lower bound of and .

Proof: If then each . By the construction of , there is a set partition on and a set partition on such that the subspaces in the meet are:


and where no subsets of smaller than and subsets of smaller than have that property. Since each is contained in some , if , then . Since both and are direct sums of some , then by the Lemma the nonzero subspace is also a direct sum of some ’s. Symmetrically, since the same and are direct sums of some ’s, then by the Lemma the nonzero subspace is also a direct sum of some ’s. But since is the smallest direct sum of both ’s and ’s, , i.e., , and thus is the greatest (in the refinement partial ordering) lower bound on and .

As the blob is compatible with all DSDs, it is the minimum element in the ordering: for any . Hence any two DSDs and always have a common lower bound, so they always have a meet , i.e., is a meet-semilattice. Thus the partial partition algebra could also be called the meet-semi-lattice of DSDs on a vector space .

The binary DSDs are the atoms of the meet-semi-lattice . A meet-semi-lattice is said to be atomistic if every element is the join of the atoms below it.

Proposition 3

The meet-semi-lattice is atomistic.

Proof: Consider a non-blob DSD . If , then for . Thus for any other atom , the join is defined and , and each nonzero subspace is the direct sum of some ’s. If a join of atoms had a subspace , , then the join with the atom would split apart , so the join of the atoms below gives .

is the quantum partition logic determined by . To be more specifically ”quantum”, could be a finite-dimensional Hilbert space.

3 Partition logics in a partial partition algebra

3.1 The implication DSD in partition logics

Just as a partial Boolean algebra is made up of overlapping Boolean algebras, so a partial partition algebra is made up of overlapping partition logics or algebras. There is no maximum DSD, only maximal DSDs. Each maximal element in the partial ordering is a discrete (or ”non-degenerate”) DSD of one-dimensional subspaces (rays) of (so is the dimension of ). A partition logic is determined by the set of DSDs compatible with a maximal element with the induced ordering and operations (which is analogous to the way in a partial Boolean algebra, a complete set of one-dimensional subspaces determines a Boolean algebra).

Figure 3: Partial Partition Logic or Meet-Semi-Lattice of DSDs of

with partition logics and .

For any , so is (by construction) the maximum or top DSD in and thus might be symbolized as the discrete DSD (). Each subspace has so absorbs what it joins and is the unit element for meets within :

and .

All the DSDs and compatible with , i.e., , are compatible with each other since they have a common upper bound. Explicitly, each in determines a set partition on (the index set for ), and thus , the Bell number for the dimension of . Each would be contained in some block of the set-partition join and thus those corresponding subspaces would span so .

In order to be properly called a ”logic,” each partition lattice has a naturally defined implication inherited from the logic of set partitions (so ”partition logic” refers to a partition lattice plus the implication operation). For , and for each , the corresponding subspaces of the implication are:


Since each , the implication is still a DSD in spite of some of the being ”discretized” into the contained in it. In the implication DSD , each either remains whole like a mini-zero-blob on the space (if is not contained in any ) or it is discretized into the ”atoms” which in effect assigns a ”” to if such that . In other words, the implication acts like an indicator or characteristic function assigning a or to each depending on whether or not such that . Thus trivially:

iff .

If we just take as a set of entities (forgetting about any vector space structure), then each DSD in defines a set partition on where each subspace determines a block .

Indeed, given any DSD , each subspace of determines a block so defines a set partition on . Thus the interval is isomorphic to the set-based partition logic (join, meet, and implication operations) on that set [13]. As a partition lattice, has many of well-known properties ([27]; [5]; [16, Chapter IV, section 4]). However, the late development of partition logic was in part retarded by the practice of referring to the lattice of equivalence relations as the ”lattice of partitions” where the partial order however ”corresponds to set inclusion for the corresponding equivalence relations” [16, p. 251] so instead of being refinement it is actually ”reverse refinement” [21, p.30]. The partial order on the partition lattice (as defined here) corresponds to set inclusion of the binary relations that are the complements of equivalence relations and are called partition relations [13] or apartness relations. In the lattice of equivalence relations, the top is the biggest (indiscrete) equivalence relation (where everything is identified) and the bottom is the smallest (discrete) equivalence relation where each element is identified only with itself–whereas the partition lattice uses the opposite partial order.111Instead of the usual duality relation within a Boolean algebra, there is a duality relation between the logic of partitions and the ”logic” of equivalence relations [13]. Either way the lattice is complete and relatively complemented but not distributive. But the reversed order reverses the join and meet, the top and bottom, and the atoms and coatoms.

3.2 DSDs, CSCOs, and measurement

Given a self-adjoint operator , the projections can be reconstructed from the DSD of eigenspaces and then the operator can reconstructed–given the eigenvalues–from the spectral decomposition . What information about self-adjoint operators is lost by dealing only with their DSDs of eigenspaces? The information about which eigenvalues for eigenvectors are the same or different is retained by the distinct eigenspaces in the DSD. It is only the specific numerical values of the eigenvalues that is lost, and those numerical values are of little importance in QM. Any transformation into other real numbers that is one-to-one (thus avoiding ”accidental” degeneracy) would do as well. Thus we can say that the essentials of the measurement process in QM can be translated into the language of the quantum logic of direct-sum decompositions. Kolmogorov referred to the set partition given by the inverse-image of a random variable as the ”experiment” [20, p. 6], so it is natural to abstractly represent the direct-sum decomposition of eigenspaces given by a self-adjoint operator as the ”measurement.”

Thus unlike the quantum logic of subspaces, the quantum logic of direct-sum decompositions can directly represent the process of measurement for any self-adjoint operators (rather than just projection operators). Given a state and a self-adjoint operator on a finite dimensional Hilbert space, the operator determines the DSD of eigenspaces for the eigenvalues . The measurement operation uses the eigenspace DSD to decompose into the unique parts given by the projections into the eigenspaces , where is the outcome of the projective measurement with probability .

The eigenspace DSD of is refined by one or more maximal eigenvector DSDs, . For each , there is a set partition on the index set such that is the direct sum of the for , i.e., .

If some of the have dimension larger than one (”degeneracy”), then more measurements by commuting operators will be necessary to further decompose down to single eigenvectors. If two operators commute, that means that their eigenspace DSDs are compatible. Given another self-adjoint operator commuting with , its eigenspace DSD (for eigenvalues of ) is compatible with and thus has a join DSD in which is also in for one or more maximal each representing an orthonormal basis of simultaneous eigenvectors. The combined measurement by the two commuting operators is just the single measurement using the join DSD .

Dirac’s notion of a Complete Set of Commuting Operators (CSCO) [9] translates into the language of the quantum logic of DSDs as a set of compatible DSDs whose join is a maximal DSD in and thus is the maximum DSD in . As the join, that DSD refines each of the compatible DSDs. The combined measurement of the CSCO of commuting operators is the single (non-degenerate) measurement by the maximal DSD that is the join of their eigenspace DSDs.

In addition to being able to naturally represent measurement, the quantum logic of DSDs in useful for quite different reasons. There is a pedagogical model of quantum mechanics using vector spaces over , called ”quantum mechanics over sets” (QM/Sets) ([12], [14]), whose probability calculus is a non-commutative version of the classical Laplace-Boole finite probability theory with real-valued random variables. Such real-valued random variables on a finite sample space cannot be represented or ”internalized” as operators on –but they can be represented by DSDs on . This allows the quantum logic of DSDs’ treatment of measurement in QM to be reproduced in an appropriate form in the pedagogical model of QM/Sets, and that in turn allows simplified pedagogical versions of quantum results such as the two-slit experiment, the indeterminacy principle, Bell’s Theorem, and so forth.

In the remainder of this introductory treatment of quantum partition logic, we will focus on this pedagogical model of QM/Sets using vector spaces over –together with an Appendix on the combinatorics of DSDs over finite vector spaces over since QM/Sets uses the special case of .

4 Review of QM/Sets

4.1 Previous attempts to model QM over sets

QM/Sets is a pedagogical or ”toy” model of quantum mechanics over sets where the quantum probability calculus is a non-commutative version of the ordinary Laplace-Boole finite logical probability theory ([22], [6]) and where the usual vector spaces over for QM are replaced with vector spaces over . Fix a basis for [i.e., a maximal DSD in ] and that basis set is the sample space or outcome space for the Laplace-Boole finite probability calculus. But there are many incompatible basis sets for so, in that sense, the probability calculus of QM/Sets is a non-commutative version of the Laplace-Boole calculus.

Quantum mechanics over sets is a bare-bones (e.g., non-physical222In full QM, the DeBroglie relations connect mathematical notions such as frequency and wave-length to physical notions such as energy and momentum. QM/sets is ”non-physical” in the sense that it is a sets-version of the pure mathematical framework of (finite-dimensional) QM without those direct physical connections.) ”logical skeleton” of QM with appropriate versions of spectral decomposition, the Dirac brackets, the norm, observable-attributes, the Born rule, commutators, and density matrices all in the simple setting of sets,333Given a basis set for , each vector is expressed as a subset of the basis set. but that nevertheless provides models of characteristically quantum results (e.g., a QM/Sets version of the double-slit experiment [14]). In that manner, QM/Sets can serve not only as a pedagogical (or ”toy”) model of QM but perhaps as an engine to better elucidate QM itself by representing the quantum features in a simple setting.

There have been at least three previous attempts at developing a version of QM where the base field of is replaced by ([28], [17], and [31]). Since there are no inner products in vector spaces over a finite field, the ”trick” is how to define the brackets, the norm, and then the probability algorithm. All these previous attempts use the aspect of full QM that the bras are dual vectors so the brackets take their values in the base field of . For instance, the Schumacher-Westmoreland model does ”not make use of the idea of probability” [28, p. 919] and have instead only a modal interpretation ( possibility and impossibility). There is a fourth category-theoretic model where the objects are sets [1] but it also has the ”brackets” taking only values and thus has only a modal or ”possibilistic” interpretation.

4.2 The Yoga of transporting vector space structures

There is a method or ”Yoga” to transport some structures from a vector space over a field to a vector space over a different field . Select a basis set for the source space and then consider a structure on that can be characterized in terms of the basis set . Then apply the free vector space over the field construction to to generate the target vector space . Since the source structure was defined in terms of the basis set , it can be carried over or ”transported” to via its basis set .

This Yoga can be stated in rigorous terms using category theory ([24]; [3]). The construction of the free vector space over a field is a functor from the category of sets and functions to the category of vector spaces over and linear transformations. The functor will only be used here on finite sets where it takes a finite set to the vector space . This paper is about direct-sum decompositions of a finite-dimensional vector space . A DSD a set of disjoint subspaces so that the whole space is their direct sum, or, in terms of category theory, is the coproduct of the subspaces . In the category , a set of disjoint subsets of a set is a set partition of if , or, in terms of category theory, is the coproduct of the disjoint subsets . The free vector space over functor is a left adjoint, ”left adjoints preserve colimits” [3, p. 197], and coproducts are a special type of colimit. Hence the free vector space functor carries a set partition to the DSD of .

Now start with the structure of a DSD on . What we previously called ”characterizing the structure in terms of a basis set ” is rigorously interpreted to mean, in this case, finding a basis and a partition on so that the given DSD is the image of the free vector space functor, i.e., . But then the free vector space functor over a different field can be applied to the same set partition of the set to generate a DSD of . That is how to rigorously describe ”transporting” a set-based structure on a vector over to a vector space over a different field .

To show that any given DSD of is in the image of the free vector space over functor, pick basis set of . The sets are disjoint and since is a DSD, the union is a basis for so and .

This method is applied to the transporting of self-adjoint operators from to that motivates QM/Sets. A self-adjoint operator has a basis of orthonormal eigenvectors and it has real distinct eigenvalues , so it defines the real eigenvalue function where for , is one of the distinct eigenvalues . For each distinct eigenvalue , there is the eigenspace of its eigenvectors and is a DSD on . The inverse-image of the eigenvalue function is a set partition on .

Thus the set-based structure we have is the set with a partition on induced by a real-value function on . That set-based structure is sufficient to reconstruct the DSD on as well as the original operator . The operator is defined on the basis by for . But it might be helpful to go through the categorical construction. Scalar multiplication in the vector space is given by the set function . There is the injection of the generators function and there is the function so by the universal mapping property (UMP) of the product , we have the factor map and thus the composition . Then we use the UMP of the free vector space over functor.

UMP of free vector space functor

That UMP is that for any function from the set to any vector space over , there is a linear transformation such that . Taking and , there is a unique linear operator on such that for any , is the scalar multiple or, where we write (and scalar multiplication by juxtaposition), for . That process of going from the function on a basis set of to an operator on might be called internalizing the function in .

Given the set-based structure of a real-valued function , which determines the set partition on , we then apply the free vector space over functor to construct the vector space . That vector space is more familiar in the form of the powerset since each function in is the characteristic function of a subset . The free vector space functor takes the coproduct to the DSD of . The attempt to internalize the real function would use the scalar multiplication function and would only work if took values in in which case would be a characteristic function for some subset . In that special case, the internalized operator would be the projection operator which in terms of the basis has the action taking any subset to .

Hence outside of characteristic functions, the real-valued functions cannot be internalized as operators on . But that is fine since the idea of the model QM/Sets is that given a basis of , the quantum probability calculus will just be the classical finite probability calculus with the outcome set or sample space where is a real-valued random variable. We have illustrated the transporting of set-based structures on to using a basis set , but in the stand-alone model QM/Sets, we cut the umbilical cord to and work with any other basis of and real-valued random variables on that sample space.

Other structures can be transported across the bridge from to . QM/Sets differs from the other four attempts to define some toy version of QM on sets by the treatment of the Dirac brackets. Starting with our orthonormal basis on a finite-dimensional Hilbert space (where the bracket is the inner product), we need to define the transported brackets applied to two subsets in . The two subsets define the vectors and in which have the bracket value . Since that value is defined just in terms of the subsets as the cardinality of their overlap, that value can be transported to as the real-valued basis-dependent brackets (see below).

4.3 Laplace-Boole finite probability theory

Since our purpose is conceptual rather than mathematical, we will stick to the simplest case of finite probability theory with a finite sample space or outcome space of equiprobable outcomes and to finite dimensional QM.444The mathematics can be generalized to the case where each point in the sample space has a probability (when it is a basis set with point probabilities that is transported) but the simpler case of equiprobable points serves our conceptual purposes. The events are the subsets , and the probability of an event occurring in a trial is the ratio of the cardinalities: . Given that a conditioning event occurs, the conditional probability that occurs is: . The ordinary probability of an event can be taken as the conditional probability with as the conditioning event so all probabilities can be seen as conditional probabilities. Given a real-valued random variable on the elements of , the probability of observing a value given an event is the conditional probability of the event given :


That is all the probability theory we will need here. Our first task is to show how the mathematics of finite probability theory can be recast using the mathematical notions of quantum mechanics with the base field of .

4.4 Vector spaces over

To show how classical Laplace-Boole finite probability theory can be recast as a quantum probability calculus, we use finite dimensional vector spaces over . The power set of is a vector space over , isomorphic to , where the vector addition is the symmetric difference of subsets. That is, for ,

so the members of are the elements that are members of or members of but not members of both.

The -basis in is the set of singletons , i.e., the set . In the context of , that basis set would correspond to the maximal element where is the one-dimensional subspace . A vector is specified in the -basis as and it is characterized by its -valued characteristic function of coefficients since .

Consider the simple case of where the -basis is , , and . The three subsets , , and also form a basis since:


; and


These new basis vectors could be considered as the basis-singletons in another equicardinal sample space where , , and refer to the same abstract vector as , , and respectively.

In the following ket table, each row is an abstract vector of expressed in the -basis, the -basis, and a -basis.

Ket table giving a vector space isomorphism: where row = ket.

In the Dirac notation [9], the ket represents the abstract vector that is represented in the -basis coordinates as . A row of the ket table gives the different representations of the same ket in the different bases, e.g., .

4.5 The brackets and the norm

In a Hilbert space, the inner product is used to define the brackets and the norm but there are no inner products in vector spaces over finite fields. The different attempts to develop a toy model of QM over a finite field ([28], [31], [17]) such as differ from this model in how they address this problem. The treatment of the Dirac brackets and norm defined here is distinguished by the fact that the resulting probability calculus in QM/Sets is (a non-commutative version of) classical finite probability theory (instead of just a modal calculus with values and ).

For a singleton basis vector , the (basis-dependent) bra is defined by the bracket:


Note that the bra and the bracket is defined in terms of the -basis and that is indicated by the -subscript on the bra portion of the bracket. Then for , (the Kronecker delta function) which is the QM/Sets-version of for an orthonormal basis of . The bracket linearly extends in the natural numbers to any two vectors :555Here takes values in the natural numbers outside the base field of just like, say, the Hamming distance function on vector spaces over in coding theory. [25] Thus the ”size of overlap” bra is not to be confused with the dual (”parity of overlap”) functional where