# Discrete Vector Models for Catalysis

Authors: Ernst-Christoph Haß; Sonja Sauerbrei; Peter Jörg Plath
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

Based on Ruch's concept of diagram lattices formed by Young
diagrams we investigated the possibility to transform incomparable
diagrams into comparable ones by means of vector catalysis. Ruch's
diagram lattices allow a very general description of comparing
frequency distributions by their mixing-character as an order
relation which is equivalent to majorisation in the mathematical
theory of inequalities. Dealing with Young diagrams or vectors
containing only integer components, respectively, vector catalysis
is strongly related to entanglement catalysis in quantum
informatics. In a very systematic way the diagram lattices of the
partitions up to the numbern=20 have been searched for
incomparable pairs which can be catalysed. This concept opens the
opportunity for regarding vector catalysis as a universal
phenomenon which is not restricted to the quantum mechanical idea
of entanglement catalysis. Such a general approach offers the possibility to compare vector
catalysis with chemical ideas of catalysis and autocatalysis in a
very fundamental sense. We emphasize that
vector catalysis is a universally valid procedure for
classification purposes, where incomparable sequences of symbols
are transformed into comparable ones in a much higher dimensional
space ignoring any physical interpretation of these symbols.


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## Body

## 1. Introduction

Catalysis is very old but still an unsolved problem in chemistry and
biochemistry, although there exist a lot of experimental and theoretical
publications on this topic. The termcatalysis was introduced 1835/36
into chemistry by Berzelius [1] to describe the ability of substances to awake
affinities, which are asleep at a particular temperature, by their mere presence and not
by their own affinity (the catalytic power seems actually
to consist in the fact that substances are able to
awake affinities, which are asleep at a particular temperature, by their mere presence
and not by their own affinity).
In other words, reactions which do not occur on their own or spontaneously take
place only  at the presence of an additional substance—the so-called catalyst. It took about
hundred years to develop the almost generally accepted idea of catalysis. Now, this classical definition of catalysis is given by Ostwald in 1894 on the basis
of the theory of chemical reaction rate [2–4]; catalysis is the acceleration of a slowly
preceding chemical reaction through the presence of a foreign substance, which
may participate intermediately but is not consumed by the reaction.Abel (85th Meeting of German Natural and Medical Scientists, Vienna 1913) claimed in a much more abstract and modern way that only reactions, but no substances are catalysing and that there do not exist
catalysts in general, “The speaker comes to the conclusion that according to the general view of current research on homogeneous catalysis one can clearly deducenot substances, but only reactions are catalysing…. Since only reactions, but not substances are
catalysing it is strange that in the presence of several catalysts they do not act additively, even though the catalysing reaction paths can be well interleaved….Catalysis emerges in case of
chemical activity of the catalyst by extension of the reaction path and in case of physical interference of the catalyst by shifting of the reaction level.”
[5].Most of the theoretical works on chemical catalysis deal with
numerical quantum-chemical treatment on the molecular level. Nonlinear dynamics
based on the ideas of Prigogine [6, 7] and Haken [8, 9] favoured the old idea of autocatalysis in order to
describe macroscopic pattern formation in chemical systems. In heterogeneous
catalysis, enormous scientific effort has been made in order to close the gap
between high vacuum investigations on molecular processes on surfaces [10–12] on the one hand and the technical level of catalysis
at normal and high pressure [13–15] on the other hand. Although the atomic structure of
elementary steps in the catalytic formation of NH3 [16–20] or
the oxidation of CO on single crystal surfaces in vacuum meanwhile has been
figured out in great detail [21–27], there still remains an unavoidable gap. Furthermore, there exists another question which rises from the assumption of the existence
of a catalytic centre (catalytic sites) which is created by separation from its
surrounding. There are a lot of fancy descriptions like “spill over effects” in
heterogeneous catalysis or “steric effects” in enzyme catalysis, which are
connected with this artificial separation [28].It is somehow a miracle that one needs large molecular systems like
enzymes or haemoglobin in order to catalyse very small molecules. In
heterogeneous catalysis, one needs surfaces or large clusters to catalyse small
molecular systems. However, if the system to be catalysed is of medium size, one only needs catalysts of medium sizes to arrange the reaction.In recent years, the term catalysis was introduced in quite another
field than chemistry, that is, in the mathematics of quantum information theory
[29]. Starting point was the observation that entangled
quantum states can be transformed with certainty into each other by local transformations, if the distribution of ordered Schmidt coefficients of one state is majorising
the corresponding distribution of the other state [30]. This connects entanglement with the linear algebraic
theory of majorisation [31–33]. As a consequence, entangled quantum states cannot
be transformed with certainty, if their distributions of Schmidt coefficients
are incomparable in terms of majorisation. But in some special cases, there
exists an additional entangled quantum state that enhances the local
transformation of two incomparable quantum states without changing itself at
the end [29]. This additional state is called catalyst in
analogy to chemistry and the corresponding transformation of quantum states as entanglement
catalysis. Although some progress was made in the last few years for
special types of incomparable states and catalysts, the problem of entanglement
catalysis is far away from being well understood [34–41].On the first view, catalysis in chemistry and entanglement catalysis
have nothing in common. But besides the name there are even more aspects
related to catalysis which suggest a connection between both fields, that is, the application of quantum logic [42, 43] and majorisation theory [31]. For this reason, we are strongly convinced that
entanglement catalysis and catalysis are much more general phenomena
than considered until now.The idea of majorisation was adopted to chemistry for the first time
when Ruch together with Schönhofer introduced agreater relation for Young
diagrams [44–47] in order to answer questions in connection with the
theory of chirality functions. Later on, Ruch generalised this concept in his
articles on diagram lattices as structural principle [48], on the principles of increasing mixing [49] and on information extent, and information distance [50]. These powerful ideas have been applied recently to
rank biodiversity indices for the comparison of water quality of lakes [51].Taking into account the concepts of Ruch, we investigated in our
previous articles the decay of beer foam and its
bubble size distributions as well as—more generally—diffusion processes by diagram lattices and
variants of majorisation using discrete partition vectors [52–55]. More precisely, the foam decay is described by a
bubble size distribution function and its variation in time. As a consequence, we consider distribution functions which are transformed into each other. Assuming
that any distribution function describes a snapshot of a certain situation or
“state” during foam decay, these states are compared by majorisation. Depending
on the kind of beer
and the conditions of decay, we observe transitions between incomparable
distributions, which are not yet fully understood. It seems that we have
similar phenomena than those observed in quantum informatics.In this paper, we will extend our
previous approaches todiscrete vector catalysis, that is, we investigate
the question under which circumstances two discrete (distribution or state)
vectors containing only integer components, which are incomparable in terms of
majorisation, can be made comparable by a third discrete state vector of
integer components, which we call catalyst vector. This corresponds in a
discrete formulation to the problem in quantum informatics of enabling a
certain transformation of two quantum states by a third quantum state which
cannot be done without this latter state.We are convinced that the results can be
transferred uniquely to entanglement catalysis in quantum informatics. Recall
that the Schmidt coefficients of the orthonormal bases of the entangled quantum
states are distributions of nonnegative real numbers that can be arranged as
ordered vectors summing up to one. On the other hand, in all numeric examples
of normalised distributions or—equivalently—proportions with sum “1” the space of real
numbersℝ is projected onto the space
of rational numbers ℚ. Furthermore, since the sets of
rational numbers ℚ and of nonnegative
integers ℕ0 are of  the same cardinality, all results obtained
for discrete vector catalysis by comparing vectors with nonnegative integer
numbers should analogously be
valid for entanglement catalysis. It is therefore our aim to treat the problem
of catalytic majorisation on this simplest possible system and to show some
fundamental aspects of discrete vector catalysis.


## 2. Mathematical Background

### 2.1. Partition Lattices and Incomparable Pairs of Partitions

Using discrete mathematical model state vectors can be described in
terms of partition diagrams. Letu1+u2+…+ud=n with d≤n and ui≥ui+1 for i=1,2,…,d−1 a partition of natural number n with d positive integers. It can be represented graphically by a partition diagram or Young
diagram [45–47] consisting of n boxes, arranged in rows and
columns in such a way that the ith row has ui boxes
and the first box in each row lies in the first column. For example, the partition
“3+2+1” of n=6 is represented by the Young diagram. (2.1)If one introduces a greater relation in the sense of majorisation [31], the set of partitions of a natural number n forms a lattice, the so-called diagram lattice ([48, 52], see Figure 1 for n=6).(a) Ruch’s lattice of the partition diagrams of the natural numbern=6 with 11 diagrams; (b) corresponding lattice of partition vectors. The diagrams no. 4 and 5 and partition vectors a and
b which are marked in magenta show examples for incomparable structures.
(a)(b)Ruch formulated this greater relation with respect toYoung
diagrams as follows. A diagramγ is called greater than a diagram γ′, if γ′ can be constructedfromγ by moving boxes upward, that is, from shorter
rows into longer or equal ones [48]. Starting from this definition, he summarised in his
fundamental article about the diagram lattice as a structural principle all
lattices up to n=10.
With regard to the following investigation we like to emphasise that diagram
lattices of n≥6 are partially ordered, that is, they exhibit
pairs of diagrams which are not comparable in sense of majorisation by the
partial order of the lattice (see Figure 1(a)).Each of the
partition diagrams can also be expressed by a (column) vectoru with trace n,  (2.2)u=(u1u2⋮ud),tr(u)=∑j=1duj=n containing d nonzero components which are ordered in
decreasing (nonincreasing) sequences (see Figure 1(b)). For comparison of different partitions, partition vectors are filled with additional zeros up to totally n components such that the numbers of components in a vector pair are equal.In the terminology of (classical) majorisation [31], a partition vector u is said to be
majorised  by a partition vector v, denoted u≺v, if (i) the partial sums ou,i of u are less or equal to the
partial sums ov,i of v for all i=1,…,n−1 and (ii) both the sums of components of u and v are equal to n (see (2.3)), which is the natural number of
the diagram or partition lattice, (2.3)u≺vif{(i)(ou,i=∑j=1iuj)≤(ov,i=∑j=1ivj),i=1,…,n−1,(ii)(ou,n=∑j=1nuj)=(ov,n=∑j=1nvj)=n. If condition (i) of (2.3) does
not hold, the vectors u and v are said to be
incomparable, denoted by (u⊀v)⋀(u⊁v), abbreviated u≉v (see also
Table 7).
As an example of an incomparable vector pair in the sense of majorisation we consider
diagrams no. 5 and no. 4, or partition vectors a and b of Figure 1. The discrete distribution function of the components
of a and b is shown in Figure 2. Forming the vectors of the partial sums oa and ob, respectively, one can easily show that a≉b, that is, there exists a component j, where oa,j<ob,j, and another component j′ with oa,j′>ob,j′,  (2.4)oa=(366),ob=(456),3<4,j=1,6>5,j′=2,6=6,a≉b. If one of the partition vectors u and v is majorising the other, there exists always a doubly stochastic matrix [53] which transforms the vector u into v or vice versa. In a physical interpretation, the vectors u and v may be any probability vectors or states and the doubly stochastic matrix
describes the transition between these states. On the other hand, if u and v are incomparable, such as a and b in the example above (a≉b, see Figure 2), it follows that there does not exist a doubly stochastic
matrix [53] which transforms the vectors  into each other, that is, there is no transition
between both states.Figure 2
The components of the two vectorsa (blue) and
b (red) of Figure 1 as a discrete function of the corresponding row numbers. Both distribution (probability) functions are incomparable.


### 2.2. Introduction of Catalyst Partition Vector

However, one may ask if there exists another vectorc (see, e.g., Figure 1(b)) which enables the vectors a and b to be transformed into each other, that is, to become comparable. This question
is usually answered by comparing the nonincreasingly ordered set of elements of
the tensor products of the vectors a and b with the
transpose cT a third vector c, respectively. The tensor product of two vectors is a rectangular matrix, where
each component of the first vector is multiplied with each component of the
second one. Arranging the elements of these matrices in a list with decreasing
(nonincreasing) order one gets ordered vectors θb⊗c and θa⊗c of the same dimension as of the rectangular
matrix which can now be investigated with respect to their comparability. For
this purpose, one has to form the vectors ob⊗c and oa⊗c of the partial sums of the components of the
new vectors θb⊗c and θa⊗c,  (2.5)b⊗cT=(411)⊗(321)=(1284321321)⇒θb⊗cT=(1284332211),a⊗cT=(330)⊗(321)=(963963000)⇒θa⊗cT=(996633000),ob⊗cT=(122024273032343536),oa⊗cT=(91824303336363636),ob⊗cT−oa⊗cT=(+3+20−3−3−4−2−10). If there is a change of the signs of the components in the
difference vector ob⊗c−oa⊗c (as shown for the actual choice of a, b, and c), the new tensor vectors θa⊗c and θb⊗c still remain incomparable. In other words, the
elements of the difference vector have to be semidefinite.To illustrate the
property of incomparableness graphically, in Figure3 the difference ob⊗c−oa⊗c is shown as a discrete function of
the position count of vector components. The existence of negative deviations
from the zero line indicates that the compared partition vectors are
incomparable.Figure 3
The components of the difference vectorob⊗c−oa⊗c as a discrete function of the position count
of the vector components.But there exist pairs of incomparable vectors which can be catalysed
by a third vector, such that the resulting tensor vectors are comparable. One
of the simplest examples is given in the following.Consider the pair of vectors(2.6)d=(54111),e=(62220)  of the partition lattice with n=12. These partition vectors
are obviously incomparable, d≉e, but can be catalysed by a third vector (2.7)fT=(211).Calculating the tensor products d⊗fT and e⊗fT, and ordering their components in decreasing sequence, respectively, one obtains
the following tensor vectors θe⊗f and θd⊗f,  (2.8)θe⊗fT=(1266444222222000),θd⊗fT=(1085544222111111). Constructing their partial sums oe⊗f and od⊗f, and building the difference oe⊗f−od⊗f yields,  (2.9)oe⊗fT=(121824283236384042444648484848),od⊗fT=(101823283236384042434445464748),oe⊗fT−od⊗fT=(+20+1000000+1+2+3+2+10). If one draws the difference oe⊗f−od⊗f as a discrete function over the position
numbers of the vector components, one obtains the graph represented in Figure 4, which is touching but not crossing the abscissa.
This indicates that the corresponding pair of ordered tensor product vectors is
comparable in the sense of majorisation.Figure 4
The components of the difference vectoroe⊗f−od⊗f as a discrete function of the position count of the vector components.The catalyst vectorf is the minimum partition vector
catalysing the incomparable pair of vectors d and e, that means it is the partition with the smallest natural number n which is the trace of the vector catalyst.
In addition to this catalyst vector one can find an infinite number of catalysing
partitions for this couple of vectors (and also for any other pair of vectors
which can be catalysed), for example, (4 2 2), (6 3 3), and so on.


## 2.1. Partition Lattices and Incomparable Pairs of Partitions

Using discrete mathematical model state vectors can be described in
terms of partition diagrams. Letu1+u2+…+ud=n with d≤n and ui≥ui+1 for i=1,2,…,d−1 a partition of natural number n with d positive integers. It can be represented graphically by a partition diagram or Young
diagram [45–47] consisting of n boxes, arranged in rows and
columns in such a way that the ith row has ui boxes
and the first box in each row lies in the first column. For example, the partition
“3+2+1” of n=6 is represented by the Young diagram. (2.1)If one introduces a greater relation in the sense of majorisation [31], the set of partitions of a natural number n forms a lattice, the so-called diagram lattice ([48, 52], see Figure 1 for n=6).(a) Ruch’s lattice of the partition diagrams of the natural numbern=6 with 11 diagrams; (b) corresponding lattice of partition vectors. The diagrams no. 4 and 5 and partition vectors a and
b which are marked in magenta show examples for incomparable structures.
(a)(b)Ruch formulated this greater relation with respect toYoung
diagrams as follows. A diagramγ is called greater than a diagram γ′, if γ′ can be constructedfromγ by moving boxes upward, that is, from shorter
rows into longer or equal ones [48]. Starting from this definition, he summarised in his
fundamental article about the diagram lattice as a structural principle all
lattices up to n=10.
With regard to the following investigation we like to emphasise that diagram
lattices of n≥6 are partially ordered, that is, they exhibit
pairs of diagrams which are not comparable in sense of majorisation by the
partial order of the lattice (see Figure 1(a)).Each of the
partition diagrams can also be expressed by a (column) vectoru with trace n,  (2.2)u=(u1u2⋮ud),tr(u)=∑j=1duj=n containing d nonzero components which are ordered in
decreasing (nonincreasing) sequences (see Figure 1(b)). For comparison of different partitions, partition vectors are filled with additional zeros up to totally n components such that the numbers of components in a vector pair are equal.In the terminology of (classical) majorisation [31], a partition vector u is said to be
majorised  by a partition vector v, denoted u≺v, if (i) the partial sums ou,i of u are less or equal to the
partial sums ov,i of v for all i=1,…,n−1 and (ii) both the sums of components of u and v are equal to n (see (2.3)), which is the natural number of
the diagram or partition lattice, (2.3)u≺vif{(i)(ou,i=∑j=1iuj)≤(ov,i=∑j=1ivj),i=1,…,n−1,(ii)(ou,n=∑j=1nuj)=(ov,n=∑j=1nvj)=n. If condition (i) of (2.3) does
not hold, the vectors u and v are said to be
incomparable, denoted by (u⊀v)⋀(u⊁v), abbreviated u≉v (see also
Table 7).
As an example of an incomparable vector pair in the sense of majorisation we consider
diagrams no. 5 and no. 4, or partition vectors a and b of Figure 1. The discrete distribution function of the components
of a and b is shown in Figure 2. Forming the vectors of the partial sums oa and ob, respectively, one can easily show that a≉b, that is, there exists a component j, where oa,j<ob,j, and another component j′ with oa,j′>ob,j′,  (2.4)oa=(366),ob=(456),3<4,j=1,6>5,j′=2,6=6,a≉b. If one of the partition vectors u and v is majorising the other, there exists always a doubly stochastic matrix [53] which transforms the vector u into v or vice versa. In a physical interpretation, the vectors u and v may be any probability vectors or states and the doubly stochastic matrix
describes the transition between these states. On the other hand, if u and v are incomparable, such as a and b in the example above (a≉b, see Figure 2), it follows that there does not exist a doubly stochastic
matrix [53] which transforms the vectors  into each other, that is, there is no transition
between both states.Figure 2
The components of the two vectorsa (blue) and
b (red) of Figure 1 as a discrete function of the corresponding row numbers. Both distribution (probability) functions are incomparable.


## 2.2. Introduction of Catalyst Partition Vector

However, one may ask if there exists another vectorc (see, e.g., Figure 1(b)) which enables the vectors a and b to be transformed into each other, that is, to become comparable. This question
is usually answered by comparing the nonincreasingly ordered set of elements of
the tensor products of the vectors a and b with the
transpose cT a third vector c, respectively. The tensor product of two vectors is a rectangular matrix, where
each component of the first vector is multiplied with each component of the
second one. Arranging the elements of these matrices in a list with decreasing
(nonincreasing) order one gets ordered vectors θb⊗c and θa⊗c of the same dimension as of the rectangular
matrix which can now be investigated with respect to their comparability. For
this purpose, one has to form the vectors ob⊗c and oa⊗c of the partial sums of the components of the
new vectors θb⊗c and θa⊗c,  (2.5)b⊗cT=(411)⊗(321)=(1284321321)⇒θb⊗cT=(1284332211),a⊗cT=(330)⊗(321)=(963963000)⇒θa⊗cT=(996633000),ob⊗cT=(122024273032343536),oa⊗cT=(91824303336363636),ob⊗cT−oa⊗cT=(+3+20−3−3−4−2−10). If there is a change of the signs of the components in the
difference vector ob⊗c−oa⊗c (as shown for the actual choice of a, b, and c), the new tensor vectors θa⊗c and θb⊗c still remain incomparable. In other words, the
elements of the difference vector have to be semidefinite.To illustrate the
property of incomparableness graphically, in Figure3 the difference ob⊗c−oa⊗c is shown as a discrete function of
the position count of vector components. The existence of negative deviations
from the zero line indicates that the compared partition vectors are
incomparable.Figure 3
The components of the difference vectorob⊗c−oa⊗c as a discrete function of the position count
of the vector components.But there exist pairs of incomparable vectors which can be catalysed
by a third vector, such that the resulting tensor vectors are comparable. One
of the simplest examples is given in the following.Consider the pair of vectors(2.6)d=(54111),e=(62220)  of the partition lattice with n=12. These partition vectors
are obviously incomparable, d≉e, but can be catalysed by a third vector (2.7)fT=(211).Calculating the tensor products d⊗fT and e⊗fT, and ordering their components in decreasing sequence, respectively, one obtains
the following tensor vectors θe⊗f and θd⊗f,  (2.8)θe⊗fT=(1266444222222000),θd⊗fT=(1085544222111111). Constructing their partial sums oe⊗f and od⊗f, and building the difference oe⊗f−od⊗f yields,  (2.9)oe⊗fT=(121824283236384042444648484848),od⊗fT=(101823283236384042434445464748),oe⊗fT−od⊗fT=(+20+1000000+1+2+3+2+10). If one draws the difference oe⊗f−od⊗f as a discrete function over the position
numbers of the vector components, one obtains the graph represented in Figure 4, which is touching but not crossing the abscissa.
This indicates that the corresponding pair of ordered tensor product vectors is
comparable in the sense of majorisation.Figure 4
The components of the difference vectoroe⊗f−od⊗f as a discrete function of the position count of the vector components.The catalyst vectorf is the minimum partition vector
catalysing the incomparable pair of vectors d and e, that means it is the partition with the smallest natural number n which is the trace of the vector catalyst.
In addition to this catalyst vector one can find an infinite number of catalysing
partitions for this couple of vectors (and also for any other pair of vectors
which can be catalysed), for example, (4 2 2), (6 3 3), and so on.


## 3. Results and Discussion

In the following, we present a
systematic approach to discrete vector catalysis by considering vectors of
integer components with sum (trace)n which can be correlated to
partition diagrams of n boxes
forming the diagram lattice according to Ruch [48]. For this purpose, we have written a computer program
in C++, which calculates all incomparable pairs of partition vectors
which occur in partition lattices up to n=20. Also, we computed for
any catalysable couple of partition vectors the list of all catalyst vectors
which can be found in partition lattices up to n=20. In addition, we
determined some more catalysts of partition lattices with n>20 using the
functional language Haskell and an empirical method. The notation being
used is summarised in Tables 7 and 8.


### 3.1. Minimal Global Properties of Discrete Vector Catalysis

Concerning minimal global
properties of discrete vector catalysis, one may ask the following questions, the answers of which are listed in Table1.Table 1
Incomparable, catalysable pairs of partition vectors and their corresponding
catalyst partition vectors with distinguished minimal global properties of discrete
vector catalysis. The significant minimal global properties are marked in bold italic letters.
nmn⋅mdu,vdcdu,v⋅dcIncomparable, catalysable pairCatalyst partition vectorof partition vectorsuTvTcT(a)11424825945(4, 4, 1, 1, 1)(5, 2, 2, 2, 0)(10, 6, 5, 5, 4,  3, 3, 3, 3)(b)124104 920438152(5, 5, 1, 1)(6, 3, 3, 0)(36, 24, 20, 20, 16, 16, 15, 15, 12, 12, 12, 12, 10, 10, 10, 10, 9, 9, 9, 9, 8, 8, 8, 8, 8, 8, 8, 8, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6)*(c)14342428(6, 6, 1, 1)(8, 3, 3, 0)(2, 1)(d)14342428(6, 6, 1, 1)(8, 3, 3, 0)(2, 1)*Catalyst partition vector may not be minimal.(a)
What is the
minimal component sum (trace)n, for which one can find an incomparable
pair u and v of partition vectors which can be
catalysed?(b)
What is the
minimal dimensiondu,v of an incomparable pair of partition vectors u and v which can be catalysed and the natural number n of
its first occurrence?(c)
What is the
smallest catalyst partition vectorc and its corresponding pair
of incomparable partition vectors u and v?(d)
What are the
smallest balanced combinations consisting of a catalysable pair of incomparable
partition vectorsu and v and its smallest catalyst c, that is, where either (n⋅m) or (du,v⋅dc) is minimal?Answering question, (a) follows that the smallest numbern for
which one can find an incomparable but catalysable vector pair is n=11, and the corresponding partition vectors are uT=(4,4,1,1,1) and vT=(5,2,2,2,0), respectively. Thus, the first diagram or partition lattice with the possibility
of vector catalysis is that of 11 boxes. The minimal catalyst vector of the
smallest catalysable pair is cT=(10,6,5,5,4,3,3,3,3), which belongs to a lattice of the integer m=42. As a
consequence, both the dimension du,v⋅dc and the trace (no. of boxes) n⋅m of the resulting tensor product space in which
vector catalysis occurs are rather high. This corresponds to the observation in
chemistry that small molecules are often catalysed by large catalysts.This analogy turns out even stronger, if one considers—as in question (b)—the tensor product space of thefirst occurrence
of an incomparable but catalysable pair u and v of
smallest dimension du,v.
Jonathan and Plenio have already proven in their fundamental article on entanglement
catalysis [29] that incomparable states with less than four
components cannot be catalysed, hence the minimal dimension of catalysable
pairs of partition vectors is du,v=4. The first such a pair is uT=(5,5,1,1) and vT=(6,3,3,0)
with n=12. It can be catalysed, but the smallest catalyst which we
found by extended calculations contains dc=38
components and has a trace of m=410 (see Table 1) yielding a tensor product space with du,v⋅dc=152 and n⋅m=4920. Such a highly dimensional space of vector catalysis may possibly be related
to enzyme catalysis in chemistry or biochemistry.On the other hand, the answers to questions (c) and (d), that is, to
the smallest catalyst partition vectorc and to the smallest
balanced combinations of u/v and c lead to a tensor product space of comparatively low dimensionality (du,v⋅dc=8,n⋅m=42).
This result corresponds in chemistry particularly to homogeneous catalysis of
medium-sized molecules by medium-sized catalysts.


### 3.2. Incomparable Vector Pairs and Necessary Conditions for Vector Catalysis

Table2 summarises all incomparable pairs u and v of partition lattices up to n=13 which may be
catalysable together with a (minimal) catalyst partition vector c, if such was found by systematic calculation or empirical investigation. Also, the number of incomparable pairs—denoted by Nc,20—which are catalysed by a catalyst partition
vector of a partition lattice with n≤20 is given in Table 2.Table 2
Incomparable pairs of partition vectors which may be catalysable from partition
lattices up ton=13 and corresponding catalyst partition vectors.
ndu,vuTvTmdccTNc,20115(4, 4, 1, 1, 1)(5, 2, 2, 2, 0)429(10, 6, 5, 5, 4, 3, 3, 3, 3)0124(5, 5, 1, 1)(6, 3, 3, 0)41038(36, 24, 20, 20, 16, 16, 15, 15, 12, 12, 12, 12, 10, 10, 10, 10, 9, 9, 9, 9, 8, 8, 8, 8, 8, 8, 8, 8, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6)*0125(5, 4, 1, 1, 1)(6, 2, 2, 2, 0)43(2, 1, 1)87126(4, 4, 1, 1, 1, 1)(5, 2, 2, 2, 1, 0)429(10, 6, 5, 5, 4, 3, 3, 3, 3)0126(3, 3, 3, 1, 1, 1)(4, 2, 2, 2, 2, 0)375(12, 8, 7, 5, 5)*0134(6, 5, 1, 1)(7, 3, 3, 0)316(10, 6, 5, 4, 3, 3)*0135(6, 4, 1, 1, 1)(7, 2, 2, 2, 0)43(2, 1, 1)120135(5, 5, 1, 1, 1)(7, 2, 2, 2, 0)43(2, 1, 1)229135(5, 5, 1, 1, 1)(6, 3, 3, 1, 0)41038(36, 24, 20, 20, 16, 16, 15, 15, 12, 12, 12, 12, 10, 10, 10, 10, 9, 9, 9, 9, 8, 8, 8, 8, 8, 8, 8, 8, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6)*0135(5, 5, 1, 1, 1)(6, 3, 2, 2, 0)?**?**?**?**136(5, 4, 1, 1, 1, 1)(6, 2, 2, 2, 1, 0)43(2, 1, 1)87136(4, 4, 2, 1, 1, 1)(5, 2, 2, 2, 2, 0)429(10, 6, 5, 5, 4, 3, 3, 3, 3)0137(4, 4, 1, 1, 1, 1, 1)(5, 2, 2, 2, 2, 0, 0)164(6, 4, 3, 3)9136(4, 3, 3, 1, 1, 1)(5, 2, 2, 2, 2, 0)43(2, 1, 1)134137(4, 4, 1, 1, 1, 1, 1)(5, 2, 2, 2, 1, 1, 0)429(10, 6, 5, 5, 4, 3, 3, 3, 3)0137(3, 3, 3, 1, 1, 1, 1)(4, 2, 2, 2, 2, 1, 0)375(12, 8, 7, 5, 5)*0*Catalyst partition vector may not be minimal for the respective pair.
**A catalyst could not yet be found by our investigations. We assume
that a catalyst vector for this pair does not exist.**A catalyst could not yet be found by our investigations. We assume
that a catalyst vector for this pair does not exist.In the case of
partition lattices withn≤10 no
incomparable pair exists which can be catalysed, as will be shown in the following.An important
question is now under which circumstances two incomparable pairs of a partition
lattice with natural numbern can be catalysed by a third partition
vector. A necessary condition was given by Jonathan and Plenio [29] which can be expressed using the terminology of
discrete partition vectors as follows.LetuT=(u1u2…ui…udu,v) and vT=(v1v2…vi…vdu,v) a pair of incomparable partition vectors of
length du,v, then
this pair can be catalysed be a third (catalytic) partition vector only if both (3.1)u1≤v1,udu,v≥vdu,v hold. If one can
find such a catalyst vector, denoted by cT=(c1c2…cdc), the
resulting ordered tensor product v⊗cT majorises
the ordered tensor product u⊗cT.
Accordingly, in the interpretation of quantum information theory the corresponding
entangled quantum states can be transformed with entanglement-assisted local
transformation and classical communication (ELQCC) [29]. As mentioned above, Jonathan and Plenio have also
shown that incomparable partitions of dimension du,v<4 cannot be
catalysed [29].In the following, we suggest a stronger necessary condition for vector catalysis which is based
on the concept of majorisation.Proposition 3.1.
A pair of incomparable partition vectors of dimensiondu,v(du,v≥4)
can be catalysed only if there exist two indices i1 and i2 with 1<i1<i2<du,v such that for the partial sums ou,i and ov,i holds (3.2)(I)ov,i≥ou,ifor1≤i<i1,(II)ov,i<ou,ifori=i1,ov,i≤ou,ifori1+1≤i<i2,(III)ov,i>ou,ifori=i2,ov,i≥ou,ifori2+1≤i≤du,v. In addition, for the du,vth
vector component the majorisation condition (3.3)ov,i=ou,i=n is fulfilled.Although we cannot give yet a strict mathematical
proof, our proposition can be confirmed by the following consideration. In the
case of the largest partition values, that is, interval (I) in (3.2), one
partition vector (v) obviously “majorises” the other (u). Since the pair
is incomparable, there must follow an interval (II) with partition values such
that the second partition vector (u)
“majorises” the first (v).
Finally, in order to enable majorisation after building the ordered tensor
products with a catalyst vector, it seems to be necessary to have a last interval
(III) with the smallest partition values, where the first partition vector (v) takes over
“majorisation” again.For a
pair of incomparable partition vectorsu and v with
these properties, we suggest to introduce the notation “v≻≺≻u” or—equivalently—“u≺≻≺v” (see also Table 7).The condition (3.2) will be illustrated at the example
of the incomparable pair uT=(4,4,1,1,1)
and vT=(5,2,2,2,0) with du,v=5 and n=11. The corresponding partial sums are ouT=(4,8,9,10,11) and ovT=(5,7,9,11,11).
Thus,  (3.4)(ov,1=5)>(4=ou,1),interval(I),(ov,2=7)<(8=ou,2),interval(II),(ov,3=9)=(9=ou,3),interval(II),(ov,4=11)>(10=ou,4),interval(III),(ov,5=11)=(11=ou,5),interval(III), and hence i1=2 and i2=4.A graphical representation of the resulting difference
vectorovT−ouT=(1,−1,0,1,0) is
drawn in Figure 5, where also the interval boundaries i1 and i1 as well as the intervals (I), (II), and (III) are
indicated, respectively. The dotted line shows the limiting curve (envelope) between
the values of the difference of partial sums. Note that this curve is crossing
twice the (dashed) zero line of the diagram indicating a twofold change of sign
of the difference vector ovT−ouT; in
other word, the envelope exhibits three extremal points (nodes).Figure 5
The components of the difference vectorovT−ouT of the
partial sums of partitions  uT=(4,4,1,1,1) and vT=(5,2,2,2,0) as a discrete function of the
position count of the vector components.Keeping in mind that the necessary condition (3.2) for
vector catalysis of the given pair u and v is
fulfilled one would like to find a suitable—possibly minimal—catalyst partition vector c.
This can be done by calculating the ordered tensor product vectors u⊗cT and v⊗cT with all possible partition vectors
of partition diagrams with n≥3 and
test them with respect to majorisation. Although one can exclude systematically
many partitions, this method of testing grows exponentially with increasing n and is only feasible up to n≈150–200 due to this problem of NP complexity.Another
way of finding catalysts is to minimise the negative deviation of the
difference vectorovT−ouT (see Figure 5) which occurs in interval (II), where ou,i≥ov,i. If a
catalyst exists, this difference must “converge” towards zero by comparing the
ordered tensor products of u and v with varying
catalyst vectors c. In
the case of the above example, a catalyst vector cT=(10,6,5,5,4,3,3,3,3)
was found for the limiting case, where the negative deviation vanishes (see Figure 6).Figure 6
The components of the difference vectorov⊗c−ou⊗cforuT=(4,4,1,1,1),  vT=(5,2,2,2,0), and cT=(10,6,5,5,4,3,3,3,3) as a discrete function of  the position count of the vector components.As mentioned above, the proposition (3.2) is more
restrictive than condition (3.1) of Jonathan and Plenio. This can be verified by
the example of the incomparable pair uT=(4,4,1,1)
and vT=(5,2,2,1) with du,v=4 and n=10 [29]. Clearly, u1<v1 and udu,v=vdu,v, and
therefore the necessary condition (3.1) for catalysis is fulfilled. On the other
hand, starting from the partial sums ov=(5,7,9,10)
and ou=(4,8,9,10) one
obtains (3.5)(ov,1=5)>(4=ou,1),interval(I),(ov,2=7)<(8=ou,2),interval(II),(ov,3=9)=(9=ou,3),interval(II),(ov,4=10)=(10=ou,4),interval(II). This means, there is no interval (III) where at least
one partial sum of v is greater than that of u and thus condition (3.2) does not hold. As a consequence, the pair (5, 2, 2, 1) and
(4, 4, 1, 1) should not be catalysable according to proposition (3.2) which agrees
with the results listed in Table 2.Concluding
the discussion of necessary conditions for vector catalysis of an incomparable
pair we would like to give an idea, how proposition (3.2) can be extended.
Starting from the concept of intervals (I), (II),…, where v and u are
alternately “majorising” each other we conjecture that an incomparable pair may
be (nontrivially) catalysable only if the number of these intervals is odd, that
is, it equals 2k+1(k=1,2,…). Conversely, if the number of such intervals is even (2k,k=1,2,…), the corresponding incomparable pair cannot
be catalysed.To get
a symbolic formulation for alternately “majorising” pairs of partition vectors, we suggest the following notation (see also Table7).Notation 1.
If the number of intervals is even, the “sequentially changing
majorisation property” should be denoted byv(≻≺)ku or u(≺≻)kv and in the case of odd intervals by v≻(≺≻)ku or u≺(≻≺)kv with k∈ℕ.
Then, the above conjecture is stated by the following.An incomparable
pair of partition vectors is potentially catalysable only ifv≻(≺≻)ku or, equivalently, u≺(≻≺)kv.A tantamount statement of this supposition is as follows.Proposition 3.2.
An incomparable pair is catalysable only if the number of crossings of the
limiting curve for the difference vectorov−ou of partial sums of the incomparable partitions u and v is even or—equivalently—the number of
nodes of this curve (envelope) is odd.This
proposition is a nice analogy to odd/even rules in chemistry which possibly may
also lead to a deeper understanding of vector catalysis.


### 3.3. Autocatalysis

#### 3.3.1. Reaction Kinetics

Beside
the classical ideas of catalysis as mentioned above, one can observe
self-accelerating chemical processes in solid-solid reactions, such as the
historical tin-refuse reaction or the decomposition of silver oxide (Lewis, 1897). In solutions the Landolt-reaction (1896) obeys a velocity dependence of
the formation of iodide on its own concentration.For such types of
reactions, the idea of autocatalysis is fully accepted. Autocatalysis became a
very powerful heuristic idea with the concept of self-organisation introduced by
Prigogine [56, 57] in order to describe chemical oscillations in
homogenous fluid systems and enzymatic reactions assuming autocatalytic
reaction steps like (3.6)A+X⟶k22X or (3.7)A+2X⟶k33X, where k2
and k3 are reaction rate constants.This symbolic
formulation of chemical reactions is expressed in the nonlinear kinetics by(3.8)dXdt=k2AX,dXdt=k3AX2,  respectively, by
introducing a dissipative term into the dynamics. The formal equations (3.6) and
(3.7) are not really chemical equations in the classical sense. Only in their
reduced form (3.9) they fulfill the rules for writing down chemical equations, (3.9)A⟶k1X. This reduced
equation will give a conservative dynamics (3.10), (3.10)dXdt=k1A, which
is not in agreement with the typical structure formation as can be observed in
dissipative chemical systems. It seems that the so-called autocatalytic
equations (3.6) and (3.7) are auxiliary constructions in order to introduce
dissipative terms in the dynamics by typical methods to derive chemical
kinetics.Both
autocatalytic reactions equations (3.6) and (3.7) seem to accentuate the classical
idea of catalysis, but the differential equations (3.8) of the correlated
chemical kinetics indicate that it is the dynamics of the reaction which is
catalysing because of the nonlinear dissipation terms. This is in total
agreement with the ideas of Abel [58] that reactions are catalysing and generating
autocatalysis as well.


#### 3.3.2. Mathematical Classification

Similarly, one
can define autocatalysis in vector catalysis, if one of the partitionsu or v of the incomparable pair u≉v is identical with the catalysing partition
vector, (3.11)u⊗u≈u⊗v,v⊗v≈u⊗v,withu≉v. As in
the discussion up to now we choose without loss of generality the partition
vector v as the one which is able to majorise u by
forming the respective tensor products.Several
kinds of autocatalysis can be distinguished.(i)
The catalyst partition is identical to the partition
vectorv, the resulting ordered tensor product vector v⊗v of which is majorising the other tensor
product vector u⊗v (3.12). We suggest to call this type Master-Autocatalysis: (3.12)Master-Autocatalysis:=(u≉v)⋀(u⊗v≺v⊗v). Our calculations
show that the first diagram or partition lattice, where Master-autocatalysis occurs
is that of natural number 16 (see Table 3).(ii)
The catalyst partition is identical
to the partition vectoru, the resulting ordered tensor product
vector u⊗u of which is majorised by the other tensor
product vector u⊗v (3.13). For this type we propose the term Slave-Autocatalysis: (3.13)Slave-Autocatalysis:=(u≉v)⋀(u⊗u≺u⊗v). Slave-Autocatalysis
occurs for the first time in the partition lattice with n=16, as can be shown
by our calculations
(see Table 3).(iii)
Both partition vectorsu and v act as catalyst partition, that is, the resulting ordered
tensor product vector v⊗v is majorising the other tensor product vector u⊗v and—simultaneously—the ordered tensor product vector u⊗u is majorised by u⊗v (3.14). In other word, Master-Autocatalysis and
Slave-Autocatalysis occur concurrently. For this type we propose the term  Mutual
Autocatalysis: (3.14)MutalAutocatalysis:=(u≉v)⋀(u⊗v≺v⊗v)⋀(u⊗u≺u⊗v). According to our
calculations, Mutual Autocatalysis can be found for the first time in partition
lattices with n=22.Table 3
Incomparable pairs of partition vectorsu and v which exhibit Master- or Slave-Autocatalysis
and are contained in partition lattices up to n=17, and incomparable pairs
showing Mutual Autocatalysis of partition lattice with n=22. In addition, the
type of autocatalysis and the minimal catalyst vector  cmin for u and v are listed.
nuTvTAutocatalytic partitionType of autocatalysiscminT15(6, 3, 3, 1, 1, 1)(7, 2, 2, 2, 2, 0)uSlave(2, 1, 1)16(6, 3, 3, 1, 1, 1, 1)(7, 2, 2, 2, 2, 1, 0)uSlave(2, 1, 1)16(4, 4, 4, 1, 1, 1, 1)(6, 3, 2, 2, 2, 1, 0)vMaster(2, 1)16(5, 3, 3, 1, 1, 1, 1, 1)(6, 2, 2, 2, 2, 2, 0, 0)uSlave(2, 1, 1)17(7, 4, 3, 1, 1, 1)(9, 2, 2, 2, 2, 0)uSlave(2, 1)17(6, 3, 3, 2, 1, 1, 1)(7, 2, 2, 2, 2, 2, 0)uSlave(2, 1, 1)17(6, 3, 3, 1, 1, 1, 1, 1)(7, 2, 2, 2, 2, 2, 0, 0)uSlave(2, 1, 1)17(5, 4, 3, 2, 1, 1, 1)(7, 2, 2, 2, 2, 2, 0)uSlave(2, 1)17(6, 3, 3, 1, 1, 1, 1, 1)(7, 2, 2, 2, 2, 1, 1, 0)uSlave(2, 1, 1)17(4, 4, 4, 1, 1, 1, 1, 1)(6, 3, 2, 2, 2, 2, 0, 0)vMaster(2, 1)17(4, 4, 4, 1, 1, 1, 1, 1)(6, 3, 2, 2, 2, 1, 1, 0)vMaster(2, 1)17(5, 3, 3, 1, 1, 1, 1, 1, 1)(6, 2, 2, 2, 2, 2, 1, 0, 0)uSlave(2, 1, 1)22(7, 5, 4, 3, 1, 1, 1)(10, 4, 2, 2, 2, 2, 0)u, vMutual(2, 1)22(8, 5, 3, 3, 1, 1, 1)(10, 3, 3, 2, 2, 2, 0)u, vMutual(2, 1, 1)22(8, 4, 4, 3, 1, 1, 1)(10, 3, 3, 2, 2, 2, 0)u, vMutual(2, 1)22(6, 5, 3, 3, 1, 1, 1, 1, 1)(9, 3, 2, 2, 2, 2, 2, 0, 0)u, vMutual(2, 1)22(6, 4, 4, 3, 1, 1, 1, 1, 1)(9, 3, 2, 2, 2, 2, 2, 0, 0)u, vMutual(2, 1)22(6, 5, 3, 3, 1, 1, 1, 1, 1)(9, 3, 2, 2, 2, 2, 1, 1, 0)u, vMutual(2, 1)Table3 lists all autocatalytic pairs which can be found in
partition lattices from n=15 to n=17 as well as all mutually
autocatalytic pairs for n=22.As can be seen from Table3, surprisingly the minimal nonautocatalytic catalyst
partition vector cminT is always (2, 1) or (2, 1, 1), that is, its
dimension dc,min and its trace mmin is much
smaller than that of the autocatalysts. This means that simple vector catalysis
with a catalyst vector c≠u,v can take place in a space of much lower
dimensionality as compared to autocatalysis.Recently, Uhlman
[59] suggested to use the term “autocatalysis” for
entanglement transformations of multiple copies of bipartite quantum states
(MLOCC or multiple-copy local operation and classical communication), a
phenomenon in quantum informatics which was discovered by Bandyopadhyay et
al. [60]. We suggest to call this type Multiple-Copy Autocatalysis (3.15), (3.15)Multiple-CopyAutocatalysis:=(u≉v)⋀{∃(p∈ℕ)>1,suchthat:u⊗p≈v⊗p}, where u⊗p and v⊗p denote tensor exponentiation
by the exponent p, that is, u and v, respectively, are p-times multiplied by tensor products with itself (3.16), (3.16)u⊗p=u⊗u⊗…⊗u︸p-times,v⊗q=v⊗v⊗…⊗v︸p-times. Note that the exponentiation with p = 0 yields the unity
vector (2.3) with one component equal to “1.”Using tensor
exponentiation our idea of Master- and Slave-Autocatalysis in vector catalysis
can be generalised by (3.17), (3.17)(u≉v)⋀{∃(p∈ℕ)>1,suchthat:u⊗p≈u⊗(p−1)⊗v}. A combination of
both concepts of autocatalysis leads to (3.18) and will be denoted by Unified
Autocatalysis. (3.18)UnifiedAutocatalysis:=(u≉v)⋀{∃p,q,r,s∈ℕ0suchthat:(p+q=r+s≥2)⋀(u⊗p⊗v⊗q≈u⊗r⊗v⊗s)}. The individual cases of
autocatalysis can be derived from (3.18) by setting the exponents p, q, r, and s as shown in Table 4.Table 4
Conditions for special cases of autocatalysis derived from unified autocatalysis.
Type of autocatalysisConditions for exponentsp, q, r, and sMaster-autocatalysis(p=1)⋀(q≥1)⋀(r=0)⋀(s=q+1)Slave-autocatalysis(p≥2)⋀(q=0)⋀(r=p−1)⋀(s=1)Multiple-copy autocatalysis(p=s≥2)⋀(q=r=0)Allowingp+q=r+s=1 for an arbitrary vector pair u and v, one gets (classical) majorisation.


#### 3.3.3. Relation to Chemistry

Let us
identify the reaction arrow “→” in formal chemical kinetics with
the majorisation relation symbol “≺.” This approach corresponds to the
observation that common spontaneous reactions occur towards higher disorder and
thus in downward direction within the diagram lattices of Ruch [48].Then
the classical autocatalytic reaction equation (3.6) is related to (3.13) of Slave-Autocatalysis
in the mathematical framework. Analogously, one can formulate formal kinetic
equations for the other types of autocatalysis: (3.19)Master-Autocatalysis:2A⟶k4A+X,(3.20)MutualAutocatalysis:Both2A⟶k4A+X,A+X⟶k22Xhold, where k2 and k4
correspond to different
reaction conditions, (3.21)Multiple-CopyAutocatalysis:pA⟶kmpX,(3.22)UnifiedAutocatalysis:pA+qX⟶kurA+sX. A
typical example for autocatalysis is the so-called tin pest, that is, the
conversion of white β-tin into grey α-tin [61, 62]. The velocity of this autocatalytic reaction is—apart from temperature—depending on size and shape of the growing
seeds. Such reactions are also called collectively autocatalytic [63], that is, the reactions produce themselves as
reaction products such that the entire set of reactions is self sustaining and
accelerating. Without going into details, we suggest that the kinetic equations
(3.6) and (3.19) to (3.22) may serve as an adequate description of such processes.A similar
phenomenon can be observed in case of heterogeneous catalysis which takes place
by processes of adsorption from the gas phase, reaction at solids or solid
surfaces and desorption into the gas phase. During these processes adsorption
clusters are formed which are the reactive units of catalysis. These units can
be local adsorption complexes at “active centres,” clusters at larger surface
domains, the adsorption layer or even the adsorbate together with the bulk [64]. The catalytic properties are strongly depending upon
the size of the reactive units which may be considered as an accelerating
autocatalytic set combined with structure formation. In other words, the
reaction is determined by the distribution function of the adsorption clusters
which catalyse theirs own formation and decay.As a
particular example, one may consider the oxidation of CO on Palladium supported
catalysts [13–15, 21, 22, 23, 24, 25, 26, 27]. By model studies one can show that the phase
transition between Pd and PdO occurs faster with decreasing size of the
catalytic Pd clusters [13–15]. As a result, it comes to resonance phenomena between
catalysts of different sizes whereupon self-affine patterns are formed the overall shape of which
depends on the particle size distribution. The synchronised CO oxidation as
well as the fast loss of catalytic activity in the oxidised state of the
catalyst particles is indicative for autocatalytic processes. A detailed
description of this model is given in [13–15].


### 3.4. Counting Properties of Partition Lattices forn=1 to n=20

To give an
impression of the complexity of  the
discrete approach to vector catalysis, in Table5 some counting properties of partition lattices up to n=20 are given. All characteristics listed in this table are increasing
exponentially due to NP complexity of the problem.Table 5
Counting values of partition vectorsu and v and their properties concerning vector catalysis depending on the natural number n of the partition lattice.
nNNu,v=(N2)Nu−vNu≉vNu(pot)vNu(c≤20)vNu(auto)vNu(Master)vNu(Slave)v11————————2211——————3332——————45104——————57216——————61155122—————715105174—————8222312815—————9304354135—————10428616285—————11561 540871701————12772 92612838541———131015 050175712115———141359 0452471 4052814———1517615 4003352 58369341—11623126 5654594 771149743121729743 9566118 3073131618261838573 92082414 7986303242772019490119 8051 08124 9761 21563178255320627196 2511 43042 8052
2861 19217157114The following
notations are used in Table5.(i)
N: No. of
partition vectors u and v with tr(u)=tr(v)=n which is identical to the no. of diagrams of
Ruch’s diagram lattices with natural number n;(ii)
Nu,v: No. of all
possible pairs of partition vectors u and v in a
partition lattice with natural no. n;(iii)
Nu−v: No. of all
adjacent pairs in a partition lattice with natural no. n, that is, couples
of partition vectors u and v, where one of them
is directly majorising the other;(iv)
Nu≉v: No. of all
incomparable pairs of partition vectors u and v in a partition lattice with natural no. n;(v)
Nu(pot)v: No. of all
incomparable pairs of partition vectors u and v in a partition lattice with natural no. n which are potentially
catalysable, that is, which fulfill the condition v≻(≺≻)ku with k∈ℕ (see Section 3.2 and Table 7);(vi)
Nu(c≤20)v: No. of all
incomparable pairs of partition vectors u and v in a partition lattice with natural number n which are catalysed by a
catalyst partition vector c of a partition lattices with n≤20;(vii)
Nu(auto)v: No.  of all incomparable pairs of partition
vectors u and v in a partition lattice with
natural no. n which are
autocatalysed according to (3.12) or (3.13) by one of the partition
vectors of the incomparable pair;(viii)
Nu(Master)v: No.  of all incomparable pairs of partition
vectors u and v in a partition lattice with natural
no. n which show Master-Autocatalysis
according to (3.12);(ix)
Nu(Slave)v: No.  of all incomparable pairs of partition
vectors u and v in a partition lattice with
natural no. n which show Slave-Autocatalysis
according to (3.13).The number series of Table5 can roughly be approximated by the power law (3.23)y=aebn0.5orlny=lna+bn0.5, where y are the least square fits of each of the counting
values as a function of n with R2
values of about
0.9999. The coefficients of these fits are listed in Table 6.Table 6
Counting values of partition vectorsu and v and their properties
concerning vector catalysis depending on the natural number n of the partition lattice.
Counting valuealn(a)bR2N5.2 e−02−3.02.10.99992Nu,v=(N2)1.0 e−03−6.94.30.99996Nu−v2.9 e−02−3.52.40.99997Nu≉v3.5 e−05−10.24.70.99996Nu(pot)v2.3 e−08−17.65.70.99992Nu(c≤20)v9.5 e−09−18.55.70.99989Nu(auto)v*2.4 e−11−24.56.60.99984Nu(Master)v*3.3 e−13−28.77.30.99997Nu(Slave)v*1.2 e−10−22.86.20.99941*Counting values were calculated up to n=24.Table 7
Notations for pairs of partition vectors and majorisation.
n∈ℕPositive integer (natural number) to be partitioned.u=(u1u2⋮udu),v=(v1v2⋮vdv)Discrete
partition vectors (column vectors) with nonnegative integersui and vi as components (i,ui,vi∈ℕ).
They are used as pair of partition vectors to be compared with respect
to majorisation and discrete vector catalysis.
The following conventions are
assumed. If a pair of different vectors u and v (u≠v) is compared with respect to majorisation, u is chosen as the vector with the first occurrence of a component ui<vi.du,dv∈ℕDimensions, that is, number of nonzero components, ofu and v, respectively. In general, du and dv are different.du,v=sup(du,dv)Supremum of dimensionsdu and dv.
The vector with the lower
dimension is filled up with zeros to du,v components.nu=tr(u)=∑j=1du,vuj,nv=tr(v)=∑j=1du,vvj,nu,nv∈ℕTrace
(sum of components) ofu and v, respectively;
(classical)
majorisation,  nu=nv=n.ou=(ou,1ou,2⋮ou,du,v),ov=(ov,1ov,2⋮ov,du,v)ou,i=∑j=1iuj,ov,i=∑j=1ivj,i=1,2,…,du,vVectors of partial sums of partition vectorsu and v, respectively.≺,≻:u≺v≡v≻u:=ou,i≤ov,i,i=1,2,…,du,v(Classical) majorisation relation.
Comparable
pair, whereu is being majorised by v and v is majorising. u
In quantum informatics the corresponding transition of entangled
states is possible with certainty by LOCC (local operation and classical communication).u≈v:=(u≺v)∨(u≻v)Pair of partition vectors which is comparable with
respect to majorisation.u≉v=¬(u≈v)Pair of partition vectors which is incomparable with respect to
majorisation.u≺≻v≡v≻≺uIncomparable pair, where bothu and v, respectively, are divided in two subsections of contiguous components, such
that in the first section all partial sums ov of v are greater or equal to ou of ou and in
the second section vice versa.
(For this and the following definitions see also Section 3.2).u(≺≻)kv≡v(≻≺)ku,k∈ℕSame asu≺≻v but changing subsections of components are repeated k times; in case of k=1 one obtains the notation above.u≺≻≺v≡v≻≺≻uIncomparable pair, where bothu and v, respectively, are divided in three subsections of contiguous components, such
that in the first and last (third) section all partial sums ov of v are greater or equal to ou of u and in the second section vice versa.u≺(≻≺)kv≡v≻(≺≻)ku,k∈ℕ0Same asu≺≻≺v but changing subsections of components are repeated
k times; in case of k=1 one obtains the notation above, in case of k=0 one
gets the majorisation relation ≻(≺).Table 8
Notations for tensor operations and discrete vector catalysis.
c=(c1c2⋮cdc)cT=(c1c2…cdc)Catalyst partition vector, represented as column vectorc or as row vector cT;
it will be used for tensor multiplication with u or v, respectively, in vector catalysis.dc∈ℕDimension (number of nonzero components) of catalyst partition
vectorc.m=tr(c)=∑j=1dccj,m∈ℕTrace (sum of components) of catalyst partition vectorc.⊗Symbol for tensor product relation.u⊗cT,v⊗cT (vector product notation)Tensor products ofu with c(cT)
and v with c(cT), respectively, in both cases represented as a (du,v×dc)-matrix.u⊗c,v⊗c (tensor product notation)θu⊗c,θv⊗cVectors of tensor productsu⊗c and v⊗c, respectively, where all components (tensor product elements) are arranged in
decreasing (nonincreasing) order.du⊗c=dv⊗c=du,v⋅dcDimension (number of components) of tensor product vectors, that
is, dimensiondu,v of partition pair vectors multiplied
with dimension dc of catalyst vector.nu⊗c=nv⊗c=n⋅mTrace (sum of components) of tensor product vectors, that is, trace
of partition vectors multiplied with trace of catalyst vector.ou⊗c,ov⊗cVectors of partial sums of tensor product vectorsθu⊗c and θv⊗c, respectively.u⊗p=u⊗u⊗…⊗u︸p-times,v⊗q=v⊗v⊗…⊗v︸q-times,p,q∈ℕ0Tensor exponentiation of partition vectorsu and v with exponents p and q, respectively. Special cases,  u⊗1=u,v⊗1=v,u⊗0=v⊗0=(1).As can be seen from coefficientsb, the counting numbers of
properties correlated to all and to the incomparable partition pairs are
growing about twice as fast as the number N of partition vectors and the
number Nu−v of directly majorising partitions. The
calculated coefficients b for properties which are attributed to vector
catalysis are even larger, but may be less significant, because a fewer number
of  values are included in the fits; if
one takes a sufficiently large number of counting values the coefficients b for catalysable or catalytic pairs cannot exceed that for incomparable pairs.


## 3.1. Minimal Global Properties of Discrete Vector Catalysis

Concerning minimal global
properties of discrete vector catalysis, one may ask the following questions, the answers of which are listed in Table1.Table 1
Incomparable, catalysable pairs of partition vectors and their corresponding
catalyst partition vectors with distinguished minimal global properties of discrete
vector catalysis. The significant minimal global properties are marked in bold italic letters.
nmn⋅mdu,vdcdu,v⋅dcIncomparable, catalysable pairCatalyst partition vectorof partition vectorsuTvTcT(a)11424825945(4, 4, 1, 1, 1)(5, 2, 2, 2, 0)(10, 6, 5, 5, 4,  3, 3, 3, 3)(b)124104 920438152(5, 5, 1, 1)(6, 3, 3, 0)(36, 24, 20, 20, 16, 16, 15, 15, 12, 12, 12, 12, 10, 10, 10, 10, 9, 9, 9, 9, 8, 8, 8, 8, 8, 8, 8, 8, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6)*(c)14342428(6, 6, 1, 1)(8, 3, 3, 0)(2, 1)(d)14342428(6, 6, 1, 1)(8, 3, 3, 0)(2, 1)*Catalyst partition vector may not be minimal.(a)
What is the
minimal component sum (trace)n, for which one can find an incomparable
pair u and v of partition vectors which can be
catalysed?(b)
What is the
minimal dimensiondu,v of an incomparable pair of partition vectors u and v which can be catalysed and the natural number n of
its first occurrence?(c)
What is the
smallest catalyst partition vectorc and its corresponding pair
of incomparable partition vectors u and v?(d)
What are the
smallest balanced combinations consisting of a catalysable pair of incomparable
partition vectorsu and v and its smallest catalyst c, that is, where either (n⋅m) or (du,v⋅dc) is minimal?Answering question, (a) follows that the smallest numbern for
which one can find an incomparable but catalysable vector pair is n=11, and the corresponding partition vectors are uT=(4,4,1,1,1) and vT=(5,2,2,2,0), respectively. Thus, the first diagram or partition lattice with the possibility
of vector catalysis is that of 11 boxes. The minimal catalyst vector of the
smallest catalysable pair is cT=(10,6,5,5,4,3,3,3,3), which belongs to a lattice of the integer m=42. As a
consequence, both the dimension du,v⋅dc and the trace (no. of boxes) n⋅m of the resulting tensor product space in which
vector catalysis occurs are rather high. This corresponds to the observation in
chemistry that small molecules are often catalysed by large catalysts.This analogy turns out even stronger, if one considers—as in question (b)—the tensor product space of thefirst occurrence
of an incomparable but catalysable pair u and v of
smallest dimension du,v.
Jonathan and Plenio have already proven in their fundamental article on entanglement
catalysis [29] that incomparable states with less than four
components cannot be catalysed, hence the minimal dimension of catalysable
pairs of partition vectors is du,v=4. The first such a pair is uT=(5,5,1,1) and vT=(6,3,3,0)
with n=12. It can be catalysed, but the smallest catalyst which we
found by extended calculations contains dc=38
components and has a trace of m=410 (see Table 1) yielding a tensor product space with du,v⋅dc=152 and n⋅m=4920. Such a highly dimensional space of vector catalysis may possibly be related
to enzyme catalysis in chemistry or biochemistry.On the other hand, the answers to questions (c) and (d), that is, to
the smallest catalyst partition vectorc and to the smallest
balanced combinations of u/v and c lead to a tensor product space of comparatively low dimensionality (du,v⋅dc=8,n⋅m=42).
This result corresponds in chemistry particularly to homogeneous catalysis of
medium-sized molecules by medium-sized catalysts.


## 3.2. Incomparable Vector Pairs and Necessary Conditions for Vector Catalysis

Table2 summarises all incomparable pairs u and v of partition lattices up to n=13 which may be
catalysable together with a (minimal) catalyst partition vector c, if such was found by systematic calculation or empirical investigation. Also, the number of incomparable pairs—denoted by Nc,20—which are catalysed by a catalyst partition
vector of a partition lattice with n≤20 is given in Table 2.Table 2
Incomparable pairs of partition vectors which may be catalysable from partition
lattices up ton=13 and corresponding catalyst partition vectors.
ndu,vuTvTmdccTNc,20115(4, 4, 1, 1, 1)(5, 2, 2, 2, 0)429(10, 6, 5, 5, 4, 3, 3, 3, 3)0124(5, 5, 1, 1)(6, 3, 3, 0)41038(36, 24, 20, 20, 16, 16, 15, 15, 12, 12, 12, 12, 10, 10, 10, 10, 9, 9, 9, 9, 8, 8, 8, 8, 8, 8, 8, 8, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6)*0125(5, 4, 1, 1, 1)(6, 2, 2, 2, 0)43(2, 1, 1)87126(4, 4, 1, 1, 1, 1)(5, 2, 2, 2, 1, 0)429(10, 6, 5, 5, 4, 3, 3, 3, 3)0126(3, 3, 3, 1, 1, 1)(4, 2, 2, 2, 2, 0)375(12, 8, 7, 5, 5)*0134(6, 5, 1, 1)(7, 3, 3, 0)316(10, 6, 5, 4, 3, 3)*0135(6, 4, 1, 1, 1)(7, 2, 2, 2, 0)43(2, 1, 1)120135(5, 5, 1, 1, 1)(7, 2, 2, 2, 0)43(2, 1, 1)229135(5, 5, 1, 1, 1)(6, 3, 3, 1, 0)41038(36, 24, 20, 20, 16, 16, 15, 15, 12, 12, 12, 12, 10, 10, 10, 10, 9, 9, 9, 9, 8, 8, 8, 8, 8, 8, 8, 8, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6)*0135(5, 5, 1, 1, 1)(6, 3, 2, 2, 0)?**?**?**?**136(5, 4, 1, 1, 1, 1)(6, 2, 2, 2, 1, 0)43(2, 1, 1)87136(4, 4, 2, 1, 1, 1)(5, 2, 2, 2, 2, 0)429(10, 6, 5, 5, 4, 3, 3, 3, 3)0137(4, 4, 1, 1, 1, 1, 1)(5, 2, 2, 2, 2, 0, 0)164(6, 4, 3, 3)9136(4, 3, 3, 1, 1, 1)(5, 2, 2, 2, 2, 0)43(2, 1, 1)134137(4, 4, 1, 1, 1, 1, 1)(5, 2, 2, 2, 1, 1, 0)429(10, 6, 5, 5, 4, 3, 3, 3, 3)0137(3, 3, 3, 1, 1, 1, 1)(4, 2, 2, 2, 2, 1, 0)375(12, 8, 7, 5, 5)*0*Catalyst partition vector may not be minimal for the respective pair.
**A catalyst could not yet be found by our investigations. We assume
that a catalyst vector for this pair does not exist.**A catalyst could not yet be found by our investigations. We assume
that a catalyst vector for this pair does not exist.In the case of
partition lattices withn≤10 no
incomparable pair exists which can be catalysed, as will be shown in the following.An important
question is now under which circumstances two incomparable pairs of a partition
lattice with natural numbern can be catalysed by a third partition
vector. A necessary condition was given by Jonathan and Plenio [29] which can be expressed using the terminology of
discrete partition vectors as follows.LetuT=(u1u2…ui…udu,v) and vT=(v1v2…vi…vdu,v) a pair of incomparable partition vectors of
length du,v, then
this pair can be catalysed be a third (catalytic) partition vector only if both (3.1)u1≤v1,udu,v≥vdu,v hold. If one can
find such a catalyst vector, denoted by cT=(c1c2…cdc), the
resulting ordered tensor product v⊗cT majorises
the ordered tensor product u⊗cT.
Accordingly, in the interpretation of quantum information theory the corresponding
entangled quantum states can be transformed with entanglement-assisted local
transformation and classical communication (ELQCC) [29]. As mentioned above, Jonathan and Plenio have also
shown that incomparable partitions of dimension du,v<4 cannot be
catalysed [29].In the following, we suggest a stronger necessary condition for vector catalysis which is based
on the concept of majorisation.Proposition 3.1.
A pair of incomparable partition vectors of dimensiondu,v(du,v≥4)
can be catalysed only if there exist two indices i1 and i2 with 1<i1<i2<du,v such that for the partial sums ou,i and ov,i holds (3.2)(I)ov,i≥ou,ifor1≤i<i1,(II)ov,i<ou,ifori=i1,ov,i≤ou,ifori1+1≤i<i2,(III)ov,i>ou,ifori=i2,ov,i≥ou,ifori2+1≤i≤du,v. In addition, for the du,vth
vector component the majorisation condition (3.3)ov,i=ou,i=n is fulfilled.Although we cannot give yet a strict mathematical
proof, our proposition can be confirmed by the following consideration. In the
case of the largest partition values, that is, interval (I) in (3.2), one
partition vector (v) obviously “majorises” the other (u). Since the pair
is incomparable, there must follow an interval (II) with partition values such
that the second partition vector (u)
“majorises” the first (v).
Finally, in order to enable majorisation after building the ordered tensor
products with a catalyst vector, it seems to be necessary to have a last interval
(III) with the smallest partition values, where the first partition vector (v) takes over
“majorisation” again.For a
pair of incomparable partition vectorsu and v with
these properties, we suggest to introduce the notation “v≻≺≻u” or—equivalently—“u≺≻≺v” (see also Table 7).The condition (3.2) will be illustrated at the example
of the incomparable pair uT=(4,4,1,1,1)
and vT=(5,2,2,2,0) with du,v=5 and n=11. The corresponding partial sums are ouT=(4,8,9,10,11) and ovT=(5,7,9,11,11).
Thus,  (3.4)(ov,1=5)>(4=ou,1),interval(I),(ov,2=7)<(8=ou,2),interval(II),(ov,3=9)=(9=ou,3),interval(II),(ov,4=11)>(10=ou,4),interval(III),(ov,5=11)=(11=ou,5),interval(III), and hence i1=2 and i2=4.A graphical representation of the resulting difference
vectorovT−ouT=(1,−1,0,1,0) is
drawn in Figure 5, where also the interval boundaries i1 and i1 as well as the intervals (I), (II), and (III) are
indicated, respectively. The dotted line shows the limiting curve (envelope) between
the values of the difference of partial sums. Note that this curve is crossing
twice the (dashed) zero line of the diagram indicating a twofold change of sign
of the difference vector ovT−ouT; in
other word, the envelope exhibits three extremal points (nodes).Figure 5
The components of the difference vectorovT−ouT of the
partial sums of partitions  uT=(4,4,1,1,1) and vT=(5,2,2,2,0) as a discrete function of the
position count of the vector components.Keeping in mind that the necessary condition (3.2) for
vector catalysis of the given pair u and v is
fulfilled one would like to find a suitable—possibly minimal—catalyst partition vector c.
This can be done by calculating the ordered tensor product vectors u⊗cT and v⊗cT with all possible partition vectors
of partition diagrams with n≥3 and
test them with respect to majorisation. Although one can exclude systematically
many partitions, this method of testing grows exponentially with increasing n and is only feasible up to n≈150–200 due to this problem of NP complexity.Another
way of finding catalysts is to minimise the negative deviation of the
difference vectorovT−ouT (see Figure 5) which occurs in interval (II), where ou,i≥ov,i. If a
catalyst exists, this difference must “converge” towards zero by comparing the
ordered tensor products of u and v with varying
catalyst vectors c. In
the case of the above example, a catalyst vector cT=(10,6,5,5,4,3,3,3,3)
was found for the limiting case, where the negative deviation vanishes (see Figure 6).Figure 6
The components of the difference vectorov⊗c−ou⊗cforuT=(4,4,1,1,1),  vT=(5,2,2,2,0), and cT=(10,6,5,5,4,3,3,3,3) as a discrete function of  the position count of the vector components.As mentioned above, the proposition (3.2) is more
restrictive than condition (3.1) of Jonathan and Plenio. This can be verified by
the example of the incomparable pair uT=(4,4,1,1)
and vT=(5,2,2,1) with du,v=4 and n=10 [29]. Clearly, u1<v1 and udu,v=vdu,v, and
therefore the necessary condition (3.1) for catalysis is fulfilled. On the other
hand, starting from the partial sums ov=(5,7,9,10)
and ou=(4,8,9,10) one
obtains (3.5)(ov,1=5)>(4=ou,1),interval(I),(ov,2=7)<(8=ou,2),interval(II),(ov,3=9)=(9=ou,3),interval(II),(ov,4=10)=(10=ou,4),interval(II). This means, there is no interval (III) where at least
one partial sum of v is greater than that of u and thus condition (3.2) does not hold. As a consequence, the pair (5, 2, 2, 1) and
(4, 4, 1, 1) should not be catalysable according to proposition (3.2) which agrees
with the results listed in Table 2.Concluding
the discussion of necessary conditions for vector catalysis of an incomparable
pair we would like to give an idea, how proposition (3.2) can be extended.
Starting from the concept of intervals (I), (II),…, where v and u are
alternately “majorising” each other we conjecture that an incomparable pair may
be (nontrivially) catalysable only if the number of these intervals is odd, that
is, it equals 2k+1(k=1,2,…). Conversely, if the number of such intervals is even (2k,k=1,2,…), the corresponding incomparable pair cannot
be catalysed.To get
a symbolic formulation for alternately “majorising” pairs of partition vectors, we suggest the following notation (see also Table7).Notation 1.
If the number of intervals is even, the “sequentially changing
majorisation property” should be denoted byv(≻≺)ku or u(≺≻)kv and in the case of odd intervals by v≻(≺≻)ku or u≺(≻≺)kv with k∈ℕ.
Then, the above conjecture is stated by the following.An incomparable
pair of partition vectors is potentially catalysable only ifv≻(≺≻)ku or, equivalently, u≺(≻≺)kv.A tantamount statement of this supposition is as follows.Proposition 3.2.
An incomparable pair is catalysable only if the number of crossings of the
limiting curve for the difference vectorov−ou of partial sums of the incomparable partitions u and v is even or—equivalently—the number of
nodes of this curve (envelope) is odd.This
proposition is a nice analogy to odd/even rules in chemistry which possibly may
also lead to a deeper understanding of vector catalysis.


## 3.3. Autocatalysis

### 3.3.1. Reaction Kinetics

Beside
the classical ideas of catalysis as mentioned above, one can observe
self-accelerating chemical processes in solid-solid reactions, such as the
historical tin-refuse reaction or the decomposition of silver oxide (Lewis, 1897). In solutions the Landolt-reaction (1896) obeys a velocity dependence of
the formation of iodide on its own concentration.For such types of
reactions, the idea of autocatalysis is fully accepted. Autocatalysis became a
very powerful heuristic idea with the concept of self-organisation introduced by
Prigogine [56, 57] in order to describe chemical oscillations in
homogenous fluid systems and enzymatic reactions assuming autocatalytic
reaction steps like (3.6)A+X⟶k22X or (3.7)A+2X⟶k33X, where k2
and k3 are reaction rate constants.This symbolic
formulation of chemical reactions is expressed in the nonlinear kinetics by(3.8)dXdt=k2AX,dXdt=k3AX2,  respectively, by
introducing a dissipative term into the dynamics. The formal equations (3.6) and
(3.7) are not really chemical equations in the classical sense. Only in their
reduced form (3.9) they fulfill the rules for writing down chemical equations, (3.9)A⟶k1X. This reduced
equation will give a conservative dynamics (3.10), (3.10)dXdt=k1A, which
is not in agreement with the typical structure formation as can be observed in
dissipative chemical systems. It seems that the so-called autocatalytic
equations (3.6) and (3.7) are auxiliary constructions in order to introduce
dissipative terms in the dynamics by typical methods to derive chemical
kinetics.Both
autocatalytic reactions equations (3.6) and (3.7) seem to accentuate the classical
idea of catalysis, but the differential equations (3.8) of the correlated
chemical kinetics indicate that it is the dynamics of the reaction which is
catalysing because of the nonlinear dissipation terms. This is in total
agreement with the ideas of Abel [58] that reactions are catalysing and generating
autocatalysis as well.


### 3.3.2. Mathematical Classification

Similarly, one
can define autocatalysis in vector catalysis, if one of the partitionsu or v of the incomparable pair u≉v is identical with the catalysing partition
vector, (3.11)u⊗u≈u⊗v,v⊗v≈u⊗v,withu≉v. As in
the discussion up to now we choose without loss of generality the partition
vector v as the one which is able to majorise u by
forming the respective tensor products.Several
kinds of autocatalysis can be distinguished.(i)
The catalyst partition is identical to the partition
vectorv, the resulting ordered tensor product vector v⊗v of which is majorising the other tensor
product vector u⊗v (3.12). We suggest to call this type Master-Autocatalysis: (3.12)Master-Autocatalysis:=(u≉v)⋀(u⊗v≺v⊗v). Our calculations
show that the first diagram or partition lattice, where Master-autocatalysis occurs
is that of natural number 16 (see Table 3).(ii)
The catalyst partition is identical
to the partition vectoru, the resulting ordered tensor product
vector u⊗u of which is majorised by the other tensor
product vector u⊗v (3.13). For this type we propose the term Slave-Autocatalysis: (3.13)Slave-Autocatalysis:=(u≉v)⋀(u⊗u≺u⊗v). Slave-Autocatalysis
occurs for the first time in the partition lattice with n=16, as can be shown
by our calculations
(see Table 3).(iii)
Both partition vectorsu and v act as catalyst partition, that is, the resulting ordered
tensor product vector v⊗v is majorising the other tensor product vector u⊗v and—simultaneously—the ordered tensor product vector u⊗u is majorised by u⊗v (3.14). In other word, Master-Autocatalysis and
Slave-Autocatalysis occur concurrently. For this type we propose the term  Mutual
Autocatalysis: (3.14)MutalAutocatalysis:=(u≉v)⋀(u⊗v≺v⊗v)⋀(u⊗u≺u⊗v). According to our
calculations, Mutual Autocatalysis can be found for the first time in partition
lattices with n=22.Table 3
Incomparable pairs of partition vectorsu and v which exhibit Master- or Slave-Autocatalysis
and are contained in partition lattices up to n=17, and incomparable pairs
showing Mutual Autocatalysis of partition lattice with n=22. In addition, the
type of autocatalysis and the minimal catalyst vector  cmin for u and v are listed.
nuTvTAutocatalytic partitionType of autocatalysiscminT15(6, 3, 3, 1, 1, 1)(7, 2, 2, 2, 2, 0)uSlave(2, 1, 1)16(6, 3, 3, 1, 1, 1, 1)(7, 2, 2, 2, 2, 1, 0)uSlave(2, 1, 1)16(4, 4, 4, 1, 1, 1, 1)(6, 3, 2, 2, 2, 1, 0)vMaster(2, 1)16(5, 3, 3, 1, 1, 1, 1, 1)(6, 2, 2, 2, 2, 2, 0, 0)uSlave(2, 1, 1)17(7, 4, 3, 1, 1, 1)(9, 2, 2, 2, 2, 0)uSlave(2, 1)17(6, 3, 3, 2, 1, 1, 1)(7, 2, 2, 2, 2, 2, 0)uSlave(2, 1, 1)17(6, 3, 3, 1, 1, 1, 1, 1)(7, 2, 2, 2, 2, 2, 0, 0)uSlave(2, 1, 1)17(5, 4, 3, 2, 1, 1, 1)(7, 2, 2, 2, 2, 2, 0)uSlave(2, 1)17(6, 3, 3, 1, 1, 1, 1, 1)(7, 2, 2, 2, 2, 1, 1, 0)uSlave(2, 1, 1)17(4, 4, 4, 1, 1, 1, 1, 1)(6, 3, 2, 2, 2, 2, 0, 0)vMaster(2, 1)17(4, 4, 4, 1, 1, 1, 1, 1)(6, 3, 2, 2, 2, 1, 1, 0)vMaster(2, 1)17(5, 3, 3, 1, 1, 1, 1, 1, 1)(6, 2, 2, 2, 2, 2, 1, 0, 0)uSlave(2, 1, 1)22(7, 5, 4, 3, 1, 1, 1)(10, 4, 2, 2, 2, 2, 0)u, vMutual(2, 1)22(8, 5, 3, 3, 1, 1, 1)(10, 3, 3, 2, 2, 2, 0)u, vMutual(2, 1, 1)22(8, 4, 4, 3, 1, 1, 1)(10, 3, 3, 2, 2, 2, 0)u, vMutual(2, 1)22(6, 5, 3, 3, 1, 1, 1, 1, 1)(9, 3, 2, 2, 2, 2, 2, 0, 0)u, vMutual(2, 1)22(6, 4, 4, 3, 1, 1, 1, 1, 1)(9, 3, 2, 2, 2, 2, 2, 0, 0)u, vMutual(2, 1)22(6, 5, 3, 3, 1, 1, 1, 1, 1)(9, 3, 2, 2, 2, 2, 1, 1, 0)u, vMutual(2, 1)Table3 lists all autocatalytic pairs which can be found in
partition lattices from n=15 to n=17 as well as all mutually
autocatalytic pairs for n=22.As can be seen from Table3, surprisingly the minimal nonautocatalytic catalyst
partition vector cminT is always (2, 1) or (2, 1, 1), that is, its
dimension dc,min and its trace mmin is much
smaller than that of the autocatalysts. This means that simple vector catalysis
with a catalyst vector c≠u,v can take place in a space of much lower
dimensionality as compared to autocatalysis.Recently, Uhlman
[59] suggested to use the term “autocatalysis” for
entanglement transformations of multiple copies of bipartite quantum states
(MLOCC or multiple-copy local operation and classical communication), a
phenomenon in quantum informatics which was discovered by Bandyopadhyay et
al. [60]. We suggest to call this type Multiple-Copy Autocatalysis (3.15), (3.15)Multiple-CopyAutocatalysis:=(u≉v)⋀{∃(p∈ℕ)>1,suchthat:u⊗p≈v⊗p}, where u⊗p and v⊗p denote tensor exponentiation
by the exponent p, that is, u and v, respectively, are p-times multiplied by tensor products with itself (3.16), (3.16)u⊗p=u⊗u⊗…⊗u︸p-times,v⊗q=v⊗v⊗…⊗v︸p-times. Note that the exponentiation with p = 0 yields the unity
vector (2.3) with one component equal to “1.”Using tensor
exponentiation our idea of Master- and Slave-Autocatalysis in vector catalysis
can be generalised by (3.17), (3.17)(u≉v)⋀{∃(p∈ℕ)>1,suchthat:u⊗p≈u⊗(p−1)⊗v}. A combination of
both concepts of autocatalysis leads to (3.18) and will be denoted by Unified
Autocatalysis. (3.18)UnifiedAutocatalysis:=(u≉v)⋀{∃p,q,r,s∈ℕ0suchthat:(p+q=r+s≥2)⋀(u⊗p⊗v⊗q≈u⊗r⊗v⊗s)}. The individual cases of
autocatalysis can be derived from (3.18) by setting the exponents p, q, r, and s as shown in Table 4.Table 4
Conditions for special cases of autocatalysis derived from unified autocatalysis.
Type of autocatalysisConditions for exponentsp, q, r, and sMaster-autocatalysis(p=1)⋀(q≥1)⋀(r=0)⋀(s=q+1)Slave-autocatalysis(p≥2)⋀(q=0)⋀(r=p−1)⋀(s=1)Multiple-copy autocatalysis(p=s≥2)⋀(q=r=0)Allowingp+q=r+s=1 for an arbitrary vector pair u and v, one gets (classical) majorisation.


### 3.3.3. Relation to Chemistry

Let us
identify the reaction arrow “→” in formal chemical kinetics with
the majorisation relation symbol “≺.” This approach corresponds to the
observation that common spontaneous reactions occur towards higher disorder and
thus in downward direction within the diagram lattices of Ruch [48].Then
the classical autocatalytic reaction equation (3.6) is related to (3.13) of Slave-Autocatalysis
in the mathematical framework. Analogously, one can formulate formal kinetic
equations for the other types of autocatalysis: (3.19)Master-Autocatalysis:2A⟶k4A+X,(3.20)MutualAutocatalysis:Both2A⟶k4A+X,A+X⟶k22Xhold, where k2 and k4
correspond to different
reaction conditions, (3.21)Multiple-CopyAutocatalysis:pA⟶kmpX,(3.22)UnifiedAutocatalysis:pA+qX⟶kurA+sX. A
typical example for autocatalysis is the so-called tin pest, that is, the
conversion of white β-tin into grey α-tin [61, 62]. The velocity of this autocatalytic reaction is—apart from temperature—depending on size and shape of the growing
seeds. Such reactions are also called collectively autocatalytic [63], that is, the reactions produce themselves as
reaction products such that the entire set of reactions is self sustaining and
accelerating. Without going into details, we suggest that the kinetic equations
(3.6) and (3.19) to (3.22) may serve as an adequate description of such processes.A similar
phenomenon can be observed in case of heterogeneous catalysis which takes place
by processes of adsorption from the gas phase, reaction at solids or solid
surfaces and desorption into the gas phase. During these processes adsorption
clusters are formed which are the reactive units of catalysis. These units can
be local adsorption complexes at “active centres,” clusters at larger surface
domains, the adsorption layer or even the adsorbate together with the bulk [64]. The catalytic properties are strongly depending upon
the size of the reactive units which may be considered as an accelerating
autocatalytic set combined with structure formation. In other words, the
reaction is determined by the distribution function of the adsorption clusters
which catalyse theirs own formation and decay.As a
particular example, one may consider the oxidation of CO on Palladium supported
catalysts [13–15, 21, 22, 23, 24, 25, 26, 27]. By model studies one can show that the phase
transition between Pd and PdO occurs faster with decreasing size of the
catalytic Pd clusters [13–15]. As a result, it comes to resonance phenomena between
catalysts of different sizes whereupon self-affine patterns are formed the overall shape of which
depends on the particle size distribution. The synchronised CO oxidation as
well as the fast loss of catalytic activity in the oxidised state of the
catalyst particles is indicative for autocatalytic processes. A detailed
description of this model is given in [13–15].


## 3.3.1. Reaction Kinetics

Beside
the classical ideas of catalysis as mentioned above, one can observe
self-accelerating chemical processes in solid-solid reactions, such as the
historical tin-refuse reaction or the decomposition of silver oxide (Lewis, 1897). In solutions the Landolt-reaction (1896) obeys a velocity dependence of
the formation of iodide on its own concentration.For such types of
reactions, the idea of autocatalysis is fully accepted. Autocatalysis became a
very powerful heuristic idea with the concept of self-organisation introduced by
Prigogine [56, 57] in order to describe chemical oscillations in
homogenous fluid systems and enzymatic reactions assuming autocatalytic
reaction steps like (3.6)A+X⟶k22X or (3.7)A+2X⟶k33X, where k2
and k3 are reaction rate constants.This symbolic
formulation of chemical reactions is expressed in the nonlinear kinetics by(3.8)dXdt=k2AX,dXdt=k3AX2,  respectively, by
introducing a dissipative term into the dynamics. The formal equations (3.6) and
(3.7) are not really chemical equations in the classical sense. Only in their
reduced form (3.9) they fulfill the rules for writing down chemical equations, (3.9)A⟶k1X. This reduced
equation will give a conservative dynamics (3.10), (3.10)dXdt=k1A, which
is not in agreement with the typical structure formation as can be observed in
dissipative chemical systems. It seems that the so-called autocatalytic
equations (3.6) and (3.7) are auxiliary constructions in order to introduce
dissipative terms in the dynamics by typical methods to derive chemical
kinetics.Both
autocatalytic reactions equations (3.6) and (3.7) seem to accentuate the classical
idea of catalysis, but the differential equations (3.8) of the correlated
chemical kinetics indicate that it is the dynamics of the reaction which is
catalysing because of the nonlinear dissipation terms. This is in total
agreement with the ideas of Abel [58] that reactions are catalysing and generating
autocatalysis as well.


## 3.3.2. Mathematical Classification

Similarly, one
can define autocatalysis in vector catalysis, if one of the partitionsu or v of the incomparable pair u≉v is identical with the catalysing partition
vector, (3.11)u⊗u≈u⊗v,v⊗v≈u⊗v,withu≉v. As in
the discussion up to now we choose without loss of generality the partition
vector v as the one which is able to majorise u by
forming the respective tensor products.Several
kinds of autocatalysis can be distinguished.(i)
The catalyst partition is identical to the partition
vectorv, the resulting ordered tensor product vector v⊗v of which is majorising the other tensor
product vector u⊗v (3.12). We suggest to call this type Master-Autocatalysis: (3.12)Master-Autocatalysis:=(u≉v)⋀(u⊗v≺v⊗v). Our calculations
show that the first diagram or partition lattice, where Master-autocatalysis occurs
is that of natural number 16 (see Table 3).(ii)
The catalyst partition is identical
to the partition vectoru, the resulting ordered tensor product
vector u⊗u of which is majorised by the other tensor
product vector u⊗v (3.13). For this type we propose the term Slave-Autocatalysis: (3.13)Slave-Autocatalysis:=(u≉v)⋀(u⊗u≺u⊗v). Slave-Autocatalysis
occurs for the first time in the partition lattice with n=16, as can be shown
by our calculations
(see Table 3).(iii)
Both partition vectorsu and v act as catalyst partition, that is, the resulting ordered
tensor product vector v⊗v is majorising the other tensor product vector u⊗v and—simultaneously—the ordered tensor product vector u⊗u is majorised by u⊗v (3.14). In other word, Master-Autocatalysis and
Slave-Autocatalysis occur concurrently. For this type we propose the term  Mutual
Autocatalysis: (3.14)MutalAutocatalysis:=(u≉v)⋀(u⊗v≺v⊗v)⋀(u⊗u≺u⊗v). According to our
calculations, Mutual Autocatalysis can be found for the first time in partition
lattices with n=22.Table 3
Incomparable pairs of partition vectorsu and v which exhibit Master- or Slave-Autocatalysis
and are contained in partition lattices up to n=17, and incomparable pairs
showing Mutual Autocatalysis of partition lattice with n=22. In addition, the
type of autocatalysis and the minimal catalyst vector  cmin for u and v are listed.
nuTvTAutocatalytic partitionType of autocatalysiscminT15(6, 3, 3, 1, 1, 1)(7, 2, 2, 2, 2, 0)uSlave(2, 1, 1)16(6, 3, 3, 1, 1, 1, 1)(7, 2, 2, 2, 2, 1, 0)uSlave(2, 1, 1)16(4, 4, 4, 1, 1, 1, 1)(6, 3, 2, 2, 2, 1, 0)vMaster(2, 1)16(5, 3, 3, 1, 1, 1, 1, 1)(6, 2, 2, 2, 2, 2, 0, 0)uSlave(2, 1, 1)17(7, 4, 3, 1, 1, 1)(9, 2, 2, 2, 2, 0)uSlave(2, 1)17(6, 3, 3, 2, 1, 1, 1)(7, 2, 2, 2, 2, 2, 0)uSlave(2, 1, 1)17(6, 3, 3, 1, 1, 1, 1, 1)(7, 2, 2, 2, 2, 2, 0, 0)uSlave(2, 1, 1)17(5, 4, 3, 2, 1, 1, 1)(7, 2, 2, 2, 2, 2, 0)uSlave(2, 1)17(6, 3, 3, 1, 1, 1, 1, 1)(7, 2, 2, 2, 2, 1, 1, 0)uSlave(2, 1, 1)17(4, 4, 4, 1, 1, 1, 1, 1)(6, 3, 2, 2, 2, 2, 0, 0)vMaster(2, 1)17(4, 4, 4, 1, 1, 1, 1, 1)(6, 3, 2, 2, 2, 1, 1, 0)vMaster(2, 1)17(5, 3, 3, 1, 1, 1, 1, 1, 1)(6, 2, 2, 2, 2, 2, 1, 0, 0)uSlave(2, 1, 1)22(7, 5, 4, 3, 1, 1, 1)(10, 4, 2, 2, 2, 2, 0)u, vMutual(2, 1)22(8, 5, 3, 3, 1, 1, 1)(10, 3, 3, 2, 2, 2, 0)u, vMutual(2, 1, 1)22(8, 4, 4, 3, 1, 1, 1)(10, 3, 3, 2, 2, 2, 0)u, vMutual(2, 1)22(6, 5, 3, 3, 1, 1, 1, 1, 1)(9, 3, 2, 2, 2, 2, 2, 0, 0)u, vMutual(2, 1)22(6, 4, 4, 3, 1, 1, 1, 1, 1)(9, 3, 2, 2, 2, 2, 2, 0, 0)u, vMutual(2, 1)22(6, 5, 3, 3, 1, 1, 1, 1, 1)(9, 3, 2, 2, 2, 2, 1, 1, 0)u, vMutual(2, 1)Table3 lists all autocatalytic pairs which can be found in
partition lattices from n=15 to n=17 as well as all mutually
autocatalytic pairs for n=22.As can be seen from Table3, surprisingly the minimal nonautocatalytic catalyst
partition vector cminT is always (2, 1) or (2, 1, 1), that is, its
dimension dc,min and its trace mmin is much
smaller than that of the autocatalysts. This means that simple vector catalysis
with a catalyst vector c≠u,v can take place in a space of much lower
dimensionality as compared to autocatalysis.Recently, Uhlman
[59] suggested to use the term “autocatalysis” for
entanglement transformations of multiple copies of bipartite quantum states
(MLOCC or multiple-copy local operation and classical communication), a
phenomenon in quantum informatics which was discovered by Bandyopadhyay et
al. [60]. We suggest to call this type Multiple-Copy Autocatalysis (3.15), (3.15)Multiple-CopyAutocatalysis:=(u≉v)⋀{∃(p∈ℕ)>1,suchthat:u⊗p≈v⊗p}, where u⊗p and v⊗p denote tensor exponentiation
by the exponent p, that is, u and v, respectively, are p-times multiplied by tensor products with itself (3.16), (3.16)u⊗p=u⊗u⊗…⊗u︸p-times,v⊗q=v⊗v⊗…⊗v︸p-times. Note that the exponentiation with p = 0 yields the unity
vector (2.3) with one component equal to “1.”Using tensor
exponentiation our idea of Master- and Slave-Autocatalysis in vector catalysis
can be generalised by (3.17), (3.17)(u≉v)⋀{∃(p∈ℕ)>1,suchthat:u⊗p≈u⊗(p−1)⊗v}. A combination of
both concepts of autocatalysis leads to (3.18) and will be denoted by Unified
Autocatalysis. (3.18)UnifiedAutocatalysis:=(u≉v)⋀{∃p,q,r,s∈ℕ0suchthat:(p+q=r+s≥2)⋀(u⊗p⊗v⊗q≈u⊗r⊗v⊗s)}. The individual cases of
autocatalysis can be derived from (3.18) by setting the exponents p, q, r, and s as shown in Table 4.Table 4
Conditions for special cases of autocatalysis derived from unified autocatalysis.
Type of autocatalysisConditions for exponentsp, q, r, and sMaster-autocatalysis(p=1)⋀(q≥1)⋀(r=0)⋀(s=q+1)Slave-autocatalysis(p≥2)⋀(q=0)⋀(r=p−1)⋀(s=1)Multiple-copy autocatalysis(p=s≥2)⋀(q=r=0)Allowingp+q=r+s=1 for an arbitrary vector pair u and v, one gets (classical) majorisation.


## 3.3.3. Relation to Chemistry

Let us
identify the reaction arrow “→” in formal chemical kinetics with
the majorisation relation symbol “≺.” This approach corresponds to the
observation that common spontaneous reactions occur towards higher disorder and
thus in downward direction within the diagram lattices of Ruch [48].Then
the classical autocatalytic reaction equation (3.6) is related to (3.13) of Slave-Autocatalysis
in the mathematical framework. Analogously, one can formulate formal kinetic
equations for the other types of autocatalysis: (3.19)Master-Autocatalysis:2A⟶k4A+X,(3.20)MutualAutocatalysis:Both2A⟶k4A+X,A+X⟶k22Xhold, where k2 and k4
correspond to different
reaction conditions, (3.21)Multiple-CopyAutocatalysis:pA⟶kmpX,(3.22)UnifiedAutocatalysis:pA+qX⟶kurA+sX. A
typical example for autocatalysis is the so-called tin pest, that is, the
conversion of white β-tin into grey α-tin [61, 62]. The velocity of this autocatalytic reaction is—apart from temperature—depending on size and shape of the growing
seeds. Such reactions are also called collectively autocatalytic [63], that is, the reactions produce themselves as
reaction products such that the entire set of reactions is self sustaining and
accelerating. Without going into details, we suggest that the kinetic equations
(3.6) and (3.19) to (3.22) may serve as an adequate description of such processes.A similar
phenomenon can be observed in case of heterogeneous catalysis which takes place
by processes of adsorption from the gas phase, reaction at solids or solid
surfaces and desorption into the gas phase. During these processes adsorption
clusters are formed which are the reactive units of catalysis. These units can
be local adsorption complexes at “active centres,” clusters at larger surface
domains, the adsorption layer or even the adsorbate together with the bulk [64]. The catalytic properties are strongly depending upon
the size of the reactive units which may be considered as an accelerating
autocatalytic set combined with structure formation. In other words, the
reaction is determined by the distribution function of the adsorption clusters
which catalyse theirs own formation and decay.As a
particular example, one may consider the oxidation of CO on Palladium supported
catalysts [13–15, 21, 22, 23, 24, 25, 26, 27]. By model studies one can show that the phase
transition between Pd and PdO occurs faster with decreasing size of the
catalytic Pd clusters [13–15]. As a result, it comes to resonance phenomena between
catalysts of different sizes whereupon self-affine patterns are formed the overall shape of which
depends on the particle size distribution. The synchronised CO oxidation as
well as the fast loss of catalytic activity in the oxidised state of the
catalyst particles is indicative for autocatalytic processes. A detailed
description of this model is given in [13–15].


## 3.4. Counting Properties of Partition Lattices forn=1 to n=20

To give an
impression of the complexity of  the
discrete approach to vector catalysis, in Table5 some counting properties of partition lattices up to n=20 are given. All characteristics listed in this table are increasing
exponentially due to NP complexity of the problem.Table 5
Counting values of partition vectorsu and v and their properties concerning vector catalysis depending on the natural number n of the partition lattice.
nNNu,v=(N2)Nu−vNu≉vNu(pot)vNu(c≤20)vNu(auto)vNu(Master)vNu(Slave)v11————————2211——————3332——————45104——————57216——————61155122—————715105174—————8222312815—————9304354135—————10428616285—————11561 540871701————12772 92612838541———131015 050175712115———141359 0452471 4052814———1517615 4003352 58369341—11623126 5654594 771149743121729743 9566118 3073131618261838573 92082414 7986303242772019490119 8051 08124 9761 21563178255320627196 2511 43042 8052
2861 19217157114The following
notations are used in Table5.(i)
N: No. of
partition vectors u and v with tr(u)=tr(v)=n which is identical to the no. of diagrams of
Ruch’s diagram lattices with natural number n;(ii)
Nu,v: No. of all
possible pairs of partition vectors u and v in a
partition lattice with natural no. n;(iii)
Nu−v: No. of all
adjacent pairs in a partition lattice with natural no. n, that is, couples
of partition vectors u and v, where one of them
is directly majorising the other;(iv)
Nu≉v: No. of all
incomparable pairs of partition vectors u and v in a partition lattice with natural no. n;(v)
Nu(pot)v: No. of all
incomparable pairs of partition vectors u and v in a partition lattice with natural no. n which are potentially
catalysable, that is, which fulfill the condition v≻(≺≻)ku with k∈ℕ (see Section 3.2 and Table 7);(vi)
Nu(c≤20)v: No. of all
incomparable pairs of partition vectors u and v in a partition lattice with natural number n which are catalysed by a
catalyst partition vector c of a partition lattices with n≤20;(vii)
Nu(auto)v: No.  of all incomparable pairs of partition
vectors u and v in a partition lattice with
natural no. n which are
autocatalysed according to (3.12) or (3.13) by one of the partition
vectors of the incomparable pair;(viii)
Nu(Master)v: No.  of all incomparable pairs of partition
vectors u and v in a partition lattice with natural
no. n which show Master-Autocatalysis
according to (3.12);(ix)
Nu(Slave)v: No.  of all incomparable pairs of partition
vectors u and v in a partition lattice with
natural no. n which show Slave-Autocatalysis
according to (3.13).The number series of Table5 can roughly be approximated by the power law (3.23)y=aebn0.5orlny=lna+bn0.5, where y are the least square fits of each of the counting
values as a function of n with R2
values of about
0.9999. The coefficients of these fits are listed in Table 6.Table 6
Counting values of partition vectorsu and v and their properties
concerning vector catalysis depending on the natural number n of the partition lattice.
Counting valuealn(a)bR2N5.2 e−02−3.02.10.99992Nu,v=(N2)1.0 e−03−6.94.30.99996Nu−v2.9 e−02−3.52.40.99997Nu≉v3.5 e−05−10.24.70.99996Nu(pot)v2.3 e−08−17.65.70.99992Nu(c≤20)v9.5 e−09−18.55.70.99989Nu(auto)v*2.4 e−11−24.56.60.99984Nu(Master)v*3.3 e−13−28.77.30.99997Nu(Slave)v*1.2 e−10−22.86.20.99941*Counting values were calculated up to n=24.Table 7
Notations for pairs of partition vectors and majorisation.
n∈ℕPositive integer (natural number) to be partitioned.u=(u1u2⋮udu),v=(v1v2⋮vdv)Discrete
partition vectors (column vectors) with nonnegative integersui and vi as components (i,ui,vi∈ℕ).
They are used as pair of partition vectors to be compared with respect
to majorisation and discrete vector catalysis.
The following conventions are
assumed. If a pair of different vectors u and v (u≠v) is compared with respect to majorisation, u is chosen as the vector with the first occurrence of a component ui<vi.du,dv∈ℕDimensions, that is, number of nonzero components, ofu and v, respectively. In general, du and dv are different.du,v=sup(du,dv)Supremum of dimensionsdu and dv.
The vector with the lower
dimension is filled up with zeros to du,v components.nu=tr(u)=∑j=1du,vuj,nv=tr(v)=∑j=1du,vvj,nu,nv∈ℕTrace
(sum of components) ofu and v, respectively;
(classical)
majorisation,  nu=nv=n.ou=(ou,1ou,2⋮ou,du,v),ov=(ov,1ov,2⋮ov,du,v)ou,i=∑j=1iuj,ov,i=∑j=1ivj,i=1,2,…,du,vVectors of partial sums of partition vectorsu and v, respectively.≺,≻:u≺v≡v≻u:=ou,i≤ov,i,i=1,2,…,du,v(Classical) majorisation relation.
Comparable
pair, whereu is being majorised by v and v is majorising. u
In quantum informatics the corresponding transition of entangled
states is possible with certainty by LOCC (local operation and classical communication).u≈v:=(u≺v)∨(u≻v)Pair of partition vectors which is comparable with
respect to majorisation.u≉v=¬(u≈v)Pair of partition vectors which is incomparable with respect to
majorisation.u≺≻v≡v≻≺uIncomparable pair, where bothu and v, respectively, are divided in two subsections of contiguous components, such
that in the first section all partial sums ov of v are greater or equal to ou of ou and in
the second section vice versa.
(For this and the following definitions see also Section 3.2).u(≺≻)kv≡v(≻≺)ku,k∈ℕSame asu≺≻v but changing subsections of components are repeated k times; in case of k=1 one obtains the notation above.u≺≻≺v≡v≻≺≻uIncomparable pair, where bothu and v, respectively, are divided in three subsections of contiguous components, such
that in the first and last (third) section all partial sums ov of v are greater or equal to ou of u and in the second section vice versa.u≺(≻≺)kv≡v≻(≺≻)ku,k∈ℕ0Same asu≺≻≺v but changing subsections of components are repeated
k times; in case of k=1 one obtains the notation above, in case of k=0 one
gets the majorisation relation ≻(≺).Table 8
Notations for tensor operations and discrete vector catalysis.
c=(c1c2⋮cdc)cT=(c1c2…cdc)Catalyst partition vector, represented as column vectorc or as row vector cT;
it will be used for tensor multiplication with u or v, respectively, in vector catalysis.dc∈ℕDimension (number of nonzero components) of catalyst partition
vectorc.m=tr(c)=∑j=1dccj,m∈ℕTrace (sum of components) of catalyst partition vectorc.⊗Symbol for tensor product relation.u⊗cT,v⊗cT (vector product notation)Tensor products ofu with c(cT)
and v with c(cT), respectively, in both cases represented as a (du,v×dc)-matrix.u⊗c,v⊗c (tensor product notation)θu⊗c,θv⊗cVectors of tensor productsu⊗c and v⊗c, respectively, where all components (tensor product elements) are arranged in
decreasing (nonincreasing) order.du⊗c=dv⊗c=du,v⋅dcDimension (number of components) of tensor product vectors, that
is, dimensiondu,v of partition pair vectors multiplied
with dimension dc of catalyst vector.nu⊗c=nv⊗c=n⋅mTrace (sum of components) of tensor product vectors, that is, trace
of partition vectors multiplied with trace of catalyst vector.ou⊗c,ov⊗cVectors of partial sums of tensor product vectorsθu⊗c and θv⊗c, respectively.u⊗p=u⊗u⊗…⊗u︸p-times,v⊗q=v⊗v⊗…⊗v︸q-times,p,q∈ℕ0Tensor exponentiation of partition vectorsu and v with exponents p and q, respectively. Special cases,  u⊗1=u,v⊗1=v,u⊗0=v⊗0=(1).As can be seen from coefficientsb, the counting numbers of
properties correlated to all and to the incomparable partition pairs are
growing about twice as fast as the number N of partition vectors and the
number Nu−v of directly majorising partitions. The
calculated coefficients b for properties which are attributed to vector
catalysis are even larger, but may be less significant, because a fewer number
of  values are included in the fits; if
one takes a sufficiently large number of counting values the coefficients b for catalysable or catalytic pairs cannot exceed that for incomparable pairs.


## 4. Conclusion and Summary

Our mathematical
model of discrete vector catalysis is based on partition vectors, the components
of which are only natural numbers including zero. The set of all these vectors
forms a special, discrete vector spaceVdisc which—in analogy to real vector spaces—is based on d-dimensional vectors u=(u1,…,ud) with ui,d∈ℕ0 and tr(u)=n∈ℕ0, a direct sum “⊕” of vector components, a multiplier μ∈ℕ0 and a scalar multiplication “⋅” such that (4.1)u,v,w,0≡(0),0d≡(00…0︸d-times)∈Vdisc,μ∈ℕ0⇒0⊕u=u,0⋅u=0d,u⊕(v⊕w)=(u⊕v)⊕w,1⋅u=u,u⊕v=v⊕u,μ⋅u=u⋅μ,μ⋅(u⊕v)=μ⋅u⊕μ⋅v. On this discrete vector space the majorisation relation “≺,≻,≈” and the tensor product “⊗” are defined as described in the previous
sections and in Tables 7 and 8. The restriction to such a discrete space
enabled us to develop some powerful rules for describing vector catalysis for
an effective search of correlated catalysts. On the other hand, this method is
of great advantage for discussing our results in the number-theoretical
framework of Young diagrams and Ruch’s diagram lattices. Particularly, with
regard to further investigations on discrete vector catalysis we would like to
represent the procedure, that a transition which is forbidden by majorisation
due to incomparable vector pairs becomes allowed by participation of a third
catalyst vector, schematically as shown in
Figure 7.Figure 7
Incomparable pairu and v corresponding to a forbidden transition, which will be allowed by a third vector c as catalyst.The results can be summarised as follows.(i)
We could verify
numerically that not all pairs of incomparable diagrams or partition vectors, respectively, can be catalysed. The smallest possible lattice, which contains a
catalytic vector pair, is the diagram lattice forn=11 with the catalysable vector pair (4, 4, 1, 1, 1) and (5, 2, 2, 2, 0), and the catalytic partition vector (10, 6, 5, 5, 4, 3, 3, 3, 3) the trace of which is m=42.(ii)
The smallest
nontrivial catalyst vector is (2, 1) belonging to the diagram lattice forn=3; and the smallest incomparable
vector pair (6,6,1,1)≉(8,3,3,0) which is catalysable by this
catalyst is contained in the diagram lattice with n=14.(iii)
We have
introduced the ideas of nonprimitively catalysable incomparable vector pairs
and of potentially catalysable vectors pairs. Moreover, we have presented a
selection rule which distinguishes between potentially catalysable and noncatalysable
incomparable vector pairs by considering alternately majorising subsections in
an incomparable pair. This can be seen as a nice analogy to odd/even rules in
chemistry.(iv)
In connection
with potentially catalysable vector pairs we have sketched a semiempirical/graphical
method for finding catalyst vectors by minimising the negative deviation of the
difference vector of the partial sums of the ordered tensor products. Using
this procedure a catalyst with 38 components and trace of value 410 could be
found for the smallest 4-dimensional pair(5,5,1,1)≉(6,3,3,0) belonging to the diagram lattice
with n=12.(v)
For any
catalysable pair of incomparable vectors, a countable infinite number of
catalysts exist. The set of the catalyst vectors assigned to the same pair can
be grouped into different classes based upon the component structure and the tensor
powers of the catalysts. This will be discussed in detail in a further paper.(vi)
In analogy to
chemistry we have adopted the term “autocatalysis” as a special type of vector
catalysis, at which at least one partition vector of the incomparable pair acts
as catalyst. Depending upon the partition vectors which are catalysing we
distinguish between Master-, Slave-, and Mutual Autocatalysis. The smallest
diagram lattice in which autocatalysis occurs is that withn=15 and
the respective incomparable pair is (6,3,3,1,1,1)≉(7,2,2,2,2,0) with (6, 3, 3, 1, 1, 1) serving as
Slave-Autocatalyst. Mutual Autocatalysis takes place at the first time in a
diagram lattice with n=22, and there exist already 6 incomparable pairs
with this property. We would like to mention that there are remarkable
connections between vector catalysis and the structure of diagram lattices
which we consider in a forthcoming publication.
Comparing the chemical ideas of catalysis and autocatalysis with the
idea of vector catalysis, it seems on a first view that vector catalysis
reflects the old chemical idea of a substance as a catalyst. But the formation
of tensor products implies that with help of a catalyst much larger vector
spaces of the entangled systems are created. These systems can be interpreted
as new pure states which are now comparable by majorisation, and therefore the
transition between these entangled states occurs in a high-dimensional space—as opposed to the forbidden noncatalysed
transition in the lower-dimensional space (see also Figure 7).But catalysis means also that there is a possibility that the
catalyst can be extracted again after a catalytic transition. The decomposition
of the new entangled vectors into the catalyst and the original vectors is
possible if one knows the one of them. In general, there are several alternatives
to disaggregate, that is, the decomposition; it is by no means unique.On the other side, recall that the process of vector catalysis is
enabled by formation of the tensor product matrices of  the partition pair vectors and the catalyst
vector, respectively, followed by downwards reorganisation of the matrix
elements into vectors with monotonously decreasing order of their components, such that the resulting vector states are comparable. As a consequence of this
rearrangement, the components of the initial partition pair vectors and the
catalyst vector are mixed within the ordered tensor product vectors, and thus
the distribution relations are changed. In other words, the new ordered vectors
of the entangled states are not at all simple linear combinations of the source
vectors but result of a complex transformation combined with the loss of
identity of the catalyst vector. Indeed, not only the formation of tensor
products in a high-dimensional space but primarily the process of
reorganisation causes catalysis. This is a nice analogy to the statement in
chemistry that not substances but only reactions are catalysing [5].We are convinced that in future the comparison of chemical catalysis
and entanglement catalysis will be fruitful for both sides.



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*Source: 590142-2008-06-04.xml*