The next stage in our comparison of the classical computational category 
and the quantum computational category focused on contrasting 
with 
We have seen already that
which is a polynomial function of 
and an exponential function of 
Of course, the cardinality of 
is infinite, but its quantum cardinality,
is a linear function of 
and a linear function of 
We now consider subobjects canonically associated to morphisms in and 
Associated to a morphism 
is its image,
which is a subset of the target 
The cardinality of 
may be called the rank of the set function 
The rank of 
is bounded by the cardinality of its source 
and we have 
if and only if is injective. The rank of is also bounded by the cardinality of its target 
, and we have 
if and only if is surjective. Thus,
A morphism is said to have full rank if
In particular, a full rank function is either injective or surjective, but we cannot say which without knowing which of 
is larger. On the other hand, the concept of rank itself is an elementary combinatorial notion, and one can find an explicit formula for the number of rank 
morphisms in in terms of Stirling numbers. Using this formula, one can show that the set of full rank morphisms in is small in comparison to the total cardinality 
Associated to a morphism 
is its image,
which is a subspace of the target space 
The rank of the linear transformation 
is the dimension of 
. The rank of is bounded by the dimension of 
and this bound is achieved if and only if is injective. The rank of is bounded by the dimension of 
, and this bound is achieved if and only if is surjective. Thus,
A morphism is said to have full rank if this upper bound is achieved, and just like in the classical situation this means either that is injective or surjective, but we cannot determine which without more information. What is very different in the quantum world is that a generic morphism in has full rank.
Theorem 4.1. The set 
of full rank morphisms in is open and dense in the norm topology.
Note that we have two norms on 
the operator norm and the Frobenius norm, and we did not specify which is being used in Theorem 4.1. This is because these two norms generate the same topology, as do all norms on a finite-dimensional vector space.
Theorem 4.1 is another way to motivate the long journey we took towards the Singular Value Decomposition. The first step on this journey is to recognize that the there is a second subobject associated to which has no classical counterpart, namely the kernel
which is a subspace of 
The dimension of 
is called the nullity of and denoted 
For an arbitrary morphism , we have two affiliated spaces: 
and 
Consider the associated orthogonal decompositions,
The rank-nullity theorem says that the restriction of to 
is an isomorphism onto 
This result is purely algebraic, and does not involve in the Hilbert space structure on 
and . The Singular Value Decomposition may be viewed as a remarkable refinement of the rank-nullity theorem which uses the geometric structure of Hilbert space to achieve a much more precise description of morphisms in
A nice observation we made is that one can guess what this mysterious enhanced decomposition looks like for morphisms 
which are quantizations of set maps, i.e. the case where 
for 
In this case, using purely combinatorial reasoning, one can construct an orthonormal basis for and an orthonormal basis for such that the restriction of to an isomorphism 
has a very explicit description in terms of the combinatorial statistics of the underlying set function It is useful to review this combinatorial “bare hands” special case of the SVD for quantized set functions before starting down the geometric road to developing it for arbitrary morphisms in 
