4.1. A Property of Connected Spaces
In this Section, we remind the notion of connectedness of a topological space (refer to the textbook [
35] for details). We prove a property of connected spaces (Lemma 2) that we need later and that we did not find in the English literature (but in the German textbook [
36]).
Definition 2. A topological space is called connected if for two non-empty open sets , , with it follows that .
In other words, a topological space is connected if it cannot be split into two disjoint, non-empty open sets. The following is well-known (e.g., [
35], Proposition 9.8):
Note 4. Let be a connected topological space and let be both, open and closed. Then or .
Definition 3. A family of open sets , with is called an open cover of .
The proof of the following lemma is from [
36].
Lemma 2. Let be a topological space. is connected For each open cover of and two points there are with:
- (i)
, for
- (ii)
, for
- (iii)
An open cover
with the properties (i), (ii), and (iii) for each pair of points
is called an
open chain connecting the points
(see
Figure 7).
Proof. “” Let be disconnected. We must show that there is an open cover of that does not satisfy conditions (i), (ii), and (iii). Because is not connected, we find open sets with , , , and . Choose and . The open cover satisfies (i) and (ii) but not (iii).
“” Let be an open cover of . Two points are called -connected (in symbols ) if there exists with the properties (i), (ii), and (iii).
Then, “” is an equivalence relation: reflexivity and symmetry are obvious. Transitivity is seen as follows: for with and choose a chain of open sets connecting and choose a chain of open sets connecting .
Set and . Then, is a chain of open sets with properties (i), (ii), (iii) connecting , i.e., .
Let be a -equivalence class. If we show that then we are done. For and a chain of open sets connecting it is by definition, and it is because each is -connected with . Thus, . This implies that , thus .
Thus, is open and closed in , but . Note 4 before implies . □
4.2. Differentiable Manifolds
The definition of a manifold with boundaries is given (the seminal textbook [
37] provides many details and proofs, and [
23] is a more modern treatment of the subject). For this purpose, we need the definition of the n-dimensional real half-space
; it is
the
boundary of
.
Note that in the context of differentiable manifolds, topological spaces are considered to be Hausdorff spaces and second countable. A
Hausdorff space requires that any two different points of the space have disjoint neighborhoods; the set of all neighborhoods of a point
is denoted by
. A space is
second countable if it has a countable basis, i.e., every open set of the space is the union of a subset of the basis. See [
38] for the detailed definitions of these terms.
Definition 4. Let be a Hausdorff and second countable topological space. If for each point there exists an open neighborhood and a homeomorphism , then is called a topological manifold with boundary of dimension (in symbol: ). The pair is called a chart of around . A set of charts with is called an atlas of .
A point with is called interior point, and a point with is called boundary point. The set of boundary points of is called the boundary of , denoted by .
For two intersecting charts , i.e., charts with the map is called the transition function between the charts.
is called a differentiable manifold of class ( -manifold for short) if all transition functions are differentiable of class ; the corresponding atlas is called -atlas. For , the manifold (and the atlas) is called smooth.
In
Figure 8 and
are interior points while
is a boundary point.
Textbooks on differential topology (e.g., [
23]) or on differential geometry (e.g., [
39]) contain many examples of differentiable manifolds as well as corresponding atlases. Standard examples include n-dimensional spheres in the (n + 1)-dimensional Euclidean space
, a torus, graphs of differentiable functions (see Lemma 9), which then include curves and surfaces in
. Especially, the Euclidean space
is a differentiable manifold (see next).
A
-manifold may have several
-atlases. For example, both,
as well as
are
-atlases (for any
) of
, and so is
. In general, when adding a chart
to a given
-atlas
and the resulting atlas
is again a
-atlas,
is said to be
compatible with
. Adding all compatible charts to
results in the (unique) maximal atlas
(that contains
). A maximal
-atlas is called a
-
differentiable structure of the corresponding manifold. Any
-atlas
determines a unique
-differentiable structure (see [
23], Proposition 1.17). Thus, we can assume that the atlases of a manifold are differentiable structures.
The following is often used, and its proof can be found in [
37].
Lemma 3. Let be a -manifold with boundary, . Then, and is a -manifold without boundary, .
Also, it is well-known that any open subset of a manifold is again a manifold:
Note 5. (a) is a smooth manifold without boundary.
(b) For a -manifold any open subset is a -manifold.
(c) The interior of a manifold with boundary is a manifold without boundary.
Proof. (a) is a smooth atlas that shows that is a manifold without a boundary.
(b) If is a chart of around , then and is a homeomorphism. Thus, is a chart of S around . Also, restrictions of transition functions are of the same differentiability class as the original functions.
(c) It is and, thus, the claim follows from (b). □
There are several ways in which manifolds can be constructed from existing manifolds, e.g., by building their sums, quotients, and products (see [
23]). In our context, the product of manifolds is of interest: if
are
-manifolds (without boundary) and
, then their Cartesian product
is another
-manifold (without boundary). For example,
The unit sphere is a smooth manifold. Thus, (the product of n copies of ) is a new smooth manifold called an n-dimensional torus.
An open interval (for ) is a smooth manifold. Thus, the open n-dimensional cuboid is a smooth manifold.
For manifolds with boundaries, building their products is a bit less straightforward. In general, there is an extensive theory behind this (see [
40]). In our context, the following facts suffice:
Let
be
-manifolds (without boundary),
, and let
N be a
-manifold with boundary,
. Then,
is a manifold with boundary,
, and
(see [
23], Proposition 1.45).
Thus, with
for
and
, the product
is a manifold with boundary
(for an example, see
Figure 9).
4.3. Singularities
In
Figure 9, the boundary of the manifold Q consists of the left and right edges without their endpoints. This is an implication of the fact that a boundary of a manifold with boundary is a manifold without boundary (see Lemma 3). If the endpoints of the left or right edges were included, they would become the boundary of the manifolds consisting of the left or right edges: contradiction.
This is a general phenomenon: manifolds exclude
singularities, i.e., non-differentiable structures. A vertex of a rectangle is an example of such a singularity: Part (a) in
Figure 10 depicts the rectangle
and one of its vertices
. Part (b) focuses on a neighborhood of
in
. Finally, part (c) extracts the boundary of this neighborhood, moves
to the origin, and rotates the boundary by 45°; movements and rotations are smooth maps, i.e., the resulting graph in (c) is diffeomorphic to the part of the boundary in (b). Obviously, the graph in (c) is the graph of the absolute function. Assume that there is a chart
of the boundary around
with
. Then
is a diffeomorphism (see Note 6 below); thus,
is differentiable, but
is the absolute function, which is known to be not differentiable at 0. This contradiction shows that
is a singularity, i.e., vertices must not be part of a rectangular “shape” in order to be a manifold.
Many other kinds of singularities exist. For example (see
Figure 11): Part (a) of the figure shows a “cusp”, which is a curve that is not differentiable at the point
. An intersection at point
in part (b) is not differentiable. Also, an edge of a cuboid (see part (c) of the figure) consists of nodes, each of which is a singularity.
The intersection point in part (b) is already a topological singularity; i.e., no reference to differentiability is needed to recognize this: Any point different from is contained in a neighborhood that is homeomorphic to an open interval in . Thus, if a chart around would exist, U could be chosen to be connected. Then, is connected. consists of four connected components. Deleting a single point from a connected subset of the real line results in two connected components, i.e., consists of two connected components—a contradiction because consists of four connected components.
Many of the results achieved in differential topology of manifolds are not valid in the presence of singularities. This is why special care must be taken when claiming that geometric objects are manifolds (with or without boundary): it must be proven that they are manifolds to avoid singularities. Singularities are extensively studied (see [
41,
42], or [
43] for example).
4.4. Differentiable Maps
In
Figure 8,
is not an open set in
; thus, the ordinary definition of differentiability (that is typically defined for open sets) does not apply. Consequently, the notion of differentiability is extended to functions with a domain that is an arbitrary subset of
.
Definition 5. Let be an arbitrary set and let be a map. is called differentiable of class , iff a map with exists that is differentiable of class in the ordinary sense and that fulfills , i.e., restricted to is identical to . is called a differentiable -extension of .
For example, is a -map with being a differentiable -extension of .
Based on this definition, we can define differentiable maps between manifolds (see also
Figure 12):
Definition 6. Let and be two -manifolds (with or without boundary) with and . A map is said to be differentiable of class , iff for every there exist a chart of around and a chart of around with , such that is differentiable of class .
Recall that is a manifold with the atlas . Thus, the definition before covers the definition of differentiability of maps .
Maps that maintain the differentiable structure of manifolds are of special interest:
Definition 7. Let and be two -manifolds. A map is said to be a -diffeomorphism iff is differentiable of class , is bijective, and has an inverse that is also differentiable of class . Furthermore, the manifolds and are called -diffeomorphic.
For example, is a -map. According to Lemma 9, is a -manifold and so is as a Euclidean space. is bijective with , as inverse map, and is also of class . Thus, is a -diffeomorphism.
Diffeomorphic manifolds cannot be distinguished based on differential topological properties. A diffeomorphism in differential topology plays the same role as homeomorphisms in general topology or isomorphisms in algebra.
Next, we show that a chart is a diffeomorphism; this is a well-known fact but the proof is instructive:
Note 6. Let be -manifold (with or without boundary) with atlas and let be a chart. Then, is a -diffeomorphism.
Proof. First, we have to show that is of class ; i.e., we have to show that is of class for a chart of and a chart of . Choose as a chart of and as a chart of . Then , which is of class .
is a homeomorphism, i.e., is bijective. Similar to before it is seen that is of class . □
The general question is about the relevance of the differentiability class
. A famous theorem by Whitney [
44] proves that studying
-manifolds suffice (a detailed proof of this theorem can be found in [
37], 2–2.10). Because of this theorem, the restriction to smooth manifolds (i.e.,
-manifolds) is justified:
Theorem 1. Let . Then: Every -manifold is -diffeomorphic to a -manifold.
4.5. Differential of a Map
A point
of a differentiable manifold
(with or without boundary) is associated with its
tangent space (see
Figure 13). This tangent space can be imagined as the set of all tangent vectors to
through
; the precise definition is much more subtle and complex (see [
23], chapter 3) but for our purpose this descriptive idea suffices.
is a vector space, and if
has dimension
it is
([
23] Proposition 3.12). The disjoint union of the tangent spaces of all points
is referred to as
tangent bundle of M:
. With
,
is a differentiable manifold with
([
23] Proposition 3.18).
If
is a differentiable map between two differentiable manifolds, then the
differential of
at
is a linear map
(see
Figure 13). As before, the precise definition is quite complex ([
23], chapter 3), but, again, a vague intuition suffices for our purpose, especially because the differential
corresponds to the Jacobian matrix of
([
23], p. 61 ff). We will need the latter representation of the differential of a map because it allows us to compute the rank of the differential as the rank of the Jacobian matrix.
In case
and
are “just” Euclidean spaces,
and
are smooth manifolds (see Note 5a), and their tangent spaces are the very same Euclidean spaces, i.e.,
and
for any points
and
([
23] proposition 3.13). The differential
of a map
becomes the total derivative
([
23] proposition C.3); thus, for
it is
where
is the directional derivative of
in the direction
.
For example, is a -map. Its differential is . For it is .
4.6. Maps of Constant Rank
For each linear map
between vector spaces
and
the rank of
is defined as the dimension of the image of
, i.e.,
. Note that in practice, the rank of a linear map is determined as the number of linear independent columns of the matrix representing the map
L (see [
34],
Section 5.3, for more details). Since the image of
is a subspace of
, it is always
. According to the Dimension Formula of Linear Algebra (see [
34] Theorem 10.9) it is
; thus,
. This proves the following:
Note 7. Let be a linear map. Then: .
The rank of differentiable maps in is the rank of the linear map . It is key to the study of local and global properties of differentiable functions .
Definition 8. Let be a differentiable map between the two differentiable manifolds M and N. The rank of at is the rank of its differential , in symbols . If has the same rank at every point then is said to have constant rank, in symbols .
According to the end of
Section 4.5, the map
has the differential
. With
it is
, i.e., it is
. For
it is
and
, i.e.,
. And for
but
is
and
, i.e.,
. For
is
, i.e.,
has a constant rank on
.
Since is linear, it is (see Note 7 before). In case , is said to have full rank at and just full rank if it has full rank at every point of . Maps of full rank have special names:
Definition 9. Let be a differentiable map between the two differentiable manifolds M and N. is called an immersion in case , i.e., is injective. is called a submersion in case , i.e., is surjective.
In our context, immersions are of special interest. Thus, give some geometric intuition of an immersion, which is helpful in what follows. Remember that the differentiability of a map at a point means that it can be locally approximated by a linear map, i.e., by its differential : for points close to it is . Thus, the properties of the differential approximate locally properties of the map. Injectivity of the linear map means that it maintains the independence of “all directions” in when mapping to : no two different vectors are “smashed” together. For the corresponding map this translates into the fact that does not “fold” or “collapse” parts of a neighborhood of the point . Since an immersion has a constant rank, which means that transforms a neighborhood of each point somehow “faithful”, not introducing any “crushing”.
An often-used well-known property is that the rank of a map can locally not decrease:
Lemma 4. Let , and . Then there is a neighborhood such that for each .
Proof. It is
; thus, there is
-submatrix
of
with
. W.l.o.g. this submatrix is
According to the Leibniz formula from linear algebra it is
i.e.,
is continuous (even differentiable) because
is differentiable and the Leibniz formula is a polynomial. Thus, with
there is an
such that
for each
. Consequently,
for each
. □
Although the rank of a map can locally not decrease, it may increase. For example, for it is . Thus, and . Arbitrary close to it is , i.e., for it is . However, if the rank is already maximal, it is locally constant since it cannot decrease locally; this proves:
Lemma 5. Let be of full rank at . Then it is of full rank in a neighborhood .
This has an important well-known implication:
Corollary 1. Let a differentiable map. Then:
- a.
If is surjective, then there exists a such that is a submersion.
- b.
If is injective, then there exists a such that is an immersion.
Proof. (a) By assumption is surjective, i.e., by definition . But implies that . Thus, because would be a contradiction. Consequently, which shows that is of full rank. The lemma before proves the claim. (b) is proven the same way. □
The next theorem (whose proof can be found in [
23], Theorem 4.5) shows that in case the differential is bijective at a point of a manifold without boundary the map is a local diffeomorphism around that point. Note, that the precondition that the manifold must have no boundary is essential here: The inclusion
is corresponding counter-example.
Theorem 2 (Inverse Function Theorem). Let be two differentiable manifolds without boundary and a differentiable map. If is bijective there are and such that is a diffeomorphism.
In the theorem before the conditions that both manifolds must have no boundary can be weakened: the codomain may have a boundary but the image of must be in the interior of the codomain, i.e., ; the reason is that is a submanifold without boundary (see Note 5c), i.e., the original inverse function theorem applies.
Special kinds of immersions play an important role:
Definition 10. Let be a differentiable map between the two differentiable manifolds M and N. is called an embedding iff is an immersion and is a homeomorphism onto in the subspace topology.
The map
is a homeomorphism,
is smooth, but because of
,
is not an immersion, thus, no embedding. Thus, not every smooth homeomorphism is automatically an embedding. However, under the following condition injective immersions are already embeddings (the proof can be found in [
23], Proposition 4.22):
Lemma 6. Let be an injective immersion, and be manifolds with or without boundary. If any of the following holds, is an embedding:
- a.
M is compact.
- b.
M has no boundary, and .
The following theorem is the basis of many other theorems in differential topology (see [
45], Theorem 3.7.5):
Theorem 3 (Rank Theorem). Let be two differentiable manifolds and a differentiable map. Let , and let be a neighborhood of such that for each it is (i.e., has constant rank in ). Then there is a chart for around and a chart for around , such that has in these charts the form .
Several key well-known properties of maps are inherited by their composition:
Note 8. Let be differentiable manifolds, and let and be maps. Then:
- a.
If and are injective or surjective or bijective then is injective or surjective or bijective.
- b.
If and are immersions then is an immersion.
- c.
If and are continuous then is continuous.
- d.
If and are homeomorphisms then is a homeomorphism.
- e.
If and are embeddings then is an embedding.
Proof. (a) implies because is injective. Next, injectivity of implies . This proves the injectivity of .
and , thus, .This proves the surjectivity of . Together, this proves the bijectivity of .
(b) and are injective. According to the chain rule it is , i.e., part (a) shows that is injective. Thus, is an immersion,
(c) Let be open. Because is continuous, is open. Because is continuous, is open. With it follows that is open. Thus, is continuous.
(d) and are bijective, so is (see part (a)). and are continuous, so is (see part (c)). and are continuous, so is (see part (c)). Thus, is a homeomorphism.
(e) and are embeddings, i.e., both are immersions as well as homeomorphisms. Because of part (b) is an immersion, and because of part (d) is a homeomorphism. Thus, is an embedding. □
4.7. Submanifolds
It is important to note that the definition of a manifold is completely independent of any surrounding space like a Euclidean space. In this sense, manifolds are abstract entities. Their concept has been introduced by Bernhard Riemann in 1854 (published 1868 [
46]). It generalizes objects like curves and surfaces that had been studied before that time. The latter are entities within a Euclidian space. Thus, it is natural to ask whether any (abstract) manifold is “equivalent” (also known as. “diffeomorphic”) to a corresponding entity in a Euclidean space: this has been proven by Hassler Whitney in 1936 [
44]. In this section, we summarize the corresponding concepts and results as relevant in our context.
A manifold maybe contained in another manifold (see
Figure 14).
Definition 11. Let be differentiable -manifold, and let be a subset of . is called an embedded submanifold of of dimension (or of codimension , respectively) and class if for any point there is a chart of around such that .
Sometimes, embedded submanifolds are also called
regular submanifolds.
Figure 14 depicts the situation. If
is an atlas of
then
is an atlas of
. Note that the latter assumes that
is a topological subspace of
, i.e., the topology of
is the subspace topology. This will become important soon.
Especially, since is an atlas of , is a -manifold in itself:
Note 9. Every embedded submanifold of class and dimension is a -manifold with .
The following lemma motivates the name “embedded” submanifold (see [
47], Theorem 11.14 for a proof):
Lemma 7. Let be differentiable -manifold, and let be an embedded submanifold of . Then, the inclusion is a -embedding (and, thus, by definition an immersion).
Vice versa, the name “embedding” of a map is justified by the following lemma (see [
47], Theorem 11.13 for a proof):
Lemma 8. Let and be differentiable -manifolds, and let be a -embedding. Then, the image is an embedded submanifold of .
A two-dimensional surface is an example of a manifold embedded in the three-dimensional Euclidian space. In general, graphs of differentiable functions are examples of such surfaces. The following lemma and definition generalizes the corresponding situation (see [
23], Proposition 5.4):
Lemma and Definition 9. Let and be -manifolds ( with or without boundary), and , and be of class . Then the graph is an embedded -manifold of dimension without boundary.
Here, the graph of is defined as .
The Euclidean space
is a manifold, so its embedded submanifolds are of interest. The following is an often-used mechanism to produce embedded submanifolds in
: For
and a
-differentiable map
the graph
is an embedded
-manifold of dimension
without boundary (according to the lemma just before). If
is of class
, the graph
is a
hypersurface in
, i.e., a
-manifold of dimension
(see
Figure 15); obviously, this generalizes the notion of a surface in
.
Another important means to obtain embedded submanifolds is via so-called level-sets and regular values:
Definition 12. Let be a -map.
a. For any point the set is called a level-set of .
b. A point is called a regular point if is surjective ( is called a critical point otherwise).
c. is called a regular value if each point of is a regular point ( is called critical value otherwise).
d. If is a regular value the level-set is called a regular level-set.
Obviously, each point of
is a critical point if
(
cannot be surjective in this case). Any regular level set is a
-manifold (see [
23], Corollary 5.14):
Lemma 10. Let and be -manifolds, and be of class . Then, any regular level-set is an embedded submanifold of with .
In
Figure 16,
,
, and
are three level-sets of the differentiable map
. Assuming that both
and
are regular values, the two level sets
and
are embedded submanifolds (the first one diffeomorphic to a circle, the second one diffeomorphic to two disjoint circles) of
of dimension 1 because of
and
. But
is not an embedded submanifold because it has the shape of an “8”, i.e., it contains a singularity in the form of a self-intersection (see
Section 4.3). This implies that
is a critical value.
The lemma before allows us to prove that the set of unitary transformations of a complex vector space is a manifold:
Lemma 11. is a smooth compact connected manifold (even an embedded submanifold) with is even a Lie-group.
Proof. It is
, and
; thus, it is
. Define
by
;
is differentiable because building the conjugate transpose of a matrix is differentiable, and the multiplication of two matrices is differentiable too. The differential of
is
(see
Appendix A).
Let
be the set of all Hermitian matrices.
is a
-vector space of dimension
([
34] Lemma 13.15). Thus,
is a smooth manifold of dimension
.
Because for each , it is , i.e., is, in fact, a map (and is differentiable).
Next, we show that any unitary map is a regular point of , i.e., that the differential is surjective for each :
Choose an arbitrary and define . Then, (hereby, (1) is because and (2) because ).
Now it is ; thus, is a regular value and is a regular level set. Consequently, is an embedded submanifold (Lemma 10), and, thus, a manifold itself (Note 9); furthermore, .
Next, we prove compactness.
(a) with is especially continuous. Now, , is a closed set, and pre-images of closed sets under continuous maps are closed, i.e., .
(b) Next, means especially that the columns of are unit vectors, i.e., for each . Thus, for all . This implies that for each , i.e., is bounded in . Thus, according to the theorem of Heine-Borel, is compact in .
Finally, we prove connectedness.
Le
. Then,
is diagonalizable, i.e., there exists a unitary matrix
such that
([
34] Theorem 18.13). Let
; then,
(because
). It is
and
, i.e., there is a path from the identity matrix
to
. Thus, any two unitary matrices can be connected by a path (e.g., via
):
is path-connected. Since every path-connected topological space is connected ([
35] Proposition 9.26),
is connected.
is a Lie group because matrix multiplication is a differentiable map , , and for two unitary matrices it is , i.e., . □
Note that the condition “regular” is key in Lemma 10. Without that condition, any closed subset of a manifold can be made the level set of a differentiable function (see [
23], Theorem 2.29 for a proof):
Lemma 12. Let be a differentiable -manifold, and let be a closed subset of . Then there exists a -function with .
In many practical situations, both, and are Euclidean spaces. For example, with and , the map , has the differential for , i.e., for ; thus, is surjective, except at the origin. This implies that each is a regular value; according to the lemma before, is a regular level-set for each . Thus, is a sphere of dimension , especially the unit sphere is an -dimensional embedded submanifold of .
There are important situations (e.g., in the context of Lie groups—see [
48]) in which the notion of a submanifold has to be generalized. Very roughly, any differentiable manifold that is a subset of another differentiable manifold and is “properly situated” there is considered a certain kind of a submanifold. More precisely,
Definition 13. Let and be -manifolds and be an injective immersion of class . Then, the image is called an immersed submanifold of .
For example, let
,
. Then,
is injective on the open interval
(see
Appendix B for more details). Also,
is an immersion because
, i.e.,
for
. Thus,
is an injective
-immersion, i.e.,
is an immersed submanifold (see
Figure 17). Furthermore,
is also surjective, thus, bijective, but
is not a homeomorphism because the image
is compact with the subspace topology (it is closed and bounded),
is not compact, while compactness is a topological invariant. Together,
is an immersed submanifold, but
is not an embedded submanifold of the manifold
. Another argument supporting the latter: any neighborhood of the origin of
contains a singularity, namely a shape like in
Figure 11b.
According to Lemma 7, for any embedded submanifold the inclusion is an embedding and, thus, an injective immersion, i.e., is an immersed submanifold of . This proves the following:
Lemma 13. Let be a -manifold (with or without a boundary), an embedded submanifold. Then, is an immersed submanifold.
The opposite is not true: the example before shows that an immersed submanifold is, in general, not an embedded submanifold. However, the following lemma ([
23], Proposition 5.21) gives two situations in which an immersed submanifold is already an embedded submanifold:
Lemma 14. Let be a -manifold (with or without a boundary), an immersed submanifold. If or if then is embedded.
Finally, we answer the question posed at the beginning of this section: Any compact (abstract) manifold is diffeomorphic to an embedded submanifold of a Euclidean space of high enough dimension—more precisely,
Theorem 4. Let and let M be a compact -manifold with boundary. M can be embedded into with .
Again, this theorem is by Whitney [
44], and a detailed proof can be found in [
37] (Theorem 1–4.3).
In summary, this section provided the necessary background of dimensional expressivity as needed in
Section 5. Before presenting the corresponding details, the next section finally focuses on the unitary approach, giving a precise definition of the corresponding notion of expressivity. This requires explaining first how volumes on manifolds can be measured.
4.8. Volumes of Manifolds and the “Uniform Approach”
Section 2.3.1 motivated to define the expressivity of a variational quantum circuit by the “volume” of
in the unitary group
: vividly, the larger this volume, the higher the likelihood that
hits the solutions
of a given problem
.
4.8.1. Linear Approximations
Here, we provide more details about the notion of “the volume” of a differentiable manifold and especially how the notion of “volume” is related to the unitary group. For this purpose, we first remember that a function
is called differentiable at a point
if it can be approximated locally by a linear function:
Here, is a point in a small neighborhood of , is a linear function, and is the small error made when considering as the value of : in a small neighborhood, the differentiable function is nearly the linear function . For a differentiable function this means that is a real number and is the tangent at at the graph of ; this tangent locally approximates the function . Thus, the graph of , i.e., the manifold (see Lemma 9), is approximated by this tangent around .
The idea of linear approximation can be used in arbitrary dimensions: in part (a) of
Figure 18, a basis
has been chosen for the tangent space
of the manifold
at point
. The basis spans a parallelepiped
. If we take small vectors
the parallelepiped approximates the manifold
around
with a small error, i.e., the manifold looks locally like a very small parallelepiped of the tangent space.
Applying this linear approximation to enough points
of the manifold, i.e., if the manifold is “covered” by parallelepipeds, the manifold is turned into a linear approximation of the whole manifold: part (b) of
Figure 18 depicts this for the upper part of a manifold which looks like small parallelepipeds glued together (in the direction of the tangent spaces).
4.8.2. Approximating Volumes
Such a linear approximation allows us to compute the approximate volume of the manifold
by computing the volume of the parallelepipeds
and summing up their volumes:
The volume of a parallelepiped is computed as follows: let
be linear independent vectors, and let
be the parallelepiped spanned by these vectors. Then, from linear algebra, it is known that the volume
of the parallelepiped is
Define
to be the matrix with columns
,
,…,
. Then:
4.8.3. Riemannian Manifolds and Volume Forms
Equation (9) reveals that computing the volume of a parallelepiped depends on a scalar product. Consequently, we need a scalar product for every tangent space of the manifold .
Definition 14. Let be a differentiable manifold, and let be a function that associates with each a scalar product “in a differentiable manner”. Then, is called a Riemannian metric on , and is called a Riemannian manifold.
Note, that the phrase “in a differentiable manner” is left vague: a precise definition would require to define differentiable vector fields which we do not need in this paper. Also, the differentiability of is not relevant in our context.
Consequently, we assume that
is a Riemannian manifold; in fact, every differentiable manifold is a Riemannian manifold ([
39], Proposition 2.4). Then, the volume of a parallelepiped
in
is as follows:
With Equations (7) and (10) we can approximate the volume of a Riemannian manifold
as follows:
By choosing infinitesimal small parallelepipeds and correspondingly more and more points from the manifold we perform a limit process. In analogy to the limit process that defines the Riemannian integral we write very informally (and only conceptually) with
:
The precise definitions behind this notion need a lot more concepts and machinery. The most fundamental concept needed is that of a volume form that abstracts our informal notation . In our context, the unitary group admits such a volume form:
Note 10. The unitary group admits a (unique) volume form .
Proof. Every differentiable manifold is a Riemannian manifold ([
39], Proposition 2.4). If a Riemannian manifold
is oriented then it admits a (unique) volume form
([
39], Proposition 2.41). According to Lemma 11,
is a differentiable manifold, thus, it is a Riemannian manifold also. Any Lie group is orientable ([
49], Lemma 6). Since
is a Lie group (Lemma 13) it admits a (unique) volume form. □
The volume form of a manifold is (in local coordinates) ; it allows (as Equation (12) indicates) us to compute the volume of , namely . It also allows us to compute the integral of functions , i.e., ([see 39], the discussion following Proposition 2.41).
4.8.4. Haar Measure
Computing volumes is tight to differentiable manifolds, not applicable to other “spaces”. For this purpose, the concept of a
measure is introduced (see [
50] for details) that is more abstract than a volume but mimics its properties. Luckily, in our context, both concepts are the same. This is roughly seen as follows:
Each open set
of a differentiable manifold
is a differentiable manifold by itself (Note 5). Thus, the inclusion map
induces a Riemannian metric on
, and
is defined. This in turn, defines a measure on the Borel sets
of
, where
is the smallest
-algebra containing all open sets of
(refer to [
50] for details about Borel sets, measure spaces, and measures). This turns the manifold
into a measure space
([
51], Section 1.7). A measure is more general than computing volumes via integrals.
As a Lie group,
admits a left-invariant measure, the so-called
Haar measure. This measure is unique up to multiplication by a positive constant ([
48], Theorem 3.1). A compact Lie group admits a bi-invariant Riemannian metric ([
52], Proposition 2.17), which induces a bi-invariant volume form. Since the Haar measure is unique, we obtain the following:
Note 11. The volume form on a compact Lie group is the Haar measure.
Especially, since is a Lie group, instead of abstractly speaking of the measure of a subset of a manifold, we can speak about its volume.
4.8.5. Volume of Submanifold
Having the ability to determine the volume of manifolds, the question of volumes of submanifolds comes up.
Note 12. Let be a manifold with volume element , and let be an (embedded or immersed) submanifold. If then .
Proof. Let
be differentiable manifolds and let
be differentiable map. If
then
has measure zero ([
23], Corollary 6.11). For an (embedded or immersed) submanifold
of
the inclusion map
is differentiable (Lemma 7), i.e., if
then
has measure zero in
. □
This implies that any submanifold of the unitary group with has measure zero. Since measure and volume coincide it is for :
Note 13. Let be a submanifold, . Then .
Thus,
in case
(if
is a submanifold at all—see next chapter). Consequently, the illustrative motivation about measuring expressivity in the “Unitary Approach” (
Section 2.3.1) requires refinement.
4.8.6. Uniform Distribution
Let be a small open square in the Euclidian plain. If a point has to be picked randomly from , the probability for any two points being picked is the same: picking is “uniformly random”. This means that the probability of a certain point being picked is identical to the probability of any other point in being picked: the corresponding distribution of probabilities is uniform.
Obviously, the probability of a point being picked from is related to the volume of : the smaller the higher the likelihood of a point within to be picked. If we move the square around in the Euclidian plain, e.g., translating it by a vector to , the probability distribution remains the same because the volume is unchanged. This is based on the left-invariance of the volume in Euclidean space: by adding from left to each vector in does not change the volume, i.e., .
This is different for arbitrary manifolds: the Euclidean space if flat, but picking points from areas on curved manifolds may behave differently. For example, let
be the unit sphere. Then
is a chart of
(leaving out both poles as well as the meridian of the sphere—otherwise
would not be a diffeomorphism).
Figure 19 depicts the images under
of different parts of the domain:
Part (a) shows the image of the whole domain, i.e., the sphere ;
Part (b) is the image of ( is a small positive number), resulting in a belt around the equator;
Part (c) is the image of resulting in a cap of the north pole;
Part (c) is the image of resulting in a cap of the south pole.
Figure 19.
Volumes indicating non-uniform distributions.
Figure 19.
Volumes indicating non-uniform distributions.
The figure indicates that the volume of the belt is larger than the volume of a cap. But a larger volume of an area means a smaller probability of a point of the area being randomly picked. Thus, points on the sphere with values of close to (i.e., points from the belt) have a smaller probability of being randomly picked than points with values of close to (i.e., points from the cap of the north pole) or close to (i.e., points from the cap of the south pole): by picking a random latitude, points close to the poles have a higher probability to be picked than points near the equator. In this sense, points on the sphere are somehow “concentrated” towards the poles. As a consequence, the corresponding probability distribution is not uniform.
The volume that represents the probability distribution of randomly picking a point must reflect this effect of concentration to become a uniform distribution. As the figure indicates, shifting the belt to a pole does change its volume, i.e., the “usual” volume on the sphere is not translation invariant. But the Haar measure is translation invariant by definition. Thus, if we take the Haar measure to compute the probability distribution (i.e., the volume of areas of the sphere), the distribution becomes uniform: every area of the sphere of the same Haar measure has equal probability of containing a particular chosen point.
Consequently, the difference between the Haar measure of an area and its “usual” volume is an indicator of how much a distribution based on the usual volume deviates from being uniform.
4.8.7. Measuring Expressivity
“Deviation” can be assessed by various means. Especially, whether “volume” is computed directly or indirectly may differ. An example of an “indirect approach” is described next and is based on [
6].
Whenever has been uniformly randomly chosen (often called “Haar random”), is a uniformly random state. Thus, if is a set of Haar random unitary matrices, then is a Haar random set of states. But is not necessary Haar random, thus, the set is not necessarily a set of uniformly random states.
Different approaches have been defined to compare
and
. One approach (see [
4,
6]) is as follows: first, the elements of these sets are turned into matrices, i.e., for
and
the density matrices
and
are taken. Note, that in fact
depends on parameters
. For
, the matrix
is computed as well as the matrix
. Finally, the deviation between these two matrices is assessed and taken as expressivity of the ansatz
. For example, based on a matrix norm, the expressivity
becomes
The smaller , the closer becomes to being Haar random. If and are two ansatzes, ansatz is called more expressive than ansatz iff .
[
31] gives a procedure how expressivity of a specific ansatz can be experimentally determined.