We start with some definitions and elementary, mostly well-known statements around maximal convex and minimal concave extensions. The second subsection is concerned with sufficient conditions for the existence of convex and concave roofs and their main properties.

#### 3.1. Convex and Concave Extensions

Let Ω be a compact convex set and g a real function defined on the set ${\Omega}^{\mathrm{pure}}$ of its extremal points.

**Definition 3.1: Convex (concave) extensions**

A convex function G is called a convex (respectively concave) extension of g if G is convex (respectively concave) and coincides on ${\Omega}^{\mathrm{pure}}$ with g.

Between any convex, concave or roof extension of a function

g the inequalities

are valid. Indeed, with an optimal decomposition (3) for

${G}^{\mathrm{roof}}$ we get

because the extensions coincide on

${\Omega}^{\mathrm{pure}}$. The right sum cannot be smaller than

${G}^{\mathrm{convex}}\left(\omega \right)$ by convexity. Similarly one argues in the concave case.

The proof of (22) is valid pointwise, leading to

**Proposition 3.1**

Let

${G}^{\mathrm{convex}}$ be a convex,

${G}^{\mathrm{concave}}$ a concave, and

G any extension of

g. If

ω is a roof point of

G, then

Let

${G}^{\mathrm{convex}}$ and

${G}^{\mathrm{concave}}$ be as in the above proposition and assume in addition equality in (23),

${G}^{\mathrm{convex}}\left(\omega \right)={G}^{\mathrm{concave}}\left(\omega \right)$. For all convex combination

we conclude

because of (22) the conclusion is

${G}^{\mathrm{convex}}\left({\omega}_{j}\right)={G}^{\mathrm{concave}}\left({\omega}_{j}\right)$ for all

j.

To express this finding we need some standard terminology. A subset $K\subset \Omega $ is called a face of Ω if $\omega \in K$ and (24) necessarily implies ${\omega}_{j}\in K$ for all j. Faces are convex subsets of Ω. If not only Ω but also ${\Omega}^{\mathrm{pure}}$ is compact then faces are compact and any face K is convexly generated by ${\Omega}^{\mathrm{pure}}\cap K$.

The intersection of faces is either empty or a face again. The smallest face containing ω will be called ω-face of Ω and denoted by face${}_{\omega}[\Omega ]$. If K is a face and ω not a point of the boundary of K, then K is the ω-face of Ω.

Now we express the finding above by

**Proposition 3.2**

Coincide a convex and a concave extension of g at $\omega \in \Omega $, then they coincide on the ω-face of Ω.

The least upper bound of a set of convex functions is convex again. Hence, given g on ${\Omega}^{\mathrm{pure}}$, there is a unique largest convex extension of g from ${\Omega}^{\mathrm{pure}}$ to Ω. Similarly there exists a smallest concave extension of g. It is convenient to introduce an extra notation for these extensions:

**Definition 3.2: ${g}^{\cap}$** and ${g}^{\cup}$

Let

g be a real function on

${\Omega}^{\mathrm{pure}}$. We denote by

${g}^{\cup}$ the largest convex and by

${g}^{\cap}$ the smallest concave extension of

g to Ω,

We also write $G={G}^{\cup}$ (or $G={G}^{\cap}$) if G is the largest convex (or the smallest concave) extension of the restriction of G onto ${\Omega}^{\mathrm{pure}}$.

**Proposition 3.3**

Let G be an extension of g and ω one of its roof points. If G is convex, then $G\left(\omega \right)={g}^{\cup}\left(\omega \right)$. If G is concave, then $G\left(\omega \right)={g}^{\cup}\left(\omega \right)$.

If a convex (resp. concave) roof extension of g exists, then it is unique. Because G is convex, $G\le {g}^{\cup}$. Because ω is a roof point, (23) asserts $G\left(\omega \right)\ge {g}^{\cup}\left(\omega \right)$.

Is there a convex extension at all for a given

g ? If there is one then there is also a largest one,

i.e. ${g}^{\cup}$ exists. The answer to the question is affirmative and has been given in [

5] by a variational characterization which is well known in quantum information theory as a recipe to construct entanglement measures:

where the “inf”, respectively “sup”, is running over all extremal convex decompositions

of

ω.

Indeed, if

G is an extension of

g which is convex, the right side of (25) must be always larger than

$G\left(\omega \right)$. On the other hand, given

${\omega}_{1}$ and

${\omega}_{2}$, one can find decompositions (27) for them differing an arbitrary small amount

$\u03f5>o$ from

${g}^{\cup}\left({\omega}_{1}\right)$ respectively

${g}^{\cup}\left({\omega}_{2}\right)$. They may be composed from the pure states

${\pi}_{i,j}$ and probabilities

${p}_{i,j}$,

$i=1,2$. Then

and this is not smaller than

${g}^{\cup}(p{\omega}_{1}+(1-p){\omega}_{2})$. Because

ϵ can be arbitrary near to zero,

${g}^{\cup}$ is convex. The concave case can be settled by a similar reasoning or by

Remark: Hulls of functions

The convex hull of G is the largest convex function which is smaller than G. The concave hull of G is the smallest concave function which is larger than G. In (25) and (26) one uses the values of G at the pure states only. Because the hull construction must respect values on the whole of Ω, there are more constraints to be fulfilled.

The expressions (25) and (26) are similarly structured as those of the convex and the concave hulls of a function

G on Ω. One mimics the proofs and gets

where, as in (24), one has to run through all convex combinations

Obviously, conv$\left[G\right]$ is smaller than ${G}^{\cup}$ and conc$\left[G\right]$ is larger than ${G}^{\cap}$. This quite simple reasoning provides also

**Proposition 3.4**

If G is a concave or a roof extension of g then ${g}^{\cup}=\mathrm{conv}\left[G\right]$.

If G is a convex or a roof extension of g then ${g}^{\cap}=\mathrm{conc}\left[G\right]$.

As a matter of fact one can do similar hull constructions with any subset of Ω which convexly generates Ω. This has been emphasized in [

26].

#### 3.2. Convex and Concave Roofs

If there is a convex roof extension of g, then it equal to ${g}^{\cup}$. There is a sufficient condition to guaranty the roof property of ${g}^{\cup}$ and of ${g}^{\cap}$.

**Proposition 3.5**

Let Ω be a convex set. Assume both, Ω and ${\Omega}^{\mathrm{pure}}$, are compact and g continuous on ${\Omega}^{\mathrm{pure}}$. Then ${g}^{\cup}$ and ${g}^{\cap}$ are roofs. According to proposition 3.3 they are the minimal respectively maximal roof extensions of g.

Remember that ${\Omega}^{\mathrm{pure}}$ is compact if $\Omega =\Omega \left(\mathcal{H}\right)$ and $\mathcal{H}$ is finite dimensional. The requirement of continuity of g is often satisfied in physically motivated applications, though not always. A counter example is the Schmidt number in bipartite quantum systems. In this case it is not known whether ${g}^{\cup}$ and ${g}^{\cap}$ are roofs. Nevertheless. the assumptions needed are rather weak ones. We met them already in proposition 2.1.

The proof will be “constructive” in a certain sense. It is arranged to sharpen “theorem 1” in [

10]: If we know an optimal decomposition with pure states

${\pi}_{1},{\pi}_{2},\cdots $, then every convex combination of them is optimal. In particular, the (convex or concave) roof is affine on the convex set generated by the pure states

${\pi}_{1},{\pi}_{2},\cdots $. Restricted to this this set, the graph of

G is a piece of an affine space. The whole graph of

G appears as composed of flat pieces [

32]. If one would know a covering of Ω by those “convex leaves”, one could compute

${g}^{\cup}$ from the values of

g at

${\Omega}^{\mathrm{pure}}$. Things are similar for

${g}^{\cap}$.

To start proving propositions 3.5 and 2.1 let us repeat the assumptions. Ω is a compact convex set in a real linear space $\mathcal{L}$ of finite dimension, the set ${\Omega}^{\mathrm{pure}}$ of all pure (i.e., extremal) points of Ω is compact. g is a real continuous function on ${\Omega}^{\mathrm{pure}}$. The dimension of Ω as a set in $\mathcal{L}$ is denoted by n. It is the dimension of the affine space generated by Ω.

Remark: The space $\Omega \left(\mathcal{H}\right)$ of density operators is embedded in Herm$\left(\mathcal{H}\right)$. The latter is of dimension ${d}^{2}$ if $dim\mathcal{H}=d$. The affine space generated by $\Omega \left(\mathcal{H}\right)$ is the hyperplane of Hermitian operators of trace one. The dimension of $\Omega \left(\mathcal{H}\right)$ is $n={d}^{2}-1$.

We enlarge

$\mathcal{L}$ to the linear space

${\mathcal{L}}^{\prime}=\mathcal{L}\oplus \mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}$. Its elements,

$X\oplus \lambda $, will be written in vector form

$\{X,\lambda \}$ with two components,

$X\in \mathcal{L}$ and

$\lambda \in \mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}$. We need the set

E is a compact set by our assumption. Hence its convex hull, denoted by

$\Omega \left[g\right]$, is a compact convex set. The set of extremal points of

$\Omega \left[g\right]$ is

E. (If one of the elements of

E would be a convex combination of the others, the same would be true for the corresponding pure states, contradicting our assumptions.)

Choose

$\omega \in \Omega $ and consider in

$\mathcal{L}\oplus \mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}$ the straight line consisting of the points

$\{\omega ,\lambda \}$,

$\lambda \in \mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{R}$. The line intersects with

$\Omega \left[g\right]$ along a compact segment

λ satisfies (32) if and only if there is an extremal decomposition

Therefore,

and there exist extremal decompositions of

ω with equality in (25) respectively (26). Therefore,

${g}^{\cup}$ and

${g}^{\cap}$ are roofs and proposition 3.5 has been proved.

If G is a roof extension of g, the point $\{\omega ,G(\omega \left)\right\}$ is contained in $\Omega \left[g\right]$. Hence it can be represented by a convex combination of elements from E. As the dimension of $\Omega \left[g\right]$ is $n+1$, there are, by a theorem of Carathèodory, pure convex decomposition of length $n+2$. This proves proposition 2.1: For $\Omega =\Omega \left(\mathcal{H}\right)$ it follows $n+2=({d}^{2}-1)+2$.

On the other hand, a point

$\{\omega ,{g}^{\cup}\left(\omega \right)\}$ belongs to a face of the boundary of

$\Omega \left[g\right]$. Its dimension cannot exceed

n. Thus there are, again by Carathèodory, pure decompositions of length

$n+1$. For

$\Omega \left(\mathcal{H}\right)$ this gives an achievable length

${d}^{2}$, an often used fact [

33].

**Definition 3.2: convex leaves**

Let G be a real function on Ω. A subset $K\subset \Omega $ is called a convex leaf of G if

(a) K is compact and convex,

(b) K is the convex hull of $K\cap {\Omega}^{\mathrm{pure}}$,

(c) G is convexly linear on

K,

i.e.K is called complete, if all pure states which can appear in an optimal decomposition of any $\rho \in K$ are contained in K.

**Proposition 3.6**

With the assumptions of proposition 3.5 it holds: For ${g}^{\cup}$ (respectively ${g}^{\cap})$ and any $\omega \in \Omega $ there are complete convex leaves containing ω.

It makes sense to call the set of all complete convex leaves of G the convex foliation of G or, shortly, the G-foliation.

Proof. The proposition will be proved for

${g}^{\cup}$. The case of

${g}^{\cap}$ is similar. There is an affine functions

l such that

We define a subset

K of Ω by

☐

**Proposition 3.6.a: (36) is a complete ${g}^{\cup}$-leaf.**

Clearly,

$\omega \in K$. Let us choose

$\rho \in K$. For an optimal pure decomposition,

$\rho =\sum {p}_{j}{\pi}_{j}$, one obtains

However,

$l\left({\pi}_{j}\right)\le {g}^{\cup}\left({\pi}_{j}\right)$ by assumption. By the equation above, all these inequalities must be equalities. Hence,

the pure composers ${\pi}_{j}$ of every optimal decomposition of any $\rho \in K$ are contained in K,i.e.,

K is complete. Now choose another

${\rho}^{\prime}\in K$ and let

${\rho}^{\prime}=\sum {p}_{k}^{\prime}{\pi}_{k}^{\prime}$ be an optimal decomposition. For

$0<p<1$ we get, applying first our assumption and then convexity of

${g}^{\cup}$,

By the assumption the right hand side can be written

and is equal to the left expression. Hence, equality must hold and

K must be a convex set. We have seen already that every

$\rho \in K$ can be represented by a convex combination of elements from

$K\cap {\Omega}^{\mathrm{pure}}$. Because

g is continuous, the set of all

π satisfying

$g\left(\pi \right)=l\left(\pi \right)$ is compact. Hence,

K is compact. Indeed, it is convexly generated by a compact set.

We repeat a further standard notation. An element ρ of a convex set K is called K-inner or “convexly inner” if for any $\nu \in K$ follows $(1+s)\rho -s\nu \in K$ and small enough positive s. Geometrically, the line segment from ν to ρ can be prolonged a bit without leaving K. There is also a topological characterization: A K-inner point is an inner point of K with respect to the affine space generated by K. (Example: The invertible density operators are the convexly inner points of $\Omega \left(\mathcal{H}\right)$.)

The intersection of complete convex leaves is either empty or it is a complete convex leaf. Hence there is a minimal complete convex leaf containing a given $\omega \in \Omega $, the ω-leaf of ${g}^{\cup}$. It is convexly generated by all those $\pi \in {\Omega}^{\mathrm{pure}}$ which can appear in an optimal decomposition of ω.

The ω-leaf is the largest convex leaf containing ω as convexly inner point.

Now let ${K}_{1}$ be the ${\omega}_{1}$-leaf and ${K}_{2}$ that of ${\omega}_{2}$. If ${\omega}_{1}$ is a convexly inner point of ${K}_{2}$, then $K{}_{1}\subset {K}_{2}$. In particular, ${K}_{1}={K}_{2}$ if and only if they contain a point which is commonly inner. Let us draw a corollary:

If ${K}_{2}$ is properly larger than ${K}_{1}$, the convex dimension of ${K}_{2}$ must be strictly larger than that of ${K}_{1}$.

A chain ${K}_{1}\subset {K}_{2}\subset \cdots $, consisting of different complete ${g}^{\cup}$-leaves, cannot contain more than $n+1$ members. As above, n denotes the convex dimension of Ω. The maximal number of different leaves in any chain is called the depth of the ${g}^{\cup}$-foliation. As said above, the ${g}^{\cup}$-foliation consists of all complete ${g}^{\cup}$-leaves.

In the case $\Omega =\Omega \left(\mathcal{H}\right)$, the depth if bounded by ${d}^{2}$. If the roof is an affine function, see example 2.3, exactly the faces of Ω are its leaves and the bound is reached.

Let us look again to the setting above in more geometric terms. We shall see that the ω-leaves of ${g}^{\cup}$ and ${g}^{\cap}$ correspond uniquely to the faces of $\Omega \left[g\right]$.

The triple

$\{\Omega [g],\Omega ,\Pi \}$ is a

fiber bundle with bundle space

$\Omega \left[g\right]$, base space Ω, and projection

In this scheme a roof

G becomes a

cross section, say

${s}_{G}$, by setting

we get

$\Pi \left({s}_{G}\left(\omega \right)\right)=\omega $, which is necessary for a bundle structure.

The boundary,

$\partial \Omega \left[g\right]$, of

$\Omega \left[g\right]$ is the union of three disjunct sets:

The cross section (38) with

$G={g}^{\cup}$ maps Ω onto

${\partial}^{0}\Omega \left[g\right]\cup {\partial}^{-}\Omega \left[g\right]$ while the cross section with

$G={g}^{\cap}$ maps the base space onto

${\partial}^{0}\Omega \left[g\right]\cup {\partial}^{+}\Omega \left[g\right]$.

The fibres degenerate to a point at the boundary part (39) and can be identified with a subset of Ω. By proposition 3.2 that subset consists of the pure point and possibly of some faces at which ${g}^{\cup}={g}^{\cap}$, and all roof extensions of g coincide and are affine.

Let us consider a face

$\tilde{K}$ contained in the “lower” part

${\partial}^{-}\Omega \left[g\right]$ of the boundary (40).

$\tilde{K}$ is convex by definition and compact because of the compactness of

$\Omega {\left[g\right]}^{\mathrm{pure}}$. Therefore, the projection

K of

$\tilde{K}$ to Ω,

$\Pi \phantom{\rule{0.166667em}{0ex}}\tilde{K}=K$, is convex and compact, see (37). We use the supposed face property:

$\tilde{\omega}\in \tilde{K}$ implies

if all

${p}_{j}>0$. Writing this out in the manner

$\tilde{\omega}=\{\omega ,{g}^{\cup}\left(\omega \right)\}$, and so on, we arrive at

Now we can state

**Proposition 3.7**

There is a one-to-one correspondence between the faces of $\Omega \left[g\right]$ contained in ${\partial}^{0}\Omega \left[g\right]\cup {\partial}^{-}\Omega \left[g\right]$ and the complete convex leaves of ${g}^{\cup}$. The cross section ${s}_{G}$, with $G={g}^{\cup}$, maps complete convex leaves of ${g}^{\cup}$ onto faces of $\Omega \left[g\right]$. The bundle projection Π returns them back to Ω.

For the concave roof things are similar.