arguably two of the most ubiquitous monotones—resource weight [2, 1] and
resource robustness [7]—within the context of resource theories with a linear
structure. By connecting them to a monotone in the resource theory of
distinguishability, we complement the results of [53], [47], and [13]. In the
first two, robustness measures are connected to discrimination tasks, while
the latter article describes a similar connection between the weight measure
and the state exclusion task.

Let’s consider a resource theory $({R}_{\rm free},{R},\boxtimes)$ with a
convex-linear structure on $R$ that is preserved by $\boxtimes$. The elements
of $R$ can thus be represented as vectors, and convex combinations are
preserved by the composition of resources $\boxtimes$. Furthermore, just like
in the previous section, we assume that $r\boxtimes s$ is a single resource
for all $r,s\in R$, so that $\boxtimes$ is a bilinear map $R\times R\to R$. A
more general scenario corresponding to a broader idea of convex-linear
resource theories is treated in [15].

We can construct the resource theory of $3$-tuples
$({R}^{\scriptscriptstyle{(3)}}_{\rm
cons},{R}^{\scriptscriptstyle{(3)}},\boxtimes)$ as described in section 4.2.1
and define the following function.

######  Definition 52.

The convex alignment is a function
$\mathsf{cva}\colon{R}^{\scriptscriptstyle{(3)}}\to\overline{\mathbb{R}}$
defined by

$\mathsf{cva}(r,s,t)\coloneqq\begin{cases}\lambda&\text{if $r=\lambda
s+(1-\lambda)t$ for $\lambda\in[0,1]$.}\\\ 1&\text{otherwise.}\end{cases}$
(93)

######  Lemma 53 (convex alignment is a $3$-contraction).

Let $({R}_{\rm free},{R},\boxtimes)$ be a resource theory with a convex-linear
structure as described at the start of this section. The convex alignment,
$\mathsf{cva}$, is a monotone in the resource theory
$({R}^{\scriptscriptstyle{(3)}}_{\rm
cons},{R}^{\scriptscriptstyle{(3)}},\boxtimes)$.

###### Proof.

Let $(r,s,t)\in{R}^{\scriptscriptstyle{(3)}}$ and
$(u,u,u)\in\mathcal{R}^{\scriptscriptstyle{(3)}}_{\rm cons}$. We aim to show
that

$\mathsf{cva}(r,s,t)\geqslant\mathsf{cva}(r\boxtimes u,s\boxtimes u,t\boxtimes
u)$ (94)

holds for all $r,s,t,u\in R$.

If $\mathsf{cva}(r,s,t)=1$, then its value cannot increase. Otherwise, if
$\mathsf{cva}(r,s,t)=\lambda$ is strictly less than 1, then $r=\lambda
s+(1-\lambda)t$. By the convex-linearity of $\boxtimes$, we have

$\bigl{(}\lambda s+(1-\lambda)t\bigr{)}\boxtimes u=\lambda s\boxtimes
u+(1-\lambda)t\boxtimes u.$ (95)

Therefore, $\mathsf{cva}(r,s,t)=\mathsf{cva}(r\boxtimes u,s\boxtimes
u,t\boxtimes u)$ whenever $\mathsf{cva}(r,s,t)<1$. Consequently, convex
alignment is a 3-contraction. ∎

Now we can use the generalized construction of monotones from $k$-contractions
(theorem 51) to get monotones for $({R}_{\rm free},{R},\boxtimes)$ by
optimizing the convex alignment in various ways. Let us focus on a
construction of $\mathsf{cva}\textup{-}\mathsf{min}\circ\mathcal{E}_{W_{\rm
dc}}$ with $W_{\rm dc}$ that is of form as in (89). Specifically, we use
$W_{\rm dc}=S_{1}\times S_{2}\times
S_{3}\in\mathcal{DC}({R}^{\scriptscriptstyle{(3)}})$, where each $S_{i}$ is
itself a downward closed subset of $R$. There are many downward closed sets
one could use for each $S_{i}$, but here we restrict our attention to the two
most obvious choices—$R_{\rm free}$ and $R$. Even with this restriction, one
can obtain 12 constructions of the form
$\mathsf{cva}\textup{-}\mathsf{min}\circ\mathcal{E}_{W_{\rm dc}}$.
Specifically, there are three possible choices of the projection $\Pi_{i}$,
which then determines $\mathcal{E}$. For each of them we let the corresponding
$S_{i}$ be $R$ without loss of generality, which leaves 4 choices for the
other two downward closed sets. Out of these 12 constructions in total, 8
produce a constant monotone and are therefore uninteresting. The other 4 are
the following.

  1. (i)

The resource weight (also known as the resource fraction) $M_{\rm w}\colon
R\to\overline{\mathbb{R}}$ is defined as
$\mathsf{cva}\textup{-}\mathsf{min}\circ\mathcal{E}_{W_{\rm dc}}$ for
$\mathcal{E}=\Pi_{1}^{-1}$ and $W_{\rm dc}=R\times R\times R_{\rm free}$.
Explicitly, its value for any resource $r\in R$ is

$\begin{split}M_{\rm
w}(r)&\coloneqq\inf\big{\\{}\mathsf{cva}(r,s,t)\nonscript\;\,\big{|}\allowbreak\nonscript\;\,\mathopen{}s\in
R,\;t\in R_{\rm free}\big{\\}}\\\
&=\inf\big{\\{}\lambda\nonscript\;\,\big{|}\allowbreak\nonscript\;\,\mathopen{}r\in\lambda
R+(1-\lambda)R_{\rm free}\big{\\}}.\end{split}$ (96)

It corresponds to the smallest weight of a resource that can be used to form
$r$ by convex mixture with some free resource.

  2. (ii)

The resource robustness $M_{\rm rob}\colon R\to\overline{\mathbb{R}}$ is
defined as $\mathsf{cva}\textup{-}\mathsf{min}\circ\mathcal{E}_{W_{\rm dc}}$
for $\mathcal{E}=\Pi_{3}^{-1}$ and $W_{\rm dc}=R_{\rm free}\times R\times R$.
Explicitly, its value for any resource $t\in R$ is

$\begin{split}M_{\rm
rob}(t)&\coloneqq\inf\big{\\{}\mathsf{cva}(r,s,t)\nonscript\;\,\big{|}\allowbreak\nonscript\;\,\mathopen{}r\in
R_{\rm free},\;s\in R\big{\\}}\\\
&=\inf\big{\\{}\lambda\nonscript\;\,\big{|}\allowbreak\nonscript\;\,\mathopen{}\lambda
s+(1-\lambda)t\in R_{\rm free},\;s\in R\big{\\}}.\end{split}$ (97)

It is the smallest weight of a resource that one needs to convexly mix with
$t$ in order to obtain a free resource.

  3. (iii)

The free robustness $M_{\rm f.\,rob}\colon R\to\overline{\mathbb{R}}$ is
defined as $\mathsf{cva}\textup{-}\mathsf{min}\circ\mathcal{E}_{W_{\rm dc}}$
for $\mathcal{E}=\Pi_{3}^{-1}$ and $W_{\rm dc}=R_{\rm free}\times R_{\rm
free}\times R$. Explicitly, its value for any resource $t\in R$ is

$\begin{split}M_{\rm
f.\,rob}(t)&\coloneqq\inf\big{\\{}\mathsf{cva}(r,s,t)\nonscript\;\,\big{|}\allowbreak\nonscript\;\,\mathopen{}r\in
R_{\rm free},\;s\in R_{\rm free}\big{\\}}\\\
&=\inf\big{\\{}\lambda\nonscript\;\,\big{|}\allowbreak\nonscript\;\,\mathopen{}\lambda
s+(1-\lambda)t\in R_{\rm free},\;s\in R_{\rm free}\big{\\}}.\end{split}$ (98)

It is the smallest weight of a free resource that one needs to convexly mix
with $t$ in order to obtain another free resource.

  4. (iv)

The resource non-convexity $M_{\rm nc}\colon R\to\overline{\mathbb{R}}$ is
defined as $\mathsf{cva}\textup{-}\mathsf{min}\circ\mathcal{E}_{W_{\rm dc}}$
for $\mathcal{E}=\Pi_{1}^{-1}$ and $W_{\rm dc}=R\times R_{\rm free}\times
R_{\rm free}$. Explicitly, its value for any resource $r\in R$ is

$\begin{split}M_{\rm
nc}(r)&\coloneqq\inf\big{\\{}\mathsf{cva}(r,s,t)\nonscript\;\,\big{|}\allowbreak\nonscript\;\,\mathopen{}s\in
R_{\rm free},\;t\in R_{\rm free}\big{\\}}\\\
&=\inf\big{\\{}\lambda\nonscript\;\,\big{|}\allowbreak\nonscript\;\,\mathopen{}r\in\lambda
R_{\rm free}+(1-\lambda)R_{\rm free}\big{\\}}.\end{split}$ (99)

It is trivial if all the sets of free resources happen to be convex. Otherwise
it tells us about the ordering of resources that are within the convex hull of
the free resources, but are not free themselves. It quantifies the relative
distance of a resource from the set of free resources in terms of its convex
decompositions into free resources. Its value is set to 0 if the resource in
question is free itself, and 1 if it is outside of the convex hull of the free
resources.

$h$$\mapsto$$h$$f$$g$$f$$\boxtimes$$g$$=$$f$$g$$\cup$$f$$g$$\cup$$g$$f$$f$$g$$\cup$$g$$f$$\cup$$=$$\cup$$\;\;g\;\;$$f$$\cup$$f$$\;\;g\;\;$$\;\;g\;\;$$f$$\cup$$f$$\;\;g\;\;$$f$$\boxtimes$$\;\;g\;\;$$\;\;g\;\;$$f$$\cup$$\cup$$\cup$$f$$\;\;g\;\;$$=$$f$$\boxtimes$$g$$h$$f\boxtimes
g$$h$$\cup$$f\boxtimes h$$g$$\quad
P\quad$$S$$T$$X$$Y$$X$$S$$T$$Y$$\Lambda$$\phi$$f\textup{-}\mathsf{yield}$$=$$\sup$$\phi$$f$
$\rho$$\;\;\psi\;\;$$\psi\in R_{\rm free}$$\rho\in R_{\rm
free}$$\phi$$f\textup{-}\mathsf{yield}$$=$$\sup$$\phi$$f$ $\rho$$\rho\in
R_{\rm free}$$\phi$$f\textup{-}\mathsf{cost}$$=$$\inf$$f$
$\rho$$\phi$$=$$\;\;\psi\;\;$ $\rho$$\text{and}\;\;\psi\in R_{\rm
free}$$\phi$$f\textup{-}\mathsf{yield}$$=$$\sup$$\phi$$f$
$\rho$$\;\;\psi\;\;$$\psi\in D$$\rho\in
D$$s_{1}$$s_{2}$$t_{1}$$t_{2}$$t_{4}$$t_{3}$$a$$b$$c$$d$$R$$R_{\rm free}$
Figure 4. A pictorial depiction of the optimal convex decompositions for each
of the four monotones mentioned in this section: (a) resource weight $M_{\rm
w}$, (b) resource robustness $M_{\rm rob}$, (c) free robustness $M_{\rm
f.\,rob}$, and (d) resource non-convexity $M_{\rm nc}$. Grey disc represents
the set $R$ of all resources, while the yellow “hourglass” witin represents
the free resources among them. In order to illustrate each of the four optimal
decompositions, we select a distinct resource (element of $X$), depicted by a
green node. These demopositions are given by the three points along one of the
line segments with an orange and purple portion. The value of each of the
monotones for these; $M_{\rm w}(a),M_{\rm rob}(b),M_{\rm f.\,rob}(c)$, and
$M_{\rm nc}(d)$; can be read off as the length of the respective orange
segment divided by the total lenth of the orange and purple segments combined.

As a consequence of lemma 50, theorem 51 and lemma 53, all four functions
above are monotones. However, being able to prove the monotonicity of these
four functions is not where the value of the generalized construction of
monotones from contractions lies. What they provide is an understanding of the
assumptions required in order for these functions to be monotones.
Furthermore, they give us a unified picture, within which we can adjust
various elements of the monotone constructions according to the question we
are interested in. In this case, there are many more monotones one can obtain
from $\mathsf{cva}$ in this way, since $R$ or $R_{\rm free}$ in the
optimization can be replaced by any other downward closed set.

###  4.3. tocGeneral Ways of Translating Monotones Between Resource
TheoriestocIn General

In section 4.1 we investigated how one can translate monotones from a resource
theory $\mathcal{Q}$ given by $(Q_{\rm free},Q,\boxtimes_{Q})$ to a resource
theory $\mathcal{R}=({R}_{\rm free},{R},\boxtimes)$ when the two are in fact
identical. Then, in section 4.2, we looked at the choice of $\mathcal{Q}$ in
the form of a resource theory of distinguishability. Here, we would like
explore what can be said in general. Can the methods introduced in sections
4.1 and 4.2 be extended to the case of arbitrary $\mathcal{Q}$?

The choices of the mediating preordered set
$(\mathcal{A},\succeq_{\mathcal{A}})$ we consider are
$(\mathcal{P}(Q),\succeq_{\rm enh})$ and $(\mathcal{P}(Q),\succeq_{\rm deg})$.
For any monotone $f$ on $\mathcal{Q}$, we again have corresponding root
monotones $f\textup{-}\mathsf{max}$ and $f\textup{-}\mathsf{min}$ as
introduced in corollary 37.

In order to find out which maps can be used as the mediating order-preserving
map ${(R,\succeq)\to(\mathcal{P}(Q),\succeq_{\rm enh})}$, we can use the
following sufficient conditions.

######  Lemma 54 (mediating maps for $\succeq_{\rm enh}$).

Let $({R}_{\rm free},{R},\boxtimes)$ and $(Q_{\rm free},Q,\boxtimes_{Q})$ be
resource theories and let $F\colon R\to\mathcal{P}(Q)$ be a function with an
extension $F\colon\mathcal{P}(R)\to\mathcal{P}(Q)$ obtained from the original
$F$ by requiring that it commutes with unions.202020The extension
$F\colon\mathcal{P}(R)\to\mathcal{P}(Q)$ maps a set $S$ to the union of images
of elements of $S$ under $F\colon R\to\mathcal{P}(Q)$. It is the unique
extension of $F\colon R\to\mathcal{P}(Q)$ to a suplattice homomorphism
$\mathcal{P}(R)\to\mathcal{P}(Q)$. If for all $r\in R$ we have

$F(R_{\rm free}\boxtimes r)\subseteq Q_{\rm free}\boxtimes_{Q}F(r),$ (100)

i.e., if
$F\bigl{(}\mathord{\downarrow}(r)\bigr{)}\subseteq\mathord{\downarrow}\bigl{(}F(r)\bigr{)}$
holds, then $F\colon(R,\succeq)\to(\mathcal{P}(Q),\succeq_{\rm enh})$ is
order-preserving.

###### Proof.

We need to show that for $F$ as above, the implication $r\succeq s\implies
F(r)\succeq_{\rm enh}F(s)$ holds for any $r,s\in R$. This fact can be broken
down as follows:

$\begin{split}r\succeq s&\iff s\in R_{\rm free}\boxtimes r\\\ &\implies
F(s)\subseteq F(R_{\rm free}\boxtimes r)\\\ &\implies F(s)\subseteq Q_{\rm
free}\boxtimes_{Q}F(r)\\\ &\iff F(r)\succeq F(s),\end{split}$ (101)

where the second implication follows from property (100). The statement of the
lemma then follows by recognizing that $\succeq$ and $\succeq_{\rm enh}$ are
identical as preorders on $\mathcal{P}(Q)$. ∎

######  Lemma 54′ (mediating maps for $\succeq_{\rm enh}$).

Alternatively, if we have

$\displaystyle F(r\boxtimes s)$
$\displaystyle=F(r)\boxtimes_{Q}F(s)\quad\forall r,s\in R,\text{ and }$ (102a)
$\displaystyle F(R_{\rm free})$ $\displaystyle\subseteq Q_{\rm
free}\boxtimes_{Q}F(0),$ (102b)

then $F\colon(R,\succeq)\to(\mathcal{P}(Q),\succeq_{\rm enh})$ is order-
preserving. Finally, if instead $F$ satisfies

$\displaystyle F(r\boxtimes s)$ $\displaystyle\subseteq
F(r)\boxtimes_{Q}F(s)\quad\forall r,s\in R,\text{ and }$ (103a) $\displaystyle
F(R_{\rm free})$ $\displaystyle\subseteq Q_{\rm free},$ (103b)

then $F\colon(R,\succeq)\to(\mathcal{P}(Q),\succeq_{\rm enh})$ is order-
preserving.

###### Proof.

Conditions (102) imply condition (100) via

$F(R_{\rm free}\boxtimes r)=F(R_{\rm free})\boxtimes_{Q}F(r)\subseteq Q_{\rm
free}\boxtimes_{Q}F(0)\boxtimes_{Q}F(r)=Q_{\rm free}\boxtimes_{Q}F(0\boxtimes
r)=Q_{\rm free}\boxtimes_{Q}F(r),$ (104)

so that the first part of lemma 54′ follows from lemma 54.

Conditions (103) imply condition (100) via

$F(R_{\rm free}\boxtimes r)\subseteq F(R_{\rm free})\boxtimes_{Q}F(r)\subseteq
Q_{\rm free}\boxtimes_{Q}F(r),$ (105)

so that the second part of lemma 54′ also follows from lemma 54. ∎

######  Example 40′ (adding a catalyst is order-preserving).

The augmentation map $\mathsf{Aug}_{C}\colon R\to\mathcal{P}(R)$ from example
40, defined for any $C\subseteq R$ by

$\mathsf{Aug}_{C}(r)\coloneqq C\boxtimes r,$ (106)

satisfies condition (100), since we have

$\mathsf{Aug}_{C}(R_{\rm free}\boxtimes r)=C\boxtimes R_{\rm free}\boxtimes
r=R_{\rm free}\boxtimes\mathsf{Aug}_{C}(r).$ (107)

Lemma 54 thus provides a way to prove that $\mathsf{Aug}_{C}$ is order-
preserving as a map $(R,\succeq)\to(\mathcal{P}(R),\succeq_{\rm enh})$.
However, it satisfies neither condition (102a) nor condition (103b) in
general.

######  Example 39′ (copying is order-preserving).

The copy map, $\mathsf{Copy}_{n}\colon R\to\mathcal{P}(R)$ was defined in
example 39 as the combination of $n$ copies of a resource,

$\mathsf{Copy}_{n}(r)\coloneqq r\boxtimes r\boxtimes\ldots\boxtimes r\equiv
r^{\boxtimes n}.$ (108)

The image of a set of resources $S$ by $\mathsf{Copy}_{n}$ cannot in general
be expressed as $S^{\boxtimes n}$. Nevertheless, one can show that
$\mathsf{Copy}_{n}(r\boxtimes
s)\subseteq\mathsf{Copy}_{n}(r)\boxtimes\mathsf{Copy}_{n}(s)$ and
$\mathsf{Copy}_{n}(R_{\rm free})\subseteq R_{\rm free}$, which corresponds to
conditions (103).

However, the map $\mathcal{E}$ defined in section 4.2.4 doesn’t satisfy these
conditions. In general, only conditions (102a) and (103a) hold for
$\mathcal{E}$. In particular, $\mathcal{E}$ is _not_ order-preserving as a
function $(R,\succeq)\to(\mathcal{P}(Q),\succeq_{\rm enh})$. The function
$\mathcal{E}$ is, nonetheless, an example of an order-preserving map
$(R,\succeq)\to(\mathcal{P}(Q),\succeq_{\rm deg})$. How could we generalize
this fact? Recall that for $k=2$, $\mathcal{E}$ maps $r$ to its preimage under
the projection $\Pi_{1}\colon(r,s)\mapsto r$. The following lemma provides
sufficient conditions for such functions to be order-preserving in general.

######  Lemma 55 (mediating maps for $\succeq_{\rm deg}$).

Let $({R}_{\rm free},{R},\boxtimes)$ and $(Q_{\rm free},Q,\boxtimes_{Q})$ be
resource theories and let $F\colon R\to\mathcal{P}(Q)$ be a function. If there
exists a map $G\colon Q\to R$ satisfying

$\displaystyle F(r)$ $\displaystyle=G^{-1}(r),$ (109a) $\displaystyle
G(p\boxtimes_{Q}q)$ $\displaystyle\supseteq G(p)\boxtimes G(q)\quad\forall
p,q\in Q,\text{ and }$ (109b) $\displaystyle G(Q_{\rm free})$
$\displaystyle\supseteq R_{\rm free},$ (109c)

then $F\colon(R,\succeq)\to(\mathcal{P}(Q),\succeq_{\rm deg})$ is order-
preserving.

###### Proof.

We want to show that for any $r,s\in R$ such that $r\succeq s$, there exists a
degradation $D\colon F(r)\to F(s)$. Firstly, note that the fact that the image
of $Q_{\rm free}$ under $G$ contains $R_{\rm free}$ (property 109c) means that
there is function $G^{\dagger}\colon R_{\rm free}\to Q_{\rm free}$ such that

$G\circ G^{\dagger}=\mathsf{Id}_{R_{\rm free}},$ (110)

where $\mathsf{Id}_{R_{\rm free}}$ is the canonical embedding of $R_{\rm
free}$ in $R$. That is, $G^{\dagger}$ is a partial right inverse of $G$.

If $r\succeq s$ holds, then there is an $x\in R_{\rm free}$ such that $s\in
r\boxtimes x$. For all $s\in F(r)=G^{-1}(r)$, we then have

$s\in r\boxtimes x=G(s)\boxtimes
G(G^{\dagger}(x))=G(s\boxtimes_{Q}G^{\dagger}(x)),$ (111)

so that there exists a resource $t$ in the set $s\boxtimes_{Q}G^{\dagger}(x)$
such that $t$ is also in $G^{-1}(s)=F(s)$. If we let $D\colon F(r)\to F(s)$ be
defined by $D(s)\coloneqq t$, then $D$ is clearly a degradation since
$G^{\dagger}(x)$ is an element of $Q_{\rm free}$. ∎

An alternative way to prove lemma 55 would be to show that conditions 109
imply

$F\bigl{(}\mathord{\uparrow}(q)\bigr{)}\subseteq\mathord{\uparrow}\bigl{(}F(q)\bigr{)}\quad\forall\;r\in
Q,$ (112)

which, by an argument analogous to the proof of lemma 54, is a sufficient
condition for the function $F\colon(R,\succeq)\to(\mathcal{P}(Q),\succeq_{\rm
deg})$ to be order-preserving. One can check that $\mathcal{E}$ indeed
satisfies conditions (109) if $G$ is chosen to be the projection
$\Pi_{i}\colon{R}^{\scriptscriptstyle{(k)}}\to R$. In fact, (109b) becomes an
equality in this case.

Lemma 55 cannot be used, however, to show that the map $\mathcal{E}_{R_{\rm
dc}}$ introduced in section 4.2.3 is order-preserving. As we have seen
explicitly in the proof of theorem 51, proving this fact is not necessary when
all we care about is the resulting monotone obtained from the root monotone
$f\textup{-}\mathsf{min}$ by the Broad Scheme. This is because we can
incorporate the restriction of the image of $\mathcal{E}$ to a downward closed
set such as $W_{\rm dc}=R\times R_{\rm free}$ into the root monotone by
restricting the domain of the contraction $f$. Corollary 37 can then still be
used to show that the function $f_{W_{\rm dc}}\textup{-}\mathsf{min}$ is a
monotone.

For completenes, we nevertheless include the following lemma which, when
combined with the other results in this section, shows that the map
$\mathcal{E}_{R_{\rm dc}}$ is itself order-preserving.

######  Lemma 56 (mediating maps with intersections).

Let $({R}_{\rm free},{R},\boxtimes)$ and $(Q_{\rm free},Q,\boxtimes_{Q})$ be
resource theories and let $W_{\rm uc}\in\mathcal{UC}(Q)$ and $W_{\rm
dc}\in\mathcal{DC}(Q)$ be upward and downward closed subsets of $Q$,
respectively.

  1. (i)

If $F\colon(R,\succeq)\to(\mathcal{P}(Q),\succeq_{\rm enh})$ is an order-
preserving map, then the map $F_{W_{\rm uc}}$ defined by

$F_{W_{\rm uc}}(r)\coloneqq F(r)\cap W_{\rm uc}$ (113)

is also order-preserving as a map $(R,\succeq)\to(\mathcal{P}(Q),\succeq_{\rm
enh})$.

  2. (ii)

If $F\colon(R,\succeq)\to(\mathcal{P}(Q),\succeq_{\rm deg})$ is an order-
preserving map, then the map $F_{W_{\rm dc}}$ defined by

$F_{W_{\rm dc}}(r)\coloneqq F(r)\cap W_{\rm dc}$ (114)

is also order-preserving as a map $(R,\succeq)\to(\mathcal{P}(Q),\succeq_{\rm
deg})$.

###### Proof.

This lemma is basically saying that the maps

$\displaystyle(\mathcal{P}(Q),\succeq_{\rm enh})$
$\displaystyle\to(\mathcal{P}(Q),\succeq_{\rm enh})$
$\displaystyle(\mathcal{P}(Q),\succeq_{\rm deg})$
$\displaystyle\to(\mathcal{P}(Q),\succeq_{\rm deg})$ $\displaystyle S$
$\displaystyle\mapsto S\cap W_{\rm uc}$ $\displaystyle S$
$\displaystyle\mapsto S\cap W_{\rm dc}$

are both order-preserving. The first one is order-preserving, because for any
$S,T\in\mathcal{P}(R)$, we have

$\begin{split}S\succeq_{\rm
enh}T&\iff\mathord{\downarrow}(S)\supseteq\mathord{\downarrow}(T)\\\
&\,\implies W_{\rm uc}\cap\mathord{\downarrow}(S)\supseteq W_{\rm
uc}\cap\mathord{\downarrow}(T)\\\
&\,\implies\mathord{\downarrow}\bigl{(}W_{\rm
uc}\cap\mathord{\downarrow}(S)\bigr{)}\supseteq\mathord{\downarrow}\bigl{(}W_{\rm
uc}\cap\mathord{\downarrow}(T)\bigr{)}\\\ &\iff\mathord{\downarrow}(W_{\rm
uc}\cap S)\supseteq\mathord{\downarrow}(W_{\rm uc}\cap T)\\\ &\iff W_{\rm
uc}\cap S\succeq_{\rm enh}W_{\rm uc}\cap T.\end{split}$ (115)

The first equivalence follows from lemma 28 and the penultimate one is a
consequence of lemma 57 presented below. All in all, this concludes the proof
of the first part of lemma 56.

The second part can be shown in an analogous way. In particular, the fact that
the map $S\mapsto S\cap W_{\rm dc}$ is order-preserving follows by

$\begin{split}S\succeq_{\rm
deg}T&\iff\mathord{\uparrow}(S)\subseteq\mathord{\uparrow}(T)\\\ &\,\implies
W_{\rm dc}\cap\mathord{\uparrow}(S)\subseteq W_{\rm
dc}\cap\mathord{\uparrow}(T)\\\ &\,\implies\mathord{\uparrow}\bigl{(}W_{\rm
dc}\cap\mathord{\uparrow}(S)\bigr{)}\subseteq\mathord{\uparrow}\bigl{(}W_{\rm
dc}\cap\mathord{\uparrow}(T)\bigr{)}\\\ &\iff\mathord{\uparrow}(W_{\rm dc}\cap
S)\subseteq\mathord{\uparrow}(W_{\rm dc}\cap T)\\\ &\iff W_{\rm dc}\cap
S\succeq_{\rm deg}W_{\rm dc}\cap T,\end{split}$ (116)

The first equivalence follows from lemma 32 and the penultimate one is again a
consequence of lemma 57. ∎

######  Lemma 57.

Let $(\mathcal{A},\succeq)$ be a preordered set and let $S$ and $T$ be two
subsets of $\mathcal{A}$. Then we have

$\displaystyle\mathord{\downarrow}\bigl{(}\mathord{\uparrow}(S)\cap\mathord{\downarrow}(T)\bigr{)}$
$\displaystyle=\mathord{\downarrow}\bigl{(}\mathord{\uparrow}(S)\cap
T\bigr{)},$ (117a)
$\displaystyle\mathord{\uparrow}\bigl{(}\mathord{\downarrow}(S)\cap\mathord{\uparrow}(T)\bigr{)}$
$\displaystyle=\mathord{\uparrow}\bigl{(}\mathord{\downarrow}(S)\cap
T\bigr{)}.$ (117b)

###### Proof.

First, let us prove equation (117a). The set on the right hand side is clearly
a subset of the one on the left, so let’s argue why also
$\mathord{\downarrow}\bigl{(}\mathord{\uparrow}(S)\cap\mathord{\downarrow}(T)\bigr{)}\subseteq\mathord{\downarrow}\bigl{(}\mathord{\uparrow}(S)\cap
T\bigr{)}$ holds. For any
$x\in\mathord{\downarrow}\bigl{(}\mathord{\uparrow}(S)\cap\mathord{\downarrow}(T)\bigr{)}$,
there exists a $y\in\mathord{\uparrow}(S)\cap\mathord{\downarrow}(T)$ such
that $y\succeq x$. Therefore, there is a $t\in T$, such that $t\succeq y$ and
$t\in\mathord{\uparrow}(S)$. Since $t\succeq x$, this means that
$x\in\mathord{\downarrow}\bigl{(}\mathord{\uparrow}(S)\cap T\bigr{)}$, thus
proving equation (117a).

Equation (117b) is the dual statement to (117a) and therefore it follows by
reversing $\succeq$. ∎

Given the choice of $(Q_{\rm
free},Q,\boxtimes_{Q})\coloneqq({R}^{\scriptscriptstyle{(k)}}_{\rm
cons},{R}^{\scriptscriptstyle{(k)}},\boxtimes)$ and $F\coloneqq\mathcal{E}$ in
lemma 56, we thus recover the fact that $\mathcal{E}_{W_{\rm
dc}}\colon(R,\succeq)\to(\mathcal{P}({R}^{\scriptscriptstyle{(k)}}),\succeq_{\rm
deg})$ is order-preserving.

######  Corollary 58 (translating monotones between resource theories).

Consider two resource theories $(Q_{\rm free},Q,\boxtimes_{Q})$ and $({R}_{\rm
free},{R},\boxtimes)$ Given

  * •

a monotone $f\colon Q\to\overline{\mathbb{R}}$,

  * •

a function $F\colon R\to\mathcal{P}(Q)$ that satisfies condition (100), and

  * •

an upward closed set $W_{\rm uc}\in\mathcal{UC}(Q)$,

we get a monotone $f\textup{-}\mathsf{max}\circ F_{W_{\rm uc}}\colon
R\to\overline{\mathbb{R}}$ given by

$f\textup{-}\mathsf{max}\circ F_{W_{\rm
uc}}(r)=\sup\big{\\{}f(q)\nonscript\;\,\big{|}\allowbreak\nonscript\;\,\mathopen{}q\in
F(r)\cap W_{\rm uc}\big{\\}}.$ (118)

Similarly, given

  * •

a monotone $f\colon Q\to\overline{\mathbb{R}}$,

  * •

a function $F\colon R\to\mathcal{P}(Q)$ that satisfies conditions (109), and

  * •

a downward closed set $W_{\rm dc}\in\mathcal{DC}(Q)$,

we get a monotone $f\textup{-}\mathsf{min}\circ F_{W_{\rm dc}}\colon
R\to\overline{\mathbb{R}}$ given by

$f\textup{-}\mathsf{min}\circ F_{W_{\rm
dc}}(r)=\inf\big{\\{}f(q)\nonscript\;\,\big{|}\allowbreak\nonscript\;\,\mathopen{}q\in
F(r)\cap W_{\rm dc}\big{\\}}.$ (119)

######  Remark 59.

Note that the generalized yield and cost constructions (theorem 24) can be
applied in succession with those from corollary 58. However, neither of these
commute in general. For example, for generic $f$ and $F$, composing the
constructions that use infima give

$\begin{split}(f\textup{-}\mathsf{min}\circ F_{W_{\rm
uc}})\textup{-}\mathsf{cost}(q)&=\inf f\bigl{(}W_{\rm uc}\cap
F(\mathord{\uparrow}(q))\bigr{)}\\\ &\neq\inf
f\bigl{(}\mathord{\uparrow}(W_{\rm uc}\cap F(q))\bigr{)}\\\
&=(f\textup{-}\mathsf{cost})\textup{-}\mathsf{min}\circ F_{W_{\rm
uc}}(q).\end{split}$ (120)

##  5\. Assessing Informativeness of Monotonestoc in General Resource Theories

The general monotone constructions (theorem 24 and corollary 58) have several
inputs that need to be specified to obtain a single resource monotone. In this
section, we would like to address the question of which choices of these input
parameters are good in the sense that they yield a useful resource monotone.
In order to assess the usefulness of monotones as far as characterizing a
preordered set $(\mathcal{A},\succeq_{\mathcal{A}})$ is concerned, we define a
preorder $\sqsupseteq_{\mathcal{A}}$ on the set of monotones itself. We denote
this set by $\mathsf{Mon}(\mathcal{A})$. It is just the collection of all
order-preserving maps from $(\mathcal{A},\succeq_{\mathcal{A}})$ to
$(\overline{\mathbb{R}},\geqslant)$. In this context, we consider a monotone
$f$ to be more “useful” than a monotone $g$ if it contains all of the
information about $(\mathcal{A},\succeq_{\mathcal{A}})$ that $g$ does and
possibly more. We now formalize what we mean by the amount of inforamtion a
monotone has about a preordered set.

A function $f\colon\mathcal{A}\to\overline{\mathbb{R}}$ is a monotone if and
only if for all pairs $(a,b)\in\mathcal{A}\times\mathcal{A}$, the following
implication holds:

$f(a)<f(b)\implies a\not\succeq_{\mathcal{A}}b.$ (121)

That is, monotones contain information about the order relation
$\succeq_{\mathcal{A}}\subseteq\mathcal{A}\times\mathcal{A}$ insofar as they
witness when pairs of elements of $\mathcal{A}$ are _not_ related by
$\succeq_{\mathcal{A}}$. Of course, if $f(a)\geqslant f(b)$, the implication
above doesn’t let us learn anything about the order relation
$\succeq_{\mathcal{A}}$. Given a monotone $f$, a pair $(a,b)$ is henceforth
called $\bm{f}$-interesting if $f(a)<f(b)$ holds. The $f$-interesting pairs
are those, for which we can learn that $a\not\succeq_{\mathcal{A}}b$ holds
from the fact that $f$ is a monotone.

The set of all $f$-interesting pairs for a monotone $f$ is denoted by

$f\textup{-}\mathsf{Interesting}\left(\mathcal{A},\succeq_{\mathcal{A}}\right)\coloneqq\big{\\{}(a,b)\in\mathcal{A}\times\mathcal{A}\nonscript\;\,\big{|}\allowbreak\nonscript\;\,\mathopen{}f(a)<f(b)\big{\\}}.$
(122)

We also refer to
$f\textup{-}\mathsf{Interesting}\left(\mathcal{A},\succeq_{\mathcal{A}}\right)$
as the $f$-interesting relation on $\mathcal{A}$.

######  Definition 60.

Let $(\mathcal{A},\succeq_{\mathcal{A}})$ be a preordered set and let
$\mathsf{Mon}(\mathcal{A})$ be the set of order-preserving maps
$(\mathcal{A},\succeq_{\mathcal{A}})\to(\overline{\mathbb{R}},\geqslant)$. We
define a preorder2121footnotemark: 21 $\sqsupseteq_{\mathcal{A}}$ on
$\mathsf{Mon}(\mathcal{A})$ by

$f\sqsupseteq_{\mathcal{A}}g\quad\iff\quad
f\textup{-}\mathsf{Interesting}\left(\mathcal{A},\succeq_{\mathcal{A}}\right)\supseteq
g\textup{-}\mathsf{Interesting}\left(\mathcal{A},\succeq_{\mathcal{A}}\right)$
(123)

and we say that $f$ is more informative about
$\bm{(\mathcal{A},\succeq_{\mathcal{A}})}$ than $g$ is if
$f\sqsupseteq_{\mathcal{A}}g$ holds. If the preordered set is $(R,\succeq)$
arising from some resource theory $({R}_{\rm free},{R},\boxtimes)$, we denote
the informativeness order relation by $\sqsupseteq$ instead of
$\sqsupseteq_{R}$.

For functions $f$ and $g$ which are monotones, we can express
$f\sqsupseteq_{\mathcal{A}}g$ also as

$\displaystyle f\sqsupseteq_{\mathcal{A}}g$
$\displaystyle\quad\iff\quad\forall a,b\in\mathcal{A}:g(a)<g(b)\implies
f(a)<f(b)$ (124a) $\displaystyle\quad\iff\quad\forall
a,b\in\mathcal{A}:f(a)\geqslant f(b)\implies g(a)\geqslant g(b).$ (124b)

We would like to compare the constructions of monotones appearing in sections
3 and 4 in terms of how useful they are depending on the input elements
thereof. One of the input elements for cost and yield constructions is a
partial function $f_{W}\colon R\to\overline{\mathbb{R}}$. Although it need not
be a monotone on its domain $W$, it can still be understood as witnessing
nonconvertibility between some resources within $W$. In proposition 62 below
we prove that whenever a partial function $f_{W}$ witnesses all the pairs of
nonconvertible resources that $g_{W^{\prime}}$ does, then $f_{W}$ is at least
as useful as $g_{W^{\prime}}$ is, when thought of as an input to the
generalized yield and cost constructions (theorem 24). That is, in such case
$f_{W}\textup{-}\mathsf{yield}_{D}$ is more informative about $(R,\succeq)$
than $g_{W^{\prime}}\textup{-}\mathsf{yield}_{D}$ is and likewise for the cost
construction. In order to make these kinds of statements more precise, we now
formalize the notion of the amount of resource nonconvertibility that a
partial function witnesses.

Let $f_{W}\colon R\to\overline{\mathbb{R}}$ be a partial function with domain
$W$. We say that $f_{W}$ witnesses the nonconvertibility of a pair of
resources $(r,s)$ if both $f(r)<f(s)$ and $r\not\succeq s$ hold. As far as
this property is concerned, we call such a pair of resources $(r,s)$
$\bm{f_{W}}$-interesting.

The set of all $f_{W}$-interesting pairs for a partial function $f_{W}$ is
denoted by

$f_{W}\textup{-}\mathsf{Interesting}\left(R,\succeq\right)\coloneqq\big{\\{}(r,s)\in
W\times
W\nonscript\;\,\big{|}\allowbreak\nonscript\;\,\mathopen{}f(r)<f(s)\;\land\;r\not\succeq
s\big{\\}}$ (125)

We also refer to $f_{W}\textup{-}\mathsf{Interesting}\left(R,\succeq\right)$
as the $f_{W}$-interesting relation on $R$. Note that this definition
coincides with the $f$-interesting relation for a monotone $f$ given by
equation (122), whenever $f_{W}$ is indeed a monotone. That is why we use the
same notation for both of these relations.

######  Definition 61.

Let $(R,\succeq)$ be a preordered set and let $f_{W},g_{W^{\prime}}\colon
R\to\overline{\mathbb{R}}$ be partial functions with domains $W$ and
$W^{\prime}$ respectively. We say that $f_{W}$ witnesses more resource
nonconvertibility in $\bm{(R,\succeq)}$ than $g_{W^{\prime}}$ does if
$f_{W}\sqsupseteq g_{W^{\prime}}$ holds, where

$f_{W}\sqsupseteq g_{W^{\prime}}\quad\iff\quad
f_{W}\textup{-}\mathsf{Interesting}\left(R,\succeq\right)\supseteq
g_{W^{\prime}}\textup{-}\mathsf{Interesting}\left(R,\succeq\right).$ (126)

######  Proposition 62 (more informative functions generate more informative
monotones).

Let $({R}_{\rm free},{R},\boxtimes)$ be a resource theory with an associated
preordered set $(R,\succeq)$ and let $D$ be a downward closed subset of $R$.
Furthermore, let $f_{W}\colon R\to\overline{\mathbb{R}}$ and
$g_{W^{\prime}}\colon R\to\overline{\mathbb{R}}$ be two partial functions with
domains $W$ and $W^{\prime}$ respectively.

If $f_{W}$ witnesses more resource nonconvertibility in $(R,\succeq)$ than
$g_{W^{\prime}}$ does, then $f_{W}\textup{-}\mathsf{yield}_{D}$ is more
informative about $(R,\succeq)$ than
$g_{W^{\prime}}\textup{-}\mathsf{yield}_{D}$ is and also
$f_{W}\textup{-}\mathsf{cost}_{D}$ is more informative about $(R,\succeq)$
than $g_{W^{\prime}}\textup{-}\mathsf{cost}_{D}$ is. That is, we have

$\displaystyle f_{W}\sqsupseteq g_{W^{\prime}}$ $\displaystyle\implies
f_{W}\textup{-}\mathsf{yield}_{D}\sqsupseteq
g_{W^{\prime}}\textup{-}\mathsf{yield}_{D},$ (127a) $\displaystyle
f_{W}\sqsupseteq g_{W^{\prime}}$ $\displaystyle\implies
f_{W}\textup{-}\mathsf{cost}_{D}\sqsupseteq
g_{W^{\prime}}\textup{-}\mathsf{cost}_{D}.$ (127b)

Moreover, if $f_{W}$ and $g_{W^{\prime}}$ are monotones on their respective
domains and their domains coincide; i.e., $W=W^{\prime}$; and if $D=R_{\rm
free}$, then the converse of both implications holds as well. That is, in such
case we have

$\displaystyle f_{W}\sqsupseteq g_{W}$ $\displaystyle\iff
f_{W}\textup{-}\mathsf{yield}_{D}\sqsupseteq
g_{W}\textup{-}\mathsf{yield}_{D},$ (128a) $\displaystyle f_{W}\sqsupseteq
g_{W}$ $\displaystyle\iff f_{W}\textup{-}\mathsf{cost}_{D}\sqsupseteq
g_{W}\textup{-}\mathsf{cost}_{D}.$ (128b)

###### Proof.

In order to prove claim (127a), we need to show that
$g_{W^{\prime}}\textup{-}\mathsf{yield}_{D}(r)<g_{W^{\prime}}\textup{-}\mathsf{yield}_{D}(s)$
implies
$f_{W}\textup{-}\mathsf{yield}_{D}(r)<f_{W}\textup{-}\mathsf{yield}_{D}(s)$
for all $r,s\in R$ such that $r\not\succeq s$. This follows via

$\displaystyle g_{W^{\prime}}\textup{-}\mathsf{yield}_{D}(r)$
$\displaystyle<g_{W^{\prime}}\textup{-}\mathsf{yield}_{D}(s)$ (129a)
$\displaystyle\iff\forall r_{2}\in
W^{\prime}\cap\mathord{\downarrow_{D}}(r),\;\exists s_{2}\in
W^{\prime}\cap\mathord{\downarrow_{D}}(s)\;:\;g_{W^{\prime}}(r_{2})<g_{W^{\prime}}(s_{2})\text{
and }r_{2}\not\succeq s_{2}$ (129b) $\displaystyle\,\implies\forall r_{2}\in
W\cap\mathord{\downarrow_{D}}(r),\;\exists s_{2}\in
W\cap\mathord{\downarrow_{D}}(s)\;:\;f_{W}(r_{2})<f_{W}(s_{2})\text{ and
}r_{2}\not\succeq s_{2}$ (129c) $\displaystyle\iff
f_{W}\textup{-}\mathsf{yield}_{D}(r)<f_{W}\textup{-}\mathsf{yield}_{D}(s).$
(129d)

In the first (and last) equivalence, we could restrict $s_{2}$ to be such that
$r_{2}\not\succeq s_{2}$ because $r_{2}\succeq s_{2}$ (together with
$r_{2}\in\mathord{\downarrow_{D}}(r)$) implies that $s_{2}$ is an element of
$\mathord{\downarrow_{D}}(r)$, which is a subset of
$\mathord{\downarrow}\circ\mathord{\downarrow_{D}}(r)$ whenever $D$ is
downward closed, as we prove in lemma 22. This in turn implies that
$g_{W^{\prime}}(s_{2})$ is bounded above by
$g_{W^{\prime}}\textup{-}\mathsf{yield}_{D}(r)$. Since
$g_{W^{\prime}}\textup{-}\mathsf{yield}_{D}(s)$ is strictly larger than
$g_{W^{\prime}}\textup{-}\mathsf{yield}_{D}(r)$, there must be such $s_{2}$
outside $\mathord{\downarrow_{D}}(r)$.

The implication (129b) $\implies$ (129c) follows from the assumption that
$f_{W}$ witnesses more resource nonconvertibility in $(R,\succeq)$ than
$g_{W^{\prime}}$ does.

In order to prove claim (127b), we need to show the analogous statement for
cost monotones.

$\displaystyle g_{W^{\prime}}\textup{-}\mathsf{cost}_{D}(r)$
$\displaystyle<g_{W^{\prime}}\textup{-}\mathsf{cost}_{D}(s)$ (130a)
$\displaystyle\iff\forall s_{2}\in
W^{\prime}\cap\mathord{\uparrow_{D}}(s),\;\exists r_{2}\in
W^{\prime}\cap\mathord{\uparrow_{D}}(r)\;:\;g_{W^{\prime}}(r_{2})<g_{W^{\prime}}(s_{2})\text{
and }r_{2}\not\succeq s_{2}$ (130b) $\displaystyle\,\implies\forall s_{2}\in
W\cap\mathord{\uparrow_{D}}(s),\;\exists r_{2}\in
W\cap\mathord{\uparrow_{D}}(r)\;:\;f_{W}(r_{2})<f_{W}(s_{2})\text{ and
}r_{2}\not\succeq s_{2}$ (130c) $\displaystyle\iff
f_{W}\textup{-}\mathsf{cost}_{D}(r)<f_{W}\textup{-}\mathsf{cost}_{D}(s).$
(130d)

In the first (and last) equivalence, we can again restrict $r_{2}$ to be such
that $r_{2}\not\succeq s_{2}$ because $r_{2}\succeq s_{2}$ (together with
$s_{2}\in\mathord{\uparrow_{D}}(s)$) implies that $r_{2}$ is an element of
$\mathord{\uparrow_{D}}(s)\subseteq\mathord{\uparrow}\circ\mathord{\uparrow_{D}}(s)$
as we show in lemma 22. In turn, this implies that $g_{W^{\prime}}(r_{2})$ is
bounded below by $g_{W^{\prime}}\textup{-}\mathsf{cost}_{D}(s)$. Since
$g_{W^{\prime}}\textup{-}\mathsf{cost}_{D}(r)$ is strictly smaller than
$g_{W^{\prime}}\textup{-}\mathsf{cost}_{D}(s)$, there must be such $r_{2}$
outside $\mathord{\downarrow_{D}}(s)$. The implication (130b) $\implies$
(130c) follows from the assumption that $f_{W}\sqsupseteq g_{W^{\prime}}$
holds. This concludes the proof of the first half of proposition 62.

Finally, in order to obtain claim (128), we can show that
$f_{W}\not\sqsupseteq g_{W}$ implies both
$f_{W}\textup{-}\mathsf{yield}\not\sqsupseteq g_{W}\textup{-}\mathsf{yield}$
and $f_{W}\textup{-}\mathsf{cost}\not\sqsupseteq
g_{W}\textup{-}\mathsf{cost}$, under the assumption that $f_{W}$ and $g_{W}$
are monotones on $W$. The statement $f_{W}\not\sqsupseteq g_{W}$ can in such
case be expressed as

$\exists r,s\in W\,:\,g_{W}(r)<g_{W}(s)\text{ and }f_{W}(r)\geqslant
f_{W}(s).$ (131)

By proposition 63 proved below, the values of
$f_{W}\textup{-}\mathsf{yield}_{D}$ and $f_{W}\textup{-}\mathsf{cost}_{D}$
coincide with the value of $f_{W}$ on $W$, and similarly for $g_{W}$.
Therefore, $f_{W}\not\sqsupseteq g_{W}$ implies the following two statements

$\displaystyle\exists r,s\in W$
$\displaystyle\,:\,g_{W}\textup{-}\mathsf{yield}_{D}(r)<g_{W}\textup{-}\mathsf{yield}_{D}(s)\text{
and }f_{W}\textup{-}\mathsf{yield}_{D}(r)\geqslant
f_{W}\textup{-}\mathsf{yield}_{D}(s)$ (132) $\displaystyle\exists r,s\in W$
$\displaystyle\,:\,g_{W}\textup{-}\mathsf{cost}_{D}(r)<g_{W}\textup{-}\mathsf{cost}_{D}(s)\text{
and }f_{W}\textup{-}\mathsf{cost}_{D}(r)\geqslant
f_{W}\textup{-}\mathsf{cost}_{D}(s).$ (133)

Since the yields and costs are also monotones, these imply that
$f_{W}\textup{-}\mathsf{yield}_{D}\not\sqsupseteq
g_{W}\textup{-}\mathsf{yield}_{D}$ and
$f_{W}\textup{-}\mathsf{cost}_{D}\not\sqsupseteq
g_{W}\textup{-}\mathsf{cost}_{D}$. Consequently, the proof of the second half
of proposition 62 is also complete. ∎

######  Proposition 63.

Let $f_{W}\colon W\to\mathbb{R}_{+}$ be a monotone. Then for all $r\in W$, we
have

$f_{W}\textup{-}\mathsf{yield}(r)=f_{W}(r)=f_{W}\textup{-}\mathsf{cost}(r).$
(134)

###### Proof.

Since $r\in\mathord{\downarrow}(r)$ and $r\in\mathord{\uparrow}(r)$, we have
$f_{W}\textup{-}\mathsf{cost}(r)\geqslant f_{W}(r)\geqslant
f_{W}\textup{-}\mathsf{yield}(r)$ for all $r\in W$. On the other hand, $f$
being a monotone on $W$ implies that for each $s,t\in W$ such that $t\preceq
r\preceq s$, we have $f_{W}(t)\leqslant f_{W}(r)\leqslant f_{W}(s)$.
Performing a supremum of the left inequality over all $t\in
W\cap\mathord{\downarrow}(r)$ yields $f_{W}\textup{-}\mathsf{cost}(r)\leqslant
f_{W}(r)$, while taking the infimum of the right inequality over all $s\in
W\cap\mathord{\uparrow}(r)$ gives $f_{W}(r)\leqslant
f_{W}\textup{-}\mathsf{yield}(r)$, so that the result follows. ∎

######  Corollary 64.

As a consequence of proposition 62, sufficient and necessary conditions for
the ordering (by $\sqsupseteq$) of generalized yields and costs relative to
$R_{\rm free}$222222In fact, the same result holds for generalized yields and
costs relative to any downward closed set $D$. However, this does not follow
directly from proposition 62. One needs to use the fact that
$f_{W}\textup{-}\mathsf{yield}_{D}$ and $f_{W}\textup{-}\mathsf{cost}_{D}$
preserve not only the convertibility relation with respect to $R_{\rm free}$
(i.e., $\succeq$), but also the convertibility relation with respect to $D$.
Note that the latter relation is not in general transitive, since $D$ may not
be closed under $\star$. are given by the ordering (by $\sqsupseteq$) of their
restrictions to $W\cup W^{\prime}$. These facts can also be expressed in terms
of the order relation with respect to informativeness about $(W\cup
W^{\prime},\succeq)$ as:

$\displaystyle f_{W}\textup{-}\mathsf{yield}\sqsupseteq_{W\cup
W^{\prime}}g_{W^{\prime}}\textup{-}\mathsf{yield}$ $\displaystyle\iff
f_{W}\textup{-}\mathsf{yield}\sqsupseteq
g_{W^{\prime}}\textup{-}\mathsf{yield}$ (135a) $\displaystyle
f_{W}\textup{-}\mathsf{cost}\sqsupseteq_{W\cup
W^{\prime}}g_{W^{\prime}}\textup{-}\mathsf{cost}$ $\displaystyle\iff
f_{W}\textup{-}\mathsf{cost}\sqsupseteq g_{W^{\prime}}\textup{-}\mathsf{cost}$
(135b)

Therefore, if one wishes to characterize the resource preorder $(R,\succeq)$
by virtue of monotones generated by the generalized yield and cost
constructions (theorem 24), then using functions $W\to\overline{\mathbb{R}}$
which are more informative about $(R,\succeq)$ according to $\sqsupseteq$
should be preferred.

A function $f_{W}$ cannot witness more resource nonconvertibility than a
complete set of monotones because such a set captures all the information in
the preordered set $(W,\succeq)$. Nonetheless, a function $f_{W}$ can witness
more resource nonconvertibility than _any single_ monotone. The simplest
example is provided by $W$ with 4 elements, two pairs of which are ordered as
in the following Hasse diagram.

$h$$\mapsto$$h$$f$$g$$f$$\boxtimes$$g$$=$$f$$g$$\cup$$f$$g$$\cup$$g$$f$$f$$g$$\cup$$g$$f$$\cup$$=$$\cup$$\;\;g\;\;$$f$$\cup$$f$$\;\;g\;\;$$\;\;g\;\;$$f$$\cup$$f$$\;\;g\;\;$$f$$\boxtimes$$\;\;g\;\;$$\;\;g\;\;$$f$$\cup$$\cup$$\cup$$f$$\;\;g\;\;$$=$$f$$\boxtimes$$g$$h$$f\boxtimes
g$$h$$\cup$$f\boxtimes h$$g$$\quad
P\quad$$S$$T$$X$$Y$$X$$S$$T$$Y$$\Lambda$$\phi$$f\textup{-}\mathsf{yield}$$=$$\sup$$\phi$$f$
$\rho$$\;\;\psi\;\;$$\psi\in R_{\rm free}$$\rho\in R_{\rm
free}$$\phi$$f\textup{-}\mathsf{yield}$$=$$\sup$$\phi$$f$ $\rho$$\rho\in
R_{\rm free}$$\phi$$f\textup{-}\mathsf{cost}$$=$$\inf$$f$
$\rho$$\phi$$=$$\;\;\psi\;\;$ $\rho$$\text{and}\;\;\psi\in R_{\rm
free}$$\phi$$f\textup{-}\mathsf{yield}$$=$$\sup$$\phi$$f$
$\rho$$\;\;\psi\;\;$$\psi\in D$$\rho\in
D$$s_{1}$$s_{2}$$t_{1}$$t_{2}$$t_{4}$$t_{3}$$a$$b$$c$$d$$R$$R_{\rm
free}$${r_{1}}$${s_{1}}$${r_{2}}$${s_{2}}$ (136)

If we let $f_{W}$ be defined as follows

$\begin{split}f_{W}(r_{1})&=0\\\
f_{W}(r_{2})&=1\end{split}\qquad\begin{split}f_{W}(s_{1})&=0\\\
f_{W}(s_{2})&=1\end{split}$ (137)

then it clearly fails to be a monotone. Note that $f_{W}$ witnesses
nonconvertibility for the two pairs of resources, $(r_{1},s_{2})$ and
$(s_{1},r_{2})$, while no single monotone can do so simultaneously. The proof
of the latter claim is that the order-preserving property of a monotone
implies that it must satisfy $M(r_{1})\geqslant M(r_{2})$ and
$M(s_{1})\geqslant M(s_{2})$. If it witnesses the nonconvertibility of the
pair $(r_{1},s_{2})$, then $M(r_{1})<M(s_{2})$ holds and these three
inequalities together imply that $M(s_{1})$ is greater than $M(r_{2})$, so
that $M$ then _cannot_ witness the nonconvertibility of the pair
$(s_{1},r_{2})$. The function $f_{W}$ is capable of witnessing the
nonconvertibility of both pairs of resources _precisely because_ it fails to
be order-preserving.

######  Remark 65.

Note that this function has another interesting property in that both
$f_{W}\textup{-}\mathsf{yield}$ and $f_{W}\textup{-}\mathsf{cost}$ are
constant; i.e., they are least informative about $(W,\succeq)$ among all
monotones $W\to\overline{\mathbb{R}}$. $f_{W}$ can thus serve as a
counterexample to various conjectures about the yield and cost constructions.

######  Example 66 (chains admit a most informative function).

If $W$ is a chain; that is, a totally ordered subset of $R$; then there _is_ a
single monotone $f_{W}$ that forms a complete set of monotones by itself.
Therefore, it is more informative about $(W,\succeq)$ than any other function
$W\to\overline{\mathbb{R}}$. As a consequence, for each downward closed set
$D\in\mathcal{DC}(R)$, there are unique most informative yield and cost
monotones with respect to $W$, namely $f_{W}\textup{-}\mathsf{yield}_{D}$ and
$f_{W}\textup{-}\mathsf{cost}_{D}$. Given the choice of $D=R_{\rm free}$,
these correspond to currencies defined in [28], as we mentioned earlier.

##  6\. Conclusions

To summarize, in this manuscript we introduced a somewhat minimal framework
for describing (universally combinable) resource theories, within which we
investigated various ways of constructing monotones through the lens of the
Broad Scheme.

Firstly, we looked at generalized resource yield and generalized resource cost
constructions. An extensive (but definitely not exhaustive) list of examples
of such constructions in the literature has been provided. We also showed how
the generalized constructions of this kind can be used to obtain monotones
that cannot be conceptualized as standard yield and cost constructions.

Secondly, we looked at monotones which can be seen as arising from another
monotone by virtue of a translation via a mediating order-preserving map.
After introducing a resource theory of $k$-distinguishability, we described a
translation of measures of distinguishability to other resource theories,
which generalizes the familiar concept of constructing monotones from
contractions. As an application of this construction, we unified resource
weight and resource robustness as arising from a single root contraction for
$3$-tuples of resources. Moreover, by varying the parameters in the general
construction, we showed how one can obtain other related monotones from the
same measure of $3$-distinguishability. General methods to translate monotones
between resource theories were then presented, culminating with corollary 58
that summarizes the results on translation of monotones.

The two main themes of the paper, generalizing yield and cost constructions
and generalizing the translation of monotones, are intricately linked. We
investigate these connections further in appendix A, by showing how the
corresponding mediating preordered sets are related to each other. One of the
main features that distinguishes the two kinds of constructions is that the
partial function $f$, a starting point in both cases, is assumed to have
different properties. The scheme for translating monotones is targeted to
functions $f$ which are monotones themselves, while yield and cost
constructions work for functions $f$ which are not monotones, for example by
virtue of a restricted domain of applicability. One could think that the
generalized yield and cost constructions are therefore superior. However,
their disadvantage is the optimization over _all_ free resources inherent in
the construction. Moreover, even though $f$ has to be a monotone if we want to
translate it, a seemingly insipid one (like the convex alignment) can still
generate interesting target monotones (like weight and robustness).

Finally, we also explored the structure of the set of all resource monotones.
The monotone constructions presented here are very general and widely
applicable, but using them in practice as a method for generating monotones
involves several choices. For example, a priori it is not clear which choices
of the downward closed set of resources $D$, the valuation function $f$, and
its domain $W$ in the generalized yield and cost constructions are the best
ones. These are the kinds of questions we made progress on by “assessing
informativeness of monotones”. In particular, we compared them in terms of how
good they are in capturing the resource ordering. With this criterion, we
investigated what are the best ways to use the monotone costructions
introduced earlier in order to get the most informative monotones.

Our work advances the studies of general structures appearing in resource
theories and has potential applications to any area where the resource-
theoretic point of view is of some use. These include the study of information
theory, both quantum and classical, but also of thermodynamics, of
renormalization, and of various other parts of physics where resource-
theoretic questions are tackled. We believe, however, that similar questions
in more distant fields can also be analyzed with the resource-theoretic
mindset, which is one of the main reasons why we choose to work in a framework
that does not presuppose the resources to be quantum processes. Indeed, one of
the benefits of working within an abstract framework for resource theories is
that there is the potential for cross-fertilization of ideas between very
different fields of study. This was one of the motivations for previous
attempts at abstract formalisms for resources theories [9, 14], which can
describe situations well beyond the scope of physics. To name a few, we can
use them to study the theory of chemical reactions, but also a kind of proof
theory wherein the free operations are compass and straight-edge and the
nontrivial resources are geometrical constructions that cannot be achieved by
compass and straight-edge. In this vein, Fritz has further shown how the
framework of ordered commutative monoids has interesting applications in
fields as diverse as graph theory and game theory [14].

There are many possible future directions for extending this work.

  1. (i)

One might aim to determine how the mathematical structures presented here
relate to other mathematical structures used in physics, mathematics, and
computer science. The study of their relation to some of the other
mathematical frameworks for resource theories can be found in [15], but other
connections are yet to be developed.

  2. (ii)

One might try to devise general techniques for constructing monotones by
considering resource theories that have more structure than we have presumed
here. One way to do so would be by strenghtening the assumptions of our
central results in order to arrive at stronger conclusions. There are many
possibilities in this direction, one of which is to assume a linear or convex
structure of resources as we did when we studied the weight and robustness
measures here. It is clear that these results will then be connected to ideas
from convex geometry [42] and convex optimization [58].
Related to this is the aim of reexpressing the results presented here in a
framework which is closer to the structure of a resource theory that one would
use in practical applications. A framework like that would capture partitioned
process theories with a restriction on the allowed types of resources for
example, which, as we argued in example 7, cannot be expressed as a
universally combinable resource theory in the sense of definition 4. This is
what we do in [15].

  3. (iii)

Last, but not least, one would hope to be able to not only unify and
generalize existing concrete results about resource theories as we have done
here, but also to find novel applications of monotones with the help of the
conceptual clarity arising from the abstract point of view.

Acknowledgements

We would like to thank Elie Wolfe, Denis Rosset, Tobias Fritz, and Bob Coecke
for useful discussions. This research was supported by a Discovery grant of
the Natural Sciences and Engineering Research Council of Canada and by the
Perimeter Institute for Theoretical Physics. Research at Perimeter Institute
is supported in part by the Government of Canada through the Department of
Innovation, Science and Economic Development Canada and by the Province of
Ontario through the Ministry of Colleges and Universities.

##  Appendix A Ordering Sets of Resources

In this appendix, we show the following isomorphisms of preordered sets:

$\displaystyle\raisebox{4.13336pt}{\leavevmode\hbox{\set@color$(\mathcal{P}(R),\succeq)$}}\mkern-5.0mu\rotatebox{-44.0}{\rule[-1.83437pt]{0.4pt}{1.83437pt}}\mkern-4.0mu\raisebox{-1.83437pt}{\leavevmode\hbox{\set@color$\sim$}}\simeq\raisebox{4.13336pt}{\leavevmode\hbox{\set@color$(\mathcal{P}(R),\succeq_{\rm
enh})$}}\mkern-5.0mu\rotatebox{-44.0}{\rule[-1.83437pt]{0.4pt}{1.83437pt}}\mkern-4.0mu\raisebox{-1.83437pt}{\leavevmode\hbox{\set@color$\sim_{\rm
enh}$}}$ $\displaystyle\simeq(\mathcal{DC}(R),\supseteq)$ (138)
$(\mathcal{P}(R),\succeq_{\rm deg})$

$\sim_{\rm deg}$ $\displaystyle\simeq(\mathcal{UC}(R),\subseteq)$ (139)

When restricted to singletons and the free images/preimages of individual
resources, all five partially ordered sets are isomorphic. In fact, it is
immediate from the definitions that $\\{r\\}\succeq_{\rm enh}\\{s\\}$ if and
only if $\\{r\\}\succeq_{\rm deg}\\{s\\}$, whence the preorders $\succeq$,
$\succeq_{\rm enh}$ and $\succeq_{\rm deg}$ are themselves identical when
restricted to singletons.

######  Theorem 67 (first isomorphism theorem for preordered sets [20]).

Let $(\mathcal{A},\succeq_{\mathcal{A}})$ and
$(\mathcal{B},\succeq_{\mathcal{B}})$ be two preordered sets and let
$\phi\colon\mathcal{A}\to\mathcal{B}$ be an order-preserving map. The kernel
of $\phi$ is an equivalence relation $\sim_{\phi}$ on $\mathcal{A}$ defined by

$a\sim_{\phi}a^{\prime}\iff\phi(a)\sim_{\mathcal{B}}\phi(a^{\prime}),$ (140)

where $\sim_{\mathcal{B}}$ is the standard equivalence relation on
$\mathcal{B}$ induced by $\succeq_{\mathcal{B}}$. Then, there is a canonical
isomorphism

$\tilde{\phi}\colon\raisebox{3.44444pt}{\leavevmode\hbox{\set@color$(\mathcal{A},\succeq_{\mathcal{A}})$}}\mkern-5.0mu\rotatebox{-44.0}{\rule[-1.83437pt]{0.4pt}{1.83437pt}}\mkern-4.0mu\raisebox{-1.83437pt}{\leavevmode\hbox{\set@color$\sim_{\phi}$}}\to\raisebox{3.44444pt}{\leavevmode\hbox{\set<EMAIL_ADDRESS>(141)

###### Proof.

$(\mathcal{A},\succeq_{\mathcal{A}})/{\sim_{\phi}}$ consists of the set of
equivalence classes $\mathcal{A}/{\sim_{\phi}}$ and the corresponding order
relation $\succeq_{\mathcal{A}}$ defined as

$[a_{1}]_{\sim_{\phi}}\succeq_{\mathcal{A}}[a_{2}]_{\sim_{\phi}}\iff
a_{1}\succeq_{\mathcal{A}}a_{2}$ (142)

for any $a_{1},a_{2}\in\mathcal{A}$, where $[a_{1}]_{\sim_{\phi}}$ is the
equivalence class of $a_{1}$ with respect to $\sim_{\phi}$. Since $\phi$ is an
order-preserving map, $\succeq_{\mathcal{A}}$ is a well-defined partial order.

We can then construct $\tilde{\phi}$ as

$\tilde{\phi}([a]_{\sim_{\phi}})\coloneqq[\phi(a)]_{\sim_{\mathcal{B}}},$
(143)

for any $a\in\mathcal{A}$. Again, $\tilde{\phi}$ is well-defined and order-
preserving because $\phi$ is order-preserving. Furthermore, the fact that it
is injective follows from the definition of $\sim_{\phi}$. ∎

######  Lemma 68.

The kernel of $\mathord{\downarrow}\colon(\mathcal{P}(R),\succeq_{\rm
enh})\to(\mathcal{DC}(R),\supseteq)$ is $\sim_{\rm enh}$ and consequently the
kernel of
$\mathord{\downarrow}\colon(R,\succeq)\to(\mathcal{DC}(R),\supseteq)$ is
$\sim$.

###### Proof.

Follows directly from lemma 28. ∎

######  Corollary 69.

The partially ordered sets $(\mathcal{P}(R),\succeq_{\rm enh})/{\sim_{\rm
enh}}$ and $(\mathcal{DC}(R),\supseteq)$ are isomorphic.

######  Lemma 70.

The kernel of $\mathord{\uparrow}\colon(\mathcal{P}(R),\succeq_{\rm
deg})\to(\mathcal{UC}(R),\supseteq)$ is $\sim_{\rm deg}$ and consequently the
kernel of $\mathord{\uparrow}\colon(R,\succeq)\to(\mathcal{UC}(R),\supseteq)$
is $\sim$.

###### Proof.

Follows directly from lemma 32. ∎

######  Corollary 71.

The partially ordered sets $(\mathcal{P}(R),\succeq_{\rm deg})/{\sim_{\rm
deg}}$ and $(\mathcal{UC}(R),\subseteq)$ are isomorphic.

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#  When are Foundation Models Effective? Understanding the Suitability for
Pixel-Level Classification Using Multispectral Imagery

Yiqun Xie, Zhihao Wang, Weiye Chen, Zhili Li, Xiaowei Jia, Yanhua Li,
Ruichen Wang, Kangyang Chai, Ruohan Li, Sergii Skakun  Y. Xie, Z. Wang, W.
Chen, Z. Li, R. Wang, K. Chai, R. Li and S. Skakun are with University of
Maryland. E-mail: {xie, zhwang1, weiyec, lizhili, r526li, ruichenw, kychai,
<EMAIL_ADDRESS>X. Jia is with University of Pittsburgh. E-mail:
<EMAIL_ADDRESS>Y. Li is with Worcester Polytechnic Institute. E-mail:
<EMAIL_ADDRESS>
###### Abstract

Foundation models, i.e., very large deep learning models, have demonstrated
impressive performances in various language and vision tasks that are
otherwise difficult to reach using smaller-size models. The major success of
GPT-type of language models is particularly exciting and raises expectations
on the potential of foundation models in other domains including satellite
remote sensing. In this context, great efforts have been made to build
foundation models to test their capabilities in broader applications, and
examples include Prithvi by NASA-IBM, Segment-Anything-Model, ViT, etc. This
leads to an important question: Are foundation models always a suitable choice
for different remote sensing tasks, and when or when not? This work aims to
enhance the understanding of the status and suitability of foundation models
for pixel-level classification using multispectral imagery at moderate
resolution, through comparisons with traditional machine learning (ML) and
regular-size deep learning models. Interestingly, the results reveal that in
many scenarios traditional ML models still have similar or better performance
compared to foundation models, especially for tasks where texture is less
useful for classification. On the other hand, deep learning models did show
more promising results for tasks where labels partially depend on texture
(e.g., burn scar), while the difference in performance between foundation
models and deep learning models is not obvious. The results conform with our
analysis: The suitability of foundation models depend on the alignment between
the self-supervised learning tasks and the real downstream tasks, and the
typical masked autoencoder paradigm is not necessarily suitable for many
remote sensing problems.

###### Index Terms:

Foundation model, remote sensing, pixel-level

##  I Introduction

Foundation models are very large deep learning models that aim to generalize
better with hundreds of millions, billions or even trillions of parameters.
These large models have recently demonstrated surprisingly well performances
in various language and vision tasks, with strong generalization abilities
across tasks. In particular, GPT-type of models showed revolutionary
capabilities in generative tasks and started to transform numerous aspects of
existing routines in education, research, government and industry. The
unprecedented impacts brought by the foundation models naturally raise high
expectations of their performances in other domains and motivate the
developments of prototypes and domain-specific customizations.

Traditionally, the training of foundation models is extremely challenging, as
the potential of the super-sized models comes together with the demand for
super-sized training datasets, which is largely unavailable at the needed
scales. Fortunately, the developments of self-supervised learning (SSL) came
into rescue for various types of tasks such as generation. SSL tackles the
data challenge for a real downstream task by creating related ”artificial”
tasks where the training data – both the input features X and artificial
targets Y – can be automatically generated at a large scale without tedious
and time-consuming manual efforts. For example, masked modeling is one of the
most common SSL tasks, where parts of sentences or images are masked (i.e.,
left empty) from the original complete ones to be used as prediction targets
for the SSL task [1]. Because such target labels can be automatically
generated in huge quantities, they can be used to pre-train the gigantic
amount of parameters in foundation models. The pre-trained models can then be
finetuned, or used with prompt-based strategies, for downstream tasks.

This paper aims to make a comparison between foundation models, regular-size
deep learning models, and traditional machine learning (ML) models to
understand the status and suitability of these models for pixel-level
classification tasks using multispectral imagery at moderate spatial
resolution (e.g., 10-30 m). Such tasks are both very common and highly
important for critical sectors including agriculture, water, natural disaster
management, etc. Several efforts have been made recently to enhance the
capabilities of foundation models for Earth science problems. For example,
prior studies such as FourCastNet [2] and GraphCast [3] have shown the
potential to accelerate weather forecasting models by hundreds of times while
maintaining a very high accuracy. As most of these methods are designed to
approximate physical Earth science models, they target physics-specific
challenges (e.g., capturing global dependencies using the Fourier neural
operator [4]) but are not applicable to pixel-level classification tasks.
Efforts have also been made to evaluate foundation models such as vision
transformers for remote sensing data [5, 6, 7], and the larger models did show
enhancements for scene classification and object detection on high-resolution
aerial imagery. The most related open-source version is the foundation model
Prithvi developed by NASA and IBM [8, 9], which is based on the multispectral
Harmonized Landsat and Sentinel-2 (HLS) dataset. However, pixel-level
classification tasks using multispectral imagery at moderate spatial
resolution exhibit differences from other typical vision tasks or remote
sensing tasks on very high spatial resolution data. In particular, class
labels may not bear a strong relationship with texture patterns in the images.
In this context, a simple yet important question that has not been addressed
is whether foundation models such as Prithvi can offer significant
improvements over traditional ML models such as random forest, which are still
the models of choice for many pillar remote sensing products at national or
global scales (e.g., [10, 11]). More importantly, there is a lack of
understanding regarding in what scenarios foundation models will have
advantages over other learning models.

To better answer these questions, we consider three foundation models
(including the most related Prithvi), three regular-size deep learning models,
and four traditional ML models. Interestingly, the results reveal that in many
scenarios traditional ML models still have a similar or better performance
than foundation models. Particularly, they often work better for tasks (e.g.,
flood mapping) where textures are less useful for classification. On the other
hand, deep learning models did show more promising results for tasks where
labels partially depend on textures (e.g., burn scar), while the difference in
performance between foundation models and regular-size models is not obvious.
These results conform with our analysis that the suitability of foundation
models is subject to the alignment between the SSL tasks and the downstream
tasks, and the typical masked autoencoder paradigm is not necessarily suitable
for many remote sensing problems.

##  II Suitability of Foundation Models for Pixel-level Tasks

While it is important to test out foundation model ideas (e.g., large-scale
self-supervised pre-training) on remote sensing as a closely related
application domain, it is equivalently important to enhance the understanding
of the suitability of the models and their current status. This paper focuses
on pixel-level classification tasks using multispectral imagery (e.g.,
Landsat-8, Sentinel-2), which are widely used in a wide range of applications
such as agriculture monitoring, flood mapping, fire detection, land cover land
use monitoring, etc. In the following, we analyze the characteristics of
foundation models and the implications of the designs for remote sensing
tasks.

###  II-A Task Alignment: Pre-training vs. Downstream Tasks

The alignment between the SSL tasks used in pre-training and the actual
downstream tasks is an important factor in the performance of the finetuned
foundation model [12, 13]. Taking large language models (LLMs) as an example,
their SSL tasks (a.k.a., pretext tasks) are often tightly aligned with the
downstream tasks. Specifically, the SSL tasks utilize masked language modeling
that masks out words or sentences as prediction targets (i.e., free labels),
and let the network train on them to develop language generation abilities.
Such SSL tasks are highly correlated to the actual downstream tasks, such as
chatting, where the response is generated based on an input sentence (e.g., a
question). Hence, the pre-training through these SSL tasks can often lead to
substantial improvements for well-aligned downstream tasks.

In vision problems, the masked autoencoding tasks [1] that are typically used
for self-supervised pre-training – including remote sensing (e.g., Prithvi) –
are quite similar in the sense that they also try to mask out local image
patches and let the network complete them. However, the difference is the
downstream task. While generating a masked-out sentence is very similar to
generating a response, generating an image patch differs substantially from
classifying the labels at the pixel-level (e.g., whether a pixel is a corn or
soybean). For high resolution images, this could still be very useful as the
ability to generate image patches corresponds to the ability to generate and
understand complex textures (e.g., the shape of a vehicle). This can help
downstream tasks such as object detection or realistic image generation.
However, for many pixel-level classification tasks on multispectral imagery
(e.g., Landsat-8), the textures at the resolution often do not carry as much
useful information to distinguish different classes such as corn vs. soybean.
In contrast, the signatures often come down to the spectral and temporal
characteristics of the single pixels. This makes many pixel-level
classification tasks similar to the classic tabular data problems.
Interestingly, for tabular problems, traditional tree-based algorithms often
still have better performance compared to deep learning models [14]. This
shows the importance of a comparison between foundation models and traditional
ML approaches. We hypothesize that traditional methods will be better suited
when texture differences are less distinguishable between different classes,
whereas foundation models may show advantages when textures do matter.

###  II-B Replicating Human Intelligence: Features vs. Labels

The difference in the alignment between SSL and real downstream tasks in
language and remote sensing problems is also related to another fundamental
difference – the data. While both pre-training data for language problems
(e.g., Wikipedia articles, books, code repositories) and remote sensing
problems (e.g., satellite images) have huge volumes, there is a key difference
when we take the goal of the models into consideration. For both problems –
and most others in AI – the goal is to imitate human intelligence, letting it
be responding to our questions in ChatGPT or automatically classifying pixels
as “flooded” or “not flooded.” With this in mind, we can see that the text
data used for language problems serve the purpose well as they are in fact
generated by humans, often with lots of thinking. Thus, when the SSL tasks
mask out some sentences as prediction targets during pre-training, the labels
(i.e., masked sentences) are human-generated labels that we want to imitate in
downstream tasks such as text generation [12]. This is fundamentally different
for satellite data, where the masked-out image patches in the SSL tasks are
not human annotations. In other words, the masked-out patches can only
represent the input features in downstream tasks (e.g., input images for flood
mapping) but not the actual human-generated labels that we need.

##  III Foundation models

This section briefly describes the architecture and training of the three
foundation models used in the evaluation. The Vision Transformer (ViT) is a
major backbone for foundation models [15, 1], which divides an image into
fixed-size patches and order them in a sequence. Then it embeds these patches
sequentially and feed the obtained embeddings into a standard transformer.
Such a learning process enables capturing long-range dependencies across the
entire image. Following [16], a simplified deconvolutional segmentation head
is used to replace the original classification head for pixel-level
segmentation. ViT is pre-trained using masked autoencoding (MAE) as the SSL
task on the ImageNet-21k. The model is finetuned for each downstream task.

SegFormer is a variant of Transformer for semantic segmentation [16].
Different from ViT, SegFormer consists of hierarchical Transformer encoders
for capturing multi-scale features and a lightweight MLP decoder head for
efficient segmentation to generate the pixel-level mask. SegFormer’s encoder
is pre-trained using an MAE on ImageNet-1k with 8 Tesla V100, and the
segmentation head is trained on ADE20k. The model is further finetuned for
each downstream task.

Prithvi is a ViT-based foundation model for remote sensing [8, 9]. It uses the
ViT backbone with 3D positional encoding and MAE to better fit the
multispectral bands and multi-temporal data. The lightweight decoder head
consists of four deconvolutional layers and a convolutional layer. Prithvi is
pre-trained over 1TB of multispectral HLS dataset across the contiguous US,
using up to 64 NVIDIA A100 GPUs, and then finetuned to each downstream task.

##  IV Evaluation

###  IV-A Pixel-level classification tasks on multispectral imagery

We use the same standard tasks and datasets as used in the evaluation of the
NASA-IBM Prithvi [8], which is the most relevant foundation model for the
scope of this paper as it is designed for multispectral imagery at moderate
spatial resolution. The tasks include multi-temporal crop classification,
wildfire scar mapping, and flood mapping. To be consistent, we adopt the same
train-test splits used in [8]. In multi-temporal crop
classification111https://huggingface.co/datasets/ibm-nasa-geospatial/multi-
temporal-crop-classification, the imagery was collected from Sentinel
satellites across the Contiguous United States in 2022, with the crop and land
cover labels from USDA’s Crop Data Layer. A total of 3,083 images were used
for training and 771 images for testing, where each image contains
224$\times$224 pixels with 6 bands over 3 time steps. Wildfire
scar222https://huggingface.co/datasets/ibm-nasa-geospatial/hls_burn_scars data
was gathered from HLS imagery aligned with the Monitoring Trends in Burn
Severity historical fire database, where each HLS image was selected in the
nearest date after one month of a fire event. The dataset included 804 HLS
image chips of 512$\times$512 pixels and 6 spectral bands with a training-
testing split of 67%-33%. The dataset for Flood
mapping333https://github.com/cloudtostreet/Sen1Floods11, Sen1Floods11 [17],
consists of raw Sentinel-1 and Sentinel-2 imagery covering 120,406 km2 and
spanning all 14 biomes with 11 flood events at global scales. The dataset
comprises 252 image chips for training, 89 for validating, and 90 for testing,
each having 512$\times$512 pixels with 6 spectral bands.

###  IV-B Experiment settings

The following are the candidate methods used in the comparison among
traditional ML models, regular-size deep learning models and foundation
models. Both foundation and deep learning models take images as inputs, while
traditional ML models use single-pixel inputs (with variants using local
neighborhood augmentation).

  * •

RF: A random forest classifier with 100 trees.

  * •

RFaug: RF with augmented domain knowledge features including spectral indices
(e.g., NDVI, NDWI) and spatial textures from a 3$\times$3 local neighborhood.

  * •

XGB: The XGBoost classifier with 100 trees [18].

  * •

XGBaug: XGB with augmented features (same as RFaug).

  * •

FCN: A fully-connected neural network with 5 layers.

  * •

U-Net: An encoder-decoder model for segmentation [19].

  * •

DeepLabv3+ (D3+): An encoder-decoder model with enhanced atrous convolution
[20].

  * •

ViT: The vision transformer foundation model in Sec. III.

  * •

SegFormer: The segmentation-driven ViT in Sec. III.

  * •

Prithvi: The foundation model designed for multispectral imagery at moderate
resolution in Sec. III.

In the training stage, traditional ML models use default parameter settings
from scikit-learn and XGBoost packages. Regular deep learning models were
trained using the Adam optimizer for 100 epochs, and the pretrained
foundational models were finetuned for 100 epochs. For Prithvi, we directly
used the official finetuned models. We evaluated the model performance using
metrics including the Intesection-over-Union (IoU) and F1 score, which is the
harmonic mean of precision and recall. Instead of using a union of no-data
pixels (areas outside the image boundaries) from the reference and the
predicted results as masks for computing the metrics[8], we calculate the
metrics in a more standard way using only no-data from the reference for
masking, making sure the results will not be biased by incorrect no-data
predictions (as a class). In addition, clouds are originally considered as
part of the no-data masks in flood and burn scar mapping. To provide a more
comprehensive view, our results (Tables III and IV) include two different
versions with clouds as no-data masks, or, as a class. In the tables, mIoU and
mF1 are the average of the metrics computed using individual classes.

TABLE I: IoU Comparison for Multi-Temporal Crop Classification

Methods Nat. Veg. Forest Corn Soybeans Wetlands Developed Water Wheat Alfalfa
Idle Cotton Sorghum Other mIoU RF 50.00 53.51 58.53 55.42 43.84 41.72 76.48
52.26 29.64 38.00 35.22 38.44 38.85 47.07 RFaug 50.25 53.76 59.56 56.51 44.20
42.00 76.10 53.24 30.64 38.68 35.60 39.80 39.72 47.70 XGB 37.92 44.55 38.34
34.00 20.78 21.44 74.56 32.61 10.95 27.23 7.09 16.53 16.29 29.41 XGBaug 35.97
44.29 39.25 36.60 27.55 23.86 73.73 34.58 11.50 29.09 7.32 22.70 18.07 31.12
FCN 37.82 42.22 33.53 29.39 21.94 14.04 73.84 29.47 7.76 21.34 0.16 13.06 8.02
25.58 U-Net 55.21 54.79 64.29 63.08 40.99 54.65 76.86 58.17 39.12 42.64 42.26
41.76 42.20 52.00 D3+ 49.06 48.18 58.77 54.30 42.33 34.89 68.29 52.74 36.07
38.66 39.55 40.18 38.63 46.28 ViT 44.39 41.63 50.07 49.22 37.69 28.33 68.34
41.44 24.74 32.22 29.47 32.68 29.78 39.23 Segformer 47.14 45.10 52.95 51.30
37.14 32.05 69.34 46.69 29.00 34.58 31.86 35.20 33.73 42.01 Prithvi 40.38
47.47 54.91 52.97 40.20 36.11 68.04 49.67 30.84 34.93 32.37 32.83 34.27 42.69

TABLE II: F1-score Comparison for Multi-Temporal Crop Classification

Methods Nat. Veg. Forest Corn Soybeans Wetlands Developed Water Wheat Alfalfa
Idle Cotton Sorghum Other mF1 RF 66.66 69.72 73.84 71.32 60.95 58.88 86.67
68.64 45.73 55.07 52.10 55.53 55.96 63.16 RFaug 66.88 69.92 74.65 72.21 61.30
59.16 86.42 69.48 46.91 55.78 52.51 56.94 56.86 63.77 XGB 54.99 61.64 55.43
50.75 34.42 35.31 85.42 49.18 19.74 42.80 13.24 28.37 28.02 43.02 XGBaug 52.91
61.39 56.37 52.59 43.20 38.53 84.88 51.39 20.62 45.07 13.63 37.01 30.61 45.32
FCN 54.88 59.37 50.22 45.43 35.99 24.62 84.95 45.52 14.41 35.17 0.32 23.11
14.85 37.60 U-Net 71.14 70.79 78.27 77.36 58.14 70.67 86.92 73.56 56.24 59.78
59.41 58.92 59.36 67.74 D3+ 65.83 65.03 74.03 70.38 59.48 51.73 81.16 69.06
53.01 55.76 56.68 57.33 55.73 62.71 ViT 61.49 58.79 66.73 65.97 54.75 44.15
81.19 58.59 39.67 48.73 45.52 49.26 45.90 55.44 Segformer 64.08 62.16 69.24
67.81 54.16 48.54 81.89 63.66 44.96 51.39 48.32 52.07 50.45 58.36 Prithvi
57.53 64.38 70.89 69.25 57.34 53.06 80.98 66.37 47.14 51.77 48.91 49.43 51.05
59.09

TABLE III: Comparative Performance of All Models in Burn Scar

Clouds as no-data masks Clouds as a class Burn Scar All classes Burn Scar All
classes Methods IoU F1 mIoU mF1 IoU F1 mIoU mF1 RF 58.86 74.1 77.12 85.87
58.85 74.1 84.69 90.55 RFaug 64.21 78.2 80.13 88.1 64.2 78.19 86.67 92.02 XGB
48.98 65.76 71.77 81.48 48.95 65.73 80.02 87.06 XGBaug 54.23 70.32 74.63 83.89
54.21 70.3 81.85 88.63 FCN 56.77 72.42 76.01 84.99 56.52 72.22 81.07 88.44
U-Net 76.95 86.97 87.22 92.85 76.87 86.92 89.65 94.29 D3+ 81.34 89.71 89.6
94.32 81.24 89.65 91.3 95.3 ViT 78.77 88.12 88.21 93.47 78.64 88.04 89.34
94.18 SegFormer 77.72 87.46 87.59 93.09 77.52 87.34 88.86 93.89 Prithvi 74.26
85.23 85.53 91.8 74.26 85.23 86.6 92.55

TABLE IV: Comparative Performance of All Models in Flood Mapping

Clouds as no-data masks Clouds as a class Water All classes Water All classes
Methods IoU F1 mIoU mF1 IoU F1 mIoU mF1 RF 79.70 88.7 88.13 93.48 78.42 87.9
78.29 87.32 RFaug 81.00 89.5 88.91 93.94 79.74 88.73 79.79 88.34 XGB 82.02
90.12 89.63 94.36 79.91 88.83 78.55 87.49 XGBaug 84.29 91.47 90.8 95.05 81.4
89.75 80.26 88.63 FCN 82.67 90.51 89.68 94.42 79.6 88.64 77.75 86.91 U-Net
81.00 89.5 87.18 93.03 79.63 88.66 77.72 86.98 D3+ 71.59 83.44 83.65 90.63
70.66 82.81 57.03 67.37 ViT 76.25 86.52 86.23 92.23 71.23 83.2 70.21 81.36
SegFormer 79.97 88.87 86.25 92.50 77.64 87.42 76.12 85.89 Prithvi 80.02 88.9
88.5 93.69 74.47 85.36 58.59 68.19

###  IV-C Results

####  IV-C1 Crop classification

Tables I and II provide comparisons of IoU and F1 scores among foundation
models, regular-size deep learning models, and traditional ML models for crop
classification. Prithvi – pre-trained with multispectral HLS data – did have
the best performance among the foundation models. Though, interestingly,
traditional ML models such as RF and RFaug outperformed it by about 4-5%. The
trend is similar across classes. The regular-size model U-Net had the best
performance, indicating that the vast SSL pre-training from masked autoencoder
in foundation models likely was not very helpful. As a result, the cons of the
larger number of parameters (e.g., harder to train) may have largely
outweighed the potential for better generalizability. Overall, U-Net
outperformed RFaug, while RFaug outperformed Prithvi by about 4%. These 3
models, as summarized in Sec. IV-D, were the most stable within their model
group (i.e., either the best or very close to the best in the group).

####  IV-C2 Burn Scar

Texture patterns and local contextual information are important for burn scar
classification as ground exposed after burns could be confused with other
normal ground if only pixel-level spectral patterns are considered. As a
result, both foundation models and regular convolutional networks showed
significant advantages over traditional models such as random forest (Table
III). DeepLabV3+ had the best results on this dataset, but the differences
among deep learning models including foundation models (except the fully-
connected FCN) are not significant. Additionally, DeepLabV3+ was less stable
across the three datasets. Traditional ML methods’ lower performance shows
their limitations in capturing texture patterns and contextual information
beyond local pixels. We also included the results where clouds are considered
as a class instead of no-data masks in the evaluation. This is because cloud
pixels differ from other types of no-data (e.g., pixels falling outside the
valid boundary of an image due to the non-orthogonal orientation of the image
tile) and they may need to be predicted as part of the model outputs. Here the
trained models are the same as before, and the only difference is in the
calculation of evaluation metrics. The results for single-class (water) also
changed slightly because clouds are no longer used as no-data masks for
computing metrics. For this burn scar task, the impact on ranking is less
significant, and all scores increased (except FCN) potentially due to higher
individual-class scores on clouds. However, the gaps between deep learning
models and traditional ML models were reduced.

####  IV-C3 Flood

For flood classification, traditional ML methods showed the best performances
(e.g., showing 2 out of the top 3 results for IoU and F1 scores; Table IV),
with XGBaug presenting the highest F1 and IoU among all models. This
observation aligns with the fact that water exhibits a pronounced spectral
difference compared to other ground types (e.g., can be well expressed by
spectral indices such as NDWI [21]). Given that pixel-based spectral
characteristics dominate – while spatial textures are less pronounced – for
flood classification, a misalignment between the downstream task and the pre-
training task becomes evident, consequently diminishing the effectiveness of
the foundation model. In Table IV, the differences among the methods are not
significant. However, the stability of the methods varied when evaluating the
scenario with clouds included as a class. Particularly, Prithvi, ViT, and
DeepLabV3+ experienced relatively larger drops in their performance as the
three methods significantly underestimated cloud pixels. The four traditional
ML models took the top-4 mIoU and mF1 scores in this scenario.

###  IV-D Summary across tasks

Fig. 1 shows the comparison across the three tasks, where the performance of
RFaug is used as a baseline. For flood and burn scar, the results shown were
the versions with clouds as no-data masks. Among the three groups of models,
RFaug, U-Net, and Prithvi, showed the most stable performances, respectively.
They were either the best models in each group or within about 1-2% compared
to the best model. In contrast, other models experienced large variations in
their performance. Other than burn scar classification where texture patterns
are important, foundation models have not outperformed traditional models that
are still widely used in major products (e.g., [10, 11]). When texture
patterns are not as important, FCN (each pixel as an input) did not outperform
traditional methods. Overall, we found U-Net to have the best performance,
which likely strikes a good balance between model size and robustness against
local noisy fluctuations (e.g., salt and pepper errors in traditional
methods). The comparison indicated that it is still premature to expect a
meaningful paradigm shift by foundation models for pixel-level classification
using multispectral imagery at moderate spatial resolution.

Figure 1: Comparison across the 3 tasks using RFaug as a baseline. The methods
are horizontally separated for better visual clarity.

##  V Conclusion

This paper aims to analyze the status and suitability of foundation models for
pixel-level classification using multispectral imagery at moderate spatial
resolution. The results conformed to our analysis: the standard masked
autoencoder methods used for self-supervised pre-training only showed
suitability when class labels have a high dependency on texture patterns from
the images. As a result, the foundation models still did not outperform (and
underperformed) traditional ML models on tasks where textures are not as
informative. However, foundation models showed effectiveness where texture
features are important, but did not show improvements over regular-size deep
learning models. This could be caused by the less complex and pronounced
texture signatures, compared to those in typical vision problems. Thus far,
the results have not indicated a paradigm shift. However, efforts such as
Prithvi are important to facilitate further understanding. Future efforts are
needed for SSL tasks that are more aligned with real downstream tasks.

## Acknowledgments

This material is based upon work supported by the National Science Foundation
under Grant No. 2105133, 2126474, 2147195, 2118329, 1942680, 1952085, and
2021871; NASA under Grant No. 80NSSC22K1164 and 80NSSC24K0354; USGS under
Grant No. G21AC10207; Google’s AI for Social Good; the Zaratan supercomputing
cluster at the University of Maryland; and Pitt Momentum Funds award and CRC
at the University of Pittsburgh.

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