choices. Under the condition in Eq. (93) it is easy to see that the choices
$C_{a|x}=\dfrac{M_{a|x}}{d},\rho_{x}=\sum_{a}\dfrac{M_{a|x}}{d}=\dfrac{\mathds{1}}{d}$,
and $\ell_{x,i}=\dfrac{p(x)}{d^{2}}$ for the dual problem in Eq. (91) lead to
$\operatorname{C_{\diamond}}(\mathcal{M}_{\mathbf{p}})\geq 1-\dfrac{1}{d}$ for
any probability distribution $p(x)$. Since
$\operatorname{\mathrm{IF}_{\diamond}}(\mathcal{M}_{\mathbf{p}})=1-\dfrac{1}{d}$
and
$\operatorname{\mathrm{IF}_{\diamond}}(\mathcal{M}_{\mathbf{p}})\geq\operatorname{C_{\diamond}}(\mathcal{M}_{\mathbf{p}})$
for any assemblage of rank-$1$ projections it has to hold
$\operatorname{\mathrm{IF}_{\diamond}}(\mathcal{M}_{\mathbf{p}})=\operatorname{C_{\diamond}}(\mathcal{M}_{\mathbf{p}})$
whenever the condition in Eq. (93) is fulfilled.

##  Appendix H More distances

In the main text, we defined general distances between measurement assemblages
in Definition 1. However, so far we only focused on one particular distance.
Here, we introduce more examples of distances for measurements and discuss
their basic properties.
We start by introducing the Schatten $\mathrm{p}-$norm functions
$\operatorname{D_{\mathrm{p}}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})$
for $\mathrm{p}\in[1,\infty)$, defined as

$\displaystyle\operatorname{D_{\mathrm{p}}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})=\sum_{a,x}p(x)\dfrac{1}{2}\lVert
M_{a|x}-N_{a|x}\rVert_{\mathrm{p}},$ (94)

where $\lVert X\rVert_{\mathrm{p}}=(\mathrm{Tr}[\lvert
X\rvert^{\mathrm{p}}])^{1/\mathrm{p}}$ is the Schatten $\mathrm{p}-$norm of
$X$. Note that the cases $\mathrm{p}=1$ and $\mathrm{p}=\infty$ correspond to
the trace norm, respectively the spectral norm. While the functions
$\operatorname{D_{\mathrm{p}}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})$
will generally not fulfil the monotonicity under Hilbert-Schmidt adjoint
channels $\Lambda^{\dagger}$ according to Definition 1, we will show in the
following that the $\mathrm{p}=\infty$ case corresponds to a proper distance.
Note that for $\mathrm{p}=1$, the monotonicity under quantum channel
$\Lambda^{\dagger}$ is not fulfilled, which can be seen by considering trivial
extensions of the form $\Lambda^{\dagger}(M_{a|x})=\mathds{1}\otimes M_{a|x}$.
Nevertheless, we also define the induced functions

$\displaystyle\operatorname{R_{\mathrm{p}}}(\mathcal{M}_{\mathbf{p}})=\min\limits_{\mathcal{F}\in\mathscr{F}}\operatorname{D_{\mathrm{p}}}(\mathcal{M}_{\mathbf{p}},\mathcal{F}_{\mathbf{p}})$
(95)

We formulate the following theorem to show that
$\operatorname{D_{\infty}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})$
is a distance between measurement assemblages.

######  Theorem 5.

The function
$\operatorname{D_{\infty}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})$
is a distance function between the WMAs $\mathcal{M}_{\mathbf{p}}$ and
$\mathcal{N}_{\mathbf{p}}$, i.e., it fulfils all the conditions stated in
Definition 1. Moreover,
$\operatorname{D_{\infty}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})$
is jointly-convex.

###### Proof.

The proof can be split into several parts. Note first, that since
$\operatorname{D_{\infty}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})$
is a weighted sum of spectral norms, it follows that
$\operatorname{D_{\infty}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})\geq
0$ with equality holding if and only if $\mathcal{M}=\mathcal{N}$. Note
further that the symmetry and triangle inequality condition in Definition 1
are fulfilled trivially.
For the monotonicity under quantum channel, we consider a more general data-
processing type inequality for the $\infty-$ distance between two POVM
elements. Namely, for $\Lambda^{\dagger}(M_{a|x})$ and
$\Lambda^{\dagger}(N_{a|x})$, where $\Lambda^{\dagger}$ is a unital completely
positive map, it follows

$\displaystyle\lVert\Lambda^{\dagger}(M_{a|x})-\Lambda^{\dagger}(N_{a|x})\rVert_{\infty}$
$\displaystyle=\max\limits_{\rho}\lvert\mathrm{Tr}[(\Lambda^{\dagger}(M_{a|x})-\Lambda^{\dagger}(N_{a|x}))\rho]\rvert$
(96)
$\displaystyle=\max\limits_{\rho}\lvert\mathrm{Tr}[(M_{a|x}-N_{a|x})\Lambda(\rho)]\rvert$
$\displaystyle\leq\max\limits_{\rho^{\prime}}\lvert\mathrm{Tr}[(M_{a|x}-N_{a|x})\rho^{\prime}]\rvert$
$\displaystyle=\|M_{a|x}-N_{a|x}\|_{\infty},$

where we used the dual representation of the Schatten-$\infty$ norm and the
fact that maximum is always achieved for a density matrix $\rho$ (more
specifically the projector onto the eigenvalue of largest absolute value of
$\Lambda^{\dagger}(M_{a|x})-\Lambda^{\dagger}(N_{a|x})$). Furthermore, we used
that the adjoint of the unital completely positive map $\Lambda^{\dagger}$ is
a CPT map $\Lambda$ which maps density matrices onto density matrices and
therefore shrinks the state-space one optimizes over. Since
$\operatorname{D_{\infty}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})$
is a sum of norms $\|M_{a|x}-N_{a|x}\|_{\infty}$, it follows that
$\operatorname{D_{\infty}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})\geq\operatorname{D_{\infty}}(\Lambda^{\dagger}(\mathcal{M})_{\mathbf{p}},\Lambda^{\dagger}(\mathcal{N})_{\mathbf{p}})$.
The monotonicity under classical simulations
$\xi(\mathcal{M}_{\mathbf{p}})_{\mathbf{q}}$ follows by direct computation,

$\displaystyle\operatorname{D_{\infty}}(\xi(\mathcal{M}_{\mathbf{p}})_{\mathbf{q}},\xi(\mathcal{N}_{\mathbf{p}})_{\mathbf{q}})$
$\displaystyle=\dfrac{1}{2}\sum_{b,y}q(y)\lVert\sum_{x,a}p(x|y)q(b|y,x,a)[M_{a|x}-N_{a|x}]\rVert_{\infty}$
(97)
$\displaystyle\leq\dfrac{1}{2}\sum_{b,y}q(y)\sum_{x,a}p(x|y)q(b|y,x,a)\lVert
M_{a|x}-N_{a|x}\rVert_{\infty}$
$\displaystyle=\dfrac{1}{2}\sum_{y,x,a}q(y)p(x|y)\lVert
M_{a|x}-N_{a|x}\rVert_{\infty}$
$\displaystyle=\dfrac{1}{2}\sum_{x,a}p(x)\lVert
M_{a|x}-N_{a|x}\rVert_{\infty}=\operatorname{D_{\infty}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}}),$

where we used the following properties.
In the first line, we used the definition of
$\operatorname{D_{\infty}}(\xi(\mathcal{M}_{\mathbf{p}})_{\mathbf{q}},\xi(\mathcal{N}_{\mathbf{p}})_{\mathbf{q}})$
by introducing the assemblages
$\mathcal{M}^{\prime}_{\mathbf{q}}=\xi(\mathcal{M}_{\mathbf{p}})_{\mathbf{q}}$
and
$\mathcal{N}^{\prime}_{\mathbf{q}}=\xi(\mathcal{N}_{\mathbf{p}})_{\mathbf{q}}$
where we inserted $M^{\prime}_{b|y}=\sum_{x}p(x|y)\sum_{a}q(b|y,x,a)M_{a|x}$
and $N^{\prime}_{b|y}=\sum_{x}p(x|y)\sum_{a}q(b|y,x,a)N_{a|x}$ directly. In
the second line, we used the triangle inequality. In the third line, we
performed the sum over $b$. Finally, in the fourth line, we used that
$\sum_{y}q(y)p(x|y)=p(x)$, which leads exactly to the definition of
$\operatorname{D_{\infty}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})$
from which the monotonicity under classical simulations $\xi$ follows.
Therefore,
$\operatorname{D_{\infty}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})$
is a distance between measurement assemblages according to Definition 1.
The proof of the joint-convexity of
$\operatorname{D_{\infty}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})$
follows exactly the same lines as the proof of the joint-convexity of
$\operatorname{D_{\diamond}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})$
in Theorem 1 and can be adapted from there. ∎

Even though they are not resource monotones generally, the functions
$\mathrm{R}_{\mathrm{p}}(\mathcal{M}_{\mathbf{p}})$ in Eq. (95) can be used to
bound the resource quantifier
$\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})$ defined in Eq. (15).
More specifically, we derive in the following the bounds on the diamond
distance based quantifier
$\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})$ given by

$\displaystyle\dfrac{1}{d}\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}})\leq\dfrac{1}{d}\mathrm{R}_{1}(\mathcal{M}_{\mathbf{p}})$
$\displaystyle\leq\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})\leq\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}})\leq\mathrm{R}_{1}(\mathcal{M}_{\mathbf{p}}),$
(98)

where $d$ is the dimension of the Hilbert space $\mathcal{H}$ the POVMs from
$\mathcal{M}$ act on. Note that due to the monotonicity of Schatten norms, it
holds $\lVert X\rVert_{\mathrm{p}}\leq\lVert X\rVert_{\mathrm{p}^{\prime}}$
for $\mathrm{p}\geq\mathrm{p}^{\prime}$ from which the bounds
$\dfrac{1}{d}\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}})\leq\dfrac{1}{d}\mathrm{R}_{1}(\mathcal{M}_{\mathbf{p}})$
and
$\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}})\leq\mathrm{R}_{1}(\mathcal{M}_{\mathbf{p}})$
follow directly.
The bound
$\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})\leq\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}})$
follows from

$\displaystyle\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})$
$\displaystyle=\min\limits_{\mathcal{F}\in\mathscr{F}}\dfrac{1}{2}\sum_{x}p(x)\max\limits_{\rho}\sum_{a}\lVert\mathrm{Tr}_{1}[(M_{a|x}\otimes\mathds{1})\rho]-\mathrm{Tr}_{1}[(F_{a|x}\otimes\mathds{1})\rho]\rVert_{1}$
(99)
$\displaystyle\leq\min\limits_{\mathcal{F}\in\mathscr{F}}\dfrac{1}{2}\sum_{x}p(x)\max\limits_{\rho}\sum_{a}\lVert(M_{a|x}\otimes\mathds{1})\rho-(F_{a|x}\otimes\mathds{1})\rho\rVert_{1}$
$\displaystyle\leq\min\limits_{\mathcal{F}\in\mathscr{F}}\dfrac{1}{2}\sum_{a,x}p(x)\max\limits_{\rho}\lVert(M_{a|x}\otimes\mathds{1})-(F_{a|x}\otimes\mathds{1})\rVert_{\infty}\lVert\rho\rVert_{1}$
$\displaystyle=\min\limits_{\mathcal{F}\in\mathscr{F}}\dfrac{1}{2}\sum_{a,x}p(x)\|(M_{a|x}\otimes\mathds{1})-(F_{a|x}\otimes\mathds{1})\|_{\infty}=\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}}),$

where we used the definition of
$\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})$ in the first line and
the monotonicity of the trace norm under partial trace in the second line. In
the third line, we used the Hölder inequality and in the last line identified
the definition of $\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}})$.

The last remaining bound
$\dfrac{1}{d}\mathrm{R}_{1}(\mathcal{M}_{\mathbf{p}})\leq\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})$
can directly be obtained by using $\rho=|\Phi^{+}\rangle\langle\Phi^{+}|$
within the optimization of the diamond norm. Here,
$|\Phi^{+}\rangle=\dfrac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|ii\rangle$ is the
maximally entangled state, where (as before) $d$ is the dimension the POVMs of
the measurement assemblage $\mathcal{M}$ act on. It follows

$\displaystyle\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})$
$\displaystyle=\min\limits_{\mathcal{F}\in\mathscr{F}}\dfrac{1}{2}\sum_{x}p(x)\max\limits_{\rho}\sum_{a}\lVert\mathrm{Tr}_{1}[(M_{a|x}\otimes\mathds{1})\rho]-\mathrm{Tr}_{1}[(F_{a|x}\otimes\mathds{1})\rho]\rVert_{1}$
(100)
$\displaystyle\geq\min\limits_{\mathcal{F}\in\mathscr{F}}\dfrac{1}{2}\sum_{x}p(x)\sum_{a}\lVert\mathrm{Tr}_{1}[(M_{a|x}\otimes\mathds{1})|\Phi^{+}\rangle\langle\Phi^{+}|]-\mathrm{Tr}_{1}[(F_{a|x}\otimes\mathds{1})|\Phi^{+}\rangle\langle\Phi^{+}|]\rVert_{1}$
$\displaystyle=\min\limits_{\mathcal{F}\in\mathscr{F}}\dfrac{1}{2}\sum_{x}p(x)\sum_{a}\dfrac{1}{d}\lVert(M_{a|x}-F_{a|x})^{T}\rVert_{1}$
$\displaystyle=\min\limits_{\mathcal{F}\in\mathscr{F}}\dfrac{1}{2}\sum_{x}p(x)\sum_{a}\dfrac{1}{d}\lVert
M_{a|x}-F_{a|x}\rVert_{1}=\dfrac{1}{d}\mathrm{R}_{1}(\mathcal{M}_{\mathbf{p}}),$

where we used in the first line the definition of
$\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})$ and in the second line
that $\rho=|\Phi^{+}\rangle\langle\Phi^{+}|$ is a feasible point within the
maximization over the quantum states within the diamond norm. In the third
line, we used that
$\mathrm{Tr}_{1}[(M_{a|x}\otimes\mathds{1})|\Phi^{+}\rangle\langle\Phi^{+}|]=\dfrac{1}{d}M_{a|x}^{T}$,
where the transposition is with respect to the computational basis. Finally,
we can use that a transposition does not change the singular values.
The monotone $\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}})$ in particular is
not only a valuable tool to bound the diamond distance
$\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})$ but is also
interesting in itself. More specifically, we show in the following that
$\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}})$ obeys also a measurement
hierarchy similar to that in Eq. (30). Let
$\operatorname{\mathrm{IF}_{\infty}}(\mathcal{M}_{\mathbf{p}}),\operatorname{C_{\infty}}(\mathcal{M}_{\mathbf{p}}),$
and $\operatorname{I_{\infty}}(\mathcal{M}_{\mathbf{p}})$ be the
informativeness, coherence, and incompatibility of ${M}_{\mathbf{p}}$ as
measured by the distance $\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}})$ in
Eq. (95) with respect to the free sets $\mathscr{F}_{\mathrm{UI}}$,
$\mathscr{F}_{\mathrm{IC}}$, and $\mathscr{F}_{\mathrm{JM}}$. It follows
directly from
$\mathscr{F}_{\mathrm{UI}}\subset\mathscr{F}_{\mathrm{IC}}\subset\mathscr{F}_{\mathrm{JM}}$
that the hierarchy

$\displaystyle\operatorname{\mathrm{IF}_{\infty}}(\mathcal{M}_{\mathbf{p}_{A}})\geq\operatorname{C_{\infty}}(\mathcal{M}_{\mathbf{p}_{A}})\geq\operatorname{I_{\infty}}(\mathcal{M}_{\mathbf{p}_{A}}),$
(101)

holds. Moreover, it can be shown that
$\operatorname{I_{\infty}}(\mathcal{M}_{\mathbf{p}_{A}})\geq\mathrm{S}(\vec{\sigma}_{\mathbf{p}_{A}})$
(remember that we already showed that
$\mathrm{S}(\vec{\sigma}_{\mathbf{p}_{A}})\geq\mathrm{N}(\mathbf{q}_{\mathbf{p}})$).
This follows from the direct computation for any quantum state $\rho$ of
appropriate dimension and the closest JM assemblage $\mathcal{F}^{*}$ to
$\mathcal{M}$ (with respect to the monotone
$\operatorname{I_{\infty}}(\mathcal{M}_{\mathbf{p}})$):

$\displaystyle\mathrm{S}(\vec{\sigma}_{\mathbf{p}_{A}})$
$\displaystyle\leq\dfrac{1}{2}\sum_{a,x}p_{A}(x)\lVert\mathrm{Tr}_{1}[((M_{a|x}-F^{*}_{a|x})\otimes\mathds{1})\rho]\rVert_{1}$
(102)
$\displaystyle\leq\dfrac{1}{2}\sum_{a,x}p_{A}(x)\lVert((M_{a|x}-F^{*}_{a|x})\otimes\mathds{1})\rho\rVert_{1}$
$\displaystyle\leq\dfrac{1}{2}\sum_{a,x}p_{A}(x)\lVert(M_{a|x}-F^{*}_{a|x})\otimes\mathds{1}\rVert_{\infty}\lVert\rho\rVert_{1}$
$\displaystyle=\dfrac{1}{2}\sum_{a,x}p_{A}(x)\|M_{a|x}-F^{*}_{a|x}\|_{\infty}=\operatorname{I_{\infty}}(\mathcal{M}_{\mathbf{p}_{A}}),$

where we first used that JM measurements always lead to unsteerable
assemblages. Second, we used that the trace norm is non-increasing under
partial traces. Third, we used the Hölder inequality. It therefore follows,
that the hierarchy

$\displaystyle\operatorname{\mathrm{IF}_{\infty}}(\mathcal{M}_{\mathbf{p}_{A}})\geq\operatorname{C_{\infty}}(\mathcal{M}_{\mathbf{p}_{A}})\geq\operatorname{I_{\infty}}(\mathcal{M}_{\mathbf{p}_{A}})\geq\mathrm{S}(\vec{\sigma}_{\mathbf{p}_{A}})\geq\mathrm{N}(\mathbf{q}_{\mathbf{p}}),$
(103)

holds.
Another distance for measurement assemblages that can be considered is based
on the $\ell_{1}$-distance between probability distributions. More
specifically, the induced $\ell_{1}$-distance between two WMAs is given by

$\displaystyle\operatorname{D_{\ell_{1}}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})=\dfrac{1}{2}\sum_{x}p(x)\max_{\rho_{A}}\sum_{a}|\mathrm{Tr}[(M_{a|x}-N_{a|x})\rho_{A}]|,$
(104)

which is the $\ell_{1}$-distance of the probability distributions
$\\{\mathrm{Tr}[M_{a|x}\rho_{A}]\\}$ and $\\{\mathrm{Tr}[N_{a|x}\rho_{A}]\\}$
maximized over all quantum states $\rho_{A}$. With the same methods as for the
distances
$\operatorname{D_{\infty}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})$
and
$\operatorname{D_{\diamond}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})$
it can be shown that
$\operatorname{D_{\ell_{1}}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})$
fulfills all the conditions in Definition 1. Moreover, the joint-convexity of
$\operatorname{D_{\ell_{1}}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})$
follows from

$\displaystyle\operatorname{D_{\ell_{1}}}(\eta\mathcal{M}^{(1)}_{\mathbf{p}}+(1-\eta)\mathcal{M}^{(2)}_{\mathbf{p}},\eta\mathcal{N}^{(1)}_{\mathbf{p}}+(1-\eta)\mathcal{N}^{(2)}_{\mathbf{p}})$
(105)
$\displaystyle=\dfrac{1}{2}\sum_{x}p(x)\max\limits_{\rho_{A}}\sum_{a}\lvert\mathrm{Tr}[\eta(M_{a|x}^{(1)}-N_{a|x}^{(1)})+(1-\eta)(M_{a|x}^{(2)}-N_{a|x}^{(2)})]\rvert$
$\displaystyle\leq\dfrac{1}{2}\sum_{x}p(x)\max\limits_{\rho_{A}}\sum_{a}\eta\lvert\mathrm{Tr}[(M_{a|x}^{(1)}-N_{a|x}^{(1)})\rho_{A}]\rvert+(1-\eta)\lvert\mathrm{Tr}[(M_{a|x}^{(2)}-N_{a|x}^{(2)})\rho_{A}]\rvert$
$\displaystyle\leq\dfrac{1}{2}\sum_{x}p(x)\max\limits_{\rho_{A}^{(1)},\rho_{A}^{(2)}}\sum_{a}\eta\lvert\mathrm{Tr}[(M_{a|x}^{(1)}-N_{a|x}^{(1)})\rho_{A}^{(1)}]\rvert+(1-\eta)\lvert\mathrm{Tr}[(M_{a|x}^{(2)}-N_{a|x}^{(2)})\rho_{A}^{(2)}]\rvert$
$\displaystyle=\eta\operatorname{D_{\ell_{1}}}(\mathcal{M}^{(1)}_{\mathbf{p}},\mathcal{N}^{(1)}_{\mathbf{p}})+(1-\eta)\operatorname{D_{\ell_{1}}}(\mathcal{M}^{(2)}_{\mathbf{p}},\mathcal{N}^{(2)}_{\mathbf{p}}),$

where we used the homogeneity and the triangle inequality of the absolute
value and the fact that the individual maximization over both terms within the
sum over $a$ can only increase the value. Hence,
$\operatorname{D_{\ell_{1}}}(\mathcal{M}_{\mathbf{p}},\mathcal{N}_{\mathbf{p}})$
is a jointly-convex distance function which induces the distance-based convex
monotone

$\displaystyle\operatorname{R_{\ell_{1}}}(\mathcal{M}_{\mathbf{p}})=\min\limits_{\mathcal{F}\in\mathscr{F}}\operatorname{D_{\ell_{1}}}(\mathcal{M}_{\mathbf{p}},\mathcal{F}_{\mathbf{p}}).$
(106)

Note that while it follows directly that
$\operatorname{R_{\ell_{1}}}(\mathcal{M}_{\mathbf{p}})$ will naturally induce
a hierarchy between the informativeness, coherence, and the incompatibility of
a WMA $\mathcal{M}_{\mathbf{p}}$, it is not clear whether there exist steering
or nonlocality monotones that are in natural correspondence to it. Note
further that in the context of coherence of single POVMs, this kind of
statistical measure has also been defined by Baek et al. [25].

Even though it is not clear whether a complete hierarchy of measurement
resources holds, the quantifier
$\operatorname{R_{\ell_{1}}}(\mathcal{M}_{\mathbf{p}})$ is important, as it
can be seen as limiting case of the quantifier
$\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})$ when the maximization
is performed only over product states. More formally, it holds

$\displaystyle\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})$
$\displaystyle\geq\min\limits_{\mathcal{F}\in
F}\dfrac{1}{2}\sum_{x}p(x)\max\limits_{\rho=\rho_{A}\otimes\rho_{B}}\sum_{a}\lVert\mathrm{Tr}_{1}[((M_{a|x}-F_{a|x})\otimes\mathds{1})\rho]\rVert_{1}$
(107) $\displaystyle=\min\limits_{\mathcal{F}\in
F}\dfrac{1}{2}\sum_{x}p(x)\max\limits_{\rho=\rho_{A}\otimes\rho_{B}}\sum_{a}\lVert\mathrm{Tr}[(M_{a|x}-F_{a|x})\rho_{A}]\rho_{B}\rVert_{1}$
$\displaystyle=\min\limits_{\mathcal{F}\in
F}\dfrac{1}{2}\sum_{x}p(x)\max\limits_{\rho=\rho_{A}\otimes\rho_{B}}\sum_{a}\lvert\mathrm{Tr}[(M_{a|x}-F_{a|x})\rho_{A}]\rvert\lVert\rho_{B}\rVert_{1}$
$\displaystyle=\min\limits_{\mathcal{F}\in
F}\dfrac{1}{2}\sum_{x}p(x)\max\limits_{\rho_{A}}\sum_{a}\lvert\mathrm{Tr}[(M_{a|x}-F_{a|x})\rho_{A}]\rvert$
$\displaystyle=\operatorname{R_{\ell_{1}}}(\mathcal{M}_{\mathbf{p}}),$

where we used in the first line that we maximize only over the set of product
states. In the second line we used the definition of the partial trace and
finally, we used that states $\rho_{B}$ are normalized in the $1-$norm and
identified the last line with the definition of the induced
$\ell_{1}$-distance quantifier
$\operatorname{R_{\ell_{1}}}(\mathcal{M}_{\mathbf{p}})$. In Appendix I, we
show that the quantifiers
$\operatorname{R_{\ell_{1}}}(\mathcal{M}_{\mathbf{p}})$ and
$\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}})$ coincide with
$\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})$ in the case of
dichotomic measurement assemblages.

##  Appendix I Dichotomic measurements

Here, we show an additional property of
$\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})$ that can be useful for
the case where we consider measurement assemblages $\mathcal{M}$ with only two
outcomes for each setting $x$. We show that in this special case, the diamond
distance quantifier $\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})$ is
equivalent to $\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}})$ and
$\operatorname{R_{\ell_{1}}}(\mathcal{M}_{\mathbf{p}})$. Consider the WMAs
$\\{M_{1|x},\mathds{1}-M_{1|x}\\}_{x}$ and
$\\{F_{1|x},\mathds{1}-F_{1|x}\\}_{x}$. Remember that we already showed
previously that
$\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})\leq\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}})$,
so we only need to show that in the case of dichotomic measurements it also
holds
$\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})\geq\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}})=\operatorname{R_{\ell_{1}}}(\mathcal{M}_{\mathbf{p}})$.
This follows directly via

$\displaystyle\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})\geq\operatorname{R_{\ell_{1}}}(\mathcal{M}_{\mathbf{p}})$
$\displaystyle=2\dfrac{1}{2}\min\limits_{\mathcal{F}\in\mathscr{F}}\sum_{x}p(x)\max_{\rho_{A}}\lvert\mathrm{Tr}[(M_{a|x}-F_{a|x})\rho_{A}]\rvert=\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}}),$
(108)

where we used the bound
$\operatorname{R_{\diamond}}(\mathcal{M}_{\mathbf{p}})\geq\operatorname{R_{\ell_{1}}}(\mathcal{M}_{\mathbf{p}})$
and the fact that for dichotomic measurements both outcomes contribute equally
towards $\operatorname{R_{\ell_{1}}}(\mathcal{M}_{\mathbf{q}})$. Finally, we
used that this holds also true for
$\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}})$. Alternatively, it is also
enough to see that the same $\rho_{A}$ is optimal for both outcomes, which
leads to the conclusion that
$\mathrm{R}_{\infty}(\mathcal{M}_{\mathbf{p}})=\operatorname{R_{\ell_{1}}}(\mathcal{M}_{\mathbf{p}})$.
Note that the above result shows that entanglement does not offer an advantage
in distinguishing two measure-and-prepare channels for dichotomic measurements
by means of the diamond norm.

## Acronyms

AGF

    average gate fidelity
AMA

    associated measurement assemblage
BOG

    binned outcome generation
CGLMP

    Collins-Gisin-Linden-Massar-Popescu
CHSH

    Clauser-Horne-Shimony-Holt
CP

    completely positive
CPT

    completely positive and trace preserving
CS

    compressed sensing
DFE

    direct fidelity estimation
DM

    dark matter
GST

    gate set tomography
GUE

    Gaussian unitary ensemble
HOG

    heavy outcome generation
JM

    jointly measurable
LHS

    local hidden-state model
LHV

    local hidden-variable model
LOCC

    local operations and classical communication
MBL

    many-body localization
ML

    machine learning
MLE

    maximum likelihood estimation
MPO

    matrix product operator
MPS

    matrix product state
MUB

    mutually unbiased bases
MW

    micro wave
NISQ

    noisy and intermediate scale quantum
POVM

    positive operator valued measure
PVM

    projector-valued measure
QAOA

    quantum approximate optimization algorithm
QML

    quantum machine learning
QMT

    measurement tomography
QPT

    quantum process tomography
QRT

    quantum resource theory
RDM

    reduced density matrix
SDP

    semidefinite program
SFE

    shadow fidelity estimation
SIC

    symmetric, informationally complete
SPAM

    state preparation and measurement
RB

    randomized benchmarking
rf

    radio frequency
TT

    tensor train
TV

    total variation
UI

    uninformative
VQA

    variational quantum algorithm
VQE

    variational quantum eigensolver
WMA

    weighted measurement assemblage
XEB

    cross-entropy benchmarking

## Code availability

Matlab codes that accompany this work are available on GitHub. These contain
in particular implementations of the SDPs presented here.

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  *[QRT]: quantum resource theory
  *[QRTs]: quantum resource theory
  *[SDP]: semidefinite program
  *[MUB]: mutually unbiased bases
  *[CPT]: completely positive and trace preserving
  *[LOCC]: local operations and classical communication
  *[POVM]: positive operator valued measure
  *[POVMs]: positive operator valued measure
  *[WMA]: weighted measurement assemblage
  *[WMAs]: weighted measurement assemblage
  *[CP]: completely positive
  *[UI]: uninformative
  *[JM]: jointly measurable
  *[LHS]: local hidden-state model
  *[LHV]: local hidden-variable model
  *[CGLMP]: Collins-Gisin-Linden-Massar-Popescu
  *[SDPs]: semidefinite program