# New additivity properties of the relative entropy of entanglement and its
generalizations

Roberto Rubboli<EMAIL_ADDRESS>Centre for Quantum Technologies,
National University of Singapore, Singapore 117543, Singapore  Marco
Tomamichel  Department of Electrical and Computer Engineering, National
University of Singapore, Singapore 117583, Singapore  Centre for Quantum
Technologies, National University of Singapore, Singapore 117543, Singapore

###### Abstract

We prove that the relative entropy of entanglement is additive when _at least
one of the two states_ belongs to some specific class. We show that these
classes include bipartite pure, maximally correlated, GHZ, Bell diagonal,
isotropic, and generalized Dicke states. Previously, additivity was
established only if both states belong to the same class. Moreover, we extend
these results to entanglement monotones based on the $\alpha$-$z$ Rényi
relative entropy. Notably, this family of monotones includes also the
generalized robustness of entanglement and the geometric measure of
entanglement. In addition, we prove that any monotone based on a quantum
relative entropy is not additive for general states. Finally, we compute the
value of the monotones based on the $\alpha$-$z$ Rényi relative entropy for
bipartite pure, Bell diagonal, isotropic, generalized Werner, generalized
Dicke, and maximally correlated Bell diagonal states. Our results rely on
proving new necessary and sufficient conditions for the optimizer of the
monotones based on the $\alpha$-$z$ Rényi relative entropy which allow us to
reduce the original optimization problem to a simpler linear one. Even though
we focus mostly on entanglement theory, we formulate some of our technical
results in a general resource theory framework and we expect that they could
be used to investigate other resource theories.

##  I Introduction

Quantifying entanglement is a central problem in quantum information theory.
Entanglement monotones have the desirable property that they are monotone
under local operations and classical communications (LOCC), so-called free
operations in the resource theory of entanglement. Furthermore, additive
entanglement monotones remain monotones even in the presence of catalysts that
can be leveraged in the procedure but need to be returned unchanged. Thus, a
fundamental problem is to establish whether a certain entanglement monotone or
measure is additive and, if not, what are the minimum constraints on the
states to ensure its additivity. While this has been an active area for many
decades, a complete understanding of the additivity properties of many
entanglement monotones is still missing. Establishing the addivity properties
of an entanglement monotone is also of particular relevance in the analysis of
transformation rates between states under LOCC where we expect operationally
relevant quantities to be (at least) weakly additive. For example, it is shown
that the regularized entropy of entanglement provides an upper bound on
transformation rates [20]. More in general, in convex resource theories, the
regularized relative entropy of resource gives an upper bound on state
transformations under free operations [23].

In this work we focus on the additivity properties of the relative entropy of
entanglement [47]. The relative entropy of entanglement is an example of a
monotone which is known to be not additive under tensor product [51].
Nevertheless, we show that additivity holds whenever one of the two states
belongs to some specific class. This first shows that for several classes of
states only one state needs to be of some specific form to ensure its
additivity. Previously, additivity was proven only in the case where both
states belong to some specific class, e.g. when both states are pure [46], and
left as an open question for the case where only one state is constrained.

We generalize our results to the monotones based on the $\alpha$-$z$ Rényi
relative entropies which provide a general framework to address different
families of entanglement monotones. Notably, this family include, among many
others, also the geometric measure [4] and the generalized robustness of
entanglement [13]. This has, for example, immediate applications in catalytic
resource theories where we are interested in the additivity of the latter
family of monotones for a pure state with any, not necessarily pure, catalyst
state [40].

Entanglement monotones based on quantum relative entropies are ubiquitous in
entanglement theory. Some notable examples include the above mentioned
relative entropy of entanglement, generalized robustness and geometric measure
of entanglement. A fundamental question is whether an entanglement monotone
based on a quantum relative entropy is additive or not. Even though it was
already known that the relative entropy of entanglement and the generalized
robustness are not additive for general states [51, 60], a complete answer to
this question was still missing. We answer this question by showing that this
is not the case, i.e., any entanglement monotone based on a quantum relative
entropy is not additive for general states.

Our results hinge on proving new necessary and sufficient conditions for the
optimizer of the monotones based on the $\alpha$-$z$ Rényi relative entropies
which allow us to recast the original optimization problem to a simpler linear
one. We formulate these condition in a general resource theory framework and
we show that they provide a powerful tool to investigate additivity questions
in any resource theory by first considering the exemplary case of the resource
theory of coherence. In this case, we can prove that the monotones based on
the $\alpha$-$z$ Rényi divergences are additive for any pair of states by
extending some known results (see the discussion in Section IV.1). In
particular, we can show that an optimizer of the tensor product of two states
is the tensor product of optimizers of the marginal problems. More
interestingly, in entanglement theory, we can show that the optimizer of the
tensor product of two states still factorizes when one state belongs to some
specific class.

In the case when the state commutes with its optimizer, we find that these
necessary and sufficient conditions considerably simplify. This allows us to
readily provide a counterexample to the additivity of any entanglement
monotone based an a quantum relative entropy. Moreover, these conditions allow
us to analytically compute the monotones in a simple fashion for several
states by extending and unifying some already known results (see Section V and
Table 1).

The paper is structured as follows:

  * •

In Section II we introduce the monotones based on $\alpha$-$z$ Rényi relative
entropies and we discuss their connections with the relative entropy of
entanglement, the geometric measure and the generalized robustness of
entanglement.

  * •

In Section III we derive necessary and sufficient conditions for the optimizer
of the resource monotones based on $\alpha$-$z$ Rényi relative entropies. We
state our result for any convex quantum resource theory.

  * •

In Section IV we prove new additivity properties of the entanglement monotones
based on $\alpha$-$z$ Rényi relative entropies. We prove that they are
additive whenever one state belongs to one of two classes. The first class
includes the states that commute with their optimizer and such that the alpha
power of their product with the inverse of their optimizer has positive
entries in a product basis. The second class is the set of maximally
correlated states, which, notably, contains the bipartite pure states.
Finally, we extend the latter result to also include the multipartite GHZ
state.

  * •

In Section V we provide some examples of states that belong to the additivity
classes mentioned above. In particular, we show that maximally correlated,
bipartite pure, GHZ, Bell diagonal, isotropic, generalized Dicke, and
separable states belong to these classes. Moreover, we compute their value for
bipartite pure, Bell diagonal, isotropic, generalized Werner, generalized
Dicke, and maximally correlated Bell diagonal states by generalizing some
known results (see Table 1).

  * •

Finally, in Section VI we prove that any monotone based on a quantum relative
entropy is not additive for general states by providing a counterexample.
Moreover, we prove that they are not additive also when the minimization over
the separable states is replaced by the minimization over the states with
positive partial transpose (PPT).

##  II Resource monotones based on $\alpha$-$z$ Rényi relative entropies

We denote with $\mathcal{P}(A)$ the set of positive operators on a Hilbert
space $A$. Moreover, we denote with $\mathcal{S}(A)$ the set of quantum
states, i.e. the subset of $\mathcal{P}(A)$ with unit trace. Let
$\sigma\in\mathcal{S}(\otimes_{j=1}^{N}A_{j})$ be a $N$-partite state shared
among N parties with Hilbert spaces $A_{1},\ldots,A_{N}$, respectively. We say
that $\sigma$ is separable if it is of the form
$\sigma=\sum_{i}p_{i}\sigma_{i}^{1}\otimes\ldots\otimes\sigma_{i}^{N}$ for
some local states $\sigma_{i}^{j}\in\mathcal{S}(A_{j})$ and a probability
distribution $\\{p_{i}\\}$. Otherwise, we call it entangled. We denote the set
of all separable states by $\operatorname{SEP}(A_{1}:\ldots:A_{N})$ or just
$\operatorname{SEP}$ if the geometry is clear from context. The extreme points
of the set of separable states are the pure product states
$\sigma=\outerproduct{\sigma}{\sigma}$ with
$|\sigma\rangle=|\sigma_{1}\rangle\otimes\ldots\otimes|\sigma_{N}\rangle$. We
denote the set of pure product states with $\text{PRO}(A_{1}:\ldots:A_{N})$ or
just PRO .

A key property that a measure of entanglement is required to satisfy is the
monotonicity under local operations and classical communications (LOCC) [25,
46, 48]. We call a function
$\mathfrak{R}:\mathcal{S}(A)\rightarrow[0,+\infty]$ an entanglement monotone
if it does not increase under LOCC, i.e., if
$\mathfrak{R}(\rho)\geq\mathfrak{R}(\mathcal{E}(\rho))$ for any state $\rho$
and any LOCC operation $\mathcal{E}$ [49]. For an entanglement monotone to be
called an entanglement measure, further properties such as faithfulness,
convexity or full monotonicity are usually required [25, 46, 48].

Here, we follow the standard convention where, given two $N$-partite states
$\rho_{1}\in\mathcal{S}(\otimes_{j=1}^{N}A_{j})$ and
$\rho_{2}\in\mathcal{S}(\otimes_{j=1}^{N}A^{\prime}_{j})$, the parties $A_{j}$
and $A^{\prime}_{j}$ for $j=1,\cdots,N$ are assumed to be in the same
laboratory and hence the tensor product $\rho_{1}\otimes\rho_{2}$ is
considered to be a $N$-partite state shared among the parties
$A_{1}A^{\prime}_{1},\ldots,A_{N}A^{\prime}_{N}$. Similarly, we extend this
convention to tensor products of more than two states. We say that
$\mathfrak{R}$ is tensor sub-additive (or just sub-additive) for the states
$\rho_{1}$ and $\rho_{2}$ if
$\mathfrak{R}(\rho_{1}\otimes\rho_{2})\leq\mathfrak{R}(\rho_{1})+\mathfrak{R}(\rho_{2})$.
Moreover, we say that $\mathfrak{R}$ is tensor additive (or just additive) for
the states $\rho_{1}$ and $\rho_{2}$ if
$\mathfrak{R}(\rho_{1}\otimes\rho_{2})=\mathfrak{R}(\rho_{1})+\mathfrak{R}(\rho_{2})$.

We now introduce the entanglement monotones based on the $\alpha$-$z$ Rényi
relative entropy. We remark that even though we formulate our results very
generally for the monotones based on the $\alpha$-$z$ Rényi relative entropy,
our results are also new for the special cases of most interest, namely the
relative entropy of entanglement, the generalized robustness of entanglement
and the geometric measure of entanglement (see the discussion below for more
details). Let $\alpha\in(0,1)\cup(1,\infty),\;z>0$, $\rho\in\mathcal{S}(A)$
and $\sigma\in\mathcal{P}(A)$. Then the $\alpha$-$z$ Rényi relative entropy of
$\sigma$ with $\rho$ is defined as [3, 59]

$D_{\alpha,z}(\rho\|\sigma):=\begin{cases}\frac{1}{\alpha-1}\log{\text{Tr}\left(\rho^{\frac{\alpha}{2z}}\sigma^{\frac{1-\alpha}{z}}\rho^{\frac{\alpha}{2z}}\right)^{z}}&\text{if}\;(\alpha<1\wedge\rho\not\perp\sigma)\vee\rho\ll\sigma\\\
+\infty&\text{else}\end{cases}\,.$ (1)

In the following, we denote
$Q_{\alpha,z}(\rho\|\sigma):=\exp\big((\alpha-1)D_{\alpha,z}(\rho\|\sigma)\big{missing})$.
In the limit points of the ranges of the parameters, we define the
$\alpha$-$z$ Rényi relative entropy by taking the corresponding pointwise
limits. In particular, we have the pointwise limits [3, 28]

$\quad D_{\min}(\rho\|\sigma)=\lim\limits_{\alpha\rightarrow
0}D_{\alpha,1}(\rho\|\sigma)\;,\quad
D(\rho\|\sigma)=\lim\limits_{\alpha\rightarrow
1}D_{\alpha,z}(\rho\|\sigma)\;,\quad
D_{\max}(\rho\|\sigma)=\lim\limits_{\alpha\rightarrow\infty}D_{\alpha,\alpha}(\rho\|\sigma)\,,$
(2)

where

$\displaystyle D_{\min}(\rho\|\sigma):=-\log{\Tr(\Pi(\rho)\sigma)}\,,$ (3)
$\displaystyle
D(\rho\|\sigma):=\Tr(\rho(\log{\rho}-\log{\sigma}))\,,\,\text{and}$ (4)
$\displaystyle D_{\max}(\rho\|\sigma):=\inf\\{\lambda\in\mathbb{R}:\rho\leq
2^{\lambda}\sigma\\}\,,$ (5)

are the min-relative entropy [39, 13], the Umegaki relative entropy, and the
max-relative entropy [45, 13, 39], respectively. Note that the second limit in
(2) holds for any $z>0$ [28]. Here, we denoted with $\Pi(\rho)$ the projector
onto the support of $\rho$.

When $z=1$, the $\alpha$-$z$ Rényi relative entropy reduces to the _Petz Rényi
divergence_ [33, 45]

$\bar{D}_{\alpha}(\rho\|\sigma):=\frac{1}{\alpha-1}\log{\Tr(\rho^{\alpha}\sigma^{1-\alpha})}\,.$
(6)

For $z=\alpha$, it reduces to the sandwiched quantum Rényi divergence [31, 56,
45],

$\widetilde{D}_{\alpha}(\rho\|\sigma):=\frac{1}{\alpha-1}\log{\Tr\big(\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}}\big{missing})^{\alpha}}\,.$
(7)

For $z=1-\alpha$, we find the reverse sandwiched Rényi relative entropy [3],

$\widetilde{D}^{\text{rev}}_{\alpha}(\rho\|\sigma):=\frac{1}{\alpha-1}\log{\Tr\big(\rho^{\frac{\alpha}{2(1-\alpha)}}\sigma\rho^{\frac{\alpha}{2(1-\alpha)}}\big{missing})^{\alpha-1}}\,.$
(8)

In [59, Theorem 1.2] the authors proved that the $\alpha$-$z$ Rényi relative
entropy satisfies the data-processing inequality (DPI) if and only if one of
the following holds

  1. 1.

$\quad 0<\alpha<1\;\;and\;\;z\geq\max\\{\alpha,1-\alpha\\},$

  2. 2.

$\quad 1<\alpha\leq 2\;\;and\;\;\frac{\alpha}{2}\leq z\leq\alpha,\;and$

  3. 3.

$\quad 2\leq\alpha<\infty\;\;and\;\;\alpha-1\leq z\leq\alpha\,.$

We denote with $\mathcal{D}$ the set of above values of the parameters
$(\alpha,z)$ for which the $D_{\alpha,z}$ satisfies the DPI. Because of the
limits in (2), the $\alpha$-$z$ Rényi relative entropy also satisfies the DPI
on the line $\alpha=1$. We include this line in the region $\mathcal{D}$. In
Fig. 1 we represent the limits discussed above in the $\alpha$-$z$ plane and
we show in blue the region for which the DPI holds.

Figure 1: $\alpha$-$z$ plane for the $\alpha$-$z$ Rényi relative entropies.
The $z=1$ and the $z=\alpha$ lines corresponds to the Petz and the sandwiched
Rényi divergences, respectively. The vertical line $\alpha=1$ for $z>0$
corresponds to the Umegaki relative entropy. The blue region (together with
the Umegaki relative entropy line) is where the data-processing inequality
holds.

We define entanglement monotones for $(\alpha,z)\in\mathcal{D}$ as

$\displaystyle\mathfrak{D}_{\alpha,z}(\rho):=\inf_{\sigma\in\operatorname{SEP}}D_{\alpha,z}(\rho\|\sigma)\,.$
(9)

In particular,

$\mathfrak{D}_{\min}(\rho):=\inf_{\sigma\in\operatorname{SEP}}D_{\min}(\rho\|\sigma),\quad\mathfrak{D}(\rho):=\inf_{\sigma\in\operatorname{SEP}}D(\rho\|\sigma),\quad\mathfrak{D}_{\max}(\rho):=\inf_{\sigma\in\operatorname{SEP}}D_{\max}(\rho\|\sigma)\,.$
(10)

In Appendix Appendix A: Lower semicontinuity and its consequences we show that
the latter monotones coincide with the corresponding pointwise limits of the
monotone (9). Namely, we have

$\mathfrak{D}_{\min}(\rho)=\lim\limits_{\alpha\rightarrow
0}\mathfrak{D}_{\alpha,1}(\rho),\quad\mathfrak{D}(\rho)=\lim\limits_{\alpha\rightarrow
1}\mathfrak{D}_{\alpha,\alpha}(\rho),\quad\mathfrak{D}_{\max}(\rho)=\lim\limits_{\alpha\rightarrow\infty}\mathfrak{D}_{\alpha,\alpha}(\rho)\,.$
(11)

In the following, we refer to the $\mathfrak{D}_{\alpha,z}$ as $\alpha$-$z$
Rényi relative entropy of entanglement or as entanglement monotone based on
$\alpha$-$z$ Rényi relative entropy. In Appendix Appendix A: Lower
semicontinuity and its consequences we show that the above infimum is always
achieved, i.e. there always exists a closest separable state and hence the
infimum can be replaced by the minimum. However, in general the minimum might
not be unique as shown in [27, Example 21] for $\alpha=z=1$.

It is straightforward to prove that, for the values of $\alpha$ and $z$ for
which the underlying $\alpha$-$z$ Rényi relative entropy satisfies the data-
processing inequality, $\mathfrak{D}_{\alpha,z}$ does not increase under LOCC,
i.e. it is indeed an entanglement monotone. Moreover, in this interval, the
function $Q_{\alpha,z}(\rho\|\sigma)$ is jointly concave for $\alpha\leq 1$
and jointly convex for $\alpha\geq 1$ [59]. This implies that any local
minimum of the problem (9) is also a global minimum. This fact has been
already pointed out for $\alpha=z=1$ in [46].

In [46], the authors propose a numerical method to compute the relative
entropy of entanglement for bipartite states. Following this approach, the
minimum of the relative entropy of entanglement can be found with a gradient
search in a multidimensional parameter space. This is done using the
Caratheodory’s theorem which provides a way to bound the number of independent
real parameters needed to parametrize a separable state. The same strategy
could be used to find the minimum of the problem (9). However, this method
becomes soon impractical even for low dimensions. More recently, a method to
compute a lower bound on the relative entropy of entanglement using
semidefinite program solvers was presented in [16].

Interestingly, the $\mathfrak{D}_{\alpha,z}$ encompass several well-known
entanglement measures. The monotone $\mathfrak{D}$ is, by definition, equal to
the relative entropy of entanglement. The relative entropy of entanglement was
the first divergence-based entanglement measure to be studied and was
introduced in [47].

In [13] it was shown that the the generalized (log) robustness is equal to the
entanglement monotone based on the $D_{\max}$ divergence, i.e.
$\mathfrak{D}_{\max}(\rho)=\log{(1+\mathfrak{R}_{g}(\rho))}$ where

$\mathfrak{R}_{g}(\rho):=\min\left\\{s\geq
0:\exists\omega\in\mathcal{S}(A)\,\,\text{s.t}\,\,\frac{1}{1+s}\left(\rho+s\omega\right)\in\operatorname{SEP}\right\\}\,.$
(12)

is the generalized robustness of entanglement. Historically, the robustness of
entanglement was introduced first in [50]. The robustness of entanglement
quantifies how much mixing with a separable state (separable noise) can take
place before an entangled state becomes a separable state. Subsequently, in
[42] and [18] the authors introduced the generalized robustness of
entanglement. The latter quantity generalizes the robustness of entanglement
and it quantifies how much mixing with a state (not necessarily separable) can
take place before an entangled state becomes separable.

The monotone $\mathfrak{D}_{1/2,1/2}(\rho)$ is linked to the fidelity of
separability and the geometric measure of entanglement through the relations
$\mathfrak{D}_{1/2,1/2}(\rho)=-\log{F_{s}(\rho)}=-\log{(1-E_{G}(\rho))}$. The
fidelity of separability is
$F_{s}(\rho)=\max_{\sigma\in\operatorname{SEP}}F(\rho,\sigma)$. Here,
$F(\rho,\sigma):=(\Tr|\sqrt{\rho}\sqrt{\sigma}|)^{2}$ is the Uhlmann’s
fidelity. The geometric measure of entanglement is defined through the convex-
roof construction as follows:

$\displaystyle
E_{G}(|\psi\rangle)=1-\max_{|\phi\rangle\in\text{PRO}}|\langle\phi|\psi\rangle|^{2}\,,$
(13) $\displaystyle
E_{G}(\rho)=\min_{\\{p_{i},|\psi_{i}\rangle\\}}\sum_{i}p_{i}E_{G}(|\psi_{i}\rangle)\,,$
(14)

where the minimization is performed over all the decomposition
$\rho=\sum_{i}p_{i}|\psi_{i}\rangle\\!\langle\psi_{i}|$ into pure states. The
geometric measure of entanglement was first proposed in [4] and subsequently
investigated in [54, 11, 26]. The proof of the above link was provided in [43]
where the authors showed the key relation $E_{G}(\rho)=1-F_{s}(\rho)$. Note
that the additivity of $\mathfrak{D}_{1/2,1/2}$ is equivalent to the
multiplicativity of the fidelity of separability $F_{s}$.

More recently, the limits $z=1$ and $z=\alpha$ have been investigated in [61].
In these cases, the $\mathfrak{D}_{\alpha,z}$ reduces to the entanglement
monotone based on the Petz Rényi divergence and to the one based on the
sandwiched Rényi divergence, respectively.

##  III Necessary and sufficient conditions for the optimizer

In this section we derive necessary and sufficient conditions for the optimal
state of the minimization problem (9). We formulate our result for any convex
resource theory [12, 8]. Resource theories offer a general framework to
quantify the usefulness of quantum states and their inconvertibility under
free operations. A convex resource theory is defined by a closed non-empty
convex subset of quantum states called free states and a set of free
operations with the property that they are closed under composition and map
free states into free states. We denote the set of free states by
$\mathcal{F}$. Some examples of resource theories include entanglement theory
[25, 34, 47], athermality in thermodynamics [9, 15, 22] , and coherence [57,
1, 5]. To this purpose, we generalize the quantity (9). We define

$\displaystyle\mathfrak{D}^{\mathcal{F}}_{\alpha,z}(\rho):=\inf_{\sigma\in\mathcal{F}}D_{\alpha,z}(\rho\|\sigma)\,.$
(15)

Note that the above minimum is always achieved for any resource theory (see
Appendix Appendix A: Lower semicontinuity and its consequences). We call a
function a resource monotone if it does not increase under free operations. It
is straightforward to prove that the above quantity is a resource monotone for
$(\alpha,z)\in\mathcal{D}$. In entanglement theory, where
$\mathcal{F}=\operatorname{SEP}$, the above quantity reduces to the
entanglement monotone in (9). We also assume that the free set contains a
state with full support. This is a consequence of the requirement that the
monotone (15) must be always finite for all $\alpha\geq 1$ and full support
states. The latter requirement is a reasonable assumption since in any
reasonable resource theory, strictly speaking, no state can have infinite
resource.

We introduce the Fréchet derivative (see e.g. [7, Sec. V.3 and Sec. X.4] ).
For $\rho,\sigma,\tau\in\mathcal{S}(\mathcal{H})$ we define

$\left.\partial_{\sigma}D_{\alpha,z}(\rho\|\tau):=\partial_{x}D_{\alpha,z}(\rho\|(1-x)\tau+x\sigma)\right|_{x=0}\,.$
(16)

We first prove two auxiliary lemmas. We use the notational convention
$A/B:=B^{-\frac{1}{2}}AB^{-\frac{1}{2}}$ for Hermitian operators $A$ and $B$
and we use the shorthand $\partial_{x}:=\partial/\partial x$. We define the
negative powers in the sense of generalized inverses; i.e., when we take
negative powers of positive operators, we simply ignore the kernel.

######  Lemma 1.

Let $\tau>0$ and $\sigma\geq 0$. Then, for $\beta\in(-1,0)\cup(0,1)$, we have

$\partial_{x}((1-x)\tau+x\sigma)^{\beta}=\frac{\sin(\pi\beta)}{\pi}\int_{0}^{\infty}\frac{\sigma-\tau}{((1-x)\tau+x\sigma+t)^{2}}t^{\beta}\mathrm{d}t\,.$
(17)

In particular, we have

$\left.\partial_{x}((1-x)\tau+x\sigma)^{\beta}\right|_{x=0}=\frac{\sin(\pi\beta)}{\pi}\int_{0}^{\infty}\frac{\sigma}{(\tau+t)^{2}}t^{\beta}\mathrm{d}t-\beta\tau^{\beta}\,.$
(18)

###### Proof.

For a positive definite operator $A>0$ we have the following integral formulas

$\displaystyle
A^{\beta}=-\frac{\sin{(\pi\beta)}}{\pi}\int_{0}^{\infty}\frac{1}{A+t}t^{\beta}\mathrm{d}t\quad\text{for
all}\quad\beta\in(-1,0)\,,\;\text{and}$ (19) $\displaystyle
A^{\beta}=\frac{\sin{(\pi\beta)}}{\pi}\int_{0}^{\infty}\frac{A}{A+t}t^{\beta-1}\mathrm{d}t\quad\text{for
all}\quad\beta\in(0,1)\,.$ (20)

We then use the above formulas to obtain

$\displaystyle\partial_{x}((1-x)\tau+x\sigma)^{\beta}=-\frac{\sin{(\pi\beta)}}{\pi}\int_{0}^{\infty}\partial_{x}\frac{1}{(1-x)\tau+x\sigma+t}t^{\beta}\mathrm{d}t\quad\beta\in(-1,0)\,,\;\text{and}$
(21)
$\displaystyle\partial_{x}((1-x)\tau+x\sigma)^{\beta}=\frac{\sin{(\pi\beta)}}{\pi}\int_{0}^{\infty}\partial_{x}\frac{(1-x)\tau+x\sigma}{(1-x)\tau+x\sigma+t}t^{\beta-1}\mathrm{d}t\quad\beta\in(0,1)\,.$
(22)

Moreover, we use that $\partial\sigma(x)^{-1}/\partial
x=-\sigma(x)^{-1}(\partial\sigma(x)/\partial x)\sigma(x)^{-1}$ to obtain

$\displaystyle\partial_{x}\frac{1}{(1-x)\tau+x\sigma+t}=-\frac{\sigma-\tau}{((1-x)\tau+x\sigma+t)^{2}}\,,\;\text{and}$
(23)
$\displaystyle\partial_{x}\frac{(1-x)\tau+x\sigma}{(1-x)\tau+x\sigma+t}=t\frac{\sigma-\tau}{((1-x)\tau+x\sigma+t)^{2}}\,.$
(24)

Combing equations (21) and (23) together with equations (23) and (24) proves
the first part of the lemma. We now evaluate the derivative at $x=0$. We
obtain

$\displaystyle\left.\partial_{x}((1-x)\tau+x\sigma)^{\beta}\right|_{x=0}$
$\displaystyle=\frac{\sin(\pi\beta)}{\pi}\int_{0}^{\infty}\frac{\sigma-\tau}{(\tau+t)^{2}}t^{\beta}\mathrm{d}t$
(25)
$\displaystyle=\frac{\sin(\pi\beta)}{\pi}\int_{0}^{\infty}\frac{\sigma}{(\tau+t)^{2}}t^{\beta}\mathrm{d}t-\frac{\sin(\pi\beta)}{\pi}\tau\int_{0}^{\infty}\frac{1}{(\tau+t)^{2}}t^{\beta}\mathrm{d}t\,.$
(26)

We now use that, for a positive operator $A>0$, we have

$\int_{0}^{\infty}\frac{1}{(A+t)^{2}}t^{\beta}\mathrm{d}t=\frac{A^{\beta-1}}{\operatorname{sinc}(\pi\beta)}\quad\text{for
all}\quad\beta\in(-1,0)\cup(0,1)\,,$ (27)

to integrate the second term in the r.h.s of the previous expression. This
proves equation (18). ∎

######  Lemma 2.

Let $A,B\geq 0$. The following two conditions are equivalent:

  1. (1)

$\exists\;0<\gamma,\beta<\infty$ such that $\gamma\Pi(A)\leq
ABA\leq\beta\Pi(A)$.

  2. (2)

$\textup{supp}(ABA)=\textup{supp}(A)$.

###### Proof.

We always have $\operatorname{supp}(ABA)\subseteq\operatorname{supp}(A)$ and
$ABA\leq\beta\Pi(A)$ with $\beta=\lambda_{\max}(B)\lambda^{2}_{\max}(A)$. We
first show $\mathit{(1)}\\!\\!\implies\\!\\!\mathit{(2)}$. Let us choose
$|\psi\rangle\in\operatorname{supp}(A)$. We have

$\langle\psi|ABA|\psi\rangle\geq\gamma\langle\psi|\Pi(A)|\psi\rangle\neq 0\,.$
(28)

We now prove that $\mathit{(2)}\\!\\!\implies\\!\\!\mathit{(1)}$. We have

$ABA\geq\lambda_{\min}(ABA)\Pi(ABA)=\lambda_{\min}(ABA)\Pi(A)\,.$ (29)

∎

We now state the necessary a sufficient conditions for the optimizer(s) of
(15). We recall that we take the negative powers in the sense of generalized
inverses; i.e. when we take negative powers of positive operators we simply
ignore the kernel. As we discuss later, to properly formulate the necessary
and sufficient conditions of the optimizer of the problem (15) (see Theorem 4
below), this leads us to introduce the following set of states. For any
$\rho\in\mathcal{S}(A)$ and $(\alpha,z)\in\mathcal{D}$, we define

$\displaystyle
S_{\alpha,z}(\rho)=\begin{cases}\left\\{\sigma\in\mathcal{S}(A):\operatorname{supp}(\rho)=\operatorname{supp}\\!\left(\Pi(\rho)\sigma\Pi(\rho)\right)\right\\}&\textup{if}\;(1-\alpha)/z=1\\\
\left\\{\sigma\in\mathcal{S}(A):\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)\right\\}&\textup{otherwise}\end{cases}\,,$
(30)

where we denoted with $\Pi(\rho)$ the projector onto the support of $\rho$ as
before. Note that the line $z=1-\alpha$ is located at the boundaries of the
DPI region for $\alpha\leq 1/2$ (see Fig. 1). We now show that the
optimizer(s) of (15) must belong to the previous set. We remark that these
support conditions are very relevant as without them the theorem could lead to
wrong conclusions. We have the following

######  Lemma 3.

Let $\rho$ be a quantum state and $(\alpha,z)\in\mathcal{D}$. If
$\tau\in\operatorname*{arg\,min}_{\sigma\in\mathcal{F}}D_{\alpha,z}(\rho\|\sigma)$
then $\tau\in S_{\alpha,z}(\rho)$.

###### Proof for $\alpha\geq 1$.

For $\alpha\geq 1$, if
$\tau\in\operatorname*{arg\,min}_{\sigma\in\mathcal{F}}D_{\alpha,z}(\rho\|\sigma)$,
it must be that $\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\tau)$
or otherwise the $\mathfrak{D}_{\alpha,z}(\rho)$, by definition, would be
infinite. However, by assumption, there always exists a free state $\sigma$
with full support such that $\mathfrak{D}_{\alpha,z}(\rho)\leq
D_{\alpha,z}(\rho\|\sigma)<\infty$. ∎

###### Proof for $z=1-\alpha$, $\alpha\in(0,1)$.

In this case, we show that a necessary condition for $\tau$ to be an optimizer
is that
$\operatorname{supp}\\!\left(\Pi(\rho)\tau\Pi(\rho)\right)=\operatorname{supp}(\rho)$.
To do that, we show that, if $\tau$ does not satisfy the latter conditions on
the support, we can always find a sufficiently small $x$ and a free state
$\sigma$ such that
$D_{\alpha,z}(\rho\|(1-x)\tau+x\sigma)<D_{\alpha,z}(\rho\|\tau)$ (or
$Q_{\alpha,z}(\rho\|(1-x)\tau+x\sigma)>Q_{\alpha,z}(\rho\|\tau)$). This
contradicts the fact that $\tau$ is a minimum. Here, we implicitly used that
the resource theory is convex, i.e. $(1-x)\tau+x\sigma\in\mathcal{F}$.

We start by calculating the Fréchet derivative along a free direction $\sigma$
with full support. We have

$\displaystyle\partial_{x}Q_{\alpha,1-\alpha}(\rho\|(1-x)\tau+x\sigma)=$
$\displaystyle z\Tr(\chi_{\alpha,z}(\rho,(1-x)\tau+x\sigma)(\sigma-\tau))$
(31) $\displaystyle=$ $\displaystyle
z\Tr(\chi_{\alpha,z}(\rho,(1-x)\tau+x\sigma)\sigma)-z\Tr(\chi_{\alpha,z}(\rho,(1-x)\tau+x\sigma)\tau)\,.$
(32)

and we defined the positive operator
$\chi_{\alpha,z}(\rho,\sigma):=\rho^{\frac{\alpha}{2z}}(\rho^{\frac{\alpha}{2z}}\tau^{\frac{1-\alpha}{z}}\rho^{\frac{\alpha}{2z}})^{z-1}\rho^{\frac{\alpha}{2z}}$.
The above function is a continuous and finite function of $x$ for $x\neq 0$.
In the following we repeatedly use that if $A,A^{\prime},B\geq 0$ and $A\leq
A^{\prime}$, then $\Tr(AB)\leq\Tr(A^{\prime}B)$. We now bound separately the
two terms of equation (32). We define $\beta:=\alpha/(2(1-\alpha))$. To bound
the first term we use that
$\rho^{\beta}((1-x)\tau+x\sigma)\rho^{\beta}\leq(1-x)\Pi(\rho^{\beta}\tau\rho^{\beta})+x\Pi(\rho)$.
Here, we used that $\rho^{2\beta}\leq\Pi(\rho)$ and that
$\Tr(\tau\rho^{\beta})\leq 1$. This implies that

$\displaystyle\Tr(\chi_{\alpha,z}(\rho,(1-x)\tau+x\sigma)\sigma)$
$\displaystyle=\Tr\left(\rho^{\beta}(\rho^{\beta}((1-x)\tau+x\sigma)\rho^{\beta})^{-\alpha}\rho^{\beta}\sigma\right)$
(33)
$\displaystyle\geq\lambda_{\min}(\sigma)\Tr\left(\rho^{2\beta}(\Pi(\rho^{\beta}\tau\rho^{\beta})+x^{-\alpha}(\Pi(\rho)-\Pi(\rho^{\beta}\tau\rho^{\beta}))\right)$
(34) $\displaystyle\geq
x^{-\alpha}\Tr\left(\Pi(\rho)(\Pi(\rho)-\Pi(\rho^{\beta}\tau\rho^{\beta}))\right)K\,,$
(35)

where $\lambda_{\min}(\rho)$ denotes the the minimum non-zero eigenvalue of
$\rho$, and we defined
$K:=\lambda_{\min}(\sigma)\lambda^{2\beta}_{\min}(\rho)$. In (34) we used that
$t\mapsto t^{s}$ is operator antimonotone for $s\in[-1,0)$. We now bound the
second term in (32). We first note that $Q_{\alpha,z}(\rho\|\sigma)\leq 1$ for
any $\rho,\sigma\in\mathcal{S}(A)$. Indeed, we have

$Q_{\alpha,z}(\rho\|\sigma)=\int Q_{\alpha,z}\left(U\rho
U^{\dagger}\middle\|U\sigma U^{\dagger}\right)\mathrm{d}U\leq
Q_{\alpha,z}\left(\int U\rho U^{\dagger}\middle\|\int U\sigma
U^{\dagger}\right)\mathrm{d}U=Q_{\alpha,z}\left(\frac{\mathds{1}}{d}\middle\|\frac{\mathds{1}}{d}\right)=1\,,$
(36)

where we denoted with $\mathrm{d}U$ the invariant Haar measure. In the first
equality we used the unitary invariance of $Q_{\alpha,z}$ and in the
inequality we used that $Q_{\alpha,z}$ is jointly concave.

We then use that $\tau\leq\tau+(x/(1-x))\sigma$ to get

$\displaystyle\Tr(\chi_{\alpha,z}(\rho,(1-x)\tau+x\sigma)\tau)$
$\displaystyle=\Tr\left(\rho^{\beta}(\rho^{\beta}((1-x)\tau+x\sigma)\rho^{\beta})^{-\alpha}\rho^{\beta}\tau\right)$
(37) $\displaystyle\leq\frac{1}{1-x}Q_{\alpha,z}(\rho\|(1-x)\tau+x\sigma)$
(38) $\displaystyle\leq 2\,.$ (39)

In the last inequality we used that $Q_{\alpha,z}(\rho\|\sigma)\leq 1$ and we
assumed that $x\leq 1/2$. Combining the above results we obtain, for $x\leq
1/2$,

$\partial_{x}Q_{\alpha,1-\alpha}(\rho\|(1-x)\tau+x\sigma)\geq
x^{-\alpha}\Tr\left(\Pi(\rho)(\Pi(\rho)-\Pi(\rho^{\beta}\tau\rho^{\beta}))\right)K-2\,.$
(40)

The trace
$\Tr\left(\Pi(\rho)(\Pi(\rho)-\Pi(\rho^{\beta}\tau\rho^{\beta}))\right)$ is
different from zero only if
$\operatorname{supp}\\!\left(\rho^{\beta}\sigma\rho^{\beta}\right)\subset\operatorname{supp}(\rho)$.
In this case, the derivative (40) goes to infinity as $x^{-\alpha}$ as $x$
vanishes. This means that in the neighborhood of $x=0$ the function
$Q_{\alpha,z}(\rho\|(1-x)\tau+x\sigma)$ is a strictly increasing (and
continuous) function of $x$ and hence we can always find a sufficiently small
$x$ such that
$Q_{\alpha,z}(\rho\|(1-x)\tau+x\sigma)>Q_{\alpha,z}(\rho\|\tau)$. This implies
that if
$\tau\in\operatorname*{arg\,min}_{\sigma\in\mathcal{F}}D_{\alpha,z}(\rho\|\sigma)$
then
$\operatorname{supp}\\!\left(\rho^{\beta}\sigma\rho^{\beta}\right)=\operatorname{supp}(\rho)$.

We finally use Lemma 2 to show that
$\operatorname{supp}\\!\left(\rho^{\beta}\tau\rho^{\beta}\right)=\operatorname{supp}(\rho)$
if and only if
$\operatorname{supp}\\!\left(\Pi(\rho)\tau\Pi(\rho)\right)=\operatorname{supp}(\rho)$.
Indeed, we have that
$\gamma\Pi(\rho)\leq\rho^{\beta}\tau\rho^{\beta}\leq\beta\Pi(\rho)$ implies

$\displaystyle\lambda_{\min}\left(\rho^{-2\beta}\right)\Pi(\rho)\leq\gamma\rho^{-2\beta}\leq\Pi(\rho)\tau\Pi(\rho)\leq\beta\rho^{-2\beta}\leq\beta\lambda_{\max}\left(\rho^{-2\beta}\right)\Pi(\rho)\,,$
(41)

where we used that, since the negative power is taken only outside the kernel
of $\rho$, it holds $\rho^{-\beta}\rho^{\beta}=\Pi(\rho)$. Conversely, the
relation $\gamma\Pi(\rho)\leq\Pi(\rho)\tau\Pi(\rho)\leq\beta\Pi(\rho)$ implies
that

$\displaystyle\gamma\lambda_{\min}\left(\rho^{2\beta}\right)\Pi(\rho)\leq\gamma\rho^{2\beta}\leq\rho^{\beta}\tau\rho^{\beta}\leq\beta\rho^{2\beta}\leq\beta\lambda_{\max}\left(\rho^{2\beta}\right)\Pi(\rho)\,.$
(42)

This means that the condition
$\operatorname{supp}\\!\left(\Pi(\rho)\tau\Pi(\rho)\right)=\operatorname{supp}(\rho)$
is a necessary condition for $\tau$ to be a minimum of
$\mathfrak{D}_{\alpha,z}(\rho)$. ∎

###### Proof for $|(1-\alpha)/z|\neq 1\wedge\alpha<1$.

The proof for this range is similar to the one provided for the case
$z=1-\alpha$. We first calculate the Fréchet derivative along the direction
$\sigma$ with full support

$\displaystyle\partial_{x}Q_{\alpha,z}(\rho\|(1-x)\tau+x\sigma)=z\Tr\left(\chi_{\alpha,z}(\rho,(1-x)\tau+x\sigma)\partial_{x}((1-x)\tau+x\sigma)^{\frac{1-\alpha}{z}}\right)\,.$
(43)

We now use Lemma 1 to get

$\displaystyle\partial_{x}Q_{\alpha,z}(\rho\|(1-x)\tau+x\sigma)$
$\displaystyle\qquad\qquad\qquad=(1-\alpha)K_{\alpha,z}\Tr\left(\chi_{\alpha,z}(\rho,(1-x)\tau+x\sigma)\int_{0}^{\infty}\frac{\sigma-\tau}{((1-x)\tau+x\sigma+t)^{2}}t^{\frac{1-\alpha}{z}}\mathrm{d}t\right)\,,$
(44)

where we defined the positive constant
$K_{\alpha,z}:=\operatorname{sinc}\left(\pi\frac{{1-\alpha}}{z}\right)$. We
first bound the first term in the sum on the r.h.s of equation (III). We have

$\displaystyle\Tr\left(\chi_{\alpha,z}(\rho,(1-x)\tau+x\sigma)\int_{0}^{\infty}\frac{\sigma}{((1-x)\tau+x\sigma+t)^{2}}t^{\beta}\mathrm{d}t\right)$
(45)
$\displaystyle\qquad\qquad\qquad\qquad\geq\lambda_{\min}(\sigma)K^{-1}_{\alpha,z}\Tr\left(\chi_{\alpha,z}(\rho,(1-x)\tau+x\sigma)((1-x)\tau+x\sigma)^{\frac{1-\alpha}{z}-1}\right)\,,$
(46)

where we used that for a positive operator $A>0$ it holds

$\int_{0}^{\infty}\frac{1}{(A+t)^{2}}t^{\beta}\mathrm{d}t=\frac{A^{\beta-1}}{\operatorname{sinc}{(\pi\beta)}}\;\quad\text{for}\;\;\beta\in(-1,0)\cup(0,1)\,.$
(47)

Let us denote $\beta:=\alpha/(2z)$. We distinguish two cases. In the following
we repeatedly use that for $A,B\geq 0$ such that $[A,B]=0$ it holds $A\leq B$
if and only if $A^{\alpha}\leq B^{\alpha}$ for any $\alpha>0$. Moreover, if
$\alpha<0$, the latter inequality goes in the opposite direction.

For $\alpha<1\wedge z<1$ we have that
$((1-x)\tau+x\sigma)\leq\Pi(\tau)+x\Pi^{c}(\tau)$ and
$\rho^{\beta}((1-x)\tau+x\sigma)^{\frac{1-\alpha}{z}}\rho^{\beta}\leq\Pi(\rho)$.
This implies that

$\displaystyle\Tr\left(\chi_{\alpha,z}(\rho,(1-x)\tau+x\sigma)((1-x)\tau+x\sigma)^{\frac{1-\alpha}{z}-1}\right)$
$\displaystyle\geq\Tr\left(\rho^{2\beta}(\Pi(\tau)+x^{\frac{1-\alpha}{z}-1}\Pi^{c}(\tau))\right)$
(48) $\displaystyle\geq
x^{\frac{1-\alpha}{z}-1}\Tr(\Pi(\rho)\Pi^{c}(\tau))K_{1}\,,$ (49)

where we denoted $\Pi^{c}(\tau):=\mathds{1}-\Pi(\rho)$ and
$K_{1}:=\lambda^{2\beta}_{\min}(\rho)$ is a positive constant. In (48) we used
that for $(\alpha,z)\in\mathcal{D}\wedge\alpha<1$ we have
$(1-\alpha)/z-1\in(-1,0)$.

For $\alpha<1\wedge z>1$ we use that
$\rho^{\beta}((1-x)\tau+x\sigma)^{\frac{1-\alpha}{z}}\rho^{\beta}\geq
x^{\frac{1-\alpha}{z}}\lambda_{\min}(\sigma)\lambda^{2\beta}_{\min}(\rho)\Pi(\rho)$.
This implies that

$\displaystyle\Tr\left(\chi_{\alpha,z}(\rho,(1-x)\tau+x\sigma)((1-x)\tau+x\sigma)^{\frac{1-\alpha}{z}-1}\right)$
(50)
$\displaystyle\qquad\qquad\qquad\geq\left(x^{\frac{1-\alpha}{z}}\lambda_{\min}(\sigma)\lambda^{2\beta}_{\min}(\rho)\right)^{z-1}\Tr\left(\rho^{2\beta}(\Pi(\tau)+x^{\frac{1-\alpha}{z}-1}\Pi^{c}(\tau))\right)$
(51) $\displaystyle\qquad\qquad\qquad\geq
x^{-\alpha}\Tr(\Pi(\rho)\Pi^{c}(\tau))K_{2}\,,$ (52)

where $K_{2}:=\lambda^{z-1}_{\min}(\sigma)\lambda^{2\beta z}_{\min}(\rho)$ is
a positive constant.

We now bound the second term in the sum on the r.h.s of equation (III). We use
that $\tau\leq\tau+(x/(1-x))\sigma$ to obtain

$\displaystyle\Tr\left(\chi_{\alpha,z}(\rho,(1-x)\tau+x\sigma)\int_{0}^{\infty}\frac{\tau}{((1-x)\tau+x\sigma+t)^{2}}t^{\beta}\mathrm{d}t\right)$
$\displaystyle\leq
K^{-1}_{\alpha,z}\frac{1}{1-x}Q_{\alpha,z}(\rho\|(1-x)\tau+x\sigma)$ (53)
$\displaystyle\leq 2K^{-1}_{\alpha,z}\,,$ (54)

where we assumed $x\leq 1/2$. Combining the above results we get

$\partial_{x}Q_{\alpha,z}(\rho\|(1-x)\tau+x\sigma)\geq\begin{cases}x^{\frac{1-\alpha}{z}-1}\Tr(\Pi(\rho)\Pi^{c}(\tau))C_{1}-2D&z<1\\\
x^{-\alpha}\Tr(\Pi(\rho)\Pi^{c}(\tau))C_{2}-2D&z>1\end{cases}\,,$ (55)

where $C_{i}:=K_{i}\lambda_{\min}(\sigma)K^{-1}_{\alpha,z}$ and $D:=1-\alpha$
are two positive constants. The trace $\Tr(\Pi(\rho)\Pi^{c}(\tau))$ is
different from zero if
$\operatorname{supp}\\!\left(\tau\right)\subset\operatorname{supp}(\rho)$. In
this case, the above derivative in would diverge $x$ vanishes in both cases.
As we discussed above for the case $z=\alpha-1$, this means that in the
neighborhood of $x=0$ the function $D_{\alpha,z}(\rho\|(1-x)\tau+x\sigma)$ is
a strictly increasing (and continuous) function of $x$ and hence we could
always find a sufficiently small $x$ such that
$D_{\alpha,z}(\rho\|(1-x)\tau+x\sigma)<D_{\alpha,z}(\rho\|\tau)$. This means
that, as for the range $\alpha\geq 1$, the condition
$\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\tau)$ is a necessary
condition for
$\tau\in\operatorname*{arg\,min}_{\sigma\in\mathcal{F}}D_{\alpha,z}(\rho\|\sigma)$
in the range $|(1-\alpha)/z|\neq 1\wedge\alpha<1$. ∎

To formulate the theorem below in a compact way, we introduce the positive
operator

$\displaystyle\Xi_{\alpha,z}(\rho,\tau):=\begin{dcases}\chi_{\alpha,1-\alpha}(\rho,\sigma)&\textup{if}\;|(1-\alpha)/z|=1\\\
K_{\alpha,z}\int_{0}^{\infty}\frac{\chi_{\alpha,z}(\rho,\sigma)}{(\tau+t)^{2}}t^{\frac{1-\alpha}{z}}\mathrm{d}t&\textup{if}\;|(1-\alpha)/z|\neq
1\end{dcases}\,,$ (56)

where $K_{\alpha,z}:=\operatorname{sinc}\left(\pi\frac{{1-\alpha}}{z}\right)$
is a positive constant. Note that the two lines $z=1-\alpha$ and $z=\alpha-1$
are located at the boundaries of the DPI region for $\alpha\leq 1/2$ and
$\alpha\geq 2$, respectively (see Fig. 1). We are now ready to state the
necessary and sufficient conditions for the optimizer(s) of (15).

######  Theorem 4.

Let $\rho$ be a quantum state and $(\alpha,z)\in\mathcal{D}$. Then
$\tau\in\operatorname*{arg\,min}_{\sigma\in\mathcal{F}}D_{\alpha,z}(\rho\|\sigma)$
if and only if $\tau\in S_{\alpha,z}(\rho)$ and
$\textup{Tr}(\sigma\,\Xi_{\alpha,z}(\rho,\tau))\leq Q_{\alpha,z}(\rho\|\tau)$
for all $\sigma\in\mathcal{F}$.

We remark that this condition is much easier to perform than the original
optimization in (15) since, by linearity of the trace, the above optimization
can be restricted to the set of pure free states.

###### Proof.

Since the function $Q_{\alpha,z}$ is jointly concave/convex in the range
$(\alpha,z)\in\mathcal{D}$ where $D_{\alpha,z}$ satisfies the data-processing
inequality, as we discussed above, it follows that the condition of local
minimum is both sufficient and necessary.

The condition $\textup{Tr}(\sigma\,\Xi_{\alpha,z}(\rho,\tau))\leq
Q_{\alpha,z}(\rho\|\tau)$ for all $\sigma\in\mathcal{F}$ then follows by the
condition of local minimum for $\tau$, i.e.
$\partial_{\sigma}Q_{\alpha,z}(\rho\|\tau)\leq 0$ and
$\partial_{\sigma}Q_{\alpha,z}(\rho\|\tau)\geq 0$ any $\sigma\in\mathcal{F}$
for $\alpha<1$ and $\alpha>1$, respectively.

Indeed, let $\tau\in S_{\alpha,z}(\rho)$. For $z=1-\alpha$ ($\alpha<1$), we
have

$\displaystyle\partial_{\sigma}Q_{\alpha,1-\alpha}(\rho\|\tau)=z\Tr(\chi_{\alpha,1-\alpha}(\rho,\tau)(\sigma-\tau))=z\Tr(\chi_{\alpha,1-\alpha}(\rho,\tau)\sigma)-zQ_{\alpha,1-\alpha}(\rho\|\tau)\,.$
(57)

For $z=\alpha-1$ ($\alpha>1$) we use that $\partial\sigma(x)^{-1}/\partial
x=-\sigma(x)^{-1}(\partial\sigma(x)/\partial x)\sigma(x)^{-1}$ to obtain

$\displaystyle\partial_{\sigma}Q_{\alpha,\alpha-1}(\rho\|\tau)$
$\displaystyle=-z\Tr\left(\chi_{\alpha,\alpha-1}\tau^{-1}(\sigma-\tau)\tau^{-1}\right)$
(58)
$\displaystyle=-z\Tr\left(\tau^{-1}\chi_{\alpha,\alpha-1}\tau^{-1}\sigma\right)+zQ_{\alpha,\alpha-1}(\rho\|\tau)\,.$
(59)

From the equality
$\tau^{-1}\chi_{\alpha,\alpha-1}\tau^{-1}=\chi_{\alpha,1-\alpha}(\rho,\tau)$
it then follows that
$\Xi_{\alpha,1-\alpha}(\rho,\sigma)=\Xi_{1-\alpha,\alpha}(\rho,\sigma)$.

For $(1-\alpha)/z\in(-1,0)\cup(0,1)$ we use Lemma 18 to obtain

$\displaystyle\partial_{\sigma}Q_{\alpha,z}(\rho\|\tau)=(1-\alpha)K_{\alpha,z}\Tr\left(\sigma\int_{0}^{\infty}\frac{\chi_{\alpha,z}(\rho,\tau)}{(\tau+t)^{2}}t^{\frac{1-\alpha}{z}}\mathrm{d}t\right)-(1-\alpha)Q_{\alpha,z}(\rho\|\tau)\,.$
(60)

Note that in the range $(\alpha,z)\in\mathcal{D}\wedge|(1-\alpha)/z|\neq 0,1$
, the condition $(1-\alpha)/z\in(-1,0)\cup(0,1)$ is always satisfied. The
theorem for $(\alpha,z)\in\mathcal{D}\wedge\alpha\neq 1$ then follows by
combining the above results with the local minimum conditions above mentioned.
For the Umegaki relative entropy, i.e. $\alpha=1$, the result has already been
obtained in [46] and can be recovered by taking the limit $\alpha,z\rightarrow
1$ of the result for the range $|1-\alpha)/z|\neq 1$ (see the discussion
below). ∎

As we show below, many states we consider commute with their optimizer. In
this case, the above theorem considerably simplifies.

######  Corollary 5.

Let $\rho$ be a quantum state and $(\alpha,z)\in\mathcal{D}$. Then, a state
$\tau$ satisfying $[\rho,\tau]=0$ belongs to the set
$\tau\in\operatorname*{arg\,min}_{\sigma\in\mathcal{F}}D_{\alpha,z}(\rho\|\sigma)$
if and only if $\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\tau)$
and $\textup{Tr}(\sigma\Xi_{\alpha}(\rho,\tau))\leq Q_{\alpha}(\rho\|\tau)$
for all $\sigma\in\mathcal{F}$ where
$\Xi_{\alpha}(\rho,\tau)=\rho^{\alpha}\tau^{-\alpha}$ and
$Q_{\alpha}(\rho\|\tau)=\textup{Tr}(\rho^{\alpha}\tau^{1-\alpha})$.

###### Proof.

If $[\rho,\tau]=0$, we use that for a positive operator $A>0$

$\int_{0}^{\infty}\frac{1}{(A+t)^{2}}t^{\beta}\mathrm{d}t=\frac{A^{\beta-1}}{\operatorname{sinc}{(\pi\beta)}}\quad\text{for}\quad\beta\in(-1,0)\cup(0,1)\,,$
(61)

to further simplify the expression (56). We obtain for $(1-\alpha)/z\in(-1,1)$

$\textup{Tr}(\sigma\Xi_{\alpha,z}(\rho,\tau))=\textup{Tr}(\sigma{\Xi}_{\alpha}(\rho,\tau))\,,$
(62)

where $\Xi_{\alpha}(\rho,\tau)=\rho^{\alpha}\tau^{-\alpha}$. The cases
$|(1-\alpha)/z|=1$ follow easily by direct computation of (56) under the
assumption $[\rho,\tau]=0$. Finally, the support condition
$\operatorname{supp}(\rho)=\operatorname{supp}\\!\left(\Pi(\rho)\sigma\Pi(\rho)\right)$
which holds for $z=1-\alpha$ reduces to
$\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\tau)$ for
$[\rho,\tau]=0$. ∎

######  Remark 1.

The necessary and sufficient conditions for the relative entropy case
($\alpha=z=1$) have already been pointed out in [46]. In this case, $\tau$ is
the solution of the minimization problem (15) if and only if
$\textup{Tr}(\sigma\,\Xi_{1,1}(\rho,\tau))\leq 1$ for all
$\sigma\in\mathcal{F}$ where

$\displaystyle\Xi_{1,1}(\rho,\tau)=\int_{0}^{\infty}\frac{\rho}{(\tau+t)^{2}}\mathrm{d}t\,.$
(63)

Note that we can recover this result by taking the limit $\alpha,z\rightarrow
1$ of the result for the range $|1-\alpha)/z|\neq 1$ in Theorem 4. Therefore,
we have that $\lim_{\alpha\rightarrow
1}\partial_{\sigma}D_{\alpha,z}(\rho\|\tau)=\partial_{\sigma}\lim_{\alpha\rightarrow
1}D_{\alpha,z}(\rho\|\tau)$ for all $z>0$, i.e. we can interchange the limit
with the Frechét derivative. Moreover, if $[\rho,\tau]=0$, we get
$\Xi_{1,1}(\rho,\tau)=\rho\tau^{-1}$. We again recover the latter result by
taking the limit $\alpha,z\rightarrow 1$ in Corollary 5.

Another interesting limit is the limit $\alpha\rightarrow 0$ with $z=1$. In
this case we obtain the min-relative entropy (see equation (3)). Using the
explicit form of the min-relative entropy it is easy to see that
$\Xi_{0,1}=\Pi(\rho)$. Note that we can recover this result by taking the
limit $\alpha\rightarrow 0$ with $z=1$ of the result for the range
$|(1-\alpha)/z|=1$ in Theorem 4.

##  IV Additivity of the $\alpha$-$z$ Rényi relative entropies of entanglement

In this section we address our main results. We prove that the monotones
$\mathfrak{D}_{\alpha,z}$ are additive whenever one state belongs to a special
class. In the next section we show below that bipartite pure, maximally
correlated, GHZ, Bell diagonal, generalized Dicke, isotropic and separable
states are some examples that are included in these classes. To the best of
our knowledge, such additivity results have previously only been established
for the case where both states belong to some class of states. Some examples
include bipartite pure states [46], maximally correlated states [61], Bell
diagonal [60] or generalized Dicke states [60].

Before focusing on entanglement theory, we first show how the conditions that
we derived in Section III provide a powerful tool to investigate additivity
questions in any convex resource theory by considering the resource theory of
coherence. For this resource theory the monotone based on the $\alpha$-$z$
Rényi relative entropy is additive for any states. This result follows
immediately for the case $\alpha=z=1$ from the closed form expression of the
relative entropy of coherence [57, Theorem 6] and has been extended for all
$z=\alpha$ and $z=1$ in [61, Theorem 3]. We generalize this result to all
$(\alpha,z)\in\mathcal{D}$ by providing a simple and independent proof which
hinges on the necessary and sufficient conditions derived in the previous
section.

###  IV.1 Additivity of the $\alpha$-$z$ Rényi relative entropies of coherence

Coherence is defined with respect a particular basis dictated by the physical
problem under consideration [57]. If $\\{|i\rangle,i=1,...,d\\}$ is such a
basis, a state is called free if it is diagonal in this basis, namely if it is
of the form $\sum p_{i}|i\rangle\\!\langle i|$ with $\sum p_{i}=1$. We call
these states incoherent states and we denote this set (free set) by
$\mathcal{I}$. States that are not incoherent states are resourceful and we
refer to them as coherent states. In this specific resource theory, the
monotones based on the $\alpha$-$z$ Rényi relative entropy are additive. In
the following, we state the result for the points $(\alpha,z)\in\mathcal{D}$.
The result for the limiting cases follow by taking the corresponding limits
(see Appendix Appendix A: Lower semicontinuity and its consequences for a more
detailed discussion).

######  Theorem 6.

Let $\rho_{1},\rho_{2}\in\mathcal{S}(A)$ be two states and
$(\alpha,z)\in\mathcal{D}$. Then we have

$\mathfrak{D}^{\mathcal{I}}_{\alpha,z}(\rho_{1}\otimes\rho_{2})=\mathfrak{D}^{\mathcal{I}}_{\alpha,z}(\rho_{1})+\mathfrak{D}^{\mathcal{I}}_{\alpha,z}(\rho_{2})\,.$
(64)

###### Proof.

To prove additivity we show that
$\tau_{1}\otimes\tau_{2}\in\operatorname*{arg\,min}_{\sigma\in\mathcal{F}}D_{\alpha,z}(\rho_{1}\otimes\rho_{2}\|\sigma)$
if $\tau_{1}$ and $\tau_{2}$ are some optimizers of the marginal problems,
i.e.
$\tau_{1}\in\operatorname*{arg\,min}_{\sigma\in\mathcal{F}}D_{\alpha,z}(\rho_{1}\|\sigma)$
and
$\tau_{2}\in\operatorname*{arg\,min}_{\sigma\in\mathcal{F}}D_{\alpha,z}(\rho_{2}\|\sigma)$.
Since the $\alpha$-$z$ Rényi relative entropy are additive [3], the latter
condition implies the additivity of the related monotones. Indeed, we have
$\mathfrak{D}^{\mathcal{I}}_{\alpha,z}(\rho_{1}\otimes\rho_{2})=D_{\alpha,z}(\rho_{1}\otimes\rho_{2}\|\tau_{1}\otimes\tau_{2})=D_{\alpha,z}(\rho_{1}\|\tau_{1})+D_{\alpha,z}(\rho_{2}\|\tau_{2})=\mathfrak{D}^{\mathcal{I}}_{\alpha,z}(\rho_{1})+\mathfrak{D}^{\mathcal{I}}_{\alpha,z}(\rho_{2})$.
Therefore, to prove additivity, according to Theorem 4, we need to show that

$\Tr(\sigma\Xi_{\alpha,z}(\rho_{1}\otimes\rho_{2},\tau_{1}\otimes\tau_{2}))\leq
Q_{\alpha,z}(\rho_{1}\otimes\rho_{2}\|\tau_{1}\otimes\tau_{2})=Q_{\alpha,z}(\rho_{1}\|\tau_{1})Q_{\alpha,z}(\rho_{2}\|\tau_{2})\;\forall\sigma\in\mathcal{I}\,.$
(65)

In the last equality we used that $Q_{\alpha,z}$ is multiplicative. Note that
we always have $\tau_{1}\otimes\tau_{2}\in
S_{\alpha,z}(\rho_{1}\otimes\rho_{2})$ since, by assumption, the states
$\tau_{1}$ and $\tau_{2}$ are optimizers of the marginal problems and hence
they satisfy $\tau_{1}\in S_{\alpha,z}(\rho_{1})$ and $\tau_{2}\in
S_{\alpha,z}(\rho_{2})$.

By linearity of the trace, we can restrict $\sigma$ above to be a pure
incoherent state. We can write any pure incoherent state as
$\sigma=|i,j\rangle$. In the following, we use the decomposition for the
optimizers $\tau_{i}=\sum t_{i,j}|j\rangle\\!\langle j|$. We have for
$|(1-\alpha)|/z\neq 1$

$\displaystyle\langle i,j|\Xi_{\alpha,z}$
$\displaystyle(\rho_{1}\otimes\rho_{2},\tau_{1}\otimes\tau_{2})|i,j\rangle$
(66)
$\displaystyle=K_{\alpha,z}\int_{0}^{\infty}\left(t_{1,i}t_{2,j}+k\right)^{-1}\left(t_{1,i}t_{2,j}+k\right)^{-1}k^{\frac{1-\alpha}{z}}\mathrm{d}k\langle
i|\chi_{\alpha,z}(\rho_{1},\tau_{1})|i\rangle\\!\langle
j|\chi_{\alpha}(\rho_{2},\tau_{2})|j\rangle$
$\displaystyle=\big{(}t_{1,i}t_{2,j}\big{)}^{\frac{1-\alpha}{z}-1}\langle
i|\chi_{\alpha,z}(\rho_{1},\tau_{1})|i\rangle\\!\langle
j|\chi_{\alpha,z}(\rho_{2},\tau_{2})|j\rangle\leq
Q_{\alpha,z}(\rho_{1}\|\tau_{1})Q_{\alpha,z}(\rho_{2}\|\tau_{2})\,,$ (67)

where we used that, if $\tau=\sum t_{j}|j\rangle\\!\langle j|$ is an optimizer
of $\rho$, the condition $t_{l}^{\frac{1-\alpha}{z}-1}\langle
l|\chi_{\alpha,z}(\rho,\tau)|l\rangle\leq Q_{\alpha,z}(\rho\|\sigma)$ holds
for any $l$. Indeed, if we choose the pure incoherent state
$\sigma=|l,l\rangle$, we obtain

$\displaystyle\Tr(\sigma\Xi_{\alpha,z}(\rho,\tau))=\langle
l|\Xi_{\alpha,z}(\rho,\tau)|l\rangle=t_{l}^{\frac{1-\alpha}{z}-1}\langle
l|\chi_{\alpha,z}(\rho,\tau)|l\rangle\,.$ (68)

Since $\tau$ is an optimizer of $\rho$ by assumption, the above equality
together with Theorem 4 implies that $t_{l}^{\frac{1-\alpha}{z}-1}\langle
l|\chi_{\alpha,z}(\rho,\tau)|l\rangle\leq Q_{\alpha,z}(\rho\|\sigma)$ for any
$l$. Hence, the inequality (65) is satisfied and additivity follows. The
result for the case $|(1-\alpha)/z|=1$ can be obtained by taking the limits
$z\rightarrow 1-\alpha$ and $z\rightarrow\alpha-1$ (see Appendix Appendix A:
Lower semicontinuity and its consequences for a more detailed discussion). ∎

###  IV.2 Additivity of the $\alpha$-$z$ Rényi entropy of entanglement:
Commuting case

We now prove that in entanglement theory, the monotones
$\mathfrak{D}_{\alpha,z}$ are additive when one state commutes with its
optimizer and the alpha power of its product with the inverse of its optimizer
has positive entries in a product basis. We remark that the latter condition
is very important since, as we show in Section VI, there exist states which
commute with their optimizer but don’t have positive entries in a product
basis for which the $\mathfrak{D}_{\alpha,z}$ are not additive. Separable
states clearly satisfy these conditions since the optimizer coincides with the
state itself. This is not surprising since for separable states, the proof
already follows from the fact that the $\mathfrak{D}_{\alpha,z}$ are
subadditive, equal zero if and only if the state is separable and they are
non-increasing under partial trace. More interestingly, as we show later, we
have that also the Bell diagonal, generalized Dicke and the isotropic states
belong to this class.

######  Theorem 7.

Let $\rho_{1}$ be a N-partite state and $(\alpha,z)\in\mathcal{D}$ . Moreover,
let
$\tau_{1}\in\operatorname*{arg\,min}_{\sigma\in\textup{SEP}}D_{\alpha,z}(\rho_{1}\|\sigma)$.
If $[\rho_{1},\tau_{1}]=0$ and $\rho_{1}^{\alpha}\tau_{1}^{-\alpha}$ has
positive entries in a product basis, then for any N-partite state $\rho_{2}$
we have that

$\mathfrak{D}_{\alpha,z}(\rho_{1}\otimes\rho_{2})=\mathfrak{D}_{\alpha,z}(\rho_{1})+\mathfrak{D}_{\alpha,z}(\rho_{2})\,.$
(69)

###### Proof.

For what we discussed in the proof of Theorem 6, to prove additivity it is
enough to show that the optimizer of $\rho_{1}\otimes\rho_{2}$ is
$\tau_{1}\otimes\tau_{2}$ whenever $\rho_{1}$ commutes with its optimizer,
i.e. $[\rho_{1},\tau_{1}]=0$, and $\rho_{1}^{\alpha}\tau_{1}^{-\alpha}$ has
positive entries in a product basis. Here, $\tau_{1}$ and $\tau_{2}$ are some
optimizers of $\rho_{1}$ and $\rho_{2}$, respectively. According to Theorem 4,
we want to show that

$\Tr(\sigma\Xi_{\alpha,z}(\rho_{1}\otimes\rho_{2},\tau_{1}\otimes\tau_{2}))\leq
Q_{\alpha,z}(\rho_{1}\otimes\rho_{2}\|\tau_{1}\otimes\tau_{2})=Q_{\alpha}(\rho_{1}\|\tau_{1})Q_{\alpha,z}(\rho_{2}\|\tau_{2})\,\,\forall\sigma\in\operatorname{SEP}\,,$
(70)

where we denoted with $\tau_{1}$ and $\tau_{2}$ are the optimizers of
$\rho_{1}$ and $\rho_{2}$, respectively. We use the spectral decomposition
$\tau_{1}=\sum_{l}t_{1,l}|\psi_{l}\rangle\\!\langle\psi_{l}|$ and
$\tau_{2}=\sum_{r}t_{2,r}|\xi_{r}\rangle\\!\langle\xi_{r}|$. Here,
$|\psi_{l}\rangle$ is the common eigenbasis of $\rho_{1}$ and $\tau_{1}$. In
the range $|(1-\alpha)/z|\neq 1$ we have

$\displaystyle\Xi_{\alpha,z}(\rho_{1}\otimes\rho_{2},\tau_{1}\otimes\tau_{2})$
(71)
$\displaystyle\qquad=\,K_{\alpha,z}\sum\limits_{l,r_{1},r_{2}}\int_{0}^{\infty}\left(t_{1,l}t_{2,r_{1}}+k\right)^{-1}\left(t_{1,l}t_{2,r_{2}}+k\right)^{-1}k^{\frac{1-\alpha}{z}}\mathrm{d}k\;\;|\psi_{l}\rangle\\!\langle\psi_{l}|\chi_{\alpha,z}(\rho_{1},\tau_{1})|\psi_{l}\rangle\\!\langle\psi_{l}|\;$
$\displaystyle\qquad\quad\otimes|\xi_{r_{1}}\rangle\\!\langle\xi_{r_{1}}|\chi_{\alpha,z}(\rho_{2},\tau_{2})|\xi_{r_{2}}\rangle\\!\langle\xi_{r_{2}}|$
(72)
$\displaystyle\qquad=\,K_{\alpha,z}\sum\limits_{l}t_{1,l}^{\frac{1-\alpha}{z}-1}|\psi_{l}\rangle\\!\langle\psi_{l}|\chi_{\alpha,z}(\rho_{1},\tau_{1})|\psi_{l}\rangle\\!\langle\psi_{l}|\;$
$\displaystyle\qquad\quad\otimes\sum_{r_{1},r_{2}}\int_{0}^{\infty}\left(t_{2,r_{1}}+k\right)^{-1}\left(t_{2,r_{2}}+k\right)^{-1}k^{\frac{1-\alpha}{z}}\mathrm{d}k|\xi_{r_{1}}\rangle\\!\langle\xi_{r_{1}}|\chi_{\alpha,z}(\rho_{2},\tau_{2})|\xi_{r_{2}}\rangle\\!\langle\xi_{r_{2}}|$
(73)
$\displaystyle\qquad=\;\rho_{1}^{\alpha}\tau_{1}^{-\alpha}\otimes\Xi_{\alpha,z}(\rho_{2},\tau_{2})$
(74)
$\displaystyle\qquad=\;\Xi_{\alpha}(\rho_{1},\tau_{1})\otimes\Xi_{\alpha,z}(\rho_{2},\tau_{2})\,,$
(75)

where in the second equality we changed the measure $k\rightarrow t_{1,l}k$.
Moreover, in (74) we used that
$\sum_{k}t_{1,l}^{\frac{1-\alpha}{z}-1}|\psi_{l}\rangle\\!\langle\psi_{l}|\chi_{\alpha,z}(\rho_{1},\tau_{1})|\psi_{l}\rangle\\!\langle\psi_{l}|=\rho_{1}^{\alpha}\tau_{1}^{-\alpha}$.
In the last equality we used the definition
$\Xi_{\alpha}(\rho_{1},\tau_{1}):=\rho_{1}^{\alpha}\tau_{1}^{-\alpha}$
introduced in Corollary 5. We then have, for any $\sigma\in\operatorname{SEP}$
and $(\alpha,z)\in\mathcal{D}$,

$\displaystyle\Tr(\sigma\Xi_{\alpha,z}(\rho_{1}\otimes\rho_{2},\tau_{1}\otimes\tau_{2}))$
(76)
$\displaystyle\quad\quad\leq\max\limits_{\sigma\in\operatorname{SEP}}\Tr(\sigma\Xi_{\alpha,z}(\rho_{1}\otimes\rho_{2},\tau_{1}\otimes\tau_{2}))$
(77)
$\displaystyle\quad\quad=\max\limits_{\sigma\in\operatorname{SEP}}\Tr(\sigma\Xi_{\alpha}(\rho_{1},\tau_{1})\otimes\Xi_{\alpha,z}(\rho_{2},\tau_{2}))$
(78)
$\displaystyle\quad\quad=\Tr(\Xi_{\alpha}(\rho_{1},\tau_{1}))\Tr(\Xi_{\alpha,z}(\rho_{2},\tau_{2}))\max\limits_{\sigma\in\operatorname{SEP}}\Tr(\sigma\frac{\Xi_{\alpha}(\rho_{1},\tau_{1})}{\Tr(\Xi_{\alpha}(\rho_{1},\tau_{1}))}\otimes\frac{\Xi_{\alpha,z}(\rho_{2},\tau_{2})}{\Tr(\Xi_{\alpha,z}(\rho_{2},\tau_{2}))})$
(79)
$\displaystyle\quad\quad=\Tr(\Xi_{\alpha}(\rho_{1},\tau_{1}))\Tr(\Xi_{\alpha,z}(\rho_{2},\tau_{2}))\max\limits_{\sigma\in\operatorname{SEP}}\Tr(\sigma\frac{\Xi_{\alpha}(\rho_{1},\tau_{1})}{\Tr(\Xi_{\alpha}(\rho_{1},\tau_{1}))})\max\limits_{\sigma\in\operatorname{SEP}}\Tr(\sigma\frac{\Xi_{\alpha,z}(\rho_{2},\tau_{2})}{\Tr(\Xi_{\alpha,z}(\rho_{2},\tau_{2}))})$
(80)
$\displaystyle\quad\quad=\max\limits_{\sigma\in\operatorname{SEP}}\Tr(\sigma\Xi_{\alpha}(\rho_{1},\tau_{1}))\max\limits_{\sigma\in\operatorname{SEP}}\Tr(\sigma\Xi_{\alpha,z}(\rho_{2},\tau_{2}))$
(81)
$\displaystyle\quad\quad=\;Q_{\alpha}(\rho_{1}\|\tau_{1})Q_{\alpha,z}(\rho_{2}\|\tau_{2})\,,$
(82)

where in (80) we used the multiplicativity result in [60, Theorem 5] together
with the assumption that
$\Xi_{\alpha}(\rho_{1},\tau_{1})=\rho_{1}^{\alpha}\tau_{1}^{-\alpha}$ has
positive entries in a product basis. The last equality follows from the fact
that $\tau_{1}$ and $\tau_{2}$ are, by assumptions, the optimizers of the
marginal problems. Indeed, from Theorem 4, if
$\tau\in\operatorname*{arg\,min}_{\sigma\in\operatorname{SEP}}D_{\alpha,z}(\rho\|\sigma)$,
we have that $\Tr(\sigma\Xi_{\alpha,z}(\rho,\tau))\leq
Q_{\alpha,z}(\rho\|\tau)$ for any $\sigma\in\operatorname{SEP}$. Moreover,
from Corollary 5, if
$\tau\in\operatorname*{arg\,min}_{\sigma\in\operatorname{SEP}}D_{\alpha,z}(\rho\|\sigma)$,
we have that $\Tr(\sigma\Xi_{\alpha}(\rho,\tau))\leq Q_{\alpha}(\rho\|\tau)$
for any $\sigma\in\operatorname{SEP}$ if $\rho$ and $\tau$ commute. It is easy
to see that the previous inequalities are saturated for $\sigma=\tau$. The
latter chain of inequalities proves the inequality (70). The result for the
case $|(1-\alpha)/z|=1$ can be obtained by taking the limits $z\rightarrow
1-\alpha$ and $z\rightarrow\alpha-1$ (see Appendix Appendix A: Lower
semicontinuity and its consequences for a more detailed discussion) ∎

###  IV.3 Additivity of the $\alpha$-$z$ Rényi entropy of entanglement:
Maximally correlated states

In this section, we prove that the monotones $\mathfrak{D}_{\alpha,z}$ are
additive when one state is a maximally correlated state. Moreover, we
generalize the latter result to include also the GHZ state.

A maximally correlated state is a bipartite state of the form [37]

$\rho=\sum_{jk}\rho_{jk}|j,j\rangle\\!\langle k,k|\,.$ (83)

Since pure bipartite states are maximally correlated states, this means that
the monotone is also additive whenever one state is a pure bipartite state.

We first derive necessary and sufficient conditions for an optimizer of a
maximally correlated state. The following proposition is a fundamental
ingredient to derive the additivity of the $\mathfrak{D}_{\alpha,z}$ when one
state is maximally correlated. Moreover, in Section V.4 we show how it allows
to compute analytically the $\mathfrak{D}_{\alpha,z}$ for bipartite states.

Let $\rho$ be a maximally correlated state. We denote with $|i,i\rangle$ is
the basis such that $\rho$ has the form (83). We define $\mathcal{T}_{\rho}$
as the set of all separable states of the form
$\sigma=\sum_{i}s_{i}|i,i\rangle\\!\langle i,i|$.

######  Proposition 8.

Let $\rho$ be a maximally correlated state and $(\alpha,z)\in\mathcal{D}$. If
$\tau=\sum_{i}t_{i}|i,i\rangle\\!\langle i,i|\in\mathcal{T}_{\rho}$, then
$\tau\in\operatorname*{arg\,min}_{\sigma\in\textup{SEP}}D_{\alpha,z}(\rho\|\sigma)$
if and only if $\tau\in S_{\alpha,z}(\rho)$ and

$\max_{l}t^{\frac{1-\alpha}{z}-1}_{l}\langle
l,l|\chi_{\alpha,z}(\rho,\tau)|l,l\rangle\leq Q_{\alpha,z}(\rho\|\tau)\,,$
(84)

where
$\chi_{\alpha,z}(\rho,\tau):=\rho^{\frac{\alpha}{2z}}(\rho^{\frac{\alpha}{2z}}\tau^{\frac{1-\alpha}{z}}\rho^{\frac{\alpha}{2z}})^{z-1}\rho^{\frac{\alpha}{2z}}$.

###### Proof.

We first prove that if $\tau=\sum_{i}t_{i}|i,i\rangle\\!\langle
i,i|\in\mathcal{T}_{\rho}$ satisfies the inequality (84), then $\tau$
satisfies the conditions of Theorem 4, i.e.
$\tau\in\operatorname*{arg\,min}_{\sigma\in\textup{SEP}}D_{\alpha,z}(\rho\|\sigma)$.
As we already discussed, since the trace functional is linear, we can restrict
the optimization problem of Theorem 4 to pure product states. Let us denote
with $|i,j\rangle$ the basis such that $\rho$ has the form (83). We can expand
any pure product state as $\sigma=\sum_{ij}a_{i}b_{j}|i,j\rangle$ with complex
normalized coefficients $\sum_{i}|a_{i}|^{2}=1$ and $\sum_{i}|b_{i}|^{2}=1$.
We obtain

$\displaystyle\Tr(\sigma\Xi_{\alpha,z}(\rho,\tau))$ $\displaystyle\leq
K_{\alpha,z}\sum_{i,j}|a_{i}||b_{i}||a_{j}||b_{j}|\left|\int_{0}^{\infty}\langle
i,i|\frac{\chi_{\alpha,z}(\rho,\tau)}{(\tau+t)^{2}}|j,j\rangle
t^{\frac{1-\alpha}{z}}\mathrm{d}t\right|$ (85) $\displaystyle\leq
K_{\alpha,z}\sum_{i,j}|a_{i}||b_{i}||a_{j}||b_{j}|\bigg{(}\int_{0}^{\infty}\langle
i,i|\frac{\chi_{\alpha,z}(\rho,\tau)}{(\tau+t)^{2}}|i,i\rangle
t^{\frac{1-\alpha}{z}}\mathrm{d}t$ (86) $\displaystyle\hskip
120.0pt\times\int_{0}^{\infty}\langle
j,j|\frac{\chi_{\alpha,z}(\rho,\tau)}{(\tau+t)^{2}}|j,j\rangle
t^{\frac{1-\alpha}{z}}\mathrm{d}t\bigg{)}^{\frac{1}{2}}$ (87)
$\displaystyle\leq K_{\alpha,z}\max_{k}\int_{0}^{\infty}\langle
k,k|\frac{\chi_{\alpha,z}(\rho,\tau)}{(\tau+t)^{2}}|k,k\rangle
t^{\frac{1-\alpha}{z}}\mathrm{d}t\sum_{i,j}|a_{i}||b_{i}||a_{j}||b_{j}|$ (88)
$\displaystyle\leq\max_{l}t^{\frac{1-\alpha}{z}-1}_{l}\langle
l,l|\chi_{\alpha,z}(\rho,\tau)|l,l\rangle\,,$ (89)

In the first inequality we used that $\tau$ is of the form $\tau=\sum
t_{i}|i,i\rangle\\!\langle i,i|$. In the second inequality we used that for a
positive operator $A$ and vectors $\ket{v},\ket{w}$ the Cauchy-Schwarz
inequality implies that $|\langle v|A|w\rangle|=|\langle
v|\sqrt{A}\,\sqrt{A}|w\rangle|\leq\sqrt{\langle v|A|v\rangle\\!\langle
w|A|w\rangle}$. In the last inequality we used the Cauchy-Schwarz inequality.
The previous inequalities together with inequality (84) and Theorem 4 implies
$\tau\in\operatorname*{arg\,min}_{\sigma\in\textup{SEP}}D_{\alpha,z}(\rho\|\sigma)$.

We now prove that any $\tau\in\mathcal{T}_{\rho}$ such that
$\tau\in\operatorname*{arg\,min}_{\sigma\in\textup{SEP}}D_{\alpha,z}(\rho\|\sigma)$
must satisfy the inequality (84). We choose the pure product state
$\sigma=|l,l\rangle$. We have for any $l$

$\displaystyle\Tr(\sigma\Xi_{\alpha,z}(\rho,\tau))$
$\displaystyle=K_{\alpha,z}\int_{0}^{\infty}\langle
l,l|\frac{\chi_{\alpha,z}(\rho,\tau)}{(\tau+t)^{2}}|l,l\rangle
t^{\frac{1-\alpha}{z}}\mathrm{d}t$ (90)
$\displaystyle=t^{\frac{1-\alpha}{z}-1}_{l}\langle
l,l|\chi_{\alpha,z}(\rho,\tau)|l,l\rangle\,.$ (91)

Since $\tau$ is an optimizer, the above equalities together with Theorem 4
imply that

$\max_{l}t^{\frac{1-\alpha}{z}-1}_{l}\langle
l,l|\chi_{\alpha,z}(\rho,\tau)|l,l\rangle\leq Q_{\alpha,z}(\rho\|\tau)\,.$
(92)

∎

In Appendix Appendix B: Relationship between conditional entropies,
entanglement monotones and coherence monotones based on $\alpha$-$z$ Rényi
divergences we show that for any maximally correlated state $\rho$, there
always exists an optimizer
$\tau\in\operatorname*{arg\,min}_{\sigma\in\textup{SEP}}D_{\alpha,z}(\rho\|\sigma)$
such that $\tau\in\mathcal{T}_{\rho}$.

######  Theorem 9.

Let $\rho_{1}$ be a maximally correlated state and $(\alpha,z)\in\mathcal{D}$.
Then, for any state $\rho_{2}$ we have

$\mathfrak{D}_{\alpha,z}(\rho_{1}\otimes\rho_{2})=\mathfrak{D}_{\alpha,z}(\rho_{1})+\mathfrak{D}_{\alpha,z}(\rho_{2})\,.$
(93)

###### Proof.

For what we discussed in the proof of Theorem 6, according to Theorem 4, to
prove additivity it is enough to show that

$\Tr(\sigma\Xi_{\alpha,z}(\rho_{1}\otimes\rho_{2},\tau_{1}\otimes\tau_{2}))\leq
Q_{\alpha,z}(\rho_{1}\otimes\rho_{2}\|\tau_{1}\otimes\tau_{2})=Q_{\alpha,z}(\rho_{1}\|\tau_{1})Q_{\alpha,z}(\rho_{2}\|\tau_{2})\;\forall\sigma\in\operatorname{SEP}\,,$
(94)

where $\tau_{1}$ and $\tau_{2}$ are optimizers of $\rho_{1}$ and $\rho_{2}$,
respectively.

We now prove that, if we choose the optimizer $\tau_{1}\in\mathcal{T}_{\rho}$,
the above inequality holds for any
$\tau_{2}\in\operatorname*{arg\,min}_{\sigma\in\operatorname{SEP}}D_{\alpha,z}(\rho_{2}\|\sigma)$.
Since the trace functional is linear, we can restrict the above optimization
to pure product (separable) states. We can write any pure product state in the
partition $AA^{\prime}\\!:\\!BB^{\prime}$ as
$\sigma=\sum_{i}a_{i}|i,\phi_{i}\rangle_{AA^{\prime}}\otimes\sum_{j}b_{j}|j,\psi_{j}\rangle_{BB^{\prime}}$
with positive normalized coefficients $a_{i}$ and $b_{j}$ and normalized
$|\phi_{i}\rangle$ and $|\psi_{i}\rangle$. Here, we choose $|i,j\rangle_{AB}$
as the product basis such that
$\rho_{\textup{MC}}=\sum_{jk}\rho_{jk}|j,j\rangle\\!\langle k,k|_{AB}$. We
take the modulus and use the Cauchy-Schwarz inequality to get

$\displaystyle\Tr(\sigma\Xi_{\alpha,z}(\rho_{1}\otimes\rho_{2},\tau_{1}\otimes\tau_{2}))$
$\displaystyle\leq\sum\limits_{i,j,i^{\prime},j^{\prime}}a_{i}b_{j}a_{i^{\prime}}b_{j^{\prime}}|\langle
i,j|\langle\phi_{i},\psi_{j}|\Xi_{\alpha,z}(\rho_{1}\otimes\rho_{2},\tau_{1}\otimes\tau_{2})|i^{\prime},j^{\prime}\rangle|\phi_{i^{\prime}},\psi_{j^{\prime}}\rangle|$
(95)
$\displaystyle\leq\sum\limits_{i,j,i^{\prime},j^{\prime}}a_{i}b_{j}a_{i^{\prime}}b_{j^{\prime}}(\langle
i,j|\langle\phi_{i},\psi_{j}|\,\Xi_{\alpha,z}(\rho_{1}\otimes\rho_{2},\tau_{1}\otimes\tau_{2})\,|i,j\rangle|\phi_{i},\psi_{j}\rangle$
(96) $\displaystyle\qquad\qquad\times\langle
i^{\prime},j^{\prime}|\langle\phi_{i^{\prime}},\psi_{j^{\prime}}|\,\Xi_{\alpha,z}(\rho_{1}\otimes\rho_{2},\tau_{1}\otimes\tau_{2})\,|i^{\prime},j^{\prime}\rangle|\phi_{i^{\prime}},\psi_{j^{\prime}}\rangle)^{\frac{1}{2}}\,.$
(97)

We now use the spectral decomposition
$\tau_{2}=\sum_{r}t_{2,r}|\xi_{r}\rangle\\!\langle\xi_{r}|$. We have for
$|(1-\alpha)/z|\neq 1$

$\displaystyle\langle
i,j|\langle\phi_{i},\psi_{j}|\Xi_{\alpha,z}(\rho_{1}\otimes\rho_{2},\tau_{1}\otimes\tau_{2})|i,j\rangle|\phi_{i},\psi_{j}\rangle$
(98)
$\displaystyle\quad\quad=K_{\alpha,z}\delta_{ij}\sum\limits_{r_{1},r_{2}}\int_{0}^{\infty}\left(t_{1,i}t_{2,r_{1}}+k\right)^{-1}\left(t_{1,i}t_{2,r_{2}}+k\right)^{-1}k^{\frac{1-\alpha}{z}}\mathrm{d}k$
(99) $\displaystyle\qquad\qquad\qquad\qquad\qquad\times\langle
i,i|\chi_{\alpha,z}(\rho_{1},\tau_{1})|i,i\rangle\langle\phi_{i},\psi_{i}|\xi_{r_{1}}\rangle\\!\langle\xi_{r_{1}}|\chi_{\alpha,z}(\rho_{2},\tau_{2})|\xi_{r_{2}}\rangle\\!\langle\xi_{r_{2}}|\phi_{i},\psi_{i}\rangle$
$\displaystyle\quad\quad=K_{\alpha,z}\delta_{ij}t_{1,i}^{\frac{1-\alpha}{z}-1}\sum\limits_{r_{1},r_{2}}\int_{0}^{\infty}\left(t_{2,r_{1}}+k\right)^{-1}\left(t_{2,r_{2}}+k\right)^{-1}k^{\frac{1-\alpha}{z}}\mathrm{d}k$
(100) $\displaystyle\qquad\qquad\qquad\qquad\qquad\times\langle
i,i|\chi_{\alpha,z}(\rho_{1},\tau_{1})|i,i\rangle\langle\phi_{i},\psi_{i}|\xi_{r_{1}}\rangle\\!\langle\xi_{r_{1}}|\chi_{\alpha,z}(\rho_{2},\tau_{2})|\xi_{r_{2}}\rangle\\!\langle\xi_{r_{2}}|\phi_{i},\psi_{i}\rangle$
$\displaystyle\quad\quad=\,K_{\alpha,z}\delta_{i,j}t_{1,i}^{\frac{1-\alpha}{z}-1}\langle
i,i|\chi_{\alpha,z}(\rho_{1},\tau_{1})|i,i\rangle\\!\int_{0}^{\infty}\langle\phi_{i}\psi_{i}|(\tau_{2}+k)^{-1}\chi_{\alpha,z}(\rho_{2},\tau_{2})(\tau_{2}+k)^{-1}|\phi_{i}\psi_{i}\rangle
k^{\frac{1-\alpha}{z}}\mathrm{d}k\,$ (101)
$\displaystyle\quad\quad\leq\,\delta_{i,j}t_{1,i}^{\frac{1-\alpha}{z}-1}\langle
i,i|\chi_{\alpha,z}(\rho_{1},\tau_{1})|i,i\rangle
Q_{\alpha,z}(\rho_{2}\|\tau_{2})\,.$ (102)

In the first equality we used the above mentioned structure of the optimizer
of $\tau_{1}$. In the second equality we used the change of measure
$k\rightarrow t_{1,i}k$. The last inequality follows from the fact that, since
by assumption $\tau_{2}$ is an optimizer of $\rho_{2}$, we have
$\Tr(\sigma\Xi_{\alpha,z}(\rho_{2},\tau_{2}))\leq
Q_{\alpha,z}(\rho_{2}\|\tau_{2})$ for any $\sigma\in\operatorname{SEP}$. In
addition we used that any diagonal matrix element of the positive operator
$\chi_{\alpha,z}(\rho_{1},\tau_{1})$ is positive. Therefore, for any
$\sigma\in\operatorname{SEP}$ we have

$\displaystyle\Tr(\sigma\Xi_{\alpha,z}(\rho_{1}\otimes\rho_{2},\tau_{1}\otimes\tau_{2}))$
(103)
$\displaystyle\qquad\leq\sum\limits_{i,i^{\prime}}a_{i}b_{i}a_{i^{\prime}}b_{i^{\prime}}Q_{\alpha,z}(\rho_{2}\|\tau_{2})\bigg{(}(t_{1,i}t_{1,i^{\prime}})^{\frac{1-\alpha}{z}-1}\langle
i,i|\chi_{\alpha,z}(\rho_{1},\tau_{1})|i,i\rangle\\!\langle
i^{\prime},i^{\prime}|\chi_{\alpha,z}(\rho_{1},\tau_{1})|i^{\prime},i^{\prime}\rangle\bigg{)}^{\frac{1}{2}}$
(104) $\displaystyle\qquad\leq
Q_{\alpha,z}(\rho_{1}\|\tau_{1})Q_{\alpha,z}(\rho_{2}\|\tau_{2})\,,$ (105)

where in the last inequality follows from the Cauchy-Schwarz inequality and
the marginal problem condition derived in Proposition 8. Hence (94) is
satisfied and additivity holds. The result for the case $|(1-\alpha)/z|=1$ can
be obtained by taking the limits $z\rightarrow 1-\alpha$ and
$z\rightarrow\alpha-1$ (see Appendix Appendix A: Lower semicontinuity and its
consequences for a more detailed discussion) ∎

The above theorem can be easily generalized to include all the states of the
form $|\rho\rangle=\sum_{i}\sqrt{p_{i}}|i,\dots,i\rangle$. In particular, this
class contains the GHZ state (see also Section V.4). The first part is the
same as above where now
$\tau=\frac{1}{C}\sum_{i}p^{\beta}_{i}\outerproduct{i,\dots,i}{i,\dots,i}$
(see Section V.4). In the final step we use Cauchy-Schwarz inequality.

######  Remark 2.

Note that the previous results imply that whenever a state $\rho$ satisfies
the conditions of Theorem 7 or Theorem 9, we have
$\mathfrak{D}_{\alpha,z}(\rho)=\mathfrak{D}^{\infty}_{\alpha,z}(\rho)$, where
the regularized or asymptotic monotone $\mathfrak{D}^{\infty}_{\alpha,z}$ is
defined as
$\mathfrak{D}^{\infty}_{\alpha,z}(\rho)=\lim_{n\rightarrow\infty}\frac{1}{n}\mathfrak{D}_{\alpha,z}(\rho^{\otimes
n})$.

The latter results have, for example, direct application in catalytic resource
theories [40]. Here, the additivity properties of the
$\mathfrak{D}_{\alpha,z}$ are tightly connected with their role in
characterizing catalytic state conversions [40]. In this setting, different
monotones for different values of the parameters have different roles.
Therefore, it is crucial to completely characterize the additivity properties
of the $\mathfrak{D}_{\alpha,z}$ for all $(\alpha,z)\in\mathcal{D}$. In
particular, to establish fundamental limits on correlated catalytic
transformations between pure states, we are interested in the additivity of
the monotones for $\alpha=z$ with $\alpha\in[1/2,1)$ of a pure state with any,
not necessarily pure, catalyst state [40].

##  V Some examples of states belonging to the additivity classes and their
computation

In this section we provide several examples of states which belong to the
additivity classes that we discussed in the previous section. We also give a
closed-from expression for the monotones $\mathfrak{D}_{\alpha,z}$ by
extending some already known results. We summarize our results and our
contribution in Table 1. Our strategy consists of the following two steps. We
first guess the optimizer and then we check analytically if the conditions of
Theorem 4 (or Corollary 5) are satisfied. We remark that it is in general a
hard problem to find the right guess for $\tau$ for general states. However,
as we show later, the optimizer can be easily guessed in many situations as it
usually lies at the boundary of the allowed separable region. In addition,
numerical simulations could give insight on the structure of the optimizer. We
then need to solve the optimization problem
$\Lambda^{2}(\Xi):=\max_{\sigma\in\operatorname{SEP}}\Tr(\sigma\Xi)$ where
$\Xi$ is a positive operator. This optimization, as we see later, is usually
an easier problem than the original one in Eq. (9). Indeed, because of the
linearity of the trace, the optimization in $\Lambda^{2}(\Xi)$ can be
restricted to the of set pure product states and can be computed analytically
for many states [60]. Note that the optimization $\Lambda^{2}(\Xi)$ could also
be performed numerically by optimizing over the set of states with positive
partial transpose (PPT) instead of the separable states. Indeed, since the set
of PPT states is larger then the set of separable ones [32], if the conditions
of Theorem 4 are satisfied for any $\sigma\in\text{PPT}$, we can conclude that
our (separable) guess is the right optimizer.

STATES |  VALUE OF $\mathfrak{D}_{\alpha,z}$
---|---
REFERENCES
Bipartite pure | $H_{\beta}(\vec{p})\;,\quad\text{where}\;\;\frac{1-\alpha}{z}+\frac{1}{\beta}=1$
$|\rho(\vec{p})\rangle=\sum_{i}\sqrt{p_{i}}|i,i\rangle$ |  $z=\alpha=1$ [46], $z=\alpha,z=1$ [61, Corollary 2]
| Other $(\alpha,z)$: Proposition 13
Bell diagonal | $\begin{cases}0&\text{if}\;\lambda_{\max}\in\left[0,\frac{1}{2}\right]\\\ 1-H_{\alpha}(\lambda_{\max},1-\lambda_{\max})&\text{if}\;\lambda_{\max}\in\left[\frac{1}{2},1\right]\end{cases}$
$\rho_{\text{BD}}(\vec{\lambda})=\sum_{j=1}^{4}\lambda_{j}|\psi_{j}\rangle\\!\langle\psi_{j}|\;$
where $\\{|\psi_{j}\rangle\\}_{j=1}^{4}$ are the Bell states |  $\alpha=z=1$ [47, 35],$\,\infty$ [60]
| Other $(\alpha,z)$: Proposition 10
Generalized Werner | $\begin{cases}1-H_{\alpha}(p,1-p)\,\,&p\in[0,1/2]\\\ 0&p\in[1/2,1]\end{cases}$
$\rho_{W}(p)=p\frac{2}{d(d+1)}P^{\text{SYM}}_{AB}+(1-p)\frac{2}{d(d-1)}P^{\text{AS}}_{AB}$
$P^{\text{SYM}}_{AB}$ projector onto the symmetric subspace |  $\alpha=z=1/2$ [54],$\,1$ [2, 60]
$P^{\text{AS}}_{AB}$ projector onto the antisymmetric subspace | Other $(\alpha,z)$: Proposition 11
Isotropic | $\begin{cases}0&F\in\left[0,\frac{1}{d}\right]\\\ \log{d}-H_{\alpha}\left(\frac{1-F}{(d-1)^{\frac{\alpha-1}{\alpha}}},F\right)\,\,&F\in\left[\frac{1}{d},1\right]\end{cases}$
$\rho_{iso}(F)=\frac{1-F}{d^{2}-1}(\mathds{1}-|\Phi^{+}\rangle\\!\langle\Phi^{+}|)+F|\Phi^{+}\rangle\\!\langle\Phi^{+}|$
where $|\Phi^{+}\rangle=\frac{1}{\sqrt{d}}\sum_{i}|ii\rangle$ |  $\alpha=z=1/2$ [54],$\,1$ [35],$\,\infty$[60]
| Other $(\alpha,z)$: Proposition 12
Generalized Dicke | $-\log{\left(\frac{N!}{\prod_{j=0}^{d-1}k_{j}!}\prod_{j=0}^{d-1}\left(\frac{k_{j}}{N}\right)^{k_{j}}\right)}$
$S(|N,\vec{k}\rangle)=\frac{1}{\sqrt{C_{n,\vec{k}}}}\sum\limits_{\text{P}}P|\underbrace{0,...,0}_{k_{0}},\underbrace{1,...,1}_{k_{1}},...,\underbrace{d-1,...,d-1}_{k_{d}-1}\rangle$ | $\alpha=z$ [54, 52, 19]
$\vec{k}=(k_{0},...,k_{d-1}),\;\sum_{j=0}^{d-1}k_{j}=N$ | Other $(\alpha,z)$: Proposition 15
Maximally correlated Bell diagonal | $\log{d}-H_{\alpha}(\vec{p})$
$\rho_{\text{MCBD}}(\vec{p})=\sum_{k=0}^{d-1}p_{k}|\psi_{k}\rangle\\!\langle\psi_{k}|$ |  $\alpha=z=1$ [60, 35]
where $|\psi_{k}\rangle=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}e^{\frac{2\pi ik}{d}j}|jj\rangle$ | Other $(\alpha,z)$: Proposition 16
Table 1: The table contains the value of the monotones
$\mathfrak{D}_{\alpha,z}$ for some states. We denoted with
$H_{\alpha}(\vec{p})=\frac{1}{1-\alpha}\log{\left(\sum_{i}p_{i}^{\alpha}\right)}$
the $\alpha$-Rényi entropy of a vector $\vec{p}$. For more details see Section
V. The monotones $\mathfrak{D}_{\alpha,z}$ are additive when one state is
among the ones listed in the table (see the discussion below). The only
exception is for Werner states which provide a counterexample to the
additivity of any entanglement monotone based on a quantum relative entropy
(see Section VI).

###  V.1 Bell Diagonal states (BD)

We consider the Bell diagonal states

$\rho_{\text{BD}}(\vec{\lambda})=\sum_{j=1}^{4}\lambda_{j}|\psi_{j}\rangle\\!\langle\psi_{j}|\,,$
(106)

where $|\psi_{j}\rangle$ are the four Bell states. A Bell diagonal state is
separable if $\lambda_{i}\in[0,1/2]$ for all $i$ [24].

######  Proposition 10.

Let $(\alpha,z)\in\mathcal{D}$. For a Bell Diagonal state (106) we have

$\displaystyle\mathfrak{D}_{\alpha,z}(\rho_{\textup{BD}}(\vec{\lambda}))=\begin{cases}0&\text{if}\;\lambda_{\max}\in\left[0,\frac{1}{2}\right]\\\
1-H_{\alpha}(\lambda_{\max},1-\lambda_{\max})&\text{if}\;\lambda_{\max}\in\left[\frac{1}{2},1\right]\end{cases}\,.$
(107)

###### Proof.

A Bell diagonal state where all $\lambda_{i}\in[0,1/2]$ is separable and
hence, in this range, we have
$\min_{\sigma\in\operatorname{SEP}}D_{\alpha,z}(\rho_{\text{BD}}(\vec{\lambda})\|\sigma)=0$.
Without loss of generality, we assume that $\lambda_{1}\geq 1/2$ and
$\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\lambda_{4}$. We take as ansatz
$\tau_{\text{BD}}=\sum_{i=1}^{4}p_{i}|\psi_{i}\rangle\\!\langle\psi_{i}|$ with
$p_{1}=1/2$ and
$p_{i}=\lambda_{i}/(2(1-\lambda_{1}))\,\,\text{for}\,\,i\in\\{2,3,4\\}$. Using
that $\Lambda^{2}(\rho_{\text{BD}})=(\lambda_{1}+\lambda_{2})/2$ [60] it is
easy to check that
$\Tr(\rho^{\alpha}_{\text{BD}}\tau^{-\alpha}_{\text{BD}}\sigma)\leq
2^{\alpha-1}(\lambda_{1}^{\alpha}+(1-\lambda_{1})^{\alpha})=\Tr(\rho^{\alpha}_{\text{BD}}\tau^{1-\alpha}_{\text{BD}})$
and the condition of Corollary 5 is satisfied. Hence, our ansatz is an
optimizer. ∎

The Bell diagonal states commute with their optimizer and the product
$\rho^{\alpha}_{\text{MCBD}}\tau^{-\alpha}_{\text{MCBD}}$ is a (not
normalized) Bell diagonal state. Bell diagonal states have positive entries in
a product basis [60]. Therefore, from Theorem 7 it follows that the
$\mathfrak{D}_{\alpha,z}$ is additive when one state is a Bell diagonal state.

###  V.2 Werner states

A state $\rho\in\mathcal{S}(A_{1}\otimes A_{2})$ is called (generalized)
Werner state if it is invariant under the projection [55, 6, 51] ($UU$ group)

$P:\rho\rightarrow\int dU(U\otimes U)\rho(U^{\dagger}\otimes U^{\dagger})\,,$
(108)

where $\mathrm{d}U$ is the invariant Haar measure. A Werner states (bipartite
state in $d$ dimensions) can be written as

$\rho_{W}(p)=p\frac{2}{d(d+1)}P^{\text{SYM}}_{AB}+(1-p)\frac{2}{d(d-1)}P^{\text{AS}}_{AB}\,,$
(109)

where $P^{\text{SYM}}_{AB}$ and $P^{\text{AS}}_{AB}$ are the projectors into
the symmetric and antisymmetric subspace, respectively. The Werner state is
separable for $p\geq 1/2$ and entangled for $p<1/2$ [51]. We define
$\rho_{+}:=\rho_{W}(1)$ and $\rho_{-}:=\rho_{W}(0)$.

######  Proposition 11.

Let $(\alpha,z)\in\mathcal{D}$. For a generalized Werner state (109) we have

$\displaystyle\mathfrak{D}_{\alpha,z}(\rho_{\text{W}}(p))=\begin{cases}1-H_{\alpha}(p,1-p)&\text{if}\;\;p\in\left[0,\frac{1}{2}\right]\\\
0&\text{if}\;\;p\in\left[\frac{1}{2},1\right]\end{cases}\,.$ (110)

###### Proof.

We first calculate the quantity
$\Lambda^{2}(\rho_{W}(p))=\max_{\sigma\in\operatorname{SEP}}\Tr(\sigma\rho_{W}(p))$.
We first note that the optimization $\Lambda^{2}(\rho_{W}(p))$ can be
restricted to separable Werner states. Indeed,

$\displaystyle\max\limits_{\sigma\in\operatorname{SEP}}\Tr(\sigma\rho_{W}(p))$
$\displaystyle=\max\limits_{\sigma\in\operatorname{SEP}}\Tr(\sigma\int
dU(U\otimes U)\rho_{W}(p)(U^{\dagger}\otimes U^{\dagger}))$ (111)
$\displaystyle=\max\limits_{\sigma\in\operatorname{SEP}}\Tr(\int
dU(U^{\dagger}\otimes U^{\dagger})\sigma(U\otimes U)\rho_{W}(p))$ (112)
$\displaystyle=\max\limits_{\sigma\in\operatorname{SEP}\cap\text{W}}\Tr(\sigma\rho_{W}(p))\,,$
(113)

where in the last step we used that $P(\sigma)\in\operatorname{SEP}$ for any
$\sigma\in\operatorname{SEP}$. Then, by direct calculation

$\Tr(\rho_{W}(p^{\prime})\rho_{W}(p^{\prime\prime}))=p^{\prime}p^{\prime\prime}\frac{2}{d(d+1)}+(1-p^{\prime})(1-p^{\prime\prime})\frac{2}{d(d-1)}\,,$
(114)

where we used that $\Tr(P^{SYM}_{AB})=d(d+1)/2$ and
$\Tr(P^{AS}_{AB})=d(d-1)/2$. It is easy to see that the previous expression is
maximized for $p^{\prime\prime}=1/2$ if $p^{\prime}\leq\frac{d+1}{2d}$. We
therefore obtain

$\Lambda^{2}(\rho_{W}(p^{\prime}))=\frac{p^{\prime}}{d(d+1)}+\frac{1-p^{\prime}}{d(d-1)}\qquad\text{for}\;\;p^{\prime}\leq\frac{d+1}{2d}\,.$
(115)

We now calculate the values of $\mathfrak{D}_{\alpha,z}$. We choose as ansatz
$\tau_{W}:=\rho_{W}(1/2)$ which commutes with any Werner state. Then
$\rho_{W}(p)^{\alpha}\tau_{W}^{-\alpha}$ is (apart from a normalization
factor) a Werner state with

$p^{\prime}=\left(1+\left(\frac{1-p}{p}\right)^{\alpha}\frac{d-1}{d+1}\right)^{-1}<(d+1)/(2d)\,,$
(116)

where in the last inequality we assumed that the input state $\rho_{W}(p)$ is
entangled, i.e. $p<1/2$. Then from the above,
$\Tr(\sigma\rho_{W}(p)^{\alpha}\tau_{W}^{-\alpha})\leq\Tr(\rho_{W}(p)^{\alpha}\tau_{W}^{1-\alpha})$
for $p<1/2$ for any $\sigma\in\operatorname{SEP}$. Hence, from Corollary 5 it
follows that $\tau_{W}$ is an optimizer. ∎

The Werner states do not satisfy the conditions for $\rho_{1}$ of Theorem 7 or
Theorem 9. In Section VI we show that they provide a counterexample to the
additivity of the $\mathfrak{D}_{\alpha,z}$.

###  V.3 Isotropic states

A state $\rho\in\mathcal{S}(A_{1}\otimes A_{2})$ is called isotropic state if
it is invariant under the projection ($UU^{*}$ group) [44, 21]

$P:\rho\rightarrow\int dU(U\otimes U^{*})\rho(U^{\dagger}\otimes
U^{*\dagger})\,,$ (117)

where $\mathrm{d}U$ is the invariant Haar measure. Isotropic states (biparite
state in $d$ dimensions) can be written as

$\rho_{iso}(F)=\frac{1-F}{d^{2}-1}(\mathds{1}-|\Phi^{+}\rangle\langle\Phi^{+}|)+F|\Phi^{+}\rangle\langle\Phi^{+}|\,,$
(118)

where $|\Phi^{+}\rangle=1/\sqrt{d}\sum_{i}|ii\rangle$ and $F\in[0,1]$. The
isotropic state is separable for $F\leq 1/d$ and entangled for $F>1/d$ [21].

######  Proposition 12.

Let $(\alpha,z)\in\mathcal{D}$ . For an isotropic state (118) we have

$\displaystyle\mathfrak{D}_{\alpha,z}(\rho_{iso}(F))=\begin{cases}0&\text{if}\;\;F\in\left[0,\frac{1}{d}\right]\\\
\log{d}-H_{\alpha}\left(\frac{1-F}{(d-1)^{\frac{\alpha-1}{\alpha}}},F\right)&\text{if}\;\;F\in\left[\frac{1}{d},1\right]\end{cases}\,.$
(119)

###### Proof.

For $F>1/d$ we choose as ansatz $\tau_{iso}:=\rho_{iso}(1/d)$ which commutes
with any isotropic state. Then $\rho_{iso}(F)^{\alpha}\tau_{iso}^{-\alpha}$ is
(apart form a normalization factor) an isotropic state with

$F^{\prime}=\left(1+\left(\frac{1-F}{F}\right)^{\alpha}\frac{d^{2}-1}{(d-1)^{\alpha}}\right)^{-1}>1/d^{2}\,,$
(120)

where in the last inequality we assumed that the input state $\rho_{iso}(F)$
is entangled, i.e. $F>1/d$. We then use that
$\Lambda^{2}(\rho_{iso}(F^{\prime}))=\frac{F^{\prime}d+1}{d(d+1)}$ if
$F^{\prime}\geq 1/d^{2}$ [60]. Hence,
$\Tr(\sigma\rho_{iso}(F)^{\alpha}\tau_{iso}^{-\alpha})\leq
Q_{\alpha}(\rho_{iso}(F)\|\tau_{iso})$ for $F>1/d$ and any
$\sigma\in\operatorname{SEP}$. From Corollary 5 it follows that $\tau_{iso}$
is an optimizer. ∎

The isotropic states commute with their optimizer and the product
$\rho^{\alpha}_{iso}\tau^{-\alpha}_{iso}$ is an (not normalized) isotropic
state. Isotropic states have positive entries in a product basis [60].
Therefore, from Theorem 7 it follows that the $\mathfrak{D}_{\alpha,z}$ is
additive when one state is an isotropic state.

###  V.4 Bipartite pure states

The exact computation of the monotones $\mathfrak{D}_{\alpha,z}(|\rho\rangle)$
was already solved for bipartite pure states in [61] for $\alpha=z$ and $z=1$.
We extend this proof outside the former ranges and to a more general class of
multipartite states which includes the GHZ state.

######  Proposition 13.

Let $(\alpha,z)\in\mathcal{D}$. For a bipartite pure state
$|\rho(\vec{p})\rangle=\sum\sqrt{p_{i}}|i,i\rangle$ we have

$\displaystyle\mathfrak{D}_{\alpha,z}(|\rho(\vec{p})\rangle)=H_{\beta}(\vec{p})\;,\quad\text{where}\;\;\frac{1-\alpha}{z}+\frac{1}{\beta}=1\,.$
(121)

###### Proof.

Our ansatz for the optimzer is
$\tau=\frac{1}{C}\sum_{i}p^{\beta}_{i}|i,i\rangle\\!\langle i,i|$ where
$C=\sum_{i}p_{i}^{\beta}$ is the normalization constant and
$\frac{1}{\beta}+\frac{1-\alpha}{z}=1$. Here, $|i,i\rangle$ is the Schmidt
basis of the pure state, i.e. the basis for which
$|\rho(\vec{p})\rangle=\sum\sqrt{p_{i}}|i,i\rangle$. According to Proposition
8 we need to check the condition

$\max_{l}\left(\frac{p_{l}^{\beta}}{C}\right)^{\frac{1-\alpha}{z}-1}\langle
l,l|\chi_{\alpha,z}(\rho,\tau)|l,l\rangle\leq Q_{\alpha,z}(\rho\|\tau)\,.$
(122)

We have that
$\chi_{\alpha,z}(\rho,\tau)=\rho\langle\rho|\sigma^{\frac{1-\alpha}{z}}|\rho\rangle^{z-1}$,
$Q_{\alpha,z}(\rho\|\tau)=\langle\rho|\sigma^{\frac{1-\alpha}{z}}|\rho\rangle^{z}$
and $\langle\rho|\sigma^{\frac{1-\alpha}{z}}|\rho\rangle=C^{\frac{1}{\beta}}$.
It is easy to check that the ansatz satisfies the above condition. Note that
the maximum of the above quantity is achieved by $\sigma=|l,l\rangle$ for any
$l$. Therefore, Proposition 13 holds. ∎

Note that, in contrast with the previous cases, for bipartite pure states, the
closest separable state depends on $\alpha$ and $z$. The bipartite pure states
are maximally correlated states. Therefore, Theorem 9 implies that the
$\mathfrak{D}_{\alpha,z}$ are additive when one state is a bipartite pure
state.

The same results can be extended to multipartite states of the form
$|\rho\rangle=\sum_{i}\sqrt{p_{i}}|i,\dots,i\rangle$. In particular, this
class contains the GHZ state.

######  Corollary 14.

Let $(\alpha,z)\in\mathcal{D}$ and
$|\rho(\vec{p})\rangle=\sum\sqrt{p_{i}}|i,\dots,i\rangle\in\mathcal{S}(A_{1}\otimes\cdots\otimes
A_{m})$. Then we have

$\displaystyle\mathfrak{D}_{\alpha,z}(|\rho(\vec{p})\rangle)=H_{\beta}(\vec{p})\;,\quad\text{where}\;\;\frac{1}{\beta}+\frac{1-\alpha}{z}=1\,.$
(123)

###### Proof.

The first part is the same as above where now
$\tau=\frac{1}{C}\sum_{i}p^{\beta}_{i}\outerproduct{i,\dots,i}{i,\dots,i}$. In
the final optimization we optimize over
$\sigma=\sum_{k_{1},\dots,k_{N}}c^{(1)}_{k_{1}}\dotsi
c^{(n)}_{k_{N}}|k_{1},\dots,k_{N}\rangle$ and we use Cauchy-Schwarz inequality
instead of Cauchy-Schwarz. ∎

The M-partite GHZ state where each subsystem has dimension $d$ reads
$|\text{GHZ}\rangle=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}|i\rangle^{\otimes M}$.
The corollary above implies that for the GHZ state
$\mathfrak{D}_{\alpha,z}(|\rho\rangle_{\text{GHZ}})=\log{d}$. The latter
result was already obtained in [54, 53, 38] for the cases
$\alpha=z=1/2,1,\infty$ and $\alpha=0,z=1$. Since the result does not depend
on $\alpha$ and $z$, the results for $\alpha=0,z=1$ and $\alpha=z=\infty$
already implies that
$\min_{\sigma\in\operatorname{SEP}}\mathbb{D}(|\text{GHZ}\rangle\|\sigma)=\log{d}$
where $\mathbb{D}$ is any quantum relative entropy (see Theorem 3 in [17]).

###  V.5 Generalized Dicke states

Generalized Dicke states (or symmetric base state) are defined as [54, 14]

$\displaystyle
S(|N,\vec{k}\rangle)=\frac{1}{\sqrt{C_{N,\vec{k}}}}\sum\limits_{\\{P\\}}P|\overbrace{0,\dots,0}^{k_{0}},\overbrace{1,\dots,1}^{k_{1}},\dots,\overbrace{d-1,\dots,d-1}^{k_{d}-1}\rangle\,,$
(124)
$\displaystyle\vec{k}=(k_{0},\dots,k_{d-1})\;,\quad\text{and}\;\;\sum_{j=0}^{d-1}k_{j}=N\,.$
(125)

Here $\\{P\\}$ denotes the set of all permutations and
$C_{N,\vec{k}}=\frac{N!}{\prod_{j=0}^{d-1}k_{j}!}$ is the normalization
factor.

######  Proposition 15.

Let $(\alpha,z)\in\mathcal{D}$. For a generalized Dicke state (124) we have

$\displaystyle\mathfrak{D}_{\alpha,z}(S(|N,\vec{k}\rangle))=-\log{\left(C_{N,\vec{k}}\prod_{j=0}^{d-1}\left(\frac{k_{j}}{N}\right)^{k_{j}}\right)}\,.$
(126)

###### Proof.

The separable ansatz of the problem is

$\tau_{\text{GD}}=\int_{0}^{2\pi}\frac{\mathrm{d}\vec{\phi}}{(2\pi)^{d}}|\xi(\vec{\phi})\rangle\\!\langle\xi(\vec{\phi})|\;,\quad\text{where}\;\;|\xi(\vec{\phi})\rangle=\left(\sum_{j=0}^{d-1}\sqrt{\frac{k_{i}}{N}}|i\rangle\right)^{\otimes
n}\,.$ (127)

The above ansatz commutes with the input state. Using that
$\Lambda^{2}_{\max}\left(S(|N,\vec{k}\rangle)\right)=C_{N,\vec{k}}\prod_{j=0}^{d-1}\left(\frac{k_{j}}{N}\right)^{k_{j}}$
[54, 52, 60] and Corollary 5 we get Proposition 15. ∎

The generalized Dicke states commute with their optimizer and the product
$S(|N,\vec{k}\rangle)^{\alpha}\tau^{-\alpha}_{\text{GD}}$ is a (not
normalized) generalized Dicke state. The latter states are have positive
entries in a product basis [60]. Therefore, from Theorem 7 it follows that the
$\mathfrak{D}_{\alpha,z}$ are additive when one state is a generalized Dicke
state.

###  V.6 Maximally correlated Bell diagonal states (MCBD)

A closed form for the maximally correlated states is known only for the
entanglement monotone based on the Petz divergence [61]. We could not find a
closed form for the $\alpha$-$z$ Rényi divergences of entanglement for
maximally correlated states for all $(\alpha,z)\in\mathcal{D}$. Nevertheless,
we could show that the $\mathfrak{D}_{\alpha,z}$ are additive whenever one
state is maximally correlated. In the following, to obtain a closed form, we
add a further constraint, i.e. we require the Bell diagonal condition. We
consider the maximally correlated Bell diagonal states (MCBD) [60]

$\rho_{\text{MCBD}}(\vec{p})=\sum_{k=0}^{d-1}p_{k}\outerproduct{\psi_{k}}{\psi_{k}}\;,\quad\text{where}\;\;|\psi_{k}\rangle=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}e^{\frac{2\pi
ik}{d}j}|jj\rangle\,.$ (128)

######  Proposition 16.

Let $(\alpha,z)\in\mathcal{D}$. For a Maximally correlated Bell Diagonal state
(128) we have

$\displaystyle\mathfrak{D}_{\alpha,z}(\rho_{\textup{MCBD}}(\vec{p}))=\log{d}-H_{\alpha}(\vec{p})\,.$
(129)

###### Proof.

We take as ansatz
$\tau_{\text{MCBD}}=\sum_{k=0}^{d-1}\frac{1}{d}\outerproduct{\psi_{k}}{\psi_{k}}$
which satisfies $[\rho_{\text{MCBD}},\tau_{\text{MCBD}}]=0$. Using that
$\Lambda^{2}(\rho_{\text{MCBD}})=1/d$ [60] we have that
$\Tr(\rho^{\alpha}_{\text{MCBD}}\tau^{-\alpha}_{\text{MCBD}}\sigma)\leq\sum
p_{k}^{\alpha}d^{\alpha-1}=\Tr(\rho^{\alpha}_{\text{MCBD}}\tau^{1-\alpha}_{\text{MCBD}})$
for any $\sigma\in\operatorname{SEP}$ and the condition of Corollary 5 is
satisfied. Hence, our ansatz is an optimizer. ∎

The maximally correlated Bell diagonal states are maximally correlated state.
Therefore, Theorem 9 implies that the $\mathfrak{D}_{\alpha,z}$ are additive
when one state is a maximally correlated Bell diagonal state.

##  VI Counterexample to the additivity of the monotones based on a quantum
relative entropy

In this section we prove that we prove that any monotone based on a quantum
relative entropy is not additive for general states. In particular, additivity
generally does not hold also when the two states are equal, which means that
the latter monotones are also not weakly additive. To the best of our
knowledge, this results is only known for $\mathfrak{D}_{\alpha,z}$ in the
points $z=\alpha=1$ and $z=\alpha=\infty$. In particular, for these two
points, in [51] and [60] the authors showed that the tensor product of two
bipartite antisymmetric states provides a counterexample to the additivity of
these monotones for general states. Moreover, in these two points, the value
of the monotones for bipartite antisymmetric states and tensor product of
bipartite antisymmetric does not depend on the value of $z$ and $\alpha$.
Since the monotones $\mathfrak{D}_{\alpha,\alpha}$ are increasing functions of
$\alpha$, the latter results already imply that the
$\mathfrak{D}_{\alpha,\alpha}$ are not additive for the range
$\alpha\in[1,\infty]$. In the following, we show that our framework allows to
extend this result to any monotone based on a quantum relative entropy. We
remark that the non-additivity of $\mathfrak{D}_{1/2,1/2}$ implies that the
fidelity of separability is not multiplicative for general states, which to
the best of our knowledge, was not known previously.

It is easy to prove that the $\mathfrak{D}_{\alpha,z}$ are sub-additive, i.e.
$\mathfrak{D}_{\alpha,z}(\rho_{1}\otimes\rho_{2})\leq\mathfrak{D}_{\alpha,z}(\rho_{1})+\mathfrak{D}_{\alpha,z}(\rho_{1})$.
We now prove that when $\rho_{1}=\rho_{2}=\rho_{-}$ where $\rho_{-}$ is the
bipartite antisymmetric (Werner) state [51, 60], the previous inequality is
strict. From Section V.2 we have that that
$\mathfrak{D}_{\alpha,z}(\rho_{-})=1$. We now consider the the tensor product
$\rho_{-}\otimes\rho_{-}$ of two antisymmetric states. We choose the
(separable) ansatz [51]
$\tau_{-,-}=\frac{d+1}{2d}\rho_{+}\otimes\rho_{+}+\frac{d-1}{2d}\rho_{-}\otimes\rho_{-}$
which commutes with $\rho_{-}\otimes\rho_{-}$. We have

$\Tr((\rho_{-}\otimes\rho_{-})^{\alpha}\tau^{-\alpha}_{-,-}\sigma)\leq\left(\frac{d-1}{2d}\right)^{1-\alpha}=\Tr((\rho_{-}\otimes\rho_{-})^{\alpha}\tau^{1-\alpha}_{-,-})\,,$
(130)

where we used that
$\Lambda^{2}(\rho_{-}\otimes\rho_{-})=\frac{d-1}{2d}\left(\frac{2}{d(d-1)}\right)^{2}$
[60]. Hence, the condition of Corollary 5 is satisfied and the ansatz is a
solution. We then have for $(\alpha,z)\in\mathcal{D}$

$\displaystyle\mathfrak{D}_{\alpha,z}(\rho_{-})=1\,,\;\text{and}$ (131)
$\displaystyle\mathfrak{D}_{\alpha,z}(\rho_{-}\otimes\rho_{-})=1-\log{\frac{(d-1)}{d}}\,.$
(132)

Hence, additivity is violated for $d>2$. Note that for $d\gg 1$ we have that
$\mathfrak{D}_{\alpha,z}(\rho_{-})\sim\mathfrak{D}_{\alpha,z}(\rho_{-}\otimes\rho_{-})$.
From Theorem 3 in [17] it follows that any monotone defined as
$\min_{\sigma\in\operatorname{SEP}}\mathbb{D}(\rho\|\sigma)$, where
$\mathbb{D}$ is a quantum relative entropy, satisfies
$\mathfrak{D}_{\min}(\rho)\leq\min_{\sigma\in\operatorname{SEP}}\mathbb{D}(\rho\|\sigma)\leq\mathfrak{D}_{\max}(\rho)$.
We recall that we have the limits (11). Hence, since the results in equation
(131) and (132) do not depend on $\alpha$ and $z$, we have that additivity is
violated for any monotone based on a quantum relative entropy.

######  Remark 3.

In the literature the minimization in (9) over the separable states is
sometimes replaced by the minimization over the states with positive partial
transpose (PPT) [36, 2]. The latter set is usually much easier to
characterise. We define

$\mathfrak{D}^{\text{PPT}}_{\alpha,z}(\rho):=\min_{\sigma\in\text{PPT}}D_{\alpha,z}(\rho\|\sigma)\,.$
(133)

Here, PPT denotes the set of states with positive partial transpose. We have
for $(\alpha,z)\in\mathcal{D}$

$\displaystyle\mathfrak{D}^{\text{PPT}}_{\alpha,z}(\rho_{-})=1\,,\;\text{and}$
(134)
$\displaystyle\mathfrak{D}^{\text{PPT}}_{\alpha,z}(\rho_{-}\otimes\rho_{-})\leq
1-\log{\frac{(d-1)}{d}}\,.$ (135)

Indeed, following Section V.2, to calculate the value of
$\mathfrak{D}^{\text{PPT}}_{\alpha,z}$ for Werner states (and in particular
for $\rho_{-}$), we can restrict the optimization to $\text{PPT}\cap W$
states. Since all entangled Werner states have negative partial transpose, the
optimizations over PPT and SEP give the same value and hence (134) holds. The
inequality in (135) follows from SEP $\subseteq$ PPT. Hence, also in this
case, the monotones $\mathfrak{D}^{\text{PPT}}_{\alpha,z}$ for
$(\alpha,z)\in\mathcal{D}$ are not additive for $d>2$. As we discussed above,
this means that also any monotone
$\min_{\sigma\in\text{PPT}}\mathbb{D}(\rho\|\sigma)$, where $\mathbb{D}$ is a
quantum relative entropy, is not additive for general states. This result has
already been established for $\alpha=z=1$ in [35] where a different
counterexample is provided.

## ACKNOWLEDGMENTS

We thank Paolo Perinotti, Erkka Happasalo, and Ryuji Takagi for discussion.
Part of the work was conducted while R.R was hosted by the QUIT Group at the
University of Pavia and M.T. was visiting the Pauli Centre for Theoretical
studies at ETH Zurich. This research is supported by the National Research
Foundation, Prime Minister’s Office, Singapore and the Ministry of Education,
Singapore under the Research Centres of Excellence programme. MT is also
supported in part by NUS startup grants (R-263-000-E32-133 and
R-263-000-E32-731).

## Appendix A: Lower semicontinuity and its consequences

In this appendix we show that the $\alpha$-$z$ Rényi entropy of entanglement
are lower semicontinuous and hence the infimum of $\mathfrak{D}_{\alpha,z}$ is
always achieved in the set of separable states. Moreover, we show that the
monotones $\mathfrak{D}$ and $\mathfrak{D}_{\max}$ can be obtained by taking
the limits $\alpha\rightarrow 1$ ($z\neq 0$) and $\alpha=z\rightarrow\infty$
of $\mathfrak{D}_{\alpha,z}$, respectively. More in general, we prove that
latter property holds also when the set of separable states is replaced by any
set of free states. All our considerations are in finite dimensions. We follow
similar arguments to the ones given in [30] and [27].

It is a straightforward fact that the $D_{\alpha,z}$ are monotonically
decreasing in the second argument in the range where they satisfy the data-
processing inequality.

######  Lemma 17.

Let $(\alpha,z)\in\mathcal{D}$, $\rho\in\mathcal{S}(A)$, and
$\sigma,\sigma^{\prime}\in\mathcal{P}(A)$. Then if $\sigma\leq\sigma^{\prime}$
we have

$\displaystyle D_{\alpha,z}(\rho\|\sigma)\geq
D_{\alpha,z}(\rho\|\sigma^{\prime})\,.$ (136)

###### Proof.

We first notice that for $\alpha<1\wedge(\alpha,z)\in\mathcal{D}$ we have
$(1-\alpha)/z\in(0,1]$. Moreover, for $\alpha>1\wedge(\alpha,z)\in\mathcal{D}$
we have $(1-\alpha)/z\in[-1,0)$.

Let $\alpha<1\wedge(\alpha,z)\in\mathcal{D}$. We then have

$\displaystyle\sigma\leq\sigma^{\prime}\implies\sigma^{\frac{1-\alpha}{z}}\leq{\sigma^{\prime}}^{\frac{1-\alpha}{z}}\implies\Tr[(\rho^{\frac{\alpha}{2z}}\sigma^{\frac{1-\alpha}{z}}\rho^{\frac{\alpha}{2z}})^{z}]\leq\Tr[(\rho^{\frac{\alpha}{2z}}{\sigma^{\prime}}^{\frac{1-\alpha}{z}}\rho^{\frac{\alpha}{2z}})^{z}]\,.$
(137)

In the first implication we used that the power is operator monotone in the
range $(0,1]$. In the last implication we used that the trace functional
$M\rightarrow\Tr(f(M))$ inherits the monotonicity from $f$ (see e.g. [10]).
Therefore, in this range, we get $D_{\alpha,z}(\rho\|\sigma)\geq
D_{\alpha,z}(\rho\|\sigma^{\prime})$. The proof for
$\alpha>1\wedge(\alpha,z)\in\mathcal{D}$ follows in the same way by noticing
that the power is operator antimonotone in the range $[-1,0)$. The property
for the relative entropy follows by taking, for example, the limit
$\alpha=z\rightarrow 1$. Therefore, we proved the lemma for all
$(\alpha,z)\in\mathcal{D}$. ∎

######  Lemma 18.

The $\alpha$-$z$ Rényi relative entropy $(\rho,\sigma)\rightarrow
D_{\alpha,z}(\rho\|\sigma)$ is lower semicontinuous for all
$(\alpha,z)\in\mathcal{D}$.

###### Proof.

For any $\varepsilon>0$, we have the operator inequality
$\sigma\leq\sigma+\varepsilon\mathds{1}$ where $\varepsilon>0$. From the
previous lemma we have $D_{\alpha,z}(\rho\|\sigma)\geq
D_{\alpha,z}(\rho\|\sigma+\varepsilon\mathds{1})$ and hence we can write

$D_{\alpha,z}(\rho\|\sigma)=\sup\limits_{\varepsilon>0}D_{\alpha,z}(\rho\|\sigma+\varepsilon\mathds{1})\,.$
(138)

For any fixed $\varepsilon>0$, the functions $(\rho,\sigma)\rightarrow
D_{\alpha,z}(\rho\|\sigma+\varepsilon\mathds{1})$ are continuous since the
second argument has full support. Since the pointwise supremum of continuous
functions is lower semicontinuous, the function $(\rho,\sigma)\rightarrow
D_{\alpha,z}(\rho\|\sigma)$ is lower semicontinuous. ∎

A direct consequence of the above lemma is that
$\inf_{\sigma\in\mathcal{F}}D_{\alpha,z}(\rho\|\sigma)=\min_{\sigma\in\mathcal{F}}D_{\alpha,z}(\rho\|\sigma)$,
i.e. the infimum is always achieved. Indeed, since the set of quantum states
is compact (with respect, for example, to the trace norm topology), the set
$\mathcal{F}$ is also compact being a closed subset of a compact set. The
statement then follows from the fact that a lower semicontinuous function on a
compact set has a minimum on the compact set.

Moreover, lower semicontinuity implies that

$\lim\limits_{\alpha\rightarrow\infty}\min\limits_{\sigma\in\mathcal{F}}D_{\alpha,\alpha}(\rho\|\sigma)=\sup\limits_{\alpha>1}\inf\limits_{\sigma\in\mathcal{F}}D_{\alpha,\alpha}(\rho\|\sigma)=\inf\limits_{\sigma\in\mathcal{F}}\sup\limits_{\alpha>1}D_{\alpha,\alpha}(\rho\|\sigma)=\min\limits_{\sigma\in\mathcal{F}}\lim\limits_{\alpha\rightarrow\infty}D_{\alpha,\alpha}(\rho\|\sigma)\,.$
(139)

i.e. we can exchange the limit with the minimum. The first equality follows
from the fact that the sandwiched Rényi divergence is monotone in $\alpha$ and
hence we can replace the limit with the supremum over $\alpha>1$. The second
inequality follows from the minimax theorem in [29, Lemma II.1] since, as we
discussed above, the $D_{\alpha,\alpha}$ is lower semicontinuous in the second
argument. Moreover, by the minimax theorem, in the last equality, the infimum
over the free states can be replaced with the minimum. In a similar way, we
can replace the following limits with the infimum or supremum to obtain

  1. 1.

$\lim\limits_{z\rightarrow\infty}\min\limits_{\sigma\in\mathcal{F}}D_{\alpha,z}(\rho\|\sigma)=\min\limits_{\sigma\in\mathcal{F}}\lim\limits_{z\rightarrow\infty}D_{\alpha,z}(\rho\|\sigma)\;$
for $\;\alpha<1$.

  2. 2.

$\lim\limits_{z\rightarrow\alpha-1}\min\limits_{\sigma\in\mathcal{F}}D_{\alpha,z}(\rho\|\sigma)=\min\limits_{\sigma\in\mathcal{F}}\lim\limits_{z\rightarrow\alpha-1}D_{\alpha,z}(\rho\|\sigma)\;$
for $\;\alpha>1$.

  3. 3.

$\lim\limits_{z\rightarrow
1-\alpha}\min\limits_{\sigma\in\mathcal{F}}D_{\alpha,z}(\rho\|\sigma)=\min\limits_{\sigma\in\mathcal{F}}\lim\limits_{z\rightarrow
1-\alpha}D_{\alpha,z}(\rho\|\sigma)\;$ for $\;\alpha\in(0,1)$.

  4. 4.

$\lim\limits_{\alpha\rightarrow
0}\min\limits_{\sigma\in\mathcal{F}}D_{\alpha,1}(\rho\|\sigma)=\min\limits_{\sigma\in\mathcal{F}}\lim\limits_{\alpha\rightarrow
0}D_{\alpha,1}(\rho\|\sigma)$.

  5. 5.

$\lim\limits_{\alpha\rightarrow
1}\min\limits_{\sigma\in\mathcal{F}}D_{\alpha,1}(\rho\|\sigma)=\min\limits_{\sigma\in\mathcal{F}}\lim\limits_{\alpha\rightarrow
1}D_{\alpha,1}(\rho\|\sigma)$ .

The first two equalities follows from the minimax theorem [29, Lemma II.1] and
the fact that the function $z\mapsto D_{\alpha,z}$ is monotonically increasing
for $\alpha<1$ and monotonically decreasing for $\alpha>1$ [28]. The third
equality follows from the monotonicity just mentioned. The last two equalities
follows from the monotonicity in $\alpha$ of the Petz Rényi divergence [45].

## Appendix B: Relationship between conditional entropies, entanglement
monotones and coherence monotones based on $\alpha$-$z$ Rényi divergences

In this appendix we derive some relationships between the conditional entropy
and the entanglement monotone on the $\alpha$-$z$ Rényi divergence. Moreover,
we derive some constraints on the structure of an optimizer of
$\mathfrak{D}_{\alpha,z}$ for maximally correlated states which constitutes a
fundamental ingredient to establish the additivity result of Theorem 9.

Let $\rho_{AB}\in\mathcal{S}(AB)$ a bipartite state. We define the
$\alpha$-$z$ Rényi conditional entropy as

$H^{\uparrow}_{\alpha,z}(A|B)_{\rho_{AB}}:=-\min\limits_{\sigma_{B}\in\mathcal{S}(B)}D_{\alpha,z}(\rho_{AB}\|I_{A}\otimes\sigma_{B})\,.$
(140)

We also denote $H_{\alpha,z}(A)_{\rho_{AB}}:=H(\rho_{A})$ where $H(\rho_{A})$
is the von Neumann entropy of the marginal $\rho_{A}=\Tr(\rho_{AB})$. We often
omit the subscript $AB$ when it is clear from the context.

We now show that for maximally correlated states, the $\alpha$-$z$ Rényi
conditional entropy are equal to the $\alpha$-$z$ Rényi relative entropy of
entanglement. We follow similar arguments to the ones given in [61].

######  Theorem 19.

Let $\rho\in\mathcal{S}(AB)$ be a maximally correlated state and
$(\alpha,z)\in\mathcal{D}$. Then we have

$\mathfrak{D}_{\alpha,z}(\rho)=-H^{\uparrow}_{\alpha,z}(A|B)_{\rho}\,.$ (141)

###### Proof.

We define the CPTP map $\Lambda$ by $\Lambda(\rho):=P\rho P+(1-P)\rho(1-P)$
where $P=\sum_{j}|jj\rangle\\!\langle jj|$. We have for a maximally correlated
state $\rho$

$\displaystyle\mathfrak{D}_{\alpha,z}(\rho)$
$\displaystyle\geq\min\limits_{\sigma_{B}\in\mathcal{S}(B)}D_{\alpha,z}(\rho\|I_{A}\otimes\sigma_{B})$
(142)
$\displaystyle\geq\min\limits_{\sigma_{B}\in\mathcal{S}(B)}D_{\alpha,z}(\rho\|P(I_{A}\otimes\sigma_{B})P+(1-P)(I_{A}\otimes\sigma_{B})(1-P))$
(143)
$\displaystyle\geq\min\limits_{\sigma_{B}\in\mathcal{S}(B)}D_{\alpha,z}(\rho\|P(I_{A}\otimes\sigma_{B})P)$
(144)
$\displaystyle\geq\min\limits_{\sigma\in\mathcal{T}_{\rho}}D_{\alpha,z}(\rho\|\sigma)\,,$
(145)

where $\mathcal{T}_{\rho}$ is the set of all separable states of the form
$\sigma=\sum_{i}s_{i}|i,i\rangle\\!\langle i,i|$. Here, $|i,i\rangle$ is the
basis such that $\rho=\sum_{jk}\rho_{jk}|j,j\rangle\\!\langle k,k|$. The first
inequality comes from the fact that for any separable state $\sigma_{AB}\leq
I_{A}\otimes\sigma_{B}$ [21] and Lemma 17. In the second inequality we used
the data-processing inequality and that a maximally correlated state is
invariant under the projection $P$, i.e. $\Lambda(\rho)=\rho$. In (144) we
used that $(1-P)(I_{A}\otimes\sigma_{B})(1-P)$ is orthogonal to $\rho$ and
hence it does not contribute to the value of $D_{\alpha,z}$. The last
inequality is a consequence of the fact that
$P(I_{A}\otimes\sigma_{B})P=\sum_{i}(\sigma_{B})_{i,i}|i,i\rangle\\!\langle
i,i|$ belongs to $\mathcal{T}_{\rho}$. Moreover, since
$\mathcal{T}_{\rho}\subseteq\operatorname{SEP}$, we have
$\min_{\sigma\in\mathcal{T}_{\rho}}D_{\alpha,z}(\rho\|\sigma)\geq\mathfrak{D}_{\alpha,z}(\rho)$
and hence
$\min_{\sigma\in\mathcal{T}_{\rho}}D_{\alpha,z}(\rho\|\sigma)=\mathfrak{D}_{\alpha,z}(\rho)$.
The latter condition implies that the inequality (142) is an equality and
equation (141) holds. ∎

The latter proof implies that for a maximally correlated state there always
exists an optimizer of $\mathfrak{D}_{\alpha,z}$ in the set
$\mathcal{T}_{\rho}$.

######  Remark 4.

In general, the value of the coherence monotones depends on the specific
choice of the coherence basis. Note that, if we set the coherence basis equal
to the basis $\\{|i,j\rangle\\}$ for which the maximally correlated state
reads $\rho_{\textup{MC}}=\sum_{jk}\rho_{jk}|j,j\rangle\\!\langle k,k|$,
following the same arguments as above, we have that
$\mathfrak{D}_{\alpha,z}(\rho)=\mathfrak{D}^{\mathcal{I}}_{\alpha,z}(\rho)$.
Moreover, because of the invariance of the underlying relative entropy under
the isometry $|i\rangle\rightarrow|i,i\rangle$, we have that the latter
quantity is equal to
$\mathfrak{D}^{\mathcal{I}}_{\alpha,z}\big{(}\sum_{i,j}\rho_{jk}|j\rangle\\!\langle
k|\big{)}$, where now the minimization runs over the states diagonal in the
marginal basis $\\{|i\rangle\\}$.

Proposition 13 together with Theorem 19 gives

######  Lemma 20.

Let $(\alpha,z)\in\mathcal{D}$ and $\rho_{AB}\in\mathcal{S}(AB)$ a bipartite
pure state. Then we have

$H^{\uparrow}_{\alpha,z}(A|B)_{\rho}+H_{\beta}(A)_{\rho}=0\;,\quad\text{where}\;\;\frac{1}{\beta}+\frac{1-\alpha}{z}=1\,.$
(146)

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