Approximate bound state solutions of the deformed Woods-Saxon potential using
asymptotic iteration method

Babatunde J. Falaye111E-mail<EMAIL_ADDRESS>
Theoretical Physics Section, Department of Physics
University of Ilorin, P. M. B. 1515, Ilorin, Nigeria.

Majid Hamzavi222E-mail<EMAIL_ADDRESS>
Department of Basic Sciences, Shahrood Branch,
Islamic Azad University, Shahrood, Iran

Sameer M. Ikhdair333E-mail<EMAIL_ADDRESS>
Physics Department, Near East University,
922022 Nicosia, North Cyprus, Mersin 10, Turkey

###### Abstract

By using the Pekeris approximation, the Schr$\ddot{o}$dinger equation is
approximately solved for the nuclear deformed Woods-Saxon potential within the
framework of the asymptotic iteration method. The energy levels are worked out
and the corresponding normalized eigenfunctions are obtained in terms of
hypergeometric function.

Keywords: Schr$\ddot{o}$dinger equation; nuclear deformed Woods-Saxon
potential; asymptotic

iteration method.

PACs No. 03.65.Pm; 03.65.Ge; 03.65.-w; 03.65.Fd; 02.30.Gp

##  1 Introduction

The deformed Woods-Saxon (dWS) potential is a short range potential and widely
used in nuclear, particle, atomic, condensed matter and chemical physics
[1-7]. This potential is reasonable for nuclear shell models and used to
represent the distribution of nuclear densities. The dWS and spin-orbit
interaction are important and applicable to deformed nuclei [8] and to
strongly deformed nuclides [9]. The dWS potential parameterization at large
deformations for plutonium ${}^{237,239,241}Pu$ odd isotopes was analyzed
[10]. The structure of single-particle states in the second minima of
${}^{237,239,241}Pu$ has been calculated with an exactly dWS potential. The
Nuclear shape was parameterized.

The parameterization of the spin-orbit part of the potential was obtained in
the region corresponding to large deformations (second minima) depending only
on the nuclear surface area. The spin-orbit interaction of a particle in a
non-central self consistent field of the WS type potential was investigated
for light nuclei and the scheme of single-particle states has been found for
mass number $A_{o}=10$ and $25$ [8]. Two parameters of the spin-orbit part of
the dWS potential, namely the strength parameter and radius parameter were
adjusted to reproduce the spins for the values of the nuclear deformation
parameters [11].

Badalov et al. investigated Woods-Saxon potential in the framework of
Schr$\ddot{o}$dinger and Klein-Gordon equations by means of Nikiforov-Uvarov
method [12, 13]. In our recent works [14, 15], we have studied the
relativistic Duffin-Kemmer-Petiau and the Dirac equation for dWS potential. In
addition, we have also obtained the bound state solutions of the $PT-/non-
PT$-symmetric and non-Hermitian modified Woods-Saxon potential with the real
and complex-valued energy levels [16].

In this paper we will study Schr$\ddot{o}$dinger equation with dWS potential
for any arbitrary orbital quantum number $\ell$. We obtain the analytical
expressions for the energy levels and wave functions in closed form.
Therefore, this work is arranged as follows: In section $2$, a brief
introduction to asymptotic iteration method (AIM) is given. In section $3$,
the Schr$\ddot{o}$dinger equation with dWS potential is solved. Finally,
results and conclusions are presented in section $4$.

##  2 The Asymptotic Iteration Method

We briefly outline the AIM here; the details can be found in references [17,
18].

###  2.1 Energy eigenvalues

AIM is proposed to solve the homogenous linear second-order differential
equation of the form

$y_{n}^{\prime\prime}(x)=\lambda_{o}(x)y_{n}^{\prime}(x)+s_{o}(x)y_{n}(x),$
(1)

where $\lambda_{o}(x)\neq 0$ and the prime denotes the derivative with respect
to $x$, the extral parameter $n$ is thought as a radial quantum number. The
variables, $s_{o}(x)$ and $\lambda_{o}(x)$ are sufficiently differentiable. To
find a general solution to this equation, we differentiate equation (1) with
respect to $x$ as

$y_{n}^{\prime\prime\prime}(x)=\lambda_{1}(x)y_{n}^{\prime}(x)+s_{1}(x)y_{n}(x),$
(2)

where

$\displaystyle\lambda_{1}(x)$ $\displaystyle=$
$\displaystyle\lambda_{o}^{\prime}(x)+s_{o}(x)+\lambda_{o}^{2}(x),$
$\displaystyle s_{1}(x)$ $\displaystyle=$ $\displaystyle
s_{o}^{\prime}(x)+s_{o}(x)\lambda_{o}(x),$ (3)

and the second derivative of equation (1) is obtained as

$y_{n}^{\prime\prime\prime\prime}(x)=\lambda_{2}(x)y_{n}^{\prime}(x)+s_{2}(x)y_{n}(x),$
(4)

where

$\displaystyle\lambda_{2}(x)$ $\displaystyle=$
$\displaystyle\lambda_{1}^{\prime}(x)+s_{1}(x)+\lambda_{o}(x)\lambda_{1}(x),$
$\displaystyle s_{2}(x)$ $\displaystyle=$ $\displaystyle
s_{1}^{\prime}(x)+s_{o}(x)\lambda_{1}(x).$ (5)

Equation (1) can be iterated up to $(k+1)th$ and $(k+2)th$ derivatives,
$k=1,2,3...$ Therefore we have

$\displaystyle y_{n}^{(k+1)}(x)$ $\displaystyle=$
$\displaystyle\lambda_{k-1}(x)y_{n}^{\prime}(x)+s_{k-1}(x)y_{n}(x),$
$\displaystyle y_{n}^{(k+2)}(x)$ $\displaystyle=$
$\displaystyle\lambda_{k}(x)y_{n}^{\prime}(x)+s_{k}(x)y_{n}(x),$ (6)

where

$\displaystyle\lambda_{k}(x)$ $\displaystyle=$
$\displaystyle\lambda_{k-1}^{\prime}(x)+s_{k-1}(x)+\lambda_{o}(x)\lambda_{k-1}(x),$
$\displaystyle s_{k}(x)$ $\displaystyle=$ $\displaystyle
s_{k-1}^{\prime}(x)+s_{o}(x)\lambda_{k-1}(x).$ (7)

From the ratio of the (k+2)th and (k+1)th derivatives, we obtain

$\frac{d}{dx}ln\left[y_{n}^{k+1}(x)\right]=\frac{y_{n}^{(k+2)}(x)}{y_{n}^{(k+1)}(x)}=\frac{\lambda_{k}(x)\left[y_{n}^{\prime}(x)+\frac{s_{k}(x)}{\lambda_{k}(x)}y_{n}(x)\right]}{\lambda_{k-1}(x)\left[y_{n}^{\prime}(x)+\frac{s_{k-1}(x)}{\lambda_{k-1}(x)}y_{n}(x)\right]},$
(8)

if $k>0$, for sufficiently large $k$, we obtain $\alpha$ values from

$\frac{s_{k}(x)}{\lambda_{k}(x)}=\frac{s_{k-1}(x)}{\lambda_{k-1}(x)}=\alpha(x),$
(9)

with quantization condition

$\delta_{k}(x)=\left|\begin{array}[]{lr}\lambda_{k}(x)&s_{k}(x)\\\
\lambda_{k-1}(x)&s_{k-1}(x)\end{array}\right|=0\ \ ,\ \ \ k=1,2,3....$ (10)

Then equation (8) reduces to

$\frac{d}{dx}ln\left[y_{n}^{(k+1)}(x)\right]=\frac{\lambda_{k}(x)}{\lambda_{k-1}(x)},$
(11)

which yields the general solution of Eq. (1)

$y_{n}(x)=\exp\left(-\int^{x}\alpha(x^{\prime})dx^{\prime}\right)\left[C_{2}+C_{1}\int^{x}\exp\left(\int^{x^{\prime}}\left[\lambda_{o}(x^{\prime\prime})+2\alpha(x^{\prime\prime})\right]dx^{\prime\prime}\right)dx^{\prime}\right].$
(12)

For a given potential, the idea is to convert the radial Schr$\ddot{o}$dinger
equation to the form of equation (1). Then $\lambda_{o}(x)$ and $s_{o}(x)$ are
determine and $s_{k}(x)$ and $\lambda_{k}(x)$ parameters are calculated by the
recurrence relations given by equation (7). The energy eigenvalues are then
obtained by the condition given by equation (10) if the problem is exactly
solvable.

###  2.2 Energy eigenfunction

Suppose we wish to solve the radial Schr$\ddot{o}$dinger equation for which
the homogenous linear second-order differential equation takes the following
general form

$y^{\prime\prime}(x)=2\left(\frac{tx^{N+1}}{1-bx^{N+2}}-\frac{m+1}{x}\right)y^{\prime}(x)-\frac{Wx^{N}}{1-bx^{N+2}}.$
(13)

The exact solution $y_{n}(x)$ can be expressed as [18]

$y_{n}(x)=(-1)^{n}C_{2}(N+2)^{n}(\sigma)_{{}_{n}}{{}_{2}F_{1}(-n,\rho+n;\sigma;bx^{N+2})},$
(14)

where the following notations has been used

$(\sigma)_{{}_{n}}=\frac{\Gamma{(\sigma+n)}}{\Gamma{(\sigma)}}\ \ ,\ \
\sigma=\frac{2m+N+3}{N+2}\ \ and\ \ \rho=\frac{(2m+1)b+2t}{(N+2)b}.$ (15)

##  3 Any $\ell$-state solutions

The deformed Woods-Saxon potential we investigate in this study is defined as
[14, 15, 19]

$V(r)=-\frac{V_{o}}{1+q\exp{(\frac{r-R}{a})}},\ \ \ \ R=r_{o}A_{o}^{1/3},\ \ \
\ V_{o}=(40.5+0.13A_{o})MeV,\ \ \ \ R>>a,\ \ \ \ q>0,$ (16)

where $V_{o}$ is the depth of potential, $q$ is a real parameter which
determines the shape (deformation) of the potential, $a$ is the diffuseness of
the nuclear surface, $R$ is the width of the potential, $A_{o}$ is the atomic
mass number of target nucleus and $r_{o}$ is radius parameter. By inserting
this potential into the Schr$\ddot{o}$dinger equation [17, 18] as

$\left(-\frac{\hbar^{2}}{2\mu}\left[\frac{1}{r^{2}}\frac{\partial}{\partial
r}r^{2}\frac{\partial}{\partial
r}+\frac{1}{r^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}\right]+V(r)\right)\Psi_{n\ell
m}(r)=E\Psi_{n\ell m}(r),$ (17)

and setting the wave functions $\Psi_{n\ell m}(r)=R_{n\ell}(r)Y_{\ell
m}(\theta,\phi)r^{-1}$, we obtain the radial part of the equation as

$\left[\frac{d^{2}}{dr^{2}}+\frac{2\mu}{\hbar^{2}}\left(E_{n\ell}+\frac{V_{o}}{1+q\exp{(\frac{r-R}{a})}}\right)-\frac{\ell(\ell+1)}{r^{2}}\right]R_{n\ell}(r)=0.$
(18)

Because of the total angular momentum centrifugal term, equation (18) cannot
be solved analytically for $\ell\neq 0$. Therefore, we shall use the Pekeris
approximation in order to deal with this centrifugal term and we may express
it as follows[20, 21]

$U_{cent.}(r)=\frac{1}{r^{2}}=\frac{1}{R^{2}\left(1+\frac{x}{R}\right)^{2}}\cong\frac{1}{R^{2}}\left(1-2\left(\frac{x}{R}\right)+3\left(\frac{x}{R}\right)^{2}+\cdots\right),$
(19)

with $x=r-R$. In addition, we may also approximately express it in the
following way

$\tilde{U}=\frac{1}{r^{2}}\cong\frac{1}{R^{2}}\left[D_{o}+\frac{D_{1}}{1+q\exp{(\nu
x)}}+\frac{D_{2}}{\left(1+q\exp{(\nu x)}\right)^{2}}\right],$ (20)

where $\nu=1/a$. After expanding (20) in terms of $x$, $x^{2}$,
$x^{3}$,$\cdots$ and next, comparing with equation (19), we obtain expansion
coefficients $D_{o}$, $D_{1}$ and $D_{2}$ as follows:

$\displaystyle D_{o}$ $\displaystyle=$ $\displaystyle 1-\frac{(1+q)^{2}}{\nu
Rq^{2}}\left(1-\frac{3}{\nu R}\right),$ $\displaystyle D_{1}$ $\displaystyle=$
$\displaystyle\frac{(1+q)^{2}}{\nu Rq^{2}}\left(-1+3q-\frac{6(1+q)}{\nu
R}\right),$ (21) $\displaystyle D_{2}$ $\displaystyle=$
$\displaystyle\frac{(1+q)^{3}}{\nu
Rq^{2}}\left(\frac{1-q}{2}+\frac{3(1+q)}{\nu R}\right).$

Now, inserting the approximation expression (20) into equation (18) and
changing the variables $r\rightarrow z$ through the mapping function
$z=(1+q\exp{(\nu x)})^{-1}$, equation (18) turns to

$\displaystyle\frac{d^{2}R_{n\ell}(z)}{dz^{2}}+\frac{1-2z}{z(1-z)}\frac{dR_{n\ell}(z)}{dz}+\frac{1}{\left[\nu
z(1-z)\right]^{2}}$ $\displaystyle\times\left[\left(\frac{2\mu
E_{n\ell}}{\hbar^{2}}-\frac{D_{o}\ell(\ell+1)}{R^{2}}\right)-z\left(-\frac{2\mu
V_{o}}{\hbar^{2}}+\frac{D_{1}\ell(\ell+1)}{R^{2}}\right)-\frac{D_{2}z^{2}\ell(\ell+1)}{R^{2}}\right]R_{n\ell}(z)=0.$
(22)

Before applying the AIM to this problem, we have to obtain the asymptotic wave
functions and then transform equation (22) into a suitable form of the AIM.
This can be achieved by the analysis of the asymptotic behaviours at the
origin and at infinity. As a result the boundary conditions of the wave
functions $R_{n\ell}(z)$ are taken as follows:

$\displaystyle R_{n\ell}(z)$ $\displaystyle\rightarrow$ $\displaystyle 0\ \ \
\ when\ \ \ \ z\rightarrow 1,$ $\displaystyle R_{n\ell}(z)$
$\displaystyle\rightarrow$ $\displaystyle 0\ \ \ \ when\ \ \ \ z\rightarrow
0,$ (23)

thus, one can write the wave functions for this problem as

$R_{n\ell}(z)=z^{\alpha}(1-z)^{\gamma}F_{n\ell}(z),$ (24)

where we have introduced parameters $\alpha$ and $\gamma$ defined by

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{1}{\nu}\left[\frac{D_{o}\ell(\ell+1)}{R^{2}}-\frac{2\mu
E_{n\ell}}{\hbar^{2}}\right]^{\frac{1}{2}},$ $\displaystyle\gamma$
$\displaystyle=$
$\displaystyle\frac{1}{\nu}\left[-\frac{2\mu}{\hbar^{2}}(V_{o}+E_{n\ell})+\frac{\ell(\ell+1)}{R^{2}}(D_{o}+D_{1}+D_{2})\right]^{\frac{1}{2}},$
(25)

for simplicity. By substituting equation (24) into equation (22), we have the
second-order homogeneous differential equation of the form:

$F_{n\ell}^{\prime\prime}(z)+\left[\frac{(2\alpha+1)-2z(\alpha+\gamma+1)}{z(1-z)}\right]F_{n\ell}^{\prime}(z)-\left[\frac{\alpha^{2}+\gamma^{2}+\alpha+\gamma+2\alpha\gamma-\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}}{z(1-z)}\right]F_{n\ell}(z)=0,$
(26)

which is now suitable to an AIM solutions. By comparing this equation with
equation (1), we can write the $\lambda_{o}(z)$ and $s_{o}(z)$ values and
consequently; by means of equation (7), we may derive the $\lambda_{k}(z)$ and
$s_{k}(z)$ as follows:

$\displaystyle\lambda_{o}(z)$ $\displaystyle=$
$\displaystyle\frac{2z(\alpha+\gamma+1)-(2\alpha+1)}{z(1-z)},$ $\displaystyle
s_{o}(z)$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}+\gamma^{2}+\alpha+\gamma+2\alpha\gamma-\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}}{z(1-z)},$
$\displaystyle\lambda_{1}(z)$ $\displaystyle=$
$\displaystyle\frac{2+z\left(2-3\gamma+\gamma^{2}-\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}\right)-z^{2}\left(7\gamma-3\gamma^{2}-2-\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}\right)}{z^{2}(1-z)^{2}}$
$\displaystyle\frac{[a(z-1)(z(6\gamma-7)-6)]+a^{2}(4-7z+3z^{2})}{z^{2}(1-z)^{2}}$
$\displaystyle s_{1}(z)$ $\displaystyle=$
$\displaystyle\frac{2\left[(\alpha+\gamma)^{2}+\alpha+\gamma-\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}\right]\left[\gamma
z-(1+\alpha(1-z))\right]}{z^{2}(1-z)^{2}}$ $\displaystyle\ldots etc.$

The substitution of the above equations into equation (10), we obtain the
first $\delta$ values as

$\delta_{o}(z)=\frac{\left[(\alpha+\gamma)^{2}+(\alpha+\gamma)-\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}\right]\left[\alpha(3+2\gamma)+\gamma(3+\gamma)-2+\alpha^{2}-\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}\right]}{z^{2}(1-z)^{2}}.$
(28)

From the root of equation (28), we obtain the first relation between $\alpha$
and $\gamma$ as
$\gamma_{o}+\alpha_{o}=-\frac{1}{2}-\frac{1}{2}\sqrt{1+4\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}}$.
In a similar fashion, we can obtain other $\delta$ values and consequently
establish a relationship between $\alpha_{n}$ and $\gamma_{n}$, $n=1,2,3,$
$\cdots$ as

$\displaystyle\delta_{1}(z)=\left|\begin{array}[]{lr}\lambda_{2}(z)&s_{2}(z)\\\
\lambda_{1}(z)&s_{1}(z)\end{array}\right|=0\ \ \ \ \Rightarrow\ \ \ \
\gamma_{1}+\alpha_{1}=-\frac{3}{2}-\frac{1}{2}\sqrt{1+4\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}}$
(31)
$\displaystyle\delta_{2}(z)=\left|\begin{array}[]{lr}\lambda_{3}(z)&s_{3}(z)\\\
\lambda_{2}(z)&s_{2}(z)\end{array}\right|=0\ \ \ \ \Rightarrow\ \ \ \
\gamma_{2}+\alpha_{2}=-\frac{5}{2}-\frac{1}{2}\sqrt{1+4\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}}$
(34) $\displaystyle\ldots etc.$ (35)

The nth term of the above arithmetic progression is found to be

$\alpha_{n}+\gamma_{n}=-\frac{2n+1}{2}-\frac{1}{2}\sqrt{1+4\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}}.$
(36)

By substituting for $\alpha$ and $\gamma$, we obtain a more explicit
expression for the eigenvalues energy as

$E_{n\ell}=\frac{\hbar^{2}D_{o}\ell(\ell+1)}{2\mu R^{2}}-\frac{\hbar^{2}}{8\mu
a^{2}}\left[\frac{\left(\frac{1}{2}\sqrt{1+4\frac{D_{2}a^{2}\ell(\ell+1)}{R^{2}}}+\frac{2n+1}{2}\right)^{2}+a^{2}\left(\frac{2\mu
V_{o}}{\hbar^{2}}-\frac{\ell(\ell+1)}{R^{2}}(D_{1}+D_{2})\right)}{\frac{1}{2}\sqrt{1+4\frac{D_{2}a^{2}\ell(\ell+1)}{R^{2}}}+\frac{2n+1}{2}}\right]^{2}.$
(37)

Let us now turn to the calculation of the normalized wave functions. By
comparing equation (26) with equation (13) we have the following:

$t=\frac{2\gamma+1}{2},\ \ \ \ b=1,\ \ \ \ N=-1,\ \ \ \
m=\frac{2\alpha-1}{2},\ \ \ \ \sigma=2\alpha+1,\ \ \ \
\rho=2(\alpha-\gamma)+1.$ (38)

Having determined these parameters, we can easily find the wave functions as

$F_{n\ell}(z)=(-1)^{n}C_{2}\frac{\Gamma(2\alpha+n+1)}{\Gamma{(2\alpha+1)}}\
_{2}F_{1}\left(-n,2(\alpha-\gamma)+1+n;2\alpha+1;z\right),$ (39)

where $\Gamma$ and ${}_{2}F_{1}$ are the Gamma function and hypergeometric
function respectively. By using equations (24) and (39), the total radial wave
function can be written as follows:

$R_{n\ell}(r)=(-1)^{n}N_{n\ell}\frac{\left[1+q^{-1}\exp{\left(\frac{R-r}{a}\right)}\right]^{\gamma}}{\left[1+q\exp{\left(\frac{r-R}{a}\right)}\right]^{\alpha}}\
{{}_{2}F_{1}\left(-n,2(\alpha-\gamma)+1+n;2\alpha+1;\left(1+q\exp{\left(\frac{r-R}{a}\right)}\right)^{-1}\right)},$
(40)

where $N_{n\ell}$ is the normalization constant

Figure 1: The variation of energy spectrum (37) as a function of the
deformation parameter. For example we select $\mu=1fm^{-1}$,
$V_{o}=0.3fm^{-1}$, $R=7fm$ and $a=0.65fm$. The radial quantum number is fixed
to $n=0$ Figure 2: The variation of energy spectrum (37) as a function of the
deformation parameter. For example we select $\mu=1fm^{-1}$,
$V_{o}=0.3fm^{-1}$, $R=7fm$ and $a=0.65fm$. The radial quantum number is fixed
to $n=1$.

##  4 Results and Conclusion

In Figures (1) and (2) we plot the energy levels $E_{n\ell}$ versus
deformation constant $q$ for ground $n=0$ and first exited $n=1$ states,
respectively. In Figure (1), it is seen that when $q\approx 0.6$, the orbital
quantum numbers $\ell=0,1,2,$ will have the same energy eigenvalues, i.e.,
$E\approx-0.165fm^{-1}$. For the case when $q<0.6$, we noticed that
$E_{00}>E_{01}>E_{02}$, whereas when $q>0.6$, then $E_{02}>E_{01}>E_{00}$
(less negative or attractive). In Figure (2), taking same parameter set, we
noticed that when $q\approx 1.0$, the three states have energy;
$E\approx-1.04fm^{-1}$. In the case when $q<1.0$, $E_{10}>E_{11}>E_{12}$ but
when $q>1.0$, $E_{12}>E_{11}>E_{10}$. The system becomes weakly attractive. We
have seen that the energy states are sensitive to the deformation constant
$q$. There are no physical energy states for $\ell=0,1,2$ when $q\approx 0.6$
for $n=0$ and when $q<1.0$ for $n=1$

Figure 3: The variation of energy spectrum (37) as a function of the
diffuseness of the nuclear surface. For example we select $\mu=1fm^{-1}$,
$V_{o}=0.3fm^{-1}$, $R=7fm$ and $q=1.5fm$. The radial quantum number is fixed
to $n=0$. Figure 4: The variation of energy spectrum (37) as a function of the
diffuseness of the nuclear surface. For example we select $\mu=1fm^{-1}$,
$V_{o}=0.3fm^{-1}$, $R=7fm$ and $q=1.5fm$. The radial quantum number is fixed
to $n=1$.

In Figures (3) and (4), we plot the energy levels $E_{n\ell}$ versus the
diffuseness of nuclear surface $a$ for $n=0$ and $n=1$ states, respectively.
For $n=0$ case, the energy eigenvalues increasing as $a$ increases, i.e.,
becoming weakly attractive ($E_{02}>E_{01}>E_{00}$). However, for $n=1$, the
case is same as former (i.e. $n=1$) but the three curves overlap each other.

Figure 5: The variation of energy spectrum (37) as a function of the potential
depth. For example we select $\mu=1fm^{-1}$, $a=0.65fm$, $R=7fm$ and
$q=1.5fm$. The radial quantum number is fixed to $n=0$. Figure 6: The
variation of energy spectrum (37) as a function of the potential depth. For
example we select $\mu=1fm^{-1}$, $a=0.65fm$ ,$R=0.7fm$ and $q=1.5fm$. The
radial quantum number is fixed to $n=1$.

In Figures (5) and (6), we plot the energy levels $E_{n\ell}$ versus the
potential depth potential depth $V_{o}$ for $n=0$ and $n=1$ states,
respectively. For $n=0$ case, we found that there is a substantial change in
the energy eigenvalues $E_{02}>E_{01}>E_{00}$, they increase with increasing
$V_{o}$ and becoming less attractive. However, the same behaviour is seen with
slow change in three curves when $n=1$.

Figure 7: The variation of energy spectrum (37) as a function of the potential
width. For example we select $\mu=1fm^{-1}$, $V_{o}=0.3fm^{-1}$, $a=0.65fm$
and $q=1.5$. The radial quantum number is fixed to $n=0$. Figure 8: The
variation of energy spectrum (37) as a function of the potential width. For
example we select $\mu=1fm^{-1}$, $V_{o}=0.3fm^{-1}$, $a=0.65fm$ and $q=1.5$.
The radial quantum number is fixed to $n=1$.

In Figures (7) and (8) we plot the energy levels $E_{n\ell}$ versus the
potential width $R$ for the two radial quantum numbers. At $R\approx 5.1fm$,
the energy is same (coincides). For $R<5.1fm$, $E_{00}>E_{01}>E_{02}$ (less
attractive) but when $R>5.1fm$, $E_{02}>E_{01}>E_{00}$. It is obvious that the
energy levels are sensitive to potential width $R$. There are no physical
energies when $R<5.1fm$ for $n=0$, and $R<6.2fm$ for $n=1$ . For $E\approx 0$,
the particles are no longer attracted. However, for $E_{01}$ and
$E_{00}\approx-0.2$. When $n=1$, $R\approx 6.2fm$, $E_{02}=E_{01}=E_{00}$. For
different values of $R$, the behavior is seen as former for $n=0$.

Figure 9: The variation of energy spectrum (37) as a function of the particle
mass. For example we select $V_{o}=0.3fm^{-1}$, $R=7fm$, $a=0.65fm$ and
$q=1.5$. The radial quantum number is fixed to $n=0$. Figure 10: The variation
of energy spectrum (37) as a function of the particle mass. For example we
select $V_{o}=0.3fm^{-1}$, $R=7fm$, $a=0.65fm$ and $q=1.5$. The radial quantum
number is fixed to $n=1$.

In Figures (9) and (10) we plot the energy levels $E_{n\ell}$ versus the
reduced mass $\mu$ for the two radial quantum numbers. In the ground level the
energy becomes repulsive (positive) as $\mu$ increases. For example, $E_{02}$
becomes positive for $\mu>2.2fm^{-1}$, $E_{01}>0$ for $\mu>2.5fm^{-1}$ and
$E_{00}>0$ for $\mu>3.0fm^{-1}$. This means that the mass has limit. When mass
increases, then we have no attractive particles and hence no energy spectrum.
However, in the excited level, the energies of the three states are close to
each other and the system remains attractive when the reduced mass increases.

We have seen that the approximately analytical bound states solutions of the
$\ell-$wave Schr$\ddot{o}$dinger equation for the nuclear deformed Woods-Saxon
potential can be solved by proper approximation to the centrifugal term within
the framework of the AIM. By using the AIM, closed analytical forms for the
energy eigenvalues are obtained and the corresponding wave functions have been
presented in terms of hypergeometric functions.

The method presented in this paper is an elegant and powerful technique. If
there are analytically solvable potentials, it provides the closed forms for
the eigenvalues and the corresponding eigenfunctions. However, the case if the
solution is not available, the eigenvalues are obtained by using an iterative
approach [23, 24, 25].

## References

  * [1] R. D. Woods and D. S. Saxon, Phys. Rev. 95 (1954) 577.
  * [2] W. S. C. Williams, Nuclear and Particle Physics, Clarendon, Oxford, 1996.
  * [3] F. Garcia et al., Eur. Phys. J. A 6 (1999) 49.
  * [4] V. Goldberg et al., Phys. Rev. C 69 (2004) 031302.
  * [5] A. Syntfelt et al., Eur. Phys. J. A 20 (2004) 359.
  * [6] A. Diaz-Torres and W. Scheid, Nucl. Phys. A 757 (2005) 373.
  * [7] J. Y. Guo and Q. Sheng, Phys. Lett. A 338 (2005) 90.
  * [8] V. A.Chepurnov and P. E. Nemirovsky, Nucl. Phys. 49 (1963) 90.
  * [9] R. R. Chasman and B. D. Wilkins, Phys. Lett. B 149 (1984) 433.
  * [10] J. Dudek and T. Wemer, J. Phys. G: nuclear Phys. 4 (1978) 1543.
  * [11] S. Fl$\ddot{u}$gge, Practical Quantum Mechanics, Springer-Verlag, Berlin, 1974.
  * [12] V. H. Badalov, H. I. Ahmadov, S. V. Badalov, Int. J. Mod. Phys. E 18 (2010) 1463.
  * [13] V. H. Badalov, H. I. Ahmadov, S. V. Badalov, Int. J. Mod. Phys. E 18 (2009) 631.
  * [14] S. M. Ikhdair and R. Sever, Cent. Eur. J. Phys. 8 (2010) 652; Int. J. Mod. Phys. A 25 (2010) 3941.
  * [15] M. Hamzavi and S. M. Ikhdair, Few-Body Syst, DOI 10.1007/s00601-012-0452-9 (2012).
  * [16] S.M. Ikhdair and R. Sever, Int. J. Theor. Phys. 46 (2007) 1643.
  * [17] H. Ciftci, R. L. Hall and N. Saad, J. Phys. A: Math Gen. 36 (2003) 11807.
  * [18] H. Ciftci, R. L. Hall and N. Saad, Phys. Lett. A: 340 (2005) 388.
  * [19] S.M. Ikhdair and R. Sever, Ann. Phys. (Leipzig) 16 (2007) 218.
  * [20] L. I. Schiff, Quantum Mechanics 3rd edn. (McGraw-Hill Book Co., New York, 1968).
  * [21] L. D. Landau and E.M. Lifshitz, Quantum Mechanics, Non-relativistic Theory, 3rd edn. (Pergamon, New York, 1977).
  * [22] V. H. Badalov, H. I. Ahmadov, and S. V. Badalov, arXiv:0912.3890v2 math-ph.
  * [23] T. Barakat, J. Phys. A: Math. Gen. 36 (2006) 823.
  * [24] F. M. Fernandez, J. Phys. A: Math. Gen. 37 (2004) 6173.
  * [25] T. Barakat, Phys. Lett. A 344 (2005) 411.


# Optimal allocation of quantum resources

Roberto Salazar<EMAIL_ADDRESS>International Centre for Theory of
Quantum Technologies, University of Gdansk, Wita Stwosza 63, 80-308 Gdansk,
Poland  Faculty of Physics, Astronomy and Applied Computer Science,
Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland  Tanmoy
Biswas  International Centre for Theory of Quantum Technologies, University of
Gdansk, Wita Stwosza 63, 80-308 Gdansk, Poland  Institute of Theoretical
Physics and Astrophysics, National Quantum Information Centre, Faculty of
Mathematics, Physics and Informatics, University of Gdansk, Wita Stwosza 57,
80-308 Gdansk, Poland  Jakub Czartowski  Faculty of Physics, Astronomy and
Applied Computer Science, Jagiellonian University, ul. Łojasiewicza 11, 30-348
Kraków, Poland  Karol Życzkowski  Faculty of Physics, Astronomy and Applied
Computer Science, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków,
Poland  National Quantum Information Center of Gdansk, University of Gdansk,
ul. Andersa 27 81-824 Sopot, Poland  Centrum Fizyki Teoretycznej PAN, Al.
Lotników 32/46, 02-668 Warszawa, Poland  Paweł Horodecki  International Centre
for Theory of Quantum Technologies, University of Gdansk, Wita Stwosza 63,
80-308 Gdansk, Poland  National Quantum Information Center of Gdansk,
University of Gdansk, ul. Andersa 27 81-824 Sopot, Poland  Faculty of Applied
Physics and Mathematics, National Quantum Information Centre, Gdansk
University of Technology, 80-233 Gdansk, Poland

###### Abstract

The optimal allocation of resources is a crucial task for their efficient use
in a wide range of practical applications in science and engineering. This
paper investigates the optimal allocation of resources in multipartite quantum
systems. In particular, we show the relevance of proportional fairness and
optimal reliability criteria for the application of quantum resources.
Moreover, we present optimal allocation solutions for an arbitrary number of
qudits using measurement incompatibility as an exemplary resource theory.
Besides, we study the criterion of optimal equitability and demonstrate its
relevance to scenarios involving several resource theories such as nonlocality
vs local contextuality. Finally, we highlight the potential impact of our
results for quantum networks and other multi-party quantum information
processing, in particular to the future Quantum Internet.

##  1 Introduction

In our daily lives, we perform various activities to meet our needs, fulfill
our desires or achieve our goals. These activities require the use of physical
objects, which may be within our reach or require a provider. Consequently,
our physical limitations and the usefulness to perform a specific activity
determine an object’s value as a resource. Resource theories is a theoretical
framework which assigns such a value to an object in a specific physical
context [1, 2, 3]. Quantum objects are no exception, and at present, their
value as resources for different operational tasks has been quantified by
resource theories [4, 5].

After establishing an object’s value to perform a given task, a significant
technical challenge is to distribute available resources to different agents
satisfying specific criteria. The solution to the above problem is _the
optimal allocation of resources_ and defines a well-established research area
applied in various fields, including operations research, economics,
computing, communication networks, and ecology [6, 7, 8, 9, 10, 11, 12, 13].
However, to date, there has been no systematic research on the optimal
allocation of quantum resources.

Online meetings provide a timely example, in which it is possible to
understand intuitively the allocation criteria used in this work. We will use
this example to present the criteria studied, but for brevity, we will
postpone the formal definitions to section 4. In online meetings, it is
necessary to distribute the bandwidth fairly among the users (_proportional
fairness_), and we want the communication to be fault-tolerant, i.e. reliable
under devices’ failures (_reliability_). The network may also impose certain
restrictions on the amount of bandwidth allocated to transmitting images or
sound that we must consider while carrying out the meeting’s tasks in the best
possible way (_equitability_). The point raised in this work is that quantum
devices’ benefits may be subject to the same requirements as classical
networks and therefore require optimization under the same criteria.

Precisely, this article investigates the optimal allocation of resources in
multipartite quantum systems and illustrates their application in quantum
resource theories. Moreover, we show allocations for an arbitrary number of
qudits that optimize both the criteria of _proportional fairness_ and
_reliability_ for the resource theory of measurement incompatibility.
Additionally, we study the _equitability_ criteria for resource theories with
trade-offs between them and show how this applies to quantum multi-resource
scenarios.

Figure 1:  A castaway meditates on the optimal allocation of wood logs into
several useful tasks such as: prepare a bonfire, build a ship, make a house or
provide a fold for his flock.

##  2 Resource theories in general

A quantum resource theory’s essential idea is to study quantum information
processing under a restricted set of physical operations. The permissible
operations are called free and because they do not cover all physical
processes that quantum mechanics allows only certain physically realizable
objects of a quantum system can be prepared. Likewise, these accessible
objects are called free, and any objects that are not free are called a
resource. Thus a quantum resource theory identifies every physical process as
being either free or prohibited and similarly it classifies every quantum
object as being either free or a resource. The theory of entanglement forms a
representative example of a quantum resource theory. For two or more quantum
systems, entanglement can be characterized as a resource when free operations
are local quantum operations and classical communication (LOCC) and free
objects are separable states.

Another important element of a resource theory is a function $\mathcal{M}$
that maps the objects considered by the resource theory into the non-negative
real numbers. The function $\mathcal{M}$ must be non-increasing under the free
operations i.e. a _monotone_ and should be proportional to the advantage of
using the object for some operational task.

###### Definition 1

(Resource theory) A quantum resource theory is defined by a triple
$\\{\mathcal{F},\mathcal{O},\mathcal{M}\\}$ where $\mathcal{F}$ is the set of
free objects under consideration, for instance quantum states or quantum
operations, which forms a subset within the set $\mathcal{S}$ of all quantum
objects. The set $\mathcal{O}$ of free operations contains functions
$o:\mathcal{S}\rightarrow\mathcal{S}$ which preserve the set of free objects.
A function $\mathcal{M}:\mathcal{S}\rightarrow[0,\infty+[$, monotone under the
set of free operations and such that $\mathcal{M}(f)=0$ for all
$f\in\mathcal{F}$. The function $\mathcal{M}$ is said to measure the
resourcefulness of objects in the set $\mathcal{S}$.

Because our research focuses on multipartite objects, we assume monotones
$\mathcal{M}$ to be well defined when applied to every subsystem. The
behaviour on multipartite objects of several standard measures, such as
robustness, trace distance and entropic measures is already well known in the
literature from the study of convertibility tasks, for more details see [3].

In the following we introduce the basic definitions of the resource theory of
incompatibility which recently attracted the attention of the community [14,
15]. We will use this particular theory to illustrate the application of these
definitions and show how they lead to sound but tractable optimization
problems.

##  3 Resource theory of incompatibility

In quantum theory measurements are described by Positive Operator Value
Measures (POVM). We define a POVM in dimension $d$ and a number of outcomes
$n$ denoted by $\textbf{M}:=\\{M_{1},M_{2},\ldots,M_{n}\\}$ where each
$M_{i}\in\mathcal{L}(\mathscr{H}_{d})$ is called measurement operator such
that $M_{i}\geq 0$ and $\Sigma_{i}M_{i}=\textbf{I}$. Suppose a set of
measurements $\\{\textbf{M}_{x}\\}_{x}$ where $x=1;...;m$, each described by
measurement operators $M_{a|x}$ labels each of the measurement outcomes $a$
where $a=1,..,o$ such that $\Sigma_{a=1}^{o}M_{a|x}=\textbf{I}$ for every $x$.
This set is said to be jointly measurable (or compatible) if there exists a
parent POVM $\mathcal{G}$ with measurement operators $\mathcal{G}_{\lambda}$
and conditional probability distributions $p(a|x,\lambda)$ such that
$M_{a|x}=\Sigma_{\lambda}p(a|x,\lambda)\mathcal{G}_{\lambda}$. Otherwise the
set is said to be incompatible. One can introduce a resource-theoretic
framework in the case of incompatibility with jointly measurable (compatible)
POVM as a free measurement set $\mathcal{F}$ and the set of free operations
$\mathcal{O}$ is: any single compatible measurement operator, classical post-
processing and random mixing of single compatible measurements[14].

A recent important result shows that for every resourceful measurement
$\mathbf{M}$ there is an instance of minimal-error quantum state
discrimination game for which $\mathbf{M}$ gives greater success probability
than all free measurements [16, 17, 18]. It has also been shown that the
relative advantage of a resourceful measurement for state discrimination is
proportional to the robustness measure [19, 20], which quantifies the minimal
amount of noise that has to be added to a POVM to make it free. The formal
expression for the generalized robustness of a measurement $\mathbf{M}$ is:

$\mathcal{R}_{g}\left(\mathbf{M}\right)=\min\left\\{s\in\mathbbm{R}_{\geq
0}\mid\exists\mathbf{N}\,\,\textrm{s.t.}\,\,\frac{\mathbf{M}+s\mathbf{N}}{1+s}\in\mathcal{F}\right\\}$
(1)

This is a monotone measure which means that for any measurement $\mathbf{M}$
and free operation $\phi\in\mathcal{O}$ we have
$\mathcal{R}_{g}\left(\phi\left[\mathbf{M}\right]\right)\leq\mathcal{R}_{g}\left(\mathbf{M}\right)$.
We use $\mathcal{R}_{g}$ in our work since it provides a clear operational
interpretation of measurement incompatibility and is easy to define for any
subsystem: simply choose any $\mathbf{N}$ that acts on the same systems as
$\mathbf{M}$ such that the mixture is some compatible POVM on the
corresponding systems.

##  4 Allocation of resources

The allocation of resources consists of _distributing resources into a set of
tasks according to a practical criterion_ , which usually involves optimizing
a figure of merit. When the selected distribution method optimizes the figure
of merit in the criteria, we have an _optimal allocation of resources_. We can
see in Fig.1 a classic example of this problem, where an isolated man on an
island must decide the best way to use the available resources.

In this article we restrict ourselves to study the above general problem in
the case of multipartite quantum objects (a state $\rho$, a measurement
$\mathbf{M}$ or a channel $\Xi$) denoted here by a common symbol $\sigma$. In
a multipartite object, some subsets of the systems can be selected for
different operational tasks. To describe this selection of relevant sets we
shall use the notion of hypergraph
$\mathcal{H}=\left\\{\mathcal{V}_{\mathcal{H}},\mathcal{E}_{\mathcal{H}}\right\\}$,
in which hyperedges $\alpha\in\mathcal{E}_{\mathcal{H}}$ connect two or more
vertices $v\in\mathcal{V}_{\mathcal{H}}$ [21]. We define a _quantum resource
allocation_ as the distribution of a quantum multipartite resource $\sigma$
over hypergraph $\mathcal{H}$ i.e. a list
$\mathcal{A}_{\mathcal{H}}\left[\sigma\right]$ of values of the measure
$\mathcal{M}$ applied to every reduction of $\sigma$ to parties in a hyperedge
$\alpha$ of $\mathcal{H}$. The way in which the measure is applied for each
$\alpha\in\mathcal{E}_{\mathcal{H}}$ must be specified in each case, which we
exemplify later in our applications of our framework. __ To illustrate the
concept of resource allocation we present hypergraphs $\mathcal{H}_{1}$,
$\mathcal{H}_{2}$ and $\mathcal{H}_{3}$ in Fig.2 which have allocations:

$\displaystyle\mathcal{A}_{\mathcal{H}_{1}}\left[\sigma_{abcd}\right]$
$\displaystyle=$
$\displaystyle\left\\{\mathcal{M}\left(\sigma_{abcd}\right),\mathcal{M}\left(\textrm{Tr}_{d}\left(\sigma_{abcd}\right)\right)\right\\}$
$\displaystyle\mathcal{A}_{\mathcal{H}_{2}}\left[\sigma_{ab}\right]$
$\displaystyle=$
$\displaystyle\left\\{\mathcal{M}\left(\sigma_{ab}\right),\mathcal{M}\left(\textrm{Tr}_{a}\left(\sigma_{ab}\right)\right),\mathcal{M}\left(\textrm{Tr}_{b}\left(\sigma_{ab}\right)\right)\right\\}$
$\displaystyle\mathcal{A}_{\mathcal{H}_{3}}\left[\sigma_{ab}\right]$
$\displaystyle=$
$\displaystyle\left\\{\mathcal{M}\left(\sigma_{ab}\right),\mathcal{M}\left(\textrm{Tr}_{a}\left(\sigma_{ab}\right)\right)\right\\}$

Figure 2: Examples of Hypergraphs used to determine the relevant parts of the
resource allocation. Explicitly, the corresponding sets of hyperedges are
$\mathcal{E}_{\mathcal{H}_{1}}=\left\\{\left\\{a,b,c,d\right\\},\left\\{a,b,c\right\\}\right\\}$,
$\mathcal{E}_{\mathcal{H}_{2}}=\left\\{\left\\{a,b\right\\},\left\\{a\right\\},\left\\{b\right\\}\right\\}$
and
$\mathcal{E}_{\mathcal{H}_{3}}=\left\\{\left\\{a,b\right\\},\left\\{b\right\\}\right\\}$.

If the list $\mathcal{A}_{\mathcal{H}}$ satisfies an _optimality criterion_
$\mathscr{C}$ then we will say that the quantum resource allocation is
_optimal._ The relevant criterion $\mathscr{C}$ depends on the allocation
features we wish to optimize. If a task can be carried out by parts of the
entire system specified by a hypergraph $\mathcal{H}$, then the _performance_
$\Phi_{\mathscr{C}}$ of the allocation is given by a sum over hyperedges
$\alpha\in\mathcal{E}_{\mathcal{H}}$ [8, 9]:

$\Phi_{\mathscr{C}}\left(\mathcal{A}_{\mathcal{H}}\left[\sigma\right]\right)=\sum_{\alpha\in\mathcal{E}_{\mathcal{H}}}\Phi_{\mathscr{C},\alpha}\left(\mathcal{M}\left(\sigma_{\alpha}\right)\right)$
(2)

Here $\Phi_{\mathscr{C},\alpha}:\mathbbm{R}_{\geq 0}\rightarrow\mathbbm{R}$ is
some monotone function of the amount of resource allocated in hyperedge
$\alpha$ and $\sigma_{\alpha}$ is the reduction of $\sigma$ in which the
complement of $\alpha$ has been traced out. The properties of
$\Phi_{\mathscr{C},\alpha}$ depend on how the allocation of resources at
$\alpha$ contribute to the satisfaction of criterion $\mathscr{C}$. Since the
operational task has associated the resource measure $\mathcal{M}$, the
optimization problem consist in finding the multipartite resource $\sigma$
such that $\mathcal{A}_{\mathcal{H}}\left[\sigma\right]$ maximizes
$\Phi_{\mathscr{C}}$ when $\mathcal{H}$ is fixed. The first optimality
criterion $\mathscr{C}_{1}$ we will consider is the _optimal proportional
fairness_ criteria.This criterion is required when it is essential to allocate
resources fairly among the parties. For example, these include allocation of
bandwidth in telecommunication networks, takeoff and landing slots at
airports, and water resources. For this case we maximize the performance of
proportional fairness $\Phi_{\mathscr{C}_{1}}$ given by:

$\Phi_{\mathscr{C}_{1}}\left(\mathcal{A}_{\mathcal{H}}\left[\sigma\right]\right)=\sum_{\alpha\in\mathcal{E}_{\mathcal{H}}}\log\left(\mathcal{M}\left(\sigma_{\alpha}\right)\right).$
(3)

Hence, each $\Phi_{\mathscr{C}_{1},\alpha}\left(\cdot\right)$ is equal to
$\log\left(\cdot\right)$. The choice
$\Phi_{\mathscr{C}_{1},\alpha}\left(\cdot\right)=\log\left(\cdot\right)$ is
useful in the case when no set of parts $\alpha$ has priority over others for
the task performed [8]. In the case when certain priority order of the
$\alpha\in\mathcal{E}_{\mathcal{H}}$ exists, a different figure of merit is
used to ensure equitability [8] and we will discuss it later in the article.
For example, the figure of merit $\Phi_{\mathscr{C}_{1}}$ is applied when in a
communication network each demand is routed on a specified single path. From a
resource point of view, the objective is to provide a resource
$\sigma_{\alpha}$ to parties in the set $\alpha$ to communicate with an
additional party $e\notin\mathcal{V}_{\mathcal{H}}$ in a way as fair as
possible among all $\alpha\in\mathcal{E}_{\mathcal{H}}$. It can be shown that
the optimal solution $\mathcal{A}_{\mathcal{H}}\left[\sigma^{\ast}\right]$
satisfies:

$\sum_{\alpha\in\mathcal{E}_{\mathcal{H}}}\left(\frac{\mathcal{M}\left(\sigma_{\alpha}\right)-\mathcal{M}\left(\sigma_{\alpha}^{\ast}\right)}{\mathcal{M}\left(\sigma_{\alpha}\right)}\right)\leq
0$

for every other $\mathcal{A}_{\mathcal{H}}\left[\sigma\right]$. Precisely, the
solution $\mathcal{A}_{\mathcal{H}}\left[\sigma^{\ast}\right]$ is called
_proportionally fair_ since the aggregate of proportional changes with respect
to any other feasible solution is zero or negative [8].

Next, we consider the _optimal reliability_ criterion $\mathscr{C}_{2}$ which
determines an optimal performance for a task in the presence of failures of
the devices used by any one of its parts in each
$\alpha\in\mathcal{E}_{\mathcal{H}}$. The appropriate performance
$\Phi_{\mathscr{C}_{2}}$ which we maximize in this case reads:

$\Phi_{\mathscr{C}_{2}}\left(\mathcal{A}_{\mathcal{H}}\left[\sigma\right]\right)=\sum_{\alpha\in\mathcal{E}_{\mathcal{H}}}\pi_{\alpha}\mathcal{M}\left(\sigma_{\alpha}\right),$
(4)

where $\pi_{\alpha}$ denotes the prior probability that only parts in $\alpha$
will perform the task for which $\sigma$ is the resource. If every part
$\left\\{a_{1},\ldots,a_{n}\right\\}$ has a prior chance $p_{a_{k}}$to work,
then:

$\pi_{\alpha}=\prod_{a_{i}\in\alpha}p_{a_{i}}\prod_{a_{j}\notin\alpha}\left(1-p_{a_{j}}\right).$
(5)

The problem of optimal reliability is a relevant problem for different
engineering areas which is usually addressed by _redundacy models_ [9]. In (4)
we adapt the classical performance function to fit the scheme of resource
theories (see for instance the classical definition in section 1.23 in [9]) by
choosing the $\Phi_{\mathscr{C}_{2},\alpha}\left(\cdot\right)$ to be the
probability of successful performace only of parts in $\alpha$ rather than
probability of failure, because $\mathcal{M}\left(\sigma_{\alpha}\right)$ is a
measure of advantage provided by $\sigma_{\alpha}$ to the corresponding task.

The previous examples of performance and optimal reliability considered the
case of a resource that should be shared in an optimal way among parties
performing the same task. It should be noted that in principle one can define
different tasks for each set $\alpha\in\mathcal{E}_{\mathcal{H}}$ in the two
previous allocation problems. However, as we shall see on the following
examples, the natural cases that involve multiple tasks are those where the
availability of the resource is limited and - what is even more important -
competition between the tasks exist. To describe such a situation in our
notation consider for every resource $\sigma$ and all
$\alpha\in\mathcal{E}_{\mathcal{H}}$:

$\displaystyle\sum_{\alpha\in\mathcal{E}_{\mathcal{H}}}\Lambda_{\alpha}\left(\sigma\right)\mathcal{M}_{\alpha}\left(\sigma_{\alpha}\right)$
$\displaystyle\leq$ $\displaystyle\Gamma\left(\sigma\right)\>$ (6)
$\displaystyle L_{\alpha}\leq\mathcal{M}_{\alpha}\left(\sigma_{\alpha}\right)$
$\displaystyle\leq$ $\displaystyle U_{\alpha}\>.$ (7)

with $\Lambda_{\alpha}\left(\sigma\right),\Gamma\left(\sigma\right)$ real non-
negative functions of states and $L_{\alpha},U_{\alpha}$ some non-negative
constants. Here we use explicitly the index dependence to express the
possibility that the parties in each $\alpha\in\mathcal{E}_{\mathcal{H}}$
measure the resourcefulness of their $\sigma_{\alpha}$ with a different
$\mathcal{M}_{\alpha}$. The choice of each $\mathcal{M}_{\alpha}$ is
determined by the task assigned to the resource in each $\alpha$. The
equations (6) stand for the trade-offs between the resources assigned for
tasks at different $\alpha$ while the (7) determine prior limitations of the
resources for each task. This set of inequalities is known as _knapsack
constraints_ (KC) [8]. In the case of two or more tasks involved and non-
trivial knapsack constraints a typical optimality criterion $\mathscr{C}_{3}$
is the _optimal equitability_ which is defined by a recursive max-min
algorithm:

  1. 1.

If upperbounds $U_{\alpha_{i}}\leq U_{\alpha_{j}}$ then introduce the
condition
$\mathcal{M}_{\alpha_{i}}\left(\sigma_{\alpha_{i}}\right)\leq\mathcal{M}_{\alpha_{j}}\left(\sigma_{\alpha_{j}}\right)$.
The corresponding partial order determines the minimization over
$\alpha\in\mathcal{E}_{\mathcal{H}}$ in the following steps.

  2. 2.

If $\mathcal{\mathcal{E}_{\mathcal{H}}}\neq\left\\{\emptyset\right\\}$ then
find a solution with prior KC for

$\Phi_{\mathscr{C}_{3}}\left(\mathcal{A}_{\mathcal{H}}\left[\sigma\right]\right)=\sup_{\sigma}\min_{\alpha\in\mathcal{E}_{\mathcal{H}}}\mathcal{M}_{\alpha}\left(\sigma_{\alpha}\right)$

  3. 3.

Let us call $\alpha_{1}$ the $\alpha\in\mathcal{E}_{\mathcal{H}}$ such that
$\mathcal{M}_{\alpha_{1}}\left(\sigma_{\alpha_{1}}\right)$ is the minimum on
step 1. If
$\mathcal{E}_{\mathcal{H}}/\alpha_{1}=\mathcal{E}_{\mathcal{H}}^{(1)}\neq\left\\{\emptyset\right\\}$
then find a solution with prior KC for:

$\Phi_{\mathscr{C}_{3}}\left(\mathcal{A}_{\mathcal{H}^{(1)}}\left[\sigma\right]\right)=\sup_{\sigma}\min_{\alpha\in\mathcal{E}_{\mathcal{H}}^{(1)}}\mathcal{M}_{\alpha}\left(\sigma_{\alpha}\right)$

under the additional constraint:

$\Phi_{\mathscr{C}_{3}}\left(\mathcal{A}_{\mathcal{H}}\left[\sigma\right]\right)=\mathcal{M}_{\alpha_{1}}\left(\sigma_{\alpha_{1}}\right)\leq\mathcal{M}_{\alpha}\left(\sigma_{\alpha}\right)$

for all $\alpha\in\mathcal{E}_{\mathcal{H}}^{(1)}$.

  4. 4.

Iterate with corresponding $\alpha_{n-1}$ from
$\alpha\in\mathcal{E}_{\mathcal{H}}^{(n-2)}$ such that
$\mathcal{M}_{\alpha_{n-1}}\left(\sigma_{\alpha_{n-1}}\right)$ is the minimum
on step $n-1$. If
$\mathcal{E}_{\mathcal{H}}^{(n-2)}/\alpha_{n-1}=\mathcal{\mathcal{E}_{\mathcal{H}}}^{(n-1)}\neq\left\\{\emptyset\right\\}$
then find a solution with prior KC for

$\Phi_{\mathscr{C}_{3}}\left(\mathcal{A}_{\mathcal{H}^{(n-2)}}\left[\sigma\right]\right)=\sup_{\sigma}\min_{\alpha\in\mathcal{E}_{\mathcal{H}}^{(n-1)}}\mathcal{M}_{\alpha}\left(\sigma_{\alpha}\right)$

under the additional constraints:

$\mathcal{M}_{\alpha_{1}}\left(\sigma_{\alpha_{1}}\right)\leq...\leq\mathcal{M}_{\alpha_{n-1}}\left(\sigma_{\alpha_{n-1}}\right)\leq\mathcal{M}_{\alpha}\left(\sigma_{\alpha}\right)$

for all $\alpha\in\mathcal{E}_{\mathcal{H}}^{(n-1)}$ and such that:

$\Phi_{\mathscr{C}_{3}}\left(\mathcal{A}_{\mathcal{H}^{\left(m-1\right)}}\left[\sigma\right]\right)=\mathcal{M}_{\alpha_{m}}\left(\sigma_{\alpha_{m}}\right)$

for each $\mathcal{M}_{\alpha_{m}}\left(\sigma_{\alpha_{m}}\right)$.

  5. 5.

When $\mathcal{E}_{\mathcal{H}}^{(m-1)}=\left\\{\emptyset\right\\}$ stop.

In the previous recursive algorithm minimization of
$\mathcal{M}_{\alpha}\left(\sigma_{\alpha}\right)$ over
$\alpha\in\mathcal{E}_{\mathcal{H}}$ considers
$\mathcal{M}_{\alpha_{i}}\left(\sigma_{\alpha_{i}}\right)\leq\mathcal{M}_{\alpha_{j}}\left(\sigma_{\alpha_{j}}\right)$
iff $U_{\alpha_{i}}\leq U_{\alpha_{j}}$ because in this case allocating
resources in $\alpha_{j}$ is potentially more advantageous than in
$\alpha_{i}$. We also note that the above algorithm will halt in finite number
steps because the number of $\alpha\in\mathcal{\mathcal{E}_{\mathcal{H}}}$ is
finite. Solutions to optimal equitability are known to be non-unique, but
nevertheless they provide a set of solutions considered safe for the usual
practical applications [8]. If this is not satisfactory, a unique solution can
be ensured by additional requirements such as Pareto optimality [8] of the
final order
$\mathcal{M}_{\alpha_{1}}\left(\sigma_{\alpha_{1}}\right)\leq...\leq\mathcal{M}_{\alpha_{N}}\left(\sigma_{\alpha_{N}}\right)$.

It is essential to highlight that the theoretical framework we developed for
the allocation of quantum resources is entirely general and that it is
possible to apply it in any case that requires at least two agents performing
more than one instance of a task or more than one task with their quantum
devices. That said, the next section’s results illustrate the usefulness of
the theoretical framework by demonstrating concrete and relevant applications.

##  5 Results

###  5.1 Fair and reliable allocation of incompatible measurements

Before presenting a specific example of resource allocation let us recall the
notion of unbiased bases. Two orthogonal measurements with labels
$j,j^{\prime}\in\left\\{0,1\right\\}$ are _unbiased_ iff they satisfy the
following condition:

$\left|\left\langle u_{a}^{j}\mid
u_{b}^{j^{\prime}}\right\rangle\right|^{2}=\begin{cases}\delta_{ab}&;\,j=j^{\prime}\\\
\frac{1}{D}&;\,j\neq j^{\prime}\end{cases}$ (8)

where $D$ is the total dimension of the system [22, 23]. Two unbiased bases
consisting of product states only were introduced in [24], for which case we
write
$\left|u_{\mathbf{a}}^{x}\right\rangle=\left|u_{a_{1}}^{x}\right\rangle\otimes\ldots\otimes\left|u_{a_{N}}^{x}\right\rangle$
with each $\left|u_{a_{k}}^{x}\right\rangle$ a state from two unbiased bases.
Therefore, for $N$ systems of dimension $d$ two product unbiased measurements
$\left|u_{\mathbf{a}}^{x_{1}}\right\rangle\left\langle
u_{\mathbf{a}}^{x_{1}}\right|,\left|u_{\mathbf{b}}^{x_{2}}\right\rangle\left\langle
u_{\mathbf{b}}^{x_{2}}\right|$ satisfy (8) replacing
$\delta_{ab}\rightarrow\delta_{\mathbf{ab}}$ and $D=d^{N}$. It was
demonstrated in [24] that one can find at least two product bases in
$\mathscr{H}_{d}^{\otimes N}$ which are mutually unbiased and such that any of
their reductions to $M$ qudits forms a set of two unbiased product bases in
$\mathscr{H}_{d}^{\otimes M}$. The above property demonstrates that product
unbiased bases form a _nested_ measurement (in the sense of [25]), which means
that tracing out any of the parts of the POVMs the remaining measurements are
also product unbiased bases.

Our main contribution consists of two examples that demonstrate the optimal
allocation of quantum resources to be sound and tractable problems. Indeed, in
our first example we show an explicit POVM
$\widetilde{\mathbf{M}}=\left\\{\widetilde{M}_{x}\right\\}_{x}$ for which the
allocation $\mathcal{A}_{\mathcal{H}}\left[\widetilde{\mathbf{M}}\right]$
satisfies both optimality criteria $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ for
any $\mathcal{H}$ of qudits in the resource theory of incompatible
measurements. The elements of each $\widetilde{M}_{x}$ are given by the
projectors $\left|u_{\mathbf{a}}^{x=1}\right\rangle\left\langle
u_{\mathbf{a}}^{x=1}\right|,\left|u_{\mathbf{b}}^{x=2}\right\rangle\left\langle
u_{\mathbf{b}}^{x=2}\right|$, onto the unbiased bases. Then we arrive at the
following theorem:

###### Theorem 1

Consider an optimal allocation with arbitrary hypergraph $\mathcal{H}$ of a
given measurement $\mathbf{M}$ over a number $N$ of qudits of dimension $d$
and with optimality criterion $\mathscr{C}$. If the criterion $\mathscr{C}$
requires maximization of performance function $\Phi_{\mathscr{C}}$, with
allocation
$\mathcal{A}_{\mathcal{H}}\left[\mathbf{M}\right]=\left\\{\mathcal{R}_{g}\left(\mathbf{M}_{\alpha}\right)\right\\}_{\alpha\in\mathcal{E}_{\mathcal{H}}}$
defined by the generalized robustness $\mathcal{R}_{g}$, then the measurement
$\widetilde{\mathbf{M}}$ composed of two product unbiased bases is a feasible
optimal solution in the resource theory of incompatibility.

Proof: First, we need to specify in which way $\mathcal{R}_{g}$ applies to
every possible measurement $\mathbf{M}_{\alpha}$ indexed by the hyperedge
$\alpha$ of the hypergraph. We use equation (1) specified to the measurement
$\mathbf{M}_{\alpha}$, where the mixture
$(\mathbf{M}_{\alpha}+s\mathbf{N}_{\alpha})/(1+s)$ belongs to the set
$\mathcal{F}_{\alpha}$ of compatible measurements only on the composed system
$\alpha\in\mathcal{E}_{\mathcal{H}}$ and $\mathbf{N}_{\alpha}$ is some
measurement also on the systems in $\alpha$. This $\alpha$-wise application of
$\mathcal{M}_{g}$ is meaningful since it should measure the advantage of
$\mathbf{M}_{\alpha}$ over any compatible measurement of parties in
$\alpha\in\mathcal{E}_{\mathcal{H}}$. Now, we remark the recent proof in [15]
that unbiased bases are optimal incompatible measurements under the
generalized robustness monotone $\mathcal{R}_{g}$ in any dimension $d$. Since,
we can always find two product unbiased bases to define
$\widetilde{\mathbf{M}}$, then because product unbiased bases are nested
measurements the evaluation of $\mathcal{R}_{g}$ for each
$\alpha\in\mathcal{\mathcal{E}_{\mathcal{H}}}$ yields the maximal value, hence
for any monotone function $\Phi_{\mathscr{C}}$ of
$\mathcal{A}_{\mathcal{H}}\left[\widetilde{\mathbf{M}}\right]=\left\\{\mathcal{R}_{g}\left(\widetilde{\mathbf{M}}_{\alpha}\right)\right\\}_{\alpha\in\mathcal{E}_{\mathcal{H}}}$
– such as $\Phi_{\mathscr{C}_{1}}$ or $\Phi_{\mathscr{C}_{2}}$– the
$\widetilde{\mathbf{M}}$ measurement achieves the maximal value q.e.d.

As an Illustration, we present optimal allocations associated with the
hypergraphs from Fig.2. The maximal value $\Phi_{\mathscr{C}_{1}}^{max}$
achieved by
$\Phi_{\mathscr{C}_{1}}\left(\mathcal{A}_{\mathcal{H}_{1}}\left[\widetilde{\mathbf{M}}\right]\right)$
is:

$\Phi_{\mathscr{C}_{1}}^{max}=\log\left(\frac{d^{2}-1}{d^{2}+1}\right)+\log\left(\frac{d^{3/2}-1}{d^{3/2}+1}\right)$
(9)

Another example is the maximal value $\Phi_{\mathscr{C}_{2}}^{max}$ achieved
by
$\Phi_{\mathscr{C}_{2}}\left(\mathcal{A}_{\mathcal{H}_{2}}\left[\widetilde{\mathbf{M}}\right]\right)$
for each qudit pair of dimension $d$ is:

$\Phi_{\mathscr{C}_{2}}^{max}=\pi_{\left\\{a,b\right\\}}\left(\frac{d-1}{d+1}\right)+\left(\pi_{\left\\{a\right\\}}+\pi_{\left\\{b\right\\}}\right)\left(\frac{\sqrt{d}-1}{\sqrt{d}+1}\right)$
(10)

Since the resources in this particular resource theory are advantageous for
state discrimination, our result shows that the device which implements the
optimal measurement $\widetilde{\mathbf{M}}$ also has optimal performance for
such a task in the presence of failures of any subsystem. The scenario
represented by $\mathcal{H}_{2}$ is the simplest case in which “at most a
number $K$ of devices may fail”, which defines a problem of allocation
relevant to the performance of different engineering tasks [9].

###  5.2 Equitable allocation of nonlocality versus local contextuality

The equitability criterion described in section 4 is relevant in contexts
where there is some trade-off between resources distributed among agents. Such
contexts are not strange in quantum information, for example, we can mention
some cases such as extracting work from local states [26] versus obtaining
thermodynamic work from shared correlations [27, 28] or obtaining random local
states [29] versus sharing private bit states [30]. For the tasks mentioned
above, the cited articles quantify the appropriate resources with monotones
based on entropic measures to which our theoretical framework applies.
However, below we select cases of nonlocality versus local contextuality
trade-offs due to their simplicity. Our choice of a specific case allows us to
present a proof of concepts clearly and understandably.

Nonlocality and contextuality, are well known resources for communication and
randomness amplification tasks [31, 32, 33, 34, 35]. A monotone for these
state resource theories is [35, 34]:

$\mathcal{N}\left(\rho\right)=\sup_{\phi\in\Phi}\mathcal{I}\left(\phi\left[\rho\right]\right)-\mathcal{B}_{c}$
(11)

where $\mathcal{I}$ is the Bell correlation function, $\Phi$ is the set of
free operations (see Appendix A) and $\mathcal{B}_{c}$ is the classical bound.
We will use the resource theories of nonlocality and contextuality to
exemplify a resource scenario in which the optimal equitability criteria
$\mathscr{C}_{3}$ is useful. In this scenario, a hypergraph
$\mathcal{H}_{3}:\>\mathcal{V}_{\mathcal{H}_{3}}=\left\\{a,b\right\\},\mathcal{E}_{\mathcal{H}_{3}}=\left\\{\left\\{a,b\right\\},\left\\{b\right\\}\right\\}$
defines the allocation of the resource state $\rho$. The objective of party
$b$ is to perform a task that improves with the contextuality of his local
state $\rho_{b}=\textrm{Tr}_{a}\left(\rho\right)$, while parties $a,b$ should
perform a task that requires nonlocality of their bipartite state
$\rho_{a,b}=\rho$.

For our application of $\mathscr{C}_{3}$ is important the existence of a
fundamental trade-off between both resources [36] :

$\mathcal{I}_{n}\left(\rho\right)+\mathcal{I}_{m}\left(\rho\right)\leq n+m-4$
(12)

Here $\mathcal{I}_{s}\left(\rho\right)$ stands for a cyclic correlation,

$\mathcal{I}_{s}\left(\rho\right)=\sum_{k=1}^{s-1}\left\langle
B_{k}B_{k+1}\right\rangle_{\rho}-\left\langle B_{s}B_{1}\right\rangle_{\rho},$
(13)

where the average of observable $O=\left\\{O_{k}\right\\}_{k}$ given by
$\left\langle O\right\rangle_{\rho}=\sum_{k}o_{k}Tr\left(O_{k}\rho\right)$,
with $o_{k}\in\\{-1,+1\\}$ the numerical value of $O$ associated with $O_{k}$
and $\mathcal{B}_{c}^{s}=s-2$ is the classical bound. The correlation
$\mathcal{I}_{s}\left(\rho\right)$ witnesses nonlocality or contextuality
depending on the kind of constraints satisfied by the
$\left\\{B_{1},...,B_{s}\right\\}$ observables [36]. Then, if we replace
appropriately $\mathcal{I}_{s}$ and $\mathcal{B}_{c}^{s}$ in (11) to define a
monotone $\mathcal{N}_{s}\left(\rho\right)$ the inequality (12) implies the
resource relation:

$\displaystyle\Lambda_{n}\left(\rho_{a,b}\right)$
$\displaystyle\mathcal{N}_{n}\left(\rho_{a,b}\right)+\Lambda_{m}\left(\rho_{b}\right)\mathcal{N}_{m}\left(\rho_{b}\right)\leq...$
$\displaystyle\Lambda_{n}\left(\rho_{a,b}\right)$
$\displaystyle\left[\mathcal{B}_{Q}^{n}-\mathcal{B}_{c}^{n}\right]+\Lambda_{m}\left(\rho_{b}\right)\left[\mathcal{B}_{Q}^{m}-\mathcal{B}_{c}^{m}\right]\,\,\,$
$\displaystyle\Lambda_{s}\left(\rho_{\alpha}\right)$
$\displaystyle=\frac{1+\textrm{sgn}\left(\mathcal{I}_{s}\left(\rho_{\alpha}\right)-\mathcal{B}_{c}^{s}\right)}{2}$

with $\mathcal{B}_{Q}^{s}$ is the quantum Tsirelson bound (i.e. quantumly
saturable) for $\mathcal{I}_{s}\left(\rho\right)$ and $\textrm{sgn}(\cdot)$
the sign function. From an allocation of resources perspective the inequality
(5.2) defines a knapsack constraint like (6). The individual bounds (7)
correspond here to
$\mathcal{N}_{n}(\rho_{a,b})\leq\mathcal{B}^{n}_{Q}-\mathcal{B}^{n}_{c}$ and
$\mathcal{N}_{m}(\rho_{b})\leq\mathcal{B}^{m}_{Q}-\mathcal{B}^{m}_{c}$
respectively. The optimal equitable solutions in this case are simply:

Solution 1:
$\displaystyle\left\\{\begin{array}[]{c}\mathcal{N}_{n}\left(\rho_{a,b}\right)=\mathcal{B}_{Q}^{n}-\mathcal{B}_{c}^{n}\\\
\\\ \mathcal{N}_{m}\left(\rho_{b}\right)=0\end{array}\right.$ Solution 2:
$\displaystyle\left\\{\begin{array}[]{c}\mathcal{N}_{n}\left(\rho_{a,b}\right)=0\\\
\\\
\mathcal{N}_{m}\left(\rho_{b}\right)=\mathcal{B}_{Q}^{m}-\mathcal{B}_{c}^{m}\end{array}\right.$

If
$\mathcal{B}_{Q}^{n}-\mathcal{B}_{c}^{n}>\mathcal{B}_{Q}^{m}-\mathcal{B}_{c}^{m}$
then the max-min algorithm will select Solution 1, since in this case
$\mathcal{N}_{n}\left(\rho_{a,b}\right)>\mathcal{N}_{m}\left(\rho_{b}\right)$
is the choice that maximizes the overall amount of resources provided by
$\rho$. Conversely if
$\mathcal{B}_{Q}^{n}-\mathcal{B}_{c}^{n}<\mathcal{B}_{Q}^{m}-\mathcal{B}_{c}^{m}$
we will obtain Solution 2 and finally, if
$\mathcal{B}_{Q}^{n}-\mathcal{B}_{c}^{n}=\mathcal{B}_{Q}^{m}-\mathcal{B}_{c}^{m}$
both solution are acceptable, if only equitability is demanded.

A nontrivial scenario for optimal equitability arises in the case of _monogamy
activation,_ for example __ in the following relationship [36]:

$\mathcal{I}_{A}^{B}\left(\rho_{a,b}\right)+\mathcal{I}_{C}^{B}\left(\rho_{b,c}\right)+2\mathcal{I}_{5}\left(\rho_{b}\right)\leq
14$ (17)

Here, $\mathcal{I}_{5}\left(\rho_{b}\right)$ is a contextual cycle correlation
with $B_{k}$ observables as in (13), while
$\mathcal{I}_{A}^{B}\left(\rho_{a,b}\right),\mathcal{I}_{C}^{B}\left(\rho_{b,c}\right)$
stand for $I_{3322}$ inequalities [36]:

$\mathcal{I}_{A}^{B}\left(\rho_{a,b}\right)=\left\langle
B_{1}\right\rangle_{\rho}+\left\langle B_{4}\right\rangle_{\rho}+\left\langle
A_{1}\right\rangle_{\rho}+\left\langle A_{2}\right\rangle_{\rho}-\left\langle
B_{1}A_{1}\right\rangle_{\rho}\\\ -\left\langle
B_{1}A_{2}\right\rangle_{\rho}-\left\langle
B_{1}A_{3}\right\rangle_{\rho}-\left\langle
B_{4}A_{1}\right\rangle_{\rho}-\left\langle B_{4}A_{2}\right\rangle_{\rho}\\\
+\left\langle B_{4}A_{3}\right\rangle_{\rho}-\left\langle
B_{6}A_{1}\right\rangle_{\rho}+\left\langle B_{6}A_{2}\right\rangle_{\rho}\leq
4$ (18)

and analogously for $\mathcal{I}_{C}^{B}\left(\rho_{b,c}\right)$ by replacing
each $A_{k}$ by $C_{k}$. Now, we study the allocation scenario defined by
$\mathcal{H}_{3}$, and with operational tasks associated with
$\mathcal{I}_{A}^{B}\left(\rho_{a,b}\right)$ and
$\mathcal{I}_{5}\left(\rho_{b}\right)$. In this case we assume $\rho_{b,c}$ to
be resourceless, but with a fixed value
$\mathcal{I}_{C}^{B}\left(\rho_{b,c}\right)=\lambda<4$. Then, we can state an
equitability problem defined by the constraints:

$\displaystyle\mathcal{N}_{A}^{B}\left(\rho_{a,b}\right)+2\mathcal{N}_{5}\left(\rho_{b}\right)$
$\displaystyle\leq$ $\displaystyle 4-\lambda$ $\displaystyle
0\leq\mathcal{N}_{A}^{B}\left(\rho_{a,b}\right)$ $\displaystyle\leq$
$\displaystyle\mathcal{B}_{Q}^{\gamma}-4$ $\displaystyle
0\leq\mathcal{N}_{5}\left(\rho_{b}\right)$ $\displaystyle\leq$
$\displaystyle\mathcal{B}_{Q}^{5}-3$ (19)

The monotones $\mathcal{N}_{A}^{B}\left(\rho_{a,b}\right)$ and
$\mathcal{N}_{5}\left(\rho_{b}\right)$ are defined analogously as
$\mathcal{N}_{s}\left(\rho\right)$ in (11), and $\mathcal{B}_{Q}^{\gamma}$,
$\mathcal{B}_{Q}^{5}$ are the quantum bounds for $I_{3322}$ and
$\mathcal{I}_{5}\left(\rho\right)$ respectively. In this case the solution is
nontrivial and reads:

Solution 3:
$\displaystyle\left\\{\begin{array}[]{c}\mathcal{N}_{A}^{B}\left(\rho_{a,b}\right)=\frac{\mu}{3}\\\
\\\ \mathcal{N}_{5}\left(\rho_{b}\right)=\frac{\mu}{3}\end{array}\right.$ (23)

with $\mu=4-\lambda$ (for the proof see Appendix A, section A.2). As mentioned
before, nonlocal and contextual resources are useful for randomness
amplification [31, 32]. Therefore, an application of the solutions to the
problems presented is an equitable assignment of security into quantum
networks. In consequence, the above examples show that optimal equitable
allocations of resources are relevant for concrete tasks in quantum
information.

##  6 Discussion

In quantum information, a variety of approaches are used to determine the
practical value of quantum systems, such as cryptography, communication
capacity, computational complexity and thermodynamics. The common factor among
these approaches is the search for quantum resources to benefit agents in
performing specific tasks. Observed as a whole, these form an economy that
requires tools to manage such quantum resources, optimal allocation being
crucial.

This article applies three different criteria of the optimal allocation of
quantum resources and demonstrates solutions to the corresponding optimization
problems. We propose the above results for applications in the interplay
between quantum processing machines (computers) and a quantum communication
network (see [37, 38, 39]).

Indeed, our results contribute to the line of work proposed by S. Wehner et
al. [40] for developing the Quantum Internet in six functionality driven
stages. Precisely, Theorem 1 shows that there are multipartite quantum
resources for communication tasks in prepare-and-measure networks with the
optimally proportionally fair and reliable distribution. Such a result fits
the type of advances desired for quantum networks in the second functionality
stage of the Quantum Internet.

Furthermore, our example of the optimal equitability criteria may be a
starting point for analyzing network scenarios in the third stage proposed for
Quantum Internet development (Entanglement distribution networks [40]).
Indeed, nonlocality may be a self-testing benchmark for entanglement while
contextuality may be related to coherence. The former may be related to
communication between the nodes, and the latter - to quantum information
processing at the nodes. If we have limited fault-tolerant resources for
protecting both local and nonlocal quantumness, then the optimal equitability
may help find a balance between local computation and delegation of tasks,
especially if desired to be secure (see [41]). Independently the very
nonlocality at single computing node may be one day an important resource
itself (see [42]). We leave these and related topics for further research.

In addition, the scheme we present opens the possibility of finding
applications for multi-party systems that nest resources symmetrically [43].
In this sense, our results suggest that resource nesting may be potentially a
tool for distributing resources according to specific criteria relevant to the
corresponding tasks.

In conclusion, it seems natural to expect that quantum resources will need
soon the tools to allocate resources provided in this article. It may be
helpful at the level of designing quantum information processing protocols (or
- possibly - even some practical experiments with limited quantum coherence
effects). On the other hand, the methods and tools for allocating resources
can contribute to the theoretical analysis of trade-offs between different
quantum information resources. One can even expect that they can contribute to
developing some form of the _economy of quantum resources_. The administration
of quantum networks – such as the Quantum Internet – could be one of this new
economy’s first challenge.

##  7 Acknowledgements

RS, TB and PH acknowledge support by the Foundation for Polish Science (IRAP
project, ICTQT, contract no.2018/MAB/5, co-financed by EU within Smart Growth
Operational Programme). JC and KZ thank the support of National Science Center
in Poland under the grant numbers DEC-2019/35/O/ST2/01049 and
DEC-2015/18/A/ST2/00274 and by Foundation for Polish Science under the grant
TEAM-NET project (contract no. POIR.04.04.00-00-17C1/18-00). We also thank A.
de Oliveira Junior, Debashis Saha and Gabriel Muniz for advice and help with
the design of figures.

##  Appendix A Appendix

###  A.1 Resource theories of nonlocality and contextuality

The resource theories of nonlocality and contextuality usually consider as
objects boxes with at least two integer inputs $x_{1},x_{2}$ and two integer
outputs $a_{1},a_{2}$, which are characterized by the conditional
probabilities $P\left(a_{1},a_{2}\mid x_{1},x_{2}\right)$ called behaviours
[34, 35]. In our research we consider only quantum resources, hence every
behaviour can be written as:

$P\left(a_{1},a_{2}\mid x_{1},x_{2}\right)=Tr\left(\rho_{1,2}M_{a_{1}\mid
x_{1}}^{\left(1\right)}\otimes M_{a_{2}\mid x_{2}}^{\left(2\right)}\right)$
(24)

For a bipartite state $\rho_{1,2}$ and measurements $M_{a_{1}\mid
x_{1}}^{\left(1\right)}\otimes M_{a_{2}\mid x_{2}}^{\left(2\right)}$ for
systems $1,2$. Moreover, in each case we consider fixed measurements, such
that in our study the resource theories of nonlocality and contextuality can
as well be considered resource theories of quantum states. Because of this
some examples of free operations for nonlocality and contextuality can be
local operations as well mixing of states. If in both cases the set $\Phi$ is
defined as the closure of free operations under composition, the measure
$\mathcal{N}_{s}\left(\rho\right)$ defined in (11) is a monotone:

$\mathcal{N}_{s}\left(\phi\left[\rho\right]\right)\leq\mathcal{N}_{s}\left(\rho\right)\>\forall\phi\in\Phi$
(25)

due to the definition of supremum.

###  A.2 Non-trivial solutions of allocation in the monogamy activation
scenario

Here we provide in more detail the possible equitable solutions to the problem
with knapsack constraints (19). First, let’s define the auxiliary variables
$\mu=4-\lambda$, $\nu_{1}=\mathcal{B}_{Q}^{I_{3322}}-4$ and
$\nu_{2}=\mathcal{B}_{Q}^{5}-3$. In Appendix C we show that $\nu_{1}\geq 1$
and from reference [44] $v_{2}=0.9442$, hence $\nu_{1}>\nu_{2}.$ From the
above and the partial order imposed by the equitability criteria
$\mathscr{C}_{3}$ we have:

$\mathcal{N}_{5}\left(\rho_{\left\\{b\right\\}}\right)\leq\mathcal{N}_{A}^{B}\left(\rho_{\left\\{a,b\right\\}}\right)$
(26)

which means that in this scenario the advantage provided by nonlocality of
$\rho_{\left\\{a,b\right\\}}$ is potentially greater than the noncontextuality
of $\rho_{\left\\{b\right\\}}$, therefore has priority. Additionally we know
(See ref [36, 44, 45]) that all values for
$\mathcal{I}_{5}\left(\rho_{\left\\{b\right\\}}\right),\mathcal{I}_{A}^{B}\left(\rho_{\left\\{a,b\right\\}}\right),\lambda$
satisfying the trade-off (17) and Tsirelson bounds to be achievable for some
$\rho_{a,b,c}$, then the supreme in each step of the optimal equitability
criterion can be replaced by a maximization. Later, because the optimization
is over a convex set – of
$\mathcal{N}_{5}\left(\rho_{\left\\{b\right\\}}\right),\mathcal{N}_{A}^{B}\left(\rho_{a,b}\right)$–
the solution must lie at a boundary. A solution in the boundary
$\mathcal{N}_{5}\left(\rho_{\left\\{b\right\\}}\right)=\nu_{2}$ will violate
the requirements (19) or (26) because $\mu\leq 2\nu_{2}$, on the other hand a
solution with
$\mathcal{N}_{A}^{B}\left(\rho_{\left\\{a,b\right\\}}\right)=\nu_{1}$, implies
$\mathcal{N}_{5}\left(\rho_{\left\\{b\right\\}}\right)=0$ which is actually a
minimum. Then, to find a maximum for
$\mathcal{N}_{5}\left(\rho_{\left\\{b\right\\}}\right)$ under the constraints,
we search at the boundary:

$\displaystyle\mathcal{N}_{A}^{B}\left(\rho_{\left\\{a,b\right\\}}\right)+2\mathcal{N}_{5}\left(\rho_{\left\\{b\right\\}}\right)$
$\displaystyle=$ $\displaystyle\mu$
$\displaystyle\Rightarrow\mathcal{N}_{A}^{B}\left(\rho_{\left\\{a,b\right\\}}\right)$
$\displaystyle=$
$\displaystyle\mu-2\mathcal{N}_{5}\left(\rho_{\left\\{b\right\\}}\right)$ (27)

Replacing (27) in (26) we obtain:

$\mathcal{N}_{5}\left(\rho_{\left\\{b\right\\}}\right)\leq\frac{\mu}{3}<\nu_{2}$
(28)

Since the lower bound of
$\mathcal{N}_{A}^{B}\left(\rho_{\left\\{a,b\right\\}}\right)$ is zero, the
maximum value that $\mathcal{N}_{5}\left(\rho_{\left\\{b\right\\}}\right)$ can
achieve is $\mu/3$ in which case we have the equitable Solution 3 of main
text. The solution (23) maximizes the amount of resources under constraints
(19) and minimizes the difference between
$\mathcal{N}_{A}^{B}\left(\rho_{\left\\{a,b\right\\}}\right)$ and
$\mathcal{N}_{5}\left(\rho_{\left\\{b\right\\}}\right)$.

###  A.3 The quantum bound of $I_{3322}$

The Bell operator $\mathbbm{B}_{Q}^{v}$ of the $I_{3322}$ inequality was
introduced in reference [45]:

$\displaystyle\mathbbm{B}_{Q}^{v}=-A_{2}\otimes\mathbbm{I}-\mathbbm{I}\otimes
B_{1}-2\mathbbm{I}\otimes B_{4}+A_{1}\otimes B_{1}$
$\displaystyle+A_{1}\otimes B_{4}+A_{2}\otimes B_{1}+A_{2}\otimes
B_{4}-A_{1}\otimes B_{6}$ $\displaystyle+A_{2}\otimes B_{6}-A_{3}\otimes
B_{1}+A_{3}\otimes B_{4}$ (29)

in terms of binary operators
$\left\\{A_{1},A_{2},A_{3},B_{1},B_{4},B_{6}\right\\}$ with outcomes $0$ and
$1$,

$\displaystyle A_{i}$ $\displaystyle=$
$\displaystyle(1)A_{i}^{1}+(0)A_{i}^{\perp}\>i\in\left\\{1,2,3\right\\}$
$\displaystyle B_{j}$ $\displaystyle=$
$\displaystyle(1)B_{j}^{1}+(0)B_{j}^{\perp}\>j\in\left\\{1,4,6\right\\}$

However, in this article we use binary operators with outcomes $-1$ and $1$,

$\displaystyle A_{i}^{\prime}$ $\displaystyle=$
$\displaystyle(1)A_{i}^{1}+(-1)A_{i}^{\perp}\>i\in\left\\{1,2,3\right\\}$
$\displaystyle B_{j}^{\prime}$ $\displaystyle=$
$\displaystyle(1)B_{j}^{1}+(-1)B_{j}^{\perp}\>j\in\left\\{1,4,6\right\\}$

Then, in this appendix we will show how to transform $\mathbbm{B}_{Q}^{v}$
into an operator $\mathbbm{B}_{Q}^{\gamma}$ in terms of binary operators with
outcomes $-1$ and $1$. First, note that:

$\displaystyle A_{i}^{1}+A_{i}^{\perp}$ $\displaystyle=$
$\displaystyle\mathbbm{I}$ (30) $\displaystyle B_{j}^{1}+B_{j}^{\perp}$
$\displaystyle=$ $\displaystyle\mathbbm{I}$ (31)

for all $i\in\left\\{1,2,3\right\\}$ and $j\in\left\\{1,4,6\right\\}$
respectively. In consequence:

$\displaystyle A_{i}=A_{i}^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\mathbbm{I}+A_{i}^{\prime}\right)$ (32)
$\displaystyle B_{j}=B_{j}^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\mathbbm{I}+B_{j}^{\prime}\right)$ (33)

Second, we apply (32), (33) to replace each $A_{i}$ and $B_{j}$ in the
definition of $\mathbbm{B}_{Q}^{v}$ we obtain:

$\displaystyle\mathbbm{B}_{Q}^{v}=-\left(\frac{\mathbbm{I}\otimes\mathbbm{I}+A^{\prime}_{2}\otimes\mathbbm{I}}{2}\right)-\left(\frac{\mathbbm{I}\otimes\mathbbm{I}+\mathbbm{I}\otimes
B^{\prime}_{1}}{2}\right)$
$\displaystyle-2\left(\frac{\mathbbm{I}\otimes\mathbbm{I}+\mathbbm{I}\otimes
B^{\prime}_{4}}{2}\right)+\left(\frac{\mathbbm{I}\otimes\mathbbm{I}+A^{\prime}_{1}\otimes\mathbbm{I}+\mathbbm{I}\otimes
B^{\prime}_{1}+A^{\prime}_{1}\otimes B^{\prime}_{1}}{4}\right)$
$\displaystyle+\left(\frac{\mathbbm{I}\otimes\mathbbm{I}+A^{\prime}_{1}\otimes\mathbbm{I}+\mathbbm{I}\otimes
B^{\prime}_{4}+A^{\prime}_{1}\otimes
B^{\prime}_{4}}{4}\right)+\left(\frac{\mathbbm{I}\otimes\mathbbm{I}+A^{\prime}_{2}\otimes\mathbbm{I}+\mathbbm{I}\otimes
B^{\prime}_{1}+A^{\prime}_{2}\otimes B^{\prime}_{1}}{4}\right)$
$\displaystyle+\left(\frac{\mathbbm{I}\otimes\mathbbm{I}+A^{\prime}_{2}\otimes\mathbbm{I}+\mathbbm{I}\otimes
B^{\prime}_{4}+A^{\prime}_{2}\otimes
B^{\prime}_{4}}{4}\right)-\left(\frac{\mathbbm{I}\otimes\mathbbm{I}+A^{\prime}_{1}\otimes\mathbbm{I}+\mathbbm{I}\otimes
B^{\prime}_{6}+A^{\prime}_{1}\otimes B^{\prime}_{6}}{4}\right)$
$\displaystyle+\left(\frac{\mathbbm{I}\otimes\mathbbm{I}+A^{\prime}_{2}\otimes\mathbbm{I}+\mathbbm{I}\otimes
B^{\prime}_{6}+A^{\prime}_{2}\otimes
B^{\prime}_{6}}{4}\right)-\left(\frac{\mathbbm{I}\otimes\mathbbm{I}+A^{\prime}_{3}\otimes\mathbbm{I}+\mathbbm{I}\otimes
B^{\prime}_{1}+A^{\prime}_{3}\otimes B^{\prime}_{1}}{4}\right)$
$\displaystyle+\left(\frac{\mathbbm{I}\otimes\mathbbm{I}+A^{\prime}_{3}\otimes\mathbbm{I}+\mathbbm{I}\otimes
B^{\prime}_{4}+A^{\prime}_{3}\otimes B^{\prime}_{4}}{4}\right)$

After some algebraic simplifications we obtain:

$\displaystyle
4\mathbbm{B}_{Q}^{v}+4\mathbbm{I}\otimes\mathbbm{I}=A^{\prime}_{1}\otimes\mathbbm{I}+A^{\prime}_{2}\otimes\mathbbm{I}-\mathbbm{I}\otimes
B^{\prime}_{1}-\mathbbm{I}\otimes B^{\prime}_{4}+A^{\prime}_{1}\otimes
B^{\prime}_{1}+A^{\prime}_{1}\otimes B^{\prime}_{4}$
$\displaystyle+A^{\prime}_{2}\otimes B^{\prime}_{1}+A^{\prime}_{2}\otimes
B^{\prime}_{4}-A^{\prime}_{1}\otimes B^{\prime}_{6}+A^{\prime}_{2}\otimes
B^{\prime}_{6}-A^{\prime}_{3}\otimes B^{\prime}_{1}+A^{\prime}_{3}\otimes
B^{\prime}_{4}$ (34)

If now we use observables $B_{1}^{\prime\prime}=-B^{\prime}_{1}$,
$B_{4}^{\prime\prime}=-B^{\prime}_{4}$ and
$A_{3}^{\prime\prime}=-A^{\prime}_{3}$ which are just re-labeling of outcomes
for the observables $B^{\prime}_{1},B^{\prime}_{4},A^{\prime}_{3}$ we have the
following relation:

$\displaystyle
4\mathbbm{B}_{Q}^{v}+4\mathbbm{I}\otimes\mathbbm{I}=A^{\prime}_{1}\otimes\mathbbm{I}+A^{\prime}_{2}\otimes\mathbbm{I}+\mathbbm{I}\otimes
B^{\prime\prime}_{1}+\mathbbm{I}\otimes
B^{\prime\prime}_{4}-A^{\prime}_{1}\otimes
B_{1}^{{}^{\prime\prime}}-A^{\prime}_{1}\otimes B_{4}^{\prime\prime}$
$\displaystyle-A^{\prime}_{2}\otimes
B_{1}^{{}^{\prime\prime}}-A^{\prime}_{2}\otimes
B_{4}^{{}^{\prime\prime}}-A^{\prime}_{1}\otimes
B_{6}^{\prime}+A^{\prime}_{2}\otimes
B^{\prime}_{6}-A^{\prime\prime}_{3}\otimes
B_{1}^{{}^{\prime\prime}}+A_{3}^{{}^{\prime\prime}}\otimes
B^{\prime\prime}_{4}$

Now, in the article we used the alternative form of $I_{3322}$ in terms of
binary operators with outputs ${+1,-1}$ whose corresponding Bell operator is:

$\displaystyle\mathbbm{B}_{Q}^{\gamma}=A^{\prime}_{1}\otimes\mathbbm{I}+A^{\prime}_{2}\otimes\mathbbm{I}+\mathbbm{I}\otimes
B^{\prime\prime}_{1}+\mathbbm{I}\otimes
B^{\prime\prime}_{4}-A^{\prime}_{1}\otimes
B_{1}^{{}^{\prime\prime}}-A^{\prime}_{1}\otimes B_{4}^{\prime\prime}$
$\displaystyle-A^{\prime}_{2}\otimes
B_{1}^{{}^{\prime\prime}}-A^{\prime}_{2}\otimes
B_{4}^{{}^{\prime\prime}}-A^{\prime}_{1}\otimes
B_{6}^{\prime}+A^{\prime}_{2}\otimes
B^{\prime}_{6}-A^{\prime\prime}_{3}\otimes
B_{1}^{{}^{\prime\prime}}+A_{3}^{{}^{\prime\prime}}\otimes
B^{\prime\prime}_{4}$

By proper identification of terms in (A.3) and (A.3), we obtain the operator
identity:

$\mathbbm{B}_{Q}^{\gamma}=4\mathbbm{B}_{Q}^{v}+4\mathbbm{I}\otimes\mathbbm{I}$
(35)

Additionally, from the definition of Tsirelson bounds $\mathcal{B}_{Q}^{v}$
and $\mathcal{B}_{Q}^{\gamma}$ we have:

$\sup_{\begin{subarray}{c}{\rho\in\mathcal{S}_{2}}\\\
{A_{1},A_{2},A_{3},B_{1},B_{2},B_{3}\in\mathcal{L}(\mathbbm{C}^{2})}\end{subarray}}\mathrm{Tr}(\mathbbm{B}_{Q}^{v}\rho)=\mathcal{B}_{Q}^{v}$
(36) $\sup_{\begin{subarray}{c}{\rho\in\mathcal{S}_{2}}\\\
{A_{1},A_{2},A_{3},B_{1},B_{2},B_{3}\in\mathcal{L}(\mathbbm{C}^{2})}\end{subarray}}\mathrm{Tr}(\mathbbm{B}_{Q}^{\gamma}\rho)=\mathcal{B}_{Q}^{\gamma}$
(37)

but in reference [45] is shown that a lower bound for $\mathcal{B}_{Q}^{v}$ is
$0.25$ and an upper bound is $0.25085...$, then identity (35) and Tsirelson
bound definition implies:

$1\leq\mathcal{B}_{Q}^{\gamma}-4\leq 1.0034$ (38)

which are the bounds used in section A.2 of the Appendix.

## References

  * [1] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009), URL: https://doi.org/10.1103/RevModPhys.81.865.
  * [2] F. G. S. L. Brandão, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, Phys. Rev. Lett. 111, 250404 (2013), URL: https://doi.org/10.1103/PhysRevLett.111.250404.
  * [3] E. Chitambar and G. Gour, Rev. Mod. Phys. 91, 025001 (2019), URL: https://doi.org/10.1103/RevModPhys.91.025001.
  * [4] J. Barrett,  Phys. Rev. A 75, 032304 (2007), URL: https://doi.org/10.1103/PhysRevA.75.032304.
  * [5] R. Gallego and L. Aolita, Phys. Rev. X 5, 041008 (2015), URL: https://doi.org/10.1103/PhysRevX.5.041008.
  * [6] M. Allingham, M. L. Burstein,  Resource Allocation and Economic Policy, Palgrave Macmillan UK (1976), URL: https://doi.org/10.1007/978-1-349-02673-9.
  * [7] A. Kakhbod, Resource Allocation in Decentralized Systems with Strategic Agents: An Implementation Theory Approach( Springer-Verlag New York, 2013), URL: https://doi.org/10.1007/978-1-4614-6319-1.
  * [8] H. Luss, T. Russell Hsing, V. K. N. Lau, Equitable Resource Allocation, Models, Algorithms, and Applications (John Wiley and Sons, 2012).
  * [9] I. A. Ushakov, Optimal Resource Allocation: With Practical Statistical Applications and Theory (John Wiley and Sons, 2013).
  * [10] J. Bredin, R.T. Maheswaran , C. Imer, T. Basar, D. Kotz, D. Rus, 4th International Conference on Autonomous Agents .
  * [11] Q. D. Lã, Y. H. Chew, B. Soong, Potential Game Theory, Applications in Radio Resource Allocation (Springer International Publishing, 2016), URL: https://doi.org/10.1007/978-3-319-30869-2.
  * [12] L. Georgiadis, M. Neely, L. Tassiulas, Foundations and Trends in Networking: Vol. 1: No. 1, pp 1-144 (2006), URL: https://doi.org/10.1561/1300000001.
  * [13] R. Matyssek, J. Koricheva, H. Schnyder, D. Ernst, J. C. Munch, W. Oßwald, H. Pretzsch, Growth and defense in Plants: Resource Allocation at multiple scale (Springer-Verlag , 2012), URL: https://doi.org/10.1007/978-3-642-30645-7.
  * [14] P. Skrzypczyk, I. Šupić, and D. Cavalcanti, Phys. Rev. Lett. 122, 130403 (2019), URL: https://doi.org/10.1103/PhysRevLett.122.130403.
  * [15] S. Designolle, M. Farkas, J. Kaniewski. New. J. Phys. 21 113053 (2019), URL: https://doi.org/10.1088/1367-2630/ab5020.
  * [16] M. Oszmaniec and T. Biswas, Quantum 3, 133 (2019), URL: https://doi.org/10.22331/q-2019-04-26-133.
  * [17] R. Uola, T. Kraft, J. Shang, X.-D. Yu, and O. Guhne, Phys. Rev. Lett. 122, 130404 (2019), https://doi.org/10.1103/PhysRevLett.122.130404.
  * [18] R. Takagi. and B. Regula, Phys. Rev. X 9, 031053 (2019), URL: https://doi.org/10.1103/PhysRevX.9.031053.
  * [19] P. Skrzypczyk and N. Linden,  Phys. Rev. Lett. 122, 140403 (2019), URL: https://doi.org/10.1103/PhysRevLett.122.140403.
  * [20] R. Takagi, B. Regula, K. Bu, Z.-W. Liu, and G. Adesso,  Phys. Rev. Lett. 122, 140402 (2019), URL: https://doi.org/10.1103/PhysRevLett.122.140402.
  * [21] C. Berge, Graphs and Hypergraphs (North-Holland Mathematical Library, 1973)
  * [22] I. D. Ivanović, J. Phys. A: Math. Gen. 14, 3241 (1981), URL: https://doi.org/10.1088/0305-4470/14/12/019.
  * [23] W. K. Wootters and B. D. Fields, Ann. Phys. 191, 363 (1989), URL: https://doi.org/10.1016/0003-4916(89)90322-9.
  * [24] D. McNulty, B. Pammer, and S. Weigert,  J. Math. Phys. 57, 032202 (2016), URL: https://doi.org/10.1063/1.4943301.
  * [25] J. Czartowski, D. Goyeneche, K. Życzkowski, J. Phys. A: Math. Theor. 51, 305302 (2018), URL: https://doi.org/10.1088/1751-8121/aac973.
  * [26] M. Horodecki, P. Horodecki, and J. Oppenheim,  Phys. Rev. A 67, 062104 (2003), URL: https://doi.org/10.1103/PhysRevA.67.062104.
  * [27] J. Oppenheim, M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 89, 180402 (2002), URL: https://doi.org/10.1103/PhysRevLett.89.180402.
  * [28] M. Horodecki, K. Horodecki, P. Horodecki, R. Horodecki, J. Oppenheim, A. Sen(De), and U. Sen, Phys. Rev. Lett. 90, 100402 (2003), URL: https://doi.org/10.1103/PhysRevLett.90.100402.
  * [29] D. Yang, K. Horodecki, and A. Winter, Phys. Rev. Lett. 123, 170501 (2019), URL: https://doi.org/10.1103/PhysRevLett.123.170501.
  * [30] K. Horodecki, M. Horodecki, P. Horodecki, and J. Oppenheim, Phys. Rev. Lett. 94, 160502 (2005), URL: https://doi.org/10.1103/PhysRevLett.94.160502.
  * [31] F. G. S. L. Brandão, R. Ramanathan, A. Grudka, K. Horodecki, M. Horodecki, P. Horodecki, T. Szarek & H. Wojewódka,  Nat Commun 7, 11345 (2016), URL: https://doi.org/10.1038/ncomms11345.
  * [32] M. Kessler, R. Arnon-Friedman,  IEEE Journal on Selected Areas in Information Theory, vol. 1, no. 2, pp. 568-584, URL: https://doi.org/10.1109/JSAIT.2020.3012498.
  * [33] D. Saha, P. Horodecki and M. Pawłowski, New J. Phys. 21, 093057 (2019), URL: https://doi.org/10.1088/1367-2630/ab4149.
  * [34] J.I. de Vicente,  J. Phys. A: Math. Theor. 47, 424017 (2014), URL: https://doi.org/10.1088/1751-8113/47/42/424017.
  * [35] K. Horodecki, A. Grudka, P. Joshi, W. Kłobus, J. Łodyga, Phys. Rev. A 92, 032104 (2015), URL: https://doi.org/10.1103/PhysRevA.92.032104.
  * [36] D. Saha and R. Ramanathan,  Phys. Rev. A 95, 030104(R) (2017), URL: https://doi.org/10.1103/PhysRevA.95.030104.
  * [37] Kimble, H. J.  Nature. 453 (7198): 1023–1030 (2008), URL: https://doi.org/10.1038/nature07127.
  * [38] R. Van Meter (2014), Quantum Networking (John Wiley and Sons, 2014).
  * [39] N. Kalb, A. A. Reiserer, P. C. Humphreys, J. J. W. Bakermans, S. J. Kamerling, N. H. Nickerson, S. C. Benjamin, D. J. Twitchen, M. Markham,  Science. 356 (6341): 928–932 (2017), URL: https://doi.org/10.1126/science.aan0070.
  * [40] S. Wehner, D. Elkouss, R. Hanson,  Science Vol. 362, Issue 6412 (2018), URL: https://doi.org/10.1126/science.aam9288.
  * [41] E. Kashefi and A. Pappa, Cryptography 1(2), 12 (2017), URL: https://doi.org/10.3390/cryptography1020012.
  * [42] S. Bravyi, D. Gosset, R.König, Science 362, Issue 308-311 (2018), URL: https://doi.org/10.1126/science.aar3106.
  * [43] J. Czartowski, D. Goyeneche, M. Grassl, K. Życzkowski,  Phys. Rev. Lett. 124, 090503 (2020), URL: https://doi.org/10.1103/PhysRevLett.124.090503.
  * [44] M. Araujo, M. T. Quintino, C. Budroni, M. T. Cunha, and A. Cabello,  Phys. Rev. A 88, 022118 (2013), URL: https://doi.org/10.1103/PhysRevA.82.022116.
  * [45] K. F. Pál and T. Vértesi,  Phys. Rev. A 82, 022116 (2010), URL: https://doi.org/10.1103/PhysRevA.82.022116. .