# Deterministic Entanglement Transmission on Series-Parallel Quantum Networks

Xiangyi Meng  Center for Complex Network Research and Department of Physics,
Northeastern University, Boston, Massachusetts 02115, USA  Jianxi Gao
Department of Computer Science, Rensselaer Polytechnic Institute, Troy, New
York 12180, USA  Shlomo Havlin  Department of Physics, Bar-Ilan University,
52900 Ramat-Gan, Israel  Center for Polymer Studies and Department of Physics,
Boston University, Boston, Massachusetts 02215, USA  Andrei E. Ruckenstein
Center for Polymer Studies and Department of Physics, Boston University,
Boston, Massachusetts 02215, USA

###### Abstract

The performance of the entanglement transmission task, i.e., distributing
entanglement between two arbitrarily distant nodes in a large-scale quantum
network (QN), is notably benchmarked by the classical entanglement percolation
(CEP) scheme. Improvement of entanglement transmission beyond CEP can only be
achieved, with great loss of generality, by nonscalable strategies and/or on
restricted QN topologies. The possibility of finding a better scheme than CEP
that remains scalable and adaptable arises eventually from a new concurrence
percolation theory (ConPT), essentially an alternative and more effective
mapping of QN than classical percolation, that roots not in probability but in
concurrence—a fundamental measure of bipartite entanglement. ConPT suggests
using not generically probabilistic but deterministic protocols to produce a
single highly entangled state between two nodes. In light of the ConPT
mapping, here we formalize a specific deterministic entanglement transmission
(DET) scheme that is fully analogous to resistor circuit analysis,
implementing the corresponding series and parallel rules by deterministic
entanglement swapping and purification protocols, respectively. The DET is
designed for general $d$-dimensional information carriers and is scalable and
adaptable for any series-parallel QN, as it reduces the entanglement
transmission task to a purely topological problem of calculating path
connectivity. Unlike CEP, different levels of optimality in terms of
$k$-concurrences can be observed in DET, depending on specific QN topologies.
More interestingly, our results suggest that the nesting feature of the well-
known repeater protocol for qubits is not helpful for optimizing the final
concurrence, the proof of which necessarily relies on a special reverse
arithmetic mean–geometric mean (AM–GM) inequality.

##  I Introduction

Many quantum communication technologies find their roots in entanglement, one
of the most vital quantum resources [1]. However, the nonlocality of
entanglement facilitating quantum communication becomes unrealistic at large
space scales where faithful quantum information is submerged in thermal
incoherent noise [2], making it very inefficient to directly establish point-
to-point long-distance entanglement. To overcome this limit of classical
locality and _indirectly_ distribute entanglement among arbitrary parties
requires utilizing some intermediate parties as “relays” and applying to them
a variety of fundamental communication protocols, including entanglement
swapping [3, 4] and entanglement purification [5, 6], as well as composite
protocols that are usually recursive and more complex, such as the nested
purification protocol using quantum repeaters [7, 8, 9], etc.

Since all parties are physically separated from one another so that locality
holds, only local operations and classical communication (LOCC) can be applied
to the information carriers (e.g., qubits) of different parties. Such
consideration allows us to treat each party as a _node_ that consists of a
collection of information carriers, and treat each bipartite (resp.
multipartite) entangled state formed by inter-node information carriers as a
_link_ (resp. hyperlink), essentially giving rise to a complex network
representation of the structure of locality on entanglement resources, widely
known as the _quantum network_ (QN) [10]. A number of QN-based protocols have
since been introduced, e.g., $q$-swapping [11] and path routing [12],
combining communication protocols with network science [13]. In practice,
basic communication protocols on small-scale QNs have been experimentally
demonstrated using diamond nitrogen-vacancy centers [14, 15, 16] and ion traps
[17, 18]. It is believed that practical QNs are easier to _scale_ than
universal quantum computing platforms, since the nonlocality of information
carriers only needs to be maintained within each node but not across the
global QN platform. The future of a worldwide large-scale QN, namely, a
“quantum Internet” [19] is hence foreseeable. This, on the other hand,
urgently demands a more efficient design of QN-based communication protocols
that should be _scalable_ with network size and _adaptable_ to different
network topology, in accordance with the well-developed methods of complex
network analysis [20].

The particular task of entanglement distribution [21] between _two_ arbitrary
nodes—which we coin as _entanglement transmission_ in the QN context—is of
special interest. The discovery of an exact mapping between entanglement
transmission and classical bond percolation theory [22] gives rise to a
straightforward entanglement transmission scheme, called classical
entanglement percolation (CEP) [10], that greatly reduces the task of
distributing a singlet (i.e., two maximally entangled qubits) between two
nodes to a pure percolation problem. The CEP scheme is naturally scalable and
adaptable for arbitrary QN. However, immediately after it was introduced, the
CEP was proven not to produce the _optimal_ probability of obtaining such a
singlet [10]. The problem of calculating the optimal probability was later
proven to be extremely complicated even for a one-dimensional (1D) chain [23,
24]. Even just improving this probability on large-scale QN can only be
systematically done for special network topology with a few other limitations
[23], making it difficult to find a better entanglement transmission scheme
than CEP that is scalable and adaptable in the same satisfactory way.

This difficulty eventually leads to the questioning of the exclusivity of the
classical-percolation-theory-based mapping itself: it was only recently
realized that another mapping from entanglement transmission to a new
statistical theory called concurrence percolation theory (ConPT) [25] may
exist, bearing its name due to its analogy to classical percolation yet
rooting not in probability but in concurrence—a fundamental measure of
bipartite entanglement [26, 27]. The ConPT predicts a lower entanglement
transmission threshold than the classical percolation threshold, exhibiting a
“quantum advantage” that is purely structural, independent of nontopological
details. To achieve the ConPT threshold, it has been suggested that
_deterministic_ communication protocols that only produce outcomes with
certainty (up to unitary equivalence) should be used, so that the protocols
can be applied recursively without becoming more and more involved with
probabilistic distributions [25].

Figure 1:  Demonstration of the deterministic entanglement transmission (DET)
scheme on arbitrary series-parallel quantum network (QN) with $d$-dimensional
information carriers. From an initial QN (i), a final state (viii) can always
be deterministically produced between Alice (A) and Bob (B) using the series
and parallel rules (Table 2).

In light of this, here we formalize a specific _deterministic entanglement
transmission_ (DET) scheme for arbitrary two nodes in QN, provided that the
network topology between the two nodes is series-parallel, i.e., can be
decomposed into only two connectivity rules— _series and parallel rules_ , in
accordance with resistor circuit analysis. This DET scheme is built by
explicitly expressing the series and parallel rules as two functions—a
swapping function $\mathcal{S}$ and a purification function $\mathcal{P}$,
which can be implemented by two fundamental communication protocols—
_entanglement swapping and purification_ , respectively. Such an
implementation is possible since both protocols can be tuned to be
deterministic. The formulation of our scheme has two important features:

  1. 1.

The DET scheme is fully scalable and adaptable for arbitrary series-parallel
QN, as it _reduces_ the entanglement transmission task to a problem of
calculating path connectivity between arbitrary two nodes, fully analogous to
calculating the total resistance in a series-parallel resistor network.

  2. 2.

The swapping function $\mathcal{S}$ and purification function $\mathcal{P}$
are defined for _$d$ -dimensional_ information carriers (qutrits, ququarts,
etc.) in general, making the DET an all-purpose scheme for more exotic design
of quantum information devices.

In particular, we show that unlike the CEP scheme, different levels of
_optimality_ can indeed be observed in the DET, depending on specific QN
topologies (Fig. 2), given in terms of a family of $d$ concurrence monotones
[27] (see Appendix B). Using majorization theory, we calculate inequalities
for the $\mathcal{S}$ and $\mathcal{P}$ functions and their recursive
combinations, confirming that to some degree, the deterministic outcomes of
DET are still the best, even when we relax the requirement of determinacy and
detune the entanglement swapping and purification protocols to be
probabilistic. The results are summarized in Table 1.

Table 1: Optimality of DET in terms of $k$-concurrences $C_{k}$ (Appendix B) with different QN topologies (Fig. 2).  Network topology | DET optimizes…
---|---
Simple series | Average $C_{d}$
Simple parallel | Average $C_{k}$ ($1<k\leq d$)
Parallel-then-series | Average $C_{d}$
Series-then-parallel | Worst-case $C_{d}$ ($d=2$ only)
Series-parallel | Worst-case $C_{d}$ ($d=2$ only)

Interestingly, our results suggest that the well-known nested purification
protocol [7] introduced for a 1D chain of parties with multiple bipartite
entangled qubits shared in between—which is essentially a parallel-then-series
QN—is _not_ a good strategy for optimizing concurrence. Instead of applying
entanglement swapping and purification in a nested way, we argue that the best
strategy is to apply entanglement purification _once for all_ between every
two adjacent nodes, and then perform entanglement swapping along the purified
links. The proof of this result has not been derived before, to our best
knowledge, since it relies on a special reverse arithmetic mean–geometric mean
(AM–GM) inequality which we will prove in Appendix D. Note that the
unnecessity of nesting could greatly simplify communication protocol design,
suggesting that DET is practically more preferable.

##  II Quantum Network

In this paper, we focus on a “minimal” version of QN where each link is a
$d^{2}$-dimensional bipartite pure state shared by the two connected nodes
[10]. Such a bipartite pure state can always be written as
$\sum_{\mu,\nu}\Psi_{\mu\nu}\left|\mu\nu\right\rangle\in\mathbb{C}^{d\times
d}$ which allows matricization, i.e., to be mapped to a $d\times d$ matrix
$\bm{\Psi}$ with elements $\Psi_{\mu\nu}$, $\mu,\nu=1,2,\cdots,d$. Left or
right multiplication of $\bm{\Psi}$ by a unitary matrix corresponds to a local
unitary transformation performed by either of the two parties, respectively.
Thus, up to unitary equivalence, the state can always be locally transformed
into a diagonal form
$\left|\bm{\lambda}\right\rangle=\sum_{j=1}^{d}\sqrt{\lambda_{j}}\left|jj\right\rangle$
by a singular value decomposition, so that each link is exclusively
represented by $d$ positive Schmidt numbers,
$\bm{\lambda}=\left(\lambda_{1},\lambda_{2},\cdots,\lambda_{d}\right)$. Note
that $\bm{\lambda}$ is always subject to the normalization constraint
$\sum_{j=1}^{d}\lambda_{j}\equiv 1$. In particular, when $d=2$ (qubits), there
is only one degree of freedom, and thus each link can always be represented by
only one parameter w.l.o.g. [25].

This representation by a sequence of Schmidt numbers is effective, since
singular values allow a built-in majorization _preorder_ powered by
matricization (see Appendix A for basic notions of the majorization theory).
Obviously, this only works for _bipartite_ states. Tripartite states, for
example, although can be completely classified by five parameters including
one phase and four moduli [28], are not yet compatible with such a preorder,
leaving us a difficult _“(quantum) three-body problem”_.

Note that within a traditional network representation, each link is also a
“bipartite” notation since it essentially connects two nodes. Hence, it seems
a perfect match to study the statistics of a large number of bipartite states
using a “bipartite” network theory, namely a _“statistical theory of (quantum)
two-body problems”_. In contrast, a “complex” QN consisting of multipartite
entangled states as “hyperlinks” is beyond the scope of the present paper,
since the difficulties coming from both quantum many-body theory and
“hypernetwork” theory are actually twofold.

###  II.1 Quantum Communication Protocols

A quantum communication protocol is a set of LOCC that allows entanglement to
be distributed among different parties. We briefly review two fundamental
bipartite protocols which are of special interest to us:

####  II.1.1 Entanglement swapping

Suppose there are three parties, Alice–Relay–Bob (A–R–B), who share two
bipartite states,
$\left|\bm{\lambda}_{a}\right\rangle=\sum_{j=1}^{d}\sqrt{\lambda_{a,j}}\left|jj\right\rangle_{\text{AR}}$
(shared between A–R) and
$\left|\bm{\lambda}_{b}\right\rangle=\sum_{k=1}^{d}\sqrt{\lambda_{b,k}}\left|kk\right\rangle_{\text{RB}}$
(shared between R–B), respectively. This can be represented as a QN of three
nodes and two links. Although A and B do not directly share any entanglement,
a swapping protocol [4] can indeed be performed, which requires a set of
quantum measurements to be conducted on R. The result, which is probabilistic,
is an ensemble of different bipartite states directly shared between A–B,
usually entangled. To be specific, if we matricize
$\left|\bm{\lambda}_{a}\right\rangle$ and
$\left|\bm{\lambda}_{b}\right\rangle$ and denote them by two diagonal matrices
$\text{diag}(\bm{\lambda}_{a})^{1/2}$ and
$\text{diag}(\bm{\lambda}_{b})^{1/2}$, then after R performs the measurements,
the (unnormalized) result state between A and B can always be written in the
matrix form as

$\bm{\Psi}^{\alpha}=\text{diag}(\bm{\lambda}_{a})^{1/2}\mathbf{X}^{\alpha}\text{diag}(\bm{\lambda}_{b})^{1/2}$
(1)

for outcome $\alpha$, subject to probability
$p_{\alpha}=\left\|\bm{\Psi}^{\alpha}\right\|_{2}^{2}$ [29]. The quantum
measurements are encoded by a set of arbitrary matrices
$\\{\mathbf{X}^{\alpha}\\}$ constrained by the completeness relation,

$\sum_{\alpha}X^{\alpha,*}_{\mu\nu}X^{\alpha}_{\mu^{\prime}\nu^{\prime}}=\delta_{\mu\mu^{\prime}}\delta_{\nu\nu^{\prime}},$
(2)

which is sufficient for $\\{\mathbf{X}^{\alpha}\\}$ to denote a positive
operator-valued measure (POVM).

####  II.1.2 Entanglement purification

Given
$\left|\bm{\lambda}\right\rangle=\sum_{j=1}^{m}\sqrt{\lambda_{j}}\left|jj\right\rangle$
for $m\in\mathbb{Z}^{+}$, what is the maximum probability to convert it to a
new state
$\left|\mathbf{x}\right\rangle=\sum_{j=1}^{m}\sqrt{x_{j}}\left|jj\right\rangle$?
Such a probability is indeed well known and can be achieved by a purification
protocol [6]. To be specific, this maximum probability is given by

$p_{\bm{\lambda}\to\mathbf{x}}=\min_{0\leq
k<m}\frac{1-\sum_{j=1}^{k}\lambda^{\downarrow}_{j}}{1-\sum_{j=1}^{k}x^{\downarrow}_{j}},$
(3)

and the LOCC implementation has been explicitly given in Ref. [30]. It is
clear from Eq. (3) that $p_{\bm{\lambda}\to\mathbf{x}}=1$ when
$\mathbf{x}\succ\bm{\lambda}$, a result originally given by Nielsen’s theorem
[31]. Also, note that it is possible to obtain a new state
$\left|\mathbf{x}\right\rangle$ of a different dimension $d$ if $m\geq d$,
since one can always pad $\mathbf{x}$ by zeros [32] and build a new sequence
of $m$ Schmidt numbers,
$\mathbf{x}^{\prime}=\mathbf{x}\oplus(\overbrace{0,0,\cdots,0}^{m-d})$, which
inserted into Eq. (3) yields the maximum probability
$p_{\bm{\lambda}\to\mathbf{x}^{\prime}}$.

###  II.2 Entanglement Transmission Schemes

We denote a _scheme_ by a (possibly infinite) number of quantum communication
protocols that can be collectively applied to QN. One of the most important
entanglement transmission schemes is the classical entanglement percolation
(CEP) scheme for $d=2$. The CEP scheme consists of two steps [10]: (i) Use the
purification protocol to convert each link to a singlet, the probability of
which is given by Eq. (3). (ii) Find a path of links connecting two arbitrary
nodes A and B that have all been successfully converted to singlets, and apply
the swapping protocol to them one by one. The final state between A and B is
guaranteed to be a singlet by Eq. (1). The probability of finding such a path
can be considered as a measure of the entanglement transmission ability
between A and B, determined by the network topology. This probability,
however, is known to be nonoptimal, since for some special topology, e.g., a
honeycomb with double links between every two connected nodes, adding a
“preprocessing” step of entanglement swapping to the CEP scheme may change the
network topology to a triangular type and potentially increase the probability
of establishing a singlet between A and B [10].

A deterministic entanglement transmission (DET) scheme, on the other hand,
does not use generic probabilistic protocols to increase the singlet
conversion probability, but uses _deterministic_ protocols to produce a single
highly entangled state between A and B (demonstrated in Fig. 1). Note that
after performing each deterministic protocol, the new system is still an
intact QN, i.e., a pure state as a whole, not a probabilistic ensemble. This
makes DET easily scalable with network size.

As we will see, the specific DET scheme we introduce utilizes only
deterministic bipartite protocols, which can be applied to arbitrary series-
parallel QN.

##  III A Deterministic Entanglement Transmission Scheme

###  III.1 Series-Parallel Network Topology

(a) Simple series QN.

(b) Simple parallel QN.

(c) Parallel-then-series QN.

(d) Series-then-parallel QN.

(e) Series-parallel QN.

(f) General QN.

Figure 2: Different QN topologies between A and B.

Whether a network is series-parallel or not depends on which two nodes of
interest are chosen [33]. Given two nodes A and B, the network topology can be
characterized into different categories (Fig. 2). All topologies between A and
B given in Figs. 2-2 are considered series-parallel, but the topology in Fig.
2 is not due to the existing “bridge”. Note also that most realistic complex
networks can be approximated as series-parallel, since loops are rare and can
usually be ignored in infinite-dimensional systems [34], leaving a tree-like
structure which is essentially series-parallel between any two nodes.

###  III.2 Series and Parallel Rules

Table 2: Series and parallel rules for DET.  Series rule
(Swapping) |  $\bm{\lambda}=\bm{\lambda}_{1}\texttildelow\mathcal{S}\texttildelow\bm{\lambda}_{2}\texttildelow\mathcal{S}\texttildelow\cdots\texttildelow\mathcal{S}\texttildelow\bm{\lambda}_{n}$,
where
$\mathcal{S}:\left(\mathbb{R}_{+}^{d},\mathbb{R}_{+}^{d}\right)\to\mathbb{R}_{+}^{d}$
[Eq. (4)].
---|---
Parallel rule
(Purification) |  $\bm{\lambda}=\mathcal{P}(\bm{\lambda}_{1}\otimes\bm{\lambda}_{2}\otimes\cdots\otimes\bm{\lambda}_{n})$,
where $\mathcal{P}:\mathbb{R}_{+}^{d^{n}}\to\mathbb{R}_{+}^{d}$ [Eq. (10)].

The series and parallel rules of our DET scheme are given in Table 2, where
$\otimes$ stands for the Kronecker product, and, for simplicity, the infix
notation $\texttildelow$ is introduced for representing binary operations,
i.e., $x\texttildelow f\texttildelow y\equiv f(x,y)$, which is left grouping,
i.e., $x\texttildelow f\texttildelow y\texttildelow f\texttildelow
z\equiv(x\texttildelow f\texttildelow y)\texttildelow f\texttildelow z$. The
series and parallel rules are implementable by the entanglement swapping and
purification protocols (Section II.1), respectively, in a deterministic
manner.

####  III.2.1 Swapping function $\mathcal{S}(\mathbf{x},\mathbf{y})$

_Definition.—_ We define
$\mathcal{S}:\left(\mathbb{R}_{+}^{d},\mathbb{R}_{+}^{d}\right)\to\mathbb{R}_{+}^{d}$
by

$\displaystyle\mathcal{S}(\mathbf{x},\mathbf{y})=d\times\sigma^{2}(\text{diag}(\mathbf{x}^{\downarrow})^{1/2}\mathbf{V}\text{diag}(\mathbf{y}^{\downarrow})^{1/2})$
(4)

where $\sigma(\bm{\Psi})$ denotes the singular values of matrix $\bm{\Psi}$,
and $\sigma^{2}(\bm{\Psi})$ the entry-wise square of $\sigma(\bm{\Psi})$, both
arranged in descending order so that
$\mathcal{S}(\mathbf{x},\mathbf{y})\equiv\left[\mathcal{S}(\mathbf{x},\mathbf{y})\right]^{\downarrow}$.
The matrix $\mathbf{V}$ is constant and unitary, with elements
$V_{\mu\nu}=d^{-1/2}\exp\left(-2\pi i\mu\nu/d\right)$, $\mu,\nu=1,2,\cdots,d$.

_Properties.—_ $\mathcal{S}(\mathbf{x},\mathbf{y})$ has the following
properties:

  * •

Permutation invariance ($\mathbf{x}\to\text{perm}(\mathbf{x})$):

$\mathcal{S}(\mathbf{x},\mathbf{y})=\mathcal{S}(\text{perm}(\mathbf{x}),\mathbf{y}).$
(5)

  * •

Trace preserving:

$\mbox{tr}(\mathcal{S}(\mathbf{x},\mathbf{y}))=\mbox{tr}(\mathbf{x})\mbox{tr}(\mathbf{y}).$
(6)

  * •

Isotone:

$\mathcal{S}(\mathbf{x},\mathbf{z})\succ\mathcal{S}(\mathbf{y},\mathbf{z}),\forall\mathbf{z}\text{
if }\mathbf{x}\succ\mathbf{y}.$ (7)

  * •

Commutativity:

$\mathcal{S}(\mathbf{x},\mathbf{y})=\mathcal{S}(\mathbf{y},\mathbf{x}).$ (8)

  * •

Associativity:

$\mathcal{S}(\mathcal{S}(\mathbf{x},\mathbf{y}),\mathbf{z})=\mathcal{S}(\mathbf{x},\mathcal{S}(\mathbf{y},\mathbf{z}))\text{
for }d\leq 3.$ (9)

Equations (5) and (8) hold because $\sigma(\bm{\Psi})$ is invariant under
Hermitian conjugate and unitary transformation of $\bm{\Psi}$. Equation (7)
holds as a result of Theorem 12 (see Appendix C), given the facts that
$\mathcal{S}$ is permutation invariant [Eq. (5)] and convex (see Lemma 1
below). Equation (6) can be inferred from its validity for the special case
$\mbox{tr}(\mathbf{x})=\mbox{tr}(\mathbf{y})=1$.

In particular, Eq. (9) is valid for $d\leq 3$ because its left-hand side (LHS)
and right-hand side (RHS) are both governed by two constraints: trace
preserving [Eq. (6)] and determinant preserving (see Lemma 2). Additionally,
employing the duality (see Lemma 3) produces two more constraints in terms of
“dual” trace preserving and “dual” determinant preserving. The latter,
however, can be shown to be equivalent to the original determinant preserving
constraint. Therefore, there are three independent constraints in total, and
thus the LHS must be equal to the RHS when $d\leq 3$. When $d>3$,
counterexamples can be easily found.

_Implementability by LOCC.—_ It remains to be shown that
$\mathcal{S}(\bm{\lambda}_{a},\bm{\lambda}_{b})$ can be implemented by the
entanglement swapping protocol [Eq. (1)] in a deterministic manner. This
construction was originally given in Ref. [27]. In short, given a special set
of matrices $\\{\mathbf{X}^{\alpha}\\}$ each with elements
$X^{\alpha}_{\mu\nu}=d^{-1}e^{-\alpha\left(d\mu+\nu\right)2\pi i/d^{2}-2\pi
i\mu\nu/d}$ for $\alpha=1,2,\cdots,d^{2}$, one can verify that
$\\{\mathbf{X}^{\alpha}\\}$ satisfies Eq. (2), thus denoting a POVM. Hence,
performing the swapping protocol and following Eq. (1), the (unnormalized)
result state $\bm{\Psi}^{\alpha}$ is obtained, for each outcome $\alpha$, with
elements
$\Psi^{\alpha}_{jk}=d^{-1}\sqrt{\lambda^{\downarrow}_{a,j}\lambda^{\downarrow}_{b,k}}e^{-\left(\alpha
dj+\alpha k+djk\right)2\pi i/d^{2}}$. Now, as long as Alice and Bob are shared
by the Relay the classical information of which $\alpha$ is obtained, they can
always rotate their shared state $\bm{\Psi}^{\alpha}$ by some phase
accordingly and transform it locally into a new $\alpha$-independent state
$\bm{\Psi}^{\prime}$ with elements
$\Psi^{\prime}_{jk}=d^{-1}\sqrt{\lambda^{\downarrow}_{a,j}\lambda^{\downarrow}_{b,k}}e^{-\left(2\pi
i\right)jk/d}$, the Schmidt numbers of which are then given by
$\mathcal{S}(\bm{\lambda}_{a},\bm{\lambda}_{b})$ [Eq. (4)] after normalization
of $\bm{\Psi}^{\prime}$ [27]. The independence of $\bm{\Psi}^{\prime}$ on
$\alpha$ confirms that the protocol is deterministic.

####  III.2.2 Purification function $\mathcal{P}(\mathbf{x})$

_Definition.—_ We define $\mathcal{P}:\mathbb{R}_{+}^{m}\to\mathbb{R}_{+}^{d}$
($m\geq d$) by the following pseudocode:

function $\mathcal{P}$($\mathbf{x}$): $\triangleright$
$\mathbf{x}^{\downarrow}=\left(x^{\downarrow}_{1},x^{\downarrow}_{2},\cdots,x^{\downarrow}_{m}\right)$

$s\leftarrow\sum_{j=1}^{m}x_{j}$

for $l\leftarrow 1,2,\cdots,d$ do

$\chi_{l}\leftarrow\max\left\\{x^{\downarrow}_{l},s/\left(d+1-l\right)\right\\}$

$s\leftarrow s-\chi_{l}$

end for

return $\bm{\chi}$ $\triangleright$
$\bm{\chi}=\left(\chi_{1},\cdots,\chi_{d}\right)$

end function

(10)

The above definition implies that $\mathcal{P}(\mathbf{x})$ is always given in
descending order,
$\mathcal{P}(\mathbf{x})\equiv\left[\mathcal{P}(\mathbf{x})\right]^{\downarrow}$,
and

$\left[\mathcal{P}(\mathbf{x})\right]_{j}\geq x^{\downarrow}_{j},\quad
j=1,2,\cdots,d.$ (11)

Moreover, if $\left[\mathcal{P}(\mathbf{x})\right]_{l}>x^{\downarrow}_{l}$ for
some $l$, then

$\left[\mathcal{P}(\mathbf{x})\right]_{j}=\left[\mathcal{P}(\mathbf{x})\right]_{l}\text{
and }\left[\mathcal{P}(\mathbf{x})\right]_{j}>x^{\downarrow}_{j}$ (12)

for all $l\leq j\leq d$.

_Properties.—_ $\mathcal{P}(\mathbf{x})$ has the following properties:

  * •

Permutation invariance ($\mathbf{x}\to\text{perm}(\mathbf{x})$):

$\mathcal{P}(\mathbf{x})=\mathcal{P}(\text{perm}(\mathbf{x})).$ (13)

  * •

Trace preserving:

$\mbox{tr}(\mathcal{P}(\mathbf{x}))=\mbox{tr}(\mathbf{x}).$ (14)

  * •

Isotone:

$\mathcal{P}(\mathbf{x})\succ\mathcal{P}(\mathbf{y})\text{ if
}\mathbf{x}\succ\mathbf{y}.$ (15)

  * •

Commutativity:

$\mathcal{P}(\mathbf{x}\otimes\mathbf{y})=\mathcal{P}(\mathbf{y}\otimes\mathbf{x}).$
(16)

  * •

Associativity:

$\mathcal{P}(\mathcal{P}(\mathbf{x}\otimes\mathbf{y})\otimes\mathbf{z})=\mathcal{P}(\mathbf{x}\otimes\mathcal{P}(\mathbf{y}\otimes\mathbf{z})).$
(17)

Equations (13), (14) and (16) all hold by definition [Eq. (10)]. And again,
Eq. (15) holds as a result of Theorem 12 (see Appendix C), given the facts
that $\mathcal{P}$ is permutation invariant [Eq. (13)] and convex (see Lemma 5
below).

In particular, Eq. (17) is valid because, on the one hand, by Eq. (15), we
have
$\mathcal{P}(\mathcal{P}(\mathbf{x}\otimes\mathbf{y})\otimes\mathbf{z})\succ\mathcal{P}(\mathbf{x}\otimes\mathbf{y}\otimes\mathbf{z})$;
on the other hand, since
$\mathcal{P}(\mathbf{x}\otimes\mathbf{y}\otimes\mathbf{z})\oplus(\overbrace{0,0,\cdots,0}^{d^{2}-d})\succ\mathcal{P}(\mathbf{x}\otimes\mathbf{y})\otimes\mathbf{z}$,
by Lemma 4 below we have
$\mathcal{P}(\mathbf{x}\otimes\mathbf{y}\otimes\mathbf{z})\succ\mathcal{P}(\mathcal{P}(\mathbf{x}\otimes\mathbf{y})\otimes\mathbf{z})$.
Thus, we must have
$\mathcal{P}(\mathbf{x}\otimes\mathbf{y}\otimes\mathbf{z})=\mathcal{P}(\mathcal{P}(\mathbf{x}\otimes\mathbf{y})\otimes\mathbf{z})$,
since $\mathcal{P}$ is permutation invariant. But the same is also true for
$\mathcal{P}(\mathbf{x}\otimes\mathcal{P}(\mathbf{y}\otimes\mathbf{z}))$.
Hence, together we have
$\mathcal{P}(\mathcal{P}(\mathbf{x}\otimes\mathbf{y})\otimes\mathbf{z})=\mathcal{P}(\mathbf{x}\otimes\mathcal{P}(\mathbf{y}\otimes\mathbf{z}))=\mathcal{P}(\mathbf{x}\otimes\mathbf{y}\otimes\mathbf{z})$.

_Implementability by LOCC.—_ The definition [Eq. (11)] implies that

$\displaystyle\mathcal{P}(\bm{\lambda})\oplus(\overbrace{0,0,\cdots,0}^{m-d})\succ\bm{\lambda}.$
(18)

Thus, any new bipartite state of Schmidt numbers $\mathcal{P}(\bm{\lambda})$
can be produced deterministically (i.e.,
$p_{\bm{\lambda}\to\mathcal{P}(\bm{\lambda})}=1$) from a bipartite state of
Schmidt numbers $\bm{\lambda}$ by the entanglement purification protocol [Eq.
(3)].

##  IV Optimality

We have shown that both the series and parallel rules in Table 2 are
implementable by the entanglement swapping and purification protocols (Section
II.1). Naturally, we are interested in how good is the DET scheme built by
these rules. Here, we proceed to study the _optimality_ of these rules. As we
will see, the degree of optimality of our series and parallel rules actually
varies for different series-parallel network topologies. This variety and its
implications will be addressed in detail below.

When discussing the optimality of the series and parallel rules, we will
compare our specific rules to generic bipartite protocols. We will show that,
to some degree, the deterministic outcomes by the series and parallel rules
are still the best, even when we _relax the requirement of determinacy_ and
consider general probabilistic ensembles of outcomes. Since an entanglement-
efficient protocol that gives general probabilistic outcomes necessarily
requires more storage and more sophisticate algorithms to keep track of every
possible outcome, the complexity of the full scheme must inevitably scale with
the QN size. Thus, given the optimality as well as the determinacy of our
series and parallel rules, we believe that the DET scheme based on these rules
is potentially the best scheme in practice for entanglement transmission on
large-scale QN.

_Concurrence monotones.—_ To start with, it is necessary to introduce a proper
_entanglement measure_ on the Schmidt numbers $\bm{\lambda}$. Although
$\bm{\lambda}$ only admits a preorder, the entanglement measure as a real
function admits a _strict total order_ , which can be used to quantify the
amount of entanglement. We will adopt a special family of entanglement
monotones, called the _concurrence monotones_ [27], $C_{k}$, $k=1,2,\cdots,d$.
We will show that the series and parallel rules optimize the task of
entanglement transmission on series-parallel QNs when measured by the average
or worst-case $k$-concurrence, or $G$-concurrence (see Appendix B for
definitions and details), depending on the network topology.

###  IV.1 Some Basic Lemmas

We first prove some basic Lemmas for the swapping and purification functions,
$\mathcal{S}(\mathbf{x},\mathbf{y})$ and $\mathcal{P}(\mathbf{x})$:

######  Lemma 1.

(Convexity) $\mathcal{S}(\mathbf{x}_{\alpha},\mathbf{z})$ is convex, i.e.,

$\sum_{\alpha}p_{\alpha}\mathcal{S}(\mathbf{x}_{\alpha},\mathbf{z})\succ\mathcal{S}(\sum_{\alpha}p_{\alpha}\mathbf{x}_{\alpha},\mathbf{z}),\quad
p_{\alpha}\in\mathbb{R}_{+},\quad\mathbf{z}\in\mathbb{R}_{+}^{d}.$

###### Proof.

Equation (4) can be rewritten as

$\mathcal{S}(\mathbf{x},\mathbf{y})=d\times\mathrm{eig}(\text{diag}(\mathbf{x}^{\downarrow})\mathbf{V}^{\dagger}\text{diag}(\mathbf{y}^{\downarrow})\mathbf{V})$
(19)

where “$\mathrm{eig}$” denotes the eigenvalues, arranged in descending order.
Thus,

$\displaystyle\mathcal{S}(\sum_{\alpha}p_{\alpha}\mathbf{x}_{\alpha},\mathbf{z})$
$\displaystyle\prec$ $\displaystyle
d\sum_{\alpha}p_{\alpha}\mathrm{eig}(\text{diag}(\mathbf{x}^{\downarrow}_{\alpha})\mathbf{V}^{\dagger}\text{diag}(\mathbf{z}^{\downarrow})\mathbf{V})$
$\displaystyle=$
$\displaystyle\sum_{\alpha}p_{\alpha}\mathcal{S}(\mathbf{x}_{\alpha},\mathbf{z})$

by Lidskii’s theorem on eigenvalues (Theorem III.4.1 of Ref. [35]). ∎

######  Lemma 2.

(Determinant preserving)

$\det(\mathcal{S}(\mathbf{x},\mathbf{y}))=d^{d}\det(\mathbf{x})\det(\mathbf{y}).$

where $\det(\mathbf{x})\equiv\det(\mathrm{diag}(\mathbf{x}))=x_{1}x_{2}\cdots
x_{d}$.

###### Proof.

By Eq. (19), we directly have
$\det(\mathcal{S}(\mathbf{x},\mathbf{y}))=d^{d}\det(\mathbf{x})\left|\det(\mathbf{V})\right|^{2}\det(\mathbf{y})=d^{d}\det(\mathbf{x})\det(\mathbf{y})$.
∎

######  Lemma 3.

(Duality) Let $\mathbf{z}=\mathcal{S}(\mathbf{x},\mathbf{y})$. Then,

$\mathrm{adj}(\mathbf{z})^{\downarrow}=d^{d-2}\mathcal{S}(\mathrm{adj}(\mathbf{x}),\mathrm{adj}(\mathbf{y})).$

where
$\mathrm{adj}(\mathbf{x})\equiv\left(\det(\mathbf{x})/x_{1},\det(\mathbf{x})/x_{2},\cdots,\det(\mathbf{x})/x_{d}\right)$.

###### Proof.

By Eq. (19),

$\displaystyle\mathcal{S}(\mathrm{adj}(\mathbf{x}),\mathrm{adj}(\mathbf{y}))$
$\displaystyle=$ $\displaystyle
d\times\mathrm{eig}(\text{diag}(\mathrm{adj}(\mathbf{x}^{\downarrow}))\mathbf{V}^{\dagger}\text{diag}(\mathrm{adj}(\mathbf{y}^{\downarrow}))\mathbf{V})$
$\displaystyle=$ $\displaystyle
d\times\mathrm{eig}(\mathrm{adj}(\text{diag}(\mathbf{x}^{\downarrow}))\mathrm{adj}(\mathbf{V})\mathrm{adj}(\text{diag}(\mathbf{y}^{\downarrow}))\mathrm{adj}(\mathbf{V}^{\dagger}))$
$\displaystyle=$ $\displaystyle
d\times\mathrm{eig}(\mathrm{adj}(\mathbf{V}^{\dagger}\text{diag}(\mathbf{y}^{\downarrow})\mathbf{V}\text{diag}(\mathbf{x}^{\downarrow})))$
$\displaystyle=$ $\displaystyle
d\times\mathrm{adj}(\mathrm{eig}(\mathbf{V}^{\dagger}\text{diag}(\mathbf{y}^{\downarrow})\mathbf{V}\text{diag}(\mathbf{x}^{\downarrow})))^{\downarrow}$
$\displaystyle=$ $\displaystyle
d\times\mathrm{adj}(\mathcal{S}(\mathbf{x},\mathbf{y})/d)^{\downarrow}=d^{2-d}\mathrm{adj}(\mathbf{z})^{\downarrow},$

where we have used several properties of the adjugate of a matrix $\bm{\Psi}$,
e.g., $\mathrm{adj}(\bm{\Psi})=\det(\bm{\Psi})\bm{\Psi}^{-1}$ and
$\mathrm{adj}(c\bm{\Psi})=c^{d-1}\mathrm{adj}(\bm{\Psi})$ for
$c\in\mathbb{R}$. ∎

######  Lemma 4.

(Majorization extremity) Let $\mathbf{z}^{\prime}\in\mathbb{R}_{+}^{d}$. If
$\mathbf{z}^{\prime}\oplus(\overbrace{0,0,\cdots,0}^{m-d})\succ\mathbf{x}$,
then $\mathbf{z}^{\prime}\succ\mathcal{P}(\mathbf{x})$.

###### Proof.

We will prove it by induction, i.e., suppose there exists
$\left[\mathcal{P}(\mathbf{x})\right]_{l}>x^{\downarrow}_{l}$ for some $l$,
and

$\sum_{j=1}^{k}z^{\prime\downarrow}_{j}\geq\sum_{j=1}^{k}\left[\mathcal{P}(\mathbf{x})\right]_{j},\quad\forall
k=1,\cdots,l-1,$ (20)

we would like to prove

$\sum_{j=1}^{l}z^{\prime\downarrow}_{j}\geq\sum_{j=1}^{l}\left[\mathcal{P}(\mathbf{x})\right]_{j}.$
(21)

Indeed,

$\displaystyle\sum_{j=1}^{l}z^{\prime\downarrow}_{j}=\sum_{j=1}^{l-1}z^{\prime\downarrow}_{j}+z^{\prime\downarrow}_{l}\geq\sum_{j=1}^{l-1}z^{\prime\downarrow}_{j}+\frac{\sum_{j=1}^{d}z^{\prime\downarrow}_{j}-\sum_{j=1}^{l-1}z^{\prime\downarrow}_{j}}{d+1-l}$
$\displaystyle\geq\sum_{j=1}^{l-1}\left[\mathcal{P}(\mathbf{x})\right]_{j}+\frac{\sum_{j=1}^{d}\left[\mathcal{P}(\mathbf{x})\right]_{j}-\sum_{j=1}^{l-1}\left[\mathcal{P}(\mathbf{x})\right]_{j}}{d+1-l}$
$\displaystyle=\sum_{j=1}^{l-1}\left[\mathcal{P}(\mathbf{x})\right]_{j}+\left(d+1-l\right)^{-1}{\sum_{j=l}^{d}\left[\mathcal{P}(\mathbf{x})\right]_{j}}$
$\displaystyle=\sum_{j=1}^{l-1}\left[\mathcal{P}(\mathbf{x})\right]_{j}+\left[\mathcal{P}(\mathbf{x})\right]_{l}=\sum_{j=1}^{l}\left[\mathcal{P}(\mathbf{x})\right]_{j}$
(22)

where the first inequality holds because the maximum is never less than the
mean, and the second because
$\mbox{tr}(\mathcal{P}(\mathbf{x}))=\mbox{tr}(\mathbf{z}^{\prime})$ and
$1-\left(d+1-l\right)^{-1}\geq 0$. Hence Eq. (21) is proved.

Finally, choose the minimum $l$ that satisfies
$\left[\mathcal{P}(\mathbf{x})\right]_{l}>x^{\downarrow}_{l}$. In other words,
$\left[\mathcal{P}(\mathbf{x})\right]_{j}=x^{\downarrow}_{j}$ for $j<l$.
Hence,

$\displaystyle\sum_{j=1}^{k}\left[\mathcal{P}(\mathbf{x})\right]_{j}=\sum_{j=1}^{k}x^{\downarrow}_{j}\leq\sum_{j=1}^{k}z^{\prime\downarrow}_{j},\quad\forall
k=1,\cdots,l-1,\qquad$

which completes the induction. ∎

######  Lemma 5.

(Convexity) $\mathcal{P}(\mathbf{x}_{\alpha})$ is convex, i.e.,

$\sum_{\alpha}p_{\alpha}\mathcal{P}(\mathbf{x}_{\alpha})\succ\mathcal{P}(\sum_{\alpha}p_{\alpha}\mathbf{x}_{\alpha}),\quad
p_{\alpha}\in\mathbb{R}_{+}.$

###### Proof.

Similar to Lemma 4, we will prove it by induction, i.e., suppose there exists
$\left[\mathcal{P}(\sum_{\alpha}p_{\alpha}\mathbf{x}_{\alpha})\right]_{l}>\left[\sum_{\alpha}p_{\alpha}\mathbf{x}_{\alpha}\right]^{\downarrow}_{l}$
for some $l$, and

$\displaystyle\sum_{j=1}^{k}\left[\sum_{\alpha}p_{\alpha}\mathcal{P}(\mathbf{x}_{\alpha})\right]^{\downarrow}_{j}\geq\sum_{j=1}^{k}\left[\mathcal{P}(\sum_{\alpha}p_{\alpha}\mathbf{x}_{\alpha})\right]_{j},$
(23)

$\forall k=1,\cdots,l-1$, we would like to prove

$\sum_{j=1}^{l}\left[\sum_{\alpha}p_{\alpha}\mathcal{P}(\mathbf{x}_{\alpha})\right]^{\downarrow}_{j}\geq\sum_{j=1}^{l}\left[\mathcal{P}(\sum_{\alpha}p_{\alpha}\mathbf{x}_{\alpha})\right]_{j}.$
(24)

But this is true and can be proved in the same way as in Eq. (IV.1). Finally,
choose the minimum $l$ that satisfies
$\left[\mathcal{P}(\sum_{\alpha}p_{\alpha}\mathbf{x}_{\alpha})\right]_{l}>\left[\sum_{\alpha}p_{\alpha}\mathbf{x}_{\alpha}\right]^{\downarrow}_{l}$.
In other words,
$\left[\mathcal{P}(\sum_{\alpha}p_{\alpha}\mathbf{x}_{\alpha})\right]_{j}=\left[\sum_{\alpha}p_{\alpha}\mathbf{x}_{\alpha}\right]^{\downarrow}_{j}$
for $j<l$. Hence,

$\displaystyle\sum_{j=1}^{k}\left[\mathcal{P}(\sum_{\alpha}p_{\alpha}\mathbf{x}_{\alpha})\right]_{j}=\sum_{j=1}^{k}\left[\sum_{\alpha}p_{\alpha}\mathbf{x}_{\alpha}\right]^{\downarrow}_{j}$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{k}\left[\sum_{\alpha}p_{\alpha}\mathbf{x}_{\alpha}^{\downarrow}\right]^{\downarrow}_{j}\leq\sum_{j=1}^{k}\left[\sum_{\alpha}p_{\alpha}\mathcal{P}(\mathbf{x}_{\alpha})\right]^{\downarrow}_{j},$

$\forall k=1,\cdots,l-1$, which completes the induction. ∎

######  Lemma 6.

(A sum-product mixing majorization inequality) Let
$\mathbf{x},\mathbf{y},\mathbf{z}\in\mathbb{R}_{+}^{m}$. If

$\ln\mathbf{x}^{\downarrow}+\ln\mathbf{y}^{\downarrow}\succ_{w}\ln\mathbf{z}^{\downarrow},$

then

$\ln\mathcal{P}(\mathbf{x})+\ln\mathcal{P}(\mathbf{y})\succ_{w}\ln\mathcal{P}(\mathbf{z}).$

###### Proof.

To begin with, w.l.o.g. we will just assume $m=d+1$, which means that
$\mathcal{P}(\mathbf{x})$ is only one dimension less than $\mathbf{x}$.
Indeed, for any $m>d$, the purification process [Eq. (10)] can be equally
constructed by applying such a one-dimension-less $\mathcal{P}(\mathbf{x})$ to
$\mathbf{x}$ recursively $m-d$ times. Thus, if our statement holds for
$m=d+1$, then it holds for general $m$ too.

First, we would like to prove the following inequality,

$\left(\prod_{j=1}^{d}\left[\mathcal{P}(\mathbf{x})\right]_{j}\right)\left(\prod_{j=1}^{d}\left[\mathcal{P}(\mathbf{y})\right]_{j}\right)\geq\prod_{j=1}^{d}\left[\mathcal{P}(\mathbf{z})\right]_{j}.$
(25)

By the definition of $\mathcal{P}(\mathbf{x})$ [Eq. (12)], we have

$\prod_{j=1}^{d}\left[\mathcal{P}(\mathbf{z})\right]_{j}=\left(\prod_{j=1}^{l-1}z^{\downarrow}_{j}\right)\left[{\sum_{j=l}^{d+1}z^{\downarrow}_{j}}/{\left(d+1-l\right)}\right]^{d+1-l}$

where $l$ separates $\mathcal{P}(\mathbf{z})$ and satisfies
$\left[\mathcal{P}(\mathbf{z})\right]_{j}=z^{\downarrow}_{j}$ for $j<l$ and
$\left[\mathcal{P}(\mathbf{z})\right]_{j}>z^{\downarrow}_{j}$ for $l\leq j\leq
d$.

Now we take advantage of a special reverse AM–GM inequality which we will
introduce and prove in Appendix D. Let $\Delta=z^{\downarrow}_{d+1}$,
$\varepsilon_{j}=z^{\downarrow}_{d+1-j}$, and
$E_{k}=\Delta+\sum_{j=1}^{k}\varepsilon_{j}$, $k=1,2,\cdots,d+1-l$. We can see
that $\varepsilon_{k}\leq\varepsilon_{k+1}\leq{E_{k}}/k$, $\forall k$ [implied
by Eq. (10)]. Thus, using Corollary 16,

$\displaystyle\left[{\sum_{j=l}^{d+1}z^{\downarrow}_{j}}/{\left(d+1-l\right)}\right]^{d+1-l}$
$\displaystyle\leq\exp({z^{\downarrow}_{d+1}/z^{\downarrow}_{d}})\prod_{j=l}^{d}z^{\downarrow}_{j}$
$\displaystyle\leq\left(1+\sqrt{{z^{\downarrow}_{d+1}}/{z^{\downarrow}_{d}}}\right)^{2}\prod_{j=l}^{d}z^{\downarrow}_{j},\quad\quad$

where the second inequality holds since ${z^{\downarrow}_{d+1}\leq
z^{\downarrow}_{d}}$. Thus,

$\displaystyle\prod_{j=1}^{d}\left[\mathcal{P}(\mathbf{z})\right]_{j}\leq\left(1+\sqrt{{z^{\downarrow}_{d+1}}/{z^{\downarrow}_{d}}}\right)^{2}\prod_{j=1}^{d}z^{\downarrow}_{j}$
$\displaystyle=$
$\displaystyle\left(z^{\downarrow}_{d}+z^{\downarrow}_{d+1}+2\sqrt{z^{\downarrow}_{d}z^{\downarrow}_{d+1}}\right)\prod_{j=1}^{d-1}z^{\downarrow}_{j}$
$\displaystyle\leq$
$\displaystyle\left(x^{\downarrow}_{d}y^{\downarrow}_{d}+x^{\downarrow}_{d+1}y^{\downarrow}_{d+1}+2\sqrt{x^{\downarrow}_{d}y^{\downarrow}_{d}x^{\downarrow}_{d+1}y^{\downarrow}_{d+1}}\right)\prod_{j=1}^{d-1}x^{\downarrow}_{j}y^{\downarrow}_{j}$
$\displaystyle\leq$
$\displaystyle\left(x^{\downarrow}_{d}+x^{\downarrow}_{d+1}\right)\left(y^{\downarrow}_{d}+y^{\downarrow}_{d+1}\right)\prod_{j=1}^{d-1}x^{\downarrow}_{j}y^{\downarrow}_{j}$
$\displaystyle\leq$
$\displaystyle\left(\prod_{j=1}^{d}\left[\mathcal{P}(\mathbf{x})\right]_{j}\right)\left(\prod_{j=1}^{d}\left[\mathcal{P}(\mathbf{y})\right]_{j}\right),$

where in the third step we use Theorem 13 (see Appendix C), in the fourth step
the usual AM–GM inequality, and in the last step the definition of the
purification process [Eq. (10)]. Hence Eq. (25) is proved.

Now we will complete the proof by induction, i.e., suppose there exists
$\left[\mathcal{P}(\mathbf{z})\right]_{l}>z^{\downarrow}_{l}$ for some $l$,
and

$\sum_{j=1}^{k}\ln\left[\mathcal{P}(\mathbf{x})\right]_{j}+\sum_{j=1}^{k}\ln\left[\mathcal{P}(\mathbf{y})\right]_{j}\geq\sum_{j=1}^{k}\ln\left[\mathcal{P}(\mathbf{z})\right]_{j},$
(26)

$\forall k=1,\cdots,l-1$, we would like to prove

$\sum_{j=1}^{l}\ln\left[\mathcal{P}(\mathbf{x})\right]_{j}+\sum_{j=1}^{l}\ln\left[\mathcal{P}(\mathbf{y})\right]_{j}\geq\sum_{j=1}^{l}\ln\left[\mathcal{P}(\mathbf{z})\right]_{j}.$
(27)

The proof structure is the same as in Eq. (IV.1), while the only difference is
that, for the second inequality in Eq. (IV.1) to hold, we need to show that
$\mbox{tr}(\ln\mathcal{P}(\mathbf{x}))+\mbox{tr}(\ln\mathcal{P}(\mathbf{y}))\geq\mbox{tr}(\ln\mathcal{P}(\mathbf{z}))$.
But this has been proved by Eq. (25). Hence Eq. (27) is proved.

Finally, choose the minimum $l$ that satisfies
$\left[\mathcal{P}(\mathbf{z})\right]_{l}>z^{\downarrow}_{l}$. In other words,
$\left[\mathcal{P}(\mathbf{z})\right]_{j}=z^{\downarrow}_{j}$ for $j<l$.
Hence,

$\displaystyle\sum_{j=1}^{k}\ln\left[\mathcal{P}(\mathbf{z})\right]_{j}=\sum_{j=1}^{k}\ln
z^{\downarrow}_{j}\leq\sum_{j=1}^{k}\ln x^{\downarrow}_{j}+\sum_{j=1}^{k}\ln
y^{\downarrow}_{j}$
$\displaystyle\leq\sum_{j=1}^{k}\ln\left[\mathcal{P}(\mathbf{x})\right]_{j}+\sum_{j=1}^{k}\ln\left[\mathcal{P}(\mathbf{y})\right]_{j},\forall
k=1,\cdots,l-1\qquad$

which completes the induction. ∎

###  IV.2 Single Link

First, we ask: if we relax the requirement of determinacy, would a better
strategy exist if we use LOCC to convert a link into a probabilistic ensemble
of links before applying the protocols? The answer is negative, as confirmed
by the following theorem.

######  Theorem 7.

Given an arbitrary series-parallel QN, converting an inbetween link into a
probabilistic ensemble cannot increase the average $k$-concurrence produced by
the series and parallel rules (Table 2) between A and B for $k=1,2,\cdots,d$.

###### Proof.

Given a probabilistic ensemble derived from link $\bm{\lambda}$ by LOCC, let
$\bm{\lambda}_{\alpha}$ be the probabilistic outcome $\alpha$, subject to
probability $p_{\alpha}$. Then,

$\sum_{\alpha}p_{\alpha}\bm{\lambda}_{\alpha}^{\downarrow}\succ\bm{\lambda}.$
(28)

Equation (28) represents a fundamental limit of locality of LOCC. This result
can be obtained by applying Lidskii’s theorem (Theorem III.4.1 of Ref. [35])
to the ensemble of reduced density matrices produced by LOCC [32].

Now, by Lemmas 1 and 5,

$\displaystyle\sum_{\alpha}p_{\alpha}\mathcal{S}(\bm{\lambda}_{\alpha},\mathbf{z})$
$\displaystyle\succ$
$\displaystyle\mathcal{S}(\sum_{\alpha}p_{\alpha}\bm{\lambda}_{\alpha}^{\downarrow},\mathbf{z})\succ\mathcal{S}(\bm{\lambda},\mathbf{z}),$
$\displaystyle\sum_{\alpha}p_{\alpha}\mathcal{P}(\bm{\lambda}_{\alpha})$
$\displaystyle\succ$
$\displaystyle\mathcal{P}(\sum_{\alpha}p_{\alpha}\bm{\lambda}_{\alpha}^{\downarrow})\succ\mathcal{P}(\bm{\lambda});$
(29)

and by Eqs. (43) and (44) in Appendix B,

$\displaystyle\sum_{\alpha}p_{\alpha}C_{k}(\bm{\lambda}_{\alpha})\leq
C_{k}(\sum_{\alpha}p_{\alpha}\bm{\lambda}_{\alpha}^{\downarrow})\leq
C_{k}(\bm{\lambda}).$ (30)

Together, Eqs. (IV.2) and (30) imply that the average $k$-concurrence will
never be greater than what could be obtained from $\bm{\lambda}$ itself.
Hence, it never helps to convert $\bm{\lambda}$ into a probabilistic ensemble.
∎

###  IV.3 Simple Series

For simple series network topology [Fig. 2], the swapping protocol is
responsible for entanglement transmission. What is the maximum of the final
average $k$-concurrence [from Eq. (1)]

$\sum_{\alpha}p_{\alpha}C^{\alpha}_{k}=\sum_{\alpha}\left\|\bm{\Psi}^{\alpha}\right\|_{2}^{2}C_{k}(\sigma^{2}({\left\|\bm{\Psi}^{\alpha}\right\|_{2}^{-1}}{\bm{\Psi}^{\alpha}}))$

between A and B that can be obtained by the swapping protocol? And how does it
compare with the deterministic series rule? This is answered by the following
theorem.

######  Theorem 8.

Given a simple series QN, compared with generic entanglement swapping
protocols of probabilistic outcomes, the series rule (Table 2) produces the
optimal average $G$-concurrence between A and B.

###### Proof.

It suffices to prove the theorem for two links $\bm{\lambda}_{a}$ and
$\bm{\lambda}_{b}$ in series. The nice properties of
$\mathcal{S}(\bm{\lambda}_{a},\bm{\lambda}_{b})$ guarantee that the proof can
be generalized to $n$ links.

Let $\\{\mathbf{X}^{\alpha}\\}$ denote a set of quantum measurements as used
in the entanglement swapping protocol [Eq. (1)]. In general, given probability
$p_{\alpha}$ and the corresponding $k$-concurrence $C^{\alpha}_{k}$ of outcome
$\alpha$, the maximum average $k$-concurrence
$\max_{\\{\mathbf{X}^{\alpha}\\}}\sum_{\alpha}p_{\alpha}C^{\alpha}_{k}$ is
intractable, and the corresponding optimal $\\{\mathbf{X}^{\alpha}\\}$ should
implicitly depend on $\bm{\lambda}_{a}$ and $\bm{\lambda}_{b}$. However, this
is not the case for $G$-concurrence. To see this, let $k=d$, then,

$\displaystyle\sum_{\alpha}p_{\alpha}C^{\alpha}_{d}=d\sum_{\alpha}\left|\det(\bm{\Psi}^{\alpha})\right|^{{2}/{d}}$
$\displaystyle=$ $\displaystyle
d\sum_{\alpha}\det(\bm{\lambda}_{a})^{1/d}\left|\det(\mathbf{X}^{\alpha})\right|^{2/d}\det(\bm{\lambda}_{b})^{1/d}$
$\displaystyle=$ $\displaystyle
d^{-1}C_{d}(\bm{\lambda}_{a})C_{d}(\bm{\lambda}_{b})\sum_{\alpha}\left|\det(\mathbf{X}^{\alpha})\right|^{2/d}$
$\displaystyle\leq$ $\displaystyle
d^{-2}C_{d}(\bm{\lambda}_{a})C_{d}(\bm{\lambda}_{b})\sum_{\alpha}\mbox{tr}(\mathbf{X}^{\alpha\dagger}\mathbf{X}^{\alpha})=C_{d}(\bm{\lambda}_{a})C_{d}(\bm{\lambda}_{b}),\qquad$
(31)

where the AM–GM inequality is used in the second last step and Eq. (2) in the
last step. Equation (IV.3) indicates that the final average $G$-concurrence
will never be greater than the product of the $G$-concurrences of
$\bm{\lambda}_{a}$ and $\bm{\lambda}_{b}$. This proof was originally given in
Ref. [27]. As we can see, this is a very special result that comes from the
unique feature of the multiplicity of determinants, which is not applicable to
other $k$-concurrence monotones.

It remains to be shown that the equality in Eq. (IV.3) holds for
$\mathcal{S}(\bm{\lambda}_{a},\bm{\lambda}_{b})$. By Lemma 2,

$\displaystyle
C_{d}(\mathcal{S}(\bm{\lambda}_{a},\bm{\lambda}_{b}))=d\det(\mathcal{S}(\bm{\lambda}_{a},\bm{\lambda}_{b}))^{1/d}$
(32) $\displaystyle=$ $\displaystyle
d^{2}\det(\bm{\lambda}_{a})^{1/d}\det(\bm{\lambda}_{b})^{1/d}=C_{d}(\bm{\lambda}_{a})C_{d}(\bm{\lambda}_{b}).$

Thus,

$\sum_{\alpha}p_{\alpha}C_{d}^{\alpha}\leq
C_{d}(\mathcal{S}(\bm{\lambda}_{a},\bm{\lambda}_{b}))$ (33)

which completes the proof. ∎

_Remark.—_ Note that Eq. (33) says nothing about the majorization preorder. In
fact, usually there is

$\sum_{\alpha}p_{\alpha}\sigma^{2}(\bm{\Psi}^{\alpha}/\left\|\bm{\Psi}^{\alpha}\right\|_{2})^{\downarrow}=\sum_{\alpha}\sigma^{2}(\bm{\Psi}^{\alpha})^{\downarrow}\nsucc\mathcal{S}(\bm{\lambda}_{a},\bm{\lambda}_{b}).$
(34)

Would the inequality in Eq. (34) hold, one could prove Eq. (33) for not only
$k=d$ but $k<d$ as well. Unfortunately this is not true.

###  IV.4 Simple Parallel

For simple parallel network topology [Fig. 2], the purification protocol is
responsible for entanglement transmission. Similarly, we ask: what is the
maximum of the final average $k$-concurrence
$\sum_{\alpha}p_{\alpha}C^{\alpha}_{k}$ between A and B that can be obtained
by the purification protocol? And how does it compare with the deterministic
parallel rule? The answer is given by the following theorem.

######  Theorem 9.

Given a simple parallel QN, compared with generic entanglement purification
protocols of probabilistic outcomes, the parallel rule (Table 2) produces the
optimal average $k$-concurrence between A and B for $k=1,2,\cdots,d$.

###### Proof.

Again, it suffices to prove the theorem for two links $\bm{\lambda}_{a}$ and
$\bm{\lambda}_{b}$ in parallel. The nice properties of
$\mathcal{P}(\bm{\lambda})$ guarantee that the proof can be generalized to $n$
links.

By the locality of LOCC [Eq. (28)], if we can obtain an ensemble of outcomes
$\bm{\lambda}_{\alpha}^{\prime}$, each with probability $p_{\alpha}$, from
$\bm{\lambda}_{a}\otimes\bm{\lambda}_{b}$, then

$\left(\sum_{\alpha}p_{\alpha}\bm{\lambda}_{\alpha}^{\prime\downarrow}\oplus(\overbrace{0,0,\cdots,0}^{d^{2}-d})\right)\succ\bm{\lambda}_{a}\otimes\bm{\lambda}_{b}.$

Hence, by Lemma 4,

$\displaystyle\sum_{\alpha}p_{\alpha}\bm{\lambda}_{\alpha}^{\prime\downarrow}\succ\mathcal{P}(\bm{\lambda}_{a}\otimes\bm{\lambda}_{b}),$
(35)

and thus

$\displaystyle\sum_{\alpha}p_{\alpha}C_{k}(\bm{\lambda}_{\alpha}^{\prime})\leq
C_{k}(\sum_{\alpha}p_{\alpha}\bm{\lambda}_{\alpha}^{\prime\downarrow})\leq
C_{k}(\mathcal{P}(\bm{\lambda}_{a}\otimes\bm{\lambda}_{b}))\quad$

for $k=1,2,\cdots,d$. ∎

_Remark.—_ Note that unlike Eq. (34) for the series rule, Eq. (35) does hold.
Therefore, the optimality of the parallel rule is for all $k$-concurrence, not
just $G$-concurrence.

###  IV.5 Parallel-then-Series

For parallel-then-series network topology [Fig. 2], both the swapping and
purification protocols are responsible for entanglement transmission. We prove
the following theorem:

######  Theorem 10.

Given a parallel-then-series QN, compared with generic entanglement swapping
and purification protocols of probabilistic outcomes, the series and parallel
rules (Table 2) produce the optimal average $G$-concurrence between A and B.

###### Proof.

It suffices to prove the theorem for four links: $\bm{\lambda}_{a}$ and
$\bm{\lambda}_{b}$ in parallel, $\bm{\lambda}_{c}$ and $\bm{\lambda}_{d}$ in
parallel, and then the two parallel groups are connected in series. Now, let
$\bm{\Psi}^{\alpha}=\text{diag}(\bm{\lambda}_{a}\otimes\bm{\lambda}_{b})^{1/2}\mathbf{X}^{\alpha}\text{diag}(\bm{\lambda}_{c}\otimes\bm{\lambda}_{d})^{1/2}$,
as used in the entanglement swapping protocol [Eq. (1)]. The difference,
however, is that $\bm{\Psi}^{\alpha}$ is not a $d\times d$ matrix, but a
$d^{2}\times d^{2}$ matrix. This is because we are considering a generic
swapping protocol, which must be applied to the full $d^{2}$ dimensions. Then,
the final $d$-dimensional outcome $\alpha$ will be given by
$\mathcal{P}(\sigma^{2}(\bm{\Psi}^{\alpha}))$.

By the Gel’Jand-Naimark theorem on singular values (Theorem III.4.5 of Ref.
[35]), we have

$\ln\sigma^{2}(\bm{\Psi}^{\alpha})\prec\ln\left(\bm{\lambda}_{a}\otimes\bm{\lambda}_{b}\right)^{\downarrow}+\ln\sigma^{2}(\mathbf{X}^{\alpha})+\ln\left(\bm{\lambda}_{c}\otimes\bm{\lambda}_{d}\right)^{\downarrow}.$
(36)

Thus, by Lemma 6 we have, in particular,

$\displaystyle\det(\mathcal{P}(\sigma^{2}(\bm{\Psi}^{\alpha})))$
$\displaystyle\leq$
$\displaystyle\det(\mathcal{P}(\bm{\lambda}_{a}\otimes\bm{\lambda}_{b}))\det(\mathcal{P}(\sigma^{2}(\mathbf{X}^{\alpha})))\det(\mathcal{P}(\bm{\lambda}_{c}\otimes\bm{\lambda}_{d})).$

Hence, similar to Eq. (IV.3),

$\displaystyle\sum_{\alpha}p_{\alpha}C^{\alpha}_{d}=d\sum_{\alpha}\det(\mathcal{P}(\sigma^{2}(\bm{\Psi}^{\alpha})))^{{1}/{d}}$
$\displaystyle\leq$ $\displaystyle
d^{-2}C_{d}(\mathcal{P}(\bm{\lambda}_{a}\otimes\bm{\lambda}_{b}))C_{d}(\mathcal{P}(\bm{\lambda}_{c}\otimes\bm{\lambda}_{d}))\sum_{\alpha}\mbox{tr}(\mathcal{P}(\sigma^{2}(\mathbf{X}^{\alpha})))$
$\displaystyle=$ $\displaystyle
d^{-2}C_{d}(\mathcal{P}(\bm{\lambda}_{a}\otimes\bm{\lambda}_{b}))C_{d}(\mathcal{P}(\bm{\lambda}_{c}\otimes\bm{\lambda}_{d}))\sum_{\alpha}\mbox{tr}(\mathbf{X}^{\alpha\dagger}\mathbf{X}^{\alpha})$
$\displaystyle=$ $\displaystyle
C_{d}(\mathcal{P}(\bm{\lambda}_{a}\otimes\bm{\lambda}_{b}))C_{d}(\mathcal{P}(\bm{\lambda}_{c}\otimes\bm{\lambda}_{d})).\qquad$
(37)

where the AM–GM inequality is used in the second step, Eq. (14) in the third
step, and Eq. (2) in the last step. Again, Eq. (IV.5) indicates that the final
average $G$-concurrence will never be greater than the product of the
$G$-concurrences of $\mathcal{P}(\bm{\lambda}_{a}\otimes\bm{\lambda}_{b})$ and
$\mathcal{P}(\bm{\lambda}_{c}\otimes\bm{\lambda}_{d})$, which is equal to
$C_{d}(\mathcal{S}(\mathcal{P}(\bm{\lambda}_{a}\otimes\bm{\lambda}_{b}),\mathcal{P}(\bm{\lambda}_{c}\otimes\bm{\lambda}_{d})))$
[Eq. (32)]. ∎

_Remark.—_ At first glimpse, it seems that we would have more degrees of
freedom to design a swapping protocol directly on the full $d^{2}$-dimensional
$\bm{\lambda}_{a}\otimes\bm{\lambda}_{b}$ and
$\bm{\lambda}_{c}\otimes\bm{\lambda}_{d}$, and we would get a better result in
terms of $G$-concurrence. Interestingly though, Theorem 10 says that the best
strategy is to first reduce them to $d$-dimensional
$\mathcal{P}(\bm{\lambda}_{a}\otimes\bm{\lambda}_{b})$ and
$\mathcal{P}(\bm{\lambda}_{c}\otimes\bm{\lambda}_{d})$ and then apply a
swapping protocol on them. Our result implies a strong practical convenience
in entanglement transmission, that it is in fact unnecessary to keep all
$d^{m}$ dimensions for $m$ parallel links. We can safely purify and produce a
$d$-dimensional link from the $m$ parallel links and then swap it with other
links connected in series. The final $G$-concurrence is guaranteed to be
optimal.

Note particularly that the nested purification protocol [7] is designed to
swap $\bm{\lambda}_{a}\otimes\bm{\lambda}_{b}$ and
$\bm{\lambda}_{c}\otimes\bm{\lambda}_{d}$ using two independent swapping
protocols on $a,c$ and $b,d$, respectively, before applying purification
protocols on the outcomes. But the tensor product of the two swapping
protocols is just a special case of a $d^{2}$-dimensional swapping protocol.
Thus, the nested purification protocol does not optimize the final average
$G$-concurrence.

###  IV.6 Series-then-Parallel

For series-then-parallel network topology [Fig. 2], unfortunately, the final
average $k$-concurrence between A and B will not be maximized by the series
and parallel rules but instead by some generic nondeterministic swapping and
purification protocols.

A counterexample can be constructed. Let $d=2$ and
$\bm{\lambda}_{a}=\bm{\lambda}_{b}=\bm{\lambda}_{c}=\left(0.9,0.1\right)$.
$\bm{\lambda}_{a}$ and $\bm{\lambda}_{b}$ are connected in series and then
with $\bm{\lambda}_{c}$ in parallel. The series and parallel rules (Table 2)
yield a deterministic final outcome
$\bm{\lambda}=\left(9\left(25+4\sqrt{34}\right)/500,1-9\left(25+4\sqrt{34}\right)/500\right)\approx\left(0.870,0.130\right)$,
the $G$-concurrence of which is $\approx 0.673$. However, applying a special
nondeterministic swapping protocol (the ZZ basis [23]) on $\bm{\lambda}_{a}$
and $\bm{\lambda}_{b}$, we can get four probabilistic outcomes,
$\bm{\lambda}_{1}=\bm{\lambda}_{2}=\left(81/82,1/82\right)$ and
$\bm{\lambda}_{3}=\bm{\lambda}_{4}=\left(1/2,1/2\right)$, with probabilities
$p_{1}=p_{2}=0.41$ and $p_{3}=p_{4}=0.09$, respectively. The final average
$G$-concurrence is
$2p_{1}C_{d}(\mathcal{P}(\bm{\lambda}_{c}\otimes\bm{\lambda}_{1}))+2p_{3}C_{d}(\mathcal{P}(\bm{\lambda}_{c}\otimes\bm{\lambda}_{3}))\approx
0.695$. Thus, the series and parallel rules are not optimal. The deeper reason
is that the inequality in Eq. (34) does not hold in general. For this example,
we clearly have
$p_{1}\bm{\lambda}_{1}^{\downarrow}+p_{2}\bm{\lambda}_{2}^{\downarrow}+p_{3}\bm{\lambda}_{3}^{\downarrow}+p_{4}\bm{\lambda}_{4}^{\downarrow}=\left(0.819,0.181\right)\nsucc\mathcal{S}(\bm{\lambda}_{a},\bm{\lambda}_{b})\approx\left(0.966,0.034\right)$.

###  IV.7 Series-Parallel

For arbitrary series-parallel network topology [Fig. 2], we already know from
the previous discussion that no optimality can be observed for the final
average $k$-concurrence. That being said, when $d=2$, a special theorem of
optimality can indeed be made in terms of the final _worst-case_
$G$-concurrence (see Appendix B). We prove the following theorem:

######  Theorem 11.

(For $d=2$ only.) Given a series-parallel QN, compared with generic
entanglement swapping and purification protocols of probabilistic outcomes,
the series and parallel rules (Table 2) produce the optimal worst-case
$G$-concurrence between A and B.

###### Proof.

From Theorems 8 and 9 we know that, for simple series and simple parallel
network topologies, given a generic ensemble of probabilistic outcomes
$\alpha$, there must be at least one $\bm{\lambda}_{\alpha}$ of which the
$G$-concurrence is smaller than (at most equal to) the $G$-concurrence which
the series and parallel rules would produce, respectively. Since $d=2$, this
$\bm{\lambda}_{\alpha}$ majorizes what the series and parallel rules would
produce. Then, given that the series and parallel rules are isotone [Eqs. (7)
and (15)], the final probabilistic outcome related to $\bm{\lambda}_{\alpha}$
will remain as the worst case when the series and parallel rules are applied
over the full series-parallel network. ∎

_Remark.—_ This special optimality on the worst-case $G$-concurrence for
arbitrary series-parallel network topology with $d=2$ was first noticed in
Ref. [25]. When $d>2$, Theorem 11 does not hold in general. This is because
other deterministic series or parallel rules different than Table 2 may exist.
As long as the other deterministic rules can produce a $\bm{\lambda}$ with
better $G$-concurrence (which is only possible for series-then-parallel
network topology), then the worst-case $G$-concurrence is automatically larger
than what Table 2 can produce.

##  V Discussion

What truly makes the formulation of deterministic entanglement transmission
(DET) scalable with network size and adaptable to different (at least series-
parallel) network topologies is the use of deterministic quantum communication
protocols [25]. One may wonder if the specific DET scheme based on the series
and parallel rules (Table 2) that we have introduced is the only possible DET
scheme using deterministic protocols. The answer is negative, because
different series and parallel rules may exist as well:

For example, the recipe of $\mathcal{S}(\bm{\lambda}_{a},\bm{\lambda}_{b})$
[Eq. (4)] may still be implementable by LOCC if the matrix $\mathbf{V}$ is not
a constant but depends explicitly on $\bm{\lambda}_{a}$ and
$\bm{\lambda}_{b}$. When properly chosen, this new swapping function can even
produce a better $k$-concurrence ($k\neq d$) than our swapping function [27].
Moreover, if quantum catalysts [36] are allowed, then it is also possible to
have a new purification function $\mathcal{P}(\bm{\lambda})$ that is
implementable by LOCC without satisfying the majorization relation [Eq. (18)].
These new functions still correspond to deterministic protocols and therefore
can be used to build a different set of series and parallel rules.

What remains is the question of whether we can also find deterministic
protocols that are applicable to non-series-parallel QN. The complexity of
answering this comes from the fact that multipartite protocols [37] and
multilink-based QN routing [12] may have to be used for producing
deterministic outcomes. The existence of such deterministic protocols can
certainly help us study entanglement transmission on a more general QN.

Given the well-behaved scalability and adaptability of DET, we find it
particularly interesting to apply our scheme to an _infinite_ series-parallel
QN that has an infinite number of nodes. We expect that percolation-like
criticality in terms of $\bm{\lambda}$ of each link can be observed for
entanglement transmission in the thermodynamic limit. However, note that to
each QN link assigned is not a single number but a sequence of Schmidt numbers
which have more than one degree of freedom when $d>2$. This prohibits us from
establishing an exact one-to-one mapping from $\bm{\lambda}$ to a real
parameter, e.g., temperature. It is unknown how criticality behaves in such
case.

##  Appendix A Basic Notations

We briefly recall some basic notions in matrix analysis and majorization
theory [35]:

Let $\mathbf{x}=\left(x_{1},x_{2},\cdots,x_{n}\right)\in\mathbb{R}_{+}^{n}$.
Note that some notations for matrices can also be defined for $\mathbf{x}$.
Given the diagonal matrix of $\mathbf{x}$,

$\text{diag}(\mathbf{x})\equiv\begin{pmatrix}x_{1}&0&\ldots\\\
0&x_{2}&\ldots\\\ \vdots&\vdots&\ddots\end{pmatrix},$

we can simply define the trace of $\mathbf{x}$ as
$\mbox{tr}(\mathbf{x})\equiv\mbox{tr}(\text{diag}(\mathbf{x}))=\sum_{j=1}^{n}x_{j}$,
the determinant of $\mathbf{x}$ as
$\det(\mathbf{x})\equiv\det(\text{diag}(\mathbf{x}))=x_{1}x_{2}\cdots x_{n}$,
and the adjugate of $\mathbf{x}$ as
$\mathrm{adj}(\mathbf{x})\equiv\left(\det(\mathbf{x})/x_{1},\det(\mathbf{x})/x_{2},\cdots,\det(\mathbf{x})/x_{n}\right)$.

Let $\mathbf{x}^{\downarrow}$ and $\mathbf{x}^{\uparrow}$ be the sequences
given by rearranging the coordinates of $\mathbf{x}$ in decreasing order and
increasing order, respectively. In other words,
$x^{\downarrow}_{1},x^{\downarrow}_{2},\cdots,x^{\downarrow}_{n}$ as the
coordinates of $\mathbf{x}^{\downarrow}$ satisfy $x^{\downarrow}_{1}\geq
x^{\downarrow}_{2}\geq\cdots\geq x^{\downarrow}_{n}$. Similarly, for
$\mathbf{x}^{\uparrow}$ there is $x^{\uparrow}_{1}\leq
x^{\uparrow}_{2}\leq\cdots\leq x^{\uparrow}_{n}$.

Let $\mathbf{x},\mathbf{y}\in\mathbb{R}_{+}^{n}$. We say that $\mathbf{x}$ is
_weakly submajorized_ by $\mathbf{y}$, or $\mathbf{x}\prec_{w}\mathbf{y}$
[35], if

$\sum_{j=1}^{k}x^{\downarrow}_{j}\leq\sum_{j=1}^{k}y^{\downarrow}_{j},\quad\forall
k=1,\cdots,n.$ (38)

In particular, if $\mathbf{x}$ is weakly submajorized by $\mathbf{y}$ and

$\sum_{j=1}^{n}x^{\downarrow}_{j}=\sum_{j=1}^{n}y^{\downarrow}_{j},\text{ or,
}\mbox{tr}(\mathbf{x})=\mbox{tr}(\mathbf{y}),$ (39)

then we say that $\mathbf{x}$ is _majorized_ by $\mathbf{y}$, or
$\mathbf{x}\prec\mathbf{y}$ [35].

For example, when $\mathbf{x}\in\mathbb{R}_{+}^{n}$ and
$\mbox{tr}(\mathbf{x})=1$, we always have
$\left(1/n,\cdots,1/n\right)\prec\mathbf{x}\prec\left(1,0,\cdots,0\right)$. On
the other hand, given two arbitrary
$\mathbf{x},\mathbf{y}\in\mathbb{R}_{+}^{n}$ with
$\mbox{tr}(\mathbf{x})=\mbox{tr}(\mathbf{y})=1$, we may have
$\mathbf{x}\nprec\mathbf{y}$ and $\mathbf{x}\nsucc\mathbf{y}$ both. Thus, the
majorization relation is not a total order. It is not a partial order either,
because $\mathbf{x}\succ\mathbf{y}$ and $\mathbf{y}\succ\mathbf{x}$ do not
necessarily imply $\mathbf{x}=\mathbf{y}$, since they may differ by a
permutation [35]. The majorization relation is only a _preorder_ on
$\mathbb{R}_{+}^{n}$.

In this work, our attention is mostly focused on $d^{2}$-dimensional
normalized states with $d$ Schmidt numbers $\bm{\lambda}$ satisfying
$\mbox{tr}(\bm{\lambda})\equiv 1$. We exclusively use the symbol
$\bm{\lambda}$ whenever we implicitly know that it has a _trace of unity_ and
$d-1$ degrees of freedom. For general vectors, we use the symbols
$\mathbf{x},\mathbf{y},\mathbf{z},\cdots\in\mathbb{R}_{+}^{n}$ instead.

##  Appendix B A Family of Concurrence Monotones: $k$-Concurrence

Given a $d\times d$-dimensional bipartite pure state, up to unitary
equivalence,
$\left|\bm{\lambda}\right\rangle=\sum_{j=1}^{d}\sqrt{\lambda_{j}}\left|jj\right\rangle$,
there are $d$ concurrence monotones, as developed in Ref. [27],

$C_{k}(\bm{\lambda})\equiv\left[{S_{k}(\bm{\lambda})}/{S_{k}(\left(\frac{1}{d},\frac{1}{d},\cdots,\frac{1}{d}\right))}\right]^{1/k},\quad\mbox{tr}(\bm{\lambda})\equiv
1,$ (40)

$k=1,2,\cdots,d$, where $S_{k}(\bm{\lambda})$ is the $k$-th elementary
symmetric polynomial [35], e.g.,

$\displaystyle S_{0}(\bm{\lambda})$ $\displaystyle=$ $\displaystyle 1,$
$\displaystyle S_{1}(\bm{\lambda})$ $\displaystyle=$
$\displaystyle\sum\nolimits_{1\leq i\leq d}\lambda_{i},$ $\displaystyle
S_{2}(\bm{\lambda})$ $\displaystyle=$ $\displaystyle\sum\nolimits_{1\leq
i<j\leq d}\lambda_{i}\lambda_{j},$ $\displaystyle S_{3}(\bm{\lambda})$
$\displaystyle=$ $\displaystyle\sum\nolimits_{1\leq i<j<k\leq
d}\lambda_{i}\lambda_{j}\lambda_{k},$ $\displaystyle\cdots$ $\displaystyle
S_{d}(\bm{\lambda})$ $\displaystyle=$
$\displaystyle\lambda_{1}\lambda_{2}\cdots\lambda_{d}.$

Note that $C_{k}$ is named _$k$ -concurrence_ since the whole monotone family
is nothing but a generalization of $C_{2}(\bm{\lambda})$, the “concurrence”
which is more commonly referred to in the context of quantum information
theory. Another special concurrence of the family is $C_{d}$ (i.e., $k=d$),
the _$G$ -concurrence_, as it stands for the _geometric_ mean of the Schmidt
numbers [27]. The importance of $G$-concurrence can be understood from the
main text.

All $k$-concurrences have the following properties [27]:

  * •

_Permutation invariance ( $\bm{\lambda}\to\text{perm}\\{\bm{\lambda}\\}$):_

$C_{k}(\bm{\lambda})=C_{k}(\text{perm}\\{\bm{\lambda}\\}).$ (41)

  * •

_Unit measure:_

$\displaystyle 0\leq C_{k}(\bm{\lambda})$ $\displaystyle\leq$ $\displaystyle
1,\text{ and }$ $\displaystyle C_{k}(\bm{\lambda})$ $\displaystyle=$
$\displaystyle 1\text{ if
}\bm{\lambda}=\left(\frac{1}{d},\frac{1}{d},\cdots,\frac{1}{d}\right)$ (42)

  * •

_Isotone:_

$C_{k}(\bm{\lambda}_{a})\leq C_{k}(\bm{\lambda}_{b})\text{ if
}\bm{\lambda}_{a}\succ\bm{\lambda}_{b}.$ (43)

  * •

_Concavity:_

$C_{k}(\sum_{\alpha}p_{\alpha}\bm{\lambda}_{\alpha})\geq\sum_{\alpha}p_{\alpha}C_{k}(\bm{\lambda}_{\alpha}),\quad
p_{\alpha}\in\mathbb{R}_{+}.$ (44)

In particular, if $\sum_{\alpha}p_{\alpha}=1$, then $p_{\alpha}$ and
$\bm{\lambda}_{\alpha}$ can be explained as a probabilistic ensemble. The
_average_ $k$-concurrence is then defined as the RHS of Eq. (44),
$\sum_{\alpha}p_{\alpha}C_{k}(\bm{\lambda}_{\alpha})$, and the _worst-case_
$k$-concurrence is defined as $\min_{\alpha}C_{k}(\bm{\lambda}_{\alpha})$,
accordingly.

##  Appendix C Some Useful Theorems

######  Theorem 12.

Let $F:\mathbb{R}_{+}^{m}\to\mathbb{R}_{+}^{n}$ be convex, i.e.,
$\sum_{\alpha}p_{\alpha}F(\mathbf{x}_{\alpha})\succ
F(\sum_{\alpha}p_{\alpha}\mathbf{x}_{\alpha})$, $p_{\alpha}\in\mathbb{R}_{+}$.
If, for all $\mathbf{x}$,

$\displaystyle F(\text{perm}(\mathbf{x}))=F(\mathbf{x}),$ (45)

then $F$ is isotone, i.e., $F(\mathbf{x})\succ F(\mathbf{y})$ if
$\mathbf{x}\succ\mathbf{y}$.

###### Proof.

Let $\mathbf{x}\succ\mathbf{y}$ in $\mathbb{R}_{+}^{m}$. By Theorem II.1.10 of
Ref. [35] there exist a set of $m\times m$-dimensional permutation matrices
$\mathbf{P}_{1},\mathbf{P}_{2},\cdots$ and a set of
$p_{1},p_{2},\cdots\in\mathbb{R}_{+}$ with $\sum_{\alpha}p_{\alpha}=1$ such
that $\mathbf{y}=\sum_{\alpha}p_{\alpha}\mathbf{P}_{\alpha}\mathbf{x}$. Thus,

$\displaystyle
F(\mathbf{y})=F(\sum_{\alpha}p_{\alpha}\mathbf{P}_{\alpha}\mathbf{x})\prec\sum_{\alpha}p_{\alpha}F(\mathbf{P}_{\alpha}\mathbf{x})=\sum_{\alpha}p_{\alpha}F(\mathbf{x}),$

i.e., $F(\mathbf{x})\succ F(\mathbf{y})$. ∎

######  Theorem 13.

Let $\mathbf{x},\mathbf{y}\in\mathbb{R}_{+}^{n}$. If
$\ln\mathbf{x}\succ_{w}\ln\mathbf{y}$, then

$\displaystyle\left(\prod_{j=1}^{l-1}x^{\downarrow}_{j}\right)\sum_{j=l}^{k}\left(x^{\downarrow}_{j}\right)^{s}\geq\left(\prod_{j=1}^{l-1}y^{\downarrow}_{j}\right)\sum_{j=l}^{k}\left(y^{\downarrow}_{j}\right)^{s},$
(47)

$\forall k=1,2,\cdots,n$ and $\forall l=1,2,\cdots,k$, given that $0\leq s\leq
k-l+1$.

###### Proof.

Let
$c=\prod_{j=1}^{l-1}\left({y^{\downarrow}_{j}}/{x^{\downarrow}_{j}}\right)\leq
1$. Then,

$\displaystyle\prod_{j=l}^{k}x^{\downarrow}_{j}\geq
c\prod_{j=l}^{k}y^{\downarrow}_{j}\geq
c^{\left(k-l+1\right)/{s}}\prod_{j=l}^{k}y^{\downarrow}_{j}=\prod_{j=l}^{k}\left(c^{1/s}y^{\downarrow}_{j}\right),\qquad$
(48)

$\forall k=l,l+1,\cdots,n$. Therefore, By Example II.3.5(v) of Ref. [35], we
have

$\displaystyle\sum_{j=l}^{k}\left(x^{\downarrow}_{j}\right)^{s}\geq
c\sum_{j=l}^{k}\left(y^{\downarrow}_{j}\right)^{s},$ (49)

which proves Eq. (47). Note that Eq. (49) holds because
$f(e^{t})=\left(e^{t}\right)^{s}$ is convex and monotone increasing in $t$
[35]. ∎

##  Appendix D A Reverse AM–GM Inequality

We will prove the following inequality:

######  Theorem 14.

Let $\bm{\varepsilon}\in\mathbb{R}_{+}^{n}$ and $\Delta\geq 0$. Let
$E_{k}=\Delta+\sum_{j=1}^{k}\varepsilon_{j}$, $k=1,2,\cdots,n$. If
$\varepsilon_{k}\leq\varepsilon_{k+1}\leq{E_{k}}/k$, $\forall k$, then

$\left(\frac{E_{n}}{n}\right)^{n}\leq\varepsilon_{n}\varepsilon_{n-1}\cdots\varepsilon_{1}\left(1+\frac{\Delta/\varepsilon_{1}}{n}\right)^{n}$
(50)

with equality if and only if
$\varepsilon_{n}=\varepsilon_{n-1}=\cdots=\varepsilon_{1}$.

###### Proof.

To start with, note that ${E_{k}}/k={\varepsilon_{k}/k+E_{k-1}}/k\leq
E_{k-1}/\left(k-1\right)$. Therefore, the following inequality holds:

$\frac{E_{n}}{n}\leq\frac{E_{n-1}}{n-1}\leq\cdots\leq\frac{E_{1}}{1}.$ (51)

Now we will prove Eq. (50) by induction, i.e., suppose

$\left(\frac{E_{j-1}}{j-1}\right)^{j-1}\leq\varepsilon_{j-1}\varepsilon_{j-2}\cdots\varepsilon_{1}\left(1+\frac{\Delta/\varepsilon_{1}}{j-1}\right)^{j-1},$
(52)

we would like to prove

$\left(\frac{E_{j}}{j}\right)^{j}\leq\varepsilon_{j}\varepsilon_{j-1}\cdots\varepsilon_{1}\left(1+\frac{\Delta/\varepsilon_{1}}{j}\right)^{j}.$
(53)

To do so, define
$f_{j}(\varepsilon_{j},\varepsilon_{j-1},\cdots,\varepsilon_{1})$ as the
right-hand side minus the left-hand side of Eq. (53). Taking the derivative of
$f_{j}$ w.r.t. $\varepsilon_{k}$ for $k=2,\cdots,j$ yields

$\displaystyle\frac{\partial f_{j}}{\partial\varepsilon_{k}}$ $\displaystyle=$
$\displaystyle\frac{\varepsilon_{j}\varepsilon_{j-1}\cdots\varepsilon_{1}}{\varepsilon_{k}}\left(1+\frac{\Delta/\varepsilon_{1}}{j}\right)^{j}-\left(\frac{E_{j}}{j}\right)^{j-1}$
(54) $\displaystyle\geq$
$\displaystyle\varepsilon_{j-1}\cdots\varepsilon_{1}\left(1+\frac{\Delta/\varepsilon_{1}}{j}\right)^{j}-\left(\frac{E_{j-1}}{j-1}\right)^{j-1}$
$\displaystyle\geq$
$\displaystyle\varepsilon_{j-1}\cdots\varepsilon_{1}\left(1+\frac{\Delta/\varepsilon_{1}}{j-1}\right)^{j-1}-\left(\frac{E_{j-1}}{j-1}\right)^{j-1}$
$\displaystyle\geq$ $\displaystyle f_{j-1}\geq 0$

with equality if and only if
$\varepsilon_{j}=\varepsilon_{j-1}=\cdots=\varepsilon_{1}$. Thus, fixing
$\varepsilon_{1}$ and noticing that $\varepsilon_{2}$ is constrained by
$\varepsilon_{1}$ only, we conclude that, since ${\partial
f_{j}}/{\partial\varepsilon_{2}}\geq 0$, $f_{j}$ takes the minimum if and only
if $\varepsilon_{2}\geq\varepsilon_{1}$ actually takes the equality; next,
fixing $\varepsilon_{1}$ and $\varepsilon_{2}=\varepsilon_{1}$ and noticing
that $\varepsilon_{3}$ is constrained by $\varepsilon_{1}$ and
$\varepsilon_{2}$ only, we conclude that $f_{j}$ takes the minimum if and only
if $\varepsilon_{3}\geq\varepsilon_{2}$ actually takes the equality; …

Taken together, we conclude that $f_{j}$ takes the minimum if and only if
$\varepsilon_{j}=\varepsilon_{j-1}=\cdots=\varepsilon_{1}$, which put back
into $f_{j}$ yields
$f_{j}(\varepsilon_{1},\varepsilon_{1},\cdots,\varepsilon_{1})=0$. Hence
$f_{j}\geq 0$. The induction is thus completed given $f_{1}=0$. ∎

We note that, despite being a reversed inequality of the AM–GM type, Theorem
14 is tight. This is because $\Delta$ can take any nonnegative value. The two
independent constraints $\varepsilon_{k+1}\geq\varepsilon_{k}$ and
$\varepsilon_{k+1}\leq{E_{k}}/k$ in Theorem 14 together imply that the
deviations between the coordinates of $\bm{\varepsilon}$ cannot be too large,
which are controlled by $\Delta$. In particular, if $\Delta=0$, then Eq. (50)
just becomes the reverse of the usual AM–GM inequality. But $\Delta=0$ also
requires $\varepsilon_{k+1}=\varepsilon_{k}=\cdots=\varepsilon_{1}$ given the
constraints. Therefore Eq. (50) will not be violated since it can only take
the equality.

######  Corollary 15.

(Same prerequisites as in Theorem 14)

$\left(\frac{E_{j}}{j}\right)^{j}\leq\varepsilon_{j}\varepsilon_{j-1}\cdots\varepsilon_{1}\left(1+\frac{\Delta/\varepsilon_{1}}{j}\right)^{j},\quad
j=1,2,\cdots,n$ (55)

with equality if and only if
$\varepsilon_{j}=\varepsilon_{j-1}=\cdots=\varepsilon_{1}$.

###### Proof.

This has been proved in the proof of Theorem 14, which is a special case
($j=n$) of this Corollary. ∎

######  Corollary 16.

(Same prerequisites as in Theorem 14)

$\left(\frac{E_{j}}{j}\right)^{j}\leq\varepsilon_{j}\varepsilon_{j-1}\cdots\varepsilon_{1}e^{\Delta/\varepsilon_{1}},\quad
j=1,2,\cdots,n$ (56)

with equality if and only if $\Delta=0$.

###### Proof.

This is because

$\left(1+\frac{\Delta/\varepsilon_{1}}{j}\right)^{j}\leq
e^{\Delta/\varepsilon_{1}},\quad j>0$ (57)

with equality if and only if $\Delta=0$. ∎

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