# Probabilistic pure state conversion on the majorization lattice

Serge Deside  Centre for Quantum Information and Communication, École
polytechnique de Bruxelles, CP 165, Université libre de Bruxelles, 1050
Brussels, Belgium  Matthieu Arnhem  Centre for Quantum Information and
Communication, École polytechnique de Bruxelles, CP 165, Université libre de
Bruxelles, 1050 Brussels, Belgium  Department of Optics, Palacký University,
17. listopadu 12, 771 46 Olomouc, Czech Republic  Célia Griffet  Centre for
Quantum Information and Communication, École polytechnique de Bruxelles, CP
165, Université libre de Bruxelles, 1050 Brussels, Belgium  Nicolas J. Cerf
Centre for Quantum Information and Communication, École polytechnique de
Bruxelles, CP 165, Université libre de Bruxelles, 1050 Brussels, Belgium

###### Abstract

We show that the majorization lattice provides the appropriate framework in
order to characterize the allowed transformations of pure entangled states
under local operations and classical communication. The underlying notions of
meet $\land$ and join $\lor$ in the majorization lattice lead us to define,
respectively, the optimal common resource and optimal common product states.
Based on these two states, we introduce two optimal probabilistic protocols
for the (single-copy) conversion of incomparable bipartite pure states, which
we name greedy and thrifty. Both protocols reduce to Vidal’s protocol [G.
Vidal, Phys. Rev. Lett. 83, 1046 (1999)] if the initial and final states are
comparable, but otherwise the thrifty protocol can be shown to be superior to
the greedy protocol as it yields a more entangled residual state when it fails
(they both yield the same entangled state with the same optimal probability
when they succeed). Finally, we consider the generalization of these protocols
to entanglement transformations involving multiple possible initial or final
states.

##  I Introduction

Quantum entanglement has long been recognized as a necessary resource in many
quantum information protocols [1, 2, 3] and is often regarded as the paramount
quantum resource [4]. In this context, separable states are defined as
resource-free states and local operations supplemented with classical
communication (LOCC) are viewed as free operations, that is, operations that
map separable states onto separable states (hence, do not create any
resource). An important aspect of such a resource theory of entanglement
concerns the study of the allowed entanglement transformations, i.e., the
conversions of entangled states of a system shared between several parties
that can be achieved using LOCC only. Here, we focus on the case of single-
copy transformations between bipartite pure states. The most notable result on
this subject, due to Nielsen [5], consists in a simple characterization of
deterministic bipartite pure state conversions using LOCC by means of
majorization theory. A related result, due to Vidal [6], further extends this
characterization to probabilistic transformations.

The theory of majorization allows one to compare probability distributions in
terms of intrinsic disorder [7]. Given two probability vectors p and q, we say
that p is majorized by q, written $\textbf{{p}}\prec\textbf{{q}}$, if and only
if

$\displaystyle\sum_{i=1}^{k}p_{i}^{\downarrow}\leq\sum_{i=1}^{k}q_{i}^{\downarrow},\quad\forall
k\in[1,d-1],$ (1)
$\displaystyle\sum_{i=1}^{k}p_{i}^{\downarrow}=\sum_{i=1}^{k}q_{i}^{\downarrow},\quad\text{for
}k=d,$ (2)

where $p_{i}^{\downarrow}$ (resp. $q_{i}^{\downarrow}$) denotes the
$i^{\text{th}}$ greatest component of p (resp. q). Here, $d$ is the number of
non-zero coefficients of the longest vector between p and q (zeros are
appended at the end of the smallest vector if necessary), hence Eq. (2) is
obviously fulfilled due to probability normalization. Note that majorization
only defines a preorder relation on probability vectors, meaning that two
probability vectors may be incomparable under majorization. For example,
$(0.5,0.4,0.1)$ and $(0.6,0.2,0.2)$ do not fulfill Eq. (1) in either
direction. On the opposite, if Eq. (1) holds in both directions
($\textbf{{p}}\prec\textbf{{q}}$ and $\textbf{{q}}\prec\textbf{{p}}$), then
the two vectors are said to be equivalent, which means that they coincide up
to a permutation.

Majorization relations have a central role in entanglement theory, as implied
by Nielsen’s theorem [5]. Consider two parties, Alice and Bob, sharing a
bipartite pure state $\ket{\psi}$. The Schmidt decomposition of $\ket{\psi}$
is written

$\ket{\psi}_{AB}=\sum_{i=1}^{d}\sqrt{\lambda_{\psi}^{(i)}}\ket{i}_{A}\ket{i}_{B},$
(3)

where
$\lambda_{\psi}=\left(\lambda_{\psi}^{(1)},\cdots,\lambda_{\psi}^{(d)}\right)$
is the Schmidt vector of $\ket{\psi}$ made with the eigenvalues of the reduced
density matrix $\hat{\rho}_{A}$ (or $\hat{\rho}_{B}$), while
$\\{\ket{i}_{A}\\}$ (resp. $\\{\ket{i}_{B}\\}$) denotes the eigenbasis of
$\hat{\rho}_{A}$ (resp. $\hat{\rho}_{B}$). If Alice and Bob wish to convert
$\ket{\psi}_{AB}$ into another pure state

$\ket{\phi}_{AB}=\sum_{i=1}^{d}\sqrt{\lambda_{\phi}^{(i)}}\ket{i}_{A}\ket{i}_{B},$
(4)

this entanglement transformation is possible with certainty using LOCC if and
only if

$\lambda_{\psi}\prec\lambda_{\phi}.$ (5)

Of course, two states are locally unitarily equivalent, i.e., interconvertible
under local unitaries $\hat{U}_{A}\otimes\hat{U}_{B}$, if and only if they
share the same Schmidt vector up to a permutation. Hence, it is sufficient to
consider Schmidt vectors sorted by decreasing order in order to compare them
with a majorization relation (in what follows, all Schmidt vectors will always
be assumed to be sorted decreasingly, that is
$\lambda_{\psi}^{(i)\downarrow}=\lambda_{\psi}^{(i)}$, so we omit the arrow
sign).

An equivalent form of Nielsen’s theorem can be stated by using a sufficiently
large set of entanglement monotones [8], i.e., functionals of the state
$\ket{\psi}$ that cannot increase, on average, under a LOCC transformation. By
using Eqs. (1) and (2), one can easily check that

$E_{l}(\psi)=\sum_{i=l}^{d}\lambda_{\psi}^{(i)},\quad\forall l\in[1,d],$ (6)

form a set of $d$ entanglement monotones which allow us to reexpress Nielsen’s
theorem as

$\ket{\psi}\overset{\text{LOCC}}{\longrightarrow}\ket{\phi}\leavevmode\nobreak\
\leavevmode\nobreak\ \Longleftrightarrow\leavevmode\nobreak\
\leavevmode\nobreak\ E_{l}(\psi)\geq E_{l}(\phi),\quad\forall l\in[1,d],$ (7)

where $\ket{\psi}\overset{\text{LOCC}}{\longrightarrow}\ket{\phi}$ means that
$\ket{\psi}$ can be deterministically converted into $\ket{\phi}$ by using
LOCC. Protocols achieving such a transformation have been discussed, for
example, in Refs. [9, 10]. The characterization of entanglement
transformations via majorization relations has been generalized to
probabilistic LOCC transformations by Vidal in Ref. [6], and an optimal
protocol achieving the desired probabilistic transformation has also been
found (it will be detailed in Sec. III).

In this paper, we exploit the majorization lattice [11], a notion recently
shown to be relevant for addressing quantum information questions [12, 13, 14,
15, 16, 17, 18], in order to revisit the probabilistic entanglement
transformations between bipartite pure states. Our analysis builds upon two
central elements of a lattice theory, the so-called meet and join (see Fig.
1), and yields two corresponding optimal protocols (see Fig. 2). Considering
an entanglement lattice, where each node is a bipartite pure state and the
lattice structure emerges from majorization relations, we associate the meet
and join to two specific states that we call, respectively, the optimal common
resource (OCR)111This term was first coined in Ref. [19] but with no reference
to the majorization lattice. and optimal common product (OCP) states. We first
build a protocol using the OCP state and show that it is close to Vidal’s
optimal protocol. More interestingly, we then build a second protocol making
use of the OCR state and prove that, while being again optimal, it better
preserves average entanglement. Both protocols trivially reduce to Vidal’s in
the special case where $\ket{\psi}$ and $\ket{\phi}$ are comparable states in
the sense of majorization theory, so we focus on the interesting case of
incomparable states.

The protocol based on the OCP state can be viewed as a greedy protocol. In the
context of optimization, a greedy algorithm evolves towards the solution by
choosing the local optimal move at each stage. In analogy, our greedy protocol
favors the immediate gain of a deterministic move and postpones the
probabilistic move (see red arrows in Fig. 2). Our second protocol based on
the OCR state can be understood, in contrast, as a thrifty protocol. A thrifty
optimization algorithm prefers not to choose immediate gain at each stage.
Here, our thrifty protocol indeed tolerates a probabilistic move first (see
green arrows in Fig. 2). The main results of this paper consist in Theorems 1
and 2, which prove, respectively, the optimality and better average
entanglement preservation of the thrifty protocol.

The paper is organized as follows. In Sec. II, we first introduce the
majorization lattice and, specifically for bipartite entanglement, the OCR and
OCP states. Then, in Sec. III, we present Vidal’s theorem for the
probabilistic conversion of bipartite pure states as well as the corresponding
optimal protocol. This provides us with the tools for describing the greedy
and thrifty protocols in Sec. IV. We further establish in Sec. V the
generalization of the greedy and thrifty protocols to an arbitrary finite
number of initial or final states. Finally, we conclude in Sec. VI.

##  II Majorization lattice

Arising from order theory, the notion of lattice unveils a partial order
relation on the elements of a set. Hereafter, we consider a specific lattice
where the elements under comparison belong to the set of probability
distributions sorted decreasingly and where the partial order is given by the
majorization relation. More formally, the majorization lattice [11] is a
quadruple $\langle\mathcal{P}_{d},\prec,\land,\lor\rangle$, where

  * •

$\mathcal{P}_{d}$ is the set of discrete probability vectors sorted
decreasingly with, at most, $d$ non-zero coefficients, that is,
$\mathcal{P}_{d}=\\{(p_{1},\cdots,p_{d})\text{, s.t. }p_{1}\geq\cdots\geq
p_{d}\geq 0\text{ and }\sum_{i=1}^{d}p_{i}=1\\}$;

  * •

$\prec$ is the majorization relation;

  * •

$\land$ denotes the so-called meet;

  * •

$\lor$ denotes the so-called join.

The meet of two elements $\textbf{{p}},\textbf{{q}}\in\mathcal{P}_{d}$,
denoted $\textbf{{p}}\land\textbf{{q}}$, is defined as the sole element of
$\mathcal{P}_{d}$ such that, $\forall\textbf{{r}}\in\mathcal{P}_{d}$ with
$\textbf{{r}}\prec\textbf{{p}}$ and $\textbf{{r}}\prec\textbf{{q}}$, we have
$\textbf{{r}}\prec\textbf{{p}}\land\textbf{{q}}$. Analogously, the join of two
elements $\textbf{{p}},\textbf{{q}}\in\mathcal{P}_{d}$, denoted
$\textbf{{p}}\lor\textbf{{q}}$, is defined as the sole element of
$\mathcal{P}_{d}$ such that, $\forall\textbf{{r}}\in\mathcal{P}_{d}$ with
$\textbf{{p}}\prec\textbf{{r}}$ and $\textbf{{q}}\prec\textbf{{r}}$, we have
$\textbf{{p}}\lor\textbf{{q}}\prec\textbf{{r}}$.

The following definition allows one to characterize mathematically the meet of
two vectors in $\mathcal{P}_{d}$.

######  Definition 1.

[11] Let p and q $\in\mathcal{P}_{d}$, the meet of p and q, denoted
$\textbf{p}\land\textbf{q}=(m_{1},m_{2},\cdots,m_{d})$, can be expressed as

$m_{i}=\min\left\\{\sum_{j=1}^{i}p_{j},\sum_{j=1}^{i}q_{j}\right\\}-\min\left\\{\sum_{j=1}^{i-1}p_{j},\sum_{j=1}^{i-1}q_{j}\right\\},$
(8)

for all $i\in[1,d]$, with the convention that
$\sum_{j=1}^{k}p_{j}=\sum_{j=1}^{k}q_{j}=0$ when $k=0$.

The determination of the join of two vectors in $\mathcal{P}_{d}$ is more
cumbersome. As we will not use it explicitly in the following, we refer the
interested reader to Ref. [11] for an algorithm producing the join.

Figure 1 provides a useful visual insight into the majorization lattice where,
for each element x of the lattice, an upper and a lower cone are defined. The
elements $\textbf{{x}}_{\text{up}}$ of the upper cone correspond to all
elements that are majorized by x, i.e.,
$\textbf{{x}}_{\text{up}}\prec\textbf{{x}}$, while the elements
$\textbf{{x}}_{\text{down}}$ of the lower cone correspond to all elements
majorizing x, i.e., $\textbf{{x}}\prec\textbf{{x}}_{\text{down}}$. In other
words, the upper cone comprises all the elements that are more disordered than
x, whereas the lower cone comprises all the elements that are more ordered
than x. Considering now two elements p and q in the lattice, the meet
$\textbf{{p}}\land\textbf{{q}}$ can be viewed as the most ordered element that
remains more disordered than both p and q. Conversely, the join
$\textbf{{p}}\lor\textbf{{q}}$ corresponds to the most disordered element that
remains more ordered than both p and q. It is worth mentioning that the meet
and join are useful notions only when the two probability vectors p and q are
incomparable under majorization (otherwise, if
$\textbf{{p}}\prec\textbf{{q}}$, then
$\textbf{{p}}\land\textbf{{q}}=\textbf{{p}}$ and
$\textbf{{p}}\lor\textbf{{q}}=\textbf{{q}}$). This will actually be our main
concern when developing the greedy and thrifty protocols in Sec. IV for
incomparable Schmidt vectors.

Figure 1: Schematic representation of the majorization lattice. Let p and
$\textbf{{q}}\in\mathcal{P}_{d}$, the meet $\textbf{{p}}\land\textbf{{q}}$ is
located at the intersection of the two upper cones, whereas the join
$\textbf{{p}}\lor\textbf{{q}}$ is located at the intersection of the two lower
cones. Since the upper cones contain more disordered elements, the meet
$\textbf{{p}}\land\textbf{{q}}$ stands for the most ordered element among
those that are more disordered than both p and q. Since the lower cones
contain more ordered elements, the join $\textbf{{p}}\lor\textbf{{q}}$ stands
for the most disordered element among those that are more ordered than both p
and q.

In the following, we will consider an entanglement lattice whose elements are
bipartite pure states. Since the entanglement of a bipartite pure state is
univocally characterized by its Schmidt vector, we may equivalently view the
elements of the lattice as probability vectors in $\mathcal{P}_{d}$. Hence,
according to Nielsen’s theorem, a state can be obtained from any state
belonging to its upper cone by using only deterministic LOCC. Conversely, a
state can be converted into any state of its lower cone by using only
deterministic LOCC. In other words, the lattice structure translates the
allowed entanglement transformations. Accordingly, the meet of two (or more)
states denotes the least entangled state that can still be deterministically
converted into any one of them. We refer to it as the optimal common resource
(OCR)1 state. Reciprocally, the join of two (or more) states denotes the most
entangled state that can still be deterministically produced from any one of
them. We refer to it as the optimal common product (OCP) state. The OCP and
OCR states will be used in Sec. IV in order to construct the greedy and
thrifty protocols, respectively, as pictured in Fig. 2.

Figure 2: Schematic representation of the majorization lattice in the context
of bipartite entanglement. Let $\ket{\psi}$ and $\ket{\phi}$ be two bipartite
pure states that are assumed to be incomparable. Any state lying in the upper
cone of $\ket{\psi}$ is convertible into $\ket{\psi}$ via deterministic LOCC,
while any state in the lower cone of $\ket{\psi}$ is reachable from
$\ket{\psi}$ via deterministic LOCC (this is pictured with a collection of
grey arrows pointing downwards). The same holds true, of course, for the cones
associated with $\ket{\phi}$. The OCR state $\ket{\psi\land\phi}$ is located
at the intersection of the two upper cones and can be understood as the least
entangled state capable of deterministically producing either $\ket{\psi}$ or
$\ket{\phi}$. In contrast, the OCP state $\ket{\psi\lor\phi}$ is located at
the intersection of the two lower cones and can be viewed as the most
entangled state that both $\ket{\psi}$ and $\ket{\phi}$ are capable to produce
deterministically. We highlight the greedy (red) and thrifty (green) protocols
as discussed in Sec. IV for converting $\ket{\psi}$ into $\ket{\phi}$ (bold
arrows correspond to deterministic LOCC transformations and dashed arrows to
probabilistic LOCC transformations). The greedy protocol (in red) starts with
a deterministic step towards the OCP state and postpones the probabilistic
step. In contrast, the thrifty protocol (in green) starts with a probabilistic
step towards the OCR state before making a deterministic step.

##  III Probabilistic state conversion protocol

When a deterministic state conversion is impossible according to Nielsen’s
theorem, the same transformation may sometimes be achieved probabilistically.
This is the content of Vidal’s theorem [6], which we now present.

Let $\ket{\psi}$ be the initial state of a bipartite system shared by Alice
and Bob, and let $\ket{\phi}$ be the target state they wish to obtain using
only LOCC. We suppose that it is possible to perform the desired
transformation from $\ket{\psi}$ to $\ket{\phi}$ with non-vanishing
probability, i.e., the number of non-zero coefficients of $\lambda_{\psi}$
must be at least as large as that of $\lambda_{\phi}$ (in the following, we
consider, without loss of generality, that they are equal). Vidal’s theorem
states that there exists a probabilistic protocol converting $\ket{\psi}$ into
$\ket{\phi}$ with probability $p$ if and only if $E_{l}(\psi)\geq
p\,E_{l}(\phi),\forall l\in[1,d]$. Thus, the optimal (maximum) probability
with which Alice and Bob can perform the desired transformation is given by

$p_{\max}=\underset{l\in[1,d]}{\min}\frac{E_{l}(\psi)}{E_{l}(\phi)}.$ (9)

Note that $p_{\max}\leq 1$ because when $l=1$, we have
$E_{1}(\psi)=E_{1}(\phi)=1$. Vidal described a protocol achieving this optimal
probability in Ref. [6], which we recall hereafter.

First, Alice and Bob apply a deterministic LOCC in order to transform
$\ket{\psi}$ into an intermediate state $\ket{\chi}$ which maximizes the
fidelity with respect to the target state $\ket{\phi}$ while being
deterministically reachable from $\ket{\psi}$ using only LOCC [20], namely,

$\ket{\chi}=\underset{\ket{\alpha}\leavevmode\nobreak\ :\leavevmode\nobreak\
\ket{\psi}\overset{\text{LOCC}}{\longrightarrow}\ket{\alpha}}{\text{argmax}}|\langle\alpha|\phi\rangle|^{2}.$
(10)

Second, Alice performs a two-outcome measurement on her share of the system,
leading, with some probability $p_{\max}$, to the target state $\ket{\phi}$
(in which case we say that the protocol has succeeded) or, with probability
$1-p_{\max}$, to a state denoted as $\ket{\xi}$ (in which case we say that the
protocol has failed). We show below that this residual state $\ket{\xi}$
always possesses a strictly smaller number of non-vanishing Schmidt
coefficients than $\ket{\phi}$, hence the conversion from $\ket{\xi}$ to
$\ket{\phi}$ is fully impossible, even probabilistically.

The construction of the intermediate state $\ket{\chi}$ works as follows. We
make use of the $d$ entanglement monotones defined in Eq. (6) in order to
define a sequence of ratios $r_{j}$’s. First, we define $r_{1}$, which
corresponds to the maximum conversion probability from $\ket{\psi}$ to
$\ket{\phi}$, as

$r_{1}=\frac{E_{l_{1}}(\psi)}{E_{l_{1}}(\phi)}=\underset{l\in[1,d]}{\min}\frac{E_{l}(\psi)}{E_{l}(\phi)},$
(11)

where $l_{1}$ is the value of $l$ that reaches the minimum (it is chosen as
the smallest value of $l$ in case of several minima). Then, we define the next
ratios $r_{j}$ with $j=2,3,\cdots$ as follows (each $r_{j}$ is associated with
a corresponding $l_{j}$)

$r_{j}=\frac{E_{l_{j}}(\psi)-E_{l_{j-1}}(\psi)}{E_{l_{j}}(\phi)-E_{l_{j-1}}(\phi)}=\underset{l\in[1,l_{j-1}-1]}{\min}\frac{E_{l}(\psi)-E_{l_{j-1}}(\psi)}{E_{l}(\phi)-E_{l_{j-1}}(\phi)},$
(12)

until we find some value of $j$, which we call $k$, satisfying $l_{k}=1$.
Finally, we define $l_{0}=d+1$. On a side note, it can easily be shown that
the sequences of $r_{j}$’s and $l_{j}$’s satisfy $0<r_{1}<\cdots<r_{k}$ and
$l_{0}>l_{1}>\cdots>l_{k}=1$ [6].

From the ratios $r_{j}$’s, we define the Schmidt coefficients of the
intermediate state

$\lambda_{\chi}^{(i)}=r_{j}\lambda_{\phi}^{(i)},\quad\text{if\leavevmode\nobreak\
}i\in[l_{j},l_{j-1}-1],\quad j\in[1,k],$ (13)

which, by construction, satisfies $\lambda_{\psi}\prec\lambda_{\chi}$. Hence,
$\ket{\psi}$ is convertible to $\ket{\chi}$ by using a deterministic LOCC.

The generalized measurement performed by Alice is described by the two Kraus
operators

$\displaystyle\hat{M}=\begin{pmatrix}\hat{M}_{k}&&\\\ &\ddots&\\\
&&\hat{M}_{1}\end{pmatrix},$ (14)
$\displaystyle\hat{N}=\begin{pmatrix}\sqrt{\hat{I}_{[l_{k-1}-l_{k}]}-\hat{M}_{k}^{2}}&&\\\
&\ddots&\\\ &&\sqrt{\hat{I}_{[l_{0}-l_{1}]}-\hat{M}_{1}^{2}}\end{pmatrix},$
(15)

with

$\hat{M}_{j}=\sqrt{\frac{r_{1}}{r_{j}}}\,\hat{I}_{[l_{j-1}-l_{j}]},\quad\text{for\leavevmode\nobreak\
}j\in[1,k],$ (16)

where $\hat{I}_{[l_{j-1}-l_{j}]}$ is the identity operator in a
$(l_{j-1}-l_{j})$-dimensional Hilbert space. We have, of course, the
completeness relation
$\hat{M}^{\dagger}\hat{M}+\hat{N}^{\dagger}\hat{N}=\hat{I}_{[d]}$. If Alice
obtains the measurement outcome linked to $\hat{M}$, she obtains the target
state $\ket{\phi}$. This happens with probability $r_{1}$ since we have

$\hat{M}\otimes\hat{I}\ket{\chi}=\sqrt{r_{1}}\ket{\phi}.$ (17)

Otherwise, if she obtains the outcome linked to $\hat{N}$, she gets the
residual state $\ket{\xi}$ such that

$\hat{N}\otimes\hat{I}\ket{\chi}=\sqrt{1-r_{1}}\ket{\xi}.$ (18)

It is clear from Eq. (15) that $\hat{N}$ possesses at least one vanishing
diagonal element, hence $\ket{\xi}$ has strictly less non-vanishing Schmidt
coefficients than $\ket{\phi}$. Therefore, it is impossible for Alice and Bob
to obtain state $\ket{\phi}$ from $\ket{\xi}$ by using (even probabilistic)
LOCC transformations.

Note that in the special case where $\lambda_{\psi}\prec\lambda_{\phi}$,
Vidal’s protocol reduces to a deterministic ($p_{\max}=1$) protocol for
converting $\ket{\psi}$ into $\ket{\phi}$, which exists as a consequence of
Nielsen’s theorem.

##  IV Greedy and thrifty protocols on the majorization lattice

We now introduce two optimal probabilistic state conversion protocols which
take advantage of the notions of OCP and OCR states on the majorization
lattice (see Fig. 3 for a precise diagrammatic representation of both
protocols). As usual, Alice and Bob initially share a pure bipartite state
$\ket{\psi}$ that they wish to transform into a target state $\ket{\phi}$ by
using only LOCC operations. Further, we assume that $\ket{\psi}$ and
$\ket{\phi}$ are incomparable. The protocol passing through the OCP state
works as follows. First, Alice and Bob transform $\ket{\psi}$ into the OCP of
the initial and target states denoted as $\ket{\psi\lor\phi}$ via a
deterministic LOCC transformation. This conversion is possible as a
consequence of Nielsen’s theorem since, by definition,
$\lambda_{\psi}\prec\lambda_{\psi\lor\phi}$. Then, Alice and Bob apply the
probabilistic transformation presented in Sec. III to $\ket{\psi\lor\phi}$ in
order to obtain the target state $\ket{\phi}$. This clearly cannot be done
deterministically since $\lambda_{\psi\lor\phi}\nprec\lambda_{\phi}$
(actually, $\lambda_{\psi\lor\phi}\succ\lambda_{\phi}$). We name this protocol
greedy because the deterministic (rewarding) phase is given priority and
precedes the probabilistic (less rewarding) phase.

Following a similar logic, we define another protocol using the OCR state
instead. In this protocol, Alice and Bob first transform $\ket{\psi}$ into the
OCR of the initial and target states denoted as $\ket{\psi\land\phi}$ using
the probabilistic LOCC transformation presented in Sec. III. This
transformation can only be made probabilistically because
$\lambda_{\psi}\nprec\lambda_{\psi\land\phi}$ (actually,
$\lambda_{\psi}\succ\lambda_{\psi\land\phi}$). Then, the second phase consists
in Alice and Bob transforming $\ket{\psi\land\phi}$ into $\ket{\phi}$ via a
deterministic LOCC operation, which is obviously possible because
$\lambda_{\psi\land\phi}\prec\lambda_{\phi}$. We name this protocol thrifty
because the deterministic (rewarding) phase is postponed after the
probabilistic (less rewarding) phase.

Interestingly, we will prove that the greedy and thrifty protocols both reach
the optimal conversion probability [the same as that of Vidal’s protocol, see
Eq. (9)]. Furthermore, we will show that the thrifty protocol better preserves
entanglement, on average, than the greedy protocol. More precisely, if the
thrifty protocol fails, the residual state, which we call $\ket{\nu}$, is more
entangled than the residual state of the greedy protocol, called $\ket{\xi}$,
in the sense that $\lambda_{\nu}\prec\lambda_{\xi}$, or equivalently
$\ket{\nu}\overset{\text{LOCC}}{\longrightarrow}\ket{\xi}$.

Figure 3: Diagrammatic representation of the greedy (red) and thrifty (green)
protocols converting $\ket{\psi}$ into $\ket{\phi}$ on the majorization
lattice. The probabilistic phase of both protocols is itself split into two
moves: a deterministic LOCC transformation (represented by a bold arrow)
towards an intermediate state (either $\ket{\chi}$ or $\ket{\zeta}$) followed
by a probabilistic LOCC (represented by a dashed arrow) involving a local two-
outcome measurement. If the conversion is successful, the resulting state is
majorized by (is more entangled than) the initial state
($\ket{\phi}\prec\ket{\psi\lor\phi}$ and
$\ket{\psi\land\phi}\prec\ket{\psi}$). The left inset represents the residual
states if the conversion has failed, emphasizing that the residual state
$\ket{\nu}$ of the thrifty protocol is majorized by (hence, is more entangled
than) the residual state $\ket{\xi}$ of the greedy protocol. Note also that
$\ket{\zeta}\prec\ket{\chi}$.

Note that if $\ket{\psi}$ and $\ket{\phi}$ were comparable instead of
incomparable states, then the greedy and thrifty protocols would be equal and
would coincide with Vidal’s conversion protocol from $\ket{\psi}$ to
$\ket{\phi}$ (which is probabilistic when $\lambda_{\phi}\prec\lambda_{\psi}$,
or even deterministic when $\lambda_{\psi}\prec\lambda_{\phi}$). Hence, we
will only be interested in the case of incomparable Schmidt vectors
$\lambda_{\psi}$ and $\lambda_{\phi}$, i.e., situations where a deterministic
LOCC conversion is impossible in both directions, from $\ket{\psi}$ to
$\ket{\phi}$ and from $\ket{\phi}$ to $\ket{\psi}$.

Let us first consider the easiest case of the greedy protocol, which can be
brought to Vidal’s protocol simply by merging the sequence of two
deterministic moves $\ket{\psi}\to\ket{\psi\lor\phi}\to\ket{\chi}$ into a
single deterministic move $\ket{\psi}\to\ket{\chi}$. Indeed, in the second
phase of the greedy protocol (i.e., the probabilistic LOCC transformation from
$\ket{\psi\lor\phi}$ to $\ket{\phi}$), the intermediate state happens to be
the same as the intermediate state $\ket{\chi}$ of the transformation from
$\ket{\psi}$ to $\ket{\phi}$ in Vidal’s protocol, defined in Eq. (13). This is
true because this intermediate state $\ket{\chi}$ has been shown to belong to
the lower cone of $\ket{\psi\lor\phi}$ in Ref. [13], and is thus
deterministically reachable from $\ket{\psi}$. Since, by definition, the
intermediate state is the closest state to $\ket{\phi}$ that is
deterministically attainable from $\ket{\psi}$ [see Eq. (10)], both
intermediate states coincide. Hence, the greedy protocol is also optimal.

The rest of this section is devoted to the analysis of the thrifty protocol,
which is more involved. Before proving the optimality of the thrifty protocol,
let us prove the following rather intuitive lemma.

######  Lemma 1.

The entanglement monotone for the OCR of two bipartite pure states
$\ket{\psi}$ and $\ket{\phi}$, $E_{l}(\psi\land\phi)$, is equal to the maximum
between the entanglement monotones of the initial state $E_{l}(\psi)$ and
target state $E_{l}(\phi)$, i.e.,

$E_{l}(\psi\land\phi)=\max\left\\{{E_{l}(\psi),E_{l}(\phi)}\right\\},\quad\forall
l\in[1,d].$ (19)

###### Proof.

We start from the definition of the $l$th entanglement monotone for the OCR
state,

$E_{l}(\psi\land\phi)=\sum_{i=l}^{d}\lambda_{\psi\land\phi}^{(i)},$ (20)

which can be rewritten, by use of Definition 1, as

$E_{l}(\psi\land\phi)=\sum_{i=l}^{d}\left(\min\left\\{\sum_{j=1}^{i}\lambda_{\psi}^{(j)},\sum_{j=1}^{i}\lambda_{\phi}^{(j)}\right\\}\right.\\\
-\left.\min\left\\{\sum_{j=1}^{i-1}\lambda_{\psi}^{(j)},\sum_{j=1}^{i-1}\lambda_{\phi}^{(j)}\right\\}\right).$
(21)

Since all terms cancel out two by two, except for the first and last terms, we
obtain

$E_{l}(\psi\land\phi)=1-\min\left\\{\sum_{j=1}^{l-1}\lambda_{\psi}^{(j)},\sum_{j=1}^{l-1}\lambda_{\phi}^{(j)}\right\\},$
(22)

or equivalently,

$\displaystyle E_{l}(\psi\land\phi)$
$\displaystyle=\max\left\\{\sum_{j=l}^{d}\lambda_{\psi}^{(j)},\sum_{j=l}^{d}\lambda_{\phi}^{(j)}\right\\},$
(23) $\displaystyle=\max\left\\{{E_{l}(\psi),E_{l}(\phi)}\right\\},$ (24)

hence completing the proof, which holds $\forall l\in[1,d]$. ∎

We can now move to our first main result, namely the optimality of the thrifty
protocol. Since the second phase of the thrifty protocol, converting
$\ket{\psi\land\phi}$ into $\ket{\phi}$ (i.e., the third green arrow in Fig.
3), is deterministic, all we need to consider is the first phase, namely the
probabilistic LOCC transformation from $\ket{\psi}$ to $\ket{\psi\land\phi}$
(i.e., the two first green arrows in Fig. 3). In the following theorem, we
prove that the maximum probability of achieving such a transformation from
$\ket{\psi}$ to $\ket{\psi\land\phi}$ is equivalent to that of realizing the
transformation from $\ket{\psi}$ to $\ket{\phi}$. In some sense, with this
specific probability, one may as well stop the thrifty protocol halfway in
order to produce the more entangled state $\ket{\psi\land\phi}$ rather than
$\ket{\phi}$.

######  Theorem 1.

Let $\ket{\psi}$ and $\ket{\phi}$ be two bipartite pure states. The optimal
probability of conversion from $\ket{\psi}$ to $\ket{\psi\land\phi}$, which we
name $\mathfrak{r}_{1}$, is equal to that of the optimal conversion from
$\ket{\psi}$ to $\ket{\phi}$, denoted as $r_{1}$ in Eq. (11).

###### Proof.

We need to show that $\mathfrak{r}_{1}=r_{1}$, where $r_{1}$ is given by Eq.
(11) and $\mathfrak{r}_{1}$ is the optimal probability of conversion from
$\ket{\psi}$ to $\ket{\psi\land\phi}$. We know that this probabilistic
transformation can be performed optimally using Vidal’s protocol, so
$\mathfrak{r}_{1}$ can be calculated with Eq. (11), namely

$\mathfrak{r}_{1}=\frac{E_{\mathfrak{l}_{1}}(\psi)}{E_{\mathfrak{l}_{1}}(\psi\land\phi)}=\underset{l\in[1,d]}{\min}\frac{E_{l}(\psi)}{E_{l}(\psi\land\phi)},$
(25)

where $\mathfrak{l}_{1}$ is defined as the smallest $l\in[1,d]$ minimizing
$E_{l}(\psi)/E_{l}(\psi\land\phi)$. With the use of Lemma 1, we can write

$\mathfrak{r}_{1}=\underset{l\in[1,d]}{\min}\leavevmode\nobreak\
\frac{E_{l}(\psi)}{\max\left\\{{E_{l}(\psi),E_{l}(\phi)}\right\\}},$ (26)

which is to be compared with $r_{1}$. We obviously have

$\frac{E_{l}(\psi)}{\max\left\\{{E_{l}(\psi),E_{l}(\phi)}\right\\}}\leq\frac{E_{l}(\psi)}{E_{l}(\phi)},\quad\forall
l\in[1,d].$ (27)

For any value of $l$ such that $E_{l}(\phi)<E_{l}(\psi)$, we have

$1=\frac{E_{l}(\psi)}{\max\left\\{{E_{l}(\psi),E_{l}(\phi)}\right\\}}<\frac{E_{l}(\psi)}{E_{l}(\phi)},$
(28)

implying that such a value of $l$ can be disregarded in the minimization over
$l$ giving $r_{1}$ or $\mathfrak{r}_{1}$ since, when $l=1$,
$E_{1}(\psi)=E_{1}(\phi)=1$ so the minimum is guaranteed not to exceed 1.
Thus, we may restrict the minimization over the values of $l$ such that
$E_{l}(\phi)\geq E_{l}(\psi)$, for which

$\frac{E_{l}(\psi)}{\max\left\\{{E_{l}(\psi),E_{l}(\phi)}\right\\}}=\frac{E_{l}(\psi)}{E_{l}(\phi)}.$
(29)

This implies that we have equal probabilities,

$\mathfrak{r}_{1}=r_{1},$ (30)

hence we have proven that the thrifty protocol is also an optimal
probabilistic protocol.

∎

We now turn to our second main result, namely that entanglement is better
preserved, on average, by the thrifty protocol rather than the greedy
protocol. Let us first prove the following lemma.

######  Lemma 2.

For all x and $\textbf{y}\in\mathcal{P}_{d}$ such that
$\textbf{x}\prec\textbf{y}$ and for all decreasingly-ordered vectors
$\textbf{a}\in\mathbb{R}^{d}$ such that
$\sum_{i=1}^{d}a_{i}x_{i}=\sum_{i=1}^{d}a_{i}y_{i}=1$, we have

$\textbf{a}\odot\textbf{x}\prec\textbf{a}\odot\textbf{y},$ (31)

where $\odot$ denotes the Hadamard (element-wise) product.

###### Proof.

The last equality [see Eq. (2)] in the majorization relation (31) is trivially
fulfilled by hypothesis, so we need to prove the $d-1$ inequalities [see Eq.
(1)]

$\sum_{i=1}^{k}a_{i}(y_{i}-x_{i})\geq 0,\quad\forall k\in[1,d-1],$ (32)

given the majorization hypothesis $\textbf{{x}}\prec\textbf{{y}}$, which reads
as

$\sum_{i=1}^{k}(y_{i}-x_{i})\geq 0,\quad\forall k\in[1,d-1].$ (33)

The decreasing rearrangement of vector a allows us to write its components in
the form

$a_{i}=\sum_{j=i}^{d}\mu_{j},\quad\forall i\in[1,d],$ (34)

where $\mu_{j}\geq 0$, for all $j\in[1,d]$. By plugging in Eq. (34) into Eq.
(32), we have to prove

$\sum_{i=1}^{k}(y_{i}-x_{i})\sum_{j=i}^{d}\mu_{j}\geq 0,\quad\forall
k\in[1,d-1],$ (35)

which can be rewritten by interchanging the two summation signs as

$\sum_{j=1}^{d}\mu_{j}\sum_{i=1}^{\min(k,j)}(y_{i}-x_{i})\geq 0,\quad\forall
k\in[1,d-1].$ (36)

These inequalities are clearly satisfied given $\mu_{j}\geq 0$ and the
majorization hypothesis, Eq. (33), thereby completing the proof.

∎

With this in mind, we can prove our second main result, namely that the
residual state $\ket{\nu}$ of the thrifty protocol is majorized by the
residual state $\ket{\xi}$ of the greedy protocol. This is the content of the
following theorem.

######  Theorem 2.

Let $\ket{\psi}$ and $\ket{\phi}$ be two bipartite pure states and let
$\ket{\xi}$ be the residual state if the two-outcome probabilistic protocol
for converting $\ket{\psi}$ into $\ket{\phi}$ fails and $\ket{\nu}$ be the
residual state if the two-outcome probabilistic protocol for converting
$\ket{\psi}$ into $\ket{\psi\land\phi}$ fails. Both states are related via the
majorization relation

$\lambda_{\nu}\prec\lambda_{\xi}.$ (37)

###### Proof.

To show this, we proceed in three steps. First, we characterize the sequence
of ratios $\mathfrak{r}_{j}$ (and corresponding values $\mathfrak{l}_{j}$)
associated with the probabilistic conversion from $\ket{\psi}$ to
$\ket{\psi\land\phi}$ and prove that $\mathfrak{r}_{j}=r_{j}$ and
$\mathfrak{l}_{j}=l_{j}$, $\forall j$. Then, we express the intermediate state
$\ket{\zeta}$ of this conversion and show that it satisfies
$\lambda_{\zeta}\prec\lambda_{\chi}$, where $\ket{\chi}$ corresponds to the
intermediate state of the probabilistic conversion from $\ket{\psi}$ (or from
$\ket{\psi\lor\phi}$) to $\ket{\phi}$, as shown in Fig. 3. Finally, we use
$\lambda_{\zeta}\prec\lambda_{\chi}$ in order to prove the desired
majorization relation $\lambda_{\nu}\prec\lambda_{\xi}$.

From Theorem 1, we already know that for $j=1$, $\mathfrak{r}_{1}=r_{1}$ and
$\mathfrak{l}_{1}=l_{1}$. In order to prove the same relations for higher
values of $j$, we proceed by iteration. We can write $\mathfrak{r}_{2}$, using
the definition from Eq. (12), as

$\displaystyle\mathfrak{r}_{2}$
$\displaystyle=\frac{E_{\mathfrak{l}_{2}}(\psi)-E_{\mathfrak{l}_{1}}(\psi)}{E_{\mathfrak{l}_{2}}(\psi\land\phi)-E_{\mathfrak{l}_{1}}(\psi\land\phi)},$
$\displaystyle=\underset{l\in[1,\mathfrak{l}_{1}-1]}{\min}\frac{E_{l}(\psi)-E_{\mathfrak{l}_{1}}(\psi)}{E_{l}(\psi\land\phi)-E_{\mathfrak{l}_{1}}(\psi\land\phi)},$

which can be reexpressed, using $l_{1}=\mathfrak{l}_{1}$ and
$E_{\mathfrak{l}_{1}}(\psi\land\phi)=\max\left\\{E_{\mathfrak{l}_{1}}(\psi),E_{\mathfrak{l}_{1}}(\phi)\right\\}=E_{l_{1}}(\phi)$,
as

$\mathfrak{r}_{2}=\underset{l\in[1,l_{1}-1]}{\min}\frac{E_{l}(\psi)-E_{l_{1}}(\psi)}{\max\left\\{{E_{l}(\psi),E_{l}(\phi)}\right\\}-E_{l_{1}}(\phi)}.$
(38)

We choose to rewrite this last expression as

$\mathfrak{r}_{2}=\underset{l\in[1,l_{1}-1]}{\min}\mathfrak{r}_{2,l}.$ (39)

We now divide the values of $l$ in two categories. Either
$l\in\mathcal{L}^{+}$ when
$\max\left\\{{E_{l}(\psi),E_{l}(\phi)}\right\\}=E_{l}(\psi)$, or
$l\in\mathcal{L}^{-}$ when
$\max\left\\{{E_{l}(\psi),E_{l}(\phi)}\right\\}=E_{l}(\phi)$. It is possible
to prove that there always exists one value $l_{\min}\in\mathcal{L}^{-}$ such
that $\mathfrak{r}_{2,l_{\min}}\leq\mathfrak{r}_{2,l},\forall
l\in\mathcal{L}^{+}\cup\mathcal{L}^{-}$. Hence,

$\mathfrak{r}_{2}=r_{2}.$ (40)

Proceeding equivalently for each $j$, we show that

$\mathfrak{r}_{j}=r_{j}\text{ and }\mathfrak{l}_{j}=l_{j},\quad\forall
j\in[1,k].$ (41)

This allows us to write the vector of Schmidt coefficients of the intermediate
state $\ket{\zeta}$ between $\ket{\psi}$ and $\ket{\psi\land\phi}$ as

$\lambda_{\zeta}^{(i)}=r_{j}\,\lambda_{\psi\land\phi}^{(i)},\quad\text{if}\
i\in[l_{j},l_{j-1}-1],\quad j\in[1,k].$ (42)

Thus, this vector can be expressed as a Hadamard product of two vectors,
namely

$\lambda_{\zeta}=\textbf{{r}}\odot\lambda_{\psi\land\phi},$ (43)

where $(\textbf{{r}})_{i}=r_{j}$, if $i\in[l_{j},l_{j-1}-1],\forall
i\in[1,d]$. Similarly, in view of Eq. (13), the vector of Schmidt coefficients
of the intermediate state $\ket{\chi}$ between $\ket{\psi}$ (or
$\ket{\psi\lor\phi}$) and $\ket{\phi}$ can be rewritten using this notation as

$\lambda_{\chi}=\textbf{{r}}\odot\lambda_{\phi}.$ (44)

Now, we simply make use of Lemma 2. Since
$\lambda_{\psi\land\phi}\prec\lambda_{\phi}$, the majorization relation

$\lambda_{\zeta}=\textbf{{r}}\odot\lambda_{\psi\land\phi}\prec\textbf{{r}}\odot\lambda_{\phi}=\lambda_{\chi},$
(45)

holds true provided the vector r is decreasingly ordered (by increasing index
$i$). This can be easily checked from the structure of this vector, namely

$\textbf{{r}}=(\underset{\overset{\uparrow}{l_{k}}}{r_{k}}\ldots
r_{k}|\ldots\ldots|\underset{\overset{\uparrow}{l_{2}}}{r_{2}}\ldots
r_{2}|\underset{\overset{\uparrow}{l_{1}}}{r_{1}}\ldots r_{1})^{T},$ (46)

where each $l_{j}$ variable points to the index of the corresponding element
in r. Hence, $\lambda_{\zeta}\prec\lambda_{\chi}$, which means that the
intermediate state $\ket{\zeta}$ of the thrifty protocol is more entangled
than the intermediate state $\ket{\chi}$ of the greedy protocol, as can also
be seen in Fig. 3.

Finally, we deduce from Eq. (45) the fact that
$\lambda_{\nu}\prec\lambda_{\xi}$, i.e., that if the protocol fails, the
residual state of the thrifty protocol is more entangled than the residual
state of the greedy protocol (see inset of Fig. 3). To do this, first notice
that

$\hat{N}\otimes\hat{I}\ket{\chi}=\sqrt{1-r_{1}}\ket{\xi},$ (47)

and

$\hat{N}\otimes\hat{I}\ket{\zeta}=\sqrt{1-r_{1}}\ket{\nu}.$ (48)

Given that $\hat{N}$ is diagonal [see Eq. (15)], we can rewrite Eqs. (47) and
(48) using vectors of Schmidt coefficients and the Hadamard product as,

$\displaystyle\lambda_{\xi}=\textbf{{n}}\odot\lambda_{\chi},$ (49)
$\displaystyle\lambda_{\nu}=\textbf{{n}}\odot\lambda_{\zeta},$ (50)

where $(\textbf{{n}})_{i}=\langle i|\hat{N}|i\rangle/\sqrt{1-r_{1}},\forall
i\in[1,d]$. Finally, using again Lemma 2, valid because of the fact that
$(\textbf{{n}})_{i}$ is a decreasingly-ordered vector and the fact that
$\lambda_{\zeta}\prec\lambda_{\chi}$, we complete the proof that
$\lambda_{\nu}\prec\lambda_{\xi}$.

∎

##  V Generalization to a collection of initial or final states

Let us first generalize Theorem 1 to a scenario involving multiple final
states. Imagine that Alice and Bob possess a state $\ket{\psi}$ and that,
instead of a single target state, there is a collection of $m$ possible target
states $\\{\ket{\phi_{j}}\\}_{j=1}^{m}$ from which they will have to obtain a
single one (without knowing which one beforehand). The idea is that they can
first perform a probabilistic LOCC transformation to the OCR state of the
$m+1$ states $\ket{\psi\overset{m}{\underset{j=1}{\land}}\phi_{j}}$, extending
the idea of the thrifty protocol. Once they possess the OCR state, they can
wait to learn which state $\ket{\phi_{j}}$ they have to obtain before
performing the corresponding deterministic LOCC transformation from the OCR
state to $\ket{\phi_{j}}$. Interestingly, the probabilistic LOCC from
$\ket{\psi}$ to the OCR state has an optimal probability that is equal to the
minimum optimal probability of all individual conversions from $\ket{\psi}$ to
$\ket{\phi_{j}}$. In some sense, with this minimum probability corresponding
to the hardest-to-reach target state $\ket{\phi_{j}}$, we may as well stop the
protocol halfway to produce the more entangled OCR state. This is the content
of the following theorem.

######  Theorem 3.

Let $\ket{\psi}$ be a bipartite pure state and
$\\{\ket{\phi_{j}}\\}_{j=1}^{m}$ be a collection of $m$ bipartite pure states.
The optimal probability $p_{\max}^{[1]}$ of conversion from $\ket{\psi}$ to
the OCR state of the $m+1$ states
$\ket{\psi\overset{m}{\underset{j=1}{\land}}\phi_{j}}$ is equal to the minimum
optimal probability among the $m$ possible conversions from $\ket{\psi}$ to
$\ket{\phi_{j}}$, i.e.,

$p_{\max}^{[1]}=\underset{j\in[1,m]}{\min}\ p_{\max,j}^{[2]},$ (51)

where $p_{\max,j}^{[2]}$ denotes the optimal probability of conversion from
$\ket{\psi}$ to $\ket{\phi_{j}}$.

###### Proof.

From Eq. (9), we have

$p_{\max}^{[1]}=\underset{l\in[1,d]}{\min}\frac{E_{l}(\psi)}{E_{l}(\psi\overset{m}{\underset{j=1}{\land}}\phi_{j})},$
(52)

as well as

$p_{\max,j}^{[2]}=\underset{l\in[1,d]}{\min}\frac{E_{l}(\psi)}{E_{l}(\phi_{j})},$
(53)

and we want to prove that

$p_{\max}^{[1]}=\underset{j\in[1,m]}{\min}\,p_{\max,j}^{[2]}\equiv
p_{\max}^{[2]}.$ (54)

In order to do so, we can use the straightforward generalization of Lemma 1,
namely

$E_{l}(\psi\overset{m}{\underset{j=1}{\land}}\phi_{j})=\max\left\\{E_{l}(\psi),E_{l}(\phi_{1}),\cdots,E_{l}(\phi_{m})\right\\}.$
(55)

Thus we have $p_{\max}^{[1]}=\underset{l\in[1,d]}{\min}\,p_{\max,l}^{[1]}$,
where

$p_{\max,l}^{[1]}=\frac{E_{l}(\psi)}{\max\left\\{E_{l}(\psi),E_{l}(\phi_{1}),\cdots,E_{l}(\phi_{m})\right\\}},$
(56)

and, by interchanging the minima, we can also write

$\displaystyle p_{\max}^{[2]}$
$\displaystyle=\underset{l\in[1,d]}{\min}p_{\max,l}^{[2]},$ (57)

where

$p_{\max,l}^{[2]}=\underset{j\in[1,m]}{\min}\frac{E_{l}(\psi)}{E_{l}(\phi_{j})}.$
(58)

Therefore, using Eqs. (56) and (58), it is easy to see that
$p_{\max,l}^{[2]}\geq p_{\max,l}^{[1]}=1$ for any value of $l$ such that
$\max\left\\{E_{l}(\psi),E_{l}(\phi_{1}),\cdots,E_{l}(\phi_{m})\right\\}=E_{l}(\psi)$.
These values of $l$ can thus be disregarded in the minimization over $l$ since
the minimum cannot exceed 1 (indeed, $p_{\max,1}^{[2]}=p_{\max,1}^{[1]}=1$
when $l=1$). Thus, we only have to consider the values of $l$ such that
$\max\left\\{E_{l}(\psi),E_{l}(\phi_{1}),\cdots,E_{l}(\phi_{m})\right\\}=E_{l}(\phi_{k})$,
$k\in[1,m]$, implying
$p_{\max,l}^{[2]}=p_{\max,l}^{[1]}=E_{l}(\psi)/E_{l}(\phi_{k})$. Hence, the
minimum over $l$ yields $p_{\max}^{[1]}=p_{\max}^{[2]}$, which completes the
proof. ∎

In other words, as long as we do not know which target state $\ket{\phi_{j}}$
we will have to reach, we may anticipate the probabilistic step and move to
the OCR, but this can only be done with the minimum probability (corresponding
to the worst-case target state). The benefit of this procedure is that once
the target $\ket{\phi_{j}}$ is disclosed, the final step is deterministic.
Otherwise, we may wait until the target $\ket{\phi_{j}}$ is disclosed and only
then perform a probabilistic transformation from $\ket{\psi}$ to
$\ket{\phi_{j}}$, whose probability may be higher (depending on which is the
target state). In a nutshell, a compromise can be made between doing the
probabilistic transformation at first (making the rest fully deterministic
regardless of the target state) or waiting to know which target state is
wanted before doing the actual probabilistic transformation (possibly with a
higher success probability).

An analogous statement can be made about a scenario involving multiple initial
states $\\{\ket{\psi_{j}}\\}_{j=1}^{m}$ and a single final state $\ket{\phi}$,
generalizing the greedy protocol. This time, the optimal probability of
conversion $p_{\max}^{[1]}$ from the OCP state of the $m+1$ states
$\ket{\psi_{j}\overset{m}{\underset{j=1}{\lor}}\phi}$ to $\ket{\phi}$ is
equivalent to the minimum optimal probability between any of the $m$ possible
conversions from $\ket{\psi_{j}}$ to $\ket{\phi}$. This is immediately
understood because, as proven in Ref. [13], the intermediate state for the
least probable of the $m$ possible conversions is located in the lower cone of
$\ket{\psi_{j}\overset{m}{\underset{j=1}{\lor}}\phi}$. Therefore, the
transformation from $\ket{\psi_{j}\overset{m}{\underset{j=1}{\lor}}\phi}$ to
$\ket{\phi}$ is at least as likely as the least probable of the $m$ possible
conversions from $\ket{\psi_{j}}$ to $\ket{\phi}$. Here, we have a compromise
between doing the deterministic step at first (postponing the subsequent
probabilistic step) or waiting until the identity of the initial state is
known before doing the actual probabilistic transformation (possibly with a
higher success probability).

##  VI Conclusion

The lattice structure of majorization uncovers two essential states in the
context of entanglement transformations between incomparable bipartite pure
states $\ket{\psi}$ and $\ket{\phi}$, namely the optimal common resource state
(i.e., the meet $\ket{\psi\land\phi}$ of the two states) and the optimal
common product state (i.e., the join $\ket{\psi\lor\phi}$ of the two states).
We have shown that both states naturally appear when considering (single-copy)
probabilistic LOCC transformations from $\ket{\psi}$ to $\ket{\phi}$. We have
indeed defined two antipodal protocols, namely the greedy protocol, passing
through $\ket{\psi\lor\phi}$, and the thrifty protocol, passing through
$\ket{\psi\land\phi}$. Both protocols can be proven to be optimal (their
success probability is maximum). However, while the greedy protocol is very
similar to Vidal’s protocol [6], the thrifty protocol is superior in that the
entanglement resource is better preserved on average (in case of failure, its
residual state is majorized by – hence, is more entangled than – the residual
state of the greedy protocol). Note that in case $\ket{\psi}$ and $\ket{\phi}$
are comparable, both the greedy and thrifty protocols reduce to Vidal’s
protocol, which reflects that the incomparability between the states is an
essential ingredient here. Finally, we have shown that the greedy and thrifty
protocols can be generalized to scenarios involving an arbitrary number of
initial or final states.

It would be interesting to extend the current work to multipartite settings.
Indeed, to our knowledge, the notions of optimal common resource or optimal
common product have never been explored for more than two parties. For
example, we know that true tripartite entanglement can be split into two
categories, namely GHZ-type and W-type states [21], but the determination of
an optimal common resource state or optimal common product state within each
class constitutes an open problem, as well as the role of incomparability in
this context.

###### Acknowledgements.

We warmly thank Ali Asadian for pointing out to us the relevance of the
majorization lattice in the context of entanglement transformations as well as
Michael Jabbour for useful discussions on majorization theory. S.D.
acknowledges support from the French Community of Belgium through a FRIA
fellowship. M.A. acknowledges support from MEYS Czech Republic and the
European Union’s Horizon 2020 research and innovation framework programme
under Grant No. 731473 (project 8C20002 ShoQC). C.G. acknowledges support from
the Fonds de la Recherche Scientifique – FNRS. N.J.C. acknowledges support
from the Fonds de la Recherche Scientifique - FNRS and the European Union
under project ShoQC within the Horizon 2020 ERA-NET Cofund in Quantum
Technologies (QuantERA) program.

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