# Matching for probabilistic entanglement swapping

Luis Roa<EMAIL_ADDRESS>Departamento de Física, Universidad de Concepción,
Casilla 160-C, Concepción, Chile.  Torben L. Purz  Faculty of Physics, Georg-
August-Universität Göttingen, Göttingen, Germany  Ariana Muñoz  Facultad de
Ingeniería, Universidad Autónoma de Chile, 5 Poniente 1670, Talca, Chile
Sebastián Castro  Departamento de Física, Universidad de Concepción, Casilla
160-C, Concepción, Chile.  Gonzalo Hidalgo  Departamento de Física,
Universidad de Concepción, Casilla 160-C, Concepción, Chile.  David Montoya
Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción,
Chile.

###### Abstract

The basic entanglement swapping protocol allows to achieve the main aim of
deterministically projecting two qubits, which have never interacted, onto a
maximally entangled state. For deterministic swapping the key ingredient is
the maximal entanglement initially contained in two pairs of qubits and the
ability to measure onto a Bell basis. In other words, the basic and
deterministic entanglement swapping scheme involves three maximal level of
entanglement. In this work we address probabilistic entanglement swapping
processes performed with different amounts of initial entanglement. Besides,
we suggest a non Bell measuring-basis, thus we introduce a third entanglement
level in the process. Additionally, we propose the _unambiguous state
extraction_ scheme as the local mechanism for probabilistically achieving the
main aim. Amalgamating these three elements allows us to design four
strategies for performing probabilistic entanglement swapping. Surprisingly,
we encountered an twofold entanglement matching effect related to the
concurrence of the measuring-basis. Specifically, the maximal probability of
accomplishing the main aim becomes a constant for concurrences higher than or
equal to the matching entanglement values. Thus, we show that maximal
entanglement in the measuring-basis is not required for attaining the main aim
with optimal probability.

###### pacs:

03.67.Bg, 03.65.Ud, 03.67.Ac

††preprint: APS

##  I Introduction

At the beginning of quantum mechanics Erwin Schrödinger noticed the existence
of a special correlation, the entanglement, present in a superposition of
tensorial product states of two or more systems Schrodinger . Currently, it is
known that the entanglement is a purely quantum ingredient, which introduces
non-local effects in protocols for processing information on atomic and
molecule scales NielsenBook ; APeres ; Mermin ; GAlber . Thus, almost
instantly the efforts focussed on finding well defined measurements of
entanglement. The most known of these measurements is the entanglement of
formation, which quantifies the resources needed to create a given entangled
state CHBennett . Specifically, W. K. Wootters found a closed analytical
formula for the concurrence of an arbitrary state of two qubits, which is a
monotone function of the entanglement of formation Wootters1 ; Wootters2 . In
consequence, for two qubits, the concurrence can be used as a measurement of
entanglement in its own merit, which is what we consider here. For a $2\otimes
2$ pure state, its concurrence can be evaluated at a glance, if it is
represented in the Schmidt decomposition; it is two times the product of its
Schmidt coefficients Wootters1 ; Wootters2 . So, a factorized state lacks
concurrence, whereas a maximal entangled state (EPR) has a concurrence equal
to $1$.

In this context, the generation, manipulation, control, and the practical
scope of this quantum correlation become important research areas. In
particular, the capacity to distribute entanglement between distant systems
has potential applications in designing innovative protocols of quantum
information without classical counterpart GBrassard ; MZukowski ; EKnill ;
HJBriegel ; PEShor ; VMKendon ; MHillery ; ASCoelho ; JCRetamal ; CJoshi ;
UAkram ; PFacchi ; CSpee ; CasparG .

The entanglement swapping, as a mechanism for distributing the entanglement
correlation, has been extensively studied both theoretically and
experimentally MZukowski ; BYurke ; Yurke ; Weinfurter ; PKok ; CBranciard ;
Adhikari ; TieJunWang ; JCho ; AKhalique ; SMRoy ; GGruning ; ShuChengLi ;
BTKirby ; ChuanMeiXie ; YHLiu ; MNaseri .

In this article, we analyze four schemes for performing probabilistic
entanglement swapping, in which the main aim is to project two qubits, that
have never interacted, onto a Bell state. We consider, that the two pairs of
qubits can be in different partially entangled pure states. Besides, we
consider that the measuring-basis can consist of non-Bell states.
Additionally, we propose the so called _unambiguous state extraction_ (USE)
protocol Hsu ; BHe ; Hutin ; Gautam  as the mechanism for implementing the
local operations.

The article is organized as follows. In Sec. II we briefly recall the well
known, deterministic and basic entanglement swapping scheme. In Sec. III we
propose four strategies for achieving the aim of probabilistically projecting
onto an EPR state, shared by two qubits which have never interacted. In the
last Sec. IV we summarize the principal results. Additionally, we included the
Appendix A where we show that the _unambiguous state extraction_ (USE)
protocol Hsu ; BHe ; Hutin ; Gautam  can be locally applied to extract an EPR
state with optimal probability. In particular, we introduce an explicit joint
unitary transformation, which allows to accomplish the main aim, additionally
finding the optimal probability of success as a function of the initial
concurrence Bose ; Vidal ; Nielsen ; BKraus ; Guevara . The results of the
appendix are directly applied onto the first, second and fourth strategy.

##  II Deterministic and basic entanglement swapping

The basic entanglement swapping scheme considers four qubits $A$, $C_{1}$,
$B$, $C_{2}$, the pairs $AC_{1}$ and $BC_{2}$ previously prepared in Bell
states. The qubits $C_{1}$ and $C_{2}$ remain in the same laboratory
(Lab-$C$), whereas the qubits $A$ and $B$ are carried to two different
laboratories, away from each other and away from Lab-$C$. Therefore, joint
operations are only allowed between qubits $C_{1}$ and $C_{2}$. Local
operations can be applied onto each of the four qubits and classical
communication is enabled among them all. The aim of this scheme is to
maximally entangle the pair $AB$ by using classical communication, local and
joint operations. Figure 1 shows the different locations of the four qubits.

The following identity gives account of the basic entanglement swapping
procedure,

$\displaystyle|\psi_{AC_{1}}^{+}\rangle|\psi_{BC_{2}}^{+}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(|\phi_{C_{1}C_{2}}^{+}\rangle|\phi_{AB}^{+}\rangle\\!+|\psi_{C_{1}C_{2}}^{+}\rangle|\psi_{AB}^{+}\rangle\right.$
(1)
$\displaystyle\\!\left.\\!-|\psi_{C_{1}C_{2}}^{-}\rangle|\psi_{AB}^{-}\rangle-|\phi_{C_{1}C_{2}}^{-}\rangle|\phi_{AB}^{-}\rangle\\!\right)\\!,$

where

$\displaystyle|\psi^{\pm}_{ij}\rangle$ $\displaystyle=$
$\displaystyle\left(|0_{i}\rangle|1_{j}\rangle\pm|1_{i}\rangle|0_{j}\rangle\right)/\sqrt{2},$
$\displaystyle|\phi^{\pm}_{ij}\rangle$ $\displaystyle=$
$\displaystyle\left(|0_{i}\rangle|0_{j}\rangle\pm|1_{i}\rangle|1_{j}\rangle\right)/\sqrt{2},$

are the Bell-basis of a pair of qubits $i\otimes j$ NielsenBook . The right
hand side of Eq. (1) clearly shows the one-to-one correlation among the Bell
states of the pairs $C_{1}C_{2}$ and $AB$. From this, one realizes that by
measuring the Bell states in the Lab-$C$ the pair $AB$ is projected onto a
Bell state, thus deterministically achieving the main aim. It is worth noting,
that determinism demands the key ingredient of maximal entanglement in the two
initial states and also in the measuring-basis.

Figure 1: Spatial distribution of the four qubits $A$, $B$, $C_{1}$ and
$C_{2}$. Qubits $C_{1}$ and $C_{2}$ are in the same laboratory. The
laboratories of the qubits $A$ and $B$ are away from each other and away from
Lab-$C$.

##  III Probabilistic entanglement swapping

We analyze four strategies for probabilistically obtaining an EPR state in the
bipartite system $AB$, when the pairs $AC_{1}$ and $BC_{2}$ initially are in
the following partially entangled pure states,

$\displaystyle|\bar{\psi}_{AC_{1}}\rangle$ $\displaystyle=$
$\displaystyle\alpha|0_{A}\rangle|1_{C_{1}}\rangle+\beta|1_{A}\rangle|0_{C_{1}}\rangle,$
(2a) $\displaystyle|\bar{\psi}_{BC_{2}}\rangle$ $\displaystyle=$
$\displaystyle\gamma|0_{B}\rangle|1_{C_{2}}\rangle+\delta|1_{B}\rangle|0_{C_{2}}\rangle,$
(2b)

where, without loss of generality, we assume the amplitudes $\alpha$, $\beta$,
$\gamma$, $\delta$ to be real and non-negative numbers,

$\alpha\leq\beta,\text{ and }\gamma\leq\delta.$ (3)

For normalization $\alpha^{2}+\beta^{2}=1$ and $\gamma^{2}+\delta^{2}=1$.

The initial concurrences of the states (2) are
$\mathcal{C}_{AC_{1}}=2\alpha\beta$ and $\mathcal{C}_{BC_{2}}=2\gamma\delta$
respectively.

###  III.1 First strategy

The first, simplest strategy is to extract an EPR state from each bipartite
state (2) with the unitary-reduction local operations described in Appendix A.

If both processes are successful, the basic swapping scheme can be carried
out. Accordingly, by considering Eqs. (28) and (3), the success probability of
obtaining an EPR in the pair $AB$ is equal to the product of the probabilities
of extracting an EPR from each state (2), which becomes,

$\displaystyle P_{s_{1}}$ $\displaystyle=$ $\displaystyle
4\alpha^{2}\gamma^{2},$ (4) $\displaystyle=$
$\displaystyle\left(1-\sqrt{1-\mathcal{C}_{AC_{1}}^{2}}\right)\left(1-\sqrt{1-\mathcal{C}_{BC_{2}}^{2}}\right).$

Note, that $P_{s_{1}}$ is an increasing function of the initial concurrences,
$\mathcal{C}_{AC_{1}}$ and $\mathcal{C}_{BC_{2}}$, besides, the entanglement
of both initial states is necessary and sufficient for having a success
probability different from zero.

###  III.2 Second strategy

The second strategy is suggested by writing the initial tensorial product
state $|\bar{\psi}_{AC1}\rangle|\bar{\psi}_{BC2}\rangle$ in the representation
of the Bell-basis of $C_{1}C_{2}$, i.e.,

$\displaystyle|\bar{\psi}_{AC1}\rangle|\bar{\psi}_{BC2}\rangle$
$\displaystyle=$
$\displaystyle\sqrt{\frac{p}{2}}|\phi_{C1C2}^{+}\rangle|\ddot{\phi}_{AB}^{+}\rangle$
(5)
$\displaystyle+\sqrt{\frac{1-p}{2}}|\psi_{C1C2}^{+}\rangle|\ddot{\psi}_{AB}^{+}\rangle$
$\displaystyle-\sqrt{\frac{1-p}{2}}|\psi_{C1C2}^{-}\rangle|\ddot{\psi}_{AB}^{-}\rangle$
$\displaystyle-\sqrt{\frac{p}{2}}|\phi_{C1C2}^{-}\rangle|\ddot{\phi}_{AB}^{-}\rangle,$

where we have defined the states of the pair $AB$ as follows:

$\displaystyle|\ddot{\phi}_{AB}^{\pm}\rangle$ $\displaystyle=$
$\displaystyle\frac{\alpha\gamma|0\rangle|0\rangle\pm\beta\delta|1\rangle|1\rangle}{\sqrt{p}},$
(6a) $\displaystyle|\ddot{\psi}_{AB}^{\pm}\rangle$ $\displaystyle=$
$\displaystyle\frac{\alpha\delta|0\rangle|1\rangle\pm\beta\gamma|1\rangle|0\rangle}{\sqrt{1-p}},$
(6b)

and the probability

$p=\alpha^{2}\gamma^{2}+\beta^{2}\delta^{2}.$

The identity (5) shows a one-to-one correlation among the Bell states of
$C_{1}C_{2}$ and the states (6) of the pair $AB$. Thus we realize, that by
measuring Bell states in Lab-$C$, the bipartite system $AB$ is projected onto
one of the partially entangled states (6) with respective probabilities $p/2$
and $(1-p)/2$. Notice, that the minus sign in $|\ddot{\phi}_{AB}^{-}\rangle$
and $|\ddot{\psi}_{AB}^{-}\rangle$ can be removed by locally applying the
Pauli operator $\sigma_{z}$ on $A$ or $B$.

Therefore, in practice, the system $AB$ has two outcomes
$|\ddot{\phi}_{AB}^{+}\rangle$ with probability $p$ and
$|\ddot{\psi}_{AB}^{+}\rangle$ with probability $1-p$.

From each of these states, $|\ddot{\phi}_{AB}^{+}\rangle$ and
$|\ddot{\psi}_{AB}^{+}\rangle$, one of the receivers, $A$ or $B$, can
probabilistically extract a Bell state by means of the local USE scheme. If
the outcome is $|\ddot{\phi}_{AB}^{+}\rangle$, the conditional probability of
extracting a Bell state becomes $p_{ext,\phi}=2\alpha^{2}\gamma^{2}/p$,
otherwise, if the outcome is $|\ddot{\psi}_{AB}^{+}\rangle$, there are two
cases for evaluating the conditional probability:
(i) when $\alpha\delta\leq\beta\gamma$, being equivalent to
$\alpha\leq\gamma$, the conditional probability of succesful extraction is
$p_{ext,\psi}=2\alpha^{2}\delta^{2}/(1-p)$;
(ii) when $\alpha\delta\geq\beta\gamma$, which is equivalent to
$\alpha\geq\gamma$, the conditional success probability is given by
$p_{ext,\psi}=2\beta^{2}\gamma^{2}/(1-p)$.

Accordingly, the total success probability of achieving an EPR state in the
pair $AB$ becomes,

$\displaystyle P_{s_{2}}$ $\displaystyle=$ $\displaystyle
pp_{ext,\phi}+(1-p)p_{ext,\psi},$ (7) $\displaystyle=$ $\displaystyle
2\min\left\\{\alpha^{2},\gamma^{2}\right\\},$ $\displaystyle=$
$\displaystyle\min\bigg{\\{}1-\sqrt{1-\mathcal{C}_{AC_{1}}^{2}},1-\sqrt{1-\mathcal{C}_{BC_{2}}^{2}}\bigg{\\}}.$

Note, that this expression of $P_{s_{2}}$ exhibits the so called entanglement
matching effect, since it is determined by the smallest entanglement of the
initial states $|\bar{\psi}_{AC_{1}}\rangle$ and
$|\bar{\psi}_{BC_{2}}\rangle$. It is worth emphasizing, that both initial
entanglements are necessary and sufficient for having a success probability
different from zero. Besides, by comparing the success probabilities of both
strategies we find that,

$P_{s_{2}}\geq P_{s_{1}}.$ (8)

The equality $P_{s_{2}}=P_{s_{1}}$holds true, if at least one of the two
initial concurrences is equal to $1$ or equal to zero. For instance, by
considering $\mathcal{C}_{AC_{1}}$ fixed, $P_{s_{2}}$ achieves its maximal
value for $\mathcal{C}_{BC_{2}}=\mathcal{C}_{AC_{1}}$, whereas $P_{s_{1}}$
reaches the same maximal value at $\mathcal{C}_{BC_{2}}=1$. In other words,
for a fixed $\mathcal{C}_{AC_{1}}$, the maximal $P_{s_{2}}$ demands
$\mathcal{C}_{BC_{2}}=\mathcal{C}_{AC_{1}}$, while the same maximal value for
$P_{s_{1}}$ demands $\mathcal{C}_{BC_{2}}=1$. Figure 2 illustrates the
behavior of the probabilities (4) and (7) for different values of
$\mathcal{C}_{AC_{1}}$.

In consequence, if we consider entanglement as a resource, we can conclude
that the second strategy is more efficient than the first one.

Figure 2: Success probabilities for generating an EPR state, as functions of
$\mathcal{C}_{BC_{2}}$ with the first strategy (solid line) and with the
second strategy (dashed line) for different vales of $\mathcal{C}_{AC_{1}}$:
0.4 (black), 0.7(gray), and 0.97 (light-gray).

###  III.3 Strategies without Bell-basis measurement

Now, instead of projecting onto Bell states in $C_{1}C_{2}$, we propose to
measure an observable that has the following eigenstates:

$\displaystyle|\mu_{C_{1}C_{2}}^{+}\rangle$ $\displaystyle=$ $\displaystyle
x|00\rangle+y|11\rangle,$ (9a) $\displaystyle|\mu_{C_{1}C_{2}}^{-}\rangle$
$\displaystyle=$ $\displaystyle y|00\rangle-x|11\rangle,$ (9b)
$\displaystyle|\nu_{C_{1}C_{2}}^{+}\rangle$ $\displaystyle=$ $\displaystyle
x|01\rangle+y|10\rangle,$ (9c) $\displaystyle|\nu_{C_{1}C_{2}}^{-}\rangle$
$\displaystyle=$ $\displaystyle y|01\rangle-x|10\rangle,$ (9d)

where, without loss of generality, we can consider the amplitudes $x$ and $y$
to be real and non-negative numbers and

$x\leq y.$ (10)

For normalization $x^{2}+y^{2}=1$. Each state (9) has the same concurrence
$\mathcal{C}_{C_{1}C_{2}}(x)=2xy$, which is a function of the basis
determining parameter $x$.

So the natural question, if there are _special values_ for the concurrence
$\mathcal{C}_{C_{1}C_{2}}(x)$, with regard to the initial ones
$\mathcal{C}_{AC_{1}}$ and $\mathcal{C}_{BC_{2}}$, for which the main aim can
be accomplished, arises. Obviously, those special values should be different
from $1$.

The answer is affirmative, we encountered two _special values_ of the
concurrence $\mathcal{C}_{C_{1}C_{2}}(x)$ given by

$\mathfrak{C}_{C_{1}C_{2}}^{\pm}=\frac{\mathcal{C}_{AC_{1}}\mathcal{C}_{BC_{2}}}{1\pm\sqrt{(1-\mathcal{C}_{AC_{1}}^{2})(1-\mathcal{C}_{BC_{2}}^{2}})}.$
(11)

In the search for it we found two schemes, which we will describe below.

Those two schemes are suggested by representing the initial state
$|\bar{\psi}_{AC1}\rangle|\bar{\psi}_{BC2}\rangle$ in the basis (9), which
leads us to analyze the identity,

$\displaystyle|\bar{\psi}_{AC1}\rangle|\bar{\psi}_{BC2}\rangle$
$\displaystyle=$
$\displaystyle\sqrt{p_{\mu^{+}}}|\mu_{C_{1}C_{2}}^{+}\rangle|\ddot{\phi}_{AB}\rangle$
(12)
$\displaystyle+\sqrt{p_{\nu^{+}}}|\nu_{C_{1}C_{2}}^{+}\rangle|\ddot{\psi}_{A,B}\rangle$
$\displaystyle-\sqrt{p_{\nu^{-}}}|\nu_{C_{1}C_{2}}^{-}\rangle|\dddot{\psi}_{A,B}\rangle$
$\displaystyle-\sqrt{p_{\mu^{-}}}|\mu_{C_{1}C_{2}}^{-}\rangle|\dddot{\phi}_{A,B}\rangle.$

Here we have defined the possible outcome states of the qubits $AB$,

$\displaystyle|\ddot{\phi}_{AB}\rangle$ $\displaystyle=$
$\displaystyle\frac{\alpha\gamma y|00\rangle+\beta\delta
x|11\rangle}{\sqrt{p_{\mu^{+}}}},$ (13a)
$\displaystyle|\ddot{\psi}_{AB}\rangle$ $\displaystyle=$
$\displaystyle\frac{\alpha\delta y|01\rangle+\beta\gamma
x|10\rangle}{\sqrt{p_{\nu^{+}}}},$ (13b)
$\displaystyle|\dddot{\psi}_{AB}\rangle$ $\displaystyle=$
$\displaystyle\frac{\alpha\delta x|01\rangle-\beta\gamma
y|10\rangle}{\sqrt{p_{\nu^{-}}}},$ (13c)
$\displaystyle|\dddot{\phi}_{AB}\rangle$ $\displaystyle=$
$\displaystyle\frac{\alpha\gamma x|00\rangle-\beta\delta
y|11\rangle}{\sqrt{p_{\mu^{-}}}},$ (13d)

and their respective probabilities,

$\displaystyle p_{\mu^{+}}$ $\displaystyle=$
$\displaystyle\alpha^{2}\gamma^{2}y^{2}+\beta^{2}\delta^{2}x^{2},$ (14a)
$\displaystyle p_{\nu^{+}}$ $\displaystyle=$
$\displaystyle\alpha^{2}\delta^{2}y^{2}+\beta^{2}\gamma^{2}x^{2},$ (14b)
$\displaystyle p_{\nu^{-}}$ $\displaystyle=$
$\displaystyle\alpha^{2}\delta^{2}x^{2}+\beta^{2}\gamma^{2}y^{2},$ (14c)
$\displaystyle p_{\mu^{-}}$ $\displaystyle=$
$\displaystyle\alpha^{2}\gamma^{2}x^{2}+\beta^{2}\delta^{2}y^{2}.$ (14d)

It is important to take into account that $x$ is a key measuring-basis
parameter, to be strategically set in each of the following schemes. By this,
we fix the required amount of entanglement of the measuring-basis (9).

####  III.3.1 Third strategy

It is worth noting that the concurrences of the outcomes (13) can attain the
maximal value $1$, but at different values for $x$. Specifically, the state
$|\ddot{\phi}_{AB}\rangle$ is maximally entangled at $x=x_{1}$, the outcome
$|\ddot{\psi}_{AB}\rangle$ at $x=x_{2}$, the state $|\dddot{\psi}_{AB}\rangle$
at $x_{3}$, and $|\dddot{\phi}_{AB}\rangle$ at $x=x_{4}$, where

$\displaystyle x_{1}$ $\displaystyle=$
$\displaystyle\frac{\alpha\gamma}{\sqrt{\alpha^{2}\gamma^{2}+\beta^{2}\delta^{2}}},$
(15a) $\displaystyle x_{2}$ $\displaystyle=$
$\displaystyle\frac{\alpha\delta}{\sqrt{\alpha^{2}\delta^{2}+\beta^{2}\gamma^{2}}},$
(15b) $\displaystyle x_{3}$ $\displaystyle=$
$\displaystyle\frac{\beta\gamma}{\sqrt{\alpha^{2}\delta^{2}+\beta^{2}\gamma^{2}}},$
(15c) $\displaystyle x_{4}$ $\displaystyle=$
$\displaystyle\frac{\beta\delta}{\sqrt{\alpha^{2}\gamma^{2}+\beta^{2}\delta^{2}}}.$
(15d)

In general, these $x_{i}$ are different and, according to (3), they are
ordered as follows,

$\displaystyle x_{1}\leq x_{2}\leq\frac{1}{\sqrt{2}}\leq x_{3}\leq x_{4},$
$\displaystyle\quad\text{if }\alpha\leq\gamma,$ (16a) $\displaystyle x_{1}\leq
x_{3}\leq\frac{1}{\sqrt{2}}\leq x_{2}\leq x_{4},$ $\displaystyle\quad\text{if
}\alpha\geq\gamma.$ (16b)

This means that the possible EPR outcomes are displaced in the measuring-basis
parameter $x$. Besides, we observe by considering (3) with $\alpha\neq\beta$
and $\gamma\neq\delta$, that $|\dddot{\phi}_{A,B}\rangle$ does not exhibit
maximal entanglement, since $x_{4}\notin[0,1/\sqrt{2}]$ (see condition (10)).
On the other hand, the maximal entanglement outcome can be at
$|\ddot{\psi}_{AB}\rangle$ or $|\dddot{\psi}_{AB}\rangle$ depending on the
relation between $\alpha$ and $\gamma$. In addition, note that if (3) is not
satisfied, then it is $|\dddot{\phi}_{A,B}\rangle$ which exhibits maximal
entanglement instead of $|\ddot{\phi}_{A,B}\rangle$.

Therefore, to obtain the optimal result, we choose the $x_{i}$ corresponding
to the greatest probability as the measuring-basis parameter. By inserting the
$x_{i}$ in their respective probability (14), we realize that the greatest
probability $P_{s_{3}}$ is given by

$\displaystyle P_{s_{3}}$ $\displaystyle=$
$\displaystyle\left\\{\begin{tabular}[]{ll}$p_{\nu^{+}}(x_{2}),$&if
$\alpha\leq\gamma,$\\\ $p_{\nu^{-}}(x_{3}),$&if
$\alpha\geq\gamma,$\end{tabular}\right.$ $\displaystyle=$
$\displaystyle\frac{2\alpha^{2}\beta^{2}\delta^{2}\gamma^{2}}{\alpha^{2}\delta^{2}+\beta^{2}\gamma^{2}},$
$\displaystyle=$
$\displaystyle\frac{\mathcal{C}_{AC_{1}}\mathcal{C}_{BC_{2}}\mathfrak{C}_{C_{1}C_{2}}^{-}}{4}.$
(20)

This probability is smaller than, or equal to the ones found in the first and
second strategy, i.e.,

$P_{s_{3}}\leq P_{s_{1}}\leq P_{s_{2}}.$ (21)

Note also, that $P_{s_{3}}\leq 1/4$ and the equality only holds true for the
limit of the basic and determinist scheme, i.e.,
$\mathcal{C}_{AC_{1}}=\mathcal{C}_{BC_{2}}=1$ and
$x_{1}=x_{2}=x_{3}=x_{4}=1/\sqrt{2}$, meaning each outcome is maximally
entangled and each one can be obtained with probability $1/4$.

In spite of (21), it is worth realizing that in general $x_{2}$ and $x_{3}$
are different from $1/\sqrt{2}$, which stands for not requiring maximal
entanglement of the measuring-basis (9) in order to have a probability
different from zero of obtaining an EPR state in the outcome (13b) or (13c).
Specifically, the values $x_{2}$ and $x_{3}$ are associated with the same
special value $\mathfrak{C}_{C_{1}C_{2}}^{-}$ of the concurrence,

$\mathfrak{C}_{C_{1}C_{2}}^{-}=\mathcal{C}_{C_{1}C_{2}}(x_{2})=\mathcal{C}_{C_{1}C_{2}}(x_{3}).$

From Eq.(11) we can note that if one of the two initial concurrences
$\mathcal{C}_{AC_{1}}$ or $\mathcal{C}_{BC_{2}}$ is equal to $1$, to say
$\mathcal{C}_{AC_{1}}=1$, then
$\mathfrak{C}_{C_{1}C_{2}}^{-}=\mathcal{C}_{BC_{2}}$. If
$\mathcal{C}_{AC_{1}}=\mathcal{C}_{BC_{2}}$, then we can conclude
$\mathfrak{C}_{C_{1}C_{2}}^{-}=1$. Provided, one of the two initial
concurrences, $\mathcal{C}_{AC_{1}}$ or $\mathcal{C}_{BC_{2}}$, is equal to
$0$, then $\mathfrak{C}_{C_{1}C_{2}}^{-}=0$ and $P_{s_{3}}=0$.

The significance of these results is that there is a probability different
from zero of accomplishing the main aim without requiring maximal entanglement
in the measuring-basis. That motivates us to propose the following strategy.

We want to mention here that in Ref. ChuanMeiXie  the authors focus on the
study of the behaviour and the relation among the concurrences of the outcome
states, the parameter of the measuring-basis, and the respective probabilities
of the outcome states.

####  III.3.2 Fourth strategy

Once knowing the measurement result (9), one of the receivers, $A$ or $B$, can
probabilistically extract an EPR state by means of the local USE procedure
from the respective outcome (13).

According to Appendix A the conditional probabilities of successfully
extracting depend on the relation between $\alpha$ and $\gamma$ and read as
follows. If the outcome is $|\ddot{\phi}_{AB}\rangle$, the conditional
probability $p_{ext,\ddot{\phi}}$ of extracting a Bell state becomes,

$p_{ext,\ddot{\phi}}=\left\\{\begin{tabular}[]{ll}$\frac{2\beta^{2}\delta^{2}x^{2}}{p_{\mu^{+}}}$,&if
$x\leq x_{1}$,\\\ $\frac{2\alpha^{2}\gamma^{2}y^{2}}{p_{\mu^{+}}}$,&if $x\geq
x_{1}$.\end{tabular}\right.$ (22a) When the outcome is
$|\ddot{\psi}_{A,B}\rangle$, the conditional probability of success
$p_{ext,\ddot{\psi}}$ is,
$p_{ext,\ddot{\psi}}=\left\\{\begin{tabular}[]{ll}$\frac{2\beta^{2}\gamma^{2}x^{2}}{p_{\nu^{+}}}$,&if
$x\leq x_{2}$,\\\ $\frac{2\alpha^{2}\delta^{2}y^{2}}{p_{\nu^{+}}},$&if $x\geq
x_{2}.$\end{tabular}\right.$ (22b) While, if the outcome is
$|\dddot{\psi}_{A,B}\rangle$, then the conditional success probability
$p_{ext,\dddot{\psi}}$ reads,
$p_{ext,\dddot{\psi}}=\left\\{\begin{tabular}[]{ll}$\frac{2\alpha^{2}\delta^{2}x^{2}}{p_{\nu^{-}}}$,&if
$x\leq x_{3}$,\\\ $\frac{2\beta^{2}\gamma^{2}y^{2}}{p_{\nu^{-}}}$,&if $x\geq
x_{3}$.\end{tabular}\right.$ (22c) If the outcome is
$|\dddot{\phi}_{A,B}\rangle$, the conditional probability
$p_{ext,\dddot{\phi}}$ of extracting an EPR state is given by,
$p_{ext,\dddot{\phi}}=\frac{2\alpha^{2}\gamma^{2}x^{2}}{p_{\mu^{-}}}.$ (22d)

Since from each possible outcome (13) an EPR state can be probabilistically
extracted, the total success probability is given by the sum of the
probabilities (14), each one multiplied by its respective conditional
probability of extracting (22), i.e.,

$P_{s_{4}}(x)=p_{\mu^{+}}p_{ext,\ddot{\phi}}+p_{\nu^{+}}p_{ext,\ddot{\psi}}+p_{\nu^{-}}p_{ext,\dddot{\psi}}+p_{\mu^{-}}p_{ext,\dddot{\phi}}.$

Similarly, for evaluating $P_{s_{4}}(x)$, we must take the relation between
$\alpha$ and $\gamma$ into account. In consequence, the total success
probability becomes,

$P_{s_{4}}(x)=\left\\{\begin{tabular}[]{ll}$2x^{2}$,&\quad if $x\leq
x_{1}$,\\\
$2\alpha^{2}\gamma^{2}+2\left(\alpha^{2}\delta^{2}+\beta^{2}\gamma^{2}\right)x^{2}$,&\quad
if $x_{1}\leq x\leq\min\left\\{x_{2},x_{3}\right\\}$,\\\
$2\min\left\\{\alpha^{2},\gamma^{2}\right\\}$,&\quad if
$\min\left\\{x_{2},x_{3}\right\\}\leq x\leq 1/\sqrt{2}$.\end{tabular}\right.$
(23)

From the expression (23) we notice the following effects:

  * •

The slope of $P_{s_{4}}(x)$ exhibits two discontinuities, at $x_{1}$ and
$\min\left\\{x_{2},x_{3}\right\\}$.

  * •

The probability increases for $x\in[0,\min\left\\{x_{2},x_{3}\right\\}]$,
hence for $x\in[\min\left\\{x_{2},x_{3}\right\\},1/\sqrt{2}]$ the probability
$P_{s_{4}}(x)$ becomes constant and equal to the maximal value obtained in the
second strategy.

  * •

There is the same entanglement matching, as found in the second strategy,
between $\mathcal{C}_{AC_{1}}$ and $\mathcal{C}_{BC_{2}}$ for all
$x\in[\min\left\\{x_{2},x_{3}\right\\},1/\sqrt{2}]$.

  * •

The maximal probability $2\min\left\\{\alpha^{2},\gamma^{2}\right\\}$ is
achieved at $x=\min\left\\{x_{2},x_{3}\right\\}$, which in general is smaller
than $1/\sqrt{2}$. Therefore, the maximal entanglement in the measuring-basis
(9) is not required for reaching the optimal success probability.

  * •

There are two _special values_ , $\mathfrak{C}_{C_{1}C_{2}}^{\pm}$ of the
measuring-basis concurrence, for which $P_{s_{4}}(x)$ changes its behavior; at
$\mathcal{C}_{C_{1}C_{2}}=\mathfrak{C}_{C_{1}C_{2}}^{\pm}$ the probability
(23) _abruptly changes_ its slope, but for all
$\mathcal{C}_{C_{1}C_{2}}(x)\geq\mathfrak{C}_{sv-}$ the probability
$P_{s_{4}}(x)$ remains constant at its maximum value.

Accordingly, although $\mathfrak{C}_{C_{1}C_{2}}^{-}$ differs from
$\mathcal{C}_{AC_{1}}$ and $\mathcal{C}_{BC_{2}}$, it plays the role of an
_entanglement matching level_ , which is a function of the initial
concurrences.

Figure 3: Success probability $P_{s_{4}}$ as a function of
$\mathcal{C}_{BC_{2}}$ and $\mathcal{C}_{C_{1}C_{2}}$ for
$\mathcal{C}_{AC_{1}}=0.7$.

Finally, we would like to point out that the application of the USE procedure
allows us to increase the probability of achieving an EPR state in the pair
$AB$, with respect to $P_{s_{3}}$, while demanding the same entanglement
$\mathfrak{C}_{C_{1}C_{2}}^{-}$ in the measuring-basis. In other words, this
strategy combines the best characteristics of the second and third strategy,
i.e., it has the highest probability, obtained in the second strategy, and
demands the special value for the concurrence of the measuring-basis, found
for the third strategy. Figure 3 illustrates the behavior of (23) as a
function of $\mathcal{C}_{BC_{2}}$ and $\mathcal{C}_{C_{1}C_{2}}$ for
$\mathcal{C}_{AC_{1}}=0.7$; the increasing surface clearly shows the _abrupt
change_ of the slope, and the plateau in the surface corresponds to the
twofold matching effect.

Therefore, by considering the greatest probability of success and the
entanglement as a resource, the fourth strategy is more efficient than the
three others.

##  IV Conclusions

In summary, we have addressed unambiguous and indeterministic entanglement
swapping, having projection of two qubits onto an EPR state, which have never
interacted, as main aim. In our analysis, a second aim becomes the
maximization of the probability of success and the minimization of the
required entanglement in the measuring-basis.

We have introduced three levels of entanglement, two of them in the initial
states and another one by considering non-Bell state in the measuring-basis.
Additionally, we proposed the _unambiguous state extraction_ scheme as an
mechanism for implementing the unitary-reduction local operator.

These considerations allow us to design four strategies for achieving the main
aim. The first one enables to accomplish the main aim, but with non-optimal
probability of success. In the second strategy, we found the optimal
probability of success, related to an entanglement matching effect between the
two concurrences of the initial states, i.e., the maximal success probability
is determined by the smallest entanglement of the initial states. The third
strategy has a success probability smaller than the first strategy, but gives
account that the main aim can be accomplished without maximal entanglement in
the measuring-basis. Thus, the three schemes lead us to suggest the fourth
strategy. We consider the fourth strategy as the optimal one, because it
combines the best characteristics of the second and third strategy, i.e., it
has the highest probability, founded in the second strategy, and demands the
non-maximal _special value_ for the concurrence of the measuring-basis, found
for the third strategy. We realized that the concurrences special value
$\mathfrak{C}_{C_{1}C_{2}}^{-}$ plays the role of a matching effect between it
and the initial concurrences, showing thus that there is a scheme with optimal
success probability, which does not require maximal entanglement in the
measurement process. In other words, the entanglement matching effect between
the two initial concurrences matches with $\mathfrak{C}_{C_{1}C_{2}}^{-}$;
therefore, any concurrence value greater than $\mathfrak{C}_{C_{1}C_{2}}^{-}$
does not affect the success probability, i.e., it remains constant.

Besides, we found another special value $\mathfrak{C}_{C_{1}C_{2}}^{+}$, for
which the increasing success probability changes its slope abruptly.

Accordingly, here we have shown that, in order to accomplish the main aim of
the entanglement swapping with maximal probability, the maximal entanglement
in the measuring-basis is not required, equivalently, the _special value_
$\mathfrak{C}_{C_{1}C_{2}}^{-}$ is necessary and sufficient.

##  Appendix A Optimal EPR extraction

Here we succinctly show that the locally applied USE protocol Hsu ; Hutin
allows us to extract an EPR state, with optimal probability, from a partially
entangled state $|\bar{\psi}_{ij}\rangle$, shared by the qubits $i$ and $j$,
i.e.,

$|\bar{\psi}_{ij}\rangle\overset{\text{local
USE}}{\rightarrow}\frac{|0_{i}\rangle|1_{j}\rangle+|1_{i}\rangle|0_{j}\rangle}{\sqrt{2}},$
(24)

with

$|\bar{\psi}_{ij}\rangle=u|0_{i}\rangle|1_{j}\rangle+v|1_{i}\rangle|0_{j}\rangle.$
(25)

The $\\{|0\rangle,|1\rangle\\}$ are eigenstates of the Pauli operator
$\sigma_{z}$ of the qubit labeled by the subindex. Without loss of generality,
we assume that $u$ and $v$ are real and non-negative numbers and due to
normalization $u^{2}+v^{2}=1$. The entanglement of the initial state (25) can
be valued by the concurrence $\mathcal{C}_{\bar{\psi}}=2uv$ Wootters1 ;
Wootters2 .

The local operation can be indistinctly applied onto qubit $i$ or $j$. For
instance, let us consider the qubit $i$ and an auxiliary qubit $a$, initially
in the state $|0_{a}\rangle$. Because the probability amplitude must have a
module smaller than or equal to $1$, we have to consider two cases:
(i) If $u\leq v$, we apply the joint unitary $U_{ia}$ onto the tensorial
product state $|\bar{\psi}_{ij}\rangle|0_{a}\rangle$, with

$\displaystyle U_{ia}$ $\displaystyle=$ $\displaystyle|0_{i}\rangle\langle
0_{i}|\otimes I_{a}+|1_{i}\rangle\langle 1_{i}|\otimes U_{a},$ (26)
$\displaystyle U_{a}|0_{a}\rangle$ $\displaystyle=$
$\displaystyle\frac{u}{v}|0_{a}\rangle+\sqrt{1-\frac{u^{2}}{v^{2}}}|1_{a}\rangle,$

where $I_{a}$ is the identity operator of the auxiliary qubit $a$. Thus,
$U_{i,a}$ transforms $|\bar{\psi}_{ij}\rangle|0_{a}\rangle$ as follows:

$\displaystyle U_{ia}|\bar{\psi}_{ij}\rangle|0_{a}\rangle$ $\displaystyle=$
$\displaystyle\sqrt{2}u\frac{|0_{i}\rangle|1_{j}\rangle+|1_{i}\rangle|0_{j}\rangle}{\sqrt{2}}|0_{a}\rangle$
$\displaystyle+\sqrt{v^{2}-u^{2}}|1_{i}\rangle|0_{j}\rangle|1_{a}\rangle,$

from where we realize, that by measuring $\sigma_{z}$ of the auxiliary qubit,
the pair $ij$ is projected onto an EPR state with probability
$p_{ext}=2u^{2}$, otherwise the correlation is lost.
(ii) If $u\geq v$, we must apply the joint unitary

$\displaystyle U_{ia}$ $\displaystyle=$ $\displaystyle|0_{i}\rangle\langle
0_{i}|\otimes U_{a}+|1_{i}\rangle\langle 1_{i}|\otimes I_{a},$ (27)
$\displaystyle U_{a}|0_{a}\rangle$ $\displaystyle=$
$\displaystyle\frac{v}{u}|0_{a}\rangle+\sqrt{1-\frac{v^{2}}{u^{2}}}|1_{a}\rangle,$

to obtain

$\displaystyle U_{ia}|\bar{\psi}_{ij}\rangle|0_{a}\rangle$ $\displaystyle=$
$\displaystyle\sqrt{2}v\frac{|0_{i}\rangle|1_{j}\rangle+|1_{i}\rangle|0_{j}\rangle}{\sqrt{2}}|0_{a}\rangle$
$\displaystyle+\sqrt{u^{2}-v^{2}}|0_{i}\rangle|1_{j}\rangle|1_{a}\rangle.$

Similarly, by measuring $\sigma_{z}$ of the auxiliary qubit, the pair $ij$ is
projected onto an EPR state with probability $p_{ext}=2v^{2}$, otherwise the
correlation is lost.

Therefore, for an given bipartite pure state $|\bar{\psi}_{ij}\rangle$ the
probability $p_{ext}$ of extracting an EPR state by means of local operators
and one way classical communication becomes

$\displaystyle p_{ext}$ $\displaystyle=$ $\displaystyle
2\min\left\\{u^{2},v^{2}\right\\},$ (28) $\displaystyle=$ $\displaystyle
1-\sqrt{1-\mathcal{C}_{\bar{\psi}}^{2}},$

where $u$ and $v$ are its Schmidt coefficients.

Noteworthy, that the probability (28) agrees with the optimal ones found in
Refs. Bose ; Vidal ; Nielsen ; Guevara  and the results does not depend on
which of the two qubits, $1$ or $2$, the USE scheme was applied. It is worth
to mention here that, the joint unitaries (26) and (27) can be implemented
experimentally in different physical systems JICirac ; FDMartini ; LIsenhower
; KMMaller  by composition of local unitaries and CNOT gates TSleator ;
DPDiVincenzo ; VBuzek .

Additionally, if the initial state is
$|\bar{\phi}_{ij}\rangle=u|0_{i}\rangle|0_{j}\rangle+v|1_{i}\rangle|1_{j}\rangle$,
then it can be transformed to $|\bar{\psi}_{ij}\rangle$ by applying the local
unitary rotation $U_{j}=e^{-i(I-\sigma_{x})pi/2}$ onto qubit $j$, and the
above describe scheme can be implemented.

###### Acknowledgements.

Authors thank grants FONDECyT 1161631, and DAAD RISE Worldwide CL-PH-1736.

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# Better than BERT but Worse than Baseline

Boxiang Liu
Baidu Research
<EMAIL_ADDRESS>
Jiaji Huang
Baidu Research
<EMAIL_ADDRESS>
Xingyu Cai
Baidu Research
<EMAIL_ADDRESS>
Kenneth Church
Baidu Research
<EMAIL_ADDRESS>
###### Abstract

This paper compares BERT-SQuAD and Ab3P on the Abbreviation Definition
Identification (ADI) task. ADI inputs a text and outputs short forms
(abbreviations/acronyms) and long forms (expansions). BERT with reranking
improves over BERT without reranking but fails to reach the Ab3P rule-based
baseline. What is BERT missing? Reranking introduces two new features:
charmatch and freq. The first feature identifies opportunities to take
advantage of character constraints in acronyms and the second feature
identifies opportunities to take advantage of frequency constraints across
documents.

##  1 Introduction: Opportunities

Transformers such as BERT Devlin et al. (2019) and ERNIE Sun et al. (2019)
have been extremely successful on a wide range of tasks. Nevertheless, there
are opportunities to improve BERT on numbers Wallace et al. (2019), negation
Ettinger (2020), and more.

This paper compares BERT to Ab3P111https://github.com/ncbi-nlp/Ab3P Sohn et
al. (2008) on the Abbreviation Definition Identification (ADI) task described
in Section 1.1. ADI inputs texts and outputs pairs of short forms (SFs) and
long forms (LFs). A number of ADI systems were developed more than a decade
ago; some use rules Sohn et al. (2008); Schwartz and Hearst (2002) and others
machine learning Kuo et al. (2009).

Section 1.2 discusses similarities between ADI and question answering (QA).
The QA dataset, SQuAD Rajpurkar et al. (2016), includes many types of
questions, some of which are similar to ADI: What does X stand for? X is a SF
(abbreviation) and the answer is a LF (expansion). A simple program based on
BERT-SQuAD222https://huggingface.co/bert-large-uncased-whole-word-masking-
finetuned-squad performs remarkably well on ADI benchmarks, though not as well
as Ab3P, a strong rule-based baseline.

Ab3P uses a set of 17 rules to extract SF-LF pairs. Rules were created
iteratively. Each iteration finds a rule that reduces the majority of missing
cases. The iteration stops when the desired recall has been achieved. One of
these rules favors pairs with matching characters. That is, it is common for
each character in an acronym to match each word in the expansion.

Why is BERT not doing better? What is BERT missing? Section 2.1 uses a
reranking approach to improve BERT by adding two features that are easy to
interpret:

  1. 1.

charmatch compares the first letter of LF to the first letter of SF, and

  2. 2.

freq counts instances of “LF (SF” in a corpus of PubMed abstracts.

It has been suggested that deep nets are so powerful that feature engineering
is no longer necessary. Reranking suggests this may not be correct, especially
for tasks like ADI where rule-based systems outperform BERT.

###  1.1 The ADI Task

Abbreviations and acronyms are especially common in technical writing such as
PubMed and arXiv333https://arxiv.org/help/bulk_data Veyseh et al. (2020),
though they can be found in many other corpora such as Wikipedia. The first
time a SF is used in a paper, there is usually a definition that connects the
dots between the SF and the LF. It is more common for definitions to
parenthesize the SF, though both types of definitions are common:

  1. 1.

LF (SF): heat shock protein (HSP)

  2. 2.

SF (LF): HSP (heat shock protein)

The ADI task takes a text as input, and outputs pairs of SFs and LFs that are
defined in the input text. Four benchmarks have become standard in the ADI
literature: Ab3P Sohn et al. (2008), BIOADI Kuo et al. (2009), MEDSTRACT Wren
et al. (2005) and SH Schwartz and Hearst (2002). Standard train, validation
and test splits are available for
download.444http://bioc.sourceforge.net/BioCresources.html All of these
benchmarks are based on PubMed abstracts in ASCII without markup.555In other
experiments, we have run Ab3P on 2 TBs of arXiv papers, and found LaTeX markup
to be helpful because definitions often make use of italics.

###  1.2 Related Work, Tasks & Tools

The ADI task is similar to a number of other tasks such as Question Answering
(QA), Named Entity Recognition (NER), Acronym Disambiguation (AD), etc. SQuAD
Rajpurkar et al. (2016) is a popular benchmark for QA systems. SQuAD examples
consist of questions, answers and contexts. There are many types of questions,
including some that are similar to ADI such as:

  1. 1.

Question: What does AFC stand for?

  2. 2.

Context: The American Football Conference (AFC) champion Denver Broncos
defeated the National Football Conference (NFC) champion Carolina Panthers
24–10 to earn their third Super Bowl title.

Systems output a span, a substring of the context that answers the question
such as: American Football Conference. This example suggests BERT-SQuAD may be
a more modern alternative to older rule-based approaches to ADI such as Ab3P.

Other studies have applied QA-like technology to a number of other tasks such
as: event extraction Du and Cardie (2020); Liu et al. (2020); Feng et al.
(2020); Sun et al. (2020), NER Li et al. (2020), entity linking Gu et al.
(2021), coreference resolution Wu et al. (2020), and more. These studies
suggest that a QA model with fine-tuning can achieve state-of-the-art (SOTA)
results on a range of tasks.

ADI is also similar to NER. NER is used for a number of tasks that extract
spans, substrings of input text, and labeling them with tags such as person,
organization, location, etc. Doddington et al. (2004), as in
ACE.666https://www.ldc.upenn.edu/collaborations/past-projects/ace Some
benchmarks use BIO tags to label spans. Each word in the input text is tagged
as B (begins a span), I (inside a span) or O (otherwise). Some benchmarks
introduce tags such as B-disease and B-chemical to distinguish disease
entities from chemical entities. Lee et al. (2020) show BioBERT is effective
on a number of such NER benchmarks:777https://github.com/dmis-lab/biobert-
pytorch/tree/master/named-entity-recognition NCBI Disease Doğan et al. (2014)
i2b2/VA Uzuner et al. (2011), BC5CDR Li et al. (2016), BC4CHEMD Krallinger et
al. (2015), BC2GM Smith et al. (2008), JNLPBA Kim et al. (2004), LINNAEUS
Gerner et al. (2010) and Species-800 Pafilis et al. (2013), as well as related
tasks such as relation extraction and QA.

PubTator888https://www.ncbi.nlm.nih.gov/research/pubtator/tutorial.html Wei et
al. (2012, 2013, 2019); Leaman and Lu (2016) is a practical NER tool. PubTator
annotations for millions of PubMed abstracts are available for
download.999https://ftp.ncbi.nlm.nih.gov/pub/lu/PubTatorCentral/ PubTator
identifies spans and tags them with six bioconcepts: genes, diseases,
chemicals, mutations, species and cell lines. PubTator also links entities to
ontologies such as MeSH. PubTator links the gene p53, for example, to
different points in the ontology for different species: humans, mice, fruit
flies, etc.

There have been concerns that work based on PubMed may not generalize well to
other domains. Benchmarks based on arXiv will be used in competitions at
AAAI-2021 for AI101010https://github.com/amirveyseh/AAAI-21-SDU-shared-
task-1-AI and AD111111https://github.com/amirveyseh/AAAI-21-SDU-shared-
task-2-AD tasks Veyseh et al. (2020). The AI task uses NER-like labels with 5
tags: B-short, B-long, I-short, I-long and O, where B-short and I-short are
used for SF spans, and B-long and I-long are used for LF spans.

This paper will use the ADI task which inputs definitions, as opposed to AI
and AD tasks which include subsequent mentions taken out of context. Here is
an example from the AI benchmark of a subsequent mention of MRC.121212We also
compare the performance of MRC cooperative NOMA with MRC cooperative
orthogonal multiple access (OMA), and we show that NOMA has a better
performance than OMA. Using Google, we found the
document.131313https://arxiv.org/abs/2003.07299 The definition of MRC appears
a few sentences earlier.

The AD task is based on word sense disambiguation (WSD). The input sentence
contains a SF, such as: The MRC technique employs a single copper loop of
small radius both at the energy transmitter end and sensor node’s receiving
end. The AD task is to choose the appropriate LF from a short list of
candidates:141414Ambiguity is more common across documents than within
documents. MRC, for example, is unlikely to expand to two different LFs within
the same document. In this respect, SFs obey a so-called “one sense per
discourse” constraint like word senses Gale et al. (1992). That is, the word
bank is ambiguous. In one document, it may refer to a money bank, and in
another document, it may refer to a river bank, but it is unlikely to be used
both ways within the same document.

  1. 1.

machine reading comprehension

  2. 2.

maximal ratio combining

  3. 3.

magnetic resonance coupling

More context would be helpful. With Google, we found the
document;151515https://arxiv.org/pdf/1805.07795.pdf the definition,
immediately before the input sentence,161616The wireless transfer is performed
by using the so called Magnetic Resonance Coupling (MRC) technique. resolves
the ambiguity.

Benchmark | Method
---|---
| Ab3P | BERT-SQuAD
Ab3P | 0.889 | 0.794
BIOADI | 0.838 | 0.698
MEDSTRACT | 0.943 | 0.844
SH | 0.858 | 0.769
Table 1: Ab3P has better F-scores on 4 benchmarks. Benchmark | $Pr(correct|charmatch)$
---|---
| not charmatch | charmatch
Ab3P | 0.15 | 0.96
BIOADI | 0.07 | 0.94
MEDSTRACT | 0.18 | 0.98
SH | 0.10 | 0.94
Table 2: BERT-SQuAD does not capture charmatch.

##  2 BERT-SQuAD: An Alternative to Ab3P for ADI

The Ab3P method takes a text as input and outputs pairs of SFs and LFs that
are defined in the input text. BERT-SQuAD takes a question and context as
input, and outputs a span from the document that answers the question. For the
comparisons in Table 1, we give BERT-SQuAD the SF from Ab3P output. These SFs
are turned into questions of the form: What does $<$SF$>$ stand for? Even with
this unfair hint, BERT-SQuAD is less effective than Ab3P, as shown in Table
1.171717BERT and Ab3P differ in many ways in addition to F-scores. BERT
generalizes to many tasks, but it is bigger and slower and limited to inputs
under 512 tokens.

Many of the errors are “off-by-one,” where the candidate LF has one word too
many or one too few, especially at the left edge of the LF. The right edge
tends to be easier because the right edge of the LF is often delimited by a
parenthesis between the LF and the SF.

###  2.1 Two More Features: Charmatch & Freq

What is BERT-SQuAD missing? Consider the example: healthy controls (HC). In
this case, BERT-SQuAD drops the first word from the LF, returning controls
instead of healthy controls. BERT’s candidate violates a constraint on
characters, where the bold characters in healthy controls are likely to match
the characters in the SF (HC). Ab3P uses old-fashioned rules to capture this
constraint.

Rank | Benchmark
---|---
| Ab3P | BIOADI | MESTRACT | SH
0 | 920 | 1129 | 128 | 698
1 | 85 | 103 | 11 | 58
2 | 19 | 124 | 3 | 25
3 | 8 | 68 |  | 4
4 | 6 | 33 |  |
Table 3: # correct by rank (position in n-best list)

It appears this character constraint is missing from BERT-SQuAD. To test this
hypothesis, we introduce a simple boolean feature, charmatch, that compares
the first character of the SF to the first character of the candidate LF.
Table 2 shows that candidates from BERT-SQuAD are more likely to be correct
when these characters match.

In addition to charmatch, we identified another promising feature that we call
freq. Consider the example: Latent herpes simplex virus (HSV) has been
demonstrated in… Again, BERT-SQuAD is off by one, but this time, the candidate
LF is too long: Latent herpes simplex virus. The freq feature takes advantage
of the fact that many of these SFs are defined in thousands of PubMed
abstracts. The freq feature uses suffix arrays Manber and Myers (1993) to
count the number of matches of: SF + ‘ (’ + LF in PubMed. In this example, we
found 6075 instances of “herpes simplex virus (HSV”, but only 8 instances of
“Latent herpes simplex virus (HSV.” Of course, raw frequencies need to be
normalized appropriately because shorter strings tend to be more frequent than
longer strings.

###  2.2 12 Models: Rank + Charmatch + Freq

To side-step difficult normalization and feature combination questions, we
make use of reranking and machine learning. BERT-SQuAD was modified to output
top-k candidates instead of just the top candidate. Table 3 shows there are
more correct candidates in top position (rank 0), but there are also many
correct candidates in other positions.

Coef | Model
---|---
| 1 | 2 | 3 | 4 | 5 | 6
$\beta_{0}$ | 1.6 | 0.7 | 1.9 | 1.4 | -1.2 | -2.5
$\beta_{1}$ | -3.3 | -1.6 | -3.9 | -3.3 | -3.2 | -1.5
$\beta_{2}$ | 0 | 0 | 0 | 0 | 3.5 | 3.8
$\beta_{3}$ | 0 | 0 | 0 | 0 | 0 | 0
| Model
| 7 | 8 | 9 | 10 | 11 | 12
$\beta_{0}$ | -1.0 | -1.9 | -2.7 | -5.2 | -3.2 | -3.1
$\beta_{1}$ | -4.0 | -3.2 | -2.9 | -1.5 | -3.8 | -2.9
$\beta_{2}$ | 3.9 | 4.1 | 3.7 | 5.2 | 4.7 | 4.3
$\beta_{3}$ | 0 | 0 | 0.3 | 0.5 | 0.4 | 0.3
Table 4: Coefficients for 12 logistic regression models: $z=\beta_{0}+\beta_{1}~{}rank+\beta_{2}~{}charmatch+\beta_{3}~{}log(1+freq)$ Features | Benchmark
---|---
| Ab3P | BIOADI | MEDS. | SH
1 (M 1-4) | 0.829 | 0.660 | 0.865 | 0.796
2 (M 5-8) | 0.908 | 0.787 | 0.945 | 0.901
3 (M 9-12) | 0.936 | 0.948 | 0.975 | 0.949
Table 5: Models with more features are more confident when they are correct.
Models 1-4 use a single feature (rank). Models 5-8 add charmatch. Models 9-12
add freq. Confidence is estimated as median $\sigma(z)$ for correct
candidates.

A dozen logistic regression models are used to rerank the top 5 candidates.
Models 1-4 use eq (1), models 5-8 use eq (2) and models 9-12 use eq (3). $y$
comes from the 4 gold sets: models 1,5 & 9 use the Ab3P benchmark for $y$,
models 2, 6 & 10 use BIOADI, models 3, 7 & 11 use MEDSTRACT, and models 4, 8 &
12 use SH. Coefficients are shown in Table 4. All coefficients are
significant. Reranking sorts candidates by $z$, as defined in Table 4.

$y\sim rank$ (1) $y\sim rank+charmatch$ (2) $y\sim rank+charmatch+log(1+freq)$
(3)

Figure 1: F-score by model and benchmark Figure 2: More features increase
F-scores.

##  3 Results

Scores for 12 BERT-SQuAD models and 4 benchmarks are shown in Figure 1. These
results are summarized in Figure 2. Adding more features to BERT-SQuAD
improves F-scores. Table 5 shows that models with more features are more
confident.

##  4 Conclusions

This paper compared BERT-SQuAD to a rule-based baseline, Ab3P, on the ADI
task. We proposed a reranking improvement to BERT that takes advantage of two
features: charmatch and freq. F-scores for the proposed solution are better
than BERT but worse than baseline, suggesting the two features shed light on
opportunities for improving BERT-like models.

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