# The role of local and global geometry in quantum entanglement percolation

Gerald John Lapeyre Jr  ICFO–Institut de Ciències Fotòniques, Mediterranean
Technology Park, 08860 Castelldefels, Spain

###### Abstract

We prove that enhanced entanglement percolation via lattice transformation is
possible even if the new lattice is more poorly connected in that: i) the
coordination number (a local property) decreases, or ii) the classical
percolation threshold (a global property) increases. In searching for
protocols to transport entanglement across a network, it seems reasonable to
try transformations that increase connectivity. In fact, all examples that we
are aware of violate both conditions i and ii. One might therefore conjecture
that all good transformations must violate them. Here we provide a counter-
example that satisfies both conditions by introducing a new method, partial
entanglement swapping. This result shows that a transformation may not be
rejected on the basis of satisfying conditions i or ii. Both the result and
the new method constitute steps toward answering basic questions, such as
whether there is a minimum amount of local entanglement required to achieve
long-range entanglement.

entanglement, quantum networks, entanglement percolation, entanglement
distribution

###### pacs:

03.67.Bg, 03.67.Hk, 64.60.ah, 03.67.Pp

##  I Introduction

Distribution of quantum entanglement on networks has been studied vigorously
over the past few years. This has been driven by the fact that entanglement is
the fundamental resource in quantum information, but it is created locally via
interaction, while it is often consumed in systems with widely separated
components. In an ideal description, each node of the network represents a
collection of qubits, and each edge or link represents entangled states of
qubits in different nodes.

But, even in the case of transporting entanglement along a chain of partially
entangled pure states, using perfect quantum operations, the resulting
entanglement decays exponentially in the number of links. Unfortunately,
technical and fundamental limits on effectively moving entanglement over even
a single link further complicate the ideal picture and have led to elaborate
protocols involving the distribution, storage, and purification of entangled
states. The most direct approach is the quantum repeater which has been
proposed to overcome these limitations on a one-dimensional chain of nodes
Briegel  _et al._ (1998); Dür  _et al._ (1999); Childress  _et al._ (2005);
Hartmann  _et al._ (2007); Sangouard  _et al._ (2011); Meter  _et al._ (2012).
There are examples of practical, deployed quantum networks, such as quantum
key distribution networks. But the technical challenges in implementing
quantum repeaters remain too great to be useful in contemporary quantum key
distribution networks Scarani  _et al._ (2009). Typically, entanglement is
established over only a single link, while at each node information is
processed classically and re-encoded in a quantum state.

A different approach is to use the entire network, rather than a linear chain,
to distribute entanglement. The availability of multiple paths is used to
overcome the the inevitable decay of entanglement. This leads to models that
are immediately more interesting because it is not obvious how to prove which
of two protocols is better, let alone which protocol is optimal. In fact
percolation theory Stauffer and Aharony (1991); Grimmett (1999) has provided
powerful tools for evaluating protocols. The best protocols use quantum
operations to transform the initial lattice into a different lattice Acín  _et
al._ (2007); Lapeyre Jr.  _et al._ (2009); Perseguers  _et al._ (2010).

As in the one-dimensional case, more realistic studies of multi-dimensional
networks have been done, for instance by considering mixed states and
imperfect quantum operations Cuquet and Calsamiglia (2009); Broadfoot  _et
al._ (2009, 2010); Lapeyre Jr.  _et al._ (2012); Cuquet and Calsamiglia
(2011). But sharp questions, say in the thermodynamic limit, are difficult to
pose in these dirtier situations because of the decay of entanglement.
Furthermore, questions about asymptotic behavior remain that are not only of
intrinsic interest, but address fundamental limits on entanglement
distribution. These are the questions that we address here.

This paper has two main goals. The first goal is to show that enhanced
entanglement percolation (defined below) via lattice transformation is
possible even if the coordination number of the transformed lattice decreases
or the classical percolation threshold increases. The second goal is to
introduce a new tool that we call partial entanglement swapping. In partial
swapping, we simply stop the swapping procedure after the first step, the
projection, and evaluate whether the output state and the new geometry may be
more profitably used in a different operation. In fact, the usefulness of the
tool is demonstrated by using it to accomplish the first goal. Although the
idea behind partial swapping is simple, it introduces a complication. In
previous entanglement percolation protocols, the Bell measurement in the
computational basis is optimal. But, the optimal basis for partial swapping is
not obvious and depends on the amount of initial entanglement.

##  II Entanglement Percolation

Entanglement percolation is described in detail in several sources Acín  _et
al._ (2007); Perseguers  _et al._ (2008); Lapeyre Jr.  _et al._ (2009);
Perseguers  _et al._ (2013); Perseguers (2010). Here we give only a brief
description. We consider the following class of entanglement percolation
models. Each node consists of a collection of qubits. Each edge, or link,
consists of a partially entangled pure state between two qubits, each on a
different node. These states
$\,|\alpha\rangle\in\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ are written in a
Schmidt basis as

$\,|\alpha\rangle=\sqrt{\alpha_{0}}\,|00\rangle+\sqrt{\alpha_{1}}\,|11\rangle,$

where the Schmidt coefficients $\alpha_{0},\alpha_{1}$ satisfy
$\alpha_{0}\geq\alpha_{1}$ and $\alpha_{0}+\alpha_{1}=1$. If
$\alpha_{0}=\alpha_{1}=1/2$, the state is maximally entangled, and is called a
Bell pair or singlet. If either $\alpha_{0}$ or $\alpha_{1}$ vanishes, then
the state is separable and is useless for quantum information tasks. The
smallest Schmidt coefficient may be used as a measure of entanglement with the
amount of entanglement increasing with $\alpha_{1}$.

Figure 1: Transformation of kagome to square lattice. Circles represent
qubits. Lines represent partially entangled bi-partite states. Left) Full
entanglement swapping is performed for each pair of links marked with a (blue)
loop. Right) The result is the square lattice, where the vertical (dashed)
links are the outcome of the swap and the horizontal links remain in the state
$\,|\alpha\rangle$. The remainder of the QEP protocol is described in the
text.

The lattice is initialized with identical states $\,|\alpha\rangle$ on each
link. So we have one free parameter $\alpha_{1}$. The goal is to design a
protocol to maximally entangle two arbitrary nodes $A$ and $B$. The utility of
the protocol is measured by the probability of success
$\mathop{\mathbf{Pr}}\nolimits(A\leftrightarrow B)$ as the distance between
$A$ and $B$ tends to infinity. We require that the protocols use only local
operations and classical communication (LOCC) Nielsen and Chuang (2000). This
means that quantum operations that include interaction between qubits on
different nodes are not allowed. But classical communication between all nodes
is allowed.

###  II.1 Classical Entanglement Percolation

The simplest entanglement distribution protocol is called classical
entanglement percolation (CEP). For some lattices, better protocols have been
found, the so-called quantum entanglement percolation (QEP) protocols. The
reason for this distinction and the relation between CEP and QEP will be made
clear below. For now, we note that it is QEP that uses lattice transformation.
We introduce the CEP and QEP using the kagome lattice shown in Fig. 1, because
it allows a concise exposition. First we describe CEP. For the moment,
consider choosing fixed $A$ and $B$. In step $1$ we perform an LOCC operation
on each link, optimally converting it with probability $p=2\alpha_{1}$ to a
Bell pair, and probability $1-p$ to a separable state. This operation is
called a singlet conversion and $p$ is the singlet conversion probability
(SCP). After step $1$, we have a lattice in which each link is either open (a
Bell pair), or closed (separable). In step $2$, we search for an unbroken path
of open links between $A$ and $B$. If no such path exists, then
$\mathop{\mathbf{Pr}}\nolimits(A\leftrightarrow B)=0$. If a path does exist,
then at each intermediate node we perform an entanglement swapping operation.
Because the input links are singlets, each swap succeeds with probability $1$,
so that $\mathop{\mathbf{Pr}}\nolimits(A\leftrightarrow B)=1$. We call a
protocol that succeeds with probability $1$ deterministic. This description
corresponds exactly to classical bond percolation, with density of open bonds
$p=2\alpha_{1}$.

Lattice |  $p_{c}$ for bond percolation
---|---
triangular | $2\sin(\pi/18)\approx 0.347$
square | $1/2$
kagome |  $\approx 0.5244053$ MC estimate
hexagonal | $1-2\sin(\pi/18)\approx 0.653$
Table 1:  $p_{c}$ for bond percolation on some lattices. All critical
densities are exactGrimmett (1999) except for $p_{c}(\text{kagome})$Ziff and
Suding (1997).

The critical bond density for the kagome lattice is
${p^{\text{kag}}_{c}}\approx 0.52$. Thus
$\mathop{\mathbf{Pr}}\nolimits(A\leftrightarrow B)=0$ if
$p<{p^{\text{kag}}_{c}}$ and $\mathop{\mathbf{Pr}}\nolimits(A\leftrightarrow
B)>0$ if $p>{p^{\text{kag}}_{c}}$.

###  II.2 Quantum Entanglement Percolation

A QEP scheme for the kagome lattice is shown in Fig. 1. We first perform
swapping on all pairs of qubits enclosed in loops. Each of the input states is
$\,|\alpha\rangle$, so the probability of obtaining a singlet in the resulting
vertical link is $p=2\alpha_{1}$. We then perform a singlet conversion on the
remaining horizontal bonds, resulting in a square lattice where each link is a
Bell pair with probability $p$ and is separable otherwise. Finally we perform
step $2$ of CEP (swapping with singlets) on this square lattice. This is
successful precisely when $p>{p^{\square}_{c}}$, where ${p^{\square}_{c}}$ is
the critical density for bond percolation on the square lattice. Since
${p^{\square}_{c}}=1/2$, it follows that long-distance entanglement on the
kagome lattice is possible with this QEP scheme, but not with CEP, if
$\alpha_{1}$ satisfies ${p^{\square}_{c}}<2\alpha_{1}<{p^{\text{kag}}_{c}}$.

CEP always gives an easily computable upper bound on the minimum initial
entanglement required for long-distance entanglement. Thus, CEP serves as a
benchmark to compare with any QEP protocol. Because we are not interested in
QEPs that perform worse than CEP, we will call any advantageous QEP simply a
QEP. However, the measure by which the QEP is advantageous may vary. We call
the smallest value of $\alpha_{1}$ such that long-range entanglement is
possible the lower threshold or percolation threshold $\hat{\alpha}_{c}$. We
call the smallest value of $\alpha_{1}$ such that long-range entanglement is
achieved with probability $1$ the upper threshold $\hat{\alpha}^{*}_{c}$. Note
that $\hat{\alpha}_{c}$ marks a phase transition, but $\hat{\alpha}^{*}_{c}$
does not. For every lattice, CEP gives $\hat{\alpha}^{*}_{c}=1/2$. We call a
QEP robust if it satisfies at least one of two conditions. 1) that it lowers
the percolation threshold $\hat{\alpha}_{c}$, and 2) that the upper threshold
satisfies $\hat{\alpha}^{*}_{c}<1/2$ . We are interested in isolating the
effect of the geometry of the transformed lattice on the performance of the
QEP. We therefore emphasize that we will compare the geometry of the classical
transformed lattice to that of the the initial lattice with no reference to
quantum states.

###  II.3 Lattice structure and entanglement distribution

For any lattice, CEP is defined and the relevant quantities can be taken
directly from percolation theory. But there is no generic prescription for
constructing a QEP. In previous work, QEP protocols have been identified by
choosing a lattice and searching for good lattice transformations.

In the example above, the initial lattice was transformed into one new
lattice. However, in general, transformations may take the initial lattice
${\cal L}$ to multiple, decoupled lattices $\\{{\cal L}^{\prime}_{i}\\}$
Lapeyre Jr.  _et al._ (2009). It is reasonable to search for $\\{{\cal
L}^{\prime}_{i}\\}$ that are more highly connected than ${\cal L}$. In fact,
in all of the examples of QEP given in refs Acín  _et al._ (2007); Perseguers
_et al._ (2008); Lapeyre Jr.  _et al._ (2009); Perseguers  _et al._ (2010) one
${\cal L}^{\prime}_{i}$ has average coordination number greater than or equal
to that of ${\cal L}$ (condition i). Furthermore, one ${\cal L}^{\prime}_{i}$
has a classical percolation threshold that is less than or equal to that of
${\cal L}$ (condition ii). In Refs. Acín  _et al._ (2007); Perseguers  _et
al._ (2008); Lapeyre Jr.  _et al._ (2009) this is easy to see because the
lattices involved are well-known111These are: double-bond hexagonal to
triangular; square to two square; kagome to square; bowtie to square and
triangular; asymmetric triangular to square and triangular.. The protocols in
Ref Perseguers  _et al._ (2010) generate multi-partite entanglement from the
initial bi-partite states. The multi-partite swapping was explicitly designed
to increase connectivity. These protocols give the best performance to date,
and often result in less common or unclassified lattices including non-planar
graphs and lattices whose sites have different coordination numbers.

Given that all known protocols satisfy conditions i and ii, a natural question
is whether this must always be the case. Must these properties, one local and
one global, that are associated with high connectivity, be non-decreasing in
an advantageous QEP? In the following section we present a counter-example
demonstrating that the answer to this question is “no”. In fact both
conditions are violated, and the improvement is robust. To achieve this, we
introduce a new ingredient into the lattice transformation protocols.

##  III QEP for the triangular lattice

CEP on the triangular lattice corresponds to classical bond percolation. With
CEP, long-range entanglement is only possible for
$\alpha_{1}>\hat{\alpha}_{c}(\text{CEP})=p_{c}^{\triangle}/2\approx 0.1736$,
and deterministic long-range entanglement is only possible for maximally
entangled initial states, i.e
$\alpha_{1}=\hat{\alpha}^{*}_{c}(\text{CEP})=1/2$. Here we present a QEP that
transforms the triangular lattice into the hexagonal lattice on which a
singlet can be created between any two nodes with probability $1$ if
$\alpha_{1}\gtrsim 0.3246$. That is, the upper threshold
$\hat{\alpha}^{*}_{c}$ is lowered. This is possible even though, classically,
the hexagonal lattice has larger critical density $p_{c}$ and smaller
coordination number than the triangular lattice.

###  III.1 Partial entanglement swapping

Figure 2: Entanglement swapping. (a) a measurement is performed on qubits $1$
and $2$. (b) after the measurement, qubits $3$ and $4$ are in one of four
states $\\{{\,|\phi_{m}\rangle}\\}$, which are partially or maximally
entangled. In full entanglement swapping, a singlet conversion is performed on
the pair $(3,4)$ which results in either a maximally entangled state, or a
separable state. In partial entanglement swapping, only the measurement is
performed. (c) A distillation is then performed on the output state together
with another entangled pair, here in the state $\,|\alpha\rangle$, to produce
(d) either a more highly entangled pair, or a separable state.

In order to show the counter-example promised in the introduction, it is
enough to consider one of the outcomes from the same swapping measurement used
in previous studies on entanglement percolation. However, we consider here
more general measurements that allow us to optimize for certain figures of
merit. In this paper we consider entanglement swapping using Bell measurements
on two qubits, one from each pair, as shown in Fig. 2). For brevity, we omit
referring to any necessary local unitaries. We call the usual entanglement
swapping in any of these bases full entanglement swapping. We shall always
assume that the two input pairs are in the same state $\,|\alpha\rangle$.
Following Ref. Perseguers  _et al._ (2008), we define an orthonormal basis
$\\{\,|\negmedspace\uparrow\rangle,\,|\negmedspace\downarrow\rangle\\}_{j}$
for each qubit $j=1,2$

$\begin{pmatrix}\,|\negmedspace\uparrow\rangle\\\
\,|\negmedspace\downarrow\rangle\end{pmatrix}_{j}=U_{j}\begin{pmatrix}\,|0\rangle\\\
\,|1\rangle\end{pmatrix}_{j},\quad U_{j}\in\mathcal{U}(2),$

and the Bell vectors

$\,|\Phi^{\pm}\rangle=\frac{\,|\negmedspace\uparrow\uparrow\rangle\pm\,|\negmedspace\downarrow\downarrow\rangle}{\sqrt{2}}\quad\text{and}\quad\,|\Psi^{\pm}\rangle=\frac{\,|\negmedspace\uparrow\downarrow\rangle\pm\,|\negmedspace\downarrow\uparrow\rangle}{\sqrt{2}}.$

The four measurement outcomes are

$\\{{\,|\phi_{m}\rangle}\\}\quad\text{ with probabilities }\quad\\{p_{m}\\}.$

Furthermore, $p_{\text{min}}=\min\\{p_{m}\\}$, and
$p_{\text{max}}=\max\\{p_{m}\\}$ are given by

$p_{\text{min}}=\alpha_{0}\alpha_{1}\quad\text{ and }\quad
p_{\text{max}}=\frac{1}{2}-\alpha_{0}\alpha_{1}.$

There is a bijective mapping between the probabilities $\\{p_{m}\\}$ and
$\\{U_{j}\\}$. In particular, every (orderless) choice of $\\{p_{m}\\}$
satisfying $p_{\text{min}}\leq p_{m}\leq p_{\text{max}}$ and $\sum_{m}p_{m}=1$
corresponds to a Bell measurement. The smallest Schmidt coefficients of the
output states are given by

$\lambda_{m}=\frac{1}{2}\left(1-\sqrt{1-\frac{\alpha_{0}^{2}\alpha_{1}^{2}}{p_{m}^{2}}}\right).$

In full entanglement swapping, we first perform the Bell measurement, and then
perform a singlet conversion on the output state. Since a singlet conversion
succeeds with probability equal to twice the smallest Schmidt coefficient, the
average SCP for full entanglement swapping is given by
$S_{M}=2\sum_{m}p_{m}\lambda_{m}$.

However, in partial entanglement swapping, we perform the Bell measurement
only, and not the singlet conversion. Instead of immediately doing a singlet
conversion we take advantage of the new geometry of the output state. We
attempt to distill a singlet from the output state and another entangled pair.
Although this is a simple idea, it is quite useful, and it has not been used
in previous work on entanglement distribution. From majorization theory
Nielsen (1999); Nielsen and Vidal (2001), we find that we can distill a Bell
pair from two partially entangled pairs with optimal probability

$p_{\text{distill}}=\min\\{1,2\left[1-(1-\beta_{1})(1-\gamma_{1})\right]\\},$
(1)

where $\beta_{1}$ and $\gamma_{1}$ are the smallest Schmidt coefficients of
the input states Lapeyre Jr.  _et al._ (2009). In the example below, the
second state used in the distillation will be $\,|\alpha\rangle$. Thus, the
input states to the distillation have $\beta_{1}=\alpha_{1}$ and
$\gamma_{1}=\lambda_{m}$. The average SCP from combining the partial swapping
with distillation is then

$S_{M}=\sum_{m}p_{m}\min\left\\{1,2-\alpha_{0}\left(1+\sqrt{1-\frac{\alpha_{0}^{2}\alpha_{1}^{2}}{p_{m}^{2}}}\right)\right\\}.$
(2)

####  III.1.1 Swapping in ZZ basis

Suppose the measurement is in the ZZ basis, $U_{1}=U_{2}=\openone_{2}$. This
is the measurement that maximizes the average SCP in full swapping. Thus, it
is the one used in all previous entanglement percolation schemes (with a
modified version for multi-partite entanglement percolation). In this case,
$p_{1}=p_{2}=p_{\text{min}}$ and $p_{3}=p_{4}=p_{\text{max}}$, with
corresponding smallest Schmidt coefficients

$\lambda(p_{\max})=\frac{\alpha_{1}^{2}}{\alpha_{0}^{2}+\alpha_{1}^{2}},\quad\lambda(p_{\min})=\frac{1}{2}.$

Two of the outcomes are already singlets. Each of the other two may be
distilled together with $\,|\alpha\rangle$ into a singlet with probability

$p=\min\left\\{1,2\left(1-\frac{\alpha^{3}_{0}}{\alpha_{0}^{2}+\alpha_{1}^{2}}\right)\right\\},$
(3)

given by (1). The average SCP using partial swapping in the ZZ basis is then

$S_{ZZ}=\alpha_{0}\alpha_{1}+(1-2\alpha_{0}\alpha_{1})\min\left\\{1,2\left(1-\frac{\alpha^{3}_{0}}{\alpha_{0}^{2}+\alpha_{1}^{2}}\right)\right\\}.$
(4)

####  III.1.2 Swapping in XZ basis

Suppose the measurement is in the XZ basis. Then $p_{m}=1/4$ and
$\lambda_{m}=\frac{1}{2}(1-\sqrt{1-16\alpha_{0}^{2}\alpha_{1}^{2}})$ for all
$m$. The average SCP using partial swapping in the XZ basis is then

$S_{XZ}=\min\left\\{1,2-\alpha_{0}\left(1+\sqrt{1-16\alpha_{0}^{2}\alpha_{1}^{2}}\right)\right\\}.$
(5)

###  III.2 The protocol

The QEP protocol proceeds as follows. Consider the triangular lattice with
each bond consisting of a single, partially entangled pure state. In step $1$,
we perform partial entanglement swappings on selected bonds as shown in Fig.
3.

Figure 3: Transformation of triangular to hexagonal lattice. a) triangular
lattice. A partial swap is applied to the dotted (red) lines. In following
frames, outcomes of partial swaps are shown as dashed (green) lines. Partial
swaps are applied to pairs of links shown as dotted lines. f) portion of
hexagonal lattice with double links. One link of each double link is in the
state $\,|\alpha\rangle$, the other link is one of the four outcomes of the
partial swap $\\{{\,|\phi_{m}\rangle}\\}$.

At the end of step $1$ we have a hexagonal lattice with double links. In each
pair, one link is the initial state $\,|\alpha\rangle$ and one link is one of
$\\{{\,|\phi_{m}\rangle}\\}$. Step $2$ consists of the following. For each
double link, if the outcome ${\,|\phi_{m}\rangle}$ is already a Bell pair,
then we do nothing. Otherwise, we attempt to distill a singlet from the two
links ${\,|\phi_{m}\rangle}$ and $\,|\alpha\rangle$.

####  III.2.1 Protocol in ZZ basis

Suppose we do the partial swap in the ZZ basis. Two outcomes are singlets and
two are partially entangled. From (3) we see that we create a singlet on every
bond of the hexagonal lattice deterministically if $\alpha_{0}$ is less than
the real root $\alpha_{0}^{*}\approx 0.6478$ of
$\alpha_{0}^{3}-\alpha_{0}^{2}+\alpha_{0}-1/2=0$. Equivalently, the condition
is $\hat{\alpha}^{*}_{c}\approx 0.3522$. The critical threshold for this
protocol is found by using (4) and solving
$S_{ZZ}(\alpha_{1})={p^{\hexagon}_{c}}$, where ${p^{\hexagon}_{c}}$ is the
classical threshold on the hexagonal lattice, with the result
$\hat{\alpha}_{c}\approx 0.1988$.

####  III.2.2 Protocol in XZ basis

Suppose we do the partial swap in the XZ basis. The smallest value of
$\alpha_{1}$ for which (5) equals $1$ is $\hat{\alpha}^{*}_{c}\approx 0.3246$.
The solution of $S_{XZ}(\alpha_{1})={p^{\hexagon}_{c}}$ is
$\hat{\alpha}_{c}\approx 0.2200$.

Protocol | $\hat{\alpha}_{c}$ | $\hat{\alpha}^{*}_{c}$
---|---|---
CEP | 0.1736 | 1/2
QEP ZZ | 0.1988 | 0.3522
QEP XZ | 0.2200 | 0.3246
QEP optimal | 0.1961 | 0.3246
Table 2:  Percolation thresholds $\hat{\alpha}_{c}$ and upper thresholds
$\hat{\alpha}^{*}_{c}$ for entanglement protocols on the triangular lattice.
Figure 4: Average SCP distilled from double links on the hexagonal lattice.
Solid curve, ZZ basis. Dash-dot curve, XZ basis. Dashed curve, optimal basis.
Dotted line $p_{c}$ for bond percolation on the hexagonal lattice.

####  III.2.3 Protocol in other Bell bases

We optimized over all Bell measurements with only two distinct values of
$p_{m}$. Visual inspection showed that the optimum average SCP occurs when the
second argument to $\min$ in (5) is equal to $1$, which occurs for
$p_{1}=\alpha_{0}^{2}\alpha_{1}/\sqrt{1-2\alpha_{1}}$. Inserting this into
(2), and solving $S_{M}(\alpha_{1})={p^{\hexagon}_{c}}$, we find the lower
threshold $\hat{\alpha}_{c}\approx 0.1961$, which is a small improvement over
the ZZ basis. Optimizing for the upper threshold $\hat{\alpha}^{*}_{c}$, we
find $p_{m}=1/4$, which is the XZ basis. A numerical search for more general
Bell measurements strongly suggests that the optimum Bell protocol has only
two distinct values of $p_{m}$ for all $\alpha_{1}$.

In summary, we found that the optimal Bell basis has exactly two distinct
values of $p_{m}$, which depend on $\alpha_{1}$. At the lower threshold, the
optimal basis gives only a slight improvement over the ZZ basis. As the upper
threshold is approached, the optimal basis approaches the XZ basis. These
results are summarized in Fig. 4 and Table 2. We did not investigate non-Bell
measurements.

##  IV Discussion

We have introduced a new tool, partial entanglement swapping, for entanglement
percolation via lattice transformation. This adds flexibility in optimally
combining the quantum and the geometric aspects of QEPs. We have demonstrated
the utility of partial swapping by using it to design a QEP that transforms
the triangular lattice to the hexagonal lattice. Partial entanglement swapping
allows sufficient concentration of entanglement to overcome lowered
connectivity in the transformed lattice. In particular, there is a least
initial amount of entanglement above which long-distance entanglement is
deterministic. Thus, we have proven that non-decreasing connectivity, as
measured by coordination number and percolation threshold, is not required for
QEP. However, in the present example, we find that CEP still provides the
optimal percolation threshold. It is interesting to note that the only other
known QEP for the triangular lattice uses multi-partite entanglement to
enhance the connectivity of the lattice by creating a non-planar graph
Perseguers  _et al._ (2010). Thus, the question of whether a transformed
lattice with lower connectivity can give a lower critical threshold remains
open. Also unknown is whether the critical threshold of the triangular lattice
can be lowered via a transformation to a planar graph, or whether the
triangular lattice is, in a sense, a maximally connected planar graph.

In addition to answering a conjecture on the geometrical constraints on QEP,
partial swapping enlarges the toolbox for QEP. It may be combined with other
techniques to push the initial entanglement thresholds lower. However, even in
this simple example, the search becomes more complicated because we find that
the optimal measurement basis for partial swapping depends on the amount of
initial entanglement. Still more interesting than each new protocol would be a
proof, constructive or otherwise, of the existence of a minimum threshold for
a particular lattice or class of lattices.

##  V Acknowledgments

The author thanks Jan Wehr for discussions and for asking a question that led
to the present work. This work was supported in part by the Spanish MICINN
(TOQATA, FIS2008-00784), by the ERC (QUAGATUA, OSYRIS), and EU projects SIQS,
EQUAM, and the Templeton Foundation.

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# Sparsity Emerges Naturally in Neural Language Models

Naomi Saphra  Adam Lopez
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
Institute for Language, Cognition, and Computation
University of Edinburgh

###### Abstract

Concerns about interpretability, computational resources, and principled
inductive priors have motivated efforts to engineer sparse neural models for
NLP tasks. If sparsity is important for NLP, might well-trained neural models
naturally become roughly sparse? Using the Taxi-Euclidean norm to measure
sparsity, we find that frequent input words are associated with concentrated
or sparse activations, while frequent target words are associated with
dispersed activations but concentrated gradients. We find that gradients
associated with function words are more concentrated than the gradients of
content words, even controlling for word frequency.

##  1 Introduction

Researchers in NLP have long relied on engineering features to reflect the
sparse structures underlying language. Modern deep learning methods promised
to relegate this practice to history, but have not eliminated the interest in
sparse modeling for NLP. Along with concerns about computational resources
Chen et al. (2016); Narang et al. (2017b) and interpretability Murphy et al.
(2012); Subramanian et al. (2018), human intuitions continue to motivate
sparse representations of language. For example, some work applies assumptions
of sparsity to model latent hard categories such as syntactic dependencies
Padó and Lapata (2007) or phonemes Cotterell and Eisner (2018). Niculae and
Blondel (2017) found that a sparse attention mechanism outperformed dense
methods on some NLP tasks; Narang et al. (2017a) found sparsified versions of
LMs that outperform dense originals. Attempts to engineer sparsity rest on an
unstated assumption that it doesn’t arise naturally when neural models are
learned. Is this true?

Using a simple measure of sparsity, we analyze how it arises in different
layers of a neural language model in relation to word frequency. We show that
the sparsity of a word representation increases with exposure to that word
during training. We also find evidence of syntactic learning: gradient updates
in backpropagation depend on whether a word’s part of speech is open or closed
class, even controlling for word frequency.

##  2 Methods

### Language model.

Our LM is trained on a corpus of tokenized, lowercased English Wikipedia
(70/10/20 train/dev/test split). To reduce the number of unique words (mostly
names) in the corpus, we excluded any sentence with a word which appears fewer
than 100 times. Those words which still appear fewer than 100 times after this
filter are replaced with `<UNK>`. The resulting training set is over 227
million tokens of around 19.5K types.

embedding$\textrm{LSTM}_{1}$$\textrm{LSTM}_{2}$softmax$x_{t-1}$$x_{t}$$\hat{x}_{t}$$h_{t}^{1},c_{t}^{1}$$h_{t}^{2},c_{t}^{2}$$\bar{x}_{t}$$h_{t}^{\emph{e}}$$\bar{h}_{t}^{\emph{e}}$$h_{t}^{1}$$\bar{h}_{t}^{1}$$h_{t}^{2}$$\bar{h}_{t}^{2}$
Figure 1: LM architecture for target word distribution $\hat{x}_{t}$, showing
gradient updates from observed word $x_{t}$.

We use a standard 2-layer LSTM LM trained with cross entropy loss for 50
epochs. The pipeline from input $x_{t-1}$ at time step $t-1$ to predicted
output distribution $\hat{x}$ for time $t$ is described in Figure 1,
illustrating intermediate activations $h_{t}^{e}$, $h_{t}^{1}$, and
$h_{t}^{2}$. At training time, the network observes $x_{t}$ and backpropagates
the gradient updates $\bar{h}_{t}^{e}$, $\bar{h}_{t}^{1}$, $\bar{h}_{t}^{2}$,
and $\bar{x}_{t}$.

The embeddings produced by the encoding layer are 200 units, and the recurrent
layers have 200 hidden units each. The batch size is set to forty, the maximum
sequence length to 35, and the dropout ratio to 0.2. The optimizer is standard
SGD with clipped gradients at $\ell_{2}=0.25$, where the learning rate begins
at 20 and is quartered whenever loss fails to improve.

### Measuring sparsity.

We measure the sparsity of a vector $v$ using the reciprocal of the Taxicab-
Euclidean norm ratio (Repetti et al., 2015). This measurement has a long
history as a measurement of sparsity in natural settings (Zibulevsky and
Pearlmutter, 2001; Hoyer, 2004; Pham et al., 2017; Yin et al., 2014) and is
formally defined as $\chi(v)=\|v\|_{2}/\|v\|_{1}$. The relationship between

sparsity and this ratio is illustrated in two dimensions in the image on the
right, in which darker blue regions are more concentrated. The pink circle
shows the area where $\ell_{2}\leq 1$ while the yellow diamond depicts
$\ell_{1}\leq 1$. For sparse vectors $\langle 1,0\rangle$ or $\langle
0,1\rangle$, the norms are identical so $\chi$ is 1, its maximum. For a
uniform vector like $\langle 1,1\rangle$, $\chi$ is at its smallest. In
general, $\chi(v)$ is higher when most elements of $v$ are close to 0; and
lower when the elements are all similar in value.

Figure 2: Average sparsity $\chi(\bar{h_{t}^{2}})$ over all training epochs
(x-axis), for target words $x_{t}$ occurring more than 100k times in training.
Target words are sorted from most frequent (bottom) to least frequent (top).

(a) $\rho_{\leftarrow}$ correlation with target word frequency

(b) $\rho_{\rightarrow}$ correlation with input word frequency

Figure 3: Correlation between mean sparsity of a word’s representation and
word frequency. Vertical dashed lines indicate when the optimizer has rescaled
the step size.

##  3 Experiments

Sparsity is closely related to the behavior of a model: If only a few units
hold most of the mass of a representation, the activation vector will be
highly concentrated. If a neural network relies heavily on a small number of
units in determining its predictions, the gradient will be highly
concentrated. A highly concentrated gradient is mainly modifying a few
specific pathways. For example, it might modify a neuron associated with
particular inputs like parentheses Karpathy et al. (2015), or properties like
sentiment Radford et al. (2017).

### Representations of Target Words.

Our first experiments look at the relationship of sparsity to target word
$x_{t}$. Gradient updates triggered by the target are often used to identify
units that are relevant to a prediction (Li et al., 2015), and as shown in
Figure 2, gradient sparsity increases with both the frequency of a word in the
corpus and the overall training time. In other words, more exposure leads to
sparser relevance. Because the sparsity of $\bar{h}^{2}$ increases with target
word frequency, we measure not sparsity itself but the Pearson correlation,
over all words $w$, between word frequency and mean $\chi(h)$ over
representations $h$ where $w$ is the target:

$\rho_{\leftarrow}(h)=\textrm{corr}_{w}(\mu_{t:x_{t}=w}(\chi(h_{t})),\textrm{freq}(w))$

Here (Figure 3(a)) we confirm that concentrated gradients are not a result of
concentrated activations, as activation sparsity $\chi(h^{2})$ is not
correlated with target word frequency.

The correlation is strong and increasing only for
$\rho_{\leftarrow}(\bar{h}^{2})$. The sparse structure being applied is
therefore particular to the gradient passed from the softmax to the top LSTM
layer, related to how a word interacts with its context.

Figure 4: $\rho_{\leftarrow}(\bar{h}^{2})$, evaluated over vocabulary from
open and closed classes of POS.

### The Role of Part of Speech.

Figure 4 shows that $\rho_{\leftarrow}(\bar{h}^{2})$ follows distinctly
different trends for open POS classes111ADJ, ADV, INTJ, NOUN, PROPN, VERB and
closed classes222ADP, AUX, CCONJ, DET, PART, PRON, SCONJ. To associate words
to POS, we tagged our training corpus with spacy333https://spacy.io/; we
associate a word to a POS only if the majority (at least 100) of its
occurrences are tagged with that POS. We see that initially, frequent words
from closed classes are highly concentrated, but soon stabilize, while
frequent words from open classes continue to become more concentrated
throughout training. Why?

Closed class words clearly signal POS. But open classes contain many ambiguous
words, like “report”, which can be a noun or verb. Open classes also contain
many more words in general. We posit that early in training, closed classes
reliably signal syntactic structure, and are essential for shaping network
structure. But open classes are essential for predicting specific words, so
their importance in training continues to increase after part of speech tags
are effectively learned.

The high sparsity of function word gradient may be surprising when compared
with findings that content words have a greater influence on outputs (Kádár et
al., 2016). However, those findings were based on the impact on the vector
representation of an entire sentence after omitting the word. Khandelwal et
al. (2018) found that content words have a longer window during which they are
relevant, which may explain the results of Kádár et al. (2016). Neither of
these studies controlled for word frequency in their analyses contrasting
content and function words, but we believe this oversight is alleviated in our
work by measuring correlations rather than raw magnitude. Because
$\rho_{\leftarrow}(\bar{h}^{2})$ is higher when evaluated over more frequent
words, which also tend to be function words (see Figure 5), we further control
for the effect of frequency by including a measurement of trends in a sample
of 120 words each from open and closed classes (Figure 4). This sample was
selected by sorting all open and closed class words by frequency, then
choosing a range of each sorted list with a similar average frequency.

Figure 5: Mean sparsity of $\chi(\bar{h}^{2})$ after 50 epochs, for words
occurring more than 1k times in the train set.

### Representations of Input Words.

We next looked at the vector representations of each step in the word sequence
as a representation of the input word $x_{t-1}$ that produced that step. We
measure the correlation with input word frequency:

$\rho_{\rightarrow}(h)=\textrm{corr}_{w}(\mu_{t:x_{t-1}=w}(\chi(h_{t})),\textrm{freq}(w))$

Here (Figure 3(b)) we find that the view across training sheds some light on
the learning process. While the lower recurrent layer quickly learns sparse
representations of common input words, $\rho_{\rightarrow}(h^{1})$ increases
more slowly later in training and is eventually surpassed by
$\rho_{\rightarrow}(h^{\textrm{e}})$, while gradient sparsity never becomes
significantly correlated with word frequency. Li et al. (2016) studied the
activations of feedforward networks in terms of the importance of individual
units by erasing a particular dimension and measuring the difference in log
likelihood of the target class. They found that importance is concentrated
into a small number of units at the lowest layers in a neural network, and is
more dispersed at higher layers. Our findings suggest that this effect may be
a natural result of the sparsity of the activations at lower layers.

We relate the trajectory over training to the Information Bottleneck
Hypothesis of Shwartz-Ziv and Tishby (2017). This theory, connected to
language model training by Saphra and Lopez (2018), proposes that the earlier
stages of training are dedicated to learning to effectively represent inputs,
while later in training these representations are compressed and the optimizer
removes input information extraneous to the task of predicting outputs. If
extraneous information is encoded in specific units, this compression would
lead to the observed effect, in which the first time the optimizer rescales
the step size, it begins an upward trend in $\rho_{\rightarrow}$ as extraneous
units are mitigated.

##  4 Potential Explanations

Why do common target words have such concentrated gradients with respect to
the final LSTM layer? A tempting explanation is that the amount of information
we have about common words offers high confidence and stabilizes most of the
weights, leading to generally smaller gradients. If this were true, the
denominator of sparsity, gradient $\ell_{1}$, should be strongly anti-
correlated with word frequency. In fact, it is only ever slightly anti-
correlated (correlation $>-.1$). Furthermore, the sparsity of the softmax
gradient $\chi(\bar{x})$ does not exhibit the strong correlation seen in
$\chi(\bar{h}^{2})$, so sparsity at the LSTM gradient is not a direct effect
of sparse logits.

However, the model could still be “high confidence” in terms of how it assigns
blame for error during common events, even if it is barely more confident
overall in its predictions. According to this hypothesis, a few specialized
neurons might be responsible for the handling of such words.

Perhaps common words play a prototyping role that defines clusters of other
words, and therefore have a larger impact on these clusters by acting as
attractors within the representation space early on. Such a process would be
similar to how humans acquire language by learning to use words like ‘dog’
before similar but less prototypical words like ‘canine’ (Rosch, 1999). As a
possible mechanism for prototyping with individual units, Dalvi et al. (2019)
found that some neurons in a translation system specialized in particular word
forms, such as verb inflection or comparative and superlative adjectives. For
example, a common comparative adjective like ‘better’ might be used as a
reliable signal to shape the handling of comparatives by triggering
specialized units, while rarer words have representations that are more
distributed according to a small collection of specific contexts.

There may also be some other reason that common words interact more with
specific substructures within the network. For example, it could be related to
the use of context. Because rare words use more context than common words and
content words use more context than function words (Khandelwal et al., 2018),
the gradient associated with a common word would be focused on interactions
with the most recent words. This would lead common word gradients to be more
concentrated.

It is possible that frequent words have sparse activations because frequency
is learned as a feature and thus is counted by a few dimensions of
proportional magnitude, as posited by Li et al. (2016).

##  5 Potential Applications

Understanding where natural sparsity emerges in dense networks could be a
useful guide in deciding which layers we can apply sparsity constraints to
without affecting model performance, for the purpose of interpretability or
efficiency. It might also explain why certain techniques are effective: for
example, in some applications, summing representations together works quite
well Hill et al. (2016). We hypothesize that this occurs when the summed
representations are sparse so there is often little overlap. Understanding
sparsity could help identify cases where such simple ensembling approaches are
likely to be effective.

Future work may develop ways of manipulating the training regime, as in
curriculum learning, to accelerate the concentration of common words or
incorporating concentration into the training objective as a regularizer. We
would also like to see how sparsity emerges in models designed for specific
end tasks, and to see whether concentration is a useful measure for the
information compression predicted by the Information Bottleneck.

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