# Quantum Wiretap Channel Coding
Assisted by Noisy Correlation

Minglai Cai1 1Grup d’Informació Quàntica
Departament de Física
Universitat Autònoma
de Barcelona, Spain
<EMAIL_ADDRESS>2ICREA–Institució
Catalana de Recerca
i Estudis Avançats
Barcelona, Spain 3Institute for
Advanced Study
Technische Universität
München, Germany Andreas Winter1,2,3,4 4QUIRCK–Quantum
Independent Research
Center Kessenich
Bonn, Germany
<EMAIL_ADDRESS>
###### Abstract

We consider the private classical capacity of a quantum wiretap channel, where
the users (sender Alice, receiver Bob, and eavesdropper Eve) have access to
the resource of a shared quantum state, additionally to their channel inputs
and outputs. An extreme case is maximal entanglement or a secret key between
Alice and Bob, both of which would allow for one-time padding the message. But
here both the wiretap channel and the shared state are general. In the other
extreme case that the state is trivial, we recover the wiretap channel and its
private capacity [N. Cai, A. Winter and R. W. Yeung, _Probl. Inform. Transm._
40(4):318-336, 2004]. We show how to use the given resource state to build a
code for secret classical communication. Our main result is a lower bound on
the assisted private capacity, which asymptotically meets the multi-letter
converse and which encompasses all sorts of previous results as special cases.

###### Index Terms:

Quantum information; communication via quantum channels; wiretap channels

##  I Introduction

Entanglement shared between sender and receiver of a transmission is a useful
resource that generically increases channel capacity. In this scenario, the
two parties may be given some number of quantum bits jointly prepared in a
fixed superposition. The advantages of a shared quantum entanglement resource
prior to communication over a noisy quantum channel has been considered
extensively in [6] and [4, 5], where the entanglement-assisted classical
capacity theorem has been derived; see also [28]. These works showed how to
increase the classical capacity of quantum channels by the assistance of
unlimited pure entanglement. Since then, more recent works have studied
extensions of the use of shared entanglement as an assisting quantum resource:
entanglement-assisted communication over quantum multiple-access channels
[25]; entanglement-assisted communication over compound and arbitrarily
varying quantum channels [7]; entanglement-assisted communication over quantum
broadcast channels [24]. For these results, it is essential that the shared
state be maximally entangled. In a somewhat dual setting, in [8, 9, 19, 20,
31, 21], Alice and Bob are connected by a noiseless quantum channel, but
instead share a general mixed quantum state (decoupled from Eve). The case of
a noisy quantum channel has been investigated in [3]. These papers showed that
sufficiently entangled states, or any entanglement in conjunction with
suitable channels, are useful resources for communication.

Secure communication over a classical channel with an eavesdropper was first
introduced by Wyner [32]. The secrecy capacity for quantum wiretap channels
has been determined in [16] and [10]. Chen _et al._ [12] demonstrated that
correlation between Alice and Bob is able to help secure message transmission
over a classical wiretap channel. Inspired by these results, we analyze secure
communication when the channel users (Alice, Bob and Eve) share a general
correlated states, which are “given by nature”, and assumed to be known to all
parties. In this work, we show a way to increase the secure capacity of
quantum channels by the assistance of such noisy correlation. Our proof works
by considering the shared resource as a component of the channel. Technically,
we employ quantum wiretap channel codes, and to incorporate the correlated
resource a variant of Gel’fand-Pinsker coding as in [13] and [2], which has
been used similarly in secure “writing on dirty paper” codes [14].

Figure 1: Communication diagramme of the wiretap coding problem over
$\mathcal{N}$ with assistance by a tripartite resource state shared by sender
(Alice), receiver (Bob) and eavesdropper (Eve). Unlike the plain wiretap
channel, here each message is encoded not into a state, but rather a
modulation $\mathcal{E}_{m}:A^{\prime}\rightarrow A$.

In the next section we introduce the communication scenario and the assisted
private capacity, followed by a section recalling certain basic notions. After
that we state and prove our main result, followed by a discussion of special
cases of interest.

##  II Communication scenario

Before starting, we review some basic notions from quantum information theory.
Let $\rho_{1}$ and $\rho_{2}$ be Hermitian operators on a finite-dimensional
complex Hilbert space $A$. We say $\rho_{1}\geq\rho_{2}$ and
$\rho_{2}\leq\rho_{1}$ if $\rho_{1}-\rho_{2}$ is positive semidefinite. We
denote the set of density operators (states) on $A$ by
$\mathcal{S}(A):=\\{\rho\in\mathcal{L}(A):\rho\geq 0,\
\operatorname{Tr}(\rho)=1\\}$, where $\mathcal{L}({A})$ is the set of linear
operators on ${A}$. For finite-dimensional complex Hilbert spaces $A$ and $B$,
a _quantum channel_ is a linear, completely positive and trace preserving
(cptp) map $\mathcal{N}:\mathcal{L}(A)\rightarrow\mathcal{L}(B)$, that acts a
$\mathcal{L}(A)\ni\rho\mapsto\mathcal{N}\rho)\in\mathcal{L}(B)$, which accepts
input quantum states in $\mathcal{S}(A)$ and produces output quantum states in
$\mathcal{S}(B)$. We shall often simplify the notation of a quantum channel as
$\mathcal{N}:A\rightarrow B$. Likewise, we will often suppress the tensor
product sign in $BE=B\otimes E$ and $A^{n}=A^{\otimes n}$, if there is no
danger of confusion.

######  Definition 1.

A _quantum wiretap channel_ is a cptp map $\mathcal{N}:A\rightarrow B\otimes
E$, with finite-dimensional quantum systems $A$, $B$ and $E$, representing
Alice’s input and Bob’s and Eve’s outputs, respectively.

Note that in [10], a wiretap channel was more generally a pair of cptp maps,
one from $A$ to $B$ and one from $A$ to $E$. We can reproduce that notion by
letting $\mathcal{N}_{B}=\operatorname{Tr}_{E}\circ\mathcal{N}$ and
$\mathcal{N}_{E}=\operatorname{Tr}_{B}\circ\mathcal{N}$, but not every pair of
cptp maps arises in this way. However, the generality offered by [10] comes in
handy later on.

######  Definition 2.

A _quantum wiretap channel assisted by a correlation_ is a pair
$(\mathcal{N},\zeta)$ consisting of a wiretap channel
$\mathcal{N}:A\rightarrow B\otimes E$ and a quantum state
$\zeta\in\mathcal{S}(A^{\prime}\otimes B^{\prime}\otimes E^{\prime})$, with
finite-dimensional quantum systems $A^{\prime}$, $B^{\prime}$ and $E^{\prime}$
in the possession of Alice, Bob and Eve, respectively, before the
transmission.

######  Definition 3.

An _$(n,\lambda,\mu)$ -wiretap code_ for $(\mathcal{N},\zeta)$ is a collection
$\\{(\mathcal{E}_{m},D_{m}):m\in[M]=\\{1,\ldots,M\\}\\}$ of cptp maps
$\mathcal{E}_{m}:{A^{\prime}}^{n}\rightarrow A^{n}$ (the “modulations”) and
operators $D_{m}\geq 0$ on $B^{n}{B^{\prime}}^{n}$,
$\sum_{m=1}^{M}D_{m}\leq\mathbf{1}$, such that there exists a state $\sigma$
on $E^{n}{E^{\prime}}^{n}$ with

$\displaystyle\frac{1}{M}\sum_{m}\operatorname{Tr}\left((\mathcal{N}^{\otimes
n}\\!\circ\mathcal{E}_{m})\zeta^{\otimes
n}\\!\cdot\\!(D_{m}\\!\otimes\mathbf{1}_{E^{n}{E^{\prime}}^{n}})\right)$
$\displaystyle\geq 1-\lambda,$ (1)
$\displaystyle\frac{1}{M}\left\|\operatorname{Tr}_{B^{n}{B^{\prime}}^{n}}(\mathcal{N}^{\otimes
n}\\!\circ\mathcal{E}_{m})\zeta^{\otimes
n}-\sigma^{E^{n}{E^{\prime}}^{n}}\right\|_{1}$ $\displaystyle\leq\mu.$ (2)

See Fig. 1 for the communication diagramme.

The _rate_ of the wiretap code is $\frac{1}{n}\log M$, and the largest number
$M$ of messages of an $(n,\lambda,\mu)$-wiretap code is denoted
$M(n,\lambda,\mu)$. This allows us to define the (weak) _assisted private
capacity_ as

$P(\mathcal{N},\zeta):=\inf_{\lambda,\mu>0}\liminf_{n\rightarrow\infty}\frac{1}{n}\log
M(n,\lambda,\mu),$ (3)

which is the main objective for the rest of the paper.

##  III Preliminaries

For a finite set $\mathcal{X}$, we denote the set of probability distributions
on $\mathcal{X}$ by ${\mathbb{P}}(\mathcal{X})$. For a discrete random
variable $X$ on a finite set $\mathcal{X}$ and a discrete random variable $Y$
on a finite set $\mathcal{Y}$ we denote the Shannon entropy of $X$ by
$H(X)=-\sum_{x\in\mathcal{X}}p(x)\log p(x)$ and the mutual information between
$X$ and $Y$ by

$I(X;Y)=\sum_{x\in\mathcal{X}}\sum_{y\in\mathcal{Y}}p(x,y)\log{\frac{p(x,y)}{p(x)p(y)}}\text{
,}$

Here $p(x,y)$ is the joint probability distribution function of $X$ and $Y$,
and $p(x)$ and $p(y)$ are the marginal probability distribution functions of
$X$ and $Y$ respectively. Throughout the paper the logarithm base is $2$.

For quantum states acting on $A$ or composite systems $A\otimes B$, these
concepts generalise. For instance for $\rho^{AB}\in\mathcal{S}(A\otimes B)$ we
have the marginals (reduced states) $\rho^{A}=\operatorname{Tr}_{B}\rho$ and
$\rho^{B}=\operatorname{Tr}_{a}\rho$ given by the partial trace. To a finite
set $\mathcal{X}$, we associate the Hilbert space $X$ with orthonormal basis
$\\{\left|x\right\rangle:x\in\mathcal{X}\\}$, such that the possible
distributions $p(x)$ on $\mathcal{X}$ correspond to diagonal density matrices
$\rho=\sum_{x}p(x)\left|x\right\rangle\\!\left\langle x\right|$.

An ensemble $\\{p(x),\rho_{x}\in\mathcal{S}(A)\\}_{x\in\mathcal{X}}$ of states
on $A$ is faithfully represented by the _classical-quantum (cq-)state_

$\gamma=\sum_{x\in\mathcal{X}}p(x)\left|x\right\rangle\\!\left\langle
x\right|^{X}\otimes\rho_{x}^{A}\in\mathcal{S}(X\otimes A).$

The entropy for quantum states $\rho\in\mathcal{S}(A)$ is the von Neumann
entropy, $S(\rho^{A})=S(A)_{\rho}=-\operatorname{Tr}\rho\log\rho$, and by
analogy with the Shannon entropy we have also the conditional entropy
$S(A|B)_{\rho}=S(AB)_{\rho}-S(B)_{\rho}$ and the mutual information
$I(A:B)_{\rho}=S(A)_{\rho}+S(B)_{\rho}-S(AB)_{\rho}$.

A special class of quantum channels are _classical-quantum (cq-)channels_.
These are given by $W:X\rightarrow B$, with
$W(\left|x\right\rangle\\!\left\langle
x^{\prime}\right|)=\delta_{xx^{\prime}}W_{x}$ for states
$W_{x}\in\mathcal{S}(B)$, $x\in\mathcal{X}$. As the channel is entirely
described by this family of states, we shall identify a cq-channel with the
map $W:\mathcal{X}\rightarrow\mathcal{S}(B)$, mapping $x\mapsto W_{x}$.

We shall nned two commonly used measures of distance and similarity of quantum
states. The fidelity of two quantum states $\rho$ and $\sigma$ is defined as
$F(\rho,\sigma)=\|\sqrt{\rho}\sqrt{\sigma}\|_{1}=\operatorname{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}$,
where $\|X\|_{1}=\operatorname{Tr}\sqrt{XX^{\dagger}}$ is the trace norm.
Then, according to Fuchs and van de Graff,

$1-F(\rho,\sigma)\leq\frac{1}{2}\|\rho-\sigma\|_{1}\leq\sqrt{1-F(\rho,\sigma)^{2}}.$

We end this section by recalling the formula for the private capacity of a
quantum wiretap channel [10, 16] when no correlation is present, i.e.
$\zeta^{A^{\prime}B^{\prime}E^{\prime}}=\left|0\right\rangle\\!\left\langle
0\right|^{A^{\prime}}\otimes\left|0\right\rangle\\!\left\langle
0\right|^{E^{\prime}}\otimes\left|0\right\rangle\\!\left\langle
0\right|^{E^{\prime}}=:\emptyset$:

$P(\mathcal{N})=P(\mathcal{N},\emptyset)=\sup_{n}\frac{1}{n}\sup_{q(u),\rho_{u}}\left(I(U:B^{n})_{\gamma}-I(U:E^{n})_{\gamma}\right),$
(4)

where the inner maximisation is over ensembles of states
$\rho_{u}\in\mathcal{S}(A^{n})$ with probabilities $q(u)$, $u\in\mathcal{U}$,
and

$\gamma^{UB^{n}E^{n}}=\sum_{u{\mathbf{n}}\mathcal{U}}q(u)\left|u\right\rangle\\!\left\langle
u\right|^{U}\otimes\left[\mathcal{N}^{\otimes
n}(\rho_{u})\right]^{B^{n}E^{n}}.$

##  IV Main result

Looking again at the definitions and the communication diagramme in Fig. 1, we
note that not just a code but any given modulations $\mathcal{E}_{u}$
($u\in\mathcal{U}$) give rise to a cq-wiretap channel
$\mathcal{M}:\mathcal{U}\rightarrow(BB^{\prime})^{n}(EE^{\prime})^{n}$,
mapping $u\in\mathcal{U}$ to $\mathcal{M}(u)=(\mathcal{N}^{\otimes
n}\\!\circ\mathcal{E}_{u})\zeta^{\otimes n}$. We can thus apply the coding
theorem and weak converse from [10] to get the following expression for the
assisted private capacity:

$P(\mathcal{N},\zeta)=\sup_{n}\frac{1}{n}\sup_{\\{q(u),\mathcal{E}_{u}\\}}\\!\\!\left(I(U\\!:\\!B^{n}{B^{\prime}}^{n})_{\gamma}-I(U\\!:\\!E^{n}{E^{\prime}}^{n})_{\gamma}\right)\\!,$
(5)

with respect to the cq-state

$\gamma^{U(BB^{\prime})^{n}(EE^{\prime})^{n}}=\sum_{u}q(u)\left|u\right\rangle\\!\left\langle
u\right|^{U}\otimes(\mathcal{N}^{\otimes
n}\\!\circ\mathcal{E}_{u})\zeta^{\otimes n},$

where the inner supremum is over all alphabets $\mathcal{U}$, probability
distribution $q$ on $\mathcal{U}$ and modulations $\mathcal{E}_{u}$. Our goal
will be to find a better achievability bound, which thus speeds up the
convergence of Eq. (5).

For the subsequent analysis, we observe that w.l.o.g. the reduced state
$\zeta^{A^{\prime}}$ has full rank (i.e. equal to the Hilbert space dimension
$|A^{\prime}|$), because otherwise we can shrink $A^{\prime}$ to the support
of $\zeta^{A^{\prime}}$ and restrict the modulation maps $\mathcal{E}_{u}$
accordingly.

######  Remark 4.

The key to our coding theorem is the following equivalent description of the
code. Consider a Hilbert space $A^{\prime\prime}\simeq A^{\prime}$ and a pure
state vector $\left|\phi_{0}\right\rangle\in A^{\prime}\otimes
A^{\prime\prime}$ such that
$\phi_{0}^{A^{\prime}}=\phi_{0}^{A^{\prime\prime}}=\zeta^{A^{\prime}}$. By the
generalised Choi theorem, there is a (unique) cptp map
$\mathcal{Z}:A^{\prime}\rightarrow B^{\prime}\otimes E^{\prime}$ such that
$\zeta^{A^{\prime}B^{\prime}E^{\prime}}=(\operatorname{id}_{A^{\prime}}\otimes\mathcal{Z}^{A^{\prime\prime}\rightarrow
B^{\prime}E^{\prime}})\phi_{0}$. Likewise, the generalised Choi states
$\eta_{u}=(\mathcal{E}_{u}\otimes\operatorname{id}_{{A^{\prime\prime}}^{n}})\phi_{0}^{\otimes
n}$ of the modulation maps have the property
$\operatorname{Tr}_{A^{n}}\eta_{u}=(\zeta^{A^{\prime}})^{\otimes n}$ and by
these two equations determine $\mathcal{E}_{u}$ uniquely as cptp maps.

Now we make the elementary but crucial observation that

$\begin{split}(\mathcal{E}_{u}\otimes\operatorname{id}_{{B^{\prime}}^{n}{E^{\prime}}^{n}})\zeta^{\otimes
n}&=(\operatorname{id}_{A^{n}}\otimes\mathcal{Z}^{\otimes n})\eta_{u},\\\
(\mathcal{N}^{\otimes
n}\circ\mathcal{E}_{u}\otimes\operatorname{id}_{{B^{\prime}}^{n}{E^{\prime}}^{n}})\zeta^{\otimes
n}&=(\mathcal{N}\otimes\mathcal{Z})^{\otimes n}\eta_{u},\end{split}$ (6)

which means that our assisted wiretap code for $(\mathcal{N},\zeta)$ turns out
to be equivalent to a regular wiretap code for
$\mathcal{N}\otimes\mathcal{Z}:AA^{\prime}\rightarrow BB^{\prime}\otimes
EE^{\prime}$, albeit with the restriction
$\operatorname{Tr}_{A^{n}}\eta_{m}=(\zeta^{A^{\prime}})^{\otimes n}$ for all
messages $m$ (see Fig. 2), and similarly for arbitrary modulations
$\mathcal{E}_{u}$.

Figure 2: Reformulation of the assisted wiretap code for $(\mathcal{N},\zeta)$
in terms of the tensor product wiretap channel $\mathcal{N}\otimes\mathcal{Z}$
with a restriction on the marginals of the signal states $\eta_{m}$ on
${A^{\prime}}^{n}$.

We can now relax the condition
$\operatorname{Tr}_{A}\eta_{u}=\zeta^{A^{\prime}}$ for all $u\in\mathcal{U}$
to an average one,
$\sum_{u}q(u)\operatorname{Tr}_{A}\eta_{u}=\zeta^{A^{\prime}}$, to obtain the
following coding theorem. This is where the Gel’fand-Pinsker and “dirty paper”
coding idea is used [17, 14].

######  Theorem 5.

Let $(\mathcal{N},\zeta)$ be an assisted quantum wiretap channel with the
wiretap channel $\mathcal{Z}:A^{\prime}\rightarrow B^{\prime}E^{\prime}$
defined as above. Assume furthermore a cq-channel
$\mathcal{C}:\mathcal{U}\rightarrow A\otimes A^{\prime}$ and probabilities
$q(u)$ such that
$\sum_{u}q(u)\operatorname{Tr}_{A}\mathcal{C}(u)=\zeta^{A^{\prime}}$. Then,

$P(\mathcal{N},\zeta)\geq
I(U:BB^{\prime})_{\gamma}-\max\left(I(U:EE^{\prime})_{\gamma},I(U:A^{\prime})_{\beta}\right),$
(7)

where

$\displaystyle\beta^{UAA^{\prime}}$
$\displaystyle\\!=\sum_{u}q(u)\left|u\right\rangle\\!\left\langle
u\right|^{U}\\!\otimes\mathcal{C}(u)^{AA^{\prime}},$ (8)
$\displaystyle\gamma^{UBB^{\prime}EE^{\prime}}$
$\displaystyle\\!=\sum_{u}q(u)\left|u\right\rangle\\!\left\langle
u\right|^{U}\\!\otimes\left((\mathcal{N}\otimes\mathcal{Z})\mathcal{C}(u)\right)^{BB^{\prime}EE^{\prime}}\\!.$
(9)

Before proving this, let us remark that it includes the achievability part of
Eq. (3), since we may choose
$\mathcal{C}(u)=\eta_{u}=(\mathcal{E}_{u}\otimes\operatorname{id}_{A^{\prime\prime}})\phi_{0}$
as in Remark 4, which results in $I(U:A^{\prime})_{\beta}=0$.

###### Proof.

We will construct $\eta_{m}$ on block length $n$ as in Remark 4 by employing
the wiretap coding strategy of Devetak [16] for the cq-wiretap channel
$\mathcal{M}=(\mathcal{N}\otimes\mathcal{Z})\circ\mathcal{C}:\mathcal{U}\rightarrow
BB^{\prime}\otimes EE^{\prime}$, and the additional eavesdropper channel
$\mathcal{U}\ni u\mapsto\operatorname{Tr}_{A}\mathcal{C}(u)$. This amounts to
choosing code words $u^{n}\in\mathcal{U}^{n}$ independently according to the
distribution $q^{\otimes n}$, and binning them randomly so as to average over
bins.

Concretely, let $\mathcal{U}^{n}\ni u_{(m,s)}\sim q^{\otimes n}$ be sampled
i.i.d. for $m\in[M]$ and $s\in[S]$, where
$S=2^{n\max(I(U:EE^{\prime}),I(U:A^{\prime}))+n\epsilon}$ and
$MS=2^{nI(U:BB^{\prime})-n\epsilon}$ for an arbitrary $\epsilon>0$. (Devetak
restricts the sampling to the typical sequences of $q^{\otimes n}$, but by
typicality the difference to the present prescription is small.)

Devetak’s analysis [16] implies that for sufficiently large $n$, with high
probability Bob’s output states
$\operatorname{Tr}_{E^{n}{E^{\prime}}^{n}}\mathcal{M}^{\otimes n}(u_{(m,s)})$
are distinguishable reliably by a decoding POVM, say with error probability
$\epsilon$. Furthermore, the averaged input states

$\widetilde{\eta}_{m}:=\frac{1}{S}\sum_{s=1}^{S}\mathcal{C}^{n}(u_{(m,s)})\in\mathcal{S}(A^{n}\otimes{A^{\prime}}^{n})$

with high probability have the properties

$\displaystyle\frac{1}{M}\\!\sum_{m=1}^{M}\left\|\widetilde{\eta}_{m}^{{A^{\prime}}^{n}}\\!-(\zeta^{A^{\prime}})^{\otimes
n}\right\|_{1}$ $\displaystyle\leq\epsilon,$ (10)
$\displaystyle\frac{1}{M}\\!\sum_{m=1}^{M}\left\|(\operatorname{Tr}_{BB^{\prime}}\circ(\mathcal{N}\otimes\mathcal{Z}))^{\otimes
n}\widetilde{\eta}_{m}\\!-(\gamma^{EE^{\prime}})^{\otimes n}\right\|_{1}$
$\displaystyle\leq\epsilon,$ (11)

where $\gamma$ is as in Eq. (9). The first comes from the Holevo-Schumacher-
Westmoreland theorem for classical information transmission over a quantum
channel, the second from the matrix tail bounds and the matrix covering lemma
in [1].

By the Fuchs-van-de-Graaf relations between trace distance and fidelity, and
Uhlmann’s theorem (cf. [22, 29]), we can find
$\eta_{m}\in\mathcal{S}(A^{n}\otimes{A^{\prime}}^{n})$ such that
$\eta_{m}^{{A^{\prime}}^{n}}=(\zeta^{A^{\prime}})^{\otimes n}$ for all $m$ and

$\frac{1}{M}\sum_{m=1}^{M}\left\|\widetilde{\eta}_{m}-\eta_{m}\right\|_{1}\leq
4\sqrt{\epsilon},$ (12)

and furthermore

$\frac{1}{M}\\!\sum_{m=1}^{M}\left\|(\operatorname{Tr}_{BB^{\prime}}\circ(\mathcal{N}\otimes\mathcal{Z}))^{\otimes
n}\eta_{m}\\!-(\gamma^{EE^{\prime}})^{\otimes
n}\right\|_{1}\leq\epsilon+4\sqrt{\epsilon}.$ (13)

According to Remark 4, this is equivalent to an assisted wiretap code for
$(\mathcal{N},\zeta)$ with block length $n$,
$\lambda=\mu=\epsilon+4\sqrt{\epsilon}$, concluding the proof.

##  V Conclusion

We have presented a coding strategy for the quantum wiretap channel assisted
by a general quantum correlation, which shows a systematic improvement of the
private communication rate over the unassisted private capacity of the channel
due to the exploitation of the correlation. Our Theorem 5 alos directly
generalises the main result of [12], since classical channels are a special
case of cptp maps and a classical correlation between random variables $X$,
$Y$ and $Z$ is naturally represented by the diagonal state
$\zeta^{A^{\prime}B^{\prime}E^{\prime}}=\sum_{xyz}P_{XYZ}(xyz)\left|x\right\rangle\\!\left\langle
x\right|^{A^{\prime}}\otimes\left|y\right\rangle\\!\left\langle
y\right|^{B^{\prime}}\otimes\left|z\right\rangle\\!\left\langle
z\right|^{E^{\prime}}$. As the proposed code uses a method derived from the
(quantum) Gel’fand-Pinsker wiretap channel with side information, it is also
always at least as good as, or superior to, the “trivial” incorporation of the
correlation resource into an augmented wiretap channel. However, due to the
general need for regularisation, both our main result (Theorem 5) and the
“trivial” achievable rate [Eq. (5)] lead to multi-letter formulas for the same
quantity $P(\mathcal{N},\zeta)$.

Of course, there are extreme cases of useless correlation: for instance any
state $\zeta$ such that Alice and Bob are uncorrelated,
$\zeta^{A^{\prime}B^{\prime}}=\zeta^{A^{\prime}}\otimes\zeta^{B^{\prime}}$,
clearly leads to $P(\mathcal{N},\zeta)=P(\mathcal{N})$, irrespective of the
wiretap channel $\mathcal{N}$. On the other hand, any
$\zeta^{A^{\prime}B^{\prime}}\otimes\left|0\right\rangle\\!\left\langle
0\right|^{E^{\prime}}$ that is not a tensor product between Alice and Bob
offers an advantage for some channel, indeed we can take the quantum broadcast
channel $\mathcal{B}:A\rightarrow B\otimes E$ with qubits $A$, $B$ and $E$,
acting as $\mathcal{B}(\rho)=B\rho B^{\dagger}$,
$B\left|x\right\rangle=\left|x\right\rangle^{B}\left|x\right\rangle^{E}$ for
$x=0,1$. Because $\mathcal{B}$ is both degradable and anti-degradable,
$P(\mathcal{B})=0$. However, $P(\mathcal{B},\zeta)>0$, since $\mathcal{B}$ can
be used to communicate classically and thus Alice and Bob can extract shared
randomness that is automatically a secret key from
$\zeta^{A^{\prime}B^{\prime}}$ [15], which then can be used to one-time-pad a
subsequent classical message.

As indicated in the introduction, various communication problems considered in
the past are special cases of our model, and it is interesting to see to which
extent our main theorem reproduces known results or sheds new light on them.
For instance, the channel without wiretapper, $\mathcal{N}^{A\rightarrow
BE}=\mathcal{N}^{A\rightarrow B}\otimes\left|0\right\rangle\\!\left\langle
0\right|^{E}$, gives rise to all sorts of problems of classical communication
assisted by entanglement. For general point-to-point channel
$\mathcal{N}:A\rightarrow B$ but arbitrary pure state
$\zeta^{A^{\prime}B^{\prime}}\otimes\left|0\right\rangle\\!\left\langle
0\right|^{E^{\prime}}$, we recover the entanglement-assisted classical
capacity $C_{E}(\mathcal{N})$ [4, 5], even though some additional work is
still required to see how Theorem 5 and Eq. (3) give rise to the familiar
maximum quantum mutual information formula from those papers.

If instead our correlation resource is a mixed state, albeit tensor product
with Eve, i.e.
$\zeta^{A^{\prime}B^{\prime}}\otimes\left|0\right\rangle\\!\left\langle
0\right|^{E^{\prime}}$, we get the classical capacity $C(\mathcal{N},\zeta)$
of $\mathcal{N}:A\rightarrow B$ assisted by $\zeta^{A^{\prime}B^{\prime}}$
considered in [3]. In that paper it was observed that a separable
$\zeta^{A^{\prime}B^{\prime}}$ cannot increase the classical capacity of any
channel, $C(\mathcal{N},\zeta)=C(\mathcal{N})$, and the hypothesis was
advanced that for every entangled $\zeta$ there might be a channel such that
$C(\mathcal{N},\zeta)>C(\mathcal{N})$. Noticing that in our present setting,
and due to the triviality of Eve in both the channel and the resource state,
the classical capacities (assisted and unassisted) are equal to private
capacities, suggests a broader interpretation of the question from [3]:
namely, for any state $\rho^{A^{\prime}B^{\prime}}$ and a purification
$\zeta^{A^{\prime}B^{\prime}E^{\prime}}$ of it, is it true that
$\rho^{A^{\prime}B^{\prime}}$ is entangled if and only if there exists a
wiretap channel $\mathcal{N}:A\rightarrow BE$ such that
$P(\mathcal{N},\zeta)>P(\mathcal{N})$?

The special case of the ideal channel
$\mathcal{N}=\operatorname{id}_{A}:A\rightarrow B=A$ merits a separate
mention, too. Its assistance by a general product state
$\zeta^{A^{\prime}B^{\prime}}\otimes\left|0\right\rangle\\!\left\langle
0\right|^{E^{\prime}}$ was considered in multiple works [8, 9, 19, 20, 31,
21], and it was found that
$C(\operatorname{id}_{A},\zeta)\geq\log|A|+I(A\rangle B^{\prime})_{\omega}$,
where
$\omega^{AB^{\prime}}=(\Omega\otimes\operatorname{id}_{B^{\prime}})\zeta$ with
a cptp map $\Omega:A^{\prime}\rightarrow A$. The optimisation of the coherent
information $I(A\rangle B^{\prime})_{\omega}$ over all systems $A$ and
channels $\Omega$ leads to the _dense coding advantage_
$\Delta(A^{\prime}\rangle B^{\prime})_{\zeta}$ of the resource state:

$\Delta(A^{\prime}\rangle B^{\prime})_{\zeta}:=\max_{\Omega\text{
cptp}}I(A\rangle B^{\prime})_{\omega}\text{ s.t.
}\omega^{AB^{\prime}}=(\Omega\otimes\operatorname{id}_{B^{\prime}})\zeta.$

For sufficiently large $A$ system, it follows that
$C(\operatorname{id}_{A},\zeta)=\log|A|+\Delta^{(\infty)}(A^{\prime}\rangle
B^{\prime})_{\zeta}$, where $\Delta^{(\infty)}$ is the regularised dense
coding advantage, cf. [21]:

$\Delta^{(\infty)}(A^{\prime}\rangle
B^{\prime})_{\zeta}=\sup_{n}\frac{1}{n}\Delta({A^{\prime}}^{n}\rangle{B^{\prime}}^{n})_{\zeta^{\otimes
n}}.$

Now this achievability bound and expression for the assisted capacity
correspond exactly to Eq. (3), and interestingly it provides and instance
where Theorem 5 is better, at least on the single-letter level (naturally, in
the regularisation both expression result in the same number). Namely, in [21]
a certain duality relation was observed between the dense coding advantage and
the so-called _entanglement of purification_ $E_{P}$ [27], which extends to
the regularisations of the two quantities:

$E_{P}(C:D)_{\rho}:=\min_{\mathcal{T}:E\rightarrow F}S(BF)_{\omega},$

for a bipartite state $\rho^{CD}$ with purification $\psi^{CDE}$, where the
minimisation is over cptp maps $\mathcal{T}:E\rightarrow F$ and
$\omega^{BF}=(\operatorname{id}_{B}\otimes\mathcal{T})\psi^{BE}$. Namely, for
any purification $\psi^{A^{\prime}B^{\prime}C^{\prime}}$ of
$\zeta^{A^{\prime}B^{\prime}}$,

$\Delta(A^{\prime}\rangle
B^{\prime})+E_{P}(C^{\prime}:B^{\prime})=S(B^{\prime}).$

It remains unknown whether $\Delta=\Delta^{(\infty)}$ and
$E_{P}=E_{P}^{(\infty)}$, but in [11] numerical evidence to the contrary was
provided for the entanglement of purification Showing
$E_{P}(\rho)>E_{P}^{(\infty)}(\rho)$ for a certain state in [11] required
calculating the l.h.s. numerically (which is the evidence for which
mathematical proof is lacking), and finding a new upper bound for the r.h.s.
via an asymptotic protocol using the covering lemma. Concretely, it was shown
for an ensemble $\\{q(u),\rho_{u}^{CD}\\}$ with $\rho=\sum_{u}q(u)\rho_{u}$
that

$E_{P}^{(\infty)}(C:D)_{\rho}\leq\sum_{u}q(u)E_{P}(C:D)_{\rho_{u}}+I(U:CD)_{\gamma},$

where $\gamma^{UCD}=\sum_{u}q(u)\left|u\right\rangle\\!\left\langle
u\right|^{U}\otimes\rho_{u}^{CD}$ is the cq-state of the ensemble. If we us
the duality relation from [21] to translate this protocol to
$\Delta^{(\infty)}$, we can see that it corresponds to the achievability bound
from Theorem 5 in the case $\mathcal{N}=\operatorname{id}_{A}$ and
$\zeta^{A^{\prime}B^{\prime}}$.

## Acknowledgments

MLC was supported by the German Research Foundation (DFG) under the Walter
Benjamin Fellowship CA-2779/1-1. AW is supported by the European Commission
QuantERA grant ExTRaQT (Spanish MICIN project PCI2022-132965), by the Spanish
MICIN (project PID2022-141283NB-I00) with the support of FEDER funds, the
Spanish MICIN with funding from European Union NextGenerationEU (PRTR-C17.I1)
and the Generalitat de Catalunya, by the Alexander von Humboldt Foundation,
and the Institute for Advanced Study of the Technical University Munich.

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# The nonlinear motion of cells subject to external forces

Aondoyima Ioratim-Uba  Aurore Loisy  Silke Henkes  Tanniemola B. Liverpool
School of Mathematics, University of Bristol - Bristol BS8 1UG, UK

###### Abstract

To develop a minimal model for a cell moving in a crowded environment such as
in tissue, we investigate the response of a liquid drop of active matter
moving on a flat rigid substrate to forces applied at its boundaries. Our
model incorporates active stresses due to a prescribed orientation profile of
the cytoskeleton, coupling with the substrate, surface tension and imposed
boundary forces. We find a highly non-linear response to forces that we
characterise using the drop velocity, its shape, and the traction between the
drop and the substrate. There are two main modes of motion: a long and thin
drop with zero traction in the bulk, mostly occurring under strong stretching
forces, and a parabolic drop with finite traction in the bulk, mostly
occurring under strong squeezing forces. There is a sharp transition between
these two modes as a function of the applied forces and indications of drop
break-up where large forces stretch the drop in opposite directions.

Cells are highly adaptable and move themselves around in a variety of
different conditions and environments Alberts (2017); Paluch  _et al._ (2016);
Lämmermann and Sixt (2009). This is essential for biological functions such as
wound repair Chiou and Collins (2018), organ development Lecuit and Lenne
(2007), and in pathological processes such as cancer metastasis Chaffer and
Weinberg (2011). Understanding individual cell motility and how it affects
collective cell migration is key to understanding these processes. Cell
motility is powered by the cytoskeleton, a dynamic network of interlinking
protein filaments inside the cell Mitchison and Cramer (1996); Fletcher and
Mullins (2010); Juelicher  _et al._ (2007). These filaments can collectively
form anisotropic liquid crystalline (LC) phases Marchetti  _et al._ (2013a).

There are a number of mechanisms by which cell motility occurs. The most
studied is cell crawling, which combines the treadmilling
(polymerisation/depolymerisation) of cytoskeletal actin filaments, with strong
adhesion to the substrate, which has been analysed extensively both
experimentally Barnhart  _et al._ (2011); Verkhovsky  _et al._ (1999); Yam
_et al._ (2007); Keren  _et al._ (2008); Anon  _et al._ (2012) and
theoretically using hydrodynamic mechanical models Tjhung  _et al._ (2015);
Blanch-Mercader and Casademunt (2013); Callan-Jones  _et al._ (2008),
reaction-diffusion chemical models Camley  _et al._ (2017); Doubrovinski and
Kruse (2011), and in confinement Lavi  _et al._ (2020); Hawkins  _et al._
(2009). Even in the absence of actin treadmilling, spontaneous motion is still
possible in LC active matter systems Tjhung  _et al._ (2012); Khoromskaia and
Alexander (2015); Giomi and DeSimone (2014); Hawkins  _et al._ (2011); Sanchez
_et al._ (2012). In cells the likely source of this type of motion is acto-
myosin contractility, where myosin II molecular motors cause actin filaments
to slide relative to each other Murrell  _et al._ (2015); Pandya  _et al._
(2017) and generate an active stress that can be contractile, i.e. positive
(or extensile, i.e. negative). Contractility also plays a role in the crawling
of cells Wolgemuth  _et al._ (2011).

A minimal description of cell motility is thus provided by the motion of a
drop of anisotropic active LC matter. The possible modes of a such a drop
freely moving on a flat rigid substrate have recently been classified by some
of us in Loisy  _et al._ (2020). We identified three modes: motion due to
self-advection of active units along their direction of orientation, motion
due to active stresses, and motion due to contact angle mismatch. We also
showed that a drop moving purely due to active stresses can do so without
exerting traction on the substrate Loisy  _et al._ (2019). This type of motion
is relevant to fast migration in crowded environments Paluch  _et al._ (2016);
Liu  _et al._ (2015), where cellular adhesions are unstable at high strain
rates Schwarz and Safran (2013). Crowded environments also lead to significant
forces on cells. However, many of the effects of external forces on cell-
motility and migration remain a mystery. These forces, either coming from
cell-cell tugging or from outside the cell, are important for the
functionality of cells and tissues. Experiments show that external forces can
alter cell stiffness, induce migration, alter cell shape, induce remodelling,
and alter cell phenotype Li  _et al._ (2019); Saez  _et al._ (2007); Ricca
_et al._ (2018). Hence a better handle on them promises to have a significant
impact on our understanding of multicellular systems.

Figure 1: (a) Schematic showing a drop on a flat surface being pushed/pulled
at its boundaries by external forces $f_{\pm}$ (which can also stretch or
squeeze the drop), positive $f_{+}$ or $f_{-}$ means that the force is
directed to the right. The boundaries are set to be at a height $h_{0}$ above
the solid substrate ($z=0$) with corresponding contact angles $\phi_{\pm}$.
The nematic director is denoted $\mathbf{n}$. Inset we indicate regions of the
phase plane that correspond to stretching, squeezing, or pushing in different
directions, where we define push: $\mathcal{F}=f_{+}+f_{-}$ and squeeze:
$\mathcal{S}=-(f_{+}-f_{-})$. (b) Schematic phase diagram for a passive drop.
The light blue region corresponds to RP(L) right-parabola left-moving drops,
green corresponds to LP(R) left-parabola right-moving drops, while droplet
breakup DB is grey. (c) Schematic phase diagram for an active drop. New phases
include the RP(R) right-parabola right-moving state (dark blue), the DH(R)
double-hump right-moving (coral) and the PC(R) pancake right-moving (yellow)
states. Characteristic drop shapes are superimposed in white.

In this letter, therefore, we study the dynamics of an active LC drop on a
flat surface under external forces applied at its two ends. We model the
anisotropy of the cytoskeleton though an imposed LC director field, and the
active stress that this generates. The behaviour is controlled by the ratio
$\mathcal{A}$ of the active stress to the winding number of the LC director,
as measured from the surface to the top of the drop. We classify the motion of
the drop according to (1) the difference between the forces on each end, i.e.
whether it is being squeezed or stretched and (2) the sum of the forces
applied to its two ends, i.e. if it is being pushed to the right (R) or to the
left (L). For a passive drop with zero active stress, $\mathcal{A}=0$, we find
parabolic shapes with simple symmetric behaviour: it moves to the right (or
left) if it is pushed in the right (or left) direction unless it is stretched
above a critical stretching force where it tends to break up into smaller
droplets. For an active drop, $\mathcal{A}>0$ we find a much richer response
to external forces. First, it moves to the R almost all the time except when
the sum of forces is large in the L direction and it is being squeezed. We
also find a wide variety of shapes. Parabolic shapes are observed only when
the the drop is being squeezed. When the drop is being stretched, we find (i)
a double humped shape when the sum of forces is strongly to the R, (ii) a flat
pancake shape which exerts almost no traction on the surface when the sum of
forces is small and (iii) droplet break up when the sum of forces is strongly
in the L direction. Changing the sign of $\mathcal{A}$, equivalent to either
flipping the winding number or moving from contractile to extensile stresses,
flips the directions L $\leftrightarrow$ R.

We model a single cell as a two dimensional incompressible active nematic drop
moving on a flat rigid substrate subject to external forces at its boundaries
(Fig. 1 (a)). We work in the $(\tilde{x},\tilde{z})$ plane, where the drop is
characterised by the height $\tilde{h}(\tilde{x},\tilde{t})$ of its free
surface above the substrate, and moves with velocity $\tilde{V}$. We use the
well established equations of active liquid crystal hydrodynamics Marchetti
_et al._ (2013b); Jülicher  _et al._ (2018) where the motion of the coarse
grained orientation of elongated units $\mathbf{n}=(\cos\theta,\sin\theta)$,
known as the director, is coupled to the fluid velocity $\tilde{\mathbf{u}}$
inside the drop. The velocity satisfies force balance equations at vanishing
Reynolds number:

$\partial_{j}\tilde{\sigma}_{ij}+\tilde{f}_{i}=0,$ (1a)
$\tilde{\sigma}_{ij}=-\tilde{p}\delta_{ij}+\eta(\partial_{i}\tilde{u}_{j}+\partial_{j}\tilde{u}_{i})-\tilde{\alpha}n_{i}n_{j},$
(1b)

where $\tilde{\mathbf{f}}=\tilde{\mathbf{f}}(\tilde{x})$ is the external force
per unit height, $\tilde{p}$ is the pressure, $\tilde{\mathbf{u}}$ is the
fluid velocity inside the drop, and $\tilde{\alpha}$ is the active stress
which represents the coarse-grained stresses generated when cytoskeletal actin
filaments slide relative to each other Hatwalne  _et al._ (2004); Liverpool
and Marchetti (2006). Following Loisy  _et al._ (2019, 2020), we keep the
lowest order gradients in the stress tensor and we work in the strong elastic
limit, where dynamic equation for the director reduces to
$\mathbf{\nabla}^{2}\theta$. We anchor the director parallel to the substrate,
i.e. $\theta(\tilde{z}=0)=0$, and parallel to the free surface with an imposed
winding, i.e. $\theta(\tilde{z}=\tilde{h})=\omega\pi+\arctan(h^{\prime})$. The
winding number $\omega\in\mathbb{Z}^{+}$ counts the number of half turns of
the director across the drop height (Fig. 1). At fixed activity, the winding
of the director, through its chirality, breaks the left-right symmetry and
sets the preferred direction of motion. With no external forces, at positive
activity (extensile drop), the drop moves right for positive winding and left
for negative winding. There is no active contribution to the drop velocity
when $\omega=0$.

The external forces
$\tilde{f}_{\tilde{x}}=\tilde{f}_{+}\delta(\tilde{x}-\tfrac{\tilde{L}}{2})+\tilde{f}_{-}\delta(\tilde{x}+\tfrac{\tilde{L}}{2})$,
$\tilde{f}_{\tilde{z}}=0$ are localised at the left and right drop borders,
which leads to boundary conditions on the pressure
$\tilde{p}(\pm\tilde{L}/2)=\tilde{\pi}_{0}\mp\tilde{f}_{\pm}$, where
$\tilde{\pi_{0}}=2\gamma\tilde{\phi}/\tilde{L}_{0}$ is the Laplace pressure
generated by the surface tension $\gamma$ of a symmetric passive drop of
length $\tilde{L}_{0}$ and contact angle $\tilde{\phi}$. For an active drop
with no external forces, $\pi_{0}$ is a constant shift in the pressure. These
boundary conditions can be derived by integrating the $\tilde{x}$ component of
(1a) with respect to $\tilde{x}$ from $\tilde{L}/2-\Delta$ to
$\tilde{L}/2+\Delta$ and from $-\tilde{L}/2-\Delta$ to $-\tilde{L}/2+\Delta$,
with $\Delta>0$, and taking the limit $\Delta\to 0^{+}$. In addition to these
boundary conditions, we use a free surface boundary condition for the stress
at the free surface: $\tilde{\bm{\sigma}}\cdot\bm{m}=\gamma\kappa\bm{m}$,
where $\gamma$ is the surface tension, $\bm{m}$ is the unit normal vector (see
Fig 1(a)), and $\kappa=\bm{\nabla}\cdot\bm{m}$ is the curvature of the free
surface. The interaction of the drop with the rigid substrate is modelled by a
partial slip boundary condition:
$\tilde{u}_{\tilde{x}}=\tilde{l}_{u}\tilde{\sigma}_{\tilde{x}\tilde{z}}/\eta$,
where $\tilde{l}_{u}$ is a slip length. We perform calculations in the
lubrication approximation Oron  _et al._ (1997) for which changes in the
height are much smaller than the width.

Figure 2: (a) Velocity of a passive drop. (b) Velocity of an active drop. (c)
2nd moment $\int_{-L/2}^{L/2}(x-\mu_{1})^{2}h(x)dx$ of an active drop, where
$\mu_{1}=\int_{-L/2}^{L/2}xh(x)dx$. (d) Traction between the drop and the
substrate integrated over $70\%$ of the drop length. Each square represents a
single simulation and its colour corresponds to the legend. The solid line in
each plot is an isoline corresponding to zero velocity, the dashed line is an
isoline corresponding to the the 1st moment $\mu_{1}=0$, and the dotted line
is an isoline corresponding to the drop having equal contact angles. (e) 2nd
moment and velocity $\mathcal{V}$ across the RP(R)/RH(R) boundary, indicating
a first order transition. (f) Stable numerical solutions for $h$ (top) and the
corresponding traction $\sigma_{xz}|_{z=0}$ (bottom) for $\mathcal{A}=1$,
$\mathcal{S}$, (g) $\mathcal{A}=1$, $\mathcal{F}=0$, (h) $\mathcal{V}<2\times
10^{-4}$. The colour of each curve matches the colour of its phase in Fig. 1
(c). The legend labels are coordinates in the $(\mathcal{F},\mathcal{S})$
phase plane.

We derive an integro-differential equation for the steady state drop height
$\tilde{h}$ by seeking travelling wave solutions
$\tilde{h}(\tilde{x}-\tilde{V}t)$ which must satisfy, from mass conservation,
$\int_{0}^{\tilde{h}}\tilde{u}_{\tilde{x}}d\tilde{z}=\tilde{V}\tilde{h},$ and
then using the relation $\tilde{p}=-\gamma\tilde{h}^{\prime\prime}$, which can
be derived by taking the leading order terms of the normal component of the
stress boundary condition at the drop free surface. We then convert the
resulting integro-differential equation into a non-linear ODE. After non-
dimensionalisation, we retain the salient parameters activity $\mathcal{A}$
and drop velocity $\mathcal{V}$,

$\mathcal{A}=\frac{\tilde{\alpha}\tilde{L}_{x}}{4\pi\omega\gamma\epsilon^{2}},\quad\mathcal{V}=\frac{\eta\tilde{V}}{\gamma\epsilon^{3}},$
(2)

and also height $h=\tilde{h}/\epsilon\tilde{L}_{x}$, coordinates
$x=\tilde{x}/\tilde{L}_{x}$, $z=\tilde{z}/\epsilon L_{x}$, slip length
$l_{u}=\tilde{l}_{u}/\epsilon L_{x}$, pressure
$\pi_{0}=\tilde{\pi}_{0}\tilde{L}_{x}/\gamma\epsilon$, and forces
$f_{\pm}=\tilde{f}_{\pm}\tilde{L}_{x}/\gamma\epsilon$, where $\tilde{L}_{x}$
is a characteristic length scale in the $x$ direction. Then the drop shape
satisfies

$-h^{\prime\prime}=C+\frac{1}{2}\bigg{(}\int_{\tfrac{-L}{2}}^{x}\frac{\mathcal{A}h+\mathcal{V}}{\tfrac{h^{2}}{3}+l_{u}h}d\xi-\int_{x}^{\tfrac{L}{2}}\frac{\mathcal{A}h+\mathcal{V}}{\tfrac{h^{2}}{3}+l_{u}h}d\xi\bigg{)},$
(3)

where $C$ and the drop velocity $\mathcal{V}$ are integration constants
determined by the boundary conditions on pressure to be
$C=\pi_{0}+\mathcal{S}/2$, and

$\mathcal{V}=\left[\int_{\tfrac{-L}{2}}^{\tfrac{L}{2}}\frac{\mathcal{A}\>dx}{\tfrac{h}{3}+l_{u}}+\mathcal{F}\right]\bigg{/}\left[\int_{\tfrac{-L}{2}}^{\tfrac{L}{2}}\frac{dx}{\tfrac{h^{2}}{3}+l_{u}h}\right],$
(4)

where we have introduced squeeze $\mathcal{S}=-(f_{+}-f_{-})$ and push
$\mathcal{F}=f_{+}+f_{-}$ (pull corresponds to negative push, stretch to
negative squeeze). Because equation (3) is a second order equation, we must
impose two boundary conditions. Here, we choose to set the height of the drop
at its boundary to a finite $h_{0}=h(\pm L/2)$, which represents a finite
contact area with neighbouring cells or with obstacles. Then equation (3) can
be differentiated with respect to $x$ to obtain

$h^{\prime\prime\prime}\bigg{(}\frac{1}{3}h^{3}+l_{u}h^{2}\bigg{)}+\mathcal{A}h^{2}=\mathcal{V}h.$
(5)

The drop length $L$ is determined as part of the solution by requiring that
the drop have constant volume $\Omega$: $\int_{-L/2}^{L/2}hdx=\Omega$. The
tangential traction exerted by the drop on the substrate is
$\sigma_{xz}|_{z=0}=hh^{\prime\prime\prime}$. In terms of $h$ alone, using
(5), the traction can be written
$\sigma_{xz}|_{z=0}=(\mathcal{V}-\mathcal{A}h)/(\tfrac{1}{3}h+l_{u})$.

We obtain solutions to (5) by numerically solving the time dependent version
of the force balance equation, $\partial_{t}h+\partial_{x}I=0$, where
$I=h^{\prime\prime\prime}(\tfrac{1}{3}h^{3}+l_{u}h^{2})+\mathcal{A}h^{2}-\mathcal{V}h$,
which represents overdamped dynamics of the height evolution. Solving the time
dependent problem (starting from a known initial condition) allows us to
iteratively evaluate the integrals in equation (4) and gives us the potential
to investigate the temporal dynamics of the drop. The steady state of the time
evolution, $\partial_{t}h=0$, is equivalent to equation (5). We simulate the
dynamics until steady state using a Crank-Nicholson algorithm with adaptive
time steeping, where the time step is increased as steady state approaches
Loisy  _et al._ (2020). In the numerics, we apply both height and pressure
boundary conditions (converted to boundary conditions on $h^{\prime\prime}$)
because the time evolution is a fourth order PDE:

$h(\pm L/2)=h_{0},\quad h^{\prime\prime}(\pm L/2)=\pm f_{\pm}-\pi_{0}.$ (6)

The boundary conditions on $h^{\prime\prime}$ are required for consistency
with force balance, however we are in principle free to change the boundary
conditions on $h$. The third derivative term is discretised implicitly, while
the other terms are explicitly discretised using finite difference
coefficients (see SI at SI  for further details). In all simulations
$l_{u}=0.05$, $h_{0}=0.1$, $\Omega=1$, $\pi_{0}=0.9226$, $\omega=1$, and the
spatial step size is $0.002$.

Figure 3: (a) Relative error between numerical solution and analytic solution
at $x=0$ as a function of $\mathcal{A}$ for
$(\mathcal{F},\mathcal{S})=(-0.007,0.005)$ (circles), $(-0.070,0.050)$
(diamonds), and $(-0.700,0.500)$ (squares). The diamond coloured in blue at
$\mathcal{A}=0.004$ corresponds to the profile plotted in (b).

The phase diagrams in Fig. 1b-c schematically show the regimes of drop shapes
and motility that we find for the passive $\mathcal{A}=0$ and the active
$\mathcal{A}=1$ case. Figure 2 provides full results for drop velocity, drop
shapes and substrate traction.

We first study a passive drop $\mathcal{A}=0$ (Fig. 1 (b)) and observe two
phases of motion, both with finite traction in the bulk: a left-moving right-
parabola RP(L) for $\mathcal{F}<0$ and a right-moving left-parabola LP(R) for
$\mathcal{F}>0$. These passive drops are qualitatively close in shape to an
upside down parabola but are asymmetric with a first moment of $h$ that is
non-zero (characteristic shapes shown in Fig. 1 (b)). The passive solutions
are antisymmetric along the push/pull $\mathcal{F}$ axis, and they move with
opposite velocities that increase as the drop is squeezed (see Fig. 2(a)). All
drops become longer and thinner as we move along the stretch/squeeze axis from
squeeze to stretch, eventually reaching the region of drop breakup (DB), where
the drop free surface reaches $h<0$. For passive drops, we are able to show
analytically (see SI at SI ) that for $\mathcal{F}=0$, in this region no
steady solutions exist that satisfy volume conservation. This argument can be
extended to linear order in activity.

With activity, as shown in shown in Fig. 1 (c), we find, in addition to RP(L),
LP(R) and DB, a right-moving right-parabola RP(R) state with finite traction,
and then two long and thin states (see 2nd moment in Figs. 2 (c)), the right-
moving double hump DH(R) state, which has a small dip followed by a prominent
frontal hump, and the right-moving pancake PC(R) drop, which is flat almost
everywhere with the average height $\bar{h}<h_{0}$. Both DH(R) and PC(R) drops
have zero traction in the bulk (Fig. 2 (d)). The tractionless drops that we
studied in Loisy  _et al._ (2019) can be found in the DH(R) region of the
phase plane along the line of equal contact angle (dotted line, Figs. 2 (b) -
(d)). The symmetry about the push/pull axis is broken because the drop is
motile at zero force due to the imposed winding $\omega=1$ (Figs. 2 (b)). Both
the RP(L) and LP(R) regions become smaller with activity and are pushed
towards the top-left (push-left/squeeze) and top-right (push-right/squeeze) of
the phase plane respectively by the emergence of the RP(R) phase, which
largely occupies the middle squeeze region that lies between RP(L) and LP(R).

The stretch region of the phase plane is populated by DH(R) and PC(R), as well
as DB. The region containing DH(R) and PC(R) shares a phase boundary with
RP(R) determined by the second moment of $h$ as a measure of the spread of the
drop, and the distribution of substrate traction. The phase boundary is
indicated in the schematic Fig. 1, as determined from the data in Figs. 2 (c)
and (d). Strikingly, the transition between RP(R) and DH(R)/PC(R) is sharp
suggesting a first order transition. The behaviour of the second moment and
the drop velocity across the phase boundary is shown in Fig. 2 (e). There is a
jump in both quantities going from RP(R) to DH(R) and also from DH(R) to
RP(R). The change in drop shape induced by moving from RP(R) to DH(R) and back
to RP(R) at constant squeeze $\mathcal{S}=2$ is shown if Fig. 2 (f). The drop
shape changes continuously within the DH(R) region but changes sharply at the
phase boundary. There are also indications of bi-stability in the region: we
have obtained two stable solutions for two different initial conditions at the
same point in the phase plane (see SI at SI ). In contrast, the transition
between DH(R) and PC(R) is smooth and happens under increased stretching for
constant push (Fig. 2 (g)).

The drastic changes in drop shape that we have observed are not associated
with changes in direction of motion. The active drop remains a right-parabola
when its velocity is reversed at the RP(L)/RP(R) boundary. Stationary drop
profiles are shown in Fig. 2 (h). Both passive and active drops are fastest
for strong push and strong squeeze, towards the top corners of the phase
plane. The drop slows down dramatically on entering the DH(R) region and
continues to slow down as it enters the PC(R) region and approaches DB,
however it does not stop before reaching DB.

For values of $\mathcal{A}$ and $f_{\pm}$ close to zero, the numerical
solution is in good agreement with the analytic solution at linear order in
$\mathcal{A}$ and $f_{\pm}$. For constant squeeze and constant push, the
relative error between the numerical solution and analytic solution decreases
with activity (Fig. 3). The asymptotic solution remains a good approximation
($\sim 10\%$ relative error) when $\mathcal{A}$ is of $\mathcal{O}(1)$ but
only if drop is in one of the parabolic phases. The analytic solution ceases
to be a good approximation for the more complex DH(R) and PC(R) drops.

In conclusion, we have studied the dynamics of a cell (an active LC drop) on a
flat surface under external forces applied at its two ends. We have produced a
phase diagram in terms of the sum and differences of these forces and have
shown how cell shape and motile behaviour is nonlinearly modulated by external
forces. Our analysis has focused on one mechanism of motility driven by
contractile/extensile active stresses. We expect that including other
mechanisms Loisy  _et al._ (2020) will change the locations of boundaries
between the different phases we have observed. Clearly the changes in shape
that we have observed here can be exploited to control behaviour of different
cell phenotypes and tissue remodelling. The relation that we have derived
between cell-substrate traction and forces, and between cell velocity and
traction could be investigated by measuring the cell-substrate force using
e.g. traction force microscopy and pushing/pulling the cell with a
micropipette.

###### Acknowledgements.

Part of this work was funded by a Leverhulme Trust Research Project Grant
RPG-2016-147. TBL acknowledges support of BrisSynBio, a BBSRC/EPSRC Advanced
Synthetic Biology Research Centre (grant number BB/L01386X/1). SH acknowledges
support of BBSRC grant BB/N009150/2.

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The nonlinear motion of cells subject to external forces: Supplemental
Material

##  I Additional phase diagrams

Figure S1: First row: passive drop ($\mathcal{A}=0$), second row: intermediate
activity ($\mathcal{A}=0.3$), third row: high activity ($\mathcal{A}=1$).
First column: drop velocity. Second column: 2nd moment
$\int_{-L/2}^{L/2}(x-\mu_{1})^{2}h(x)dx$, where
$\mu_{1}=\int_{-L/2}^{L/2}xh(x)dx$. Third column: raction between the drop and
the substrate integrated over $70\%$ of the drop length. Fourth column: left
contact angle $\phi_{-}=h^{\prime}(-L/2)$, fifth column: right contact angle
$\phi_{+}=h^{\prime}(L/2)$. The solid line in each plot is an isoline
corresponding to zero velocity, the dotted line is an isoline corresponding to
the drop having equal contact angles, and the dashed line is an isoline
corresponding to the 1st moment $\mu_{1}=0$. The high activity phase diagrams
have a $\mu_{1}=0$ isoline in the DH(R)/PC(R) region but this is not included
here.

##  II Deriving the height equation

We begin with the force balance equation in the lubrication approximation

$\displaystyle\partial_{\tilde{x}}\tilde{p}-\tilde{f}_{\tilde{x}}$
$\displaystyle=\eta\partial^{2}_{\tilde{z}}\tilde{u}-\tilde{\alpha}\partial_{\tilde{z}}(n_{x}n_{z})=\partial_{\tilde{z}}\tilde{\sigma}_{\tilde{x}\tilde{z}},$
(S1a) $\displaystyle\partial_{\tilde{z}}\tilde{p}$ $\displaystyle=0,$ (S1b)

where we have chosen

$\displaystyle\tilde{f}_{\tilde{x}}=\tilde{f}_{+}\delta(\tilde{x}-\tfrac{\tilde{L}}{2})+\tilde{f}_{-}\delta(\tilde{x}+\tfrac{\tilde{L}}{2}),$
$\displaystyle\tilde{f}_{\tilde{z}}=0.$ (S2)

For here, we will omit the tildes, but note that the quantities in this
section are not non-dimensionalised. The strategy will be to first integrate
(S1a) at $\pm L/2$ to get boundary conditions on $p(\pm L/2)$ imposed by the
Dirac deltas. We will then solve (S1a) away from $x=\pm L/2$, and apply the
derived boundary conditions at $\pm L/2$.

###  II.1 Integrating at the boundaries

At $x=L/2$, we have

$\int_{\tfrac{L}{2}-\Delta}^{\tfrac{L}{2}+\Delta}\partial_{x}pdx-
f_{+}\int_{\tfrac{L}{2}-\Delta}^{\tfrac{L}{2}+\Delta}\delta(x-\tfrac{L}{2})dx-
f_{-}\int_{\tfrac{L}{2}-\Delta}^{\tfrac{L}{2}+\Delta}\delta(x+\tfrac{L}{2})dx=\int_{\tfrac{L}{2}-\Delta}^{\tfrac{L}{2}+\Delta}\partial_{z}\sigma_{xz}dx.$

The last term on the LHS vanishes because of the definition of the delta
function and the term on the RHS vanishes as we shrink the integration region
($\Delta\rightarrow 0^{+}$) because the integrand is continuous. The procedure
is identical at $x=-L/2$. Thus we have

$\displaystyle p(\tfrac{L}{2}+\Delta)-p(\tfrac{L}{2}-\Delta)-f_{+}=0,$
$\displaystyle p(-\tfrac{L}{2}+\Delta)-p(-\tfrac{L}{2}-\Delta)-f_{-}=0.$

We model the system to have a uniform pressure $\pi_{0}+\pi_{ref}$, where
$\pi_{ref}$ is an arbitrary reference pressure while $\pi_{0}$ is the uniform
pressure that is present in the absence of activity $\alpha$ and external
forces $F_{\pm}$, where the drop is stationary and symmetric. Because the
external force is localised to $x=\pm L/2$ and we take the pressure to be
uniform everywhere else outside the drop, it must be the case that
$p(\tfrac{L}{2}+\Delta)=p(-\tfrac{L}{2}-\Delta)=\pi_{0}+\pi_{ref}$. We also
define

$\displaystyle p(L/2)=\lim_{\Delta\to 0^{+}}p(L/2-\Delta),$ $\displaystyle
p(-L/2)=\lim_{\Delta\to 0^{+}}p(-L/2+\Delta),$

resulting in the boundary conditions

$\displaystyle p(\tfrac{L}{2})$ $\displaystyle=\pi_{0}+\pi_{ref}-f_{+},$ (S3a)
$\displaystyle p(-\tfrac{L}{2})$ $\displaystyle=\pi_{0}+\pi_{ref}+f_{-}.$
(S3b)

Solving equation (S1a) is now equivalent to solving

$\partial_{x}p=\partial_{z}\sigma_{xz}$ (S4)

with boundary conditions (S3).

###  II.2 Pressure as a functional of drop height

The pressure $p(x)$ inside the drop can be related to the drop height $h(x)$
using the normal component of the free surface boundary condition (see main
text)

$\bm{m}\cdot\bm{\sigma}\cdot\bm{m}=\gamma\kappa-\pi_{ref}.$

Using the scalings from the lubrication approximation,
$\alpha\sim\epsilon^{-1}$, $p\sim\epsilon^{-2}$, $u\sim 1$, and
$w\sim\epsilon$ (from incompressibility),
$\sigma_{xx}=\sigma_{zz}\approx-p\sim\epsilon^{-2}$,
$\sigma_{xz}\sim\epsilon^{-1}$, we can write the LHS of the above equation as

$\displaystyle\begin{split}\bm{m}\cdot\bm{\sigma}\cdot\bm{m}&=m_{x}\sigma_{xz}m_{z}+m_{z}\sigma_{xz}m_{x}+m_{z}\sigma_{zz}m_{z}+m_{x}\sigma_{xx}m_{x}\\\
&\approx\frac{1}{1+(h^{\prime})^{2}}(-p(h^{\prime})^{2}-p-2h^{\prime}\sigma_{xz})\\\
&=-p-2\frac{h^{\prime}\sigma_{xz}}{1+(h^{\prime})^{2}}\\\
&\approx-p.\end{split}$

Approximating $\kappa$ to leading order, we have

$\displaystyle\begin{split}\kappa&=\frac{h^{\prime\prime}}{(1+(h^{\prime})^{2})^{\frac{3}{2}}}\\\
&\approx h^{\prime\prime}.\end{split}$

Thus

$p(x)=-\gamma h^{\prime\prime}+\pi_{ref}.$ (S5)

###  II.3 Height equation

To get and ODE for $h(x)$ we apply a travelling wave condition to the
kinematic boundary condition at the free surface of the drop. Because of the
nature of the problem, we first need to integrate equation (S4) in order to
apply the boundary conditions on the pressure (S3). After doing this, we use
the relation between the pressure and the second derivative of $h(x)$ (S5) to
obtain an integro-differential equation for $h(x)$. Finally, we differentiate
said integro-differential equation to obtain an ordinary differential equation
for $h(x)$.

The kinematic boundary boundary condition is written

$\partial_{t}h+u\partial_{x}h-w=0,$

and it comes from requiring $\tfrac{D}{Dt}(h(x,t)-z)$ = 0, where
$\tfrac{D}{Dt}=\partial_{t}+\mathbf{u}\cdot\mathbf{\nabla}$. We impose a
travelling wave solution $h=h(x-Vt)$, where $V$ is the unknown constant drop
velocity. Substituting the travelling wave condition into the kinematic
boundary condition gives the expression

$\int_{0}^{h(x)}u\,dz=Vh,$ (S6)

where we have made the transformation $x\leftarrow x-Vt$, so that now $x$ is
the centre of mass coordinate. To obtain the fluid velocity field $u$, we
integrate (S4) twice with respect to $z$ and use the partial slip boundary
condition at the substrate and the tangential component of the free surface
boundary condition (see main text) (which can be re-written as a condition on
$\partial_{z}u(z=h)$). The fluid velocity field $u$ is then given by

$u=\frac{\alpha h}{4\pi\omega\eta}\bigg{(}1-\cos{\frac{2\omega\pi
z}{h}}\bigg{)}+\frac{\partial_{x}p}{\eta}\bigg{(}\frac{z^{2}}{2}-h(z+l_{u})\bigg{)}.$
(S7)

Substituting (S7) into (S6) and isolating the pressure gradient yields

$\partial_{x}p=\frac{\eta(\tilde{\alpha}h-V)}{\frac{1}{3}h^{2}+l_{u}h},$ (S8)

where $\tilde{\alpha}=\frac{\alpha}{4\pi\omega\eta}$. From here the strategy
will be to integrate (S8) and apply the boundary conditions (S3) in order to
determine the integration constant and the unknown drop velocity $V$. We find
that the drop velocity is given by

$\eta
V=\frac{\int_{-\tfrac{L}{2}}^{\tfrac{L}{2}}\frac{\eta\tilde{\alpha}}{\frac{1}{3}h+l_{u}}\,dx+(f_{+}+f_{-})}{\int_{-\tfrac{L}{2}}^{\tfrac{L}{2}}\frac{1}{\tfrac{1}{3}h^{2}+l_{u}h}\,dx}.$
(S9)

As a sanity check, we see that the second term in the numerator vanishes when
the forces are equal and opposite, which means that the drop velocity in
unchanged by the forces when they cancel each other out. This is good. The
drop velocity also vanishes when the activity $\alpha$ and both forces vanish,
which is good. After some algebra, and taking advantage of the fact that we
can obtain two equivalent expressions for pressure by integrating (S8) from
$-\tfrac{L}{2}$ to $x$ or from $x$ to $\tfrac{L}{2}$, we arrive find that the
pressure is given by

$\begin{split}p(x)&=\pi_{0}+\pi_{ref}-\frac{f_{+}-f_{-}}{2}+\frac{1}{2}\int_{-L/2}^{x}\frac{\eta\tilde{\alpha}}{\frac{1}{3}h+l_{u}}\,dy+\frac{1}{2}\int_{L/2}^{x}\frac{\eta\tilde{\alpha}}{\frac{1}{3}h+l_{u}}\,dy\\\
&-\frac{1}{2}\bigg{(}\frac{f_{+}+f_{-}+\eta\tilde{\alpha}I_{1}}{I_{2}}\bigg{)}\bigg{(}\int_{-L/2}^{x}\frac{1}{\frac{1}{3}h^{2}+l_{u}h}\,dy+\int_{L/2}^{x}\frac{1}{\frac{1}{3}h^{2}+l_{u}h}\,dy\bigg{)},\end{split}$
(S10)

where

$\displaystyle I_{1}=\int_{-L/2}^{L/2}\frac{1}{\frac{1}{3}h+l_{u}}\,dy,$
$\displaystyle I_{2}=\int_{-L/2}^{L/2}\frac{1}{\frac{1}{3}h^{2}+l_{u}h}\,dy.$

Using the relation between the pressure and the second derivative of the drop
height (S5), we can write (S10) as an integro-differential equation that is
independent of the reference pressure

$\begin{split}-\gamma
h^{\prime\prime}&=\pi_{0}-\frac{f_{+}-f_{-}}{2}+\frac{1}{2}\int_{-L/2}^{x}\frac{\eta\tilde{\alpha}}{\frac{1}{3}h+l_{u}}\,dy+\frac{1}{2}\int_{L/2}^{x}\frac{\eta\tilde{\alpha}}{\frac{1}{3}h+l_{u}}\,dy\\\
&-\frac{1}{2}\bigg{(}\frac{f_{+}+f_{-}+\eta\tilde{\alpha}I_{1}}{I_{2}}\bigg{)}\bigg{(}\int_{-L/2}^{x}\frac{1}{\frac{1}{3}h^{2}+l_{u}h}\,dy+\int_{L/2}^{x}\frac{1}{\frac{1}{3}h^{2}+l_{u}h}\,dy\bigg{)}.\end{split}$
(S11)

Finally, we can obtain an ordinary differential equation by differentiating
(S11) to obtain

$\frac{\gamma
h^{\prime\prime\prime}}{\eta}\bigg{(}\frac{1}{3}h^{3}+l_{u}h^{2}\bigg{)}+\frac{\alpha
h^{2}}{4\pi\omega\eta}=\frac{1}{\eta}\bigg{(}\frac{I_{1}+(f_{+}+f_{-})}{I_{2}}\bigg{)}h,$
(S12)

where the drop velocity $V$ is the pre-factor of $h$ on the RHS.

##  III Solving the height equation with the Crank-Nicholson method

The equation that is fed into the Crank-Nicholson algorithm is

$\partial_{t}h+\partial_{x}(-\mathcal{V}h+h^{\prime\prime\prime}(\tfrac{1}{3}h^{3}+l_{u}h^{2})+\mathcal{A}h^{2})=0,$
(S13)

where the drop velocity $\mathcal{V}$ is defined in the main text. Equation
(S13) is subject to the constraints

$\displaystyle\int_{-L/2}^{L/2}h(x)\,dx=\Omega,$ $\displaystyle h(\pm
L/2)=h_{0},$ (S14)

where $\Omega$ is the dimensionless drop volume, and

$\displaystyle h^{\prime\prime}(-L/2)+\pi_{0}+f_{-}=0,$ $\displaystyle
h^{\prime\prime}(L/2)+\pi_{0}-f_{+}=0,$ (S15)

with

$\pi_{0}=\frac{2\phi}{-\frac{3h_{0}}{\phi}+\sqrt{\frac{9h_{0}^{2}}{\phi^{2}}+\frac{6\Omega}{\phi}}},$

where $\phi$ is the re-scaled contact angle.

Equation (S13) is of the form $\partial_{t}h=I$, where $I$ represents the
second term on the LHS of (S13). The Crank-Nicholson scheme advances in time
according to

$\frac{h_{i}^{n+1}}{\Delta
t}-\frac{1}{2}I_{i}(\mathbf{X}^{n+1})=\frac{h_{i}^{n}}{\Delta
t}+\frac{1}{2}I_{i}(\mathbf{X}^{n}),$ (S16)

where the subscripts refer to the spatial discretisation and the superscripts
refer to the time discretisation. Note that in the second term on the LHS,
$I_{i}(\mathbf{X}^{n+1})$, means $I$ evaluated at spatial point $i$ at time-
step $n+1$.

###  III.1 Spatial discretisation

For the discretisation we use the substitution $x=Ly$, with $L$ being the drop
length, and discretise the domain $y\epsilon[-0.5,0.5]$ with uniform grid
spacing $\Delta y$. The stiff term is discretised as follows

$\partial_{x}\bigg{(}h^{\prime\prime}\bigg{(}\frac{1}{3}h^{3}+l_{u}h^{2}\bigg{)}\bigg{)}\rightarrow\frac{\bigg{[}(\frac{1}{3}h_{i}^{3}+l_{u}h_{i}^{2})+(\frac{1}{3}h_{i+1}^{3}+l_{u}h_{i+1}^{2})\bigg{]}(h_{i+2}-3h_{i+1}+3h_{i}-h_{i-1})}{2L^{4}\Delta
y^{4}}\\\
-\frac{\bigg{[}(\frac{1}{3}h_{i-1}^{3}+l_{u}h_{i-1}^{2})+(\frac{1}{3}h_{i}^{3}+l_{u}h_{i}^{2})\bigg{]}(h_{i+1}-3h_{i}+3h_{i-1}-h_{i-2})}{2L^{4}\Delta
y^{4}},$

and all other terms are discretised as

$\partial_{x}g(h)\rightarrow\frac{g(h_{i+1})-g(h_{i-1})}{2L\Delta y},$

where $g(h)=Ah^{2}$ for the active term, and
$g(h)=\frac{AI_{1}+(f_{+}+f_{-})}{I_{2}}h$ for the velocity term.

###  III.2 Algorithm and boundary conditions

Equation (S16) leads to a set of nonlinear algebraic equations for
$\\{h_{i}^{n+1}\\}$, where $\\{h_{i}^{n}\\}$ are known, which is solved using
the Matlab fsolve algorithm. For $N$ spatial grid points, define
$\mathbf{X}^{n+1}=(h_{1},...,h_{N})^{n+1}$. The algorithm solves
$\mathbf{F}(\mathbf{X}^{n+1})=\mathbf{0}$ with

$\displaystyle\mathbf{F}(\mathbf{X}^{n+1})=\frac{\mathbf{X}^{n+1}}{\Delta
t}-\frac{1}{2}\mathbf{I}(\mathbf{X}^{n+1})-\frac{\mathbf{X}^{n}}{\Delta
t}-\frac{1}{2}\mathbf{I}(\mathbf{X}^{n})$ $\displaystyle i=3,...,N-2,$ (S17)

where $\mathbf{I}=(I_{1},...,I_{N})$, with $I_{i}$ being the spatial
differential operator evaluated at spatial point $i$. The boundary conditions
are implemented using grid points $1,2,N-1,N$. The condition on the drop
height at $\pm L/2$ is implemented as

$\displaystyle F_{1}=h_{1}^{n+1}-h_{0}$ (S18a) $\displaystyle
F_{N}=h_{N}^{n+1}-h_{0}.$ (S18b)

The boundary conditions on the second derivative (S15) are implemented using
finite difference coefficients to approximate the second derivative:

$\displaystyle
F_{2}=\frac{2h_{1}^{n+1}-5h_{2}^{n+1}+4h_{3}^{n+1}-h_{4}^{n+1}}{\Delta
y^{2}}+L^{2}\bigg{(}\frac{\pi_{0}+f_{-}}{\tilde{C}}\bigg{)},$ (S19a)
$\displaystyle
F_{N-1}=\frac{2h_{N}^{n+1}-5h_{N-1}^{n+1}+4h_{N-2}^{n+1}-h_{N-3}^{n+1}}{\Delta
y^{2}}+L^{2}\bigg{(}\frac{\pi_{0}-f_{+}}{\tilde{C}}\bigg{)}.$ (S19b)

Once equation (S13) is discretised, we use a nonlinear solver (MATLAB’s
“fsolve”) for the simultaneous equations
$\mathbf{F}(\mathbf{X}^{n+1})=\mathbf{0}$, with the components of $\mathbf{F}$
given by (S17), (S18), and (S19). We also calculate the Jacobian $\partial
F_{i}/\partial_{h_{j}}$ explicitly and supply it to the nonlinear solver. The
algorithm begins with a user set initial condition $\mathbf{X}^{1}$ which is
chosen to satisfy the boundary conditions on drop height but not necessarily
the boundary conditions on the second derivative. The drop velocity
$\mathcal{V}$ as well as the drop length $L$ are calculated iteratively
starting from the initial condition (the drop length is derived from the
constraint that the drop has a constant volume) and plugged into the Crank-
Nicholson evolution equation which is solved for $\mathbf{X}^{2}$. This
process is repeated until steady state is reached i.e. the difference between
$\mathbf{X}^{n+1}$ and $\mathbf{X}^{n}$ is less than some tolerance.

##  IV Asymptoics

We expand the equation

$h^{\prime\prime\prime}(\tfrac{1}{3}h^{3}+l_{u}h^{2})+\mathcal{A}h^{2}=\bigg{(}\frac{\mathcal{A}I_{1}+(f_{+}+f_{-})}{I_{2}}\bigg{)}h,$
(S20)

where $I_{1}=\int_{-L/2}^{L/2}(\tfrac{1}{3}h+l_{u})^{-1}\,dx$ and
$I_{2}=\int_{-L/2}^{L/2}(\tfrac{1}{3}h^{2}+l_{u}h)^{-1}\,dx$, for small
activity and small forces.

###  IV.1 Small forces and small activity

We consider small perturbations to a symmetric passive drop by expanding
$h(x)$ to linear order in $f_{\pm}$ and $\mathcal{A}$ around the passive
solution, obtained by setting $f_{+}=f_{-}=0$ and $\mathcal{A}=0$ in (S20) and
either imposing $h^{\prime\prime}(x)=\pi_{0}$ or $h^{\prime}(\tfrac{\pm
L}{2})=\mp\phi$ along with $h(\tfrac{\pm L}{2})=h_{0}$:

$h_{p}=-\frac{\phi L}{4}\bigg{(}\frac{4x^{2}}{L^{2}}-1\bigg{)}+h_{0}.$ (S21)

We also expand $L$ to linear order in $f_{\pm}$ and $\mathcal{A}$:

$L=L_{p}-f_{+}L_{+}-f_{-}L_{-}+\mathcal{A}L_{\alpha}+\cdots$ (S22)

which leads to

$\begin{split}h(y)=&H_{0}+f_{+}\bigg{(}\frac{\phi
L_{+}}{4}(y^{2}-1)-h_{+}\bigg{)}+f_{-}\bigg{(}\frac{\phi
L_{-}}{4}(y^{2}-1)-h_{-}\bigg{)}\\\ &+\mathcal{A}\bigg{(}h_{\alpha}-\frac{\phi
L_{\alpha}}{4}(y^{2}-1)\bigg{)}+\cdots\end{split}$ (S23)

where $y=2x/L$, $H_{0}=-\tfrac{\phi L_{p}}{4}(y^{2}-1)+h_{0}$, and $L_{p}$ is
the length of the passive drop given by
$L_{p}=\sqrt{\tfrac{6\Omega}{\phi}+\tfrac{9h_{0}^{2}}{\phi^{2}}}-\tfrac{3h_{0}}{\phi}$
for a drop of volume $\Omega$. Substituting (S23) into (S20) and keeping terms
only to linear order yields three differential equations at
$\mathcal{O}(f_{+})$, $\mathcal{O}(f_{-})$, and $\mathcal{O}(\mathcal{A})$
that can be integrated for $h_{\pm}$ and $h_{\alpha}$. At $\mathcal{O}(f_{+})$
we have

$\displaystyle
h_{+}^{\prime\prime\prime}(x)(\tfrac{1}{3}H_{0}^{2}+l_{u}H_{0})=-\frac{1}{I_{2}},$
$\displaystyle h_{+}(\pm 1)=0,$ $\displaystyle
h^{\prime\prime}(1)=-\frac{L_{p}^{2}}{4}-\frac{\phi L_{+}}{2}.$ (S24)

For consistency, we must have also $h_{+}^{\prime\prime}(-1)=-\phi L_{+}/2$
but we cannot impose this on the equation, as there are already three boundary
conditions. Fortunately, it falls out automatically because the integrals
$I_{1}$ and $I_{2}$ encode both boundary conditions on $h^{\prime\prime}$. At
$\mathcal{O}(f_{-})$ we have

$\displaystyle
h_{-}^{\prime\prime\prime}(x)(\tfrac{1}{3}H_{0}^{2}+l_{u}H_{0})=-\frac{1}{I_{2}},$
$\displaystyle h_{-}(\pm 1)=0,$ $\displaystyle
h_{-}^{\prime\prime}(-1)=\frac{L_{p}^{2}}{4}-\frac{\phi L_{-}}{2}.$ (S25)

Again, for consistency, we must have $h_{-}^{\prime\prime}(1)=-\phi L_{-}/2$,
which again falls out automatically. At $\mathcal{O}(\mathcal{A})$ we have

$\displaystyle
h_{\alpha}^{\prime\prime\prime}(x)(\tfrac{1}{3}H_{0}^{2}+l_{u}H_{0})=\frac{I_{1}}{I_{2}}-H_{0},$
$\displaystyle h_{\alpha}(\pm 1)=0,$ $\displaystyle
h_{\alpha}^{\prime\prime}(1)=-\frac{\phi L_{\alpha}}{2}.$ (S26)

For consistency we must have $h_{\alpha}^{\prime\prime}(-1)=-\phi
L_{\alpha}/2$, which falls out automatically. To perform the integration, we
use the variable $y=\tfrac{2x}{L}$. It is useful to define the following
function

$\displaystyle G_{f}(y)=$
$\displaystyle\bigg{(}1+\tfrac{y}{\sqrt{1+4h_{0}/\phi
L_{p}}}\bigg{)}^{2}\log{\bigg{(}1+\tfrac{y}{\sqrt{1+4h_{0}/\phi
L_{p}}}\bigg{)}}-\bigg{(}1-\tfrac{y}{\sqrt{1+4h_{0}/\phi
L_{p}}}\bigg{)}^{2}\log{\bigg{(}1-\tfrac{y}{\sqrt{1+4h_{0}/\phi
L_{p}}}\bigg{)}}$
$\displaystyle+\frac{1}{\beta\phi}\bigg{[}\bigg{(}\beta-\tfrac{\phi
y}{\sqrt{1+4h_{0}/\phi L_{p}}}\bigg{)}^{2}\log{\bigg{(}\beta-\tfrac{\phi
y}{\sqrt{1+4h_{0}/\phi L_{p}}}\bigg{)}}-\bigg{(}\beta+\tfrac{\phi
y}{\sqrt{1+4h_{0}/\phi L_{p}}}\bigg{)}^{2}\log{\bigg{(}\beta+\tfrac{\phi
y}{\sqrt{1+4h_{0}/\phi L_{p}}}\bigg{)}}\bigg{]},$

where

$\beta=\sqrt{\frac{1+\tfrac{4h_{0}}{\phi L_{p}}+\tfrac{12l_{u}}{\phi
L_{p}}}{1+\tfrac{4h_{0}}{\phi L_{p}}}}.$

We also need

$\displaystyle G_{\alpha}(y)=$
$\displaystyle\frac{\phi}{4\beta}\bigg{[}\bigg{(}\frac{\beta}{\phi}+\frac{y}{\sqrt{1+4h_{0}/\phi
L_{p}}}\bigg{)}^{2}\log\bigg{(}\frac{\beta}{\phi}+\frac{y}{\sqrt{1+4h_{0}/\phi
L_{p}}}\bigg{)}$
$\displaystyle-\bigg{(}\frac{\beta}{\phi}-\frac{y}{\sqrt{1+4h_{0}/\phi
L_{p}}}\bigg{)}^{2}\log\bigg{(}\frac{\beta}{\phi}-\frac{y}{\sqrt{1+4h_{0}/\phi
L_{p}}}\bigg{)}\bigg{]}.$

In terms of these functions, the leading order contributions to the drop
height are

$\displaystyle h_{+}(y)=\frac{-L_{p}^{2}(1+\tfrac{4h_{0}}{\phi
L_{p}})}{8G_{f}^{\prime\prime}(1)}\bigg{[}G_{f}(y)+\tfrac{1}{2}G_{f}^{\prime\prime}(1))\bigg{(}1+\frac{4\phi}{L}_{+}{L_{p}^{2}}\bigg{)}y^{2}-\sqrt{1+\tfrac{4h_{0}}{\phi
L_{p}}}G_{f}(1)y-\frac{G_{f}^{\prime\prime}(1)(1+4\phi
L_{+}/L_{p}^{2})}{2(1+4h_{0}/\phi L_{p})}\bigg{]},$ (S27a) $\displaystyle
h_{-}(y)=\frac{-L_{p}^{2}(1+\tfrac{4h_{0}}{\phi
L_{p}})}{8G_{f}^{\prime\prime}(1)}\bigg{[}G_{f}(y)-\tfrac{1}{2}G_{f}^{\prime\prime}(1)\bigg{(}1-\frac{4\phi
L_{-}}{L_{p}^{2}}\bigg{)}y^{2}-\sqrt{1+\tfrac{4h_{0}}{\phi
L_{p}}}G_{f}(1)y+\frac{G_{f}^{\prime\prime}(1)(1-4\phi
L_{-}/L_{p}^{2})}{2(1+4h_{0}/\phi L_{p})}\bigg{]},$ (S27b) $\displaystyle
h_{\alpha}(y)=\frac{3L_{p}^{2}\sqrt{1+\tfrac{4h_{0}}{\phi
L_{p}}}}{2\phi}\bigg{[}\frac{G_{\alpha}^{\prime\prime}(1)}{G_{f}^{\prime\prime}(1)}G_{f}(y)-G_{\alpha}(y)+\sqrt{1+\tfrac{4h_{0}}{\phi
L_{p}}}\bigg{(}G_{\alpha}(1)-\frac{G_{\alpha}^{\prime\prime}(1)}{G_{f}^{\prime\prime}(1)}G_{f}(1)\bigg{)}y\bigg{]},$
(S27c)

where

$\displaystyle L_{+}=\frac{-L_{p}^{3}}{24}\bigg{(}\frac{\phi
L_{p}}{2}+h_{0}\bigg{)}^{-1},$ (S28a) $\displaystyle
L_{-}=\frac{L_{p}^{3}}{24}\bigg{(}\frac{\phi L_{p}}{2}+h_{0}\bigg{)}^{-1},$
(S28b) $\displaystyle L_{\alpha}=0.$ (S28c)

##  V Criterion for drop break-up at zero net force

Consider a passive drop with $\mathcal{A}=0$ and $\mathcal{F}=0$. The equation
for drop height becomes

$h^{\prime\prime\prime}(\tfrac{1}{3}h^{3}+l_{u}h^{2})=0$ (S29)

and has the solution

$h=-\frac{1}{2}\bigg{(}\pi_{0}+\frac{\mathcal{S}}{2}\bigg{)}\bigg{(}x^{2}-\frac{L^{2}}{4}\bigg{)}+h_{0},$
(S30)

where we have used $h(\pm L/2)=h_{0}$ and $h^{\prime\prime}=f_{+}-\pi_{0}$.
The drop length $L$ is determined by the volume constraint
$\int_{-L/2}^{L/2}h\,dx=\Omega>0$. The drop length diverges at
$\mathcal{S}=-2\pi_{0}\approx-1.99$ ($\pi_{0}$ is chosen so that the contact
angle of a passive drop is one), and goes negative for
$\mathcal{S}<-2\pi_{0}$. This means that volume conservation is not truly
satisfied even though the condition $\int_{-L/2}^{L/2}h\,dx=\Omega$ is
satisfied mathematically by a negative $L$. This is clear in the left plot in
figure S2, where $h<0$ for $\mathcal{S}<-2\pi_{0}$ and thus
$\int_{-|L|/2}^{|L|/2}h\,dx<0$, with $|L|$ being the actual length of the
drop. The critical value of $\mathcal{S}=-2\pi_{0}$ is consistent with what we
see on the phase diagrams: the DB region for a passive drop begins between
$\mathcal{S}=-1$ and $\mathcal{S}=-2$ along the line $\mathcal{F}=0$ (see main
text).

Figure S2: Left: droplet height profiles at $\mathcal{F}=0$ and
$\mathcal{A}=0$. Here, $\pi_{0}=0.9226$ and $\Omega=1$. Right: drop length $L$
as a function of $\mathcal{S}$.

##  VI Different steady states resulting from different initial conditions

Our model is capable of producing different steady state drop profiles for the
same activity and applied forces starting from different height profiles in
the iterative Crank-Nicholson algorithm. This is shown in figure S3.

Figure S3: Left: Initial height profiles (dashed lines) and steady state
profiles (solid lines) for Right: Only the initial height profiles.