# Accessing inaccessible information via quantum indistinguishability

Sebastian Horvat<EMAIL_ADDRESS>University of Vienna, Faculty
of Physics & Vienna Doctoral School in Physics & Vienna Center for Quantum
Science and Technology, Boltzmanngasse 5, 1090 Vienna, Austria  Borivoje Dakić
<EMAIL_ADDRESS>University of Vienna, Faculty of Physics, Vienna
Center for Quantum Science and Technology, Boltzmanngasse 5, 1090 Vienna,
Austria  Institute for Quantum Optics and Quantum Information (IQOQI),
Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria.

###### Abstract

In this Letter we present an information-theoretic task that consists in
learning an apparently perfectly isolated bit of information by merely
spatially translating the object that encodes it. We show that the task can be
solved with the use of independently prepared indistinguishable particles, but
cannot be solved with distinguishable ones, unless they are entangled. Our
results thus corroborate the view that the entanglement that seems to be
necessarily present in systems comprised of indistinguishable particles is
more than a mere representational artefact and can indeed be used as a
resource for information processing. Apart from its foundational merit, our
results are of practical importance for cryptography.

##  I Introduction

There are two ways in which indistinguishable particles can be represented in
quantum-mechanical models: the first-quantization formalism models their state
vectors as elements of the (anti)symmetric subspace of a multi-particle
Hilbert space, whereas the second-quantization formalism takes their state
space to be given by a Fock space of modes. Interestingly, the former option
represents indistinguishable particles as necessarily entangled, while
according to the latter option, the modes’ entanglement depends on the
particles’ past interactions. Since entanglement is a paradigmatic resource
for quantum information processing, the question arises whether the
entanglement that is apparently intrinsic according to the first-quantization
formalism can in principle be accessed and used, or whether it is a mere
artefact stemming from the choice of representation: possible answers are
still being actively debated ghirardi2002entanglement ; ghirardi2004general ;
cavalcanti2007useful ; franco2018indistinguishability ;
barghathi2019operationally ; morris2020entanglement ; johann2021locality ;
benatti2021entanglement  (for a recent review see benatti2020entanglement ).
In this Letter we corroborate the former take on this question by proposing an
information-theoretic task that can be solved with the use of independently
prepared indistinguishable particles. We thereby show that the particles’
indistinguishability can be used as an information-theoretic resource, and
that there is a sense in which the entanglement present according to the
first-quantization notation is more than a mathematical artefact.

We will start by providing an informal presentation of our task, before
putting it on abstract grounds. We will then present the quantum-mechanical
analysis of the task and expand on the requirements necessarily satisfied by
any quantum-mechanical protocol that solves it. The last section contains a
discussion on the broader merit of our results and an outlook on possible
future developments.

##  II Informal presentation

Consider the following scenario. An agent named Alice is given a box, which
contains a physical object (e.g. an atom) prepared in one of two perfectly
distinguishable states, thereby encoding one bit of information. The box is
sealed in such a way that Alice does not have the means of opening it and thus
cannot access the object directly. Suppose that the only action she can
perform on the box is to translate it in space, e.g. she can move the box from
some initial position $\vec{x}$ to another position $\vec{x}^{\prime}$. Alice
may also have available various experimental devices and other physical
objects (as for example other boxes). Lastly, we will assume that the
translation of the box through space leaves the state of the object within it
invariant, and that Alice cannot infer the state by merely translating the box
(e.g. via some state-dependent back-reaction acting on Alice’s experimental
devices or additional objects). An example of this scenario, where the
additional objects consist of other boxes, is pictured in Fig. 1.

Figure 1: Alice is given a box containing an object prepared in state
$k\in\left\\{0,1\right\\}$. She also has at disposal other boxes, with their
pertaining objects set in state $0$. Alice is challenged to learn the value
$k$ by solely moving the box that contains the hidden object (without opening
it), while also being allowed to implement transformations on the other boxes:
for example, as represented in the diagram, she can swap the locations of the
first two boxes.

Now we are ready to formulate the task: can Alice learn the state of the
object within the box, given the above restrictions on what she is allowed to
do? At the current stage, this question may appear nonsensical: if (i) Alice
is only able to translate the box, and (ii) a mere translation of the box
cannot reveal the state of the object within it, then it trivially follows
that Alice cannot accomplish the task! However, if the box can be treated
quantum-mechanically and if Alice has at disposal additional identical quantum
systems, then, as we will show in the next section, she can in fact learn the
state of the object, while still in some sense only moving the box.
Let us now provide a formalized version of the task and the quantum-mechanical
protocol that can be used to solve it.

##  III Formalization of the task

Here we will put the scenario presented in the previous section on more formal
and abstract grounds. Consider that Alice is given a localized physical
object, henceforth named $\cal{T}$. $\cal{T}$ can be characterized to have two
degrees of freedom (d.o.f.): an intrinsic one, which we will label with $k$,
and a position, labelled with $\vec{x}$. We will thus represent the overall
state of the object with the ordered pair $(\vec{x},k)_{T}$. Throughout the
paper, we attach only operational meaning to states, i.e. the sentence “the
object is in state $(\vec{x},k)_{T}$” is hereafter synonymous with “a position
measurement on the object would output value $\vec{x}$, and an appropriate
measurement of the internal d.o.f. would output value $k$ ”. We furthermore
assume that $k$ can take only two possible values, 0 or 1. Let us now suppose
that Alice is only allowed to translate the object through space, while
keeping the internal d.o.f. intact. We will model this restriction by assuming
that she has at disposal a device $\cal{D}$ that can implement “translation
maps” $M:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$, the latter being linear
operators that map position vectors $\vec{x}$ into position vectors
$\vec{x}^{\prime}=M\vec{x}$. We will associate states $(M)_{D}$ to the device,
where $M$ indicates which map the device is set to implement on a potential
target object. The device is constructed in such a way that the dynamical
interaction with the localized target object $\cal{T}$, which is initially in
state $(\vec{x},k)_{T}$, is given by

$(M)_{D}(\vec{x},k)_{T}\rightarrow(M)_{D}(M\vec{x},k)_{T}.$ (1)

Furthermore, Alice may have at disposal other localized objects, which cannot
interact directly with the initial object $\cal{T}$, but can interact with the
device $\cal{D}$. Labelling collectively these additional objects with
$\cal{A}$, and their pertaining state with $(\alpha)_{A}$, the latter
constraint means that the overall interaction between $\cal{D}$, $\cal{A}$ and
$\cal{T}$ can be represented as:

$(M)_{D}(\alpha)_{A}(\vec{x},k)_{T}\rightarrow(M)_{D}(\tilde{\alpha}(M,\alpha))_{A}(M\vec{x},k)_{T},$
(2)

where the final state $\tilde{\alpha}(M,\alpha)$ of $\cal{A}$ is in principle
any function insensitive to the internal d.o.f. pertaining to $\cal{T}$.
Moreover, we assume that if the initial state of $\cal{D}$ is not equal to any
$(M)_{D}$ (but is e.g. a probabilistic mixture or a quantum superposition
state), we assume that it is still the case that, were one to find the final
state of $\cal{D}$ to be $(M)_{D}$, one would find the final state of
$\cal{T}$ and $\cal{A}$ to be
$(\tilde{\alpha}(M,\alpha))_{A}(M\vec{x},k)_{T}$. In other words, for any
initial state of $\cal{D}$, postselecting the final state on definite states
$(M)_{D}$ of $\cal{D}$ results in the postselected joint state being equal to
the RHS of Eq. (2).
Now suppose that Alice is allowed only to interact with her device, i.e. she
is solely able to control and to read out the state of $\cal{D}$. She can thus
interact only indirectly with objects $\cal{T}$ and $\cal{A}$, via mediation
through $\cal{D}$. Furthermore, Alice has perfect knowledge of the initial
state $(\alpha)_{A}$ associated to the additional objects $\cal{A}$, but does
not have any prior knowledge of the value $k$ pertaining to $\cal{T}$. The
information-theoretic task can now be formulated as follows:

Task: can Alice learn the state $k$ pertaining to $\cal{T}$, solely by
manipulating the given device $\cal{D}$?

As can be immediately seen from Eq. (2), the state of $\cal{D}$ is insensitive
to $k$, so the answer to the latter question seems to be an immediate ‘No’.
Also, at the current stage, it is admittedly not clear at all what role, if
any, the additional objects $\cal{A}$ could play for the accomplishment of the
required task. However, in what follows, we will show that if $\cal{D}$,
$\cal{A}$ and $\cal{T}$ can be treated as quantum systems, and if Alice is
able to coherently manipulate the device $\cal{D}$, then she can in fact learn
the required value $k$, given that $\cal{A}$ and $\cal{T}$ can be modelled as
indistinguishable quantum particles.

### The quantum protocol

We will now assume that $\cal{D}$, $\cal{A}$ and $\cal{T}$ can be treated as
quantum systems that can be modelled via the usual rules of quantum mechanics.
Therefore, we can associate a quantum state

$\ket{\psi}\in\cal{H}\equiv H_{D}\otimes H_{A}\otimes H_{T}$ (3)

to the joint system comprised of $\cal{D}$, $\cal{A}$ and $\cal{T}$. The
Hilbert space $\cal{H}_{D}$ associated to the device is spanned by vectors
$\left\\{\ket{M},\forall M:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}\right\\}$,
whereas the Hilbert space $\cal{H}_{T}$ associated to $\cal{T}$ is spanned by
$\left\\{\ket{\vec{x},k}\equiv\ket{\vec{x}}\otimes\ket{k},\forall\vec{x}\in\mathbb{R}^{3},\forall
k=0,1\right\\}$. Vectors $\ket{M}$ are eigenstates of the observable $\hat{M}$
that corresponds to the “measurement” of the device in the intended classical
basis, whereas $\ket{\vec{x}}$ and $\ket{k}$ are respectively eigenstates of
observables $\hat{\vec{x}}$ and $\hat{k}$ corresponding to measurements of the
position and internal d.o.f. of $\cal{T}$. The structure of the Hilbert space
$\cal{H}_{A}$ pertaining to $\cal{A}$ remains for now unspecified. We now
assume, in accord with Eq. (2), that the interaction between $\cal{D}$,
$\cal{A}$ and $\cal{T}$ is given by the following unitary evolution:

$\ket{M}_{D}\ket{\alpha}_{A}\ket{\vec{x},k}_{T}\rightarrow\ket{M}_{D}\ket{\tilde{\alpha}(M,\alpha)}_{A}\ket{M\vec{x},k}_{T}.$
(4)

Notice that the above corresponds to a control gate, where the device
$\cal{D}$ acts as the control system, and the objects $\cal{T}$ and $\cal{A}$
act as targets. We stress that we are assuming that $\cal{D}$, $\cal{A}$ and
$\cal{T}$ are not interacting with any further environment, i.e. that they
constitute an isolated system; in the next subsection we will analyze
protocols that violate this assumption.
Let us now specify the additional objects $\cal{A}$ to consist only of one
object localized at position $\vec{x}_{A}$, and possessing a binary internal
d.o.f., whose value is for simplicity set to $k_{A}=0$. The Hilbert space
associated to $\cal{A}$ is thus isomorphic to the one associated to $\cal{T}$,
and the object $\cal{A}$ is assigned state $\ket{\vec{x}_{A},0}$. We also
assume that the device $\cal{D}$ can be used to translate $\cal{A}$ through
space. Consequently, Alice is now able to swap the two objects by using her
device, by moving $\cal{T}$ to position $\vec{x}_{A}$, and $\cal{A}$ to
position $\vec{x}_{T}$. In order to simplify the discussion, let us introduce
an effective state of $\cal{D}$, which we will label with ‘$S$’, and that is
constructed in such a way that it effectively swaps the two objects upon
interaction, i.e.

$\ket{S}_{D}\ket{\vec{x}_{A},0}_{A}\ket{\vec{x}_{T},k_{T}}_{T}\rightarrow\ket{S}_{D}\ket{\vec{x}_{T},0}_{A}\ket{\vec{x}_{A},k_{T}}_{T},$
(5)

where we introduced more indices in order to avoid confusion.
Under the assumption that $\cal{A}$ and $\cal{T}$ are distinguishable systems,
Eq. (4) implies that Alice cannot accomplish the task, because the internal
degree of freedom pertaining to $\cal{T}$ is isolated from the other
subsystems. Let us now suppose that objects $\cal{A}$ and $\cal{T}$ are
indistinguishable, i.e. that they can be modelled as indistinguishable quantum
particles: here we assume the bosonic case, even though the protocol would be
equally valid in the fermionic case as well. This warrants us to introduce the
“second quantization” notation via the following recipe:

$\ket{\vec{x}_{A},0}_{A}\ket{\vec{x}_{T},k_{T}}_{T}\Rightarrow\ket{0}_{\vec{x}_{A}}\ket{k_{T}}_{\vec{x}_{T}},$
(6)

where the latter state means that at location $\vec{x}_{A}$ there is an object
with internal state $0$, and at location $\vec{x}_{T}$ an object with internal
state $k_{T}$, with no further labels that may distinguish the objects.
Mathematically, the quantum state is now an element of
$\mathcal{H}_{\vec{x}_{T}}\otimes\mathcal{H}_{\vec{x}_{A}}$, where
$\mathcal{H}_{\vec{x}_{T/A}}$ is associated to spatial mode $\vec{x}_{T/A}$
and is spanned by vectors
$\left\\{\ket{0}_{\vec{x}_{T/A}},\ket{1}_{\vec{x}_{T/A}}\right\\}$. The
interaction between the device $\cal{D}$ set in state ‘$S$’ and the two
indistinguishable objects is thus given by:

$\ket{S}_{D}\ket{0}_{\vec{x}_{A}}\ket{k_{T}}_{\vec{x}_{T}}\rightarrow\ket{S}_{D}\ket{k_{T}}_{\vec{x}_{A}}\ket{0}_{\vec{x}_{T}}.$
(7)

Now comes the crucial observation: unlike as in the case of distinguishable
objects (see Eq. (5)), the overall quantum state in Eq. (7) is invariant upon
interaction if and only if $k_{T}=k_{A}$, i.e. $k_{T}=0$. This enables the
construction of the following protocol that can be used by Alice to learn
$k_{T}$ with probability higher than $\frac{1}{2}$, given that she knows the
value $k_{A}=0$ of the additional system $\cal{A}$. The procedure goes as
follows.
(a) Alice prepares the device in state
$\ket{\phi}\equiv\frac{1}{\sqrt{2}}\left(\ket{\mathbb{1}}_{D}+\ket{S}_{D}\right)$,
where ‘$\mathbb{1}$’ is the state of the device that implements the identity
transformation.
(b) She then lets $\cal{D}$, $\cal{A}$ and $\cal{T}$ interact as:

$\begin{split}&\frac{1}{\sqrt{2}}\left(\ket{\mathbb{1}}_{D}+\ket{S}_{D}\right)\ket{0}_{\vec{x}_{A}}\ket{k_{T}}_{\vec{x}_{T}}\rightarrow\\\
&\frac{1}{\sqrt{2}}\left(\ket{\mathbb{1}}_{D}\ket{0}_{\vec{x}_{A}}\ket{k_{T}}_{\vec{x}_{T}}+\ket{S}_{D}\ket{k_{T}}_{\vec{x}_{A}}\ket{0}_{\vec{x}_{T}}\right).\end{split}$
(8)

After the interaction, the reduced density state $\rho$ associated to
$\cal{D}$ is:

$\rho=\begin{cases}\ket{\phi}\bra{\phi}&\text{if $k_{T}=0$}\\\
\frac{1}{2}\mathbb{1}_{2}&\text{if $k_{T}=1$},\end{cases}$ (9)

where $\mathbb{1}_{2}$ is the identity operator on $\cal{H}_{D}$ restricted to
the subspace spanned by $\left\\{\ket{\mathbb{1}}_{D},\ket{S}_{D}\right\\}$.
Notice that, were the objects $\cal{T}$ and $\cal{A}$ distinguishable, the
device would end up in a maximally mixed state for all combinations of
$k_{T},k_{A}$.
(c) Finally, Alice measures $\cal{D}$ with projectors
$\left\\{\Pi_{0}=\ket{\phi}\bra{\phi},\Pi_{1}=\mathbb{1}_{2}-\ket{\phi}\bra{\phi}\right\\}$.
If she obtains outcome ‘0’, she guesses $k_{T}=0$; conversely, if she obtains
‘1’ she guesses $k_{T}=1$: the probability of a correct guess is
$\frac{3}{4}$, thereby beating a random guess. Can the probability of success
be raised closer to unity? The answer is affirmative, and is given by a
straightforward extension of the protocol, where $\cal{A}$ now consists of
$(N-1)$ additional identical quantum objects, each of them set in the
reference state $k_{A}=0$. The extended protocol is presented in Appendix A,
where it is shown that Alice’s probability $P_{W}$ of correctly guessing the
required bit is

$P_{W}=1-\frac{1}{2N},$ (10)

that asymptotically reaches unity for large $N$. Note that if Alice at the end
of the process wants to guess which of the objects is the one containing
$k_{T}$ (i.e. $\cal{T}$), she is able to do so only with probability
$\frac{1}{N}$. Alternatively, instead of implementing measurement
$\left\\{\Pi_{0},\Pi_{1}\right\\}$, she could have measured $\cal{D}$ in the
$\left\\{\ket{\mathbb{1}}_{D},\ket{S}_{D}\right\\}$ basis and found out with
certainty the location of $k_{T}$, which would however not have provided her
with any knowledge of the value of $k_{T}$. There is thus a trade-off between
the possibility of acquiring knowledge of the value of $k_{T}$ and of
retaining knowledge of its location, which is inherited from the non-
commutativity of the observables on $\cal{D}$ that would correspondingly need
to be measured.

### The necessity of entanglement and indistinguishability

In the protocol presented in the previous subsection, the interaction between
the target objects ($\cal{T}$ and $\cal{A}$) and the device $\cal{D}$ produces
a $k_{T}$-dependent back-reaction on the latter: in particular, the device
gets entangled with the target objects if and only if $k_{T}=1$. This suggests
that the possibility of establishing entanglement between the device and the
targets, along with the targets being indistinguishable, may be a necessary
ingredient for the protocol to work. However, it is not yet clear whether this
is the case, since the transformation defined abstractly in Eq. (2) admits
other quantum-mechanical realizations besides the ideal quantum control gate
that we assumed in Eq. (4): we thus cannot yet exclude the possibility of
there being a noisy control gate that produces a $k_{T}$-dependent
transformation on the device even without the latter getting entangled with
the target objects. Nevertheless, as we will sketch here (while leaving the
proof for Appendix B), the possibility of establishing entanglement between
$\cal{D}$ and the targets turns out after all to be a necessary condition for
any quantum-mechanical protocol to solve the task.
Let us assume that $N=2$ and that the two target objects are
indistinguishable. As before, states $\ket{\mathbb{1}}_{D}$ and $\ket{S}_{D}$
pertaining to $\mathcal{D}$ correspond to the ‘identity’ and ‘swap’
operations. The assumption of our task is that the interaction between
$\mathcal{D}$ and the targets is such that, postselecting the final state on
$\ket{\mathbb{1}}_{D}$ leaves the targets’ state invariant, whereas
postselecting on $\ket{S}_{D}$ leaves the targets in a ‘swapped’ state. More
precisely, the most general allowed interaction $G$ is a CPTP-map that
satisfies the following: for any quantum states $\rho^{(D)}$ and $\rho^{(T)}$
of the device $\cal{D}$ and the targets $\cal{T}$ and $\cal{A}$, the post-
interaction state $\tilde{\rho}\equiv G(\rho^{(D)}\otimes\rho^{(T)})$
satisfies

$\begin{split}\operatorname{Tr}\left[(\Pi_{1}\otimes\mathbb{1})\tilde{\rho}\right]=p_{1}\rho^{(T)}\\\
\operatorname{Tr}\left[(\Pi_{S}\otimes\mathbb{1})\tilde{\rho}\right]=p_{S}\hat{S}\rho^{(T)}\hat{S}^{\dagger},\end{split}$
(11)

where $\Pi_{1/S}\equiv\ket{\mathbb{1}/S}_{D}\bra{\mathbb{1}/S}$, and $\hat{S}$
is the ‘swap’ operator acting on the targets. The factors
$p_{1}\equiv\operatorname{Tr}\left(\Pi_{1}\rho^{(D)}\right)$ and
$p_{S}\equiv\operatorname{Tr}\left(\Pi_{S}\rho^{(D)}\right)$ are determined by
the requirement on how $G$ acts in the case of the initial state of
$\mathcal{D}$ being $\ket{\mathbb{1}}_{D}$ or $\ket{S}_{D}$.
Let us label with $\rho^{(T)}_{k_{T}}$ the initial state of the targets when
the unknown bit’s value is $k_{T}$. For any gate $G$ that satisfies Eq. (11)
and any initial state $\rho^{(D)}$ of $\mathcal{D}$, our task can be solved
with probability higher than $\frac{1}{2}$ only if the final reduced state of
$\mathcal{D}$ depends on $k_{T}$, i.e.

$\operatorname{Tr}_{T}\left(\tilde{\rho}_{0}\right)\neq\operatorname{Tr}_{T}\left(\tilde{\rho}_{1}\right),$
(12)

where $\tilde{\rho}_{k_{T}}\equiv G(\rho^{(D)}\otimes\rho^{(T)}_{k_{T}})$. The
permutation invariance of state $\rho^{(T)}_{0}$ implies that
$\tilde{\rho}_{0}$ is a separable state. On the other hand, Eq. (12) holds if
and only if $\tilde{\rho}_{1}$ is an entangled state, as we prove in Appendix
B, wherein we also prove a generalization of the latter condition for
arbitrary $N>2$.

Therefore, the necessary conditions for a quantum-mechanical protocol to
outperform a random guess are that (i) it involves indistinguishable target
objects, and (ii) the final quantum state of the device and the target objects
is entangled if and only if $k_{T}=1$.

##  IV Discussion

We have seen in the previous section that Alice can accomplish her task thanks
to the indistinguishability of the target objects and the possibility of
establishing entanglement between the latter and the device. We now want to
emphasize that the task and its solution are reminiscent of the swap test
buhrman2001quantum , which enables one to check whether two quantum systems
are prepared in equal states by performing a control-swap operation on them,
with the ‘swap’ operation acting as
$\ket{\psi}\ket{\phi}\rightarrow\ket{\phi}\ket{\psi}$. This offers another
angle on how to understand the necessity of the targets’ indistinguishability
for the quantum protocol to solve our task. Namely, when applied to
indistinguishable particles, a spatial swap acts on spatial modes exactly as
the ‘swap’ operation that is required in the swap-test, whereas for
distinguishable particles the two operations are strictly different:
indistinguishability thus enables the allowed operations to be used to
implement a standard swap-test. Therefore our task presents an instance of
quantum indistinguishability serving as an information-theoretic resource,
which leads us back to the ongoing debate on the possible physical merit of
the entanglement apparently intrinsic to indistinguishable particles, as
mentioned in the Introduction. Indeed, notice that our task cannot be solved
with distinguishable particles under the requirement that the target objects
$\cal{T}$ and $\cal{A}$ be prepared independently. However, if we drop this
assumption, the task can be equally well solved with mutually entangled
distinguishable particles, as can be noticed by a mere re-interpretation of
the quantum state associated to indistinguishable particles in the first-
quantization notation. Indistinguishability and entanglement thus represent
equivalent information-theoretic resources in our task, which may be taken to
support the claim that the entanglement that appears in the first quantization
notation is more than a mere representational artefact. However, we do not
want to delve here into a more detailed discussion on whether
indistinguishable particles need to (or can) be considered as necessarily
entangled or not, the answer of which would starkly depend on particular
definitions and measures of entanglement johann2021locality : our aim is just
to point out that there exists an information-theoretic task in which
indistinguishability and entanglement serve as equivalent resources.
Let us now comment on possible future developments of our results. In the
current manuscript we have provided only a quantum-mechanical analysis of our
task; however, we believe that the latter, as presented abstractly at the
beginning of Section III, can be transposed into the framework of generalized
probabilistic theories barrett2007information ; plavala2021general , which
would enable the assessment of how much our results hinge on the specificities
of quantum theory. We may thereby gain understanding of the general relation
between indistinguishability and entanglement, by analyzing questions such as:
in which other operational theories are the indistinguishability of the
targets and the possibility of entangling the latter and the device, both
necessary to solve the task? In which theories do indistinguishability and
entanglement of the targets constitute equivalent resources for our task?
Finally, moving on to potential practical aspects of our findings, it is
commonly expected that, following Moore’s law, hardware components used in
information processing may soon reach a regime in which quantum-mechanical
effects cannot be neglected powell2008quantum . When (and if) that becomes the
case, our results may prove relevant for hardware-security modules that
incorporate physical protection against tempering. Examples of this type of
devices can already be found both in classical and quantum computing, where
the security of one-time programs essentially relies on hardware components,
i.e. on one-time memories broadbent2013quantum ; roehsner2018quantum . Other
potential applications of our finding may be expected to be found in
cryptography and hardware security in general.

###### Acknowledgements.

S.H. and B.D. acknowledge financial support from the Austrian Science Fund
(FWF) through BeyondC-F7112.

## References

  * (1) Ghirardi, G., Marinatto, L. and Weber, T., 2002. Entanglement and properties of composite quantum systems: a conceptual and mathematical analysis. Journal of statistical Physics, 108(1), pp.49-122.
  * (2) Ghirardi, G. and Marinatto, L., 2004. General criterion for the entanglement of two indistinguishable particles. Physical Review A, 70(1), p.012109.
  * (3) Cavalcanti, D., Malard, L.M., Matinaga, F.M., Cunha, M.T. and Santos, M.F., 2007. Useful entanglement from the Pauli principle. Physical Review B, 76(11), p.113304.
  * (4) Franco, R.L. and Compagno, G., 2018. Indistinguishability of elementary systems as a resource for quantum information processing. Physical review letters, 120(24), p.240403.
  * (5) Barghathi, H., Casiano-Diaz, E. and Del Maestro, A., 2019. Operationally accessible entanglement of one-dimensional spinless fermions. Physical Review A, 100(2), p.022324.
  * (6) Morris, B., Yadin, B., Fadel, M., Zibold, T., Treutlein, P. and Adesso, G., 2020. Entanglement between identical particles is a useful and consistent resource. Physical Review X, 10(4), p.041012.
  * (7) Johann, T.J.F. and Marzolino, U., 2021. Locality and entanglement of indistinguishable particles. Scientific Reports, 11(1), pp.1-10.
  * (8) Benatti, F., Floreanini, R. and Marzolino, U., 2021. Entanglement and non-locality in quantum protocols with identical particles. Entropy, 23(4), p.479.
  * (9) Benatti, F., Floreanini, R., Franchini, F. and Marzolino, U., 2020. Entanglement in indistinguishable particle systems. Physics Reports, 878, pp.1-27.
  * (10) Carl W Helstrom, “Quantum detection and estimation theory,” Journal of Statistical Physics 1, 231–252 (1969).
  * (11) Harry Buhrman, Richard Cleve, John Watrous, and Ronald De Wolf, “Quantum fingerprinting,” Physical Re- view Letters 87, 167902 (2001).
  * (12) Jonathan Barrett, “Information processing in generalized probabilistic theories,” Physical Review A 75, 032304 (2007).
  * (13) Martin Plavala, “General probabilistic theories: An in- troduction,” arXiv preprint arXiv:2103.07469 (2021).
  * (14) James R Powell, “The quantum limit to moore’s law,” Proceedings of the IEEE 96, 1247–1248 (2008).
  * (15) Anne Broadbent, Gus Gutoski, and Douglas Stebila, “Quantum one-time programs,” in Annual Cryptology Conference (Springer, 2013) pp. 344–360.
  * (16) Marie-Christine Roehsner, Joshua A Kettlewell, Tiago B Batalhao, Joseph F Fitzsimons, and Philip Walther, “Quantum advantage for probabilistic one-time pro- grams,” Nature communications 9, 1–8 (2018).
  * (17) Peres, A., 1996. Separability criterion for density matrices. Physical Review Letters, 77(8), p.1413.

## Appendix A

Suppose that Alice is given the object $\mathcal{T}$ in state
$\ket{k_{T}}_{\vec{x}_{T}}$ and wants to find out the value $k_{T}$ by using
her device $\mathcal{D}$. Let us assume that $\mathcal{A}$ now consists of
$(N-1)$ additional identical quantum objects, each of them set in the
reference state $k_{A}=0$ (which is known to Alice), and located at positions
$\vec{x_{2}}$,…,$\vec{x}_{N}$. The joint state of $\mathcal{T}$ together with
the $(N-1)$ additional objects is thus
$\ket{k_{T}}_{\vec{x}_{1}}\ket{0}_{\vec{x}_{2}}...\ket{0}_{\vec{x}_{N}}$,
where we have for simplicity renamed position $\vec{x}_{T}$ into
$\vec{x}_{1}$. Alice’s device $\mathcal{D}$ can now be used to swap any pair
of the target objects: let us label with ‘$S_{i}$’ the state of $\mathcal{D}$
that is set to swap $\mathcal{T}$ with the $i$-th object. The protocol is then
partitioned in three steps:
(a) Alice prepares $\mathcal{D}$ in state
$\ket{\phi}=\sum_{i=1}^{N}\ket{S_{i}}_{D}$, where
$\ket{S_{1}}_{D}\equiv\ket{\mathbb{1}}_{D}$.
(b) She lets $\mathcal{D}$, $\mathcal{T}$ and the $(N-1)$ additional objects
interact as:

$\frac{1}{\sqrt{N}}\sum_{i=1}^{N}\ket{S_{i}}_{D}\ket{k_{T}}_{\vec{x}_{1}}\ket{0}_{\vec{x}_{2}}...\ket{0}_{\vec{x}_{N}}\rightarrow\frac{1}{\sqrt{N}}\sum_{i=1}^{N}\ket{S_{i}}_{D}\ket{0}_{\vec{x}_{1}}\ket{0}_{\vec{x}_{2}}...\ket{k_{T}}_{\vec{x}_{i}}...\ket{0}_{\vec{x}_{N}}.$
(13)

After the interaction, the reduced density state $\rho$ associated to
$\mathcal{D}$ is:

$\rho=\begin{cases}\ket{\phi}\bra{\phi}&\text{if $k_{T}=0$}\\\
\frac{1}{N}\mathbb{1}_{N}&\text{if $k_{T}\neq 0$},\end{cases}$ (14)

where $\mathbb{1}_{N}$ is the identity operator on $\mathcal{H}_{D}$ when
restricted to the corresponding subspace.
(c) Finally, Alice measures $\mathcal{D}$ with projectors
$\left\\{\Pi_{0}=\ket{\phi}\bra{\phi},\Pi_{1}=\mathbb{1}_{D}-\ket{\phi}\bra{\phi}\right\\}$.
If she obtains outcome ‘0’, she guesses $k_{T}=0$; conversely, if she obtains
‘1’ she guesses $k_{T}=1$. The probability of a correct guess $P_{W}$ is

$P_{W}=1-\frac{1}{2N},$ (15)

which coincides with the maximum possible value given by the Helstrom bound
helstrom1969quantum , thereby showing that the above measurement is optimal.

## Appendix B

Here we will provide the proof that in all quantum-mechanical protocols that
solve our task with probability higher than $\frac{1}{2}$, the device and the
targets get entangled upon interaction in the case that $k=1$. We will delve
immediately into the general case, where $\mathcal{A}$ consists of $(N-1)$
identical quantum systems set in state $0$ and located at positions
$\vec{x}_{2}$,…,$\vec{x}_{N}$, whereas $\mathcal{T}$ is set in state $k$ and
located at position $\vec{x}_{1}$. The joint initial state of the targets is
thus $\ket{k}_{\vec{x}_{1}}\ket{0}_{\vec{x}_{2}}...\ket{0}_{\vec{x}_{N}}$. Let
us for simplicity introduce the following notation:

$\begin{split}&\ket{0}_{T}\equiv\ket{0}_{\vec{x}_{1}}\ket{0}_{\vec{x}_{2}}...\ket{0}_{\vec{x}_{N}}\\\
&\ket{i}_{T}\equiv\ket{0}_{\vec{x}_{1}}\ket{0}_{\vec{x}_{2}}...\ket{1}_{\vec{x}_{i}}...\ket{0}_{\vec{x}_{N}},\quad
i=1,...,N.\end{split}$ (16)

Since by assumption Alice is only able to coherently swap the targets, the
effective Hilbert space $\mathcal{H}_{T}$ that can be associated to the latter
is spanned by the vectors in Eq. (LABEL:a1), i.e.
$\mathcal{H}_{T}=\text{Span}(\ket{i}_{T},i=0,...,N)$. A swap gate $S_{j}$ that
swaps the first and $j$-th object thus acts as $S_{j}\ket{0}_{T}=\ket{0}_{T}$,
and $S_{j}\ket{1}_{T}=\ket{j}_{T}$ (notice that $S_{1}$ is the identity
operator). The initial targets’ state is equal to $\ket{k}_{T}$, with $k$
being 0 or 1. Finally, the Hilbert space $\mathcal{H}_{D}$ associated to the
device $\mathcal{D}$ is spanned by vectors
$\left\\{\ket{j}_{D},j=1,...,N\right\\}$, with each $\ket{j}_{D}$ representing
the setting corresponding to swap gate $S_{j}$.
Now we want to characterize the most general interaction between $\mathcal{D}$
and the targets that is allowed by the assumptions of our task. Let
$\mathbf{St}_{DT}$, $\mathbf{St}_{D}$ and $\mathbf{St}_{T}$ be respectively
sets of density operators on $\mathcal{H}_{D}\otimes\mathcal{H}_{T}$,
$\mathcal{H}_{D}$ and $\mathcal{H}_{T}$. We will label with $\mathbf{Ch}_{DT}$
the set of completely positive trace preserving (CPTP) superoperators that map
$\mathbf{St}_{DT}$ into itself. Let $\mathcal{G}$ be the subset of CPTP maps
that represent the device-targets interactions implementable by Alice. The
abstract formulation of the task implies that, for any $G\in\mathcal{G}$, if
the initial state of $\mathcal{D}$ is some $\ket{i}_{D}$, then the interaction
is given by

$G\left(\ket{i}_{D}\bra{i}\otimes\rho^{(T)}\right)=\ket{i}_{D}\bra{i}\otimes
S_{i}\rho^{(T)}S_{i},$ (17)

for any $\rho^{(T)}\in\mathbf{St}_{T}$, where we have used the property
$S^{\dagger}_{i}=S_{i}$. More generally, if the initial state of $\mathcal{D}$
is an arbitrary density operator $\rho^{(D)}\in\mathbf{St}_{D}$, then the
interaction is such, that postselecting the final state on the device’s state
$\ket{i}_{D}$ implements gate $S_{i}$ on the targets, i.e.

$\operatorname{Tr}_{D}\left[\left(\Pi_{i}\otimes\mathbb{1}\right)G\left(\rho^{(D)}\otimes\rho^{(T)}\right)\right]=\alpha_{i}S_{i}\rho^{(T)}S_{i},$
(18)

where $\Pi_{i}\equiv\ket{i}_{D}\bra{i}$, and $\alpha_{i}$ are arbitrary real
coefficients. Since $G$ is linear in its arguments, Eqs. (17) and (18) imply
the following condition:

$\operatorname{Tr}_{D}\left[\left(\Pi_{i}\otimes\mathbb{1}\right)G\left(\rho^{(D)}\otimes\rho^{(T)}\right)\right]=\operatorname{Tr}\left(\Pi_{i}\rho^{(D)}\right)S_{i}\rho^{(T)}S_{i},$
(19)

thereby fixing the coefficients $\alpha_{i}$. Therefore, the assumptions of
the task imply that the set $\mathcal{G}$ of allowed device-targets
interactions can be characterized as follows:

$\mathcal{G}=\left\\{G\in\mathbf{Ch}_{DT}|\quad\forall\rho^{(D)}\in\mathbf{St}_{D},\rho^{(T)}\in\mathbf{St}_{T},i=1,...,N:\quad\operatorname{Tr}_{D}\left[\left(\Pi_{i}\otimes\mathbb{1}\right)G\left(\rho^{(D)}\otimes\rho^{(T)}\right)\right]=p_{i}S_{i}\rho^{(T)}S_{i}\right\\},$
(20)

where $p_{i}\equiv\operatorname{Tr}\left(\Pi_{i}\rho^{(D)}\right)$.
$\mathcal{G}$ can be understood as the class of generalized control gates,
with $\mathcal{D}$ being the control system and $\mathcal{A}$ and
$\mathcal{T}$ constituting the target system. In the main text we have seen
that in the case of the ideal control gate, which is a particular element of
$\mathcal{G}$, the device and the targets get entangled upon interaction if
$k=1$; in what follows we will prove that this is the case for any element of
$\mathcal{G}$ that can be used to outperform a random guess at solving our
task. Let us first slightly generalize the discussion by defining a broader
class of generalized control gates for which we will prove a Lemma that will
be of use later.

######  Definition 1.

Consider $\mathcal{H}_{D}\equiv\mathbb{C}^{N_{D}}$ and
$\mathcal{H}_{T}\equiv\mathbb{C}^{N_{T}}$ for some finite natural numbers
$N_{D}$ and $N_{T}$. Let $\mathbf{St}_{DT}$, $\mathbf{St}_{D}$ and
$\mathbf{St}_{T}$ be sets containing all and only density operators on
respectively $\mathcal{H}_{D}\otimes\mathcal{H}_{T}$, $\mathcal{H}_{D}$ and
$\mathcal{H}_{T}$. Let $\mathbf{Ch}_{DT}$ be the set of completely positive
trace preserving (CPTP) superoperators that map $\mathbf{St}_{DT}$ into
itself.
Let $\vec{U}\equiv\langle U^{(1)},...,U^{(N_{D})}\rangle$ be a list of $N_{D}$
unitary operators on $\mathcal{H}_{T}$, and let
$\vec{\Pi}\equiv\langle\Pi_{1},...,\Pi_{N_{D}}\rangle$ be a list of projectors
that form a projection-valued measure on $\mathcal{H}_{D}$. Define a 4-tuple
$X$ as $X\equiv\langle N_{D},N_{T},\vec{U},\vec{\Pi}\rangle$.
The set of generalized control gates relative to $X$ is a set
$\mathcal{G}_{X}$ defined as

$\mathcal{G}_{X}=\left\\{G\in\mathbf{Ch}_{DT}|\quad\forall\rho^{(D)}\in\mathbf{St}_{D},\rho^{(T)}\in\mathbf{St}_{T},i=1,...,N_{D}:\quad\operatorname{Tr}_{D}\left[\left(\Pi_{i}\otimes\mathbb{1}\right)G\left(\rho^{(D)}\otimes\rho^{(T)}\right)\right]=p_{i}U^{(i)}\rho^{(T)}U^{(i)\dagger}\right\\},$

where $p_{i}=\operatorname{Tr}\left(\Pi_{i}\rho^{(D)}\right)$.

######  Lemma 1.

Consider a 4-tuple $X\equiv\langle N_{D},N_{T},\vec{U},\vec{\Pi}\rangle$,
where $N_{D},N_{T}$ are natural numbers, $\vec{U}\equiv\langle
U^{(1)},...,U^{(N_{D})}\rangle$ is a list of unitary operators on
$\mathcal{H}_{T}\equiv\mathbb{C}^{N_{T}}$, and
$\vec{\Pi}\equiv\langle\Pi_{1},...,\Pi_{N_{D}}\rangle$ is a projection-valued
measure on $\mathcal{H}_{D}\equiv\mathbb{C}^{N_{D}}$. Let $\mathcal{G}_{X}$ be
the set of generalized control gates relative to $X$.
Then, for any $G\in\mathcal{G}_{X}$, there exists a natural number $M$ and a
set $\left\\{\vec{v}^{(1)},...,\vec{v}^{(N_{D})}\right\\}$ of unit vectors in
$\mathbb{C}^{M}$, such that, for all $\rho^{(D)}\in\mathbf{St}_{D}$ and
$\rho^{(T)}\in\mathbf{St}_{T}$:

$G\left(\rho^{(D)}\otimes\rho^{(T)}\right)=\sum_{k,l=1}^{N_{D}}\vec{v}^{(k)}\cdot\vec{v}^{(l)}\rho^{(D)}_{kl}\ket{k}\bra{l}\otimes
U^{(k)}\rho^{(T)}U^{(l)\dagger},$

where $\vec{v}^{(k)}\cdot\vec{v}^{(l)}=\sum_{j=1}^{M}v_{j}^{(k)}v_{j}^{(l)*}$
is the dot product in $\mathbb{C}^{M}$, and $\rho^{(D)}_{kl}$ are the
components of $\rho^{(D)}$ in the basis
$\left\\{\ket{1},...,\ket{N_{D}}\right\\}$ associated to the list of
projectors $\vec{\Pi}$.

###### Proof.

Take arbitrary elements $G\in\mathcal{G}_{X}$, $\rho^{(D)}\in\mathbf{St}_{D}$
and $\rho^{(T)}\in\mathbf{St}_{T}$. Since $G$ is a CPTP operator, then, for
some $M\geq 1$, there exists a set of Kraus operators
$\left\\{K_{1},...,K_{M}\right\\}$ on $\mathcal{H}_{D}\otimes\mathcal{H}_{T}$
that satisfy $\sum_{j}K_{j}^{\dagger}K_{j}=\mathbb{1}$ and

$G\left(\rho^{(D)}\otimes\rho^{(T)}\right)=\sum_{j=1}^{M}K_{j}\left(\rho^{(D)}\otimes\rho^{(T)}\right)K_{j}^{\dagger}.$
(21)

For each Kraus operator $K_{j}$ let us introduce a set of $(N_{D})^{2}$
operators $\left\\{B_{j}^{(kl)}\right\\}$ on $\mathcal{H}_{T}$, such that the
following holds:

$K_{j}=\sum_{k,l=1}^{N_{D}}\ket{k}\bra{l}\otimes B_{j}^{(kl)},$ (22)

where the basis vectors $\left\\{\ket{1},...,\ket{N_{D}}\right\\}$ are the
ones associated to the projectors $\vec{\Pi}$. Plugging in decomposition (22)
into the Kraus representation (21) we obtain

$G\left(\rho^{(D)}\otimes\rho^{(T)}\right)=\sum_{j=1}^{M}\sum_{k,l,n,m=1}^{N_{D}}\rho_{lm}^{(D)}\ket{k}\bra{l}\otimes
B_{j}^{(kl)}\rho^{(T)}B_{j}^{(nm)\dagger}.$ (23)

Since $G$ is an element of $\mathcal{G}$, it satisfies

$\operatorname{Tr}_{D}\left[\left(\Pi_{s}\otimes\mathbb{1}\right)G\left(\rho^{(D)}\otimes\rho^{(T)}\right)\right]=\operatorname{Tr}_{D}\left(\Pi_{s}\rho^{(D)}\right)U^{(s)}\rho^{(T)}U^{(s)\dagger},$
(24)

for any $s=1,...,N_{D}$. Together with Eq. (23), this implies that the
following holds for all $\rho^{(D)},\rho^{(T)}$:

$\sum_{j=1}^{M}\sum_{l,m=1}^{N_{D}}\rho_{lm}^{(D)}\otimes
B_{j}^{(sl)}\rho^{(T)}B_{j}^{(sm)\dagger}=\rho_{ss}^{(D)}U^{(s)}\rho^{(T)}U^{(s)\dagger}.$
(25)

Since the latter is valid for all density operators $\rho^{(D)}$, then it must
also hold for for all operators in
$\left\\{\ket{a}\bra{b},a,b=1...,N_{D}\right\\}$, as the latter constitutes a
basis for the space of density operators on $\mathcal{H}_{D}$. Therefore, for
all $s,a,b=1...,N_{D}$, and all $\rho^{(T)}$:

$\sum_{j=1}^{M}B_{j}^{(sa)}\rho^{(T)}B_{j}^{(sb)\dagger}=\delta_{s,a}\delta_{s,b}U^{(s)}\rho^{(T)}U^{(s)\dagger}.$
(26)

It can be easily seen that Eq. (26) implies that, for all $j$ and $s\neq a$:

$B_{j}^{(sa)}=0,$ (27)

which motivates us to define a new set of operators
$\left\\{B^{(s)}_{j}\right\\}$ on $\mathcal{H}_{T}$, such that
$B_{j}^{(sa)}=\delta_{s,a}B_{j}^{(s)}$. Eq. (26) then implies that for all
$s=1,...,N_{D}$:

$\sum_{j=1}^{M}B_{j}^{(s)}\rho^{(T)}B_{j}^{(s)\dagger}=U^{(s)}\rho^{(T)}U^{(s)\dagger}.$
(28)

Getting back to the initially introduced Kraus operators, it is simple to
deduce from the normalization condition
$\sum_{j}K_{j}^{\dagger}K_{j}=\mathbb{1}$ that the auxiliary operators
$B_{j}^{(kl)}$ satisfy

$\sum_{j=1}^{M}\sum_{k=1}^{N_{D}}B_{j}^{(kl)\dagger}B_{j}^{(kn)}=\delta_{l,n}\mathbb{1},$
(29)

for all $l,n=1,...,N_{D}$. This in turn implies that the newly defined
operators $\left\\{B^{(s)}_{j}\right\\}$ satisfy

$\sum_{j=1}^{M}B_{j}^{(s)\dagger}B_{j}^{(s)}=\mathbb{1}.$ (30)

Eqs. (28) and (30) entail that for each $s$, the set
$\left\\{B_{j}^{(s)},j=1,...,M\right\\}$ is a Kraus representation of the CPTP
map $\mathcal{C}^{(s)}:\textbf{St}_{T}\rightarrow\textbf{St}_{T}$ that acts as

$\mathcal{C}^{(s)}\left(\rho^{(T)}\right)=U^{(s)}\rho^{(T)}U^{(s)\dagger},$
(31)

for all $\rho^{(T)}\in\textbf{St}_{T}$. One Kraus representation of
$\mathcal{C}^{(s)}$ is trivially given by $B_{j}^{(s)}=\delta_{j,1}U^{(s)}$.
Now recall that all Kraus representations of a CPTP map are unitarily
equivalent, meaning that for any two representations $\left\\{B_{j}\right\\}$
and $\left\\{A_{j}\right\\}$ of the same map, there exists a unitary operator
$u$ on $\mathbb{C}^{M}$, such that $A_{i}=\sum_{j=1}^{M}u_{ij}B_{j}$.
Therefore, for all $s$, all sets that satisfy Eqs. (28) and (30) are unitarily
equivalent to $\left\\{\delta_{j,1}U^{(s)}\right\\}$, which means that a set
$\left\\{B_{j}^{(s)},j=1,...,M\right\\}$ satisfies Eqs. (28) and (30) if and
only if there exists a unitary operator $u^{(s)}$, such that
$B_{j}^{(s)}=u^{(s)}_{j1}U^{(s)}$. Since the column
$\left(u^{(s)}_{11},...,u^{(s)}_{M1}\right)$ of any unitary operator $u^{(s)}$
on $\mathbb{C}^{M}$ is just a unit vector in $\mathbb{C}^{M}$, it follows that
$\left\\{B_{j}^{(s)}\right\\}$ is a viable solution if and only if there
exists a unit vector $\vec{v}\in\mathbb{C}^{M}$ such that
$B_{j}^{(s)}=v^{(s)}_{j}U^{(s)}$, with $v^{(s)}_{j}$ being the $j$-th
component of vector $\vec{v}^{(s)}$ in some basis.
It thus follows from the defining Eq. (22) that for any $G\in\mathcal{G}_{X}$,
and any $M$-element-set of Kraus operators $\left\\{K_{1},...,K_{M}\right\\}$
of $G$, there exists a set $\left\\{\vec{v}_{1},...,\vec{v}_{N_{D}}\right\\}$
of unit vectors in $\mathbb{C}^{M}$, such that

$K_{j}=\sum_{s=1}^{N_{D}}v^{(s)}_{j}\ket{s}\bra{s}\otimes U^{(s)},$ (32)

for all $j=1,...,M$. Eq. (32), together with the definition of the Kraus
representation, entails that for all $G\in\mathcal{G}$, there exists a natural
number $M\geq 1$ and a set $\left\\{\vec{v}_{1},...,\vec{v}_{N_{D}}\right\\}$
of unit vectors in $\mathbb{C}^{M}$, such that for all
$\rho^{(D)}\in\mathbf{St}_{D}$ and $\rho^{(T)}\in\mathbf{St}_{T}$:

$G\left(\rho^{(D)}\otimes\rho^{(T)}\right)=\sum_{k,l=1}^{N_{D}}\vec{v}^{(k)}\cdot\vec{v}^{(l)}\rho^{(D)}_{kl}\ket{k}\bra{l}\otimes
U^{(k)}\rho^{(T)}U^{(l)\dagger},$ (33)

where $\vec{v}^{(k)}\cdot\vec{v}^{(l)}=\sum_{j=1}^{M}v_{j}^{(k)}v_{j}^{(l)*}$
is the dot product in $\mathbb{C}^{M}$. ∎

Let us briefly comment on two special cases of Eq. (33). Consider a
transformation $G$, which is such that all its corresponding vectors
$\vec{v}^{(i)}$ are equal, i.e. $\vec{v}^{(i)}\cdot\vec{v}^{(j)}=1$ for all
$i,j$. Then it is easy to see that $G$ represents an ideal unitary gate, such
as the one that we have used in the main text’s protocol, i.e.

$G(\rho^{(D)}\otimes\rho^{(T)})=U(\rho^{(D)}\otimes\rho^{(T)})U^{\dagger},\quad
U=\sum_{k=1}^{N_{D}}\ket{k}\bra{k}\otimes U^{(k)}.$ (34)

On the other hand, consider a gate $G$ whose vectors are all mutually
orthogonal, i.e. $\vec{v}^{(i)}\cdot\vec{v}^{(j)}=\delta_{i,j}$; then $G$ acts
as

$G(\rho^{(D)}\otimes\rho^{(T)})=\sum_{k=1}^{N_{D}}\rho^{(D)}_{kk}\ket{k}\bra{k}\otimes
U^{(k)}\rho^{(T)}U^{(k)\dagger}.$ (35)

The latter gate erases the off-diagonal terms of $\rho^{(D)}$ in the basis
defined by projectors $\vec{\Pi}$. In other words, it decoheres the device in
the aforementioned basis, and may thus as well be understood as a “classical
control gate”.

Let us now use the obtained results to prove that any protocol that solves our
task necessarily establishes entanglement between the target and the devices
if $k=1$. Applying Definition 1 to Eq. (20), we see that the set
$\mathcal{G}_{Y}\equiv\mathcal{G}$ of generalized control gates that are
allowed in our task is defined relative to the 4-tuple $Y\equiv\langle
N_{D},N_{T},\vec{U},\vec{\Pi}\rangle$, with the following identifications:
$N_{D}=N$ and $N_{T}=N+1$, where $N$ is the number of target objects;
$\vec{U}=\langle S_{1},...,S_{N}\rangle$ where the unitary operators $S_{i}$
are swap-gates; and $\vec{\Pi}=\langle\Pi_{1},...,\Pi_{N}\rangle$ is the set
of projectors on states representing settings associated to the corresponding
swap-gates. Lemma 1 then implies that for any $G\in\mathcal{G}_{Y}$, there
exists a natural number $M\geq 1$ and a set
$\left\\{\vec{v}^{(1)},...,\vec{v}^{(N_{D})}\right\\}$ of unit vectors in
$\mathbb{C}^{M}$, such that for all $\rho^{(D)}\in\mathbf{St}_{D}$ and
$\rho^{(T)}\in\mathbf{St}_{T}$:

$G\left(\rho^{(D)}\otimes\rho^{(T)}\right)=\sum_{k,l=1}^{N}\vec{v}^{(k)}\cdot\vec{v}^{(l)}\rho^{(D)}_{kl}\ket{k}\bra{l}\otimes
S_{k}\rho^{(T)}S_{l},$ (36)

where $\vec{v}^{(k)}\cdot\vec{v}^{(l)}=\sum_{j=1}^{M}v_{j}^{(k)}v_{j}^{(l)*}$
is the dot product in $\mathbb{C}^{M}$. If the unknown bit $k$ is equal to 0,
then $\rho^{(T)}$ is equal to $\rho^{(T)}_{0}=\ket{0}\bra{0}$. Therefore,
since $S_{k}\ket{0}\bra{0}S_{l}=\ket{0}\bra{0}$, then

$G\left(\rho^{(D)}\otimes\rho^{(T)}_{0}\right)=\sum_{k,l=1}^{N}\vec{v}^{(k)}\cdot\vec{v}^{(l)}\rho^{(D)}_{kl}\ket{k}\bra{l}\otimes\rho^{(T)}_{0}.$
(37)

On the other hand, if $k=1$, then $\rho^{(T)}$ is equal to
$\rho^{(T)}_{1}=\ket{1}\bra{1}$. Since
$S_{k}\ket{1}\bra{1}S_{l}=\ket{k}\bra{l}$, then

$G\left(\rho^{(D)}\otimes\rho^{(T)}_{1}\right)=\sum_{k,l=1}^{N}\vec{v}^{(k)}\cdot\vec{v}^{(l)}\rho^{(D)}_{kl}\ket{k}\bra{l}\otimes\ket{k}\bra{l}.$
(38)

If we define the corresponding reduced density states of $\mathcal{D}$ as
$\tilde{\rho}^{(D)}_{k}\equiv\operatorname{Tr}_{T}\left[G\left(\rho^{(D)}\otimes\rho^{(T)}_{k}\right)\right]$,
then Eqs. (37) and (38) imply

$\begin{split}&\tilde{\rho}^{(D)}_{0}=\sum_{i,j=1}^{N}\alpha_{ij}\ket{i}\bra{j},\\\
&\tilde{\rho}^{(D)}_{1}=\sum_{i}^{N}\alpha_{ii}\ket{i}\bra{j},\end{split}$
(39)

where $\alpha_{ij}\equiv\vec{v}^{(i)}\cdot\vec{v}^{(j)}\rho^{(D)}_{ij}$.
In order to be able to acquire some information about the value $k$ via a
measurement on $\mathcal{D}$, the two states $\tilde{\rho}^{(D)}_{0}$ and
$\tilde{\rho}^{(D)}_{1}$ cannot be equal. Therefore, Eq. (39) implies that the
probability $P_{W}$ of successfully solving the task can be higher than
$\frac{1}{2}$ if and only if there exists at least one pair $n\neq m$, such
that $\alpha_{nm}\neq 0$. The latter is equivalent to the requirement that
$\rho^{(D)}_{nm}\neq 0$ and $\vec{v}^{(n)}\cdot\vec{v}^{(m)}\neq 0$, which
means that the initial state of $\mathcal{D}$ needs to have some coherence in
the preferred basis chosen by projectors $\vec{\Pi}$, and that the gate cannot
fully decohere the control in the latter basis, i.e. it cannot be a “classical
control gate”. This establishes a sense in which the task cannot be solved
classically: the initial state of the control needs to be “non-classical” in
the sense of having coherence in the aforementioned basis, and the interaction
between the device and the targets need to be non-classical, in the sense of
preserving the device’s coherence terms.
Now we are finally ready to inspect whether it is possible to gain some
information about $k$ without establishing entanglement between the device and
the targets when $k=1$. In other words, if $\alpha_{nm}\neq 0$ for at least
one pair $n\neq m$, is the state
$G\left(\rho^{(D)}\otimes\rho^{(T)}_{1}\right)$ necessarily entangled? We are
going to answer this question by employing the PPT criterion, which states
that a necessary condition for a quantum state $\rho$ defined on
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ to be separable is that the partially
transposed state $\rho^{T_{B}}\equiv(\mathbb{1}\otimes T)\rho$ does not have
negative eigenvalues, where the transpose $T$ acts as
$T\ket{i}\bra{j}=\ket{j}\bra{i}$ peres1996separability . Consequently, a
sufficient (but generally not necessary) condition for a state to be entangled
is that its partial transpose has at least one negative eigenvalue.
Let us apply the partial transpose operation on $\rho_{1}\equiv
G\left(\rho^{(D)}\otimes\rho^{(T)}_{1}\right)$, thereby obtaining

$(\mathbb{1}\otimes
T)\rho_{1}=\sum_{j}\alpha_{jj}\ket{j}\bra{j}\otimes\ket{j}\bra{j}+\sum_{m<n}\alpha_{mn}\ket{m}\bra{n}\otimes\ket{n}\bra{m}+\sum_{m<n}\alpha_{mn}^{*}\ket{n}\bra{m}\otimes\ket{m}\bra{n}.$
(40)

Now suppose that there exist $k\neq l$, for which $\alpha_{kl}\neq 0$. Then it
is simple to check that the vector $\ket{\psi}_{kl}$ defined as

$\ket{\psi}_{kl}\equiv\ket{k}\otimes\ket{l}-\frac{\alpha_{kl}^{*}}{|\alpha_{kl}|}\ket{l}\otimes\ket{k}$
(41)

is an eigenvector of $(\mathbb{1}\otimes T)\rho_{1}$ with its corresponding
eigenvalue equal to $-|\alpha_{kl}|$, which is negative. Therefore, if
$\alpha_{kl}\neq 0$ for some $k\neq l$, then the state $\rho_{1}$ is
necessarily entangled. We thereby completed the proof that a quantum-
mechanical protocol that involves indistinguishable particles can outperform a
random guess if and only if the device and the targets get entangled upon
interaction when $k=1$.


* (71) M. J. D. Powell, _The convergence of variable metric methods for nonlinearly constrained optimization calculations_ in: O.L. Mangasarian, R. R. Meyer and S. M. Robinson, eds., _Nonlinear Programming 3_ , Academic Press, New York, pp. 27-63, 1978.
  * (72) K. Schittkowski, _NLPQL: A fortran subroutine solving constrained nonlinear programming problems_ , Ann Oper Res 5 (1986), 485-500, available at https://doi.org/10.1007/BF02739235.
  * (73) J. Schropp, _A dynamical systems approach to constrained minimization_ , Numer Funct Anal Optim 21 (2000), 537-551.
  * (74) J. Schropp, _One and multistep discretizations of index 2 differential algebraic systems and their use in optimization_ , J Comput Appl Math 150 (2003), 375-396.
  * (75) L. F. Shampine, I. Gladwell and S. Thompson, _Solving ODEs with MATLAB_ , Cambridge University Press, Cambridge, 2003.
  * (76) D. F. Shanno, _Conditioning of quasi-Newton methods for function minimization_ , Math Comput 24 (1970), 647-656.
  * (77) G. Steidl and T. Teuber, _Removing multiplicative noise by Douglas-Rachford splitting methods_ , J Math Imaging Vis 36 (2010), 168-184.
  * (78) S. Surjanovic and D. Bingham, _Virtual library of simulation experiments: Test functions and datasets_ , retrieved from http://www.sfu.ca/~ssurjano, January 2020.
  * (79) W. Y. Sun and Y. X. Yuan, _Optimization Theory and Methods: Nonlinear Programming_ , Springer, New York, 2006.
  * (80) K. Tanabe, _A geometric method in nonlinear programming_ , J Optim Theory Appl 30 (1980), 181-210.
  * (81) A. N. Tikhonov, _The stability of inverse problems_ , Dokl Akad Nauk SSRR 39 (1943), 176-179.
  * (82) A. N. Tikhonov and V. Y. Arsenin, _Solutions of Ill-posed Problems_ , John Wiley & Sons, New York-Toronto-London, 1977.
  * (83) R. Vanderbei, K. Lin, H. Liu and L. Wang, _Revisiting compressed sensing: exploiting the efficiency of simplex and sparsification methods_ , Math Prog Comp 8 (2016), 253-269, http://doi.org/10.1007/s12532-016-0105-y.
  * (84) Z.-W. Wen and W.-T. Yin, _A feasible method for optimization with orthogonality constraints_ , Math Program 142 (2013), 397-434.
  * (85) R. B. Wilson, _A Simplicial Method for Convex Programming_ , Ph.D. thesis, Harvard University, 1963.
  * (86) D. M. Witten, R. Tibshirani and T. Hastie, _A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis_ , Biostatistics 10 (2009), 515-534.
  * (87) H. Yamashita, _A differential equation approach to nonlinear programming_ , Math Program 18 (1980), 155-168.
  * (88) Y. Yuan, _Recent advances in trust region algorithms_ , Math Program 151 (2015), 249-281.


# Natural convection above circular disks of evaporating liquids

Benjamin Dollet  Institut de Physique de Rennes, UMR 6251 CNRS and Université
Rennes 1, Campus Beaulieu, Bâtiment 11A, 35042 Rennes Cedex, France  François
Boulogne  Laboratoire de Physique des Solides, CNRS, Univ. Paris-Sud,
Université Paris-Saclay, Orsay 91405, France

###### Abstract

We investigate theoretically and experimentally the evaporation of liquid
disks in the presence of natural convection due to a density difference
between the vapor and the surrounding gas. From the analogy between thermal
convection above a heated disk and our system, we derive scaling laws to
describe the evaporation rate. The local evaporation rate depends on the
presence of a boundary layer in the gas phase such that the total evaporation
rate is given by a combination of different scaling contributions, which
reflect the structure of the boundary layer. We compare our theoretical
predictions to experiments performed with water in an environment controlled
in humidity, which validate our approach.

##  1 Introduction

The evaporation of small liquid disks takes its roots in botany, in particular
with some studies on the transpiration of plants published in the early
twentieth century [1, 2, 3, 4]. Beyond this original inspiration, evaporation
is an ubiquitous phenomenon in nature as for the evaporation of liquids from
water drops [5] to lakes or oceans [6] and in the industry, in particular for
coating processes. Indeed, the transport of solutes is often driven by
evaporation and dictates the self-organization of particles at micro and
macroscopic lengthscales [7]. Therefore, a correct modeling of the evaporation
dynamics is crucial for understanding evaporation kinetics and colloidal
deposition.

When the evaporation is limited by the diffusion in the vapor phase [8], the
derivation of the evaporative flux shows that the total evaporation rate is
linear with the radius of the liquid surface [9]. In 2006, Shahidzadeh-Bonn et
al. reported experimental observations of evaporating drops of water and
hexane [10]. They observed that while a drop of hexane evaporates as predicted
by a diffusive model, a drop of water has an anomalous evaporation rate above
a certain radius. They attributed this different behavior to the natural
convection that takes place, as water vapor is less dense than air. The
dimensionless number representing the balance between the buoyant forces in
favor of convection and the viscous forces in favor of diffusion, is the
Grashof number defined as

${\rm
Gr}=\left|\frac{\rho_{s}-\rho_{\infty}}{\rho_{\infty}}\right|\frac{gR^{3}}{\nu^{2}},$
(1)

where $\rho_{s}$ and $\rho_{\infty}$ are respectively the vapor density at
saturation and at infinity, $g$ the gravity constant, $R$ the radius of the
drop and $\nu$ the kinematic viscosity of air. For small Grashof numbers,
evaporation is limited by diffusion while for large Grashof numbers, buoyancy
creates a convective flow.

Additional experimental questioning [11] and evidences [12, 13, 14, 15, 16,
17] of the importance of convective effects have been reported more recently
in the literature. Direct visualizations of evaporating drops have been
achieved by X-ray imaging method [12], IR absorption [14], Schlieren technique
[18] or by interferometric measurements [19] and these studies concluded that
evaporation can be enhanced by convection. The transition between diffusive
and convective evaporation regimes has been reported and the convective
evaporation rate can be captured as ${\rm Gr}^{\beta}$ with $\beta\approx
0.20$ [13, 14, 15].

As we expect from the definition of the Grashof number (Eq. (1)), the
threshold for convective evaporation is a function of the radial lengthscale
of the evaporating surface. Recently, Carrier et al. investigated the mutual
influence of closely deposited drops on the evaporation rate [20] as it can be
encountered for a sprayed liquid. If the drops are separated by a distance
comparable to their radius, they found that a cooperative effect induces
convective evaporation. In addition, they studied the evaporation of circular
evaporating surfaces of different radii and they concluded that for large
Grashof numbers, convection dominates with an evaporation rate that scales as
$R^{2}$.

From the point of view of the governing equations, the evaporation of a liquid
is similar to the dissolution of a liquid into another. Recent attention has
been devoted to the sessile drop dissolution [21, 22, 23] and bubble growth in
supersaturated solutions [24] with a combination of experimental, numerical
and theoretical investigations. These studies show that natural convection is
also observed above dissolving drops. The main difference between evaporation
and dissolution is the magnitude of the Schmidt number $\mathrm{Sc}=\nu/{\cal
D}$ defined as the ratio of the kinematic viscosity $\nu$ of the surrounding
phase and the diffusivity ${\cal D}$ of the molecules. For evaporation, the
Schmidt number is close to unity as diffusivity and kinematic viscosity are
similar for gases. In contrast, the Schmidt number is large for dissolution
[22]. This difference of Schmidt number is important for the structure of the
boundary layer of the convective flow [25].

In this paper, we propose to derive scaling laws for a flat circular
evaporating surface in a regime dominated by convection. We base our analysis
on an analogy between convective evaporation of flat circular surfaces and the
thermal convection that takes place above a heated disk. We show that our
prediction is in agreement with our measurements and with some empirical
predictions available in the literature.

##  2 Model

As stated in the introduction, the evaporation of the liquid changes the
composition of the surrounding atmosphere and thus, its local density.
Therefore, we consider the natural convection that can take place above a
circular disk of liquid. In this Section, we establish a scaling law between
the evaporation flux and the radius of the flat drop. By analogy between heat
and mass transfer, this derivation closely follows that of the heat transfer
above a horizontal heated surface, which is a classical problem [26, 27, 28,
29]. Dehaeck et al. have also performed a detailed study of the role of
natural convection on an evaporating pendant droplet, see the Supporting
Information of [19]. In particular, they discussed the role of the varying
slope of the droplet, and of thermal Marangoni flows induced in the droplet by
the latent heat released during evaporation. However, these papers either
considered a plate geometry instead of a disk [26, 27], or started from
dimensionless or simplified equations [28, 29, 19], which is why we prefer to
present the derivation in details.

Figure 1: Sketch presenting the notations for the convective evaporation above
a circular disk of volatile liquid.

###  2.1 Diffusion-convection equations

We establish the equations to describe the flow in the gas phase and our
analysis requires the following assumptions. (i) The liquid-vapor interface is
horizontal. (ii) The flow of vapor is axisymmetric and in a steady state.
(iii) Thermal effects are negligible. (iv) We use the Boussinesq
approximation: the air density is assumed constant, except in the terms where
it acts as a driving force for the flow; in particular, the flow is considered
incompressible.

Under these conditions, the velocity field of the vapor above the drop writes
$\mathbf{u}=u\,\mathbf{e}_{r}+w\,\mathbf{e}_{z}$ and depends only on the
cylindrical coordinates $r$ and $z$, with origin at the center of the drop
interface (Fig. 1). The continuity equation is

$\frac{1}{r}\frac{\partial(ru)}{\partial r}+\frac{\partial w}{\partial z}=0.$
(2)

With $p$ the pressure field, $c$ the mass concentration field, $\rho(c)$ the
gas density field and $\mu$ its dynamic viscosity, the radial and vertical
components of the Navier-Stokes equation are respectively

$\displaystyle\rho_{\infty}\left(u\frac{\partial u}{\partial
r}+w\frac{\partial u}{\partial z}\right)$ $\displaystyle=-\frac{\partial
p}{\partial r}+$ $\displaystyle\mu\left(\frac{\partial^{2}u}{\partial
r^{2}}+\frac{1}{r}\frac{\partial u}{\partial
r}-\frac{u}{r^{2}}+\frac{\partial^{2}u}{\partial z^{2}}\right),$ (3a)
$\displaystyle\rho_{\infty}\left(u\frac{\partial w}{\partial
r}+w\frac{\partial w}{\partial z}\right)$
$\displaystyle=-\rho(c)g-\frac{\partial p}{\partial z}+$
$\displaystyle\mu\left(\frac{\partial^{2}w}{\partial
r^{2}}+\frac{1}{r}\frac{\partial w}{\partial r}+\frac{\partial^{2}w}{\partial
z^{2}}\right).$ (3b)

In these equations, according to the Boussinesq approximation, the gas density
is taken as constant and equal to its value at infinity $\rho_{\infty}$,
except in the driving force term of natural convection, $\rho(c)g$ in the
equation (3b) of vertical motion. From the ideal gas law, the gas density
varies linearly with the vapor concentration:

$\rho(c)=\rho_{0}-\Delta\rho\frac{c}{c_{s}},$ (4)

where $\rho_{0}$ is the density of dry air, $\Delta\rho=\rho_{0}-\rho_{s}$ the
density difference between pure air and air saturated with vapor, and $c_{s}$
the saturation concentration of vapor in air. Finally, the diffusion-
convection equation for the concentration field is

$u\frac{\partial c}{\partial r}+w\frac{\partial c}{\partial z}={\cal
D}\left[\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial
c}{\partial r}\right)+\frac{\partial^{2}c}{\partial z^{2}}\right],$ (5)

where ${\cal D}$ is the diffusion coefficient of vapor in air.

Let $p=p_{\mathrm{stat}}+\varpi$ with $p_{\mathrm{stat}}$ the hydrostatic
pressure, such that $-\rho_{\infty}g-\partial p_{\mathrm{stat}}/\partial z=0$.
The Navier-Stokes equations (3) become:

$\displaystyle\rho_{\infty}\left(u\frac{\partial u}{\partial
r}+w\frac{\partial u}{\partial z}\right)$
$\displaystyle=-\frac{\partial\varpi}{\partial r}+$
$\displaystyle\mu\left(\frac{\partial^{2}u}{\partial
r^{2}}+\frac{1}{r}\frac{\partial u}{\partial
r}-\frac{u}{r^{2}}+\frac{\partial^{2}u}{\partial z^{2}}\right),$ (6a)
$\displaystyle\rho_{\infty}\left(u\frac{\partial w}{\partial
r}+w\frac{\partial w}{\partial z}\right)$
$\displaystyle=-\left[\rho(c)-\rho_{\infty}\right]g-\frac{\partial\varpi}{\partial
z}+$ $\displaystyle\mu\left(\frac{\partial^{2}w}{\partial
r^{2}}+\frac{1}{r}\frac{\partial w}{\partial r}+\frac{\partial^{2}w}{\partial
z^{2}}\right).$ (6b)

We now nondimensionalize equations (2), (5) and (6) by the characteristic
length scale $R$, the characteristic velocity $\nu/R$ and the characteristic
pressure $\rho_{\infty}\nu^{2}/R^{2}$. We thus introduce the following
dimensionless quantities, denoted by a tilde: $\tilde{r}=r/R$,
$\tilde{z}=z/R$, $\tilde{u}=Ru/\nu$, $\tilde{w}=Rw/\nu$ and
$\tilde{p}=R^{2}p/\rho_{\infty}\nu^{2}$. We also define a dimensionless
concentration field $\tilde{c}=(c-c_{\infty})/(c_{s}-c_{\infty})$, where
$c_{\infty}$ is the vapor concentration far from the drop. We then obtain:

$\displaystyle\frac{1}{\tilde{r}}\frac{\partial(\tilde{r}\tilde{u})}{\partial\tilde{r}}+\frac{\partial\tilde{w}}{\partial\tilde{z}}$
$\displaystyle=0,$ (7a)
$\displaystyle\tilde{u}\frac{\partial\tilde{u}}{\partial\tilde{r}}+\tilde{w}\frac{\partial\tilde{u}}{\partial\tilde{z}}$
$\displaystyle=-\frac{\partial\tilde{\varpi}}{\partial\tilde{r}}+\frac{\partial^{2}\tilde{u}}{\partial\tilde{r}^{2}}+\frac{1}{\tilde{r}}\frac{\partial\tilde{u}}{\partial\tilde{r}}-\frac{\tilde{u}}{\tilde{r}^{2}}+\frac{\partial^{2}\tilde{u}}{\partial\tilde{z}^{2}},$
(7b)
$\displaystyle\tilde{u}\frac{\partial\tilde{w}}{\partial\tilde{r}}+\tilde{w}\frac{\partial\tilde{w}}{\partial\tilde{z}}$
$\displaystyle=-\frac{\partial\tilde{\varpi}}{\partial\tilde{z}}+\mathrm{Gr}\,\tilde{c}+\frac{\partial^{2}\tilde{w}}{\partial\tilde{r}^{2}}+\frac{1}{\tilde{r}}\frac{\partial\tilde{w}}{\partial\tilde{r}}+\frac{\partial^{2}\tilde{w}}{\partial\tilde{z}^{2}},$
(7c)
$\displaystyle\tilde{u}\frac{\partial\tilde{c}}{\partial\tilde{r}}+\tilde{w}\frac{\partial\tilde{c}}{\partial\tilde{z}}$
$\displaystyle=\frac{1}{\mathrm{Sc}}\left[\frac{1}{\tilde{r}}\frac{\partial}{\partial\tilde{r}}\left(\tilde{r}\frac{\partial\tilde{c}}{\partial\tilde{r}}\right)+\frac{\partial^{2}\tilde{c}}{\partial\tilde{z}^{2}}\right],$
(7d)

with two dimensionless numbers, the Grashof number defined by equation (1),
and the Schmidt number $\mathrm{Sc}=\nu/{\cal D}$, which is of order one
because all diffusivities have the same order of magnitude in gases.

Since the liquid is out of equilibrium with the atmosphere, a mass transfer
occurs at the liquid-vapor interface. From Fick’s law, the local flux normal
at the liquid-vapor interface is given by

$j(r)=-{\cal D}\left.\frac{\partial c(r,z)}{\partial z}\right|_{z=0},$ (8)

where the concentration gradient is taken at the liquid-vapor interface, i.e.
$z=0$. The total evaporating flux is given by

$Q=\int_{S}j(r)\,{\rm d}S,$ (9)

where $S$ is the surface of the liquid-vapor interface.

To determine these local and total fluxes, we must establish the vapor
concentration gradient at the interface. Therefore, we analyze in the next
paragraphs the set of equations (7) to derive this concentration gradient.

###  2.2 Diffusion-limited evaporation

For small Grashof numbers, the unique driving term of the gas flow, namely
$\mathrm{Gr}\,\tilde{c}$ in equation (7c), vanishes. From the definition of
the Grashof number given by equation (1), and from (4), the condition ${\rm
Gr}<1$ is equivalent to a drop radius smaller than $R^{\star}$ defined as

$R^{\star}=\left(\frac{c_{s}-c_{\infty}}{c_{\infty}}\frac{\nu^{2}}{g}\right)^{1/3}.$
(10)

Hence, in this condition, there is no gas flow i.e. $\tilde{u}=\tilde{w}=0$,
and the set of equations (7) reduces to the Laplace equation for the
concentration field. From equation (7d), the Laplace equation writes

$\frac{1}{\tilde{r}}\frac{\partial}{\partial\tilde{r}}\left(\tilde{r}\frac{\partial\tilde{c}}{\partial\tilde{r}}\right)+\frac{\partial^{2}\tilde{c}}{\partial\tilde{z}^{2}}=0.$
(11)

The boundary conditions are a saturated vapor concentration at the surface of
the disk $c(r<R,z=0)=c_{s}$ and a vapor concentration $c_{\infty}$ far from
the disk. Introducing the oblate spheroidal coordinates ($\kappa,\sigma$)
defined as $\tilde{r}^{2}=(1-\kappa^{2})(1+\sigma^{2})$ and
$\tilde{z}=\kappa\sigma$, the vapor concentration can be written in the simple
form [30, 31]

$\tilde{c}(\kappa,\sigma)=1-\frac{2}{\pi}\arctan(\sigma).$ (12)

This solution is represented in Fig. 2.

Figure 2: Dimensionless vapor concentration map above an evaporating disk in
the dimensionless space $(\tilde{r},\tilde{z})$ obtained from equation (12).
White and cyan lines are the oblate spheroidal coordinates $(\kappa,\sigma)$,
for respectively constant $\kappa$ and $\sigma$ values.

Thus, for a diffusion-limited evaporation, the local flux calculated from
equation (8), is

$j_{\rm diff}(r)=\frac{2}{\pi}\frac{{\cal
D}(c_{s}-c_{\infty})}{\sqrt{R^{2}-r^{2}}},$ (13)

which is the well-known flux for a drop of a small contact angle. This flux
presents a divergence at the edge that can be interpreted as a tip effect by
analogy with electrostatic problems. Substituting equation (13) in equation
(9), we derive the total flux

$Q_{\rm diff}=4{\cal D}(c_{s}-c_{\infty})R.$ (14)

Equations (13) and (14) have been largely commented in the literature,
particularly in the frames of evaporating droplet lifetime [32, 33] and the
so-called coffee stain effect [9, 34, 35, 36, 17]. Therefore, we do not
develop further the diffusion-limited case and we analyze in the next
paragraph the convective evaporation.

###  2.3 Convective evaporation

Figure 3:  (a) For a large Grashof number, sketch representing the three
characteristic zones of the boundary layer depicted in red above an
evaporating disk of liquid. The thickness of the boundary layer is given by
Eq. (17). Blue arrows indicate the direction of the gas flow for a light
vapor, $\Delta\rho>0$. (b) Relative surface area of the center, edge and
intermediate zone as a function of the Grashof number. Each area is calculated
from the scaling presented in (a). The corresponding radius $R$ is indicated
for water.

####  2.3.1 Spatial structure of the boundary layer

For large Grashof numbers, a flow is established in the gas phase. The
direction and the spatial structure of this flow depends on the sign of the
density difference $\Delta\rho$, as well as the spatial structure of the flow,
especially at the edge and at the center of the liquid disk. As the
experiments conducted in Section 3 are for a positive density difference, we
henceforth assume that $\Delta\rho>0$. Under this assumption, the flow is
mostly horizontal inwards. Due to this convective flow, unsaturated air is
brought toward the edge of the evaporating film. At some distance from the
edge of the film, a slender boundary layer is established and progressively
thickens. At the center of the disk, the flow field converges, and thus moves
upwards to form a rising plume of wet air.

Consequently, we distinguish three zones as depicted in Fig. 3(a): the _edge_
and the _center_ zones, that do not satisfy the conditions for a slender
boundary layer, in contrast to the _intermediate_ zone, which is slender in
the sense that vertical variations are much sharper than horizontal ones. We
first focus on this intermediate zone where the approximation of a thin
horizontal boundary layer can be applied to determine its thickness variation.
Then, from this first analysis, we precise the horizontal extension of the
intermediate zone and, by consequence, we also define the sizes of the edge
and center zones.

####  2.3.2 Intermediate zone

We assume that the flow is almost horizontal and that the vertical variations
occur over lengthscales mush smaller than the horizontal ones. To retain the
necessary inertial, viscous and pressure terms describing such a flow, the
following rescaling is introduced [26]: $\hat{z}=\mathrm{Gr}^{1/5}\tilde{z}$,
$\hat{u}=\mathrm{Gr}^{-2/5}\tilde{u}$, $\hat{w}=\mathrm{Gr}^{-1/5}\tilde{w}$
and $\hat{\varpi}=\mathrm{Gr}^{-4/5}\tilde{\varpi}$. Substituting these new
variables in equations (7), we obtain:

$\displaystyle\frac{1}{\tilde{r}}\frac{\partial(\tilde{r}\hat{u})}{\partial\tilde{r}}+\frac{\partial\hat{w}}{\partial\hat{z}}=$
$\displaystyle 0,$ (15a)
$\displaystyle\hat{u}\frac{\partial\hat{u}}{\partial\tilde{r}}+\hat{w}\frac{\partial\hat{u}}{\partial\hat{z}}=$
$\displaystyle-\frac{\partial\hat{\varpi}}{\partial\tilde{r}}+$
$\displaystyle\mathrm{Gr}^{-2/5}\left(\frac{\partial^{2}\hat{u}}{\partial\tilde{r}^{2}}+\frac{1}{\tilde{r}}\frac{\partial\hat{u}}{\partial\tilde{r}}-\frac{\hat{u}}{\tilde{r}^{2}}\right)$
$\displaystyle+\frac{\partial^{2}\hat{u}}{\partial\hat{z}^{2}},$ (15b)
$\displaystyle\mathrm{Gr}^{-2/5}\left(\hat{u}\frac{\partial\hat{w}}{\partial\tilde{r}}+\hat{w}\frac{\partial\hat{w}}{\partial\hat{z}}\right)=$
$\displaystyle-\frac{\partial\hat{\varpi}}{\partial\hat{z}}+\tilde{c}+$
$\displaystyle\mathrm{Gr}^{-4/5}\left(\frac{\partial^{2}\hat{w}}{\partial\tilde{r}^{2}}+\frac{1}{\tilde{r}}\frac{\partial\hat{w}}{\partial\tilde{r}}\right)$
$\displaystyle+\mathrm{Gr}^{-2/5}\frac{\partial^{2}\hat{w}}{\partial\hat{z}^{2}},$
(15c)
$\displaystyle\hat{u}\frac{\partial\tilde{c}}{\partial\tilde{r}}+\hat{w}\frac{\partial\tilde{c}}{\partial\hat{z}}=\frac{1}{\mathrm{Sc}}$
$\displaystyle\left[\mathrm{Gr}^{-2/5}\frac{1}{\tilde{r}}\frac{\partial}{\partial\tilde{r}}\left(\tilde{r}\frac{\partial\tilde{c}}{\partial\tilde{r}}\right)+\frac{\partial^{2}\tilde{c}}{\partial\hat{z}^{2}}\right].$
(15d)

Hence, within corrections of relative order $\mathrm{Gr}^{-2/5}$,

$\displaystyle\frac{1}{\tilde{r}}\frac{\partial(\tilde{r}\hat{u})}{\partial\tilde{r}}+\frac{\partial\hat{w}}{\partial\hat{z}}$
$\displaystyle=0,$ (16a)
$\displaystyle\hat{u}\frac{\partial\hat{u}}{\partial\tilde{r}}+\hat{w}\frac{\partial\hat{u}}{\partial\hat{z}}$
$\displaystyle=-\frac{\partial\hat{\varpi}}{\partial\tilde{r}}+\frac{\partial^{2}\hat{u}}{\partial\hat{z}^{2}},$
(16b) $\displaystyle 0$
$\displaystyle=-\frac{\partial\hat{\varpi}}{\partial\hat{z}}+\tilde{c},$ (16c)
$\displaystyle\hat{u}\frac{\partial\tilde{c}}{\partial\tilde{r}}+\hat{w}\frac{\partial\tilde{c}}{\partial\hat{z}}$
$\displaystyle=\frac{1}{\mathrm{Sc}}\frac{\partial^{2}\tilde{c}}{\partial\hat{z}^{2}},$
(16d)

which are the analogue equations that describe natural convection above a
heated disk [29].

Although these equations can only be solved numerically, the full solution is
not necessary to obtain the scaling of the evaporative flux. Starting from the
edge of the drop ($\tilde{r}=1$), a boundary-layer solution emerges. To see
this, we set $\tilde{r}=1-\tilde{x}$ in equations (16), and consider their
behavior at small $\tilde{x}$. We then obtain equations similar to those
describing natural convection above a horizontal heated plate with a straight
edge. As shown by Stewartson [26], such equations admit a self-similar
solution depending on the rescaled variable111Notice that there is a typo in
the definition (9) of $\eta$ in [26]. $\hat{\eta}=\hat{z}/\tilde{x}^{2/5}$.
Merkin [29] showed that, although such a self-similarity is lost when
considering equations (16), it is still possible to use $\hat{\eta}$ as a
rescaling variable for $\hat{z}$. Hence, the concentration gradient is
localized within a boundary layer of dimensionless thickness
$\tilde{x}^{2/5}$. In dimensional units, from the definitions of $\hat{z}$,
$\tilde{z}$ and $\tilde{x}$, the scaling of the boundary layer thickness is
thus

$\delta(r)\approx\frac{(R-r)^{2/5}R^{3/5}}{\mathrm{Gr}^{1/5}}.$ (17)

####  2.3.3 Spatial extension of the three zones

Now, we estimate the spatial extension of the different zones. From equation
(17), the slenderness of the boundary layer scales as

$\frac{\delta}{x}=\frac{1}{\mathrm{Gr}^{1/5}}\left(\frac{R}{x}\right)^{3/5}.$
(18)

As a consequence, starting from the drop edge, the boundary layer becomes
slender only for distances larger than $x=R/\mathrm{Gr}^{1/3}=R^{\star}$ from
Eqs. (1) and (10), which defines the extent of the edge zone; in this domain,
horizontal and vertical variations occur over a similar lengthscale
$R^{\star}$. This is similar to the breakdown of the boundary layer
approximation, for the flow of a fluid at high Reynolds number, at the
immediate vicinity of the leading edge of a solid, which is well known in
fluid mechanics [37]. This effect is often negligible in the estimation of the
friction force on a solid. However, as we discuss later, the contribution of
the edge on the total evaporation rate is significant in our situation.

Close to the center of the drop, the solution given by Eq. (22) also breaks
down, and must match the rising plume. As discussed by Merkin [29], the
situation is complex close to the center, where a strong pressure builds up in
response to the converging horizontal flow. Indeed, the flow structure is
still described by Eqs. (16) but, approaching the center, the boundary layer
splits into two layers: an inner layer in contact with the drop where viscous
and inertial effects balance the pressure buildup while concentration
gradients are not significant, and an outer inviscid layer where the
concentration gradients are significant. The inner layer has a dimensionless
thickness $\tilde{r}^{2/3}$ [29]. Similarly to equation (18), the slenderness
of the inner layer scales as $\mathrm{Gr}^{-1/5}(R/r)^{1/3}$. Thus, the inner
layer cannot be considered as slender for $r\lesssim\mathrm{Gr}^{-3/5}R$,
which defines the extent of the central zone where the boundary-layer
approximation underlying equation (16) breaks down (Fig. 3(a)).

####  2.3.4 Evaporative flux

A proper estimate of the evaporative flux must consider in principle the three
zones (Fig. 3(a)). In the edge zone $x<R^{\star}$, we showed that the boundary
layer has both horizontal and vertical extensions of lengthscale $R^{\star}$.
Thus, we can estimate the scaling of the local flux defined by equation (8) as

$j_{\rm edge}\approx{\cal D}\frac{c_{s}-c_{\infty}}{{R^{\star}}},$ (19)

where ${R^{\star}}$ is the characteristic lengthscale of the vertical vapor
concentration gradient. The domain $x<R^{\star}$ has a surface $2\pi
RR^{\star}$ in the limit of large Grashof numbers, i.e. $R^{\star}\ll R$.
Hence, the evaporative flux in this zone has the following order of magnitude:

$Q_{\rm edge}\approx 2\pi RR^{\star}\frac{{\cal
D}(c_{s}-c_{\infty})}{R^{\star}}=2\pi{\cal D}R(c_{s}-c_{\infty}).$ (20)

Interestingly, this is exactly the same scaling as the evaporation flux of the
whole drop in the purely diffusive regime as shown in Eq. (14).

In the intermediate zone, for $r<R-R^{\star}$, the flux scales as

$j_{{\rm int}}(r)\approx{\cal D}\frac{c_{s}-c_{\infty}}{\delta(r)},$ (21)

where $\delta(r)$ is the boundary layer thickness that corresponds to the
lengthscale of the vapor concentration gradient. From the scaling of
$\delta(r)$ given by Eq. (17), we have

$j_{{\rm int}}(r)\approx\frac{{\cal D}(c_{s}-c_{\infty}){\rm
Gr}^{1/5}}{R^{3/5}(R-r)^{2/5}},$ (22)

and using equation (9) with ${\rm d}S=2\pi r{\rm d}r$, we obtain

$Q_{{\rm int}}\approx 2\pi{\cal
D}R(c_{s}-c_{\infty})\mathrm{Gr}^{1/5}=2\pi\frac{{\cal
D}(c_{s}-c_{\infty})^{6/5}}{c_{\infty}^{1/5}\nu^{2/5}}R^{8/5}.$ (23)

The main prediction of this analysis is that the evaporation flux must be
proportional to $R^{8/5}$.

A corrective term must be established for the center zone. However, the ratio
of the central zone area to the total drop area is $\mathrm{Gr}^{-6/5}$. As
shown in Fig. 3(b), the center zone represents a small portion of the drop.
Thus, we neglect the contribution of the center zone on the total evaporative
flux.

Consequently, from Eqs. (23) and (20), our analysis yields the following law
for the evaporation flux:

$\displaystyle Q_{\rm conv}$ $\displaystyle\approx a_{1}Q_{{ \rm
int}}+a_{2}Q_{{\rm edge}}$ (24a) $\displaystyle\approx 2\pi{\cal
D}R(c_{s}-c_{\infty})(a_{1}\mathrm{Gr}^{1/5}+a_{2}),$ (24b)

where $a_{1}$ and $a_{2}$ are two constant numbers. By inspection of Eq.
(24b), we notice that in a convective regime, the total flux is a combination
of a term reminiscent of a diffusive evaporation regime and a second term
depending on the Grashof number, a signature of the convective flow. In the
following, we perform experiments to measure the evaporation rate for large
Grashof numbers to validate our prediction given by equation (24).

##  3 Experiments

Figure 4:  Sketch of the experimental setup with a scale in a box for which
the atmosphere is controlled in humidity.

Our experiments of controlled evaporation are performed in a box made in
polycarbonate ($50\times 50\times 50$ cm3). A precision scale (Ohaus Pioneer
210 g) with a precision of $0.1$ mg is placed at the center of the box and is
interfaced with a Python code using the pyserial library to record the time
evolution of the weight. The humidity is regulated with a PID controller based
on an Arduino Uno and a humidity sensor (Honeywell HIH-4021-003) positioned
far from the evaporating surface (Fig. 4). Dry air is produced by circulating
ambient air with an air pump (Tetra APS 300) in a container filled with
desiccant made of anhydrous calcium sulfate (Drierite). Moist air is obtained
by bubbling air in water. The relative humidity is set to $R_{H}=50$% in all
of our experiments.

Figure 5: (a) Time evolution of the weight of water for different radii of
circular troughs. The solid lines corresponds to linear fit of the
experimental data points. (b) Measured evaporation rate as a function of the
surface radius and the Grashof number. The dotted blue line is a fit with
equation (24) with a Grashof number to the power $0.18$, and with the
coefficients $a_{1}=0.31$ and $a_{2}=0.48$. The solid line is the diffusive
prediction $Q_{\rm diff}$. (c) Sherwood number ${\rm Sh}$ defined by equation
(26) as a function of the Grashof number. Lines are equivalent to the plot
(b). The inset shows the compensated plot ${\rm Sh}/{\rm Gr}^{1/5}$ vs ${\rm
Gr}$.

Circular troughs of different radii are filled with pure water right to the
brim and a particular attention is paid to get a flat surface. The diffusion
coefficient of water vapor in air is ${\cal D}\approx 2\times 10^{-5}$ m2/s,
and the kinematic viscosity of the gas is $\nu\approx 1.5\times 10^{-5}$ m2/s
in our experimental conditions [17]. The time variation of the weight of these
containers is reported in Fig. 5(a) over a time duration between $6000$ to
$9000$ s. Data points are fitted with a linear function to get the mass
evaporation rate $Q$ such that $\Delta m(t)=-Qt$. In Fig. 5(b), we report the
evaporation rates $Q$ as a function of the radius of the liquid patch and the
equivalent Grashof number as defined by equation (1).

To estimate the Grashof number, we evaluate the density gradient in the
atmosphere due to the difference of vapor concentration. The air density
$\rho$ at a pressure $P_{0}$, a temperature $T$ and a relative humidity
$R_{H}$ is given by [38]

$\rho=\frac{1}{z_{m}}\frac{P_{0}}{{\cal
R}T}M_{d}\left[1-fR_{H}\left(1-\frac{M_{w}}{M_{d}}\right)\frac{P_{s}}{P_{0}}\right],$
(25)

where $z_{m}$ and $f$ are the compressibility and enhancement factors, ${\cal
R}$ the ideal gas coefficient, $M_{d}$ and $M_{w}$ the molar density of dry
and saturated air, and $P_{s}$ the saturated pressure. At room temperature, we
have $f=1$, $z_{m}=1$ [38] and a saturated pressure $P_{s}=2.3$ kPa [39].
Therefore, the density variation at a relative humidity $R_{H}=0.5$ is
$|\rho_{s}-\rho_{\infty}|/\rho_{\infty}\approx 5\times 10^{-3}$. In the next
section, we discuss the evaporation model that we proposed in Section 2 in
comparison with our experimental results.

##  4 Discussion

In Fig. 5(b), we show a fit from equation (24b) that correctly captures our
data. From our experimental results, we determine the values of $a_{1}$ and
$a_{2}$ by fitting equation (24b) with a variable Grashof exponent. The
resulting exponent is 0.18 and coefficient values are $a_{1}=0.31$ and
$a_{2}=0.48$ (Fig. 5b). The two dimensionless coefficients $a_{1}$ and $a_{2}$
are close to unity and the power of the Grashof number is close to $1/5$, in
agreement with the theoretical prediction. The fact that $a_{1}\simeq a_{2}$
and ${\rm Gr}^{1/5}$ is between 1 and 10, shows that the contribution of the
edge ($a_{2}\,Q_{{\rm edge}}$), although smaller, remains significant compared
to the intermediate zone ($a_{1}\,Q_{{\rm int}}$) in our experimental
conditions.

To non-dimensionalize the convective flux, we introduce the Sherwood number
${\rm Sh}$ defined as the ratio of the convective and diffusive fluxes,

${\rm Sh}=\frac{Q_{{\rm conv}}}{Q_{{\rm
diff}}}\approx\frac{\pi}{2}\left(a_{1}{\rm Gr}^{1/5}+a_{2}\right),$ (26)

from equations (14) for $Q_{{\rm diff}}$ and (24b) for $Q_{{\rm conv}}$. In
Fig. 5c, we plot the Sherwood number as a function of the Grashof number.
Thus, we can estimate for instance that convection becomes significant for
${\rm Sh=1.5}$, i.e. the convective flux is 50% higher than the predicted
diffusive flux, which corresponds to ${\rm Gr}\approx 20$.

The evaporation regime in the experiments is convective. To follow the first
data analysis performed by Kelly-Zion et al. [13], we fit our data with a
power law against the disk radius. We obtain that $Q\propto R^{1.39}$, in
agreement with their experimental results on heptane and 3-methylpentane where
the radius exponents are $1.37$ and $1.43$, respectively. Nevertheless, we
must underline that the vapors of heptane and 3-methylpentane are denser than
air, contrary to water vapor. This difference in vapor density can change the
total evaporation rate, especially at the edge of the disk. The similar values
for the exponent can be attributed to the fact that this scaling mainly probes
the effect of the radius due to the intermediate zone, which is insensitive to
the sign of $\Delta\rho$.

In their study, Kelly-Zion et al. [13] also considered that the total
evaporation flux is the result of two significant contributions, one based on
diffusion, and the other on convection. In our theoretical analysis in Section
2, we saw that the evaporative flux is always a diffusive flux evaluated right
at the interface of the evaporating drop. However, it turns out that the
analysis of the spatial structure of the flow of vapor above the drop, and its
separation into two convective-diffusive zones: one slender far from the edge,
and one more isotropic close to the edge, justifies that the evaporative flux
separates into two contributions, one scaling as a purely diffusive flux, the
other one as a convective one in a regime of large Grashof number. Kelly-Zion
et al. [13] studied volatile compounds with vapor heavier than air, contrary
to water vapor. Although this leads to a different flow structure, starting
from the center outwards, this does not affect the scaling derived in Sec. 2.
Therefore, our model justifies a posteriori the hypothesis made by Kelly-Zion
et al. [13] and later by Carle et al. [15], who have added a diffusive and a
convective flux to model their data.

Carrier et al. investigated the evaporation of water with beakers filled to
the rim and they reported two regimes [20]. For radii smaller than $30$ mm,
the evaporation rate is proportional to the radius $R$. For larger radii, they
suggest that the evaporation rate scales as $R^{2}$, for radii up to $300$ mm
and they concluded that this second regime can be attributed to convection.
This power exponent is significantly larger than the value found by Kelly-Zion
et al. [13] and ours. Carrier et al. attributed the difference to measurements
by Kelly-Zion et al. to a crossover between the diffusive and a regime
characterized by the development of convection cells. In the spirit of the
classical Rayleigh-Bénard instability, these authors proposed a direct
transition from a quiescent state for the gas with diffusive exchanges solely,
to an unstable convection state where the gas would be set in motion along
convection cells. However, such a transition appears in configurations where
the gas would be confined between horizontal surfaces, which is far from our
experimental configuration (Fig. 4). Unfortunately, the precise configuration
and the confinement are not reported for the experiments performed by Carrier
et al., so we cannot conclude as to whether the difference in exponent can be
ascribed to a difference in confinement. In our case, the steady single-plume
situation becomes unstable above a certain Grashof number, with the appearance
of multiple plumes and even of a large-scale circulation [40]. However, to the
best of our knowledge, the onset of such an instability is unknown.
Nevertheless, our measurements suggest that, even if such an instability
appears in our range of Grashof numbers, it does not lead to a significant
deviation from the scaling (24).

##  5 Conclusion

In this paper, we used an analogy between thermal convection that occurs above
a uniformly heated disk and the convective evaporation of a circular patch of
liquid. The dimensionless form of the flow equations in the gas phase reveals
a dimensionless parameter called the Grashof number ${\rm Gr}$, which relates
the opposite effects of buoyant and viscous forces. This number increases with
the radius $R$ of the disk as $R^{3}$. Therefore, natural convection is
expected for large Grashof numbers.

In our theoretical analysis, we recalled the vapor concentration field
surrounding an evaporating disk and the related evaporative flux for small
Grashof numbers, i.e. a diffusion-limited evaporation. In addition, we
investigated the evaporation dynamics under natural rising convection. The
evaporating surface must be split in three domains. Near the edge of the disk,
over a distance $R^{\star}$, the boundary layer is not slender and the
evaporative flux is diffusive with a concentration gradient across a distance
$R^{\star}$. In an intermediate zone, the boundary layer can be considered as
slender with a thickness that scales as $(R-r)^{2/5}$ and the evaporative flux
scales as ${\rm Gr^{1/5}}$. Near the center of the disk, the flow is
converging and is directed upward with a more complex boundary layer
structure. The characteristic area ratio of this zone is ${\rm Gr}^{-6/5}$,
such that the precise contribution of the center zone may be neglected in a
first approach, although further experimental and theoretical investigation of
this zone could be performed to clarify its fine structure. Therefore, the
total evaporative flux under natural convection results in a combination of a
diffusive-like flux at the edge and a flux across the boundary layer in the
intermediate zone, and is correctly described by equation (24). The Sherwood
number, which compares the convective flux $Q_{\rm conv}$ to the diffusive
flux $Q_{\rm diff}$ shows that the convective flow has a significant effect
for ${\rm Gr>20}$.

With our experimental measurements, we successfully verify our prediction
under natural convection. The scaling laws of our theoretical developments
provide the different evaporating mechanisms that occur above a circular
liquid disk. In particular, we explain the origin of the convective
evaporation dynamics observed experimentally [13, 15], where a combination of
diffusion and convection-like contributions has been noticed.

Further refinements would require numerical calculations to provide a full
resolution of the gas flow and, therefore, a prediction of the evaporation
rate without unknown prefactors. In addition, it would be interesting to
investigate also the case of drops with a non-zero contact angle to evaluate
how the interface slope can modify the spatial extension of the intermediate
zone. Furthermore, it would be interesting to revisit the case of vapor
heavier than air, for which the flow is expected to be outwards above the
evaporating disk. In such a case, we expect a significant effect of the
geometrical configuration at the edge. If the disk is placed at height, a
downwards gravity current starting from the edge can evacuate the vapor
collected from the disk, playing the same role as the rising plume in the case
of light vapor. In contrast, if the disk is placed on an infinite plane or in
a trough, vapor cannot be evacuated, and accumulates, probably leading to a
reduction of the evaporation rate. From a general perspective, the convective
flow strongly depends on the geometrical aspects, which can lead to complex
expressions of the evaporative flux with effects of boundaries.

##  6 Appendix

In Section 2.1, we assume that the flow in the gas phase is steady in order to
neglect the time derivatives that would appear in equations (3). This
assumption can be justified as follows. For large Grashof numbers, the
characteristic velocity in the gas phase is

${\cal U}=\frac{\nu}{R}{\rm Gr}^{2/5},$ (27)

as suggested by the rescaling $\hat{u}={\rm Gr}^{-2/5}\tilde{u}$ with
$\tilde{u}=Ru/\nu$. Therefore, the characteristic timescale associated to the
natural convection is

$\tau_{\rm conv}=\frac{R}{{\cal U}}=\frac{R^{2}}{\nu}{\rm Gr}^{-2/5}.$ (28)

For a disk radius of water of $R=20$ mm, the Grashof number is ${\rm
Gr}\approx 1.5\times 10^{3}$ and the timescale is $\tau_{\rm conv}\approx 1.4$
s, which is much smaller than the observation timescales.

Equivalently, we can estimate the timescale for small Grashof numbers. For a
diffusion-limited evaporation, the timescale is simply [41]

$\tau_{\rm diff}=\frac{R^{2}}{{\cal D}}.$ (29)

For a millimeter water drop radius, $\tau_{\rm diff}\approx 5\times 10^{-2}$
s, which is also much smaller than the lifetime of such droplet [11].
Consequently, this supports the steady state assumption made in Section 2.1.

## Acknowledgments

We thank F. Ingremeau and H.A. Stone for stimulating discussions. F.B. thanks
M. Decraene for her assistance. This study has been carried out with a funding
support by the ANR (ANR-11-BS04-0030-WAFPI project).

## References

  * [1] B. E. Livingston and W. H. Brown.  Relation of the daily march of transpiration to variations in the water content of foliage leaves.  Botanical Gazette, 53(4):309–330, 1912.
  * [2] N. Thomas and A. Ferguson.  On evaporation from a circular water surface.  Philosophical Magazine Series 6, 34(202):308–321, 1917.
  * [3] N. Thomas and A. Ferguson.  On the reduction of transpiration observations.  Annals of Botany, 31(122):241–255, 1917.
  * [4] H. Jeffreys.  Some problems of evaporation.  The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 35(207):270–280, 1918.
  * [5] A.-M. Cazabat and G. Guéna.  Evaporation of macroscopic sessile droplets.  Soft Matter, 6(12):2591–2612, 2010.
  * [6] E. Bodenschatz, W. Pesch, and G. Ahlers.  Recent developments in Rayleigh-Bénard convection.  Annual Review of Fluid Mechanics, 32(1):709–778, 2000.
  * [7] A. F. Routh.  Drying of thin colloidal films.  Reports on Progress in Physics, 76(4):046603, 2013.
  * [8] I. Langmuir.  The evaporation of small spheres.  Phys. Rev., 12(5):368–370, 1918.
  * [9] R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, and T. A. Witten.  Capillary flow as the cause of ring stains from dried liquid drops.  Nature, 389(6653):827–829, 1997.
  * [10] N. Shahidzadeh-Bonn, S. Rafaï, A. Azouni, and D. Bonn.  Evaporating droplets.  Journal of Fluid Mechanics, 549:307–313, 2006.
  * [11] G. J. Dunn, S. K. Wilson, B. R. Duffy, S. David, and K. Sefiane.  The strong influence of substrate conductivity on droplet evaporation.  Journal of Fluid Mechanics, 623:329–351, 2009.
  * [12] B. M. Weon, J. H. Je, and C. Poulard.  Convection-enhanced water evaporation.  AIP Advances, 1(1):012102, 2011.
  * [13] P.L. Kelly-Zion, C.J. Pursell, S. Vaidya, and J. Batra.  Evaporation of sessile drops under combined diffusion and natural convection.  Colloids and Surfaces A, 381(1–3):31 – 36, 2011.
  * [14] P.L. Kelly-Zion, C.J. Pursell, N. Hasbamrer, B. Cardozo, K. Gaughan, and K. Nickels.  Vapor distribution above an evaporating sessile drop.  International Journal of Heat and Mass Transfer, 65:165 – 172, 2013\.
  * [15] F. Carle, B. Sobac, and D. Brutin.  Experimental evidence of the atmospheric convective transport contribution to sessile droplet evaporation.  Applied Physics Letters, 102(6):061603, 2013.
  * [16] S. Somasundaram, T. N. C. Anand, and S. Bakshi.  Evaporation-induced flow around a pendant droplet and its influence on evaporation.  Physics of Fluids, 27(11), 2015.
  * [17] F. Boulogne, F. Ingremeau, and H. A. Stone.  Coffee-stain growth dynamics on dry and wet surfaces.  Journal of Physics: Condensed Matter, 29(7):074001, 2017.
  * [18] P.L. Kelly-Zion, J. Batra, and C.J. Pursell.  Correlation for the convective and diffusive evaporation of a sessile drop.  International Journal of Heat and Mass Transfer, 64:278 – 285, 2013\.
  * [19] S. Dehaeck, A. Rednikov, and P. Colinet.  Vapor-based interferometric measurement of local evaporation rate and interfacial temperature of evaporating droplets.  Langmuir, 30(8):2002–2008, 2014.
  * [20] O. Carrier, N. Shahidzadeh-Bonn, R. Zargar, M. Aytouna, M. Habibi, J. Eggers, and D. Bonn.  Evaporation of water: evaporation rate and collective effects.  Journal of Fluid Mechanics, 798:774–786, 2016.
  * [21] X. Zhang, J. Wang, L. Bao, E. Dietrich, R. C. A. van der Veen, S. Peng, J. Friend, H. J. W. Zandvliet, L. Yeo, and D. Lohse.  Mixed mode of dissolving immersed nanodroplets at a solid-water interface.  Soft Matter, 11:1889–1900, 2015.
  * [22] E. Dietrich, S. Wildeman, C.W. Visser, K. Hofhuis, S. Kooij, H.J.W. Zandvliet, and D. Lohse.  Role of natural convection in the dissolution of sessile droplets.  Journal of Fluid Mechanics, 794:45–67, 2016.
  * [23] G. Laghezza, E. Dietrich, J.M. Yeomans, R.A. Ledesma-Aguilar, S. Kooij, H. Zandvliet, and D. Lohse.  Collective and convective effects compete in patterns of dissolving surface droplets.  Soft Matter, 12:5787–5796, 2016.
  * [24] O. R. Enríquez, C. Sun, D. Lohse, A. Prosperetti, and D. van der Meer.  The quasi-static growth of CO2 bubbles.  Journal of Fluid Mechanics, 741, 2014.
  * [25] A. Bejan.  Heat Transfer.  John Wiley & Sons, 1993.
  * [26] K. Stewartson.  On the free convection from a horizontal plate.  Journal of Applied Mathematics and Physics, 9:276–282, 1958.
  * [27] Z. Rotem and L. Claassen.  Natural convection above unconfined horizontal surfaces.  Journal of Fluid Mechanics, 39:173–192, 10 1969.
  * [28] M. Zakerullah and J.A.D. Ackroyd.  Laminar natural convection boundary layers on horizontal circular discs.  Journal of Applied Mathematics and Physics, 30(3):427–435, 1979\.
  * [29] J.H. Merkin.  Free convection above a heated horizontal circular disk.  Zeitschrift für angewandte Mathematik und Physik, 34(5):596–608, 1983.
  * [30] N. Lebedev.  Special Functions and their Applications.  Prentice-Hall, 1965.
  * [31] B.M. Marder and N.R. Keltner.  Heat flow from a disk by separation of variables.  Numerical Heat Transfer, 4:485–497, 1981.
  * [32] J.M. Stauber, S.K. Wilson, B.R. Duffy, and K. Sefiane.  On the lifetimes of evaporating droplets.  Journal of Fluid Mechanics, 744, 2014.
  * [33] J. M. Stauber, S. K. Wilson, B. R. Duffy, and K. Sefiane.  On the lifetimes of evaporating droplets with related initial and receding contact angles.  Physics of Fluids, 27(12), 2015.
  * [34] R. Deegan, O. Bakajin, T. Dupont, G. Huber, S. Nagel, and T. Witten.  Contact line deposits in an evaporating drop.  Phys. Rev. E, 62:756–765, 2000.
  * [35] Y. Popov.  Evaporative deposition patterns: Spatial dimensions of the deposit.  Phys. Rev. E, 71:036313, 2005.
  * [36] A. G. Marín, H. Gelderblom, D. Lohse, and J. H. Snoeijer.  Order-to-disorder transition in ring-shaped colloidal stains.  Phys. Rev. Lett., 107:085502, 2011.
  * [37] L. G. Leal.  Advanced Transport Phenomena.  Cambridge University Press, 2007.
  * [38] P.T. Tsilingiris.  Thermophysical and transport properties of humid air at temperature range between 0 and 100 ∘C.  Energy Conversion and Management, 49(5):1098 – 1110, 2008.
  * [39] R.M. Tennent.  Science Data Book.  Oliver and Boyd, 1971.
  * [40] S. Grossmann and D. Lohse.  Scaling in thermal convection: a unifying theory.  Journal of Fluid Mechanics, 407:27–56, 2000.
  * [41] J. Crank.  The Mathematics of Diffusion.  Oxford, 1975.