††thanks: This work was done prior to joining AWS.

# Metrology-assisted entanglement distribution in noisy quantum networks

Simon Morelli<EMAIL_ADDRESS>Institute for Quantum Optics and
Quantum Information — IQOQI Vienna, Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria  Atominstitut, Technische Universität
Wien, 1020 Vienna, Austria  David Sauerwein<EMAIL_ADDRESS>Amazon
Web Services Europe, Zürich, Switzerland  Michalis Skotiniotis
<EMAIL_ADDRESS>Física Teòrica: Informació i Fenòmens Quàntics,
Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra,
Spain  Nicolai Friis<EMAIL_ADDRESS>Institute for Quantum Optics
and Quantum Information — IQOQI Vienna, Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria  Atominstitut, Technische Universität
Wien, 1020 Vienna, Austria

###### Abstract

We consider the distribution of high-dimensional entangled states to multiple
parties via noisy channels and the subsequent probabilistic conversion of
these states to desired target states using stochastic local operations and
classical communication. We show that such state-conversion protocols can be
enhanced by embedded channel-estimation routines at no additional cost in
terms of the number of copies of the distributed states. The defining
characteristic of our strategy is the use of those copies for which the
conversion was unsuccessful for the estimation of the noise, thus allowing one
to counteract its detrimental effect on the successfully converted copies.
Although this idea generalizes to various more complex situations, we focus on
the realistic scenario, where only finitely many copies are distributed and
where the parties are not required to process multiple copies simultaneously.
In particular, we investigate the performance of so-called one-successful-
branch protocols, applied sequentially to single copies and an adaptive
Bayesian estimation strategy. Finally, we compare our strategy to more general
but less easily practically implementable strategies involving distillation
and the use of quantum memories to process multiple copies simultaneously.

##  1 Introduction

Entanglement between two or more parties is an important ingredient for the
development of many quantum technologies such as, e.g., fault-tolerant quantum
computation [1, 2, 3, 4], quantum simulation [5, 6, 7], and, in particular,
communication via quantum networks [8, 9, 10, 11, 12]. A landmark goal of
quantum technologies is the establishment of a ‘quantum internet’ [13, 14,
15]—a highly interconnected quantum network able to distribute and manipulate
entangled quantum states via fibres and optical links. Recent efforts [16] go
in the direction of understanding the resources, challenges and opportunities
that come along with such an endeavour. In this context, entanglement is a
crucial resource that allows spatially separated parties to overcome the
restriction of local operations and classical communication (LOCC) thereby
implementing classically impossible tasks, such as quantum teleportation [17,
18, 19]. However, the technological requirements for quantum processors and
networks on a large scale are immense and will have to involve significant
improvements in the efficient manipulation and control of quantum systems. A
deeper comprehension of the possibilities and limitations for multipartite
entanglement distribution and its manipulation via LOCC is hence necessary.
The present work aims to contribute to this endeavour by exploring the
synergies between stochastic LOCC (SLOCC) protocols—that aim to convert
multiparty quantum states—and quantum parameter estimation protocols whose
goal is to improve entanglement distribution by precisely identifying the
noise suffered by such states when distributed through quantum networks.

On the theoretical side, a lot of progress has been made with regards to the
conversion of multipartite entanglement via LOCC (cf. [20, 21, 22, 23, 24, 25,
26]), despite the fact that LOCC is notoriously difficult to handle
mathematically [27], while practical developments towards efficient
entanglement certification techniques (see, e.g., [28, 29, 30, 31, 32, 33])
and real-world quantum networks are rapidly progressing (cf. [34]). Yet, basic
theoretical questions about multipartite entanglement structures arising in
networks have only begun to be explored [35, 36, 37, 38, 39]. In particular,
apart from recent examples [40, 41], mostly transformations of pure states in
the single-copy regime or the asymptotic limit of infinitely many copies have
been considered.

Here, we consider a scenario that more closely resembles situations expected
to be encountered in future real-world quantum networks: the distribution of
multiple but finitely many copies of multipartite quantum states via imperfect
or varying channels available only for a restricted period of time. There, one
central node, the vendor, is assumed to be capable of implementing the
entangling operations necessary to create a highly entangled quantum state
that is subsequently distributed to spatially separated parties via imperfect
but local quantum channels. In general, the states provided by such a vendor
will not match the target state that the parties might wish to employ in their
communication or computation protocol of choice. For multiple parties, there
is no unique resource state that can be deterministically converted to any
other state via LOCC; in fact, the so-called maximally entangled set (MES) of
states from which any state can be reached in this way has infinitely many
elements [22]. In addition, the vendor might only be able to create a limited
variety of such states and may therefore be unable to offer the desired state.
Indeed, this might even be in the interest of the users, as they may not want
to divulge information on the specific form (and hence the intended use) of
their desired target state. Since deterministic LOCC transformations between
multi-party pure states (with fixed local dimensions) almost never exist [42,
26], the distributed state will have to be converted to the target state
_probabilistically_ , i.e., via SLOCC. Two states that can be converted into
each other via SLOCC are said to belong to the same _entanglement class_.

Optimal SLOCC conversion protocols will generally require acting on many
copies simultaneously. Beside the fact that little is known about the
structure of such protocols yet, the implementation of complex multi-copy
operations also represents a substantial technological challenge. Therefore we
focus on sequential strategies that process a single copy at a time. Compared
to general protocols the difficulty to implement a sequential strategy does
not increase with the number of copies. To employ a finite number,
$k_{\text{s}}$, of copies for multipartite quantum communication or
computation, users must purchase, on average, a larger number
$k=k_{\text{s}}/p_{\text{s}}$ of copies from the vendor to compensate for the
nonzero failure probability $p_{\text{f}}=1-p_{\text{s}}>0$ of the conversion
protocol.

Here, we demonstrate that this apparent disadvantage has important redeeming
qualities in the presence of noise. We show that, for certain ‘benign’ types
of noise, those copies for which the state conversion was unsuccessful can
nevertheless be useful for obtaining information about the noise and hence
partially compensate its effects on the successfully converted copies. Our
results thus establish that multipartite entanglement distribution protocols
can be assisted by quantum metrology at no additional resource costs in terms
of additionally distributed states, and thus feature built-in noise robustness
‘for free’.

To reach this conclusion, we first consider a particular situation as
described above, i.e., a family of specific noisy multipartite state
conversion protocols and show how the use of parameter estimation tools leads
to an improvement in the quality of the resulting final states. Based on this
exemplary situation, we then discuss the applicability of this approach to
more general entanglement distribution protocols and contrast it with
alternative strategies based on distillation. Finally, we discuss the merits
of metrology-assisted entanglement conversion protocols and argue for their
default integration in future quantum networks.

##  2 Noisy entanglement distribution

We consider a situation where $N$ parties require several copies of a highly
entangled $N$-qudit state $\rho_{\text{target}}$ for a specific task. They
purchase a number of copies from a vendor, where the purchased state
$\rho_{\text{vendor}}$ differs from the target state $\rho_{\text{target}}$,
but belongs to the same entanglement class and can thus be converted to the
target state via SLOCC. The vendor distributes the state
$\rho_{\text{vendor}}$ to the $N$ parties via local quantum channels
represented by completely-positive, trace-preserving (CPTP) maps,
$\\{\Lambda_{i}\\}_{i=1}^{N}$, so that the state shared by the $N$ parties is
given by

$\rho_{\text{received}}=\Lambda(\rho_{\text{vendor}})=\bigotimes_{i=1}^{N}\Lambda_{i}(\rho_{\text{vendor}}).$
(1)

The parties now wish to transform the received state $\rho_{\text{received}}$
into the target state $\rho_{\text{target}}$ within the same entanglement
class. This can be done either deterministically or probabilistically. We
focus on transformations of fully entangled (i.e., with full local rank) pure
states via a so-called one-successful-branch protocol (OSBP) [43, 44]. The
OSBP is in fact the optimal SLOCC protocol for almost all $N$-qudit pure-state
transformations [44]. However, the idea of using unsuccessful branches of
SLOCC protocols for parameter estimation presented in this paper applies in
equal measure to any other probabilistic transformations of pure and mixed
states of arbitrary dimensions.

In an OSBP that transforms $\left|\right.\\!{\Psi}\\!\left.\right\rangle$ into
$\left|\right.\\!{\Phi}\\!\left.\right\rangle$, each party $k$ performs a
single two-outcome measurement $\\{M_{\text{s}}^{k},\,M_{\text{f}}^{k}\\}$
with $\sum_{i=\text{s,f}}(M_{i}^{k})^{\dagger}M_{i}^{k}=\mathds{1}$. The
successful outcome $s$ leads to a state that can still be transformed to the
final state, while the failure outcome $f$ leads to a state that is no longer
fully entangled and can therefore no longer be transformed into
$\left|\right.\\!{\Phi}\\!\left.\right\rangle$. Hence, the final state is only
reached if the successful measurement outcome is realised for all parties.
That is, only one branch of the OSBP is successful, as suggested by its name,
and realized with some probability $p_{\text{s}}$.

The transformation performed by the $N$ parties to convert the received state
into the target state depends on the channel used to distribute the state. For
the general case when the channel is not sufficiently known, the task for the
parties becomes more challenging. They have to estimate the channel, i.e.,
identify the channel $\Lambda$ from a discrete or continuous family of
possible candidates, before transforming the received state
$\rho_{\text{received}}$. The goal of the $N$ parties is now to obtain the
largest number of copies $\rho_{\text{corr}}$ that are $\epsilon$-close in
trace distance to the desired target state $\rho_{\text{target}}$, i.e.,

$\displaystyle
D(\rho_{\text{corr}},\rho_{\text{target}})=\frac{1}{2}\mathrm{Tr}\Bigl{[}\sqrt{(\rho_{\text{corr}}-\rho_{\text{target}})^{2}}\,\Bigr{]}\,\,\leq\,\epsilon.$
(2)

If an asymptotically large number of copies of the state is available, the
parties can always sacrifice a finite number of such states to estimate the
channel to any desired precision before commencing with the state
transformation. Consequently, the incurred cost for estimation has no bearing
on the final yield of the target state. On the other hand, if only a finite
number of copies is available, then the above strategy can be quite wasteful,
and protocols that estimate the channel whilst implementing the OSBP at the
same time may provide better yields. This is due to the observation that the
states resulting from the failure branches of the OSBP potentially carry
information about the noisy channel and can thus be used to estimate the
latter at no extra cost.

A particular class of channels for which simultaneous channel estimation and
state conversion may be beneficial is the family of channels that admit a
decomposition into Kraus operators that commute with the POVM corresponding to
the OSBP. Such channels include dephasing and phase drift [45] noise which are
typical types of noise in quantum communication using optical fibres [46].
Alternatively, such channels also describe a situation often encountered in
satellite communication where the sender (vendor) and receivers ($N$ parties)
do not share a common phase reference during the transmission stage [47]. In
this latter case the receivers know that the state they receive is local-
unitarily (LU) equivalent to the original state of the vendor and thus is in
the same entanglement class. However, the lack of knowledge of the relative
phase between the sender’s and receivers’ phase reference makes channel
estimation indispensable, especially if the task they have in mind requires a
particular state $\rho_{\text{target}}$, and not any other local-unitarily
equivalent state.

In what follows, we shall consider the above class of channels as a special
case of more general parameter estimation problems and employ a Bayesian
estimation strategy to estimate the parameter (or set of parameters) $\theta$
that characterizes the channel, illustrated in Fig. 1. Such a strategy
involves one or more rounds of subsequent measurements on different copies of
the received states. The knowledge of the parameter, $\theta$, is updated in
every round depending on the measurement outcome, $m$, according to Bayes’
law, i.e.,

$\displaystyle p(\theta|m)$ $\displaystyle=\,\frac{p(m|\hskip
0.5pt\theta)\,p(\theta)}{p(m)}.$ (3)

Based on the posterior distribution of $\theta$ one can then assign an
estimator $\hat{\theta}(m)$ after each round and adapt the next measurement to
this estimate. This makes Bayesian parameter estimation more flexible and
better suited for a limited number of probes compared to frequentist
estimation. For an increasing number of copies, also a frequentist strategy
can be employed to estimate the channel from the failure branches of the OSBP.
Here, however, we will focus on scenarios with a very restricted number of
copies. In particular, we consider the exemplary situation of local dephasing
noise, to provide quantitative support for our strategy of metrology-assisted
entanglement distribution.

Figure 1:  Schematic of the metrology-assisted entanglement distribution
protocol. Each copy of the distributed multi-party state
$\left|\right.\\!{\Psi}\\!\left.\right\rangle$ is subject to a (local) noise
channel (here represented by an unknown unitary $U_{\theta}$) parameterized by
$\theta$, which is subsequently measured by an OSBP with measurement operators
$M_{\text{s}}$ and $M_{\text{f}}$, corresponding to successful (green) and
failed (orange) state conversion, respectively. The copies in the failure
branch can be used to perform further measurements with outcomes $m_{i}$ to
estimate the noise channel, and apply corresponding corrections (represented
here by $U_{\hat{\theta}}^{\dagger}$) to obtain partially corrected states
$\rho_{\text{corr}}$.

##  3 Entanglement distribution with dephasing noise

Assume the state distributed by the vendor is a pure $N$-qudit state from the
GHZ class of the form

$\left|\right.\\!{\Psi}\\!\left.\right\rangle=\sum_{l=0}^{d-1}\sqrt{\lambda_{l}}\left|\right.\\!{l}\\!\left.\right\rangle^{\otimes
N},$ (4)

where we assume without loss of generality that
$\lambda_{m}\geq\lambda_{m+1}$. Let us further assume that the vendor
distributes these states via $N$ local dephasing channels, i.e., each qudit
undergoes a unitary transformation parameterized by an unknown angle $\theta$,

$U(\theta)=\sum_{l=0}^{d-1}e^{\mathrm{i}l\theta}\left|\right.\\!{l}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{l}\\!\left.\right|.$
(5)

The parties wish to obtain the maximally-entangled qudit state

$\left|\right.\\!{\mathrm{GHZ}_{d}}\\!\left.\right\rangle=\sqrt{\tfrac{1}{d}}\sum_{l=0}^{d-1}\left|\right.\\!{l}\\!\left.\right\rangle^{\otimes
N}.$ (6)

To this end, they estimate the unknown phase $\theta$ of the dephasing noise
acting on the received states and recover the target state via SLOCC-
transformations from the received state.

More specifically, let us assume the vendor produces the pure two-parameter
qutrit state

$\displaystyle\left|\right.\\!{\Psi(\alpha,\beta)}\\!\left.\right\rangle=$
$\displaystyle\sin\alpha\cos\beta\left|\right.\\!{0}\\!\left.\right\rangle^{\otimes
N}+\sin\alpha\sin\beta\left|\right.\\!{1}\\!\left.\right\rangle^{\otimes N}$
(7)
$\displaystyle+\cos\alpha\left|\right.\\!{2}\\!\left.\right\rangle^{\otimes
N},$

where $\tan^{-1}(\csc\beta)\leq\alpha\leq\pi/2$ and $0\leq\beta\leq\pi/4$.
This choice of angles ensures that
$\sin\alpha\cos\beta\geq\sin\alpha\sin\beta\geq\cos\alpha\geq 0$. The parties
wish to obtain the maximally entangled state of Eq. (6), for which the optimal
OSBP comprises measurement operators

$\displaystyle M_{\mathrm{s}}$
$\displaystyle=\operatorname{diag}(\cot\alpha\sec\beta,\cot\alpha\csc\beta,1),$
(8) $\displaystyle M_{\mathrm{f}}$
$\displaystyle=\operatorname{diag}(\sqrt{1-\cot^{2}\alpha\sec^{2}\beta},\sqrt{1-\cot^{2}\alpha\csc^{2}\beta},0)$

for one party and the identity for the remaining parties. The conversion is
successful with probability $p_{\text{s}}=3\cos^{2}\alpha$, which is already
the optimal conversion rate for any protocol acting on a single copy of the
state. This follows from the fact that the optimal conversion rate from a
bipartite pure state $\left|\right.\\!{\Psi}\\!\left.\right\rangle$ to another
bipartite pure state $\left|\right.\\!{\tilde{\Psi}}\\!\left.\right\rangle$
can be calculated [48] from the respective Schmidt coefficients $\lambda_{i}$
and $\tilde{\lambda}_{j}$ (real, non-negative,and in decreasing order) as
$P(\Psi\rightarrow\tilde{\Psi})=\min_{j}\sum_{l=j}\lambda^{2}_{l}/\sum_{l^{\prime}=j}\tilde{\lambda}^{2}_{l^{\prime}}$,
and from the observation that the optimal conversion rate for any splitting of
the $N$ parties into two groups gives an upper bound on the conversion rate.
By splitting the state in Eq. (7) into one party versus the remaining $N-1$
parties, one immediately sees that the conversion rate for a single state is
bounded by $P(\Psi(\alpha,\beta)\rightarrow\mathrm{GHZ}_{3})=3\cos^{2}\alpha$
for the parameter range we have chosen for $\alpha$ and $\beta$.

With probability $p_{\text{f}}=1-3\cos^{2}\alpha$ the conversion protocol
fails and results in the state

$\left|\right.\\!{\Phi_{\text{f}}}\\!\left.\right\rangle=a\left|\right.\\!{0}\\!\left.\right\rangle^{\otimes
N}+\sqrt{1-a^{2}}\left|\right.\\!{1}\\!\left.\right\rangle^{\otimes N},$ (9)

where

$a=\frac{\sqrt{\sin^{2}\alpha\cos^{2}\beta-\cos^{2}\alpha}}{\sqrt{1-3\cos^{2}\alpha}}.$
(10)

Observing that $[M_{x},U(\theta)]=0$, $\forall\,x\in\\{\text{s, f}\\}$, it
follows that the resulting states for the successful and failure branches of
the OSBP are given by

$\displaystyle\left|\right.\\!{\mathrm{GHZ}_{3}(\theta)}\\!\left.\right\rangle$
$\displaystyle=\sqrt{\frac{1}{3}}\sum_{l=0}^{2}e^{\mathrm{i}lN\theta}\left|\right.\\!{l}\\!\left.\right\rangle^{\otimes
N},$ (11a)
$\displaystyle\left|\right.\\!{\Phi_{\text{f}}(\theta)}\\!\left.\right\rangle$
$\displaystyle=a\left|\right.\\!{0}\\!\left.\right\rangle^{\otimes
N}+\sqrt{1-a^{2}}e^{\mathrm{i}N\theta}\left|\right.\\!{1}\\!\left.\right\rangle^{\otimes
N},$ (11b)

respectively, with the corresponding probabilities invariant under the
presence of dephasing noise.

In the case that the OSBP is successful, the parties obtain the state

$\displaystyle\Lambda\bigl{[}\rho_{\mathrm{GHZ}_{3}}\bigr{]}$
$\displaystyle=\int_{0}^{2\pi}p(\theta)U(\theta)^{\otimes
N}\rho_{\mathrm{GHZ}_{3}}U(\theta)^{\dagger\,\otimes N}\mathrm{d}\theta,$

where
$\rho_{\mathrm{GHZ}_{3}}=\left|\right.\\!{\mathrm{GHZ}_{3}}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\mathrm{GHZ}_{3}}\\!\left.\right|$
and $p(\theta)$ is a probability distribution over $\theta\in[0,2\pi)$
describing the belief/knowledge the parties have of111Actually the estimation
only gives us information about $N\theta$, which allows us to draw conclusions
about $\theta$ only up to a shift by $2\pi/N$. But this is not a problem,
since the final state also only depends on $N\theta$. $\theta$. Loosely
speaking, the more peaked the distribution becomes, the closer the state gets
to the target state. The goal of the $N$-parties now is to apply correction
operations to the successfully converted copies to obtain the highest yield of
$N$-partite highly entangled qutrit states $\rho_{\text{corr}}$, that are
$\epsilon$-close in trace distance to the ideal state
$\left|\right.\\!{\mathrm{GHZ}_{3}}\\!\left.\right\rangle$.

In the case of a flat prior, $p(\theta)=(2\pi)^{-1}$, we can analytically
calculate (see Appendix A for details) the average trace distance to the
target state after the first measurement to be

$\displaystyle\bar{D}(\rho_{\text{corr}},\rho_{\text{target}})=\tfrac{1}{6}\bigl{(}1+\sqrt{9+8a^{2}(1-a^{2})-16a\sqrt{1-a^{2}}}\bigr{)}.$
(12)

The average distance after one measurement hence only depends on the parameter
$a$. Since $a\in[0,1]$, the maximal value $\tfrac{2}{3}$ is obtained for $a=0$
or $a=1$. This coincides with the distance based on the prior, i.e., ignoring
measurement data. The minimal value of $\bar{D}=\frac{1}{6}(1+\sqrt{3})$ is
obtained for $a=1/\sqrt{2}$.

Given a particular measurement outcome the posterior distribution is updated
using Bayes’ law, [Eq. (3)]. In general the parties can optimize their
measurement directions at every step but this amount of generality quickly
renders the problem analytically intractable. We thus resort to numerical
analysis henceforth, and we focus on a strategy where the measurement
direction on the Bloch sphere of the two-dimensional subspace relevant for the
copies in the failure branch (see Appendix A) is chosen on the equator but
perpendicular to the estimator of $\theta$ of the previous round. This
measurement is particularly sensitive in the direction of $\hat{\theta}$,
since both outcomes are equally likely in this case. The simplest strategy of
fixed local measurements is treated in Appendix B, where also strategies that
change the measurement direction according to some pre-established rule are
considered.

Figure 2:  Average distance to the target state. The curves show the average
trace distance (vertical axis) between the target state and the successfully
converted copies for the exemplary metrology-assisted entanglement
distribution protocol discussed in Sec. 3 for different values of $a$ as a
function of the number $k_{\text{f}}$ of copies (horizontal axis) in the
failure branch.

In Fig. 2 we see how the trace distance $\bar{D}$ of the corrected state
resulting from the successful branch to the target state reduces with the
number of failure branches used for the estimation. After 20 measurement
rounds the average trace distance drops to $\bar{D}\approx 0.038$ for
$a=1/\sqrt{2}$. This is a remarkable improvement considering that the failure
states would be discarded in the naive protocol that we will discuss in Sec.
4.

One could also think of more sophisticated strategies, where a swap operation
is applied to the state by the vendor before it is distributed, changing the
basis vectors
$\left|\right.\\!{1}\\!\left.\right\rangle\leftrightarrow\left|\right.\\!{2}\\!\left.\right\rangle$.
Since the state $\left|\right.\\!{2}\\!\left.\right\rangle$ reacts more
sensitive to the transformation, i.e., picks up a phase of
$e^{2\mathrm{i}N\theta}$, this can be advantageous for estimation. Note,
however, that in this case one can draw conclusions about $\theta$ only up to
a $\pi/N$ phase shift, whereas the successful branch still depends on
$N\theta$. Thus, this strategy can only be employed once the interval of
$\theta$ is sufficiently narrowed down.

Although we focus on the distribution of qutrit GHZ states under local
dephasing noise in our example, we want to emphasize that this situation can
be straightforwardly generalized to the distribution of any $d$-dimensional
state in the GHZ entanglement class of the form of Eq. (4) with the same noise
model. Here an OSBP converting the received state to the GHZ state in Eq. (6)
leaves the failure-branch in a subspace of dimension $2\leq d_{\text{f}}<d$
and one can use estimation strategies similar to that used in Appendix C.
Additionally, the GHZ class is broad in the sense that, according to Ref.
[20], almost all pure multipartite entangled three-qubit states belong to this
entanglement class, whereas states from the W class form a subset of measure
zero. While this may not necessarily hold for higher dimensions or more
parties, our example remains relevant as states in the GHZ class are often
used (see, e.g., [49, 50, 51, 52]) and have desirable properties such as their
symmetry with respect to the exchange of parties.

##  4 Comparison with ’naive’ protocols

In this section we compare the performance of our strategy to what we call
’naive’ strategies. These are strategies that first sacrifice a number of
copies of the received state for the estimation of the LU-channel and
subsequently perform the transformation to the target state for all following
copies. For this comparison we are interested in determining the number $k$ of
copies of received states required to obtain a certain number $k_{\text{s}}$
of successfully converted states whose average trace distance from the desired
target state is below a given value $\epsilon$. Moreover, to make the
comparison fair, first note that our previous metrology-assisted entanglement
conversion protocol processes the received multipartite systems _sequentially_
, i.e., one copy of each multipartite state at a time, and only uses local
measurements and operations both for the estimation and the state
transformation. Although this drastically simplifies potential practical
implementations, it is known that processing multiple copies simultaneously —
via the use of quantum memories — can lead to a higher success rate for
entanglement distillation [53, 54]. For a fair comparison we therefore impose
the same restrictions on the ’naive’ strategy, i.e. we assume that also any
’naive’ protocol is restricted to process each copy of the multipartite state
sequentially via LOCC.

Within this framework for the ’naive’ approach we consider a Bayesian strategy
that estimates the phase parameter based on measurements performed on a
certain number $k_{\text{e}}$ of individual received states, while the
$k-k_{\text{e}}$ subsequent copies are individually transformed via the
previously considered SLOCC protocol. We calculate the average trace distance
of the successfully converted copies from the target state as a function of
the initially sacrificed $k_{\text{e}}$ copies used for estimation. Since the
chosen Bayesian strategy might not be optimal in general, we supplement this
analysis with an estimate of the lower bound on the achievable trace distance
based on the Bayesian quantum Cramér-Rao bound. Comparing these results to the
performance of our previously presented metrology-assisted protocol, we
conclude that there exist combinations of the parameters $k$ and $\epsilon$
for which our protocol outperforms any ‘naive’ protocol since the latter
always sacrifices some of the copies that can be successfully converted within
an approach that solely uses the failure-branch states for estimation.

After these ’fair’ comparisons, we also relax the condition of sequential
processing and consider protocols based on distillation where each party can
use quantum memories and is able to process multiple copies at the same time.
For this scenario, we derive upper bounds on the obtainable number of
successfully converted copies valid for any distillation strategy and conclude
that such strategies potentially outperform our sequential strategy. As these
bounds are only asymptotically tight, however, it remains unclear how well the
optimal strategy actually performs on a finite number of copies.

Let us now consider the task of the previous example, i.e., $k$ copies of the
state in Eq. (7) are distributed via a local dephasing channel, described in
Eq. (5), and the goal of the parties is to obtain the highest number of states
(at least) $\epsilon$-close to
$\left|\right.\\!{\mathrm{GHZ}_{3}}\\!\left.\right\rangle$. They therefore use
$k_{\text{e}}$ copies of
$\left|\right.\\!{\Psi(\alpha,\beta)}\\!\left.\right\rangle$ to estimate the
phase $\theta$ and assign an estimator $\hat{\theta}(\mathbf{m})$ based on the
measurement outcome $\mathbf{m}=(m_{1},\dots,m_{k_{\text{e}}})$. Using a
Bayesian estimation strategy as before, we have to choose an appropriate
measurement. As our probe is a three-level system now, we perform a three-
outcome projective measurement. Since we assume not to have any knowledge of
$\theta$ initially, we consider a flat prior, for which a reasonable choice of
measurement basis presents itself in the form of any basis that is mutually
unbiased with respect to the eigenbasis of the generating Hamiltonian of the
local dephasing noise $U(\theta)$. Since no further restrictions seem to apply
to the measurement basis we therefore use the discrete Fourier transform of
the eigenbasis of the generator [55]. However, note that for the measurement
on the second and subsequent copies, we rotate the measurement basis by
$U(\hat{\theta}+\pi/3)$ to take into account our estimate $\hat{\theta}$. The
obtained basis still is mutually unbiased with respect to the eigenbasis of
the encoding Hamiltonian. A more detailed calculation of this estimation can
be found in Appendix C.

Let us compare this strategy to the numerical results of the previously
employed estimation of the failure branch for the exemplary parameters
$k=100$, $\alpha=\arccos(2/\sqrt{15})$ and $\beta=\pi/4$. This leads to
$p_{\text{s}}=4/5$, and so to $k_{\text{s}}=80$ and $k_{\text{f}}=20$
successful and failure branches, respectively, on average. Twenty rounds of
estimation with the failure branch lead to an average trace distance of
$\bar{D}\approx 0.038$, see Fig. 2. To achieve a comparable distance with the
received state requires $k_{\text{e}}=9$ copies on average, see Fig. 4 in
Appendix C. The observation that fewer copies of the received state are needed
to estimate the phase to the same precision as compared to the failure-branch
state comes as no surprise, since any processing of the state after the
transmission can only reduce (on average) the amount of information about
$\theta$ encoded in the system. This means that the received states will
generally be more advantageous for the estimation than states in the failure
branch. While this is true, it is essentially the same strategy that was also
used for the failure branch in the previous protocol adapted to the different
probe state. Hence we argue that this is indeed a fair comparison, as we do
not know the optimal strategy for finitely many copies.

This notwithstanding, we can calculate a lower bound on the number of copies
needed for any estimation strategy to reach a given trace distance based on
the Bayesian quantum Cramér-Rao bound, see Appendix D for a detailed
calculation. Comparing this to the numerical results of the previously
employed strategy for the same exemplary parameters $k=100$,
$\alpha=\arccos(2/\sqrt{15})$ and $\beta=\pi/4$, one finds that achieving a
comparable trace distance, any estimation needs on average at least
$k_{\text{e}}\geq 7.5$ copies of the received state, see Fig. 4 in Appendix C.
However, note that $k_{\text{e}}\geq 7.5$ is only a lower bound, and it is not
clear if a strategy exists that saturates this bound for finitely many copies.

Once the phase $\theta$ has been estimated sufficiently well, the states
resulting from the successful branches can be transformed via SLOCC. We have
already argued in the previous section that any sequential strategy
transforming individual copies of the state in Eq. (7) to the state
$\left|\right.\\!{\text{GHZ}_{3}}\\!\left.\right\rangle$ is bounded by
$p_{\text{s}}\leq 3\cos^{2}\alpha$. For the same exemplary parameters
considered before, this leads to a success rate of $p_{\text{s}}=4/5$. For the
’naive’ Bayesian strategy we hence obtain
$(k-k_{\text{e}})p_{\text{s}}=91\,\tfrac{4}{5}=72.8$ successfully converted
copies on average from the initial $100$ copies of the distributed state.
Additionally, the lower bound based on the Bayesian quantum Cramér-Rao bound
informs us that any ’naive’ sequential strategy obtains at most
$(k-k_{\text{e}})p_{\text{s}}=92.5\,\tfrac{4}{5}=74$ copies of the target
state on average, proving our failure-branch estimation considerably superior
with $80$ successfully converted copies on average in this example.

Indeed, for any noise particular SLOCC conversion protocol with success
probability $p_{\text{s}}$ (in which a channel is estimated whose Kraus
operators commute with the SLOCC protocol) one may find pairs of $k$ and
$\epsilon$ for which our strategy outperforms naive protocols since the latter
always sacrifice some of the copies that can otherwise be successfully
converted. If the average trace distance $\bar{D}$ after $k$ copies is equal
to or smaller than the desired distance $\epsilon$, then our protocol cannot
be beaten by a naive strategy for this combination of parameters since any
naive strategy would have to sacrifice some number $k_{\text{e}}>0$ of the
received states, thus ultimately being left with only
$p_{\text{s}}(k-k_{\text{e}})<k_{\text{s}}=p_{\text{s}}k$ copies within the
desired distance $\epsilon$. However, note that this does not imply that our
strategy outperforms naive protocols for all values of $k$ and $\epsilon$.
There also exist pairs of $k$ and $\epsilon$ where our failure branch
estimation is not sufficiently precise and good copies need to be sacrificed
for estimation to reach the desired precision. In such a case the performance
of such a ’hybrid’ strategy compared to a ’naive’ strategy depends on the
specific parameters of the considered example.

In addition, we note that naive protocols based on distillation, i.e., which
act jointly on all $k-k_{\text{e}}$ states available after the initial
estimation could in principle perform better than sequential protocols based
on failure-branch estimation only, even in cases where the latter outperform
sequential naive protocols. For pure bipartite states the asymptotic
distillation rate from $\left|\right.\\!{\psi}\\!\left.\right\rangle$ to
$\left|\right.\\!{\phi}\\!\left.\right\rangle$ is bounded by the entropy ratio
of the reduced states for any party $i$, i.e.,

$\displaystyle\mathcal{D}(\left|\right.\\!{\psi}\\!\left.\right\rangle\rightarrow\left|\right.\\!{\phi}\\!\left.\right\rangle)=\frac{S(|{\psi}\rangle\\!\langle{\psi}|_{i})}{S(|{\phi}\rangle\\!\langle{\phi}|_{i})},$
(13)

and this bound is tight [56]. For multipartite states this immediately gives
the, generally not tight, asymptotic bound

$\displaystyle\mathcal{D}(\left|\right.\\!{\psi}\\!\left.\right\rangle\rightarrow\left|\right.\\!{\phi}\\!\left.\right\rangle)=\min\limits_{i}\Bigl{(}\frac{S(|{\psi}\rangle\\!\langle{\psi}|_{i})}{S(|{\phi}\rangle\\!\langle{\phi}|_{i})}\Bigr{)}.$
(14)

In our case, however, the latter bound is tight asymptotically [57, 58] and we
can calculate

$\displaystyle\mathcal{D}(\left|\right.\\!{\Psi(\alpha,\beta)}\\!\left.\right\rangle\rightarrow$
$\displaystyle\left|\right.\\!{\text{GHZ}}\\!\left.\right\rangle)=-\cos^{2}\alpha\log_{3}(\cos^{2}\alpha)$
(15)
$\displaystyle-\sin^{2}\alpha\cos^{2}\beta\log_{3}(\sin^{2}\alpha\cos^{2}\beta)$
$\displaystyle-\sin^{2}\alpha\sin^{2}\beta\log_{3}(\sin^{2}\alpha\sin^{2}\beta).$

For our choice of parameters this means there might exist a distillation
protocol acting on all available copies at once able to successfully distill
(on average) $91.6$ copies of the desired GHZ-state out of the
$k-k_{\text{e}}=92.5$ remaining copies after an optimal estimation. This
clearly exceeds the $80$ successful states obtained by our adaptive strategy.
Here we again want to emphasise that these $91.6$ copies are only an upper
bound. Both an optimal estimation and distillation strategy are only known to
exist asymptotically and it is not clear that any strategy can attain this
bound for finitely many copies. Indeed, even if such a strategy exists, it
would generally require acting on all remaining copies simultaneously. Even if
the presented metrology-assisted protocol does not saturate these ’optimal’
bounds, it nonetheless shows the potential of practically easily implementable
adaptive strategies acting on single copies. In particular, the task of
designing and implementing optimal strategies for estimation and especially
distillation becomes extremely challenging for larger numbers of copies and
may be out of reach for experimental realizations in the medium term.
Therefore it is essential to find feasible strategies that perform well and
whose implementation remains tractable with growing number of copies.

##  5 Metrology-assisted protocols beyond dephasing noise

A key assumption in our previous example was that the parties know that the
measurement operators of the OSBP commute with the Kraus operators of the
noisy channel. If this does not hold, as is the case for depolarizing,
channels with high loss, or for dephasing channels with unknown rotation axis,
the order in which the POVM measurements of the OSBP and those for the
estimation protocol are carried out matter. If the channel is not known to
commute with the POVM operators of the OSBP, this impacts the performance of
the presented strategy. Nonetheless one could design strategies relying on the
failure branch for the channel estimation. We now present a protocol that
integrates the estimation of general unknown unitary transformations
$\text{SU}(d)$ into an OSBP. Possible strategies to estimate a general unitary
transformation are given by [59, 60]. We focus on how an estimation strategy
can incorporate the failure branch of an OSBP for a potentially more efficient
estimation, without restricting ourselves to a specific strategy.

The naive way to perform this task would again be to first use a number of
copies to estimate the unknown unitary sufficiently well, such that the
received state can be corrected up to within trace distance
$\bar{D}\leq\epsilon$ of the target. After that, some distillation procedure
is applied to recover the desired state.

input : $k$ copies of the state $\left|\right.\\!{\psi}\\!\left.\right\rangle$

output : $p_{(k)}(U)$, $\hat{U}_{(k)}$ and all saved copies
$|\tilde{\psi}_{s}^{(i)}\rangle$

begin

start with uniform distribution $p_{(0)}(U)$ over $SU(d)$

set estimator $\hat{U}_{(0)}=\mathds{1}$

$i=1$

while  _$i\leq k$ , i.e., there are copies left_ do

send copy number $i$ of the state $|\psi\rangle$

obtain $U^{\otimes N}|\psi\rangle$

correct to $|\tilde{\psi}^{(i)}\rangle=(\hat{U}^{\dagger}_{(i-1)}U)^{\otimes
N}|\psi\rangle$

construct the POVM for the OSBP:

  * •

define $M_{\mathrm{s}}^{V}$ as the operator s.t. $M_{\mathrm{s}}^{V}V^{\otimes
N}\left|\right.\\!{\psi}\\!\left.\right\rangle=\left|\right.\\!{\mathrm{GHZ}_{d}}\\!\left.\right\rangle$

* •

$M_{\mathrm{s}}^{(i)}=\int\limits_{SU(d)}p_{(i-1)}(V)M_{\mathrm{s}}^{\hat{U}^{\dagger}_{(i-1)}V}\mathrm{d}V$

* •

$M_{\mathrm{f}}^{(i)}$ s.t.
$(M_{\mathrm{s}}^{(i)})^{\dagger}M_{\mathrm{s}}^{(i)}+(M_{\mathrm{f}}^{(i)})^{\dagger}M_{\mathrm{f}}^{(i)}=\mathds{1}$

perform OSBP with $\\{M_{\mathrm{s}}^{(i)},M_{\mathrm{f}}^{(i)}\\}$

if  _success_ then

get
$|\tilde{\psi}_{s}^{(i)}\rangle=M_{\mathrm{s}}^{(i)}|\tilde{\psi}^{(i)}\rangle$

if
_$\scaleto{\int\limits_{\scaleto{SU(d)}{5pt}}p_{\scaleto{(i-1)}{5pt}}(U)|\langle{\mathrm{GHZ}_{d}}|{M_{\mathrm{s}}^{(i)}(\hat{U}^{\dagger}_{\scaleto{(i-1)}{5pt}}U)^{\otimes
N}}|{\psi}\rangle|^{2}\mathrm{d}U\geq 1-\epsilon}{15pt}$_ then

keep $|\tilde{\psi}_{s}^{(i)}\rangle$ and save corresponding
$M_{\mathrm{s}}^{(i)}$ and $\hat{U}_{(i-1)}$

leave probability distribution and estimator of the unitary unchanged

$p_{(i)}(U)=p_{(i-1)}(U)$

$\hat{U}_{(i)}=\hat{U}_{(i-1)}$

else

perform a measurement on $|\tilde{\psi}_{s}^{(i)}\rangle$ to estimate $U$

update distribution $p_{(i)}(U)$ and estimator $\hat{U}_{(i)}$

end if

else

perform a measurement on
$|\tilde{\psi}_{f}^{(i)}\rangle=M_{\mathrm{f}}^{(i)}|\tilde{\psi}^{(i)}\rangle$
to estimate $U$

update distribution $p_{(i)}(U)$ and estimator $\hat{U}_{(i)}$

end if

$i++$

end while

return $p_{(k)}(U)$, $\hat{U}_{(k)}$ and all saved copies
$|\tilde{\psi}_{s}^{(i)}\rangle$

end

Algorithm 1 Metrology-integrated OSBP

In our approach we combine both tasks into one. The idea is to construct the
POVM operators for the OSBP based on the knowledge of the unknown unitary at
that specific stage of the protocol. We start with the POVM
$\\{\mathds{1},0\\}$, that will always result in the success branch. Then we
check if the state is sufficiently close to the target state, based on our
knowledge of the unitary. This will not be the case if the distribution $p(U)$
is sufficiently broad. If the state is not close enough to the target, we make
a measurement and update our believe and the estimator. With increasing
knowledge of the unitary transformation, also the POVM of the OSBP will change
and converge towards the optimal POVM for the received state. During this
process the copies in the successful branch close to the target (according to
the knowledge at the time) are stored, while all the others are used for
estimation. With an increasingly peaked distribution $p(U)$ also the ratio of
stored copies to discarded copies (used for estimation) will increase. Given
that enough copies were purchased, we can assume that the unitary
transformation is known sufficiently well at the end of this protocol. For all
stored copies it is checked if they are indeed close enough to the target
according to the updated knowledge of $U$. If not, they are transformed into
the target state via SLOCC. The exact estimation and updating procedure are
left unspecified here on purpose because these will strongly depend on the
exact state and noise model. In Algorithm 1 we consider the exemplary
situation where the desired state is an $N$-partite GHZ-state of local
dimension $d$ and the distributed state is of the form of Eq. (4) with
corresponding dimension and party number. But the idea works for general
scenarios and Algorithm 1 can be adapted for broader classes of vendor and
target states.

Once the first round is completed, all states that were saved, i.e., the
states with a successful OSBP that were sufficiently close to the target state
(with the information at the time), are checked to be sufficiently close with
the final distribution. If not, one may try to transform them into the target
via unitary transformations and an OSBP. The eventual failure branch copies
can be used for estimation.

For the first rounds this protocol will do almost the same as the naive
protocol. Since the distribution $p(U)$ is broad, the POVM will be close to
$\\{\mathds{1},0\\}$, resulting most likely in the success branch (not
altering the state). But as the state will be far from the target, it will be
used for estimation. After many rounds the distribution $p(U)$ will be very
peaked, all copies in the successful branch are now close to the target state
and only the failure branches are used for estimation, resulting in a protocol
similar to the one presented above.

##  6 Conclusion

Potential future implementations of large-scale quantum networks, such as the
envisioned ‘quantum internet’ [13, 14, 15], can be expected to merge idealized
protocols for quantum communication and information processing with real-world
conditions arising from noise and technological constraints which will
necessitate the development of pragmatic and flexible solutions. Here, we have
considered a quantum communication protocol that will be relevant in this
context: the distribution and conversion of high-dimensional entangled states
among multiple parties using noisy channels. Procedures for quantum state-
conversion employed under these circumstances will involve probabilistic
transformations via SLOCC. As we have shown, such scenarios are naturally
amenable to enhancements via embedded channel estimation routines.

While the ideas presented here apply quite generally, we have illustrated the
improvements one may expect in such metrology-assisted entanglement
distribution by focusing on a particular example: probabilistic conversion of
finitely many copies of $N$-qutrit states distributed via a local dephasing
channels to GHZ target states. For this example, we have shown that already
practically easily implementable Bayesian estimation strategies based on local
measurements of individual copies can lead to excellent performance comparable
to optimal distillation procedures. The latter, however, might involve
significantly more complex and technologically challenging joint operations on
multiple copies.

We believe that metrology-assisted entanglement distribution has the potential
to become a relevant factor in future quantum networks. We thus expect the
study and optimization of such protocols for different target states and noisy
models, as well as their comparison to relevant alternative protocols (e.g.,
based on distillation) to open up a broad range of interesting questions
beyond this initial pilot study.

We thank Cornelia Spee for feedback on the manuscript. S.M. acknowledges
support from the Austrian Science Fund (FWF) through projects Y879-N27 (START)
and P 31339-N27 (Stand-Alone). N.F. acknowledges support from the Austrian
Science Fund (FWF) through the projects Y879-N27 (START) and P 31339-N27
(Stand-Alone), and the joint Czech-Austrian project MultiQUEST (I 3053-N27 and
GF17-33780L). M.S. ackowledges support from Spanish MINECO reference
FIS2016-80681-P (with the support of AEI/FEDER,EU), the Catalan Government for
the project QuantumCAT 001-P-001644 (RIS3CAT comunitats) co-financed by the
European Regional Development Fund (FEDER), and the Generalitat de Catalunya,
project CIRIT 2017-SGR-1127.

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##  Appendix A Details on single measurement

We consider an initial measurement where each of the $N$ parties measures in
the basis
$\\{\left|\right.\\!{\pm}\\!\left.\right\rangle_{i}=\tfrac{1}{\sqrt{2}}\bigl{(}\left|\right.\\!{0}\\!\left.\right\rangle_{i}\pm\left|\right.\\!{1}\\!\left.\right\rangle_{i}\bigr{)}\\}$
for $i=1,2,\dots,N$ of the two-dimensional subspace relevant for the copies in
the failure branch. Since the setup is symmetric with respect to the exchange
of any of the $N$ parties, only the parity of the overall measurement matters,
i.e., if one obtains an even or odd number of "$-$" outcomes. To see this note
that the probability to get an outcome "$n$" corresponding to a possible
operator $E_{n}$ is

$\displaystyle\mathrm{Tr}(\left|\right.\\!{\Phi_{\hskip
0.5pt\text{f}}(\theta)}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\Phi_{\hskip
0.5pt\text{f}}(\theta)}\\!\left.\right|E_{n})=\left\langle\right.\\!{\Phi_{\hskip
0.5pt\text{f}}(\theta)}\\!\left.\right|E_{n}\left|\right.\\!{\Phi_{\hskip
0.5pt\text{f}}(\theta)}\\!\left.\right\rangle$
$\displaystyle=a^{2}\left\langle\right.\\!{0}\\!\left.\right|^{\otimes
N}E_{n}\left|\right.\\!{0}\\!\left.\right\rangle^{\otimes
N}+(1-a^{2})\left\langle\right.\\!{1}\\!\left.\right|^{\otimes
N}E_{n}\left|\right.\\!{1}\\!\left.\right\rangle^{\otimes N}$ $\displaystyle\
\ \ \ +a\sqrt{1-a^{2}}\,\mathrm{e}^{-i\theta
N}\left\langle\right.\\!{1}\\!\left.\right|^{\otimes
N}\\!\\!E_{n}\left|\right.\\!{0}\\!\left.\right\rangle^{\otimes N}$ (16)
$\displaystyle\ \ \ \ +a\sqrt{1-a^{2}}\,\mathrm{e}^{i\theta
N}\left\langle\right.\\!{0}\\!\left.\right|^{\otimes
N}\\!\\!E_{n}\left|\right.\\!{1}\\!\left.\right\rangle^{\otimes N},$

while $E_{n}$ is one of $2^{N}$ products of projectors
$\left|\right.\\!{\pm}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\pm}\\!\left.\right|$,
for instance
$E_{+-+--}=\left|\right.\\!{+}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{+}\\!\left.\right|\otimes\left|\right.\\!{-}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{-}\\!\left.\right|\otimes\left|\right.\\!{+}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{+}\\!\left.\right|\otimes\left|\right.\\!{-}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{-}\\!\left.\right|\otimes\left|\right.\\!{-}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{-}\\!\left.\right|$.
But only the parity of the number of "$-$" projector matters. Out of all
$2^{N}$ combinations of
$\left|\right.\\!{+}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{+}\\!\left.\right|$
and
$\left|\right.\\!{-}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{-}\\!\left.\right|$
projectors for $N$ qubits, half have an even number of "$-$" projectors, so we
have

$\displaystyle p(\text{even}/\text{odd}|\theta)=\tfrac{1}{2}\pm
a\sqrt{1-a^{2}}\cos(N\theta)$ (17)

Given a prior $p(\theta)$, we then assign a probability

$\displaystyle p(\text{even}/\text{odd})$
$\displaystyle=\int\limits_{0}^{2\pi}p(\theta)\,p(\text{even}/\text{odd}|\theta)d\theta$
(18) $\displaystyle=\tfrac{1}{2}\pm
a\sqrt{1-a^{2}}\int\limits_{0}^{2\pi}p(\theta)\cos(N\theta)d\theta$

and for a flat prior $p(\theta)=\frac{1}{2\pi}$ this becomes
$p(\text{even}/\text{odd})=\frac{1}{2}$. We now use Bayes’ law to obtain a
posterior distribution. For a flat prior this becomes

$\displaystyle p(\theta|\text{even}/\text{odd})=\frac{1}{2\pi}[1\pm
2a\sqrt{1-a^{2}}\cos(N\theta)].$ (19)

Next, we nominate a suitable estimator for $N\theta$. As an estimator we use

$\displaystyle\hat{\theta}_{N}^{\raisebox{0.0pt}{\tiny{$(m)$}}}=\arg(\int\limits_{0}^{2\pi}p(\theta|m)\,\mathrm{e}^{iN\theta}\,d\theta),$
(20)

and for a flat prior we have

$\displaystyle\hat{\theta}_{N}^{(\text{even/odd})}=\arg(\mathrm{e}^{\frac{1\mp
1}{2}i\pi}),$ (21)

and so $\hat{\theta}_{N}^{(\text{even})}=0$ and
$\hat{\theta}_{N}^{(\text{odd})}=\pi$. We are then interested in evaluating
the trace distance between the updated and corrected success-branch state

$\displaystyle\rho_{\text{corr}}^{\raisebox{0.0pt}{\tiny{$(m)$}}}=\int\limits_{0}^{2\pi}p(\theta|m)\left|\right.\\!{\psi_{\text{s}}(\theta-\tfrac{\hat{\theta}_{N}^{\raisebox{0.0pt}{\tiny{$(m)$}}}}{N})}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\psi_{\text{s}}(\theta-\tfrac{\hat{\theta}_{N}^{\raisebox{0.0pt}{\tiny{$(m)$}}}}{N})}\\!\left.\right|,$
(22)

with

$\displaystyle\left|\right.\\!{\psi_{\text{s}}(\theta)}\\!\left.\right\rangle=\frac{1}{\sqrt{3}}\sum\limits_{l=0}^{2}\mathrm{e}^{ilN\theta}\left|\right.\\!{l}\\!\left.\right\rangle^{\otimes
N}$ (23)

and the desired target state
$\left|\right.\\!{\psi_{\text{s}}(0)}\\!\left.\right\rangle$. The trace
distance is

$\displaystyle
D(\rho_{\text{corr}}^{\raisebox{0.0pt}{\tiny{$(m)$}}},\rho_{\text{target}})=\frac{1}{2}\mathrm{Tr}\bigg{(}\sqrt{(\rho_{\text{corr}}^{\raisebox{0.0pt}{\tiny{$(m)$}}}-\rho_{\text{target}})^{2}}\bigg{)},$
(24)

and we can calculate

$\displaystyle\rho_{\text{corr}}^{\raisebox{0.0pt}{\tiny{$(m)$}}}-\rho_{\text{target}}=\frac{1}{3}\sum\limits_{j\neq
l}c_{jl}^{\raisebox{0.0pt}{\tiny{$(m)$}}}(|{j}\rangle\\!\langle{l}|)^{\otimes
N},$ (25)

with

$\displaystyle
c_{jl}^{\raisebox{0.0pt}{\tiny{$(m)$}}}=\int\limits_{0}^{2\pi}p(\theta|m)\,\mathrm{e}^{i(N\theta-\hat{\theta}_{N}^{\raisebox{0.0pt}{\tiny{$(m)$}}})(j-l)}d\theta-1.$
(26)

For $\hat{\theta}_{N}^{\raisebox{0.0pt}{\tiny{$(m)$}}}=0$ and
$\hat{\theta}_{N}^{\raisebox{0.0pt}{\tiny{$(m)$}}}=\pi$ the coefficients
$c_{jl}^{\raisebox{0.0pt}{\tiny{$(m)$}}}$ are real and therefore

$\displaystyle
D(\rho_{\text{corr}}^{\raisebox{0.0pt}{\tiny{$(m)$}}},\rho_{\text{target}})=\frac{1}{6}(|c_{02}|+\sqrt{8c_{01}^{2}+c_{02}^{2}}).$
(27)

For the flat prior we find

$\displaystyle c_{0,1}^{(\text{even}/\text{odd})}$
$\displaystyle=\,c_{1,2}^{(\text{even}/\text{odd})}\,=\,a\sqrt{1-a^{2}}-1,$
(28a) $\displaystyle c_{0,2}^{(\text{even}/\text{odd})}$ $\displaystyle=\,-1,$
(28b)

and with this the trace distance becomes

$\displaystyle
D(\rho_{\text{corr}}^{\raisebox{0.0pt}{\tiny{$(m)$}}},\rho_{\text{target}})=\tfrac{1}{6}\Bigl{(}1\\!+\\!\sqrt{9+8a^{2}(1-a^{2})-16a\sqrt{1-a^{2}}}\,\Bigr{)},$
(29)

and since this is independent of the outcome it already matches the average
trace distance
$\bar{D}(\rho_{\text{corr}},\rho_{\text{target}})=\sum\limits_{m}p(m)D(\rho_{\text{corr}}^{\raisebox{0.0pt}{\tiny{$(m)$}}},\rho_{\text{target}})$.

##  Appendix B Multiple copies without updating

Now let us see how measurements on more than one copy, in particular on $k$
copies, can improve this value by considering a simple scenario where the
measurement directions are not updated. This means, each copy is locally
measured in the basis $\\{\left|\right.\\!{\pm}\\!\left.\right\rangle_{i}\\}$
by every party, resulting in $k$ individual even/odd measurement outcomes.
Since there is no updating in-between, the particular order of these outcomes
is irrelevant and only the overall number of even/odd outcomes matter. We can
thus view the entire procedure as one $(k+1)$-outcome measurement with
outcomes $m=0,1,2,\dots,k$, i.e., the number of "even" outcomes. We hence
assign the probabilities

$\displaystyle
p(m|\theta)=p(\text{even}|\theta)^{m}p(\text{odd}|\theta)^{k-m}\begin{pmatrix}k\\\
m\end{pmatrix}.$ (30)

The corresponding unconditional probabilities are

$\displaystyle p(m)=$ $\displaystyle\frac{1}{2^{k}}\begin{pmatrix}k\\\
m\end{pmatrix}\sum\limits_{i=0}^{m}\sum\limits_{j=0}^{k-m}\begin{pmatrix}m\\\
i\end{pmatrix}\begin{pmatrix}k-m\\\ j\end{pmatrix}(-1)^{j}$ $\displaystyle\
\times\,(2a\sqrt{1-a^{2}})^{i+j}\,\int\limits_{0}^{2\pi}p(\theta)\,\cos^{i+j}(N\theta)d\theta.$
(31)

For a flat prior we calculate

$\displaystyle p(m)=$ $\displaystyle\frac{1}{2^{k}}\begin{pmatrix}k\\\
m\end{pmatrix}\sum\limits_{i=0}^{m}\sum\limits_{j=0}^{k-m}\begin{pmatrix}m\\\
i\end{pmatrix}\begin{pmatrix}k-m\\\ j\end{pmatrix}(-1)^{j}$ $\displaystyle\
\times\,(2a\sqrt{1-a^{2}})^{i+j}\,\frac{(-1)^{i}+(-1)^{j}}{2^{i+j+1}}\begin{pmatrix}i+j\\\
\frac{i+j}{2}\end{pmatrix}.$ (32)

The posterior given outcome $m$ is then

$\displaystyle p(\theta|m)=$
$\displaystyle\frac{1}{2^{k}p(m)}\begin{pmatrix}k\\\
m\end{pmatrix}\sum\limits_{i=0}^{m}\sum\limits_{j=0}^{k-m}\begin{pmatrix}m\\\
i\end{pmatrix}\begin{pmatrix}k-m\\\ j\end{pmatrix}$
$\displaystyle\times\,(-1)^{j}\,(2a\sqrt{1-a^{2}})^{i+j}\,p(\theta)\,\cos^{i+j}(N\theta).$
(33)

For the estimator we have to calculate

$\displaystyle\bar{z}_{N,k}^{\raisebox{0.0pt}{\tiny{$(m)$}}}=\int\limits_{0}^{2\pi}p(\theta|m)\mathrm{e}^{iN\theta}d\theta.$
(34)

Figure 3:  Average distance to target state for different ways of adapting the
measurement direction. The curves show the average trace distance between the
target state and the successfully converted copies for different measurement-
updating strategies for the exemplary metrology-assisted entanglement
distribution protocol discussed in Sec. 3 for fixed value of
$a=\tfrac{1}{\sqrt{2}}$ as a function of the number $k_{\text{f}}$ of copies
in the failure branch. The blue curve represents the strategy with constant
measurement directions discussed in Appendix B, for the yellow curve
measurements are alternated between $\sigma_{x}$ and $\sigma_{y}$, for green
the measurement directions are rotated by a constant angle $\pi/8$ after each
measurement, and the red curve corresponds to an adaptive strategy updating
the measurement direction using the maximum likelihood estimator from Eq. (21)
of the previous round.

For the flat prior the estimator becomes the argument of

$\displaystyle\bar{z}_{N,k}^{\raisebox{0.0pt}{\tiny{$(m)$}}}=$
$\displaystyle\frac{1}{2^{k}p(m)}\begin{pmatrix}k\\\
m\end{pmatrix}\sum\limits_{i=0}^{m}\sum\limits_{j=0}^{k-m}\begin{pmatrix}m\\\
i\end{pmatrix}\begin{pmatrix}k-m\\\ j\end{pmatrix}(-1)^{j}$ $\displaystyle\
\times\,(2a\sqrt{1-a^{2}})^{i+j}\,\frac{(-1)^{i}+(-1)^{j}}{2^{i+j+1}}\begin{pmatrix}i+j\\\
\frac{i+j}{2}\end{pmatrix}.$ (35)

We can distinguish two cases, as long as $m<k/2$ we have
$\bar{z}_{N,k}^{\raisebox{0.0pt}{\tiny{$(m)$}}}<0$ and therefore
$\arg(\bar{z}_{N,k}^{\raisebox{0.0pt}{\tiny{$(m)$}}})=\pi=\hat{\theta}_{N,m}^{\raisebox{0.0pt}{\tiny{$(m)$}}}$.
For $m\geq k/2$ we have $\bar{z}_{N,k}^{\raisebox{0.0pt}{\tiny{$(m)$}}}\geq 0$
and therefore
$\arg(\bar{z}_{N,k}^{\raisebox{0.0pt}{\tiny{$(m)$}}})=0=\hat{\theta}_{N,m}^{\raisebox{0.0pt}{\tiny{$(m)$}}}$.
What remains is calculating the distance to our target state, and the result
is illustrated in Fig. 3.

In the same figure we numerically investigate the performance of different
measurement-updating strategies. We notice that, perhaps unsurprisingly, the
strategy with constant measurement direction performs worst of all considered
strategies, whereas strategies with changing measurement directions perform
considerably better. More specifically, we compare an alternating strategy
that switches between two mutually unbiased measurements, and a rotating
strategy, where the measurement direction is rotated by a fixed angle, to the
adaptive strategy. In the latter strategy, which is also considered in the
main text, the measurement direction is updated based on the outcome of the
previous measurement rounds and chosen perpendicular (in Bloch representation)
to the estimator for the angle $\theta$ from the previous round. While the
adaptive strategy performs best, as expected, also the strategies that change
the measurement direction according to some pre-established rule perform
comparably well. This gives viable alternatives that are possibly easier to
implement, as they do not depend on previous outcomes.

##  Appendix C Estimation with received copies

Figure 4:  Average distance to target state for estimation with received
states. The curve shows the average trace distance between the target state
and the successfully converted copies as a function of the number $k$ of
copies used. The blue line shows the ’naive’ protocol discussed in Sec. 4,
where the received states
$\left|\right.\\!{\Psi(\alpha,\beta)}\\!\left.\right\rangle$ from Eq. (7) are
used directly for the estimation. The yellow curve shows the estimation
performed on copies of the failure branch described by Eq. (9). The gray line
shows the trace distance of $\epsilon\approx 0.038$ after measuring 20 copies
of the failure branch and we see that 9 copies of the received state are
necessary to achieve the same distance. The dotted line represents the lower
bound calculated in Appendix D based on the Bayesian Cramér-Rao bound for the
respective probe state. The parameters used are $\alpha=\arccos(2/\sqrt{15})$
and $\beta=\pi/4$.

In this appendix we briefly explain how to estimate the phase $\theta$ using
the state $\left|\right.\\!{\Psi(\alpha,\beta)}\\!\left.\right\rangle$ given
in Eq. (7) along with a Bayesian estimation strategy. The strategy is
essentially the same as that employed before, adapted to the qutrit probe
states. First, we choose an appropriate three-outcome projective measurement
in a basis that is mutually unbiased with respect to the eigenbasis of the
generating Hamiltonian of the transformation $U(\theta)$ from Eq. (5), e.g.
the discrete Fourier transform of the standard basis

$\displaystyle\left|\right.\\!{\xi_{0}}\\!\left.\right\rangle$
$\displaystyle=\tfrac{1}{\sqrt{3}}(\left|\right.\\!{0}\\!\left.\right\rangle+\left|\right.\\!{1}\\!\left.\right\rangle+\left|\right.\\!{2}\\!\left.\right\rangle),$
(36a) $\displaystyle\left|\right.\\!{\xi_{1}}\\!\left.\right\rangle$
$\displaystyle=\tfrac{1}{\sqrt{3}}(\left|\right.\\!{0}\\!\left.\right\rangle+\omega\left|\right.\\!{1}\\!\left.\right\rangle+\omega^{2}\left|\right.\\!{2}\\!\left.\right\rangle),$
(36b) $\displaystyle\left|\right.\\!{\xi_{2}}\\!\left.\right\rangle$
$\displaystyle=\tfrac{1}{\sqrt{3}}(\left|\right.\\!{0}\\!\left.\right\rangle+\omega^{2}\left|\right.\\!{1}\\!\left.\right\rangle+\omega\left|\right.\\!{2}\\!\left.\right\rangle),$
(36c)

where $\omega=\mathrm{e}^{2\pi i/3}$. Each party performs a measurement in
this basis. Let $E_{n}$ denote the $3^{N}$ possible products of $N$ projectors
$|{\xi_{j}}\rangle\\!\langle{\xi_{j}}|$. We then calculate

$\displaystyle\mathrm{Tr}(\left|\right.\\!{\Psi(\alpha,\beta)}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\Psi(\alpha,\beta)}\\!\left.\right|E_{n})=\langle{\Psi(\alpha,\beta)}|{E_{n}}|{\Psi(\alpha,\beta)}\rangle$
$\displaystyle=\tfrac{1}{3^{N}}\big{[}1+2\sin^{2}\alpha\cos\beta\sin\beta\cos(\tfrac{2\pi}{3}[n_{1}+2n_{2}]-\theta
N)$ $\displaystyle\ \
+2\sin\alpha\cos\alpha\cos\beta\cos(\tfrac{2\pi}{3}[2n_{1}+n_{2}]-2\theta N)$
$\displaystyle\ \
+2\sin\alpha\sin\beta\cos\alpha\cos(\tfrac{2\pi}{3}[n_{1}-n_{2}]-\theta
N)\big{]},$ (37)

where $n_{j}$ denotes the number of projectors
$|{\xi_{j}}\rangle\\!\langle{\xi_{j}}|$ in $E_{n}$. The probabilities for the
different outcomes of the measurement hence only depend on the respective
number of projectors constituting the measurement, which also follows from the
symmetry with respect to the exchange of the $N$ parties. Moreover, the
probabilities of the outcomes only depend on $n_{1}+2n_{2}$, $2n_{1}+n_{2}$
and $n_{1}-n_{2}$ modulo $3$. We therefore only need to distinguish between
three possible measurement outcomes $(0,0,0)$, $(1,2,1)$, and $(2,1,2)$, which
we can rewrite as $m^{l}=(l,2l,l)$. It is clear that there are $3^{(N-1)}$
sequences resulting in a given outcome, and hence it follows that

$\displaystyle p(m^{l}|\theta)$
$\displaystyle=\,\tfrac{1}{3}\big{[}1+2\sin^{2}\alpha\cos\beta\sin\beta\cos(\tfrac{2\pi}{3}l-\theta
N)$ $\displaystyle\
+2\sin\alpha\cos\alpha\cos\beta\cos(\tfrac{4\pi}{3}l-2\theta N)$
$\displaystyle\ +2\sin\alpha\sin\beta\cos\alpha\cos(\tfrac{2\pi}{3}l-\theta
N)\big{]}.$ (38)

Here, we again want to mention that we actually estimate $\theta N$ and that
the estimation precision is therefore independent of the number of parties. By
Bayes’ law we can now calculate $p(\theta|m)$ and use the estimator from Eq.
(20). In each subsequent measurement round we adapt the measurement direction
based on the outcome of the previous round, that is we rotate our measurement
basis by $U(\hat{\theta}+\pi/3)$. In Fig. 4 we compare the resulting average
trace distance for this strategy to that obtained from an estimation strategy
based only on the failure-branch states.

##  Appendix D A lower bound for the number of copies for any estimation
protocol

In this appendix we derive a lower bound for the number of copies needed for
any estimation to achieve a given trace distance. Since we assume that no
prior information about the phase is available initially, we rely on Bayesian
estimation with a flat prior distribution. After $k$ measurement rounds the
posterior distribution of the parameter $\theta$ is
$p(\theta|m_{1},\dots,m_{k})=p(\theta|\mathbf{m})$ and the corrected state
becomes

$\displaystyle\rho_{\text{corr}}^{\raisebox{0.0pt}{\tiny{$(\mathbf{m})$}}}=\int\limits_{0}^{2\pi}p(\theta|\mathbf{m})\left|\right.\\!{\psi_{\text{s}}(\theta-\tfrac{\hat{\theta}_{N}^{\raisebox{0.0pt}{\tiny{$(\mathbf{m})$}}}}{N})}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\psi_{\text{s}}(\theta-\tfrac{\hat{\theta}_{N}^{\raisebox{0.0pt}{\tiny{$(\mathbf{m})$}}}}{N})}\\!\left.\right|d\theta,$
(39)

for some estimator
$\hat{\theta}_{N}^{\raisebox{0.0pt}{\tiny{$(\mathbf{m})$}}}$. Now we can bound
the average trace distance from below using its convexity, i.e., using the
notation
$\rho_{\text{target}}=\left|\right.\\!{\psi_{\text{s}}(0)}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\psi_{\text{s}}(0)}\\!\left.\right|$
and
$\rho_{\text{s}}(\theta)=\left|\right.\\!{\psi_{\text{s}}(\theta)}\\!\left.\right\rangle\\!\\!\left\langle\right.\\!{\psi_{\text{s}}(\theta)}\\!\left.\right|$
we have

$\displaystyle\bar{D}(\rho_{\text{corr}}^{\raisebox{0.0pt}{\tiny{$(\mathbf{m})$}}},\rho_{\text{target}})=\sum\limits_{\mathbf{m}}p(\mathbf{m})D(\rho_{\text{corr}}^{\raisebox{0.0pt}{\tiny{$(\mathbf{m})$}}},\rho_{\text{target}})$
(40) $\displaystyle\ \geq
D(\sum\limits_{\mathbf{m}}p(\mathbf{m})\rho_{\text{corr}}^{\raisebox{0.0pt}{\tiny{$(\mathbf{m})$}}},\rho_{\text{target}})$
$\displaystyle\ \
=D\bigl{(}\,\int\limits_{0}^{2\pi}\sum\limits_{\mathbf{m}}p(\mathbf{m})p(\theta|\mathbf{m})\rho_{\text{s}}(\theta-\tfrac{\hat{\theta}_{N}^{\raisebox{0.0pt}{\tiny{$(\mathbf{m})$}}}}{N})d\theta,\,\rho_{\text{target}}\,\bigr{)}.$

To continue the calculation and to actually compute a useful lower bound we
need to make an approximation. We replace the average posterior distribution
by a Gaussian distribution with the same mean $\mu$ and variance $\sigma^{2}$.
Since the trace distance
$D(\left|\right.\\!{\psi_{\text{s}}(\theta)}\\!\left.\right\rangle,\left|\right.\\!{\psi_{\text{s}}(0)}\\!\left.\right\rangle$
grows sublinearly with $\theta$, we expect that the difference in the higher-
order moments of the distributions will not influence the result
significantly. Additionally, after some number of measurements the estimation
is sufficiently precise such that the numerical support of the distribution is
within the interval $[0,2\pi)$ and we can therefore integrate over the real
line. With this we finally conclude

$\displaystyle\bar{D}(\rho_{\text{corr}}^{\raisebox{0.0pt}{\tiny{$(\mathbf{m})$}}},\rho_{\text{target}})\geq
D(\sum\limits_{\mathbf{m}}p(\mathbf{m})\rho_{\text{corr}}^{\raisebox{0.0pt}{\tiny{$(\mathbf{m})$}}},\rho_{\text{target}})$
$\displaystyle\approx
D\bigl{(}\int\limits_{-\infty}^{\infty}\\!\\!\frac{1}{\sqrt{2\pi}\sigma}\mathrm{e}^{-\frac{(\theta-\mu)^{2}}{2\sigma^{2}}}\rho_{\text{s}}(\theta-\tfrac{\hat{\theta}_{N}^{\raisebox{0.0pt}{\tiny{$(\mathbf{m})$}}}}{N})d\theta,\,\rho_{\text{target}}\,\bigr{)}$
$\displaystyle\geq
D\bigl{(}\int\limits_{-\infty}^{\infty}\\!\\!\frac{1}{\sqrt{2\pi}\sigma}\mathrm{e}^{-\frac{(\theta-\hat{\theta}_{N}^{\raisebox{0.0pt}{\tiny{$(\mathbf{m})$}}}/N)^{2}}{2\sigma^{2}}}\rho_{\text{s}}(\theta-\tfrac{\hat{\theta}_{N}^{\raisebox{0.0pt}{\tiny{$(\mathbf{m})$}}}}{N})d\theta,\,\rho_{\text{target}}\,\bigr{)}$
$\displaystyle=\tfrac{\mathrm{e}^{-2N^{2}\sigma^{2}}}{6}\bigl{(}\mathrm{e}^{\tfrac{N^{2}\sigma^{2}}{2}}\\!-\\!1\bigr{)}\Bigl{[}1\\!+\\!\mathrm{e}^{\tfrac{N^{2}\sigma^{2}}{2}}\\!+\\!\mathrm{e}^{N^{2}\sigma^{2}}+\mathrm{e}^{\tfrac{3N^{2}\sigma^{2}}{2}}$
$\displaystyle\ \
+\sqrt{8\mathrm{e}^{3N^{2}\sigma^{2}}+(1+\mathrm{e}^{N^{2}\sigma^{2}/2})^{2}(1+\mathrm{e}^{N^{2}\sigma^{2}})^{2}}\,\Bigr{]},$
(41)

where we have used the fact that the trace distance is minimal if the mean of
the Gaussian distribution equals the value of the estimator, i.e., using the
mean of the posterior as estimator. The average variance of any estimator is
bounded from below by the van Trees inequality
$V[\hat{\theta}(\mathbf{m})]\geq
1/(I[p(\theta)]+\bar{I}[p(\mathbf{m}|\theta)])$, where $I[p(\theta)]$ is the
Fisher information (FI) of the prior, and $\bar{I}[p(\mathbf{m}|\theta)])$ is
the average FI of the likelihood over all values of $\theta$ [61]. Now we
bound the FI from above by the quantum FI (QFI),
$p(\mathbf{m}|\theta)]\geq\mathcal{I}[\left|\right.\\!{\psi(\theta)}\\!\left.\right\rangle]$,
and notice that the QFI is independent of $\theta$ for a unitary evolution and
hence equals its average [62, 63]. One can thus get a Bayesian analogue of the
quantum Cramér-Rao bound for the mean of the posterior, bounding the average
posterior variance by $\bar{V}[p(\theta|\mathbf{m})]\geq
1/(I[p(\theta)]+\mathcal{I}[\left|\right.\\!{\psi(\theta)}\\!\left.\right\rangle])$
(see, e.g., [64, 65]). For the state in Eq. (7) the FI is

$\displaystyle\mathcal{I}\bigl{[}\left|\right.\\!{\psi(\theta)}\\!\left.\right\rangle\bigr{]}$
$\displaystyle=\,4N^{2}[\sin^{2}\alpha\sin^{2}\beta+4\cos^{2}\alpha$
$\displaystyle\ \ \ \ -(\sin^{2}\alpha\sin^{2}\beta+2\cos^{2}\alpha)^{2}]$
(42)

and for a flat prior the FI vanishes. Since the expression in the last line of
(41) increases monotonically in $\sigma^{2}$ we can further bound the average
trace distance from below by inserting $\sigma^{2}=1/(k\
\mathcal{I}\bigl{[}\left|\right.\\!{\psi(\theta)}\\!\left.\right\rangle\bigr{]})$.
This leaves us with an expression that is decreasing monotonically with the
number $k$ of copies used for estimation but which is independent of the
number of parties $N$. With this we can determine a lower bound on the number
of copies needed to reach a given trace distance.