# One-Shot Coherence Distillation: Towards Completing the Picture

Qi Zhao, Yunchao Liu, Xiao Yuan, Eric Chitambar and Andreas Winter Qi Zhao and
Yunchao Liu are with Center for Quantum Information, Institute for
Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084,
China. Email<EMAIL_ADDRESS>Yuan is with Department of Materials,
University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom.Eric
Chitambar is with the Department of Electrical and Computer Engineering at the
University of Illinois, Urbana-Champaign, Illinois 61801, USA. Email:
<EMAIL_ADDRESS>Winter is with ICREA—Institució Catalana de
Recerca i Estudis Avançats, Pg. Lluis Companys, 23, ES-08001 Barcelona, Spain
and also with Física Teòrica: Informació i Fenòmens Quàntics, Departament de
Física, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona),
Spain. Email<EMAIL_ADDRESS>January 2019.

###### Abstract

The resource framework of quantum coherence was introduced by Baumgratz,
Cramer and Plenio [_Phys. Rev. Lett._ 113, 140401 (2014)] and further
developed by Winter and Yang [_Phys. Rev. Lett._ 116, 120404 (2016)]. We
consider the one-shot problem of distilling pure coherence from a single
instance of a given resource state. Specifically, we determine the distillable
coherence with a given fidelity under incoherent operations (IO) through a
generalisation of the Winter-Yang protocol. This is compared to the
distillable coherence under maximal incoherent operations (MIO) and dephasing-
covariant incoherent operations (DIO), which can be cast as a semidefinite
programme, that has been presented previously by Regula _et al._ [_Phys. Rev.
Lett._ 121, 010401 (2018)]. Our results are given in terms of a smoothed min-
relative entropy distance from the incoherent set of states, and a variant of
the hypothesis-testing relative entropy distance, respectively. The one-shot
distillable coherence is also related to one-shot randomness extraction.
Moreover, from the one-shot formulas under IO, MIO, DIO, we can recover the
optimal distillable rate in the many-copy asymptotics, yielding the relative
entropy of coherence. These results can be compared with previous work by some
of the present authors [Zhao _et al._ , _Phys. Rev. Lett._ 120, 070403 (2018)]
on one-shot coherence formation under IO, MIO, DIO and also SIO. This shows
that the amount of distillable coherence is essentially the same for IO, DIO,
and MIO, despite the fact that the three classes of operations are very
different. We also relate the distillable coherence under strictly incoherent
operations (SIO) to a constrained hypothesis testing problem and explicitly
show the existence of bound coherence under SIO in the asymptotic regime.

##  I Introduction

Coherence, as the signature of non-classicality, has applications in many
quantum information processing tasks, including cryptography [1], metrology
[2], randomness generation [3, 4], biological systems [5, 6] and
thermodynamics [7, 8, 9, 10, 11]. Recently, a resource theory framework of
quantum coherence has been introduced by Baumgratz, Cramer and Plenio [12]
(after prior work of Åberg [13], as well as Braun and Georgeot [14]). A
general quantum resource theory consists of two objects: a set of free states
and a set of free operations, with the latter acting invariantly on the
former. In the resource theory of coherence, the free states $\mathcal{I}$ for
a $d$-dimensional Hilbert space is the set of density matrices that are
invariant under conjugation by the phase unitary

$Z=\left(\begin{array}[]{cccc}1&0&\cdots&0\\\ 0&e^{2\pi i/d}&&\vdots\\\
\vdots&&\ddots&0\\\ 0&\cdots&0&e^{2\pi i(d-1)/d}\end{array}\right),$ (1)

with $Z$ being expressed in a priori fixed computational basis
$\\{|x\rangle\\}_{x=1}^{d}$. Furthermore, different physical and mathematical
motivations have led to the proposal and study of different classes of free
operations, most notably the maximally incoherent operations (MIO) [13], the
dephasing-covariant incoherent operation (DIO) [15, 16], the incoherent
operations (IO) [12], and the strictly incoherent operations (SIO) [17, 18].
For instance, SIO emerges as a natural class of operations to consider when
quantifying visibility in interferometer experiments [19]. Several coherence
measures have been introduced to quantify the amount of coherence in a state,
including the relative entropy and $\ell_{1}$-norm of coherence [12], the
coherence of formation [3, 13], and the robustness of coherence [20], which
have all been shown to possess different operational meanings [3, 17, 20, 21].
An overview on recent developments of the resource theory of coherence can be
found in Ref. [22].

An important problem in the resource theory is convertibility of states via
free operations, especially those between an arbitrary state $\rho$ and a
maximal resource state $|\Psi_{M}\rangle$. In an $M$-level system, the
superposition state
$|\Psi_{M}\rangle=\frac{1}{\sqrt{M}}\sum_{i=1}^{M}|i\rangle$ has the maximal
coherence with respect to the aforementioned, and indeed all, coherence
measures, and the qubit state $|\Psi_{2}\rangle$ serves as resource unit in
the theory, the “cosbit” [23]. The process of converting a given state $\rho$
to $|\Psi_{M}\rangle$ is referred to as _coherence distillation_ and the
reverse process as _coherence dilution_. In the asymptotic case where an
unbounded number of independent and identically distributed (i.i.d.) copies of
the initial state are provided, the optimal rates of asymptotic distillation
and dilution are quantified by the relative entropy of coherence and the
coherence of formation, respectively [17].

In practical scenarios, i.i.d. resources are not available and an analysis of
non-asymptotic tasks becomes crucial. In recent work [24], the formation
problem was addressed in the one-shot setting under all four of the above
operational classes. Subsequently, the converse question of one-shot coherence
distillation was solved for the classes MIO and DIO [25]. In the present
paper, we close the remaining gap by providing an analysis of general one-shot
coherence distillation under IO and SIO. Our results show that, while not
being exactly identical, the three classes MIO, DIO, and IO all lead to
essentially the same expression for the distillable coherence in terms of a
min-relative entropy distance from the incoherent states; the differences are
in the smoothing parameters and universal additive terms. When extended to the
asymptotic case, our one-shot results imply that MIO, DIO, and IO all have
distillable coherence given by the relative entropy, thereby recovering
earlier work found in [17]. In contrast, we show that the coherence
distillation by SIO can behave quite differently than its more general
counterparts, and we characterize the one-shot distillable coherence as a
constrained hypothesis testing problem. We further show that coherence can be
a bound resource under SIO, meaning that states exist with zero distillable
coherence but nonzero coherence cost. We also show the relationship among the
extractable randomness, IO and DIO distillable rate, which are are indeed
closely related. The reason is that the distillation proceeds by a kind of
decoupling process, and can thus be related to randomness extraction.

The structure of the remainder of the paper is as follows: In Section II we
review the four mentioned classes of incoherent operations and introduce the
dilution and distillation tasks for coherence in the one-shot setting. We
present the prior results on dilution in all four cases, and in Section III
review the prior results on distillation under MIO and DIO. In Section IV we
come to the first main result of the present paper, a tight one-shot
characterization of distillation with IO, showing in particular a distillation
protocol achieving the lower bound with an operation that is at the same time
IO and DIO. Then, in Section V we give a one-shot formula for distillation
under SIO, and most importantly, show the existence of bound coherence in this
model. In Section VI we show how the obtained one-shot formulas for IO, DIO
and MIO imply the previously known asymptotic distillation rate $C_{r}(\rho)$,
and finally Section VII contains the discussion on the relation between
randomness extraction and coherence distillation, after which we conclude.

##  II Coherence Dilution
and Distillation

###  II-A Classes of incoherent operations

A general resource theory of coherence is constructed by imposing restrictions
on the allowed completely positive trace-preserving (CPTP) maps
$\Lambda\mathrel{\mathop{\mathchar 58\relax}}\mathcal{L}(A)\to\mathcal{L}(B)$,
where $\mathcal{L}(A)$ denotes the space of linear operators acting on a
finite-dimensional Hilbert space $A$ and likewise for $\mathcal{L}(B)$. One
necessary restriction is that $\Lambda$ acts invariantly on the set of
incoherent states $\mathcal{I}$; i.e. $\Lambda(\delta)\in\mathcal{I}$ whenever
$\delta\in\mathcal{I}$. The completely dephasing map
$\Delta(\rho)=\sum_{x=1}^{d}|x\rangle\langle x|\rho|x\rangle\langle x|$ plays
an important role in this theory as it destroys all coherence in a state, and
it therefore maps any resource state to a free one. The following operations
classes have been proposed in [12, 13, 15, 16, 17, 18], each motivated by
different physical considerations.

_Maximal Incoherent Operations_ [13] are characterized by
$\Lambda(\delta)\in\mathcal{I}$ for all $\delta\in\mathcal{I}$, which may be
expressed as the identity $\Lambda\circ\Delta=\Delta\circ\Lambda\circ\Delta$;

_Dephasing-Covariant Incoherent Operations_ [15, 16] satisfy the stronger
condition $\Lambda\circ\Delta=\Delta\circ\Lambda$;

_Incoherent Operations_ [12] are those MIO with a Kraus decomposition
$\Lambda(\rho)=\sum_{\alpha}K_{\alpha}\rho K_{\alpha}^{\dagger}$, such that
$K_{\alpha}\delta K_{\alpha}^{\dagger}/\operatorname{Tr}(K_{\alpha}\delta
K_{\alpha}^{\dagger})\in\mathcal{I}$ for all $\alpha$ and
$\delta\in\mathcal{I}$;

_Strictly Incoherent Operations_ [17] are those IO with a Kraus decomposition
$\Lambda(\rho)=\sum_{\alpha}K_{\alpha}\rho K_{\alpha}^{\dagger}$ such that
$K_{\alpha}=\sum_{x}c_{\alpha,x}|f_{\alpha}(x)\rangle\langle x|$, and
$f_{\alpha}\mathrel{\mathop{\mathchar 58\relax}}[d]=\\{1,\cdots,d\\}\to[d]$ is
a one-to-one function for all $\alpha$.

The relationships among these classes are

$\displaystyle\text{SIO}\varsubsetneq\text{IO}\varsubsetneq\text{MIO},$ (2)
$\displaystyle\text{SIO}\varsubsetneq\text{DIO}\varsubsetneq\text{MIO};$

note that IO is not included in DIO and vice versa.

For later use, and because it will turn out to be significant, we also give a
name to the intersection of IO and DIO,

$\text{DIIO}=\text{IO}\cap\text{DIO},$

which we dub _dephasing-covariant incoherent IO_.

|  | MIO====== | DIO ====== | IO | SIO
---|---|---|---|---|---
=One-shot $\biggl{\\{}$ | distillation= |  $\widetilde{C}_{H}^{\varepsilon}$ [25] |  $\widetilde{C}_{H}^{\varepsilon}$ [25] |  $\mathbf{C_{\min}^{\varepsilon^{\prime}}}$ [$\ast$] |  $\mathbf{Thm.~{}\ref{th:SIO}}$ [$\ast$]
formation |  $C_{\max}^{\varepsilon}$ [24] |  $C_{\Delta,\max}^{\varepsilon}$ [24] |  $C_{0}^{\varepsilon}$ [24] |  $C_{0}^{\varepsilon}$ [24]=
=Asymptotic $\biggl{\\{}$ | distillation |  $C_{r}$ [25] |  $C_{r}$ [25] |  $C_{r}$ [17] |  $\mathbf{\neq C_{r}}$ [$\ast$]
formation |  $C_{r}$ [24, 26] |  $C_{r}$ [24, 26] |  $C_{f}$ [17] |  $C_{f}$ [17]=
TABLE I: Coherence distillation and formation rates, with attributions where
the corresponding result was first obtained. Our new contributions (in bold,
and marked by asterisks) are the IO and SIO distillation rates in the one-shot
setting; furthermore, we prove that MIO, DIO and IO one-shot distillation
rates differ only by varying the smoothing parameter and by universal additive
terms. The one-shot distillation rate under SIO is converted into an
optimization problem in Theorem 10, and in the asymptotic case, we prove the
existence of bound coherence under SIO, showing that the rate is different
from the relative entropy of coherence in general.

###  II-B Coherence dilution

The one-shot coherence dilution problem characterizes the minimal resource
required for the formation of a target state with an allowed error
$\varepsilon$. The definition of one-shot coherence dilution is as follows.

######  Definition 1.

Let $\mathcal{O}\in\\{\text{MIO},\,\text{DIO},\,\text{IO},\,\text{SIO}\\}$
denote some class of incoherent operations. Then for a given state $\rho$ and
$\varepsilon\geq 0$, the one-shot coherence formation cost with error
$\varepsilon$ under $\mathcal{O}$ is defined as

$C_{c,\mathcal{O}}^{\varepsilon}(\rho)=\min_{\Lambda\in\mathcal{O}}\bigl{\\{}\log
M\,\mathrel{\mathop{\mathchar 58\relax}}\,F(\Lambda(\Psi_{M}),\rho)^{2}\geq
1-\varepsilon\bigr{\\}}.$ (3)

where $F(\rho,\sigma)=\operatorname{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}$ is
the usual mixed state fidelity. By default, here and throughout the paper,
$\log\equiv\log_{2}$ is the binary logarithm, in accordance with information
theoretic use.

In Ref. [24], several coherence monotones were proposed to estimate the one-
shot coherence cost. The first two are based on the max relative entropy
$D_{\max}(\rho\|\sigma)=\log\min\\{\lambda\;|\;\rho\leq\lambda\sigma\\}$.
Alternatively, one can introduce this quantity using the sandwiched quantum
$\alpha$-Rényi divergence

$\tilde{D}_{\alpha}(\rho\|\sigma)=\frac{1}{\alpha-1}\log\left(\operatorname{Tr}\left[\left(\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}}\right)^{\alpha}\right]\right),$
(4)

letting $\alpha\to\infty$ [27]. One then defines

$\displaystyle C_{\max}(\rho)$
$\displaystyle=\min_{\delta\in\mathcal{I}}D_{\max}(\rho\|\delta)=\log\min\\{\lambda\;|\;\rho\leq\lambda\delta\\},$
(5) $\displaystyle C_{\Delta,\max}(\rho)$
$\displaystyle=\min_{\sigma\in\overline{A_{\rho}}}D_{\max}(\rho\|\sigma)=\log\min\\{\lambda\;|\;\rho\leq\lambda\Delta(\rho)\\},$
(6)

where $A_{\rho}=\\{\sigma\geq 0\;|\;\sigma+t\rho=(1+t)\Delta(\rho),\;t>0\\}$
[28] and $\overline{A_{\rho}}$ is its closure. The quantity $C_{\max}(\rho)$
is a monotone under MIO, while $C_{\Delta,\max}(\rho)$ is a monotone under
DIO, but not IO in general. However, for a pure state $|\psi\rangle$,
$C_{\Delta,\max}$ reduces to its incoherent rank [28], defined as

$C_{0}(\psi)\mathrel{\mathop{\mathchar
58\relax}}=S_{0}(\Delta(\psi))=\log\text{rank}[\Delta(\psi)],$ (7)

and the incoherent rank is a monotone under IO even for stochastic pure state
transformations [12]. Thus, one can obtain a general mixed state monotone for
IO by taking a convex roof extension:

$\displaystyle C_{0}(\rho)$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\min_{p_{i},|\psi_{i}\rangle}\max_{i}C_{0}(\psi_{i}),$ (8)

where the minimization is over all pure state ensembles satisfying
$\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$. Note, the minimization
is followed by a maximization rather than an averaging over the
$|\psi_{i}\rangle$ (as typically done in such measures) because $C_{0}(\psi)$
is a stochastic IO monotone.

Since we allow $\varepsilon$ error in our one-shot tasks, we likewise apply an
$\varepsilon$ smoothing to our coherence measures. Depending on the quantity,
this is done by either minimizing or maximizing the measure over all states
$\rho^{\prime}$ lying in an $\varepsilon$-ball around $\rho$. For example, if
$C$ represents any one of the measures $C_{\max}$, $C_{\Delta,\max}$ or
$C_{0}$, its $\varepsilon$-smoothed variant is given by

$C^{\varepsilon}(\rho)=\min_{\rho^{\prime}\in
B_{\varepsilon}(\rho)}C(\rho^{\prime}).$ (9)

Here $B_{\varepsilon}(\rho)=\\{\rho^{\prime}\mathrel{\mathop{\mathchar
58\relax}}P(\rho^{\prime},\rho)<\varepsilon\\}$ is defined with respect to the
purified distance $P(\rho^{\prime},\rho)=\sqrt{1-F(\rho^{\prime},\rho)^{2}}$.
Note that the purified distance is related to the trace distance by [29]

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

As shown in [24], the measures $C^{\varepsilon}_{\max}$,
$C^{\varepsilon}_{\Delta,\max}$ and $C^{\varepsilon}_{0}$ precisely
characterize the one-shot coherence formation for MIO, DIO, and IO/SIO,
respectively (see Table I). Note that our definition of
$B_{\varepsilon}(\rho)$ differs from [24] by replacing
$\varepsilon\leftrightarrow\sqrt{\varepsilon}$.

The regularized (many-copy) coherence cost of formation can then be defined
using the one-shot quantities. For
$\mathcal{O}\in\\{\text{MIO},\,\text{DIO},\,\text{IO},\,\text{SIO}\\}$, the
asymptotic rate of coherence cost for a state $\rho$ under operations
$\mathcal{O}$ is defined as

$C_{c,\mathcal{O}}^{\infty}(\rho)\mathrel{\mathop{\mathchar
58\relax}}=\limsup_{\varepsilon\to
0^{+}}\limsup_{n\to\infty}\frac{1}{n}C_{c,\mathcal{O}}^{\varepsilon}(\rho^{\otimes
n}).$ (10)

Remarkably, for all four operational classes, the coherence cost has a single-
letter characterization. For MIO and DIO, the regularized coherence cost is
given by the relative entropy of coherence, which we call the asymptotic
coherence cost under $\text{MIO},\,\text{DIO}$ [24, 26],

$C_{r}(\rho)=\min_{\delta\in\mathcal{I}}D(\rho\|\delta)=S(\Delta(\rho))-S(\rho),$
(11)

where $D(X\|Y)=-\operatorname{Tr}[X(\log Y-\log X)]$ is the relative entropy,
and we have

$\lim_{\varepsilon\to
0^{+}}\lim_{n\to\infty}\frac{1}{n}C_{c,MIO/DIO}^{\varepsilon}\left(\rho^{\otimes
n}\right)=C_{r}(\rho).$ (12)

In the following, we use $\lim$ instead of $\limsup$ or $\liminf$ for
simplicity. On the other hand, for IO and SIO, the regularized coherence cost
is given by the so-called coherence of formation [17], which we call the
asymptotic coherence cost under $\text{IO},\,\text{SIO}$ and is defined as

$C_{f}(\rho)=\min_{p_{i},|\psi_{i}\rangle}\sum_{i}p_{i}C_{r}(\psi_{i}),$ (13)

where

$\lim_{\varepsilon\to
0^{+}}\lim_{n\to\infty}\frac{1}{n}C_{c,IO/SIO}^{\varepsilon}\left(\rho^{\otimes
n}\right)=C_{f}(\rho).$ (14)

In contrast to the entanglement of formation, the coherence of formation is an
additive function [17].

###  II-C Coherence distillation

We next turn to the main focus of this paper, which is the task of
transforming a given state $\rho$ into a maximally coherent pure state. The
one-shot distillation of the problem with $\varepsilon$ error is stated as
follows.

######  Definition 2.

Let $\mathcal{O}\in\\{\text{MIO},\,\text{DIO},\,\text{IO},\,\text{SIO}\\}$
denote some class of incoherent operations. Then for a given state $\rho$ and
$\varepsilon\geq 0$, the one-shot coherence distillation rate with error
$\varepsilon$ under $\mathcal{O}$ is defined as

$C_{d,\mathcal{O}}^{\varepsilon}(\rho)=\max_{\Lambda\in\mathcal{O}}\bigl{\\{}\log
M\,\mathrel{\mathop{\mathchar 58\relax}}\,F(\Lambda(\rho),\Psi_{M})^{2}\geq
1-\varepsilon\bigr{\\}}.$ (15)

In order to quantify the one-shot distillable rates, we first introduce two
additional functions. The first one is based on the min quantum Rényi
divergence $D_{\min}(\rho\|\sigma)=-\log
F(\rho,\sigma)^{2}=\tilde{D}_{1/2}(\rho\|\sigma)$ [27], $\alpha=\frac{1}{2}$
in Eq. (4). Using this quantity, the min-entropy of coherence $C_{\min}(\rho)$
can be defined as

$\displaystyle C_{\min}(\rho)$
$\displaystyle=\min_{\delta\in\mathcal{I}}D_{\min}(\rho\|\delta).$ (16)

The properties of $C_{\min}$ and its relationship between other coherence
monotones are discussed in [30].

The second quantifier of coherence relevant to our study is based on the
hypothesis testing relative entropy [31, 32]. In the task of hypothesis
testing, a positive operator valued measure (POVM) with two elements
$\\{W,\mathbbm{1}-W\\}$ is used to distinguish two possible states $\rho$ and
$\sigma$. The probability of obtaining a correct guess on $\rho$ is
$\mathrm{Tr}(\rho W)$ and the probability of obtaining a wrong guess on
$\sigma$ is $\mathrm{Tr}(\sigma W)$. The hypothesis testing relative entropy
characterizes the minimal wrong guessing probability on $\sigma$, with the
constraint that the probability of the correct guess on $\rho$ is no less than
$1-\varepsilon$:

$\begin{split}D_{H}^{\varepsilon}(\rho\|\sigma)=-\log\min\bigl{\\{}\operatorname{Tr}\sigma{W}\,\mathrel{\mathop{\mathchar
58\relax}}\,0\leq{W}&\leq\mathbbm{1},\bigr{.}\\\
\bigl{.}\operatorname{Tr}\rho{W}&\geq 1-\varepsilon\bigr{\\}}.\end{split}$
(17)

This serves as a parent quantity for two other coherence functions. Namely, we
have

$\displaystyle C^{\varepsilon}_{H}(\rho)$
$\displaystyle=\min\\{D^{\varepsilon}_{H}(\rho\|\delta)\,\mathrel{\mathop{\mathchar
58\relax}}\,\delta\in\mathcal{I}\\},$ (18)
$\displaystyle\widetilde{C}^{\varepsilon}_{H}(\rho)$
$\displaystyle=\min\\{D^{\varepsilon}_{H}(\rho\|\sigma)\,\mathrel{\mathop{\mathchar
58\relax}}\,\sigma=\Delta(\sigma),\ \operatorname{Tr}\sigma=1\\}.$ (19)

The function $\widetilde{C}^{\varepsilon}_{H}(\rho)$ characterizes the one-
shot distillable distillable coherence under MIO and DIO.

In the many-copy scenario, the regularized coherence distillation rate by
operations
$\mathcal{O}\in\\{\text{MIO},\,\text{DIO},\,\text{IO},\,\text{SIO}\\}$ is
defined as

$C_{d,\mathcal{O}}^{\infty}(\rho)=\liminf_{\varepsilon\to
0^{+}}\liminf_{n\to\infty}\frac{1}{n}C_{d,\mathcal{O}}^{\varepsilon}(\rho^{\otimes
n}).$ (20)

For MIO, DIO, and IO, the distillable coherence of $\rho$ is given by the
relative entropy of coherence, which we call the asymptotic coherence
distillation rate [17], where

$\lim_{\varepsilon\to
0^{+}}\lim_{n\to\infty}\frac{1}{n}C_{d,MIO/DIO/IO}^{\varepsilon}\left(\rho^{\otimes
n}\right)=C_{r}(\rho).$ (21)

Combining Eq. (12), (21), we can conclude that coherence is strongly converse
under MIO/DIO. In contrast, as we show below, certain coherent states are
undistillable under SIO. A summary of the distillation/formation rates for
different operations is given in Table I.

##  III One-shot distillation
under MIO and DIO

In this section, we review the results of [25] on MIO and DIO distillation.

For a quantum channel $\Lambda\mathrel{\mathop{\mathchar
58\relax}}\mathcal{L}(A)\to\mathcal{L}(B)$, consider its Choi operator

$\Gamma=({\operatorname{id}}\otimes\Lambda)\Phi=\sum_{i,j}|i\rangle\langle
j|_{R}\otimes\Lambda(|i\rangle\langle j|_{A}),$ (22)

where $R$ is isomorphic to $A$. Let $\Gamma_{ij}=\Lambda(|i\rangle\langle
j|_{A})$. By Choi’s theorem, $\Lambda$ is completely positive if and only if
$\Gamma\geq 0$. Also, $\Lambda$ is trace preserving if and only if
$\operatorname{Tr}\Gamma_{ij}=\delta_{ij}$.

Now consider $\Lambda\in\text{MIO}$, meaning $\Lambda(\delta)\in\mathcal{I}$
for all $\delta\in\mathcal{I}$, which is equivalent to
$\Lambda(|i\rangle\langle i|)\in\mathcal{I}$ for all $i$. For the Choi matrix
this says $\Gamma_{ii}\in\mathcal{I}$ for all $i$. For $\Lambda\in\text{DIO}$,
from $\Lambda(\Delta(|i\rangle\langle j|))=\Delta(\Lambda(|i\rangle\langle
j|))$ for all $i$ and $j$, we get
$\delta_{ij}\Gamma_{ij}=\Delta(\Gamma_{ij})$.

Since $\Lambda(\rho)=\sum_{i,j}\rho_{ij}\Gamma_{ij}$ (by linearity), we have
now,

$F(\Lambda(\rho),\Psi_{M})^{2}=\operatorname{Tr}[\Lambda(\rho)\Psi_{M}]=\sum_{i,j}\rho_{ij}\operatorname{Tr}\Gamma_{ij}\Psi_{M}.$
(23)

The one-shot distillation under MIO/DIO can thus be expressed as

$\begin{split}\max M\text{ s.t.
}\sum_{i,j}\rho_{ij}\operatorname{Tr}\Gamma_{ij}\Psi_{M}&\geq
1-\varepsilon,\\\ \sum_{i,j}|i\rangle\langle j|\otimes\Gamma_{ij}&\geq 0,\\\
\operatorname{Tr}\Gamma_{ij}&=\delta_{ij},\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\begin{cases}\phantom{\delta_{ij}}\Gamma_{ii}\in\mathcal{I}&\text{[MIO]},\\\
\delta_{ij}\Gamma_{ij}=\Delta(\Gamma_{ij})&\text{[DIO]}.\end{cases}\end{split}$
(24)

Suppose that the $\Gamma_{ij}$ satisfy the above constraints. Consider the
twirling transformation

$\widetilde{\Gamma}_{ij}=\frac{1}{M!}\sum_{\pi\in
S_{M}}U_{\pi}\Gamma_{ij}U_{\pi}^{\dagger},$ (25)

where $S_{M}$ denotes the permutation group on $M$ objects, and $U_{\pi}$ is
the unitary representation of $\pi$ as a permutation matrix. It is easy to see
that then the $\widetilde{\Gamma}_{ij}$ also satisfy the constraints, and lead
to the same objective function, due to the permutation invariance of
$\Psi_{M}$. Therefore, without loss of generality, we can assume that each of
the $\Gamma_{ij}$ is permutation-invariant and has hence the form

$\Gamma_{ij}=\alpha_{ij}\Psi_{M}+\beta_{ij}\frac{\mathbbm{1}-\Psi_{M}}{M-1}.$
(26)

Furthermore, $\operatorname{Tr}\Gamma_{ij}=\alpha_{ij}+\beta_{ij}$. We can see
that both MIO and DIO lead to the following constraints: When $i\neq j$, we
have $\beta_{ij}=-\alpha_{ij}$ and when $i=j$, we have
$\alpha_{ii}=\frac{1}{M}$, $\beta_{ii}=1-\frac{1}{M}$. For the matrices
$A=(\alpha_{ij})_{i,j}$ and $B=(\beta_{ij})_{i,j}$, this translates into the
simple relation $B=\mathbbm{1}-A$. Notice that

$\Gamma=\sum_{i,j}\alpha_{ij}|i\rangle\langle
j|\otimes\Psi_{M}+\sum_{i,j}\beta_{ij}|i\rangle\langle
j|\otimes\frac{\mathbbm{1}-\Psi_{M}}{M-1},$ (27)

and we must satisfy $\langle\psi|\Gamma|\psi\rangle\geq 0$ for all
$|\psi\rangle$. Consider the special case
$|\psi\rangle=|u\rangle\otimes|v\rangle$; by choosing
$|v\rangle=|\Psi_{M}\rangle$, we learn that $A\geq 0$. By choosing $|v\rangle$
orthogonal with $|\Psi_{M}\rangle$, we see likewise that $B\geq 0$.
Conversely, $A,B\geq 0$ implies that $\Gamma\geq 0$.

In conclusion, one-shot distillation under MIO/DIO can be solved by the
following SDP: $C_{d,MIO/DIO}^{\varepsilon}(\rho)=\log M_{\text{opt}}$, with

$\begin{split}M_{\text{opt}}=\max M\text{ s.t.
}\operatorname{Tr}\rho^{T}A&\geq 1-\varepsilon,\\\ 0\leq A&\leq\mathbbm{1},\
A_{ii}=\frac{1}{M}\,\forall i.\end{split}$ (28)

To express this result in terms of a suitable relative-entropy distance, we
recall the _hypothesis testing relative entropy_ introduced above [31, 32],

$\begin{split}D_{H}^{\varepsilon}(\rho\|\sigma)=-\log\min\bigl{\\{}\operatorname{Tr}\sigma
W\,\mathrel{\mathop{\mathchar 58\relax}}\,0\leq W&\leq\mathbbm{1},\bigr{.}\\\
\bigl{.}\operatorname{Tr}\rho W&\geq 1-\varepsilon\bigr{\\}},\end{split}$ (29)

and the corresponding coherence measure

$C_{H}^{\varepsilon}(\rho)=\min_{\delta\in\mathcal{I}}D_{H}^{\varepsilon}(\rho\|\delta).$
(30)

We show that the coherence measure $C_{H}^{\varepsilon}(\rho)$ can be computed
by SDP. Notice that

$\begin{split}C_{H}^{\varepsilon}(\rho)&=-\log\max_{\delta\in\mathcal{I}}\min_{\begin{subarray}{c}W\mathrel{\mathop{\mathchar
58\relax}}0\leq W\leq\mathbbm{1}\\\ \operatorname{Tr}\rho W\geq
1-\varepsilon\end{subarray}}\operatorname{Tr}\delta W\\\
&=-\log\min_{\begin{subarray}{c}W\mathrel{\mathop{\mathchar 58\relax}}0\leq
W\leq\mathbbm{1}\\\ \operatorname{Tr}\rho W\geq
1-\varepsilon\end{subarray}}\max_{\delta\in\mathcal{I}}\operatorname{Tr}\delta
W\\\ &=-\log\min_{\begin{subarray}{c}W\mathrel{\mathop{\mathchar
58\relax}}0\leq W\leq\mathbbm{1}\\\ \operatorname{Tr}\rho W\geq
1-\varepsilon\end{subarray}}\max_{i}W_{ii},\end{split}$ (31)

where in the second line we have appealed to the minimax theorem [33], noting
that the objective function, $\operatorname{Tr}\delta W$, is linear in each of
the two arguments, and that the domains of optimization are closed convex
sets.

The third line in Eq. (31) can now be expressed as an SDP as follows:

$\begin{split}\mu_{\text{opt}}=\max\mu\text{ s.t. }\operatorname{Tr}\rho
W&\geq 1-\varepsilon,\\\ 0\leq W&\leq\mathbbm{1},\
W_{ii}\leq\frac{1}{\mu}\,\forall i,\end{split}$ (32)

where $C_{H}^{\varepsilon}(\rho)=\log\mu_{\text{opt}}$. Notice that in Eq.
(28) we have a transpose on $\rho$, while in Eq. (32) we don’t. We can simply
change $A$ into $A^{T}$ in the SDP (28) without changing its value, so that
the two SDP has similar form, except that one has an equality sign where the
other other has a “$\leq$”. In [25], it was shown that the r.h.s. of Eq. (28)
can be expressed in terms of $D_{H}^{\varepsilon}$, as well:

$\begin{split}C_{d,MIO/DIO}^{\varepsilon}(\rho)=\max\log M\\\ \text{ s.t.
}\operatorname{Tr}\rho^{T}A&\geq 1-\varepsilon,\\\ 0\leq A&\leq\mathbbm{1},\
A_{ii}=\frac{1}{M}\,\forall i\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\min
D_{H}^{\varepsilon}(\rho\|\delta)\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\text{
s.t. }\delta\text{ diagonal and }\operatorname{Tr}\delta=1\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\mathrel{\mathop{\mathchar
58\relax}}\widetilde{C}_{H}^{\varepsilon}(\rho),\end{split}$ (33)

where the minimization is crucially over Hermitian, but not necessarily
positive semidefinite matrices $\delta$.

We record these findings in the following theorem.

######  Theorem 3 (Regula _et al._ [25]).

For any state $\rho$, the one-shot MIO- and DIO-distillable coherence is given
by

$C_{d,\text{DIO}}^{\varepsilon}(\rho)=C_{d,\text{MIO}}^{\varepsilon}(\rho)=\widetilde{C}_{H}^{\varepsilon}(\rho)\leq
C_{H}^{\varepsilon}(\rho).$ (34)

##  IV One-shot distillation under IO

Now we come to the main contribution of the present paper, the extension of
the one-shot distillation protocols to the class IO.

To start, recall the definition of the min-relative entropy and its smoothed
version,

$\begin{split}D_{\min}(\rho\|\sigma)&=-\log F(\rho,\sigma)^{2},\\\
D_{\min}^{\varepsilon}(\rho)&=\max_{\rho^{\prime}\in
B^{\varepsilon}(\rho)}D_{\min}(\rho^{\prime}\|\sigma).\end{split}$ (35)

The _smooth min-relative entropy of coherence_ is defined as

$C_{\min}^{\varepsilon}(\rho)=\max_{\rho^{\prime}\in
B^{\varepsilon}(\rho)}\min_{\delta\in\mathcal{I}}D_{\min}(\rho\|\delta).$ (36)

One might wonder why we do not interchange min and max here, but this is the
version of the quantity that appears naturally both in the achievability bound
we will derive on Subsection IV-A, and in the upper bound in Subsection IV-B.

###  IV-A An achievable lower bound

We will generalize the protocol in [17, Thm. 6] to obtain a min-relative-
entropic lower bound on the distillable coherence. In the process, the privacy
amplification aspect will become even more apparent.

######  Theorem 4.

For an arbitrary state $\rho$ and $0<\varepsilon<1$,

$C^{\varepsilon}_{d,IO}(\rho)\geq
C^{\frac{\varepsilon}{2}-\eta}_{\min}(\rho)-2\log\frac{1}{\eta},$ (37)

for any $0<\eta<\frac{\varepsilon}{2}$.

Proof. In order to accomplish our proof, we need to introduce the conditional
min/max entropy and their smoothed versions. For a bipartite quantum state
$\rho^{AB}$, the min-entropy of $A$ conditioned on $B$ is defined as

$H_{\min}(A|B)_{\rho^{AB}}\mathrel{\mathop{\mathchar
58\relax}}=-\min_{\sigma^{B}}D_{\max}(\rho^{AB}\|\mathbbm{1}^{A}\otimes\sigma^{B}),$
(38)

and the max-entropy of $A$ conditioned on $B$ is defined as

$H_{\max}(A|B)_{\rho^{AB}}\mathrel{\mathop{\mathchar
58\relax}}=-H_{\min}(A|C)_{\rho^{AC}},$ (39)

where $\rho^{ABC}$ is a purification of $\rho^{AB}$ and
$\rho^{AC}=\operatorname{Tr}_{B}\rho^{ABC}$. It is proven that the max-entropy
has an alternative form [34]

$H_{\max}(A|B)_{\rho^{AB}}\mathrel{\mathop{\mathchar
58\relax}}=-\min_{\sigma^{B}}D_{\min}(\rho^{AB}\|\mathbbm{1}^{A}\otimes\sigma^{B}).$
(40)

Furthermore, the smoothed conditional min- and max-entropy are defined by

$\displaystyle H_{\min}^{\varepsilon}(A|B)_{\rho^{AB}}$
$\displaystyle\mathrel{\mathop{\mathchar 58\relax}}=\max_{\rho^{\prime}\in
B^{\varepsilon}(\rho^{AB})}H_{\min}^{\varepsilon}(A|B)_{\rho^{\prime}},$ (41)
$\displaystyle H_{\max}^{\varepsilon}(A|B)_{\rho^{AB}}$
$\displaystyle\mathrel{\mathop{\mathchar 58\relax}}=\min_{\rho^{\prime}\in
B^{\varepsilon}(\rho^{AB})}H_{\max}^{\varepsilon}(A|B)_{\rho^{\prime}}.$

In the following proof, we denote the system of interest as $A$, i.e.
$\rho^{A}\mathrel{\mathop{\mathchar 58\relax}}=\rho$, and the incoherent basis
as $\\{|x\rangle\\}$. Denote $\psi\mathrel{\mathop{\mathchar
58\relax}}=|\psi\rangle\langle\psi|$. Choose a purification of $\rho^{A}$,

$|\psi\rangle^{AE}=\sum_{x}\sqrt{p_{x}}|x\rangle^{A}|\psi_{x}\rangle^{E},$

where $\operatorname{Tr}_{E}\psi^{AE}=\rho^{A}$, and introduce the dephased
cq-state

$\omega^{AE}=(\Delta\otimes{\operatorname{id}})\psi=\sum_{x}p_{x}|x\rangle\langle
x|^{A}\otimes\psi_{x}^{E}.$

According to [35, Cor. 5.6.1] (actually, a slight adaptation to get rid of a
factor of $2$), for every $\log M\leq
H_{\min}^{\frac{\varepsilon}{2}-\eta}(A|E)_{\omega}-2\log\frac{1}{\eta}$ there
exists a function $G\mathrel{\mathop{\mathchar
58\relax}}\mathcal{X}\rightarrow[M]=\\{1,2,\ldots,M\\}$ such that the state

$\Omega^{KE}\mathrel{\mathop{\mathchar
58\relax}}=(G\otimes{\operatorname{id}})\omega^{AE}=\sum_{x}p_{x}|G(x)\rangle\langle
G(x)|^{K}\otimes\psi_{x}^{E}$

satisfies

$\frac{1}{2}\left\|\Omega^{KE}-\tau_{K}\otimes\sigma^{E}\right\|_{1}\leq\frac{\varepsilon}{2},$

for $\tau^{K}=\frac{1}{M}\mathbbm{1}_{K}$ and a suitable state $\sigma^{E}$
(which may be equal to $\omega^{E}$, but it doesn’t have to be). Hence,
because of the well-known relation between trace distance and fidelity [29],

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

we have $F(\Omega^{KE},\tau_{K}\otimes\sigma^{E})\geq
1-\frac{\varepsilon}{2}$.

To turn this property, which encapsulates the uniformity and independence of
the “key” $K=G(X)$ from $E$, into a coherence distillation protocol, consider
the following purification of $\Omega^{KE}$:

$|\Omega\rangle^{KAE}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{x}\sqrt{p_{x}}|G(x)\rangle^{K}|x\rangle^{A}|\psi_{x}\rangle^{E},$

which can be obtained at from $|\psi\rangle^{AE}$ by applying the isometry

$U\mathrel{\mathop{\mathchar
58\relax}}|x\rangle^{A}\longmapsto|G(x)\rangle^{K}|x\rangle^{A}.$

Crucially, $U$ is incoherent (even SIO). For $\tau_{K}\otimes\sigma^{E}$, on
the other hand, we choose a purification
$|\Phi\rangle^{KL}\otimes|\zeta\rangle^{EF}$, with the standard maximally
entangled state
$|\Phi\rangle=\frac{1}{\sqrt{M}}\sum_{i=0}^{M-1}|i\rangle|i\rangle$. By
Uhlmann’s theorem [36], there exists an isometry $V\mathrel{\mathop{\mathchar
58\relax}}A\hookrightarrow LF$ such that

$|\langle\Phi|\langle\zeta|\cdot(\mathbbm{1}_{KE}\otimes V)|\Omega\rangle|\geq
1-\frac{\varepsilon}{2}.$ (42)

If we had $|\Phi\rangle^{KL}|\zeta\rangle^{EF}$, we could clearly create a
maximally coherent state $\Psi_{M}$ on system $K$ by tracing out $F$, and
destructively measuring $L$ in the Fourier conjugate basis
$|\widehat{\alpha}\rangle\mathrel{\mathop{\mathchar
58\relax}}=Z^{\alpha}|\Psi_{M}\rangle$. Here $Z$ is the phase unitary defined
in Eq. (1).

With this, the protocol is clear: starting from $\rho^{A}$, first apply $U$;
then apply $V$, followed by tracing out $F$, measuring $L$ in the Fourier
basis $\\{|\widehat{\alpha}\rangle\\}_{\alpha=0,\ldots,M-1}$; finally, apply
$Z^{\alpha}$ on $K$. The first and the third step clearly are incoherent (even
SIO); the second seems suspicious, and indeed $V$ may not be incoherent at
all, but notice that we follow it by a destructive measurement, and these are
all IO. We can write Kraus operators of the resulting map
$\Lambda\mathrel{\mathop{\mathchar 58\relax}}A\rightarrow K$ as follows:

$M_{\alpha\beta}\mathrel{\mathop{\mathchar
58\relax}}=\bigl{(}Z^{\alpha}_{K}\otimes\langle\widetilde{\alpha}|^{L}\langle\beta|^{F}V\bigr{)}U,$

where $\\{|\beta\rangle\\}$ is an arbitrary basis of $F$.

To analyze the fidelity of the protocol, we pass to the purifications and look
at eq. (42); by the monotonicity of the fidelity under the CPTP maps $\Lambda$
and $\operatorname{Tr}_{E}$, we obtain
$F\bigl{(}\Lambda(\rho),\Psi_{M}\bigr{)}\geq 1-\frac{\varepsilon}{2}$ and
hence $F\bigl{(}\Lambda(\rho),\Psi_{M}\bigr{)}^{2}\geq 1-\varepsilon$.

This shows that $\log M\approx
H_{\min}^{\frac{\varepsilon}{2}-\eta}(A|E)_{\omega}-2\log\frac{1}{\eta}$ is
achievable. Now, introducing the purification
$|\omega\rangle^{ABE}=\sum_{x}\sqrt{p_{x}}|x\rangle^{A}|x\rangle^{B}|\psi_{x}\rangle$,
we have

$\begin{split}H_{\min}^{\alpha}(A|E)_{\omega}&=-H_{\max}^{\alpha}(A|B)_{\omega}\\\
&=-\log\min_{\omega^{\prime}\in
B^{\alpha}(\omega)}\max_{\sigma}F(\omega^{\prime},\mathbbm{1}\otimes\sigma)^{2}\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\geq-\log\min_{\begin{subarray}{c}\omega^{\prime}\in
B^{\alpha}(\omega)\\\
\text{max.\,correlated}\end{subarray}}\max_{\sigma}F(\omega^{\prime},\mathbbm{1}\otimes\sigma)^{2}\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=-\log\min_{\begin{subarray}{c}\omega^{\prime}\in
B^{\alpha}(\omega)\\\
\text{max.\,correlated}\end{subarray}}\max_{\sigma\in\mathcal{I}}F(\omega^{\prime},\mathbbm{1}\otimes\sigma)^{2}\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=-\log\min_{\rho^{\prime}\in
B^{\alpha}(\rho)}\max_{\delta\in\mathcal{I}}F(\rho^{\prime},\delta)^{2}\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\max_{\rho^{\prime}\in
B^{\alpha}(\rho)}\min_{\delta\in\mathcal{I}}D_{\min}(\rho^{\prime}\|\delta)\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=C_{\min}^{\alpha}(\rho).\end{split}$ (43)

Here, the second line follows from Eq. (40), and the third line is motivated
by the observation that
$\omega^{AB}=\sum_{xy}\sqrt{p_{x}p_{y}}\langle\psi_{y}|\psi_{x}\rangle|xx\rangle\langle
yy|$ is a maximally correlated state, so it is natural to impose the same
structure on $\omega^{\prime}$; the fourth line follows from the $Z\otimes
Z^{\dagger}$-invariance of maximally correlated states, so by the concavity of
the fidelity we can impose w.l.o.g. the same structure on
$\mathbbm{1}\otimes\sigma$, meaning that $\sigma$ is diagonal. The rest is
straightforward algebra. $\blacksquare$

Note that the equality in Eq. (43) can also be achieved by directly using the
results by Coles [37], or [30]:

$H_{\min}(X|E)_{\rho_{XE}}=\min_{\delta\in\mathcal{I}}D_{\min}(\rho_{A}\|\delta).$
(44)

######  Remark 5.

The CPTP map
$\Lambda(\cdot)=\sum_{\alpha,\beta}M_{\alpha\beta}(\cdot)M_{\alpha\beta}^{\dagger}$
we constructed in the proof is not only IO but also DIO. To see this, first
expand

$\displaystyle\Lambda(|x\rangle\langle
y|)=\sum_{\alpha,\beta}Z^{\alpha}|G(x)\rangle\langle
G(y)|Z^{\alpha\dagger}\langle\widetilde{\alpha}\beta|V|x\rangle\langle
y|V^{\dagger}|\widetilde{\alpha}\beta\rangle$

for any incoherent basis states $|x\rangle$ and $|y\rangle$ of system $A$. The
key observation is that fully dephased states are invariant under conjugation
by $Z^{\alpha}$, and moreover this conjugation commutes with $\Delta$, i.e.
$\Delta[Z^{\alpha}(\cdot)Z^{\alpha\dagger}]=\Delta(\cdot)$. Hence if we
dephase system $K$ after applying the map $\Lambda$, we find

$\displaystyle\Delta[\Lambda(|x\rangle\langle y|)]$
$\displaystyle=\delta_{G(x)G(y)}|G(x)\rangle\langle G(y)|$
$\displaystyle\quad\times\sum_{\alpha,\beta}\langle\widetilde{\alpha}\beta|V|x\rangle\langle
y|V^{\dagger}|\widetilde{\alpha}\beta\rangle$
$\displaystyle=\delta_{xy}|G(x)\rangle\langle
G(y)|=\Lambda[\Delta(|x\rangle\langle y|)],$ (45)

where the second equality follows from the fact that
$\\{|\widetilde{\alpha}\beta\rangle\\}_{\alpha,\beta}$ forms a complete basis.
Thus, $\Lambda$ is a DIO map.

######  Corollary 6.

For an arbitrary state $\rho$ and $0<\varepsilon<1$,

$C^{\varepsilon}_{d,DIIO}(\rho)\geq
C^{\frac{\varepsilon}{2}-\eta}_{\min}(\rho)-2\log\frac{1}{\eta},$ (46)

for any $0<\eta<\frac{\varepsilon}{2}$, where DIIO refers to the intersection
of IO and DIO. $\blacksquare$

###  IV-B Upper bound and comparison with MIO distillation

We have a partial converse theorem to Theorem 4 which can bound
$C^{\varepsilon}_{d,IO}(\rho)$ from both sides, as follows.

######  Theorem 7.

For an arbitrary state $\rho$ and $0<\varepsilon<1$,

$C^{\varepsilon}_{d,IO}(\rho)\leq
C_{\min}^{\sqrt{\varepsilon(2-\varepsilon)}}(\rho).$ (47)

Proof. Due to the inclusion of the classes of operations, and Theorems 3 and
4, we have the first four of the following (in)equalities:

$\begin{split}C_{\min}^{\frac{\varepsilon}{2}-\eta}(\rho)-2\log\frac{1}{\eta}&\leq
C_{d,\text{IO}}^{\varepsilon}(\rho)\\\ &\leq
C_{d,\text{M/D\,IO}}^{\varepsilon}(\rho)=\widetilde{C}_{H}^{\varepsilon}(\rho)\\\
&\leq C_{H}^{\varepsilon}(\rho)\\\ &\leq
C_{\min}^{\sqrt{\varepsilon(2-\varepsilon)}}(\rho).\end{split}$ (48)

The last one also follows essentially from known facts, namely [32, Prop.
4.2]. Note only that our definition of $D_{H}^{\varepsilon}$ differs from [32]
by $\varepsilon\leftrightarrow 1-\varepsilon$, and an additional term of
$\log(1-\varepsilon)$ added. We made this choice for easier comparison with
the results from [25]. With this in mind, [32, Eq. (51)] reads

$D_{H}^{\varepsilon}(\rho\|\sigma)\leq
D_{\min}^{\sqrt{2\varepsilon}}(\rho\|\sigma),$

and looking at the last step of the proof, one observes that
$\sqrt{2\varepsilon}$ can be improved to $\sqrt{\varepsilon(2-\varepsilon)}$.
We can adapt the proof in [32] to include the minimization over
$\delta\in\mathcal{I}$, according to the following Lemma 9. By applying it
with $\mathcal{S}=\mathcal{I}$, and maximizing over the
$\sqrt{\varepsilon(2-\varepsilon)}$-ball on the right hand side, we precisely
obtain the last inequality in Eq. 48. $\blacksquare$

######  Remark 8.

Combining Theorem 4 and Theorem 7, we conclude that
$C_{d,IO}^{\varepsilon}(\rho)\approx C_{\min}^{\varepsilon^{\prime}}(\rho)$,
with $\varepsilon^{\prime}\in[\frac{\varepsilon}{2},\sqrt{(2-\varepsilon)}]$.

######  Lemma 9.

Let $\mathcal{S}$ be a closed convex set of states on a Hilbert space $A$, and
$\rho$ a state. Then, for every $0<\varepsilon<1$ there exists a subnormalized
density matrix $\rho^{\prime}$ with
$P(\rho,\rho^{\prime})\leq\sqrt{\varepsilon(2-\varepsilon)}$, such that

$\min_{\sigma\in\mathcal{S}}D_{H}^{\varepsilon}(\rho\|\sigma)\leq\min_{\sigma\in\mathcal{S}}D_{\min}^{\sqrt{\varepsilon(2-\varepsilon)}}(\rho^{\prime}\|\sigma).$

Proof. The crucial observation is that due to the convexity of the sets of
operators, $\mathcal{S}$ and $\\{0\leq
W\leq\mathbbm{1}\,\mathrel{\mathop{\mathchar 58\relax}}\,\operatorname{Tr}\rho
W\geq 1-\varepsilon\\}$, we can invoke the minimax theorem [33], to obtain

$\max_{\sigma\in\mathcal{S}}\min_{\begin{subarray}{c}0\leq W\leq\mathbbm{1}\\\
\operatorname{Tr}\rho W\geq
1-\varepsilon\end{subarray}}\operatorname{Tr}\sigma
W=\min_{\begin{subarray}{c}0\leq W\leq\mathbbm{1}\\\ \operatorname{Tr}\rho
W\geq
1-\varepsilon\end{subarray}}\max_{\sigma\in\mathcal{S}}\operatorname{Tr}\sigma
W.$

Thus, there exists an optimizer $W_{0}$ of the second expression, $0\leq
W_{0}\leq\mathbbm{1}$, $\operatorname{Tr}\rho W_{0}\geq 1-\varepsilon$, with

$\min_{\sigma\in\mathcal{S}}D_{H}^{\varepsilon}(\rho\|\sigma)=\min_{\sigma\in\mathcal{S}}-\log\operatorname{Tr}\sigma
W_{0}.$ (49)

Following the the example of [32, Prop. 4.2], we define a subnormalized state
$\rho^{\prime}=\sqrt{W_{0}}\rho\sqrt{W_{0}}$, which we claim to be the sought-
after object.

To start with, from optimality of $W_{0}$, we have
$\operatorname{Tr}\rho^{\prime}=\operatorname{Tr}\rho W_{0}=1-\varepsilon$,
hence from [32, Lemma A.3] (see also [38, Lemma 7]),
$P(\rho^{\prime},\rho)\leq\sqrt{1-(\operatorname{Tr}\rho
W_{0})^{2}}=\sqrt{\varepsilon(2-\varepsilon)}$.

At the same time, choosing a purification of
$\rho^{A}=\operatorname{Tr}_{B}|\varphi\rangle\langle\varphi|^{AB}$, we get a
purification of $\rho^{\prime}$ by letting
$|\varphi^{\prime}\rangle=(\sqrt{W_{0}}\otimes\mathbbm{1})|\varphi\rangle$.
Conjugating the inequality $\varphi\leq\mathbbm{1}$ by
$\sqrt{W_{0}}\otimes\mathbbm{1}$ this results in $\varphi^{\prime}\leq
W_{0}\otimes\mathbbm{1}$. Now, just as in the proof of [32, Prop. 4.2], we
employ the dual variational characterization of the fidelity,

$F(\rho^{\prime},\sigma)^{2}=\min\operatorname{Tr}\sigma Z\text{ s.t.
}\varphi^{\prime}\leq Z\otimes\mathbbm{1}.$

This implies, that $\operatorname{Tr}\sigma W_{0}\geq F(\rho^{\prime},\sigma)$
in (49), for all $\sigma\in\mathcal{S}$, and so
$\min_{\sigma\in\mathcal{S}}D_{H}^{\varepsilon}(\rho\|\sigma)\leq\min_{\sigma\in\mathcal{S}}-\log
F(\rho^{\prime},\sigma)^{2}$, as claimed. $\blacksquare$

##  V Distillation under SIO

###  V-A Characterizing one-shot SIO distillation

We now turn to coherence distillation under strictly incoherent operations
(SIO). Ever since [17], it has been an open question whether coherence
distillation such as the protocol in Sec. IV-A, or in [17, Thm. 6], really
requires IO, or can be performed within the smaller class of SIO (as all other
protocols discussed in [17] can). The crucial object in this setting turns out
to be the incoherent rank. Recall that the incoherent rank of a positive
operator $\Omega$ is defined by

$C_{0}(\Omega)=\min_{\\{\lambda_{j},|\phi_{j}\rangle\\}}\max_{j}\log\text{rank}[\Delta(\phi_{j})],$
(50)

where the minimization is taken over all positive rank-one decompositions of
$\Omega$.

######  Theorem 10.

For any state $\rho$, the one-shot distillable coherence under SIO is given by

$\displaystyle C_{d,SIO}^{\varepsilon}(\rho)=\max\log M\text{ s.t.
}\operatorname{Tr}\rho A$ $\displaystyle\geq 1-\varepsilon,$ $\displaystyle
0\leq A$ $\displaystyle\leq\mathbbm{1},\ A_{ii}=\frac{1}{M}\,\forall i$
$\displaystyle C_{0}(A)$ $\displaystyle\leq\log M.$ (51)

Proof. Suppose that $\operatorname{Tr}[\Lambda(\rho)\Psi_{M}]\geq
1-\varepsilon$ for some SIO map $\Lambda$ and
$|\Psi_{M}\rangle=\frac{1}{\sqrt{M}}\sum_{i=1}^{M}|i\rangle$. Let
$\Pi_{>M}=\frac{1}{M}\sum_{x=M+1}^{d_{A}}|x\rangle\langle x|$. Notice that

$\displaystyle 1-\varepsilon$
$\displaystyle\leq\operatorname{Tr}[\Lambda(\rho)\Psi_{M}]$
$\displaystyle\leq\operatorname{Tr}[\Lambda(\rho)(\Psi_{M}+\Pi_{>M})]$
$\displaystyle=\operatorname{Tr}[\rho A],$ (52)

where $A\mathrel{\mathop{\mathchar
58\relax}}=\Lambda^{*}(\Psi_{M})+\Lambda^{*}(\Pi_{>M})$ and $\Lambda^{*}$ is
the adjoint channel of $\Lambda$. Using the form of SIO Kraus operators, we
have

$\displaystyle\Lambda^{*}(\Psi_{M})$
$\displaystyle=\frac{1}{M}\sum_{\alpha}\sum_{\begin{subarray}{c}x\;s.t.\\\
f_{\alpha}(x)\in[M]\end{subarray}}\sum_{\begin{subarray}{c}x^{\prime}\;s.t.\\\
f_{\alpha}(x^{\prime})\in[M]\end{subarray}}c_{\alpha,x}^{*}|x\rangle\langle
x^{\prime}|c_{\alpha,x^{\prime}}$
$\displaystyle=\frac{1}{M}\sum_{\alpha}|\phi_{\alpha}\rangle\langle\phi_{\alpha}|,$
(53)

and likewise

$\Lambda^{*}(\Pi_{>M})=\frac{1}{M}\sum_{\alpha}\sum_{\begin{subarray}{c}x\;s.t.\\\
f_{\alpha}(x)\not\in[M]\end{subarray}}|c_{\alpha,x}|^{2}|x\rangle\langle x|.$
(54)

Thus $A$ has a decomposition into rank-one vectors each having an incoherent
rank no greater than $M$. Also, for any $y\in\\{1,\cdots,d_{A}\\}$, we see
that

$\displaystyle\langle y|A|y\rangle$
$\displaystyle=\frac{1}{M}\left(\sum_{\begin{subarray}{c}\alpha\;s.t.\\\
f_{\alpha}(y)\in[M]\end{subarray}}+\sum_{\begin{subarray}{c}\alpha\;s.t.\\\
f_{\alpha}(y)\not\in[M]\end{subarray}}\right)|c_{\alpha,y}|^{2}=\frac{1}{M},$
(55)

i.e. $A_{yy}=\frac{1}{M}$ for all $y$.

The converse involves essentially reversing these steps. Suppose that
$\operatorname{Tr}\rho A\geq 1-\varepsilon$ for some operator $0\leq
A\leq\mathbbm{1}$ with $C_{0}(A)\leq\log M$ and $A_{ii}=\frac{1}{M}$. Then
there exists a decomposition

$\displaystyle A$
$\displaystyle=\frac{1}{M}\sum_{\alpha}|\phi_{\alpha}\rangle\langle\phi_{\alpha}|$
$\displaystyle=\frac{1}{M}\sum_{\alpha}\sum_{x,x^{\prime}=1}^{d_{A}}c_{\alpha,x}c^{*}_{\alpha,x^{\prime}}|x\rangle\langle
x^{\prime}|,$ (56)

where $(c_{\alpha,x})_{x}$ contains at most $M$ nonzero elements for each
$\alpha$. Hence we can define permutations $f_{\alpha}$ on the set
$\\{1,\cdots,|A|\\}$ such that $f_{\alpha}(x)\in[M]$ for every $x$ and
$\alpha$ with $c_{\alpha,x}\not=0$. The Kraus operators
$K_{\alpha}=\sum_{x=1}^{d_{A}}c^{*}_{\alpha,x}|f_{\alpha}(x)\rangle\langle x|$
satisfy

$\displaystyle\sum_{\alpha}K_{\alpha}^{\dagger}\Psi_{M}K_{\alpha}=\Omega.$
(57)

Furthermore,
$\sum_{\alpha}K^{\dagger}_{\alpha}K_{\alpha}=\sum_{\alpha}\sum_{x=1}^{d_{A}}|c_{\alpha,x}|^{2}|x\rangle\langle
x|=\mathbbm{1}$, since by assumption

$\frac{1}{M}=\langle x|A|x\rangle=\frac{1}{M}\sum_{\alpha}|c_{\alpha,x}|^{2}.$

Therefore, the $\\{K_{\alpha}\\}_{\alpha}$ define a CPTP SIO map $\Lambda$
satisfying $\operatorname{Tr}\Lambda(\rho)\Psi_{M}\geq 1-\varepsilon$.
$\blacksquare$

######  Remark 11.

Comparing with Eq. (33), we see that the one-shot distillable coherence under
SIO takes the form of DIO distillation with the added constraint of
$C_{0}(A)\leq\log M$.

######  Remark 12.

An explicit calculation of the incoherent rank $C_{0}$ can be made through
semi-definite programming techniques [39]. However the number of computational
constraints scales as $\binom{d}{M}$ for certifying whether a $d$-dimensional
state $\rho$ has $C_{0}(\rho)\geq\log(M+1)$.

###  V-B Bound coherence exists under SIO

The constraint on the incoherent rank of $A$ in Theorem 10 greatly diminishes
the power of SIO to distill coherence. Here we illustrate this effect by a
dramatic example. Consider the state

$\rho=\frac{1}{2}\bigl{(}|+\rangle\langle+|\otimes|+\rangle\langle+|+|-\rangle\langle-|\otimes|\widetilde{+}\rangle\langle\widetilde{+}|\bigr{)},$
(58)

where

$\displaystyle|+\rangle|+\rangle$
$\displaystyle=\frac{1}{2}(|00\rangle+|01\rangle+|10\rangle+|11\rangle)$
$\displaystyle|-\rangle|\widetilde{+}\rangle$
$\displaystyle=\frac{1}{2}(|00\rangle+i|01\rangle-|10\rangle-i|11\rangle).$

The $n$-copy state $\rho^{\otimes n}$ is then an equal mixture of states
belonging to the ensemble

$\mathfrak{E}_{n}\mathrel{\mathop{\mathchar
58\relax}}=\\{|+\rangle|+\rangle,|-\rangle|\widetilde{+}\rangle\\}^{\otimes
n}.$

We will show that not a single cosbit of coherence can be distilled from
$\rho^{\otimes n}$ by SIO with an error smaller than the minimal one-copy
error. In comparison, $n$ bits of coherence can be distilled by IO error-free:
the first system in each copy of $\rho$ is simply measured with the IO Kraus
operations $\\{|0\rangle\langle+|,|1\rangle\langle-|\\}$ followed by a
suitable controlled phase on the second qubit. Such a measurement is not
possible by SIO.

######  Theorem 13.

For the state $\rho$ defined in Eq. (58), $C_{d,SIO}^{\infty}(\rho)=0$.

The proof of this will follow by studying the structure of $\rho^{\otimes n}$
and showing that for a fixed value of $\epsilon$ (independent of $n$),
$\operatorname{Tr}\rho^{\otimes n}A<1-\epsilon$ for any operator $A$ having an
incoherent rank of two and satisfying the conditions of Theorem 10. The key
property we use is that the eigenvectors of $\rho^{\otimes n}$ will always be
maximally coherent states with complex phases belonging to
$\\{0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\\}$. For the $n$-copy analysis to be
tractable, we need to introduce some new notation. Let
$\mathbf{b}_{j}=(b_{j,0},b_{j,1},\cdots,b_{j,n-1})\in\\{0,1\\}^{n}$ denote the
$j^{\text{th}}$ binary sequence of length $n$. We then define an ensemble of
$2^{n}$ equiprobable states $\\{|\mathbf{b}_{j}\rangle\\}_{j=1}^{2^{n}}$,
where

$\displaystyle|\mathbf{b}_{j}\rangle\mathrel{\mathop{\mathchar 58\relax}}=$
$\displaystyle\frac{1}{\sqrt{4^{n}}}\sum_{m_{0},\cdots,m_{n-1}=0}^{3}\exp\left(i\frac{\pi}{2}\sum_{k=0}^{n-1}b_{j,k}m_{k}\right)\bigg{|}\sum_{k=0}^{n-1}4^{k}m_{k}\bigg{\rangle}.$
(59)

We claim that, up to relabelling, this ensemble is precisely
$\mathfrak{E}_{n}$. For example, when $n=1$ we have

$\displaystyle|\mathbf{b}_{1}\rangle$
$\displaystyle=\frac{1}{2}\left(|0\rangle+|1\rangle+|2\rangle+|3\rangle\right)$
$\displaystyle|\mathbf{b}_{2}\rangle$
$\displaystyle=\frac{1}{2}\left(|0\rangle+i|1\rangle-|2\rangle-i|3\rangle\right),$

and for $n=2$ we have

$\displaystyle|\mathbf{b}_{1}\rangle$
$\displaystyle=\frac{1}{4}(|0\rangle+|1\rangle+|2\rangle+|3\rangle+|4\rangle$
$\displaystyle+|5\rangle+|6\rangle+\cdots+|14\rangle+|15\rangle)$
$\displaystyle|\mathbf{b}_{2}\rangle$
$\displaystyle=\frac{1}{4}(|0\rangle+i|1\rangle-|2\rangle-i|3\rangle+|4\rangle$
$\displaystyle+i|5\rangle-|6\rangle-\cdots-|14\rangle-i|15\rangle)$
$\displaystyle|\mathbf{b}_{3}\rangle$
$\displaystyle=\frac{1}{4}(|0\rangle+|1\rangle+|2\rangle+|3\rangle+i|4\rangle$
$\displaystyle+i|5\rangle+i|6\rangle+\cdots-i|14\rangle-i|15\rangle)$
$\displaystyle|\mathbf{b}_{4}\rangle$
$\displaystyle=\frac{1}{4}(|0\rangle+i|1\rangle-|2\rangle-i|3\rangle+i|4\rangle$
$\displaystyle-|5\rangle-i|6\rangle+\cdots+i|14\rangle-|15\rangle).$

The case of general $n$ can be checked by induction.

Now for any two distinct vectors $\vec{m}=(m_{0},\cdots,m_{n-1})$ and
$\vec{m}^{\prime}=(m_{0}^{\prime},\cdots,m_{n-1}^{\prime})$ belonging to
$\\{0,1,2,3\\}^{n}$, let us denote the kets

$\displaystyle|\vec{m}\rangle$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\bigg{|}\sum_{k=0}^{n-1}4^{k}m_{k}\bigg{\rangle},$
$\displaystyle|\vec{m}^{\prime}\rangle$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\bigg{|}\sum_{k=0}^{n-1}4^{k}m^{\prime}_{k}\bigg{\rangle}.$ (60)

The relative phase between $|\vec{m}\rangle$ and $|\vec{m}^{\prime}\rangle$ in
any $|\mathbf{b}_{j}\rangle$ is defined to be

$\frac{\pi}{2}\sum_{k=0}^{n-1}b_{j,k}(m_{k}^{\prime}-m_{k})\in\left\\{0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\right\\}.$
(61)

We now make a crucial observation about the distribution of relative phases
among the $|\mathbf{b}_{j}\rangle$ in $\mathfrak{E}_{n}$.

######  Proposition 14.

For any fixed pair of distinct vectors $\vec{m}$ and $\vec{m}^{\prime}$, at
most half of the $|\mathbf{b}_{i}\rangle$ in $\mathfrak{E}_{n}$ have the same
relative phase between $|\vec{m}\rangle$ and $|\vec{m}^{\prime}\rangle$.

Proof. Let $k\in\\{0,\cdots,n-1\\}$ be chosen such that
$m_{k}^{\prime}-m_{k}\not=0$. Let $\Delta m=m_{k}^{\prime}-m_{k}$. Consider
all the states $|\mathbf{b}_{j}\rangle\in\mathfrak{E}_{n}$ with binary
sequence $\mathbf{b}_{j}$ such that $b_{j,k}=0$; this represents exactly half
of all states in the ensemble $\mathfrak{E}_{n}$. We partition these states
into four groups, $A_{0}$, $A_{\pi/2}$, $A_{\pi}$, and $A_{3\pi/2}$, according
to their respective relative phases between $|\vec{m}\rangle$ and
$|\vec{m}^{\prime}\rangle$. Now we consider the other half of the states in
$\mathfrak{E}_{n}$, those having $b_{j,k}=1$. We likewise partition these
states into sets $B_{0}$, $B_{\pi/2}$, $B_{\pi}$, and $B_{3\pi/2}$ of common
relative phase between $|\vec{m}\rangle$ and $|\vec{m}^{\prime}\rangle$.
Notice that for any $|\mathbf{b}_{j}\rangle$ in, say, $A_{\pi/2}$, there will
be a corresponding $|\mathbf{b}_{j^{\prime}}\rangle$ in $B_{\pi/2(1+\Delta
m})$ and vice versa, the only difference between $\mathbf{b}_{j}$ and
$\mathbf{b}_{j^{\prime}}$ being their $k^{th}$ component. Hence
$|A_{0}|=|B_{\pi/2\Delta m}|$, $|A_{\pi/2}|=|B_{\pi/2(1+\Delta m)}|$,
$|A_{\pi}|=|B_{\pi/2(2+\Delta m)}|$ and $|A_{3\pi/2}|=|B_{\pi/2(3+\Delta
m)}|$, where all arithmetic is done modular $4$. Therefore, the total number
of states in the ensemble having a relative phase of, say, $\pi/2$ is

$\displaystyle|A_{\pi/2}|+|B_{\pi/2}|$
$\displaystyle=|A_{\pi/2}|+|A_{\pi/2(1-\Delta m)}|\leq 2^{n-1}.$

The same bound likewise holds for the other three relative phases.
$\blacksquare$

Proof of Theorem 13. Consider an arbitrary vector in the complex linear span
of $|\vec{m}\rangle$, $|\vec{m}^{\prime}\rangle$,

$|\psi\rangle=\cos\theta|\vec{m}\rangle+\sin\theta
e^{i\phi}|\vec{m}^{\prime}\rangle.$ (62)

Let $N_{0}$, $N_{\pi/2}$, $N_{\pi}$, $N_{3\pi/2}$ denote the number of states
in $\mathfrak{E}_{n}$ having a relative phase between $|\vec{m}\rangle$ and
$|\vec{m}^{\prime}\rangle$ of $0$, $\pi/2$, $\pi$, and $3\pi/2$, respectively.
We can then explicitly compute

$\displaystyle\langle\psi|\rho^{\otimes
n}|\psi\rangle=\frac{1}{2^{n}4^{n}}\sum_{i=1}^{2^{n}}|\langle\psi|\mathbf{b}_{i}\rangle|^{2}$
$\displaystyle=\frac{N_{0}}{2^{n}4^{n}}|\cos\theta\\!+\\!e^{-i\phi}\sin\theta|^{2}+\frac{N_{\pi/2}}{2^{n}4^{n}}|\cos\theta\\!+\\!ie^{-i\phi}\sin\theta|^{2}$
$\displaystyle\phantom{=}+\frac{N_{\pi}}{2^{n}4^{n}}|\cos\theta\\!-\\!e^{-i\phi}\sin\theta|^{2}+\frac{N_{3\pi/2}}{2^{n}4^{n}}|\cos\theta\\!-\\!ie^{-i\phi}\sin\theta|^{2}$
$\displaystyle=\frac{1}{4^{n}}+\frac{\sin
2\theta}{2^{n}4^{n}}\left[(N_{0}\\!-\\!N_{\pi})\cos\phi+(N_{3\pi/2}\\!-\\!N_{\pi/2})\sin\phi\right],$
(63)

where the last line follows by expanding out the squared amplitudes, the
identity $2\cos\theta\sin\theta=\sin 2\theta$, and using the fact that
$N_{0}+N_{\pi/2}+N_{\pi}+N_{3\pi/2}=2^{n}$. Our goal is to maximize (63) under
the constraint that $\max\\{N_{0},N_{\pi/2},N_{\pi},N_{3\pi/2}\\}\leq
2^{n-1}$. This constraint implies that $|N_{0}-N_{\pi}|\leq 2^{n-1}$ and
$|N_{3\pi/2}-N_{\pi/2}|\leq 2^{n-1}$, and so

$\displaystyle\langle\psi|\rho^{\otimes n}|\psi\rangle$
$\displaystyle\leq\frac{1}{4^{n}}\left(1+\cos\theta\sin\theta(|\cos\phi|+|\sin\phi|)\right)$
$\displaystyle\leq\frac{1}{4^{n}}\left(1+\frac{\sqrt{2}}{2}\right).$ (64)

Suppose now that $A$ has an incoherent rank of two and satisfies
$\operatorname{Tr}[A]=\frac{4^{n}}{2}$. Then by the previous calculation we
have the fidelity bound

$\displaystyle\operatorname{Tr}\rho^{\otimes
n}A\leq(\operatorname{Tr}A)\frac{1}{4^{n}}\left(1+\frac{\sqrt{2}}{2}\right)$
$\displaystyle=\frac{1}{2}\left(1+\frac{\sqrt{2}}{2}\right)$
$\displaystyle=1-\varepsilon,$ (65)

where $\varepsilon=\frac{1}{2}-\frac{\sqrt{2}}{4}$ is independent of $n$. This
is precisely the single-copy error bound. As a consequence, it follows that
$C_{d,SIO}^{\infty}(\rho)=0$, proving the theorem. $\blacksquare$

######  Remark 15.

This result should be compared with the recent proof, by Marvian [40], that
coherence distillation is generally impossible in the resource theory of
energy conservation, which is characterized by the class of so-called time-
translation-covariant operations (TIO). That class is difficult to compare
with DIO, as at the single-system level, TIO is contained in DIO, but since
the composition of systems works differently, it may result in TIO operations
outside DIO on the multi-system level.

The result of [40] shows that for generic mixed states, the rate of distilling
cosbit states $|\Psi_{2}\rangle$ is zero, but that at the same time it is
possible to obtain a single cosbit (or a sublinear number) with fidelity going
to $1$ as asymptotically many copies of the mixed resource become available.
In contrast, here we showed that under DIO the fidelity remains bounded away
from $1$, irrespective of the number of resource states.

##  VI Recovering the information theoretic limit

In the asymptotic limit, the coherence distillation rate under operation class
$\mathcal{O}$ is defined as

$C_{d,\mathcal{O}}^{\infty}(\rho)=\liminf_{\varepsilon\to
0^{+}}\liminf_{n\to\infty}\frac{1}{n}C_{d,\mathcal{O}}^{\varepsilon}(\rho^{\otimes
n}).$ (66)

From [17] and [25] (see also [26]) we know that
$C_{d,IO}^{\infty}(\rho)=C_{d,DIO}^{\infty}(\rho)=C_{d,MIO}^{\infty}(\rho)=C_{r}(\rho)$.
Below we show that our results on one-shot IO distillation can be used to
recover the asymptotic limit, at the same time improving the result by showing
that the limit exists and equals $C_{r}(\rho)$ for any fixed
$0<\varepsilon<1$; such a statement is known as a _strong converse_ in
information theory.

######  Theorem 16.

For any state $\rho$ and any $0<\varepsilon<1$,

$\begin{split}\lim_{n\to\infty}\frac{1}{n}C_{d,IO}^{\varepsilon}(\rho^{\otimes
n})&=\lim_{n\to\infty}\frac{1}{n}C_{d,DIO}^{\varepsilon}(\rho^{\otimes n})\\\
&=\lim_{n\to\infty}\frac{1}{n}C_{d,MIO}^{\varepsilon}(\rho^{\otimes
n})=C_{r}(\rho).\end{split}$

Proof. Recall the results in Theorems 4 and 7, which state that for any
$\eta<\frac{1}{2}\varepsilon$,

$\begin{split}C_{\min}^{\frac{\varepsilon}{2}-\eta}(\rho)-2\log\frac{1}{\eta}&\leq
C_{d,IO}^{\varepsilon}(\rho)\\\ &\leq C_{d,MIO}^{\varepsilon}(\rho)\\\
&=C_{d,DIO}^{\varepsilon}(\rho)\leq
C_{\min}^{\sqrt{\varepsilon(2-\varepsilon)}}(\rho).\end{split}$ (67)

Hence, to show the theorem, we only need to prove that for all $0<\delta<1$,

$\lim_{n\to\infty}\frac{1}{n}C_{\min}^{\delta}(\rho^{\otimes n})=C_{r}(\rho),$
(68)

which is equivalent to

$\lim_{n\to\infty}\frac{1}{n}\max_{\rho^{\prime}\in
B_{\delta}({\rho^{A}}^{\otimes
n})}H_{\min}(X^{n}|E^{n})_{\omega^{\prime}}=C_{r}(\rho).$ (69)

Here,
$\omega_{X^{n}E^{n}}^{\prime}=(\Delta_{A^{n}}\otimes{\operatorname{id}}_{E^{n}})|\psi^{\prime}\rangle\langle\psi^{\prime}|$
and $|\psi^{\prime}\rangle$ is an arbitrary purification of
$\rho_{A}^{\prime}$. Recall the quantum asymptotic equipartition theorem [41],
which states that for any $0<\eta<1$,

$\lim_{n\to\infty}\frac{1}{n}\max_{\rho_{AB}^{\prime}\in
B_{\eta}(\rho_{AB}^{\otimes
n})}H_{\min}(A|B)_{\rho_{AB}^{\prime}}=H(A|B)_{\rho_{AB}}.$ (70)

Now, in one direction, if we have a state $\rho^{\prime}\in
B_{\delta}(\rho^{\otimes n})$, then by Uhlmann’s characterization of the
fidelity there exists a purification $\psi^{\prime}\in
B_{\delta}(\psi^{\otimes n})$, hence
$\omega^{\prime}=(\Delta_{A^{n}}\otimes{\operatorname{id}}_{E^{n}})\psi^{\prime}\in
B_{\delta}(\omega^{\otimes n})$, with
$\omega^{XE}=(\Delta_{A}\otimes{\operatorname{id}}_{E})\psi$. In the other
direction, for $\omega^{\prime\prime}\in B_{\delta}(\omega^{\otimes n})$, it
is known that since $\omega$ is a cq-state, an optimal $\omega^{\prime\prime}$
for $H_{\min}(X^{n}|E^{n})$ may be assumed to be a cq-state as well [35],
hence we can find a $\psi^{\prime}\in B_{\delta}(\psi^{\otimes n})$ such that
$\omega^{\prime\prime}=(\Delta_{A^{n}}\otimes{\operatorname{id}}_{E^{n}})\psi^{\prime}$.
Thus, we can conclude that

$\max_{\rho_{A}^{\prime}\in B_{\delta}({\rho^{A}}^{\otimes
n})}H_{\min}(X^{n}|E^{n})_{\omega^{\prime}}=\max_{\omega^{\prime\prime}\in
B_{\delta}(\omega^{\otimes n})}H_{\min}(X^{n}|E^{n})_{\omega^{\prime\prime}},$

where the left hand side corresponds to Eq. (69).

But this means that we can apply the quantum AEP directly, and get

$\begin{split}\lim_{n\to\infty}\frac{1}{n}C_{\min}^{\delta}(\rho^{\otimes
n})&=\lim_{n\to\infty}\frac{1}{n}\max_{\omega^{\prime}\in
B_{\delta}({\omega}^{\otimes n})}H_{\min}(X^{n}|E^{n})_{\omega^{\prime}}\\\
&=H(X|E)_{\omega}=C_{r}(\rho),\end{split}$

as claimed. $\blacksquare$

##  VII Coherence distillation
and randomness generation

Suppose a purification of $\rho^{A}$ is written as

$|\psi\rangle^{AE}=\sum_{x}\sqrt{p_{x}}|x\rangle^{A}|\psi_{x}\rangle^{E},$

we use this state to generate randomness by first performing a computational
basis measurement. The dephased cq-state after measuring $A$ in the
computational basis is

$\omega^{AE}=(\Delta\otimes{\operatorname{id}})\psi=\sum_{x}p_{x}|x\rangle\langle
x|^{A}\otimes\psi_{x}^{E}.$

Considering the measurement as a raw randomness generation process, a
subsequent randomness extraction (via a deterministic function $G$) can
further extract a random string that is almost uniform and independent of $E$.
We think of the function as an incoherent operation, by letting
$G(|x\rangle\langle y|)=\delta_{xy}|G(x)\rangle\langle G(x)|$. This
identification is natural as every incoherent (MIO) operation $\Lambda$
defines an associated classical channel via $\Lambda(|x\rangle\langle
x|)=\sum_{y}\Lambda(y|x)|y\rangle\langle y|$.

Denote $\ell^{\varepsilon}_{\textrm{ext}}$ to be the maximum length of the
extractable randomness that is $\varepsilon$-close to a string that it is
perfectly uniform and independent of $E$, i.e.

$\ell^{\varepsilon}_{\textrm{ext}}(\rho_{A})=\max_{G}\left\\{\log
M\mathrel{\mathop{\mathchar
58\relax}}\frac{1}{2}\left\|(G\\!\otimes\\!{\operatorname{id}})\omega^{AE}\\!-\\!\tau^{K}\\!\otimes\\!\omega^{E}\right\|_{1}\leq\varepsilon\right\\}\\!,$
(71)

where we recall the notation $\tau^{K}=\frac{1}{M}\mathbbm{1}_{K}$ for the
maximally mixed state of the $M$-dimensional key system.

Note that our definition of extractable randomness differs somewhat from the
one in [42]; in that work, a model based on incoherent operations was
proposed, which is shown in Fig. 1(a). The main process consists of three
parts, incoherent operations $\Lambda$, dephasing operation $\Delta$, a random
hashing function as the extractor $\mathrm{Ext}$. Here our definition is more
straightforward and we do not need to perform the real incoherent operations.
As shown in Fig. 1(b), after the dephasing operation $\Delta$, we use a
function $G$ as an extractor to extract the secure randomness. This function
has to be deterministic, as opposed to a noisy channels, since otherwise
infinite randomness can be generated independent of $E$.

Moreover, in order to obtain the optimal $G$ in our definition, we first
consider the randomness extraction process via DIO which is shown in Fig.
1(c), where we apply the DIO distillation followed by dephasing operation
$\Delta$, a classical extractor (which may not be needed). Benefiting from the
property of DIO, we can change the order of DIO and $\Delta$, which implies
that the DIO distillation may act as a good extractor (the blue part in Fig.
1(d)). The only remaining problem here is that we would have to show that this
DIO operation gives rise to a deterministic classical channel, which is in
general not true. For instance, the optimal coherence distillation process
under DIO derived in Theorem 3 has the property that $\Lambda(|x\rangle\langle
x|)=\tau_{K}$ for all $x$. Via the permutation twirling Eq. (25), this can be
imposed equally on any optimal IO distillation process, and hence even on DIIO
= IO$\cap$DIO.

Instead, inspired by Remark 5, we know that a suboptimal but achievable IO
distillation operation is also a DIO operation and after a modification
(another dephasing channel) we can construct a valid extractor $G$ from it,
which is shown in the proof of Theorem 17.

Figure 1: The different schemes for extracting randomness. (a) Extracting
randomness using incoherent strategy in [42]. Alice is allowed to perform the
unitary IO operation on her system together with another ancilla system
prepared in $|0\rangle\langle 0|$. (b) The randomness extraction process
defined in our protocol. The extractor is implemented by a deterministic
function $G$ (c) and (d) Applying DIO distillation for extracting secure
randomness. Due to the property of DIO, we can achieve a new operation (blue
part) combining the DIO distillation operation and original extractor.

We first recall that every valid $G$ can be used for IO distillation. Here, we
consider $G$ as a deterministic function, and $\omega^{E}=\psi^{E}$ is the
reduced state of $\omega^{AE}$ on system E. For every function $G$ satisfying
$\frac{1}{2}\left\|(G\otimes{\operatorname{id}})\omega^{AE}-\tau^{K}\otimes\omega^{E}\right\|_{1}\leq\varepsilon$,
we can substitute it in the proof of Theorem 4 and obtain an incoherent
distillation channel. Thus as the maximal distillable rate
$C^{\varepsilon}_{d,IO}(\rho)$, it satisfies

$C^{2\varepsilon}_{d,IO}(\rho)\geq\ell^{\varepsilon}_{\textrm{ext}}(\rho).$
(72)

The IO map achieving $C^{2\varepsilon}_{d,IO}(\rho)$ can also be applied to
extract randomness by the following theorem.

######  Theorem 17.

For an arbitrary state $\rho$ and $0<\varepsilon<1$,

$\ell^{\sqrt{2\varepsilon}}_{\textrm{ext}}(\rho)\geq
C^{(\frac{\varepsilon}{2}-\eta)^{2}/2}_{d,IO}(\rho)-2\log\frac{1}{\eta},$ (73)

for any $0<\eta<\frac{\varepsilon}{2}$.

Proof. Recall that the distillable coherence under IO is given by

$C_{d,IO}^{\varepsilon}(\rho)=\max_{\Lambda\in\mathcal{IO}}\bigl{\\{}\log
M\,\mathrel{\mathop{\mathchar 58\relax}}\,F(\Lambda(\rho),\Psi_{M})^{2}\geq
1-\varepsilon\bigr{\\}}.$

Suppose $\Lambda$ is the IO that achieves the right hand side of Theorem 4,
that is $C^{\frac{\varepsilon}{2}-\eta}_{\min}(\rho)-2\log\frac{1}{\eta}=\log
M$ where $0<\eta<\frac{\varepsilon}{2}$ and $F(\Lambda(\rho),\Psi_{M})^{2}\geq
1-\varepsilon$. Note that $\Lambda$ is not necessarily an optimal IO coherence
distillation operation. For the purification state $|\psi\rangle^{AE}$, the
resulting state by applying $\Lambda$ on system $A$ is given by

$\Omega^{AE}=(\Lambda\otimes{\operatorname{id}})(|\psi\rangle\langle\psi|^{AE}),$

and it follows that

$F(\Omega^{AE},\Psi_{M}\otimes\Omega^{E})^{2}\geq(1-\varepsilon)^{2}.$

To prove that, suppose a purification of $\Omega^{AE}$ is
$|\Omega\rangle^{AA^{\prime}E}$. Then, considering an orthogonal basis
$\\{|\phi_{x}\rangle^{A}\\}$ of system $A$ such that
$|\phi_{0}\rangle^{A}=|\Psi_{M}\rangle$, we can write

$|\Omega\rangle^{AA^{\prime}E}=\sum_{x}\alpha_{x}|\phi_{x}\rangle^{A}|\psi_{x}\rangle^{A^{\prime}E}.$

As $F(\Lambda(\rho),\Psi_{M})^{2}\geq 1-\varepsilon$ and

$\Lambda(\rho)=\sum_{xx^{\prime}}\alpha_{x}\alpha_{x}^{\prime}\langle\psi_{x^{\prime}}|\psi_{x}\rangle^{A^{\prime}E}|\phi_{x}\rangle\langle\phi_{x^{\prime}}|^{A},$

we have $F(\Lambda(\rho),\Psi_{M})^{2}=|\alpha_{0}|^{2}\geq 1-\varepsilon$.
The fidelity between $\Omega^{AA^{\prime}E}$ and
$\Psi_{M}^{A}\otimes\psi_{0}^{A^{\prime}E}$ is

$F({\Omega}^{AA^{\prime}E},\Psi_{M}^{A}\otimes{\psi_{0}}^{A^{\prime}E})^{2}=|\alpha_{0}|^{2}\geq
1-\varepsilon.$

Denoting $\Omega^{A^{\prime}E}=\operatorname{Tr}_{A}{\Omega}^{AA^{\prime}E}$,
then the fidelity between ${\Omega}^{AA^{\prime}E}$ and
$\Psi_{M}^{A}\otimes\Omega^{A^{\prime}E}$ is

$F({\Omega}^{AA^{\prime}E},\Psi_{M}^{A}\otimes\Omega^{A^{\prime}E})^{2}=|\alpha_{0}|^{4}\geq(1-\varepsilon)^{2}.$

Then $F(\Omega^{AE},\Psi_{M}\otimes\Omega^{E})^{2}\geq(1-\varepsilon)^{2}$ can
be obtained by tracing out system $A^{\prime}$.

Applying the dephasing operation $\Delta$ on system $A$,

$F\left((\Delta\otimes{\operatorname{id}})\Omega^{AE},\tau^{K}\otimes\Omega^{E}\right)^{2}\geq
F(\Omega^{AE},\Psi_{M}\otimes\Omega^{E})^{2}.$

From the Remark 5, we know that $\Lambda$ is also a DIO which commutes with
$\Delta$, so we have equivalently

$\displaystyle
F\left((\Lambda\otimes{\operatorname{id}})\omega^{AE},\tau^{K}\otimes\Omega^{E}\right)^{2}$
$\displaystyle\geq F(\Omega^{AE},\Psi_{M}\otimes\Omega^{E})^{2}$ (74)
$\displaystyle\geq(1-\varepsilon)^{2}.$

In order to construct a deterministic $G$, we apply another dephasing
operation after the incoherent channel $\Lambda$,

$\displaystyle
F\left((\Delta\circ\Lambda\otimes{\operatorname{id}})\omega^{AE},\tau^{K}\otimes\Omega^{E}\right)^{2}\geq(1-\varepsilon)^{2},$
(75)

hence

$\frac{1}{2}\left\|(\Delta\circ\Lambda\otimes{\operatorname{id}})\omega^{AE}-\tau^{K}\otimes\Omega^{E}\right\|_{1}\leq\sqrt{2\varepsilon}.$

Note that from Remark 5, the map $\Delta\circ\Lambda$ acts on the incoherent
basis states $|x\rangle\langle x|$ as

$\displaystyle\Delta[\Lambda(|x\rangle\langle x|)]=|G(x)\rangle\langle G(x)|,$
(76)

which is deterministic. From the achievable distillation IO map, we can
construct an extractor and obtain

$\ell^{\sqrt{2\varepsilon}}_{\textrm{ext}}(\rho)\geq
C^{\frac{\varepsilon}{2}-\eta}_{\min}(\rho)-2\log\frac{1}{\eta}.$ (77)

Recall the result in Theorem 7,

$C^{\varepsilon}_{d,IO}(\rho)\leq
C_{\min}^{\sqrt{\varepsilon(2-\varepsilon)}}(\rho)\leq
C_{\min}^{\sqrt{2\varepsilon}}(\rho),$ (78)

and we obtain

$\ell^{\sqrt{2\varepsilon}}_{\textrm{ext}}(\rho)\geq
C^{(\frac{\varepsilon}{2}-\eta)^{2}/2}_{d,IO}(\rho)-2\log\frac{1}{\eta},$ (79)

finishing the proof. $\blacksquare$

Combining with Eq. (72), we have

$C^{\sqrt{8\varepsilon}}_{d,IO}(\rho)\geq\ell^{\sqrt{2\varepsilon}}_{\textrm{ext}}(\rho)\geq
C^{(\frac{\varepsilon}{2}-\eta)^{2}/2}_{d,IO}(\rho)-2\log\frac{1}{\eta}.$ (80)

In the regime of vanishingly small $\varepsilon$, the distillable coherence
rate $C_{d,IO}(\rho)$ and $\ell_{\textrm{ext}}(\rho)$ are hence essentially
the same. Whether $C_{d,DIO}(\rho)$ and $\ell_{\textrm{ext}}(\rho)$ are the
same is still an open problem. Though DIO can commute with dephasing
operation, the difficulty stems from that the combination of DIO and extractor
(the blue part in Fig. 1(d)) may be not deterministic thus not a valid
extractor.

##  VIII Discussion

We have considered the problem of one-shot coherence distillation under the
classes MIO, DIO, IO, and SIO of incoherent operations. Our results indicate
that the distillation rates under IO, MIO, DIO are roughly the same, up to
different smoothing parameters and universal additive terms. The results allow
us to recover the asymptotic (many-copy) limit, in which the distillation
rates for all these three classes tend to be relative entropy of coherence,
which is consistent with the previous results in [17, 25, 26].

The smallest class for which we have been able to show a non-trivial
distillation of coherence is that of dephasing-covariant incoherent IOs, DIIO
= IO$\cap$DIO. On the other hand, interestingly, there is a gap between
distillation rates under SIO and DIIO, both in the one-shot, and more
importantly in the asymptotic regime. As a matter of fact, we showed that
there is _bound coherence_ under SIO; no pure coherence can be distilled under
SIO from these states though they possess coherence, as shown by distillable
coherence under IO.

An interesting future direction is then to study the case involving another
system to help this distillation process, which referred as assisted coherence
distillation for such bound coherence with SIO [43, 44, 45, 46]. Furthermore,
our work also connects the distillation of coherence to randomness extraction.
The distillation process is also related to the decoupling in cryptography.
Thus our results also shed light on other quantum information processing tasks
like random number generation, extraction and cryptography.

_Note added:_ After completion of this work and circulating a preliminary
preprint, Ludovico Lami _et al._ [47] have shown that the bound coherence
under SIO is in fact a generic phenomenon, showing that the fidelity of
distilling even a single cosbit is bounded away from $1$ for all but a
measure-zero set of mixed states.

## Acknowledgment

The authors thank María García Díaz, Xiongfeng Ma, Hongyi Zhou, Ludovico Lami
and Bartosz Regula for various insightful discussions on coherence theory.

QZ acknowledges support by the National Natural Science Foundation of China
Grant No. 11674193. XY was supported by BP plc and the EPSRC National Quantum
Technology Hub in Networked Quantum Information Technology (EP/M013243/1). EC
was supported by the National Science Foundation (NSF) Early CAREER Award No.
1352326. AW acknowledges support by the Spanish MINECO (project
FIS2016-80681-P), with the support of FEDER funds, and the Generalitat de
Catalunya (CIRIT project 2017-SGR-1127).

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