One-Shot Coherence Distillation
Bartosz Regula,
1,*
Kun Fang,
2,†
Xin Wang,
2,‡
and Gerardo Adesso
1,§
1
School of Mathematical Sciences and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems,
University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
2
Centre for Quantum Software and Information, School of Software, Faculty of Engineering and Information Technology,
University of Technology Sydney, New South Wales 2007, Australia
(Received 5 December 2017; revised manuscript received 18 April 2018; published 3 July 2018)
We characterize the distillation of quantum coherence in the one-shot setting, that is, the conversion of
general quantum states into maximally coherent states under different classes of quantum ope rations. We
show that the maximally incoherent operations (MIO) and the dephasing-covariant incoherent ope rations
(DIO) have the same power in the task of one-shot coherence distillation. We establish that the one-shot
distillable coherence under MIO and DIO is efficiently computable with a semidefinite program, which we
show to correspond to a quantum hypothesi s testing problem. Further, we introduce a family of coherence
monotones generalizing the robustness of coherence as well as the modified trace distance of coherence,
and show that they admit an operational interpretation in characterizing the fidelity of distillation under
different classes of operations. By providing an explicit formula for these quantities for pure states, we
show that the one-shot distillable coherence under MIO, DIO, strictly incoherent operations, and incoherent
operations is equal for all pure states.
DOI: 10.1103/PhysRevLett.121.010401
The phenomenon of quantum coherence, emerging from
the fundamental property of quantum superposition, has
found use in a variety of physical tasks in quantum
cryptography, quantum information processing, thermody-
namics, metrology, and even quantum biology [1]. Recent
years have seen the development of the resource-theoretic
framework of quantum coherence, establishing precise
physical and mathematical laws governing the creation,
manipulation, and conversion of coherence [2–5]. The
archetypal example of a resource theory is quantum
entanglement, and although coherence and entanglement
share a large number of similarities, which allowed for
many parallels and interrelations between the two resource
theories to be established [5–16], the two are also very
different in some aspects and can require different
approaches. One particular difference is the lack of a
single, physically motivated choice of free operations that
best describe the allowed state manipulations in the
resource theory of quantum coherence, unlike the standard
choice of local operations and classical communication for
entanglement [17]. It thus becomes necessary to character-
ize the operational properties and applications of quantum
coherence under several different sets of such operations
[1,9,18–20], the most common ones being incoherent
operations (IO) [5], strictly incoherent operations (SIO)
[9], dephasing-covariant incoherent operations (DIO)
[18,19], and maximally incoherent operations (MIO) [2].
One of the most significant aspects of a resource theory
are the rules governing state transformations under the free
operations. In particular, the problem of distillation asks the
following: given a canonical unit of coherence represented
by the maximally coherent state jΨi, what is the best rate at
which we can convert copies of a state ρ into copies of jΨi
under a chosen set of free operations? The standard
approach to this problem in quantum information theory,
both in the resource theories of entanglement [21–23] and
coherence [9,24], is to consider the asymptotic limit—that
is, assume that we have access to an unbounded number of
independent and identically distributed (IID) copies of a
quantum system. In a realistic setting, however, the
resources are finite and the number of IID prepared states
is necessarily limited. More importantly, it is very difficult
to perform coherent state manipulations over large numbers
of systems. Therefore, it becomes crucial to be able to
characterize how well we can distill maximally coherent
states from a finite number of copies of the prepared states.
The study of such nonasymptotic scenarios has garnered
great interest in quantum information theory [25–32],
including work in the one-shot theory of entanglement
distillation [33–35]. More recently, one-shot results in the
resource theory of coherence [36,37] and more general
quantum resource theories [38,39] were obtained.
In this Letter, we develop the framework for nonasymp-
totic coherence distillation, in which one has access only to
a single copy of a quantum system and allows for a finite
accuracy, reflecting the realistic restrictions on state trans-
formations. In particular, we establish an exact expression
for the one-shot distillable coherence under MIO and DIO,
which can be efficiently computed as a semidefinite
program (SDP). Interestingly, we show that the two
PHYSICAL REVIEW LETTERS 121, 010401 (2018)
0031-9007=18=121(1)=010401(6) 010401-1 © 2018 American Physical Society
quantities are in fact the same, demonstrating that MIO and
DIO have the same power in the task of coherence
distillation, and together with recent results in coherence
dilution [36,40] shed light on the asymptotic reversibility of
state transformations under DIO. Further, we generalize
two fundamental quantifiers of coherence, the robustness of
coherence [41] and the modified trace distance of coher-
ence [42], establishing a family of measures of coherence,
which we show to have an operational application in
characterizing the maximal fidelity of distillation under
different sets of operations. We derive exact expressions for
these measures for all pure states, leading to a complete
characterization of pure-state one-shot distillation of coher-
ence and showing that all the considered sets of operations
—IO, SIO, DIO, and MIO—have exactly the same power
in such a task. We discuss our methods and results below
and defer more technical derivations to the Supplemental
Material [43].
A family of coherence monotones.—Consider a fixed
orthonormal basis fjiig in a d-dimensional Hilbert space
(d<∞). We will use D to denote the set of all density
matrices in this space, and for a pure state jψi, we will write
ψ ≔ jψihψj. Let Δ denote the diagonal map (fully dephas-
ing channel) in the basis fjiig. We will denote by I the set
of density matrices that are diagonal in this basis, i.e., ρ ∈
D such that ρ ¼ ΔðρÞ, and by I
the cone of diagonal
positive semidefinite matrices that are not necessarily
normalized.
The resource theory of coherence consists of the follow-
ing ingredients [5]: the set of free incoherent states,
represented by I, and the free operations, that is, a set
of quantum operations that do not generate coherence. The
largest possible set of such free operations are the max-
imally incoherent operations [2], which are given by
quantum channels E such that EðρÞ ∈ I for every ρ ∈ I.
The incoherent operations [5] are those for which there
exists a Kraus decomposition into incoherent Kraus oper-
ators, that is, fK
l
g such that K
l
ρK
†
l
∈ I
for all l and all
ρ ∈ I. The strictly incoherent operations [9] are operations
for which both fK
l
g and fK
†
l
g are sets of incoherent
operators. Finally, the dephasing-covariant incoherent oper-
ations are maps E such that ½Δ; E¼0. The following strict
inclusions hold: MIO⊋IO⊋SIO, MIO⊋DIO⊋SIO [19,47].
Throughout the development of the resource theory of
coherence, many different quantifiers of this resource have
been defined [1,5,6,41,42,48]. A particular example is the
trace distance of coherence, given by [5,48]
T
I
ðρÞ¼min fkρ − σk
1
jσ ∈ I g: ð1Þ
Although the trace distance is a fundamental quantity in
quantum information theory [49,50], the trace distance of
coherence was found to violate the property of strong
monotonicity under incoherent operations [42], which is
considered as one of the requirements that a valid measure
of coherence should satisfy [1,5]. Therefore, an alternative
measure, called the modified trace distance of coherence,
satisfying strong monotonicity under IO was proposed
[42],
T
0
I
ðρÞ¼minfkρ − λσk
1
jσ ∈ I ; λ ≥ 0g
¼ minfkρ − Xk
1
jX ∈ I
g: ð2Þ
Noting that strong duality holds [51,52], we can consider
the Lagrange dual of the above expression to obtain a
characterization of the modified trace distance of coherence
as
T
0
I
ðρÞ¼maxfhρ;Wij − 1 ≤ W ≤ 1; ΔðWÞ ≤ 0g; ð3Þ
with the Hilbert-Schmidt inner product hX; Y i¼TrðXY Þ
for Hermitian operators. We then extend the above to a
family of quantifiers given by SDPs of the form
T
ðmÞ
I
ðρÞ¼maxfhρ;Wij − 1 ≤ W ≤ m1; ΔðWÞ ≤ 0g; ð4Þ
similar to the approach of Brandão [53] for entanglement
measures. Here, we will take m ∈ N, although it can be
treated as a continuous parameter in general. The fact that
each such measure is a valid coherence monotone can be
shown by expressing T
ðmÞ
I
as a convex gauge function [51],
and we formalize it as follows.
Proposition 1.—For each m ≥ 1, T
ðmÞ
I
is a faithful and
convex coherence measure satisfying strong monotonicity
under MIO.
Note that m ¼ 1 gives T
0
I
ðρÞ.Form ¼ d − 1, one can
notice that the constraint W ≤ ðd − 1Þ1 is redundant: the
other constraints ensure that the smallest eigenvalue of W is
at least −1 and that the trace of W is at most zero, together
implying that there cannot exist an eigenvalue of W that is
larger than d − 1. Therefore, we get
T
ðd−1Þ
I
ðρÞ¼max fhρ;Wij − 1 ≤ W; ΔðWÞ ≤ 0g¼R
I
ðρÞ;
ð5Þ
where R
I
ðρÞ is the robustness of coherence [41,54]. This
shows that T
ðmÞ
I
can be thought of as a family of measures
interpolating between the modified trace distance and the
robustness of coherence. Notice that we clearly also have
T
ðmÞ
I
ðρÞ¼R
I
ðρÞ for any m>d− 1. In general, we have
that 0 ≤ T
ðmÞ
I
ðρÞ ≤ m∀ρ ∈ D. It is straightforward to see
by strong Lagrange duality that the family T
ðmÞ
I
satisfies
T
ðmÞ
I
ðρÞ¼min
X∈I
mTrðρ − XÞ
þ
þ Trðρ − XÞ
−
; ð6Þ
where ðρ − XÞ
denote the positive and negative parts of
the Hermitian operator ρ − X.
PHYSICAL REVIEW LETTERS 121, 010401 (2018)
010401-2
To characterize the values of the family of quantifiers
T
ðmÞ
I
on pure states, we consider the following quantity
kjψik
½m
¼ min
jψi¼jxiþjyi
kjxik
l
1
þ
ffiffiffiffi
m
p
kjyik
l
2
; ð7Þ
which we (for reasons that will become clear later) call the
m-distillation norm. We can then obtain the following
result.
Theorem 1.—For any pure state jψi and any m ≥ 1,
T
ðm−1Þ
I
ðψÞ¼kjψik
2
½m
− 1: ð8Þ
The proof of this Theorem relies on the fact that each
T
ðm−1Þ
I
can be viewed as a robustness measure R
Q
m
defined
with respect to the set Q
m
≔ I ∪ ð1=mÞD. Each such
quantity was shown in [51] to reduce on pure states to a
corresponding norm defined at the level of the underlying
Hilbert space—in this case, it is precisely the m-distillation
norm kjψik
½m
.
A property of the m-distillation norm that will be crucial
in the characterization of coherence distillation is that it
can, in fact, be computed exactly. In particular, the
following holds.
Theorem 2.—For a pure state jψ i, let ψ
↓
1∶k
denote the
vector consisting of the k largest (by magnitude) coef-
ficients of jψi, and analogously let ψ
↓
kþ1∶d
denote the vector
of the d − k smallest coefficients of jψi, with ψ
↓
1∶0
being the
zero vector. Then, for any pure state jψi and any integer
m ∈ f1; …;dg,wehave
kjψik
½m
¼kψ
↓
1∶m−k
⋆
k
l
1
þ
ffiffiffiffiffi
k
⋆
p
kψ
↓
m−k
⋆
þ1∶d
k
l
2
;
where k
⋆
¼ argmin
1≤k≤m
ðkψ
↓
m−kþ1∶d
k
2
l
2
=kÞ.
Theorem 2 generalizes the recent result of Johnston et al.
[52], where an explicit formula for the modified trace
distance T
ð1Þ
I
was obtained for all pure states. Notice in
particular that, if all coefficients of jψi satisfy
jψ
i
j ≤ 1=
ffiffiffiffi
m
p
, then k
⋆
¼ m is optimal and we have
kjψik
½m
¼
ffiffiffiffi
m
p
, which means that T
ðm−1Þ
I
ðψÞ reaches its
maximum value m − 1. As a consequence, T
ðm−1Þ
I
does not,
in general, admit a unique maximizer in the form of the
maximally coherent state. In [52], this was considered as a
possible indication that this quantity is not suitable as a
coherence quantifier. In the following, however, we will
instead demonstrate its operational usefulness in the char-
acterization of the fidelity of one-shot coherence
distillation.
Distillation of coherence.—We will denote by Ψ
m
¼
jΨ
m
ihΨ
m
j the m-dimensional maximally coherent state
jΨ
m
i¼
P
m
i¼1
ð1=
ffiffiffiffi
m
p
Þjii in the reference basis. The distil-
lable coherence C
∞
d;IO
ðρÞ is the asymptotic rate at which Ψ
2
can be obtained per copy of a given state ρ via incoherent
operations. Winter and Yang [9] showed that the distillable
coherence of an arbitrary mixed state coincides with the
relative entropy of coherence C
r
ðρÞ ≔ min
σ∈I
DðρkσÞ
introduced in [2], where the quantum relative entropy is
given as DðρkσÞ ≔ Trρðlog ρ − log σÞ with the logarithm
taken in base 2. For any state ρ, the distillable coherence is
then given by C
∞
d;IO
ðρÞ¼C
r
ðρÞ¼S½ΔðρÞ − SðρÞ.
We now consider the nonasymptotic setting. For any
quantum state ρ, the fidelity of coherence distillation under
the class of operations O is defined by
F
O
ðρ;mÞ ≔ max
Λ∈O
hΛðρÞ; Ψ
m
i: ð9Þ
The one-shot ε-error distillable coherence is then defined as
the maximum over all distillation rates achievable under the
given class of operations with an error tolerance of ε, that is,
C
ð1Þ;ε
d;O
ðρÞ ≔ log max fm ∈ NjF
O
ðρ;mÞ ≥ 1 − εg: ð10Þ
As a consequence, the asymptotic distillable coherence can
be given as
C
∞
d;O
ðρÞ¼lim
ε→0
lim
n→∞
1
n
C
ð1Þ;ε
d;O
ðρ
⊗n
Þ: ð11Þ
One of the main results of this Letter is that the one-shot
distillable coherence can be computed exactly, as charac-
terized in the following result.
Theorem 3.—If O ∈ fMIO; DIOg, then, for any state
ρ ∈ D, the fidelity of coherence distillation and one-shot ε-
error distillable coherence can both be written as the
following semidefinite programs
F
O
ðρ;mÞ¼max
hG;ρi
0 ≤ G ≤ 1; ΔðGÞ¼
1
m
1
;
C
ð1Þ;ε
d;O
ðρÞ¼log
max
m
hG;ρi≥ 1 − ε;
0 ≤ G ≤ 1; ΔðGÞ¼
1
m
1
: ð12Þ
The result reveals a fundamental relation between differ-
ent sets of operations in the resource theory of coherence,
showing that MIO and DIO have the same power in the task
of coherence distillation. This correspondence is in fact
surprising: not only is DIO a strict subset of MIO, it is also
known that MIO is strictly more powerful than DIO in state
transformations [19,47], that there exist entropic coherence
monotones under DIO that are not monotones under MIO
[47], and that the two sets can exhibit different operational
capabilities in tasks such as coherence dilution [36].
Furthermore, since MIO constitutes the largest class of
free operations in the resource theory of coherence, the
result is of practical relevance, as it shows that using DIO is
PHYSICAL REVIEW LETTERS 121, 010401 (2018)
010401-3
sufficient to achieve the best rates of distillation achievable
under any class of free operations.
We will now show that the quantities introduced in
Theorem 3 admit alternative characterizations. In particu-
lar, we will express the fidelity of distillation as a measure
related to the family T
ðmÞ
I
introduced before and the one-
shot distillable coherence as a quantum hypothesis testing
problem. To do so, we will need to optimize over a larger
set of matrices than the incoherent states I, namely, the set
J ≔ fXjTrðXÞ¼1; ΔðXÞ¼Xg of unit-trace diagonal
Hermitian matrices and analogously the set J
of unnor-
malized diagonal matrices. We then define the quantities
T
ðmÞ
J
ðρÞ≔ min
X∈J
mTrðρ − XÞ
þ
þTrðρ − XÞ
−
¼maxfhρ;Wij− 1 ≤ W ≤ m1;ΔðWÞ¼0g; ð13Þ
in analogy with the measures T
ðmÞ
I
. Following the proof of
Proposition 1, one can easily see that T
ðmÞ
J
are also faithful
strong monotones under MIO. Let us now consider the
hypothesis testing relative entropy D
ε
H
[25,55], defined as
D
ε
H
ðρkσÞ ≔−log minfhM; σij0 ≤ M ≤ 1; 1 − hM; ρi ≤ εg:
ð14Þ
In the setting of quantum hypothesis testing, one is
interested in distinguishing between two quantum states
(ρ and σ) by performing a test measurement fM; 1 − Mg,
where 0 ≤ M ≤ 1. The probability of incorrectly accepting
state σ as true (type-I error) is given by h1 − M; ρi, and the
probability of incorrectly accepting state ρ as true (type-II
error) is given by hM; σi [56]. The quantity D
ε
H
ðρjjσÞ then
characterizes the minimum type-II error while constraining
the type-I error to be no greater than ε. Alternatively,
D
ε
H
ðρjjσÞ can be viewed as the operator-smoothed version
of min-relative entropy [34,57]. Using this quantity, we can
obtain the following result.
Proposition 2.—The fidelity of coherence distillation
and the one-shot ε-error distillable coherence under O ∈
fMIO; DIOg admit a characterization as the semidefinite
programs
F
O
ðρ;mÞ¼
1
m
½T
ðm−1Þ
J
ðρÞþ1;
C
ð1Þ;ε
d;O
ðρÞ¼min
X∈J
D
ε
H
ðρjjX Þ − δ; ð15Þ
where δ ≥ 0 is the least number such that the solution
corresponds to the logarithm of an integer.
Although the optimization in the above problems is over
matrices that are not necessarily positive semidefinite, one
can show that, if one of the problems admits a positive
semidefinite optimal solution, then so does the other. In the
particular case of m ¼ d, not only does the fidelity of
distillation simplify to an optimization over I, but combin-
ing Proposition 2 with Theorem 1 of Ref. [37] we know
that, in fact, F
O
ðρ;dÞ is the same for any
O ∈ fMIO; DIO; SIO; IOg. However, the case of interest
is when such a property holds for any value of m—we will
now show that this is true for all pure states, significantly
simplifying the computation of the above quantities.
We first notice that each T
ðmÞ
I
provides an upper bound
on the corresponding T
ðmÞ
J
,givingF
MIO
ðρ;mÞ ≤
ð1=mÞ½T
ðm−1Þ
I
ðρÞþ1. To show that this bound is in fact
tight for all pure states, we consider different sets of
operations—SIO as well as IO. Pure-state transformations
under IO and SIO are known to be fully characterized by
majorization relations [9,14,47,58], which allow us to lower
bound the fidelity of distillation and obtain the following
result.
Theorem 4.—For any pure state jψi, any integer m ≥ 1,
and O ∈ fMIO; DIO; SIO; IOg,
F
O
ðψ;mÞ¼
1
m
kjψik
2
½m
: ð16Þ
This extends the operational equivalence between MIO
and DIO in coherence distillation to the strictly smaller set
SIO and has several important consequences. First, it shows
that the one-shot distillable coherence of pure states under
any of the classes of operations O ∈ fMIO; DIO; SIO; IOgis
exactly the same and, in fact, can be expressed as the quantum
hypothesis testing problem C
ð1Þ;ε
d;O
ðψÞ¼min
σ∈I
D
ε
H
ðψjjσÞ −
δ with δ as before. Second, we can use the properties of the
m-distillation norm to obtain exact formulas for the one-shot
distillable coherence. In particular, noting that kjψik
½m
¼
ffiffiffiffi
m
p
[or, equivalently, F
O
ðψ;mÞ¼1] if and only if
kjψik
l
∞
≤ 1=
ffiffiffiffi
m
p
, we see that the zero-error distillable
coherence is given by
C
ð1Þ;0
d;O
ðψÞ¼logbkjψik
−2
l
∞
c: ð17Þ
Relating the m-distillation norm with the fidelity of
distillation also allows us to more easily make quantitative
statements about the distillability of pure states on average.
For example, in Ref. [52], it was shown that the proportion
of pure states with respect to the Haar measure for which
kjψik
½2
¼ 1 is given by 1 − d2
1−d
, which sharply tends to
one as d increases. In light of our results, this then shows
that, with growing dimension, only an exponentially small
fraction of pure states are one-shot undistillable, while a
significant majority of pure states satisfy C
ð1Þ;0
d;O
ðψÞ ≥ 1 and
therefore allow for a zero-error one-shot distillation of at
least one bit of coherence.
The results of our work have important consequences
beyond the one-shot regime, in particular, for the asymp-
totic reversibility of state transformations in the resource
PHYSICAL REVIEW LETTERS 121, 010401 (2018)
010401-4
theory of coherence—that is, the question whether the
amount of coherence that can be distilled from a number of
copies of a state ρ (distillable coherence C
∞
d
) is the same as
the amount of coherence needed to prepare the same
number of copies (coherence cost C
∞
c
) in the asymptotic
limit of an arbitrarily large number of IID copies. It is
known that the resource theory of coherence is reversible
under MIO [9,36], but irreversible under IO as we have
C
∞
d;IO
ðρÞ <C
∞
c;IO
ðρÞ in general [9]. Recently, it has been
claimed that C
∞
c;DIO
ðρÞ¼C
∞
c;MIO
ðρÞ¼C
r
ðρÞ [36], although a
complete proof of this fact did not appear until [40].Our
result in Theorem 3, in particular, shows that C
ð1Þ;ε
d;DIO
ðρÞ¼
C
ð1Þ;ε
d;MIO
ðρÞ and therefore C
∞
d;DIO
ðρÞ¼C
∞
d;MIO
ðρÞ¼C
r
ðρÞ,
complementing the claims of Ref. [36] and strengthening
the asymptotic results of Ref. [40] by showing their appli-
cability even in the one-shot case. The fact that state trans-
formations are indeed reversible under DIO and the maximal
set of operations MIO is not necessary for full reversibility
contrasts with other resource theories such as entanglement,
where the only set of operations known to provide asymptotic
reversibility is strictly larger than the maximal set [59–62].
Conclusions.—We have characterized the operational
task of one-shot coherence distillation for several classes
of free operations, showing in particular that MIO and DIO
have the same power in this task and providing computable
expressions for the rates of distillation in terms of a
quantum hypothesis testing problem. Further, we have
introduced a family of coherence measures and related it
to the achievable fidelity of distillation. By quantifying the
introduced measures exactly on pure states and showing
that they reduce to a class of much simpler vector norms,
we have obtained a full characterization of one-shot
coherence distillation from pure states and established that
in this case all relevant sets of operations are equally useful.
Our Letter unveils several new features of the resource
theory of coherence and contributes to a better under-
standing of the properties of the different sets of free
operations, as well as generalizes and provides an opera-
tional interpretation to several coherence monotones. This
yields further insight on how quantum coherence can be
created and transformed in the realistic setting of finitely
many quantum states available.
The possible applications of one-shot coherence distil-
lation are multifold. Notably, the framework presented
herein can be used to precisely characterize the experi-
mentally feasible rates at which maximally coherent states,
often employed as “currency” in operational tasks, can be
prepared. One such application is randomness extraction
[63], which can be implemented by distilling the coherence
of a quantum state followed by a measurement generating
uniformly random bits. Another promising way of utilizing
one-shot coherence distillation is to enable coherent state
preparation for direct use in quantum key distribution and
quantum algorithms [64–66]. Furthermore, the comparison
of the operational capabilities of different classes of
operations provides, in particular, new insight about the
relatively unexplored class DIO, whose relation with the
so-called thermal operations could find use in the resource
theory of quantum thermodynamics [67,68].
We are grateful to Eric Chitambar, Min-Hsiu Hsieh,
Nathaniel Johnston, Ludovico Lami, Andreas Winter, and
Wei Xie for discussions. B. R. and G. A. acknowledge
financial support from the European Research Council
(ERC) under the Starting Grant GQCOP (Grant
No. 637352). K. F. and X. W. were partly supported by
the Australian Research Council (Grants
No. DP120103776 and No. FT120100449).
Note added.—Recently, we became aware of an indepen-
dent work by E. Chitambar [40], where the author considers
the asymptotic properties of state transformations under
DIO and in particular obtains a different proof that the
asymptotic rate of coherence distillation under MIO and
DIO is the same.
*
bartosz.regula@gmail.com
†
kun.fang-1@student.uts.edu.au
‡
xin.wang-8@student.uts.edu.au
§
gerardo.adesso@nottingham.ac.uk
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