Hierarchy of quantum operations in manipulating coherence and entanglement
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Hierarchy of quantum operations in manipulating coherence
and entanglement
Hayata Yamasaki1,2,3 , Madhav Krishnan Vijayan4 , and Min-Hsiu Hsieh4,5
1 Photon Science Center, Graduate School of Engineering, The University of Tokyo, 7–3–1 Hongo, Bunkyo-ku, Tokyo 113–8656, Japan
2 Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna,
Austria
3 Atominstitut, Technische Universität Wien, Stadionallee 2, 1020 Vienna, Austria
4 Centre for Quantum Software & Information (UTS:QSI), University of Technology Sydney, Sydney NSW, Australia
5 Hon Hai Quantum Computing Research Center, Taipei City, Taiwan
Quantum resource theory under differ- whether in computation [28, 64], communication [66],
arXiv:1912.11049v3 [quant-ph] 22 Jun 2021
ent classes of quantum operations advances or cryptography [45], arise from various inherent prop-
multiperspective understandings of inherent erties of quantum mechanics, such as quantum co-
quantum-mechanical properties, such as quan- herence and quantum entanglement. Quantum re-
tum coherence and quantum entanglement. source theories [14, 34, 35] have grown to be an im-
We establish hierarchies of different opera- portant theoretical framework for quantitative anal-
tions for manipulating coherence and entan- yses of such properties from operational perspectives
glement in distributed settings, where at least using information processing tasks. A resource the-
one of the two spatially separated parties are ory is conventionally defined by specifying a class of
restricted from generating coherence. In these allowed operations as free operations. One way to
settings, we introduce new classes of opera- choose free operations may be to use practical or ex-
tions and also characterize those maximal, i.e., perimental restrictions. For example, the resource
the resource-non-generating operations, pro- theory of entanglement can be defined by consider-
gressing beyond existing studies on incoherent ing a distributed setting for multiple parties with ac-
versions of local operations and classical com- cess only to local operations on each party’s quantum
munication and those of separable operations. system [25, 32, 46]; then, local operations and clas-
The maximal operations admit a semidefinite- sical communication (LOCC) [19, 23, 73] may arise
programming formulation useful for numeri- as a natural candidate for free operations. Entan-
cal algorithms, whereas the existing operations glement serves as a resource for distributed quan-
not. To establish the hierarchies, we prove a tum information processing where spatially separated
sequence of inclusion relations among the op- parties are restricted to LOCC, by enabling quan-
erations by clarifying tasks where separation tum teleportation [5] and allowing for the imple-
of the operations appears. We also demon- mentation of nonlocal operations to the shared sys-
strate an asymptotically non-surviving sepa- tem [73, 75, 76]. Yet importantly, to deepen our
ration of the operations in the hierarchy in understandings of entanglement, it is also crucial to
terms of performance of the task of assisted introduce and exploit larger classes of free opera-
coherence distillation, where a separation in a tions than LOCC, such as separable (SEP) opera-
one-shot scenario vanishes in the asymptotic tions [49, 58] and positive-partial-transpose (PPT)
limit. Our results serve as fundamental ana- operations [50], in analytical and numerical studies
lytical and numerical tools to investigate in- of entanglement [25, 32, 46]. Along with the studies
terplay between coherence and entanglement of entanglement, distributed settings also commonly
under different operations in the resource the- arise in other resource theories, where each party has
ory. a restricted power of manipulating given quantum re-
sources rather than performing arbitrary local opera-
tions, and needs assistance of another party in using
1 Introduction the given resources [10, 37, 43, 52, 59].
Advantages of quantum information processing In this paper, we investigate the distributed set-
over conventional classical information processing, tings that involve two prominent resource theories,
entanglement and coherence [16, 24, 41, 57, 71]. In
Hayata Yamasaki: hayata.yamasaki@gmail.com
particular, in the spirit of studying LOCC, SEP, and
Madhav Krishnan Vijayan: madhavkrishnan.vijayan@uts.edu.au
PPT operations in entanglement theory, we intro-
Min-Hsiu Hsieh: min-hsiu.hsieh@foxconn.com
duce and study different natural classes of operations
Accepted in Quantum 2021-06-15, click title to verify. Published under CC-BY 4.0. 1in the distributed settings of manipulating coherence A B
and entanglement, and compare their relative power
in performing information theoretic tasks. Coher- CC
ence, i.e., superposition of a certain set of quantum
states, has been shown to play important roles in
quantum biology [33], quantum thermodynamics [27] AB
and photonic experiments [7], where certain states are
Quantum
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easier to create than their superposition. The re-
source theory of coherence [56] is a well-established or Incoherent
resource theory that is useful for introducing classi- Incoherent
fications, partial orders, and quantifications of quan-
tum coherence. The resource theory of coherence con- Figure 1: Distributed manipulation of coherence and entan-
glement of a quantum state ψ AB shared between two spe-
siders situations where coherence cannot be created
cially separated parties A and B. The parties manipulate
on a quantum system due to a restriction of opera- their local quantum systems, while they can use classical
tions for manipulating the system. The free states communication (CC). Local operations and classical commu-
in the resource theory of coherence are states rep- nication with one party B restricted to incoherent operations
resented as diagonal density operators in some fixed are called LQICC, which can be regarded as a client-server
basis. As is the case of entanglement, several different setting where the ability of the client B to generate coher-
free operations that preserve diagonal density opera- ence is restricted while the server A can perform any local
tors have been well investigated, such as incoherent quantum operation to assist B. Local operations and classi-
operations (IO) [4, 67], maximally incoherent oper- cal communication with both parties A and B restricted to
ations (MIO) [47, 48], strictly incoherent operations incoherent operations are called LICC, where the abilities of
(SIO) [67, 71], and physically incoherent operations A and B are the same.
(PIO) [12, 13], to name a few. The resource theory of
coherence in the distributed settings has also attracted
attentions of broad interests [16, 24, 41, 57, 71], as lem, more general classes of operations than LOCC,
with the distributed settings in other resource theo- such as separable operations and PPT operations,
ries. These resource theories for the distributed ma- are vital to investigating performances of information
nipulation of coherence provide a framework for inves- processing tasks, which also yields bounds of the per-
tigating an interplay between coherence and entangle- formance under LOCC. Especially, PPT operations
ment in various information processing tasks such as provide numerical algorithms for calculating perfor-
distillation and dilution of these resources [16], as- mance of the entanglement-assisted tasks by means
sisted distillation of coherence [10, 52, 59], quantum of semidefinite programming (SDP) [65], even if the
state merging [55], quantum state redistribution [2], corresponding tasks under LOCC are hard to analyze
and multipartite state transformation [9, 39]. From due to its mathematical structure [26, 50, 53, 60–63].
a practical perspective, the distributed manipulation Similarly, in the resource theory of coherence, MIO
of coherence naturally arises in photonic systems as serves as a class of operations beyond IO, and MIO
demonstrated in recent experiments [68–70]. provides numerical algorithms based on SDP simi-
larly to PPT operations [51]. Importantly, even if the
In the distributed settings of manipulating coher- operations such as SEP, PPT, and MIO are defined
ence, especially for two parties A and B, LOCC with mathematically, these different classes of operations
one party restricted to IO are called LQICC, and provide efficiently calculable bounds in analyzing the
those with both parties restricted to IO are LICC [57], information processing tasks and crucially help us to
as depicted in Fig. 1. LQICC can be regarded as a understand the properties of resources in the study
client-server setting where the client’s ability to gen- of the resource theories. In the same way, in our dis-
erate coherence is restricted, while the abilities of two tributed settings of manipulating coherence and en-
parties in LICC are the same. The set of free states tanglement, we may suffer from the difficulty if the
for LQICC, that is, the states that can be obtained operations are restricted to LQICC and LICC. To
from any initial state by LQICC, yields the set of circumvent this difficulty, Ref. [57] introduced a class
quantum-incoherent (QI) states, which is incoherent of operations generalizing LQICC by considering SEP
on one of the parties. The set of free states for LICC with one party restricted to IO, called SQI, and that
is incoherent states on both of the two parties. generalizing LICC by considering SEP with both par-
However, to obtain deeper understandings, it is cru- ties restricted to IO, called SI. However, SQI and SI
cial to introduce and compare different classes of op- are insufficient for providing numerically tractable al-
erations beyond LQICC and LICC. For example, in gorithms in these resource theories, in contrast with
the entanglement theory, LOCC may be a conven- PPT operations for entanglement and MIO for coher-
tional and well-motivated choice of free operations, ence.
but the set of LOCC is mathematically difficult to Progressing beyond the above previous research on
characterize and analyze. To circumvent this prob- LQICC, LICC, SQI, and SI, we here introduce and an-
Accepted in Quantum 2021-06-15, click title to verify. Published under CC-BY 4.0. 2the distributed settings. In the entanglement theory,
QIP = CQIP
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MIO
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it has been a crucial question when the difference
QIP \ PPT
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MIO \ PPT
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between LOCC, SEP, and PPT operations appears
in the performance of achieving tasks, such as local
QIP \ SEP
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MIO \ SEP
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state discrimination [3, 6, 15, 18, 22, 77] and entan-
SQI SI
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glement manipulation [1, 20, 26, 50, 53, 60–63]. Also
in the resource theory of coherence, the difference be-
AAACL3icbVDLSsNAFJ34rNVqq+7cBIvgqiQqPnYFXeiuRfuAJpTJdNIOnWTCzI1QQn/Drf6C/6FrcSNu/QvzKGKsBwYO597LOXOcgDMFhvGuLSwuLa+sFtaK6xulza1yZbutRCgJbRHBhew6WFHOfNoCBpx2A0mx53DaccaXybxzT6Viwr+DSUBtDw995jKCIZYsy8MwUm5027yZ9stVo2ak0OeJOSPV+u5LgtdGv6KVrIEgoUd9IBwr1TONAOwIS2CE02nRChUNMBnjIe3F1MceVXaUhp7qB7Ey0F0h4+eDnqq/LyLsKTXxnHgzDfl3loj/zXohuOd2xPwgBOqTzMgNuQ5CTxrQB0xSAnwSE0wki7PqZIQlJhD3lHNxJB5TyP0jSvxACK7ysuN4WYEXCU5/6pon7aOaeVw7aZrV+hXKUEB7aB8dIhOdoTq6Rg3UQgQF6AE9oiftWXvTPrTPbHVBm93soBy0r2+K7q9T
LQICC
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LICC
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tween MIO, IO, and other possible restricted opera-
tions has been investigated for tasks such as coher-
ence manipulation [11, 36, 51, 67]. As for our case of
distributed manipulation of coherence and entangle-
A: Quantum A: Incoherent
ment, we identify tasks where differences of the opera-
B: Incoherent B: Incoherent tions in the hierarchy appear, leading to a proof of the
strict separation of the newly introduced operations.
Figure 2: A hierarchy of operations for distributed manip-
ulation of coherence and entanglement over two parties A In contrast with these separations of the operations
and B beyond LQICC and SQI on the left, and that beyond in the hierarchies, we also demonstrate an asymptot-
LICC and SI on the right. The classes of operations that ically non-surviving separation of these operations in
we introduce and characterize in this paper are shown by red terms of performance of a task. In the context of
and black bold circles, while the existing classes are black when the difference between different classes of oper-
dotted circles. Especially, the red bold circles represent the ations arises, the asymptotically non-surviving sep-
classes of operations that have characterizations in terms of aration refers to a phenomena that the difference
semidefinite programming (SDP), and hence can be used for
in the performance of a task in a one-shot scenario
numerical algorithms based on SDP.
disappears in the corresponding asymptotic scenario.
An asymptotically non-surviving separation has been
alyze several classes of operations illustrated in Fig. 2, recently discovered in Ref. [74] in a communication
which are also summarized in Table 1. As a general- task of quantum state merging [29, 30, 72]. Refer-
ization of LQICC and SQI to a class of operations ence [74] discloses this phenomena by proving the dif-
that transform any QI state into a QI state [40], we ference in the required amount of entanglement re-
consider the QI-preserving (QIP) operations. As for sources in achieving one-shot quantum state merging
the generalization of LICC and SI as a counterpart under one-way and two-way LOCC, while this differ-
of QIP, we use MIO that preserves incoherent states ence ceases to exist in the conventional asymptotic
on both of the two party. We show that no inclu- quantum state merging. In contrast with this com-
sion relation holds between QIP and MIO on two munication task, we here consider another resource-
parties, which contrasts with the fact that LQICC theoretic task that is known as assisted distillation
includes LICC and SQI includes SI. The QIP is of coherence [10, 52, 59]. We demonstrate the differ-
analogous to separability-preserving (SEPP) opera- ence in the maximal amount of distillable coherence
tions [8] in entanglement theory, which transforms with assistance under QIP and one-way LQICC in a
any separable state into a separable state rather than one-shot scenario, using the SDP calculation that we
QI states. In entanglement theory, separable opera- establish in this paper, and at the same time prove
tions, which preserve the separability even if applied the coincidence of them in the corresponding asymp-
to a subsystem, are known to be a strict subclass of totic scenario. This demonstration finds an applica-
separability-preserving operations [20]. In contrast, tion of the SDP-based numerical technique that we
considering a seemingly smaller class of operations introduce for investigating manipulation of coherence
that is QI-preserving even if applied to a subsystem, and entanglement, which by itself has wide applica-
which we define as completely QI-preserving (CQIP) bility beyond the assisted distillation of coherence.
operations, we prove that CQIP actually coincides Consequently, our results establish essential ana-
with QIP exactly. We introduce the new classes of lytical and numerical tools for investigating resource
operations shown in Fig. 2 by taking intersection of theories for manipulating coherence and entanglement
different sets of operations. These generalized classes in the distributed settings, by introducing different
of operations beyond LQICC, SQI, LICC, ans SI are classes of operations and clarifying relative powers
advantageous in analyzing tasks because we can char- of the operations. The significance of our results is
acterize some of them by SDP, as shown in Fig. 2, to that the operations that we prove to have different
provide SDP-based numerical algorithms for analyz- powers are useful for bounding performances of infor-
ing resource theories for distributed manipulation of mation processing tasks performed by the distributed
coherence and entanglement. manipulation of coherence and entanglement. The
Using these operations, we establish hierarchies results are also of fundamental importance in clari-
(Fig. 2) of operations in the resource-theoretic frame- fying similarity and difference between the manipula-
work of manipulating coherence and entanglement in tion of entanglement in entanglement theory and the
Accepted in Quantum 2021-06-15, click title to verify. Published under CC-BY 4.0. 3Table 1: Summary of definitions and examples of operations in resource theories for distributed manipulation of coherence and
entanglement. The examples in the table lead to our results on the strict inclusion relations among the operations shown in
Fig. 2.
Set of operations Definition Examples that lead to our results on separation
QIP (Definition 5. See also (13) (Proposition 20) An operation on A and B achieving the
for the definition of QI task of one-shot assisted distillation of coherence given
states shared between A and in Proposition 20.
B.) Operations (CPTP linear
maps) on A and B that map
any QI state shared between
A and B to a QI state.
QIP ∩ PPT Operations in the intersection (Proposition 18) A map implemented by a PPT mea-
of QIP and PPT. surement with B outputting a classical state represent-
ing the measurement outcome.
QIP ∩ SEP Operations in the intersection (Proposition 16) MIO on B with A having a one-
of QIP and SEP. dimensional system.
SQI (Definition 7) SEP where B’s (Proof of Theorem 14, Proposition 17) A map imple-
Kraus operators satisfy the mented by a separable measurement with B outputting
condition (10) of IO. a classical state representing the measurement outcome.
LQICC (Definition 6) LOCC with re- (Proof of Theorem 14) A map implemented by an LOCC
striction on B’s operations to measurement with B outputting a classical state repre-
IO. senting the measurement outcome.
(Proposition 16) IO on B with A having a one-
dimensional system.
MIO (Definition 5. See also (14) (Proposition 18) A SWAP unitary operation between A
for the definition of incoher- and B.
ent states shared between A
and B.) Operations (CPTP
linear maps) on A and B that
map any incoherent state
shared between A and B to
an incoherent state.
MIO ∩ PPT Operations in the intersection (Proposition 17) A map implemented by a PPT mea-
of MIO and PPT. surement with both A and B outputting a classical state
representing the measurement outcome.
MIO ∩ SEP Operations in the intersection (Proposition 16) MIO on one of the parties A and B
of MIO and SEP. with the other party having a one-dimensional system.
SI (Definition 7) SEP where (Proof of Theorem 14, Proposition 17) A map imple-
both A and B’s Kraus opera- mented by a separable measurement with both A and
tors satisfy the condition (10) B outputting a classical state representing the measure-
of IO. ment outcome.
LICC (Definition 6) LOCC with re- (Proof of Theorem 14) A map implemented by an LOCC
striction on A and B’s opera- measurement with both A and B outputting a classical
tions to IO. state representing the measurement outcome.
(Proposition 16) IO on one of the parties A and B with
the other party having a one-dimensional system.
Accepted in Quantum 2021-06-15, click title to verify. Published under CC-BY 4.0. 4distributed manipulation of coherence in this paper. 2.1 Operations in resource theories of entan-
This will be useful for better understanding the inter- glement
play between coherence and entanglement from the
operational viewpoint, based on resource theories un- We review here definitions of operations that are im-
der different operations. portant in the resource theory of entanglement [32].
We recall the definition of LOCC as follows.
The rest of this paper is organized as follows. In
Sec. 2, we recall operations that are known in resource Definition 1 (LOCC). Local operations with classical
theories of entanglement, coherence, and distributed communication or LOCC can be defined for a bipartite
manipulation of coherence and entanglement. In system as follows: if a completely positive and trace-
Sec. 3, we introduce QI-preserving operations and preserving (CPTP) map E AB has a Kraus decomposi-
prove the equivalence between QI-preserving opera- tion using local operations of measurements by parties
tions and completely QI-preserving operations. In A and B along with classical communication of the
Sec. 4, we establish a hierarchy of operations beyond measurement outcomes between them, then the map
LQICC and SQI and that beyond LICC and SI, clar- represents LOCC. The set of r-round LOCC maps, de-
ifying tasks that separate the power of these oper- noted as r-LOCC, consists of an LOCC map that can
ations. In Sec. 5, we demonstrate the asymptoti- be implemented using r rounds of classical communi-
cally non-surviving separation between LQICC and cation, where the case of r = 0 yields local operations
QI-preserving operations. Our conclusion is given in (without classical communication). Note that it can
Sec. 6. be shown that r-LOCC is a strict subset of (r + 1)-
LOCC [19]. LOCC can be thought to be a physical
operation in the resource theory of entanglement in
the sense that it describes all the allowed space-like
separated operations that the two classically commu-
nicating parties can perform. For a more rigorous
2 Preliminaries definition, see Refs. [19, 23, 73].
Separable operations include LOCC as special
As we are interested in manipulation of coherence and cases. While there can be separable operations that
entanglement in distributed settings, the first ques- are not LOCC, separable operations are often used as
tion to ask is what are the allowed operations in such a relaxation of LOCC since separable operations can
settings. The choices of allowed operations for spa- be mathematically easier to characterize than LOCC
tially separated parties A and B, e.g., local operations itself.
with classical communication (LOCC), separable op-
Definition 2 (SEP). A CPTP map E AB represents
erations (SEP) and positive partial transpose maps
separable operations if and only if there exists a
(PPT), have been well explored in entanglement the-
Kraus decomposition of E AB where Kraus operators
ory [32]. Similarly there are several candidates for
are product operators, i.e.,
allowed operations that do not create coherence in a
fixed reference basis [56]. We will focus our attention X †
E AB ρAB = MiA ⊗ KiB ρAB MiA ⊗ KiB ,
on two prominent ones, namely, incoherent operations i
(IO) and maximally incoherent operations (MIO). We (1)
will now define these operations formally and discuss
B †
= 1AB ,
X
MiA ⊗ Ki MiA ⊗ KiB
how they can be combined to investigate distributed (2)
manipulation of coherence and entanglement. i
In this paper, we consider quantum systems repre- where 1AB is the identity operator on the full space of
sented as finite-dimensional Hilbert spaces, which are A and B. The set of separable operations is denoted
0 0
written as HA ⊗HA ⊗· · · on A and HB ⊗HB ⊗· · · on by SEP. The separable measurement refers to a mea-
B. We fix an orthonormal basis on A and B, which is surement
A represented by the measurement operators
A
called a reference basis and denoted by {|ii }i=0,...,DA Mi ⊗ KiB i satisfying the above condition of the
B
of HA and {|ji }j=0,...,DB of HB , where we write the separable maps. The set LOCC is a strict subset of
the set of SEP operations [19].
dimension of HA and HB as DA and DB , respec-
tively. We may use superscripts of bras and kets to PPT operations include separable operations and
represent the quantum system to which the bras and LOCC as special cases. PPT operations may gener-
AB
kets belong, such as |ψi ∈ HA ⊗ HB . Superscripts ate PPT entangled states from separable states, while
of operators and maps represent the quantum system separable operations and LOCC do not. However,
on which they act, such as E AB for a linear map of PPT operations are useful for analyzing tasks since
0
operators on HA ⊗ HB and E A→A for that from HA the condition (4) in the following can be used for
to HA . The identity operator is denoted by 1, and
0
numerical algorithms based on semidefinite program-
the identity map is id. ming, unlike separable operations and LOCC.
Accepted in Quantum 2021-06-15, click title to verify. Published under CC-BY 4.0. 5Definition 3 (PPT). A CPTP map E AB represents where I denotes the set of incoherent
P states, and the
a PPT operation if and only if the map trace-preserving condition yields i Ki† B KiB = 1.
Equivalently, these Kraus operators KiB i for inco-
idA ⊗T B ◦ E AB ◦ idA ⊗T B (3) herent operations can be written in the form of
is completely positive, where idA is the identity map
X B
KiB := ci (j) |fi (j)i hj| , ∀i, (10)
on A, and T B is the transpose map on B with respect j
to the reference basis. Or equivalently, a CPTP map
E AB is a PPT map if and only if the Choi operator where fi is a function (which is not necessarily bi-
A0 B 0 AB
J (E) [65] satisfies a condition [50] jective), and ci (j) ∈ C for each j [4]. The set of
TA0 A
incoherent operations is denoted by IO.
J (E) = 0, (4)
While IO clearly preserves diagonal states in the ref-
where TA0 A is the partial transpose on A0 A with
erence basis, it is not the maximal set of CPTP maps
respect to the reference basis. A PPT measure-
that do so. In a quantum resource theory, the maxi-
ment [3, 22, 77, 78] can be
represented by a family mal set of operations refers to the largest possible set
of measurement operators KiAB i satisfying
of operations that do not generate any resource from
free states [14]. We now define this maximal set, that
Ki AB KiAB = 1,
X †
(5)
is, maximally incoherent operations MIO. Notice that
i
TA unlike IO, MIO may create coherence from an inco-
Ki† AB KiAB = 0, ∀i. (6) herent state probabilistically by post-selecting a mea-
surement outcome corresponding to a Kraus operator
Note
AB that for any PPT measurement given by that implements MIO.
Ki i
, a CPTP map implemented by this measure-
ment X AB AB † AB Definition 5 (MIO). A CPTP map E represents a
E AB ρAB = Ki ρ Ki (7) maximally incoherent operation (MIO) if and only if
i E maps any incoherent state into an incoherent state
is a PPT map. The set SEP is a strict subset of deterministically [48], that is, for any ρ ∈ I,
PPT [19].
E (ρ) ∈ I, (11)
2.2 Operations in resource theories of coher-
where I denotes the set of incoherent states.
ence
Free operations in the resource theory of coherence 2.3 Operations in resource theories of dis-
are ones that map diagonal states to diagonal states
in a particular reference basis [56], so that the diago-
tributed manipulation of coherence and entan-
nal states can be free states for the free operations in glement
the resource theory of coherence. Recall the reference In the two-party distributed settings for manipulating
A B
bases {|ii }i of HA and {|ji }j of HB . Then, the coherence and entanglement, we can look at two possi-
reference basis of a bipartite system HA ⊗ HB is the ble settings, namely, (i) a quantum-incoherent setting
product reference bases of the subsystems. The set of where one party is restricted to use incoherent opera-
incoherent states of HA is given by tions, and (ii) an incoherent-incoherent setting where
(
X
) both parties are restricted to incoherent operations,
A
p (i) |ii hi| , (8) as depicted in Fig. 1. We now recall the extensions
i of LOCC to the quantum-incoherent and incoherent-
where p is a probability distribution, and the set of incoherent settings.
incoherent states of HB can be given in the same way. Definition 6 (LQICC and LICC). The set of
We recall that the class of incoherent operations IO LQICC [57] is defined in the same way as LOCC, with
refers to those which cannot create coherent states restriction on B’s operations to incoherent operations.
from incoherent states even with post-selection. The set of LICC [57] is also defined in the same way
Definition 4 (IO). A CPTP map E represents an as LOCC, with restriction on both A and B’s opera-
incoherent operation if and only if there exists a Kraus tions to incoherent operations. Corresponding to the
decomposition satisfying r-round LOCC, we define r-LQICC and r-LICC by re-
placing LOCC with r-LOCC in the above definitions,
Ki ρKi† ,
X
E (ρ) := respectively. One-way LQICC refers to operations
i represented as CPTP maps by A and B consisting
(9)
Ki ρKi† of A’s (arbitrary quantum) measurement represented
∀i, ρ ∈ I ⇒ ∈ I, p (i) := Tr Ki ρKi† ,
p (i) by a quantum instrument EiA i with the outcome i,
Accepted in Quantum 2021-06-15, click title to verify. Published under CC-BY 4.0. 6followed by B’s incoherent operation ẼiB ∈ IO condi- While these operations combine operations in the
tioned on i, that is, CPTP maps in the form of resource theory of entanglement and that of coher-
X ence, they are not necessarily equivalent to the inter-
E 1-LQICC := EiA ⊗ ẼiB , (12) section of the sets of two different operations in these
i resource theories. We remark the subtleties of taking
which we may also write as 1-LQICC because one-way the intersection of two operations in the following.
LQICC is composed of one round of classical commu- Remark 1 (Difference between the set intersection of
nication. One-way LICC is defined in a similar way CPTP maps and the operational intersection). It is
to the the definition of one-way LQICC using incoher- not straightforward to consider intersection of two dif-
ent operations on both A and B. Note that we always ferent sets of operations because of the fact that a
consider one-way classical communication from A to CPTP map may have different Kraus operators {Mi }i
B in one-way LQICC. and {Kj }j that are related by a unitary operator
u = (ui,j )i,j as [65]
The set of free states for LQICC in the quantum-
incoherent setting are quantum-incoherent (QI) X
states [10]. A bipartite state ρAB is called a QI state Kj = ui,j Mi . (17)
if and only if ρAB is in the form of i
X B For example, consider a CPTP map
ρAB = p (j) ρA
j ⊗ |ji hj| , (13)
j 1
Mi ρMi† ,
X
E AB ρAB :=
B
(18)
where {|ji } is the incoherent basis of HB, p (j) is a i=0
probability distribution, and ρA j ∈ D H
A
. The set
of QI states is denoted by QI. Note that QI states are where
different from, but a special case of, quantum-classical
M0 := |+i h+| ⊗ 1B ,
A
(QC) states, i.e., states that have zero quantum dis- (19)
A B
cord [42], since the reference basis of B’s diagonal part M1 := |−i h−| ⊗ Z , (20)
is fixed for QI states but not for QC states. The set
I of incoherent states of a bipartite quantum system which is a one-way LOCC map that can be imple-
AB refers to that of states which are incoherent on mented by A’s measurement represented by Kraus
both A and B, i.e. operators as {|+i h+| , |−i h−|}, followed by B’s cor-
rection 1B or Z B conditioned on A’s outcome. At the
X
A B
same time, this CPTP map E AB is in IO on AB be-
I= p (i, j) |ii hi| ⊗ |ji hj| , (14) cause E AB can also be regarded using a different set
i,j of Kraus operators as
where p (i, j) is a probability distribution. These are 1
Ki ρKi† ,
X
the free states for LICC in the incoherent-incoherent E AB ρAB :=
(21)
setting. i=0
We now define a larger set of operations by consid- 1A ⊗ |0i h0|B + X A ⊗ |1i h1|B
ering separable operations in conjunction with inco- K0 := √ , (22)
2
herent operations.
X A ⊗ |0i h0| + 1A ⊗ |1i h1|
B B
Definition 7 (SQI and SI). In analogy to separable K1 := √ , (23)
2
maps, define SQI [57] to be a class of CPTP maps by
A and B as which is an incoherent map that can be implemented
X † by a nonlocal incoherent measurement on AB. How-
E SQI ρAB := MiA ⊗ KiB ρAB MiA ⊗ KiB ,
ever, it is unclear whether E is in LICC in the sense
i
that there exists an implementation by local incoher-
(15)
ent operations and classical communication. It is not
where MiA ⊗ KiB i is a family of Kraus operators
straightforward to present Kraus operators of E that
satisfying
simultaneously satisfy the conditions of LOCC and IO.
X
MiA ⊗ KiB
†
MiA ⊗ KiB = 1,
(16) We propose that a sensible way to resolve this is to
i
consider a subset of the intersection of two sets of
CPTP maps, so that CPTP maps in this subset can
and for each i, KiB satisfies the condition (10) of in- be implemented by Kraus operators satisfying condi-
coherent operations. The class SI [57] is defined in tions of both of the original two simultaneously; e.g.,
the same way as SQI where not only KiB but both for LOCC and IO, the Kraus operators are both local
MiA and KiB satisfy the condition (10) of incoherent and incoherent, which we name the operational inter-
operations. section of LOCC and IO.
Accepted in Quantum 2021-06-15, click title to verify. Published under CC-BY 4.0. 73 QI-preserving maps following refers to the class of operations in Defini-
tion 5 with the set of incoherent states I given by (14),
In this section, we investigate a class of operations which is incoherent on both A and B. We have, by
that preserve QI states, which we name QI-preserving definition, an inclusion relation between LQICC and
operations, or QIP for short. This QIP is a generaliza- LICC, and the same inclusion between SQI and SI,
tion of LQICC and SQI as we will investigate further that is,
in the next section. As for a generalization of LICC
and SI, we use MIO that preserves incoherent state LICC $ LQICC,
shared between A and B. We show that no inclusion SI $ SQI. (25)
relation holds between QIP and MIO.
The QIP is analogous to the separability-preserving In contrast, QIP and MIO do not have this inclusion
operations (SEPP) [8] in the entanglement theory, relation as we show in the following, which illumi-
i.e., operations that preserve separable states. It is nates the difference between QIP and these previously
known that SEPP is different from separable (SEP) known classes of operations.
operations, because the SWAP operation on a bi-
partite system preserves separability of the bipartite Proposition 9. It holds that
state, but the SWAP is not separable [20]. In the
QIP 6⊂ MIO, (26)
context of the separability-preserving property, SEP
is characterized as a strict subclass of separability- MIO 6⊂ QIP. (27)
preserving operations because SEP preserves separa-
Proof. We prove (26) and (27) by showing instances.
bility even if applied to a subsystem; that is, a CPTP
We have (26) by definition. In particular, a QI-
map E AB is separable if and only if for any identity
0 0 0 0 preserving map can transform an incoherent state
map idA B on a bipartite auxiliary system HA ⊗HB , A B A B
|0i ⊗ |0i into a QI state |+i ⊗ |0i , which is not
0 0
idA B ⊗E AB is separability-preserving. Similarly, a incoherent on A, and hence, this map is not maxi-
completely positive map is also characterized as a spe- mally incoherent in the sense that an incoherent state
cial subclass of positive maps that preserve positivity on the two parties is transformed into a state that is
even if applied to a subsystem [65]. Based on these not in I defined as (14).
observations, we can consider a seemingly special class To show (27), we construct a nonlocal IO that is
of QIP that preserves QI states even if applied to not QI-preserving, while IO is included in MIO by
a subsystem, which we call completely QI-preserving definition. Consider the following IO
(CQIP) operations. However, we show that QIP and
CQIP are the same set of operations, which leads to an 1
KiAB ρAB Ki† AB ,
X
essential difference between the entanglement theory E AB ρAB = (28)
and the resource theory of distributed manipulation i=0
of coherence and entanglement. In addition, the tech- where
niques for showing the equivalence of QIP and CQIP
A B A B
yield a characterization of QIP that can be used for K0AB = |0i h0| ⊗ |0i h0| + |1i h1| ⊗ |1i h0| , (29)
numerical algorithms based on semidefinite program- A B A B
K1AB = |0i h0| ⊗ |0i h1| + |1i h1| ⊗ |1i h1| . (30)
ming (SDP).
We define QI-preserving operations as a class of bi- A B A B A B
This is IO since |0i ⊗ |0i , |0i ⊗ |1i , |1i ⊗ |0i ,
partite operations that transforms any QI state into A B
a QI state, which generalizes LQICC and SQI over and |1i ⊗|1i are transformed into incoherent states
the two parties. Note that this class of operations is by these Kraus operators {K0 , K1 }. However, in-
A B
first considered in Ref. [40] in the context of study- putting a QI state |+i ⊗ |0i to this IO, we obtain
ing discord and coherence, but our key contributions an entangled state, that is, a non-QI state
in this paper are to clarify the relation between QIP
A B
AB
and the other possible classes of operations in the re- E AB |+i h+| ⊗ |0i h0| = Φ+ Φ+ , (31)
source theory of manipulating coherence and entan-
glement, and to provide the SDP-based characteriza- where
tion of QIP. The formal definition of QIP is as follows. 1 A B A B
Φ+ = √ |0i ⊗ |0i + |1i ⊗ |1i . (32)
Definition 8 (QIP). A CPTP map E AB is said to be 2
QI-preserving if and only if for any QI state ρAB ∈
Thus, E AB is not QI-preserving. Q.E.D.
QI,
E AB ρAB ∈ QI.
(24) We also define completely QI-preserving operations
The set of QI-preserving maps is denoted by QIP. as a class of operations over two parties that is QI-
preserving even if the operation is applied to subsys-
We can regard MIO over the two parties as a gen- tems shared between the two parties. The formal def-
eralization of LICC and SI. In particular, MIO in the inition is as follows.
Accepted in Quantum 2021-06-15, click title to verify. Published under CC-BY 4.0. 8Definition 10 (CQIP). A CPTP map E AB is said where the last inclusion follows from the convexity of
to be completely QI-preserving if and only if for an the set of QI states QI. Using the convexity in the
0 0
identity map idA B on any shared auxiliary system same way, for any QI state ρAB in the form of (35),
0 0 0 0
HA ⊗ HB , the map idA B ⊗E AB is QI-preserving. we obtain
The set of completely QI-preserving maps is denoted
E AB ρAB
by CQIP. (39)
(j)
X B
To show the equivalence of QIP and CQIP, we = p (j) ca,b E AB ρA
a,b ⊗ |ji hj| ∈ QI, (40)
characterize QI-preserving operations using a finite a,b,j
set of equations that can also be used for numeri-
cal algorithms based on SDP. Define a density op- which yields the conclusion. Q.E.D.
erator of a DA -dimensional system A for any a, b ∈
{0, . . . , DA − 1} As for completely QI-preserving operations, we also
A
show a characterization of completely QI-preserving
|ai ha|
if a = b, operations similarly to Proposition 11 on QIP.
A 1 A
ρa,b := 2 (|ai + |bi) (ha| + hb|) if a < b, (33)
1 A Proposition 12 (Characterization of completely
2 (|ai + i |bi) (ha| − i hb|) if a > b.
QI-preserving operations). A CPTP map E AB is com-
pletely QI-preserving if and only if for any j ∈
n o
These density operators ρA a,b serve as a basis
a,b {0, . . . , DB − 1}, it holds that
spanning the set of all the operators of this DA -
dimensional system A [65]. Then, we show the fol- A00 A
00
B
idA ⊗E AB |ΦDA i hΦDA | ⊗ |ji hj| ∈ QI,
lowing characterization of QI-preserving maps. Note
that it is straightforward to use the condition (34) in (41)
the following proposition as a linear constraint of Choi where QI in this proposition means the set of QI
operators of QI-preserving operations in SDP, as we states that are quantum on A00 A and incoherent on B,
will demonstrate in the proof of Proposition 20. and |ΦDA i is a maximally entangled state of Schmidt
rank DA
Proposition 11 (Characterization of QI-preserving
operations). A CPTP map E AB is QI-preserving if 1
DXA −1
A00 A A00 A
and only if for any a, b ∈ {0, . . . , DA − 1} and any |ΦDA i := √ |di ⊗ |di . (42)
DA d=0
j ∈ {0, . . . , DB − 1}, it holds that
.
B
E AB ρA
a,b ⊗ |ji hj| ∈ QI, (34)
Proof. Any completely QI-preserving map satisfies
where ρa,b is defined as (33). the condition (41) by definition, and hence it suf-
fices to show the converse; that is, we prove that
Proof. Any QI-preserving map satisfies the condi- 00
for any CPTP map E AB , if idA ⊗E AB satisfies (41),
tion (24) by definition, and hence it suffices to show 0 0 0 0
the converse; that is, we prove that any CPTP map then idA B ⊗E AB for any auxiliary system HA ⊗HB
satisfying (24) transforms any QI state into a QI state. transforms any QI state into a QI state. We write the
Consider any QI state QI state obtained from (41) as
A00 A
X 00
B B
ρAB = p (j) ρA
j ⊗ |ji hj| . (35) idA ⊗E AB |ΦDA i hΦDA | ⊗ |ji hj|
j
(j) 00
X B
= q (j) (k) σk A A ⊗ |ki hk| . (43)
Using density operators (33) that serve as a basis of k
HA , we can write ρA
j for each j in (35) as
Consider any QI state
(j)
X
ρA
j = ca,b ρA
a,b , (36)
0 0 0 B0 B
X
a,b ρA AB B
= p (j, j 0 ) ρA A 0 0
j,j 0 ⊗ |j ji hj j| , (44)
j,j 0
(j)
where ca,b ∈ C for each a, b is a coefficient in this
linear expansion of ρA A00 A
j . Then, due to the linearity of where we write |j 0 ji = |j 0 i ⊗ |ji. Since |ΦDA i is
AB
E , we have for each j a maximally entangled state, for each j and j 0 , there
A00 →A0
exists a CPTP linear map Ej,j 0 from A00 to A0 that
B
E AB ρAj ⊗ |ji hj| (37) A00 A 0
transforms |ΦDA i into ρA A
j,j 0 , that is,
X (j)
B
= ca,b E AB ρA a,b ⊗ |ji hj| ∈ QI, (38)
A00 A
00 0
0
A →A
a,b Ej,j 0 ⊗ idA |ΦDA i hΦDA | = ρA A
j,j 0 . (45)
Accepted in Quantum 2021-06-15, click title to verify. Published under CC-BY 4.0. 9Then, we obtain 4 Hierarchy of operations for dis-
tributed manipulation of coherence and
0 0 0 0
idA B ⊗E AB ρA AB B
X 0 0 0 0
0 B B
entanglement
= p (j, j 0 ) idA B ⊗E AB ρj,j
AA 0
0 ⊗ |j ji hj j|
j,j 0 In this section, we establish hierarchies of operations
A00 A for distributed manipulation of coherence and entan-
00 0
B0
X
= p (j, j 0 ) Ej,j
A →A
0 ⊗ id ⊗E AB
|Φ DA i hΦ DA |
j,j 0 glement, as depicted in Fig. 2. In particular, we prove
B 0
B
strict inclusion relations among the classes of opera-
⊗ |j 0 ji hj 0 j| tions that include LQICC and SQI, or LICC and SI,
X
0 (j) as special cases.
= p (j, j ) q (k)
j,j 0 ,k
We prove the following theorem establishing the hi-
h 00 0 i 0 erarchies of operations. The proof of this theorem
A →A (j) 00 B B
Ej,j 0 ⊗ idA σk A A ⊗ |j 0 ki hj 0 k| . is based on propositions that we will prove later in
(46) this section for showing each strict inclusion in (53)
and (54) in the theorem.
Since the state in the last line is a QI state, we obtain
the conclusion. Q.E.D. Theorem 14 (Hierarchy of operations in distributed
manipulation of coherence and entanglement). It
Using the characterization of QIP and CQIP shown holds that
in Propositions 11 and 12, we prove the equivalence
of QIP and CQIP as follows. 1-LQICC $ LQICC $ SQI
(53)
Theorem 13 (Equivalence of QI-preserving opera- $ (QIP ∩ SEP) $ (QIP ∩ PPT) $ QIP,
tions and completely QI-preserving operations). It 1-LICC $ LICC $ SI
holds that (54)
$ (MIO ∩ SEP) $ (MIO ∩ PPT) $ MIO,
CQIP = QIP. (47)
Proof. It is trivial by definition that any CQIP map where MIO is the set of maximally incoherent opera-
is a QIP map, that is, tions on AB.
CQIP j QIP, (48) Proof. The inclusions 1-LQICC $ LQICC and
1-LICC $ LICC follow from Proposition 15 using
and hence, it suffices to show that any QIP map E is
one-shot entanglement distillation. Reference [54]
a CQIP map, that is,
shows LQICC $ SQI and LICC $ SI using local state
QIP j CQIP. (49) discrimination. We show SQI $ (QIP ∩ SEP) and
SI $ (MIO ∩ SEP) in Proposition 16 using coherence
Consider any QIP map E AB for a bipartite system dilution. As for (QIP ∩ SEP) $ (QIP ∩ PPT) and
A and B. For any i, j, and k, Proposition 11 yields (MIO ∩ SEP) $ (MIO ∩ PPT), we prove the separa-
A B
tion in Proposition 16 using local state discrimination,
idA ⊗∆B ◦ E AB |ii hj| ⊗ |ki hk|
whereas a conventional way of separating SEP and
PPT by considering preparation of a PPT entangled
A B
= E AB |ii hj| ⊗ |ki hk| . (50)
state is insufficient. We prove (MIO ∩ PPT) $ MIO by
Then, it holds that showing an instance of MIO that is not in MIO ∩ PPT,
which is the SWAP operation. Note that the SWAP
A00
00 00
A
idA A ⊗∆B ◦ idA ⊗E AB |ii hj| ⊗ |ii hj| operation is in MIO but not in QIP, and hence it
B
is not straightforward to generalize this example to
⊗ |ki hk| (QIP ∩ PPT) $ QIP. To show (QIP ∩ PPT) $ QIP, we
00
A00 A B
will investigate assisted distillation of coherence in the
= idA ⊗E AB |ii hj| ⊗ |ii hj| ⊗ |ki hk| , next section, especially in Proposition 20. Q.E.D.
(51)
In the following, we show propositions used in
where A00 is a quantum system whose dimension is the above proof of the hierarchies. The following
the same as A. Due to linearity, we have proposition shows that the difference between r-round
A00 A LQICC/LICC and (r−1)-round LQICC/LICC for any
00
B
idA ⊗E AB |ΦDA i hΦDA | ⊗ |ki hk| ∈ QI,
r = 1, respectively, arises in one-shot entanglement
(52) distillation.
where DA is the dimension of the system A, and
A00 A
|ΦDA i is a maximally entangled state between A00 Proposition 15 (Separation between r-round
and A. Therefore, Proposition 12 implies that E AB is LQICC/LICC and (r − 1)-round LQICC/LICC). For
a CQIP map, which yields the conclusion. Q.E.D. any r ∈ {1, 2, . . .}, there exists a quantum state ρAB
r
Accepted in Quantum 2021-06-15, click title to verify. Published under CC-BY 4.0. 10shared between A and B for which one-shot entan- PPT, we use local state discrimination. Note that
glement distillation of 1 ebit is achievable by r-round the separation between SEP and PPT in the entan-
LICC but not achievable by (r−1)-round LOCC, lead- glement theory is first proven in Ref. [31] based on
ing to a separation between r- and (r−1)-round LICC, the fact that there exists a free state of PPT, a PPT
and a separation between r- and (r−1)-round LQICC. state, that is not a free state of SEP, a separable state.
However, this proof based on free states in the entan-
Proof. Consider a family of quantum states that are
glement theory is not applicable to our case because
represented as a convex combination of maximally en-
the sets of free states for QIP ∩ PPT and MIO ∩ PPT
tangled states corresponding to the origami distribu-
are included in QI and hence are always separable.
tion in Ref. [17], where each of the maximally entan-
Rather than this conventional argument based a PPT
gled states is represented in terms of the reference
entangled state, our proof generalizes a more recent
basis for A and B. Then, Ref. [17] shows that for
result on separating SEP and PPT based on local state
each r ∈ {1, 2, . . .}, there is a mixed state in this
discrimination as follows.
family for which one-shot entanglement distillation
of 1 ebit cannot be achieved by any (r − 1)-round Proposition 17 (Separations between QIP∩SEP and
LOCC, yet there exists an r-round LOCC protocol QIP ∩ PPT and between MIO ∩ SEP and MIO ∩ PPT).
that achieves one-shot entanglement distillation of 1 It holds that
ebit from the same mixed state. Since (r − 1)-round
LOCC includes (r − 1)-round LICC and (r − 1)-round (QIP ∩ SEP) $ (QIP ∩ PPT) , (57)
LQICC, this one-shot entanglement distillation can- (MIO ∩ SEP) $ (MIO ∩ PPT) . (58)
not be achieved by any (r − 1)-round LICC or any
(r − 1)-round LQICC. Proof. Recall that there exists a set of mutually or-
If each maximally entangled state for this mixed thogonal bipartite pure states for which there exists a
state is represented in terms of the reference basis PPT measurement achieving success probability 7/8
for A and B, the r-round LOCC protocol shown in in state discrimination [22], and hence a PPT map
Ref. [17] consists only of incoherent measurements; implemented by this PPT measurement can achieve
that is, this protocol is an r-round LICC protocol, this success probability. However, it is also known
which yields the separation between r-round LICC that no separable measurement can achieve success
and (r − 1)-round LICC. As for the separation be- probability greater than 3/4 in this state discrimina-
tween r-round LQICC and (r − 1)-round LQICC, tion [3], and hence by definition, no separable map
since r-round LICC is a subset of r-round LQICC, can achieve this success probability. Note that these
the above r-round LICC protocol also yields the sep- PPT and separable measurements can be regarded as
aration between r-round LQICC and (r − 1)-round CPTP maps that transform any input state into an in-
LQICC. Q.E.D. coherent state representing the probabilistic mixture
of the measurement outcomes. These CPTP maps
As for the separation between SQI and QIP ∩ SEP corresponding to the measurements are in QIP and
and that between SI and MIO ∩ SEP, we can consider MIO because the output states are always incoher-
a special case where A is one-dimensional, so that ent. Thus, this local state discrimination yields the
these operations can be regarded as IO and MIO on separations of (57) and (58) in terms of the success
B. Then, the difference between IO and MIO yields probability. Q.E.D.
the following separation.
We prove the separation between MIO ∩ PPT and
Proposition 16 (Separations between SQI and MIO by showing an instance. In general, it is not
QIP ∩ SEP and between SI and MIO ∩ SEP). It holds straightforward to identify an instance of operation
that that is in one class but is probably not in the other
SQI $ QIP ∩ SEP, (55) class, because the latter no-go theorem is generally
hard to prove. However, we here show that the SWAP
SI $ MIO ∩ SEP. (56)
operation is such an instance in this case.
Proof. The separation in (55) and (56) is a special
Proposition 18 (Separation between MIO∩PPT and
case of separation between IO and MIO, by consider-
MIO). Given any bipartite system AB, it holds that
ing A’s system to be one-dimensional. This separation
between IO and MIO can be seen in tasks such as co- (MIO ∩ PPT) $ MIO, (59)
herence dilution [79]. Therefore, when A’s system is
one-dimensional, there exists an MIO on B included where MIO is the set of maximally incoherent opera-
in QIP ∩ SEP \ SQI and MIO ∩ SEP \ SI, whereas any tions on AB.
IO on B is included in LQICC and LICC, and hence in
SQI and SI. Q.E.D. Proof. A SWAP unitary operation is an MIO map
because SWAP transforms any incoherent state into
To show separation between QIP ∩ SEP and QIP ∩ a swapped incoherent state, but this SWAP is not
PPT, and show that between MIO ∩ SEP and MIO ∩ a PPT map because the partial transpose of the
Accepted in Quantum 2021-06-15, click title to verify. Published under CC-BY 4.0. 11Choi operator of SWAP is not positive semidefinite. where {|ji}j is the fixed reference basis on B, and M
Hence, this SWAP is an example showing the conclu- B
may be called the coherence rank of |ΦM i , which
sion. Q.E.D. quantifies the coherence. To distill as much coher-
ence as possible, the parties perform the free opera-
AB
tions O to transform the initial state, i.e., |ψi , to
5 Asymptotically non-surviving sepa- a maximally coherent state on B up to a given error
ration of hierarchy in assisted distillation in the trace distance, so that M of B’s final maxi-
mally coherent state can be maximized. In a one-shot
of coherence scenario, the one-shot distillable coherence with assis-
In this section, we investigate a phenomena of asymp- tance is defined as
totically non-surviving separation [74] of the hierar- O
ψ B := max {log2 M :
Ca,1,
chy, especially between 1-LQICC and QIP, in achieving
an information-processing task, specifically, assisted AB B
E AB |ψi hψ| − |ΦM i hΦM | 5 ,
distillation of coherence. The assisted distillation of 1
coherence involves two parties A and B sharing some M ∈ N, E AB ∈ O } . (61)
initial state, and A can perform an arbitrary quan-
Correspondingly in the asymptotic scenario where the
tum operation while B has a restriction in generating
task is repeated infinitely many times within a van-
coherence. The aim of the task is to distill on B as
ishing error, the asymptotic distillable coherence with
much coherence as possible from the shared initial
assistance is defined as
state with assistance of A. While we prove difference
between the classes of operations for distributed ma- O 1 O B ⊗n
ψ B := lim lim Ca,1,
Ca,∞ ψ , (62)
nipulation of coherence in the previous section, this →0 n→∞ n
difference does not necessarily affect achievability of
We show that the asymptotic distillable coherence
certain tasks. Indeed, while 1-LQICC and SQI are
with assistance under QIP is always equal to that un-
different as the sets of operations, it is known that AB
der 1-LQICC for any pure initial state |ψi . In other
this difference does not appear in the task of asymp-
words, all the operations in the hierarchy (53) in The-
totic assisted distillation of coherence for any pure
orem 14 have the same power in this task.
state [10]. We here generalize this result to show that
even the difference between 1-LQICC and QIP does Proposition 19 (Equivalence of asymptotic dis-
not appear in this asymptotic scenario; that is, all tillable coherence with assistance under QIP and
the operations in this hierarchy have the same power AB
1-LQICC). Given any bipartite pure state |ψi , it
in achieving this asymptotic task for any initial pure holds that
state. At the same time, we consider a one-shot sce-
QIP
ψ B = Ca,∞
1-LQICC
ψ B = H ∆ ψ B , (63)
nario of this task and discover an instance of the ini- Ca,∞
tial pure state for which the amount of distillable co-
AB
herence with assistance is different under QIP ∩ PPT where ψ B = TrA |ψi hψ| is the reduced state of B
AB
and QIP; that is, the separation between 1-LQICC and for |ψi , ∆ is the completely dephasing channel, and
QIP arises in this one-shot scenario. While the proof H is the quantum entropy given by
of no-go theorem for showing the separation in one-
shot scenarios is hard in general, we identify such an H (ψ) := − Tr ψ log2 ψ. (64)
instance of separation by exploiting a numerical algo-
Proof. It suffices to prove
rithm by SDP based on the characterization of QIP
that we have proven in Proposition 11. QIP
Ca,∞ ψB 5 H ∆ ψB ,
(65)
The task of assisted distillation of coherence [10,
52, 59] is defined as follows. Consider two parties A since Theorem 14 yields
and B. While the original formulation [10] consid-
1-LQICC
ψ B 5 Ca,∞
QIP
ψB ,
ers 1-LQICC as free operations performed by A and Ca,∞ (66)
B, we here generalize the situation to other classes
of operations in the hierarchy, and let O denote a and it is shown in Ref. [10] that
class of free operations on A and B that is in the hi- 1-LQICC
ψB = H ∆ ψB .
Ca,∞ (67)
erarchy (53) in Theorem 14, such as QIP. Given an
arbitrary mixed state ψ B on B, we consider its purifi- To prove Inequality (65), we use the QI relative
AB
cation |ψi shared initially between A and B, where entropy [10]
AB
ψ B = TrA |ψi hψ| . The state to be distilled on B
CrA|B ρAB := min D ρAB ||σ AB ,
is a maximally coherent state denoted by (68)
σ AB ∈QI
M −1
B 1 X B where D (ρ||σ) := Tr ρ log2 ρ − Tr ρ log2 σ is the
|ΦM i := √ |ji , (60)
M j=0 quantum relative entropy. The QI relative entropy
Accepted in Quantum 2021-06-15, click title to verify. Published under CC-BY 4.0. 12You can also read
