This is the accepted manuscript made available via CHORUS. The article has been
published as:

Universal limitations on implementing resourceful unitaryevolutions
Ryuji Takagi and Hiroyasu Tajima
Phys. Rev. A 101, 022315 — Published 12 February 2020
DOI: 10.1103/PhysRevA.101.022315

Universal limitations on implementing resourceful unitary evolutions
Ryuji Takagi1, ∗ and Hiroyasu Tajima2, †1

Center for Theoretical Physics and Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2
Yukawa Institute for Theoretical Physics, Kyoto University,
Oibuncho Kitashirakawa Sakyo-ku, Kyoto, 606-8502, JapanWe derive a trade-off relation between accuracy of implementing a desired unitary evolution
using a restricted set of free unitaries and the size of the assisting system, in terms of the resource
generating/losing capacity of the target unitary. In particular, this relation implies that, for any
theory equipped with a resource measure satisfying lenient conditions, any resource changing unitarycannot be perfectly implemented by a free unitary applied to a system and an environment if theenvironment has finite dimensions. Our results are applicable to a wide class of resources including
energy, asymmetry, coherence, entanglement, and magic, imposing ultimate limitations inherent insuch important physical settings, as well as providing new insights into operational restrictions ingeneral resource theories.

I.

INTRODUCTION

One of the ultimate goals in quantum information science is to understand the operational enhancement madepossible by quantum phenomena as well as limitations on
the enhancement imposed by laws of quantum mechanics. This is not only an important theoretical question
but also of practical relevance, as recent years have witnessed the burgeoning development in manipulation ofsystems on small scales, in which quantum effects play
central roles.
Any quantum information processing involves time evolution of quantum states, and the most fundamentalbuilding block for the quantum dynamics is unitary evolution. Even though general quantum dynamics is described by completely positive trace preserving (CPTP)
maps, also called quantum channels, any channel acting
on a system can be simulated by an appropriate unitary
operation applied over the system and an environment[85], and thus any quantum evolution can be realized if
one has access to an arbitrary unitary. However, due totechnological limitations as well as restrictions imposedby laws of physics, physical systems usually do not allow
one to apply an arbitrary unitary. This makes it essential to consider to what extent a desired unitary dynamics
can be realized only using a limited set of accessible unitaries. This question has been specifically addressed forthe systems with additive conserved quantities, in which
only unitaries that respect the conservation laws can beapplied [1, 55, 56, 69, 70, 87, 88]. In particular, Ref. [87]has derived a lower bound for necessary amount of quantum fluctuation that the ancillary state must possess to∗ rtakagi@mit.edu† hiroyasu.tajima@yukawa.kyoto-u.ac.jp

implement a desired unitary in terms of its implementation accuracy and the amount of energy that the target
unitary can create, and they further derived lower andupper bounds that always match asymptotically in theregion where the implementation error is small [88]. The
presented bounds lead to a fundamental no-go theorem
that prohibits the perfect implementation of any unitarythat can create energy using energy conserving unitaryand finite-sized ancillary state.
However, there are various settings where other typesof quantities can play the main role, and one can ask
whether this type of trade-off relation is a general property shared by generic physical situations. This line ofthought naturally leads to the idea of resource theories,
which are general frameworks that deal with quantification and manipulation of precious quantities considered“resource” under a given setting [24]. The resource theoretic framework allows for systematic investigation on
specific physical settings [2, 4, 8, 12, 13, 38, 39, 44, 51,
52, 64, 73, 76, 86, 94, 96, 100, 103] and has turned out tobe especially useful for providing a unifying operationalview to general class of quantities [7, 14, 26, 28, 42, 50, 57–
59, 61, 62, 75, 91, 92, 97]. In this context, it can be seenthat the previous works [87, 88] dealt with a specific theory (i.e. theory of asymmetry with U (1) group [44, 64]),and it has remained elusive whether one can extend therelevant consideration to more general resources.
Here, we address the above question for the setting
where a set of “free” (i.e. accessible) unitaries is given,and one aims to implement “resourceful” (i.e. non-free)
unitaries with a free unitary and an aiding state definedin the ancillary system. Our main results are tradeoff relations between the implementation accuracy, theamount of resources that the target unitary can change,
and the size of the ancillary system, which are applicable
to a wide class of physical settings that satisfy several

2
lenient conditions. These relations immediately lead tono-go theorems that prohibit us from implementing any
resourceful unitary with perfect accuracy only using free
unitaries and aiding states defined in a system with finite size, which qualitatively reproduces the results in[87, 88] as a special case. We also apply our results toseveral important settings and discuss significance of theresults.
This paper is organized as follows. In Section II, oursetup and useful quantities as well as conditions that playmajor roles in later discussions are introduced. In Section III, our first main result on the trade-off relation
between accuracy, the amount of resources the target unitary can change, and the size of the ancillary system ispresented. In Section IV, we show our second main resultthat relaxes one of the conditions in the trade-off relation,
which significantly increases its applicability. In Section
V, we apply our results to various resources such as energy, asymmetry, coherence, entanglement, and magic.
In Section VI, we discuss possibilities of extending theno-go result to even more general settings. We finallyconclude our discussion in Section VII.

II.

FREE UNITARIES AND RESOURCE
MEASURES

Let Hd denote the Hilbert space with dimension d andD(Hd ) be the set of density operators acting on Hd . Also,
let UF (d) ⊆ U (d) be some set of unitaries acting on Hd
Sand define UF := d UF (d), which we call set of free
unitaries. The set of free unitaries is usually determinedby the system of interest, and it can be most naturally
understood as free operations in the context of resource
theories. A resource theory is specified by its set of freestates and free operations, which are considered given
for free under the interested physical setting, and an important requirement for free operations is that they are
not capable of creating any resources out of free states.
For instance, for the setting where two parties are physically separated apart, a reasonable theory comes with theset of separable states as free states and the set of local
operations and classical communication (LOCC) as free
operations. Motivated by the resource theoretic considerations, we also define resource measures as the maps fromstates to non-negative real numbers. If one assumes someunderlying resource theory of quantum states, one natural choice is to take resource monotones (which evaluate
zero for free states and do not increasing under application of free operations) defined in the theory as resource
measures.
Once some resource theory is provided, one can naturally consider UF as the set of unitaries that are also

free operations (e.g. the set of local unitaries for the caseof entanglement.) However, although considering the underlying resource theory is conceptually useful, for our
purpose as long as the set of free unitaries is given, one
does not necessarily need to assume an underlying structure of the resource theory. Indeed, as we shall see later
it is sometimes convenient to only consider the set of free
unitaries, not explicitly taking into account the underlying set of free states. In the same vein, we do not impose
the monotonicity property for resource measures in general. Instead, we consider the following properties for a
resource measure R determined by the given set of free
unitaries, which play major roles in later discussions.
Property 1: (Invariance under free unitaries) R(ρ) =R(V ρV † ), ∀V ∈ UF .
Property 2: (Continuity) There exist non-negative increasing functions f , g with limx→0 f (x) = 0,g(x) < ∞, ∀x < ∞, and a real function h withlimx→0 h(x) = 0 such that
|R(ρ) − R(σ)| ≤ f (D(ρ, σ))g(d) + h(D(ρ, σ))(1)for ρ, σ ∈ D(Hd ) where D(ρ, σ) is some distancemeasure between ρ and σ.
Property 3: (Additivity for product states) R(ρ ⊗ σ) =R(ρ) + R(σ).
Property 1 refers to that free unitaries do not change
the resource contents attributed to quantum states, and
it is especially a natural property when application ofa free unitary can be reversed by another free unitary.
Property 2 states that if two states are close to each other,
the amount of resources possessed by these states shouldbe also close. Property 3 is the property that if a stateis prepared independently of another state, the resource
contents attributed to the two states is evaluated as thesum of the amount of resources possessed by each state.
As we see in Section V, these properties are shared by anumber of known resource measures, and we shall obtain
ultimate bounds on implementation accuracy of desired
unitary in term of the resource measures satisfying these
conditions.
We also define the resource generating power and resource losing power for unitary U [29, 59, 77, 106, 111]:

GU := max R(U ρU † ) − R(ρ) ,(2)ρLU := − min R(U ρU † ) − R(ρ) .
(3)ρNote that GU , LU ≥ 0 for any U because there always
exists a state ρ that is invariant under U , for which one
can for instance take ρ = |uihu| where |ui is an eigenstateof the unitary.

3
III.

IMPLEMENTATION OF RESOURCEFULUNITARIES

Once the concept of free unitaries is introduced, one
can ask what can be done with them and what are ultimate limitations imposed on the tasks accomplished by
the given free unitaries. One of the fundamental questions that is both practically and theoretically importantis whether we can implement (or simulate) non-free unitaries, which we call resourceful unitaries, only using free
unitaries with aid of ancillary system.
More specifically, our aim is to simulate the given unitary US on the Hilbert space HS by a channel ΛS implemented by a free unitary VSE ∈ UF acting on the Hilbert
space HS ⊗ HE and some ancillary state ρE ∈ D(HE ),i.e.
†ΛS (·) := TrE [VSE (· ⊗ ρE )VSE].

(4)

The tuple I := (HE , VSE , ρE ) defines a specific implementation of the channel. A standard way of evaluating
the closeness of two quantum channels is to see how close
the output states from these channels are when the channels are allowed to act on only part of the input space.
In order to take into account the worst-case input, we
define the error for the given implementation I, which isa type of gate fidelity, asδIUS := max δIUS (ρS )(5)

δIUS (ρS ) := Le (ρS , ΛU † ◦ ΛS ),

(6)

ρS

where
S

ΛU (·) := U · U †

(7)

and
p
2(1 − Fe (ρS , Λ)),q
Fe (ρS , Λ) := hψ|SR [Λ ⊗ idR ](ψSR ) |ψiSR .

Le (ρS , Λ) :=

(8)
(9)

where |ψiSR is a purification of ρS . A related distancemeasure is the Bures distance for two quantum states:p(10)
L(ρ, σ) := 2 (1 − F (ρ, σ))√ √where F (ρ, σ) := k ρ σk1 is the Uhlman fidelity. The
choice of this distance measure is primarily due to themathematical convenience in later discussions, but because of the well-known relations with other distancemeasures, one can easily transform the results to theones with respect to other measures as well — indeed,
we will reformulate the relation in terms of the distancemeasure based on trace norm and diamond norm, whichcome with clear operational meaning in terms of distinguishability.

Then, we obtain the following trade-off relation between resourcefulness of desired unitary, implementation
accuracy, and dimension of the ancillary system with respect to any resource measure satisfying the three properties above.
Theorem 1. Let R be a resource measure satisfying
Property 1, 2, 3 and fL , gL , hL , GUS , LUS be the functions defined in (1), (2), (3) with respect to R and theBures distance: D(ρ, σ) := L(ρ, σ). Then, for any implementation I, it holds that
GUS + LUS ≤ αL (δIUS , dE ) + βL (δIUS ).
(11)√where αL (x, y)√:= fL (2 2x)gL (y) + 2fL (2x)gL (dS · y),βL (x) := hL (2 2x)+ 2hL (2x) with dE := dim HE , dS :=dim HS .
The proof of Theorem 1 can be concisely stated by utilizing the “no-correlation lemma” shown in [88], whichquantitatively clarifies the fact that in order to implement a unitary on the target system approximately, thecorrelation between the target system and the external
device must become weak. We defer a detailed proof toAppendix. Note that αL and βL are increasing functions
that approach 0 as x, y → 0. Thus, fixing the dimensionof the system of interest, Theorem 1 can be seen as atrade-off relation between the size of the device in the
ancillary system and the implementation accuracy, andin particular the result indicates that in order to implement a resourceful unitary the dimension of the ancillarysystem must grow as the implementation becomes better, and at the limit of perfect implementation the sizeof the ancillary system must diverge. Notably, Theorem
1 holds for any resource measure that satisfies Property
1, 2, 3, which ensures a wide applicability of the tradeoff relation. This observation immediately leads to thefollowing fundamental no-go theorem.
Corollary 2. Given the set of free unitaries UF and afinite dimensional ancillary system HE with dim HE <∞, it is impossible to perfectly implement any unitarythat can generate (or lose) nonzero resources in terms ofat least one resource measure satisfying Property 1, 2, 3
by means of Eq. (4).
Theorem 1 and Corollary 2 suggest an important implication — one might think that if a target operation
can only create certain amount of resource, supplying astate defined in a finite-dimensional space with roughly
the same amount of resource would be enough to accomplish the desired implementation. The above results statethat it is not the case when it comes to the unitary implementation, and Theorem 1 in particular provides aquantitative estimation of the necessary dimension evenwhen a non-zero error is allowed.

4
It is also convenient to rewrite Theorem 1 in terms ofthe trace norm and diamond norm.
Corollary 3. Suppose the implementation I =(HE , ρE , VSE ) implements channel ΛS with the errorUSmeasured by the diamond norm: δI,⋄:= kΛUS − ΛS k⋄ .
Let R be a resource measure satisfying Property 1, 2,
3 and f1 , g1 , h1 , GUS , LUS be the functions definedin (1), (2), (3) with respect to R and the trace norm:D(ρ, σ) := kρ − σk1 . Then, it holds thatUSUS
, dE ) + β1 (δI,⋄
GUS + LUS ≤ α1 (δI,⋄
)

(12)

√ 
√
where α1 (x, y) := √f1 4 2x g1 (y) + 2f1 (4 x) g1 (dS · y)√and β1 (x) := h1 4 2x + 2h1 (4 x).
This is a direct consequence from Theorem 1, but weinclude a proof in Appendix for completeness.

IV.

RELAXATION OF ADDITIVITY
CONDITION

Although a large class of resource theories possess
generic resource measures that satisfy Property 1 andProperty 2, the additivity condition (Property 3) israther a peculiar one. In fact, classes of resource measures that can be defined for any convex resource theory(e.g. relative entropy measure, robustness measure, convex roof measure etc.) are often only subadditive for
product states. Thus, relaxing the additivity conditionis highly desired in order for the results to be applicable
to more generic scenarios.
Here, we relax the additivity condition into that forpure product states. It gives us much more freedom tochoose resource measures because some important measures are additive only for pure product states. Examples for such measures include relative entropy of entanglement [98] and (logarithm of) stabilizer extent for thetheory of magic [15], which we discuss later in detail.
To this end, we introduce a relaxed version of Property 3 for resource measures.
Property 3’: (Additivity for pure product states) R(ρ⊗σ) = R(ρ) + R(σ) for any pure states ρ, σ.
We also define the following resource generating/losing
power for pure input states:
(13)
GUp := max R(U |ψihψ|U † ) − R(|ψihψ|)
|ψi

(14)
LpU := − min R(U |ψihψ|U † ) − R(|ψihψ|) .
|ψi

For the same reason that GU , LU ≥ 0, it also holds that
GUp , LpU ≥ 0 for any unitary U .
Then, we obtain the following trade-off relation.

Theorem 4. Let R be a resource measure satisfying
Property 1, 2, 3’ and fL , gL , hL , GUp S , LpUS be the functions defined in (1), (13), (14) with respect to R and theBures distance: D(ρ, σ) := L(ρ, σ). Then, for any implementation I = (HE , VSE , ρE ) with a pure state ρE , itholds that
GUp S + LpUS
√√
≤ 2 fL (2(1 + 2)δIUS )gL (dE dS ) + hL (2(1 + 2)δIUS ) .
(15)

A proof can be found in Appendix. It is worth notingthat R does not have to be defined for general mixed
states; as long as it is well-defined for pure states, thestatement holds and the continuity (Property 2) can berelaxed to that for pure states.
This Theorem leads to a variant of the aforementioned
no-go theorem on perfect implementability of resourcefulunitary.
Corollary 5. Given the set of free unitaries UF and afinite dimensional ancillary system HE with dim HE <∞, it is impossible to perfectly implement any unitarythat can generate (or lose) nonzero resources out of purestates in terms of at least one resource measure satisfying
Property 1, 2, 3’ by means of Eq. (4) with ρE being a purestate.
These results encompass a standard setup where someunit resource state (e.g. Bell state for entanglement, uniform superposition state for coherence), which is usuallypure, is prepared in the ancillary system. Although using
the unit state as a resource supply appears to be more
effective than using a mixed state, interestingly the requirement for Theorem 4 to hold is more lenient thanthat for Theorem 1, imposing more severe restriction on
the achievable accuracy for the implementation with apure ancillary state.

V.

APPLICATIONS

Here, we examine the validity of our results by applying them to specific physical settings. Although there
is no systematic way of constructing a resource measuresatisfying the three properties to our knowledge, it turns
out that many of the important settings come with suchmeasures tailored to each situation.

A.

Systems with additive conserved quantities

Consider a composite system consisting of subsystems⊗M−1{Si }M+I⊗
i=1 with an observable Htot = H1 ⊗ I

5
H2 ⊗ I⊗M−2 + . . . where Hi are local observables associated with subsystem Si . For these observables, we choose
the set of free unitaries as the ones that conserve the expectation values for any states, or equivalently, commute
with the observable. Namely, we chooseno(16)UF = US1 ...SM [Htot , US1 ...SM ] = 0 .

An important setting that fits into this formalism isthe system with conserved energy where the observable inquestion is the Hamiltonian of the system. Then, the free
unitaries can be considered time evolutions that respect
the energy conservation law, which in particular play key
roles in thermodynamics on small scales [3, 12, 13, 43, 48,
49, 65, 78, 81, 89, 95].
For this theory, natural resource measures one can takewill be the expectation value of the observable: R(ρS ) :=Tr[ρS HS ]. It is clear that this measure satisfies Property
1 and 3. Regarding Property 2, let us take the observable
PdS −1
j|jihj|. Then, we getof the form HS = j=0|R(ρ) − R(σ)| = | Tr[(ρ − σ)HS ]|X(ρjj − σjj )HS,j |
=|
j

≤
≤

X
j

X
j

|(ρjj − σjj )||HS,j |(17)|(ρjj − σjj )|kHk∞= k∆(ρ − σ)k1 (dS − 1)≤ kρ − σk1 (dS − 1)

where ρjj = hj| ρ |ji, σjj = hj| σ |ji, HS,j = hj| HS |ji, ∆is the dephasing with respect to the eigenbasis of HS , andwe used the contractivity of the trace norm under CPTP
maps in the last inequality. Thus, for this case one cantake f1 (x) = x, g1 (x) = x, and h1 (x) = −x in Corollary 3, and we conclude that finite dimensional environment does not allow for perfect implementation of unitarythat changes the energy by any energy-conserving unitary
and an energy “battery” state, which qualitatively reproduces the results in [87, 88]. Although we considered theobservable with uniform spectrum, a similar argument
can be applied to other observables with more generalform.
It will be worth pointing out that this is a situation
where our approach in which one does not necessarilyneed to assume any underlying resource theory becomes
useful, since the concept of free states and free operations
for this setting can be ambiguous — from the perspectivethat the energy is resource, one could say that the groundstate |0i is free, but in that case the set of free unitaries
defined in terms of free operations does not coincide withthe set of energy-conserving unitaries since any unitarythat can change energy but does not affect the groundstate (e.g. bit flip between |1i and |2i) also becomes freein this definition. Thus, when the focus is put on theconservation law, it is natural to just consider the set offree unitaries that meets the physical requirement.
On the other hand, by shifting our focus on the typeof resource of interest from the expectation value of theobservable to that of fluctuation, the underlying resource
theory can be naturally identified as the resource theoryof asymmetry [44, 64]. In particular, the resource theoryof asymmetry with U (1) group with unitary representation Ut = eiHS t is equipped with a family of resource
monotones that are additive for product states known
as metric-adjusted skew informations [47, 90, 112]. Oneof the examples in this family is the well-known WignerYanase skew information [63, 107] defined as
1√
I W Y (ρ, HS ) = − Tr([ ρ, HS ]2 )2√√= Tr(ρHS2 ) − Tr( ρHS ρHS ).

(18)

Since this satisfies Property 1 and 3, Theorem 1 andCorollary 2 can be applied with respect to this measure
as well, providing another way of looking at the trade-off
relation.
Finally, when the observable of interest is the Hamiltonian, the free unitaries in (16) preserve the Gibbs stateτ = exp(−HS /T )/Z where T is the temperature and Z
is the partition function of the system. This motivates us
to consider the “athermality”, a measure indicating the
distance from the Gibbs state to the given state, and especially the free energy is recovered by taking the relative
entropy as a distance measure:AR (ρ) := S(ρ||τ ) =1(F (ρ) − F (τ ))T(19)where F (ρ) := Tr[ρHS ] − T S(ρ) is the free energy. It
is then easy to see that this also satisfies all the threeproperties.

B.

Coherence

Consider the theory of coherence [2, 8, 86] where
one is interested in the degree of superposition with respect to the given preferred basis {|ii}. For this theory, the set of incoherent states I := conv({|iihi|})
is a reasonable choice for the free states, and one cannaturallychoose the relevanto free unitaries UF (d) =nPd−1 iθj
U U = j=0 e |π(j)ihj| where π is the permutation on {0, . . . , d − 1}, which is often called the set ofincoherent unitaries.

6
As a resource measure, let us consider a standard coherence measure, the relative entropy of coherence:
CR (ρ) := min S(ρ||σ) = S(∆(ρ)) − S(ρ).
σ∈I

(20)

For this measure, it is easy to see that Property 1 is satisfied. The explicit form of CR in (20) ensures Property 3
as well because of the additivity of the von Neumann entropy for product states. As for Property 2, recall thefollowing asymptotic continuity property that holds for
relative entropy measure MR (ρ) := inf σ∈F S(ρ||σ) with
F being any convex and closed set of positive semidefinite operators that contains at least one full-rank operator [108]:


ǫ
(21)
|MR (ρ) − MR (σ)| ≤ κǫ + (1 + ǫ)b
1+ǫ
for any two states 21 kρ − σk1 ≤ ǫ where κ :=supτ,τ ′ {MR (τ ) − MR (τ ′ )} and b(x) := −x log x − (1 −x) log(1 − x) is the binary entropy. For the case of theoryof coherence, (21) reduces to the following bound:ǫ, (22)|CR (ρ) − CR (σ)| ≤ ǫ log d + (1 + ǫ)b
1+ǫ
for which we find f1 (x) = x, g1 (x) = log x, and h1 (x) =(1 + x)b(x/(1 + x)). Since this measure is also faithful,i.e. CR (ρ) = 0 iff ρ ∈ I , Corollary 3 implies that any
coherence generating unitary that can create a coherent
state out of an incoherent state cannot be implemented
with zero-error with aid of any coherent state acting on
finite-dimensional ancillary system.

C.

Entanglement

Arguably, entanglement is one of the most importantresources to consider, which has a strong connection tooperational tasks in quantum information processing. In
particular, using only local operations and classical communication to implement desired global operations withhelp of preshared entanglement is a key idea of quantum
network and distributed quantum computing [30, 72],
and methodology as well as necessary entanglement costfor implementing global gates with local operations andclassical communication have been considered for various settings [21, 22, 33, 82, 104, 105]. Our formalism
addresses a more restricted scenario where the parties
only have access to local gates in order to implement adesired global gate with aid of preshared entanglement.
Our results induce necessary size of the shared entangled state and imply the impossibility of perfectly implementing any entangling gate with finite-sized aiding system. Since it is clearly possible to perfectly implement

any global unitary if classical communication is allowed(via quantum teleportation), our results clarify the significance of classical communication for the situations suchas distributed quantum computing [67].
In order to apply our results, we need to find an entanglement measure satisfying the three properties. In
particular, one needs to be careful about the additivity
property since some well-known entanglement measures(e.g. such as the (max-)relative entropy of entanglement[27, 102], robustness of entanglement [101]) are only subadditive even for product states, and it had been indeed
an important program to find an additive measure of entanglement. As a result, the squashed entanglement was
introduced as an additive entanglement measure [25], andits continuity was also shown [5]. In addition, the conditional entanglement of mutual information [109] wasintroduced as another additive and continuous measureof entanglement. Remarkably, this measure can be easily
extended to multipartite entanglement, which allows our
results to be applied to the multipartite scenarios.
On the other hand, Theorem 4 allows us to avoid thissubtlety and take even simpler entanglement measure.
For instance, the relative entropy of entanglement is additive for pure product states, as can be seen by noting that it reduces to the entanglement entropy for purestates. Since it clearly satisfies Property 1 and 2 as well,
Theorem 4 and Corollary 5 immediately follows for suchmeasure.

D.

Fault-tolerant quantum computation

To realize the quantum computation in a noise-resilient
fashion, which is so called fault-tolerant quantum computation [74, 80], encoding quantum states into quantum
error correcting codes and carrying out logical computation inside the code space is essential. Since many
promising error correcting codes allow for relatively efficient implementation of the logical Clifford gates in a
fault-tolerant manner [10, 36, 79, 83, 84], for the situations where those codes are in use, Clifford gates canbe naturally considered “free”. However, since Cliffordgates do not form a universal gate set, some non-Clifford
gate needs to be implemented fault-tolerantly, and a popular way of realizing it is via the gate teleportation [41],in which “magic states” [17] are injected as resourcesof “non-Cliffordness”. Since good logical magic states
are hard to prepare in general, a magic-state distillation
protocol [17] should be run beforehand to increase thequality of the noisy magic states. However, a large overhead cost comes with the distillation protocols and how
to reduce the overhead has been under active research[16, 18, 19, 31, 32, 34, 35, 40, 46, 54, 60, 68] (error cor-

7
recting codes that avoid using the magic-state distillation
have also been investigated [6, 9, 20, 53, 66, 71, 93, 110]),and this costly nature of magic states motivates us toconsider the resource theory of magic, which considers
the “magicness” as precious resources.
The resource theory of magic is defined by the setof free states called stabilizer states, which is the convex combinations of pure states produced by Cliffordgates [100]. By definition, non-Clifford gates are able
to create non-stabilizer sates out of stabilizer states, andas described above it is an essential building block foruniversal quantum computation. This operationally motivated framework leads us to a natural question on how
well non-Clifford gate could be implemented by Cliffordgates with aid of magic states as resources. Our results
address this question by considering appropriate resource
measures for magicness. We consider the cases of qubits(dimension 2) and quopits (qudits with odd-prime dimensions) separately.
1.

QubitsAlthough one can consider valid magic monotones defined for multiqubit states (e.g. relative entropy of magic[100], robustness of magic [52]), they are not additive
for product states in general, which prevents us from applying Theorem 1. However, Theorem 4 turns out tobe useful in this case since there indeed exists a measuredefined for pure states and additive for pure product multiqubit states. To this end, consider the stabilizer extent
introduced in [15]:


!2
 X

X
ξ(|ψi) := min|ci |ci |φi i
|ψi =
(23)


i

i

where |φi i are pure stabilizer states. The stabilizer extent was originally introduced for investigating the overhead cost for classically simulating quantum circuits, but
we find that it is also useful for our purpose, providing
a new perspective to this measure. Let us take our resource measure as R(|ψihψ|) = log ξ(|ψi). It was shownthat the stabilizer extent is multiplicative for tensor products between states supported on up to three qubits [15],and thus R satisfies Property 3’. Property 1 is also satisfied because of the monotonicity of ξ under Cliffordgates and reversibility of Clifford unitary under another
Clifford unitary (since Clifford gates constitute a group).
As for Property 2, we first remark that our measure coincides with the max-relative entropy of magic for purestates as shown in Ref. [75], where the max-relative entropy measure is defined asnoDmax (ρ) := min r ρ  2r σ, σ ∈ STAB
(24)

where STAB refers to the set of stabilizer states, and denotes the inequality with respect to the positive
semidefiniteness. Then, we prove the following continuity bound for max-relative entropy of magic, which may
be of independent interest. Using the identity between
R and (24) for pure states, the continuity of stabilizer
extent is derived as a special case of this result. It would
be also worth noting that the following result holds forthe max-relative entropy measure defined for any convex
resource theory that includes the maximally mixed state
as a free state. (One can also easily extend the relation
to the theories with at least one full-rank free state.)
Proposition 6. Let ρ, σ ∈ D(HdS ) and suppose thatkρ − σk1 < 1/(2dS ). Then, it holds that
|Dmax (ρ) − Dmax (σ)| ≤ 2kρ − σk1 dS .

(25)

The proof is presented in Appendix. Our results provide an interesting implication for implementation of nonClifford gates. Suppose we are given qubits acting on
system A and try to implement some non-Clifford gate
UNC on the subsystem A1 ⊂ A by applying Clifford gateson A. Let N be the number of qubits supported on thesubsystem A \ A1 . Then, our results imply that in orderto realize the implementation accuracy ǫ with respect tothe diamond norm, the required
 number of qubits N must
scale as Ω log

p
GU

+Lp
U

NC√

. This observation explic-NCǫ

itly tells us the importance of measurement + feedforward (adaptive) operations for quantum circuits to gain
their power.

2.

Quopits

For the case when the dimension of the system thateach qudit acts on is odd-prime, “mana” was introduced
as a magic monotone [100]:!
X
M(ρ) := log|Wρ (u)|(26)
uwhere Wρ (u) is the discrete Wigner function for state ρ[45]. The mana essentially measures the total negativity of the discrete Wigner function, which is motivatedby the fact that stabilizer states only take non-negativevalue for the discrete Wigner function. An important
property of this measure for our purpose is that it isadditive for product states, which comes from that thediscrete Wigner function for a product state is just themultiplication of the two discrete Wigner functions of thestates that constitute the product state. It is also continuous (although it is not asymptotically continuous asshown in [100]), and Property 1 can be also easily seen by

8
the monotonicity of mana under Clifford gates and thefact that the application of Clifford gate can be reversedby another Clifford gate. Thus, Theorem 1 and Corollary
2 can be applied with respect to the mana measure.
Note that the mana is not faithful: there exists a magicstate ρ with M(ρ) = 0 [99]. However, the discrete Hudson’s theorem [45] ensures that it is faithful for purestates, which is enough to show that any non-Clifford
unitary cannot be implemented with zero-error with finite number of magic states.
VI.

TOWARD FULL GENERALITY

Although Theorem 4 covers most of the known important settings, one could still argue that some theory ofinterest may not come with a resource measure that satisfies all the three properties, especially the additivity condition. Here, we focus on the qualitative no-go statement
and see that it is quite unlikely for the perfect implementation of resourceful unitary to be possible even in moregeneral settings.
Suppose that free unitary VSE and pure state|φi hallow for an exact implementationof US , i.e.
i†= US ρS US† for any ρS . By
TrE VSE (ρS ⊗ |φihφ|E )VSE

taking δIUS = 0 in (C1), we get
i
h
†
′
= σE
TrS VSE (ρS ⊗ |φihφ|E )VSE

(27)′
where σE
is a pure state. Since states with pure reduced
states are only product states, we know that the total
state must look like†′
VSE (ρS ⊗ |φihφ|E )VSE
= US ρS US† ⊗ σE.

(28)Then, we get for any ρS and any measure R that is invariant under free unitaries that†
R(ρS ⊗ |φihφ|) = R(VSE (ρS ⊗ |φihφ|)VSE)′= R(US ρS US† ⊗ σE)

(29)Thus, for the given theory, unless any resource measurewith Property 1 (but not necessarily Property 2, 3, 3’)satisfies (29) for any ρS , it is impossible to implement
the target US exactly. Note that this is a very strong
restriction, and when R is additive for product states,
Corollary 2 and 5 are reproduced.
Let us impose another natural condition on R that it
be a subadditive monotone for some resource theory inwhich composition of free states and partial trace are free
operations. For such cases, one can show that R(|φihφ|) =′R(σE) as follows. Take a free state τS and ηS = US† τS US .
Then, we getR(|φihφ|) ≥ R(τS ⊗ |φihφ|)′′= R(US τS US† ⊗ σE
) ≥ R(σE
)

(30)

and
′
′
R(σE
) ≥ R(US ηS US† ⊗ σE)

= R(ηS ⊗ |φihφ|) ≥ R(|φihφ|).

(31)

where to show both of the above relations we used thatthe composition of free states is a free operation in thefirst inequalities, the invariance of R under free unitaries
and (28) in the equalities, and that the partial trace is afree operation in the last inequalities together with theassumption that R is a monotone under free operations.
This makes it even more surprising that Eq. (29) holds
for any ρS for resourceful unitary US since it would indicate that attaching ancillary states with the same amountof resources to two states with different amount of resources would necessarily produce the states with thesame amount of resources. We leave the thorough analysis on how general the no-go statement can be made for
future work.

VII.

CONCLUSIONS

We considered a general setting where one aims to implement a target unitary with access to a restricted set ofunitaries as well as ancillary system. We derived a tradeoff relation between the implementation accuracy and thesize of the ancillary system in terms of the amount of theresources that can be changed by the target unitary withrespect to resource measures that satisfy three properties:
invariance under free unitaries, continuity, and additivity
for product states. Using this relation, we presented afundamental no-go theorem on the perfect implementation of resourceful unitaries with finite-dimensional ancillary systems. We further relaxed the subtle condition in
the above three properties, additivity for product states,and showed an analogous trade-off relation that only requires the resource measures to be additive for pure product states, in addition to the other two properties. We
exemplified the wide validity of our results by applying
them to various important settings and discussed physical significance implied by the results for specific settings.
We finally discussed the feasibility of extending our no-go
results to even more general settings that do not assume
all the properties for the resource measures we considered.
For future work, it will be intriguing to clarify whether
some of the required properties for resource measures considered in this work can be dropped to obtain a similar
trade-off relation. It will also be interesting to investigate
how good our lower bounds are in general by constructing
upper bounds with explicit protocols that approximately
implement desired unitaries.

9
Note added.
— During the completion of thismanuscript, we became aware of the independent related
work by G. Chiribella, Y. Yang, and R. Renner [23].

and corresponding final states on SE and E as
(i)

(i)

†
,
σSE := VSE (ρS ⊗ ρE )VSE

(A7)

(i)
(i)
σE := TrS [σSE ].

(A8)ACKNOWLEDGMENTS

Due to Property 1 and 3 of the resource measure R, we
have

We thank Tomoyuki Morimae for fruitful discussions.
R. T. acknowledges the support of NSF, ARO, IARPA,and the Takenaka Scholarship Foundation. H. T. acknowledges the support of JSPS (Grants-in-Aid for Scientific Research No. JP19K14610).

R(ρS ) + R(ρE ) = R(σSE ).

(i)

(i)

(A9)

Using (A3), we get
(i)

(i)

(i)

L(σSE , US ρS US† ⊗ σE ) ≤ 2δIUS .

(A10)

Due to Property 2 of R and (A9), (A10), we obtain
(i)

(i)

We first retrieve the main lemma we use for the readers’convenience.

≤ fL (2δIUS )gL (dE dS ) + hL (2δIUS ).

(i)

(i)

(A1)

(i)

(i)

(A2)

σB := TrA [σAB ].

Let ΛA be the channel implemented by the implementation I = (HE , ΛAB , ρB ), i.e. ΛA (·) := TrB [ΛAB (· ⊗ ρB )]and write the accuracy of implementation of UA with im(i)
U,(i)
(i)
plementation I for input state ρA as δI:= δIU (ρA )as in (7). Then, for any UA and I, we have the following
relations:
1. It holds that
(i)

(i)

(i)

U ,(i)L(σAB , UA ρA UA† ⊗ σB ) ≤ 2δI A′(0+1)

2. There exists a state σB(0)′(0+1)L(σB , σB′(0+1)) + L(σB.

(A3)

of B such that
√ U ,(0+1)(1), σB ) ≤ 2 2δI A.
(A4)Moreover, if ρB is a pure state and ΛAB is a unitary′(0+1).
operation, one can take a pure state for σB
We are now in a position to prove Theorem 1.
(i)
Proof. Define ρS ,

(0)

(0)

|R(ρS ) − R(US ρS US† ) − R(σE )

(1)

(1)

(1)

− R(ρS ) + R(US ρS US† ) + R(σE )|≤ 2 fL (2δIUS )gL (dE dS ) + hL (2δIUS ) .

(A12)Another use of the triangle inequality leads to
(0)

(1)

|R(σE ) − R(σE )|

(0)

(0)

(1)

(1)

≥ |R(US ρS US† ) − R(ρS ) − R(US ρS US† ) + R(ρS )|

− 2 fL (2δIUS )gL (dE dS ) + hL (2δIUS )= GUS + LUS − 2 fL (2δIUS )gL (dE dS ) + hL (2δIUS )(A13)

where we used GUS , LUS ≥ 0 in the equality. On theother hand, using (A4) together with triangle inequality
and Property 2 of R, we get
(0)

(1)

|R(σE ) − R(σE )|√√≤ fL (2 2δIUS )gL (dE ) + hL (2 2δIUS ).

(A14)

Combining (A13) and (A14), we finally obtain√√
GUS + LUS ≤ fL (2 2δIUS )gL (dE ) + hL (2 2δIUS )+ 2 fL (2δIUS )gL (dE dS ) + hL (2δIUS )= αL (δIUS , dE ) + βL (δIUS ).

(A15)

Appendix B: Proof of Corollary 3

i = 0, 1 as(0)
ρS := argmax(R(US ρS US† ) − R(ρS ))(1)
ρS := argmin(R(US ρS US† ) − R(ρS ))(A11)

Using the triangle inequality and (A11), we get(0)Lemma 7 (No-correlation lemma [88]). Let ΛAB be achannel on the composite system AB and UA be a unitary
operation on A. We consider three possible initial states
(0)
(1)
(0+1)
(0)
(1)
of A: ρA , ρA , and ρA:= (ρA + ρA )/2 and write
the initial state of B as ρB . We refer to the final states(i)of AB and B with the initial state ρA (i = 0, 1, 0 + 1) as
σAB := ΛAB (ρA ⊗ ρB ),(i)|R(ρS ) + R(ρE ) − R(US ρS US† ) − R(σE )|Appendix A: Proof of Theorem 1

(A5)
(A6)

Proof. Recall the relation between the Bures distance andthe trace distance [37]11(L(ρ, σ))2 ≤ kρ − σk1 ≤ L(ρ, σ),
2
2

(B1)

10
which also implies δIUS ≤
imply

q
US
δI,⋄
. Then, (A3) and (A4)

1 (i)
(i)
(i)
kσ − US ρS US† ⊗ σE k1 ≤ 2
2 SE
and

q
US
δI,⋄

(B2)

q
1 (0)
(1)
US
kσB − σB k1 ≤ 2 2δI,⋄(B3)
2Then, the same proof as Theorem 1 can be employed toobtain the statement.
Appendix C: Proof of Theorem 4

Proof. Lemma 7 together with the assumption that ρE′
is pure ensures that there exists a pure state σE
thatsatisfies (A4), namely
(1)
(0)
(i)
′
′
′
)
) + L(σE , σE
) ≤ L(σE , σE
L(σE , σE√ U≤ 2 2δI S .

which proves the statement.

(C1)

Appendix D: Proof of Proposition 6

Proof. We assume Dmax (ρ) ≥ Dmax (σ) without loss ofgenerality. The definition of max-relative entropy measure (24) admits the following dual form [11]:maximizelog Tr[ρX]subject to

X0

(D1)

Tr[τ X] ≤ 1, ∀τ ∈ STAB.

Then, we obtain
(i)

(i)

′
L(σSE , US ρS US† ⊗ σE
)

(i)
(i)
(i)
≤ L(σSE , US ρS US† ⊗ σE )
(i)
(i)
(i)
′
+ L(US ρS US† ⊗ σE , US ρS US† ⊗ σE
)
(i)
′
)
≤ 2δIUS + L(σE , σE
√ US
≤ 2(1 + 2)δI

Let Xρ be an optimal solution that achieves (D1) for
state ρ. Then, we obtain

Dmax (σ) ≥ log Tr[σXρ ](C2)

where in the first inequality we used the triangle inequality, in the second inequality we used (A3) and the factthat L(ρ ⊗ σ, ρ ⊗ τ ) = L(σ, τ ), and in the third inequality
we used (C1).
(0)(1)
Let ρS and ρS be pure states that achieve (13) and(14) respectively. Then, Property 1 and 3’ of R lead to
(i)

(i)

R(σSE ) = R(ρS ) + R(ρE ).

(C3)

and
(i)

(i)

′
′
).
) = R(US ρS US† ) + R(σE
R(US ρS US† ⊗ σE

(C4)

Combining Property 2, (C2), (C3), (C4), we get
(i)

(i)

′
)|
|R(ρS ) + R(ρE ) − R(US ρS US† ) − R(σE
√ U√
S
≤ fL (2(1 + 2)δI )gL (dE dS ) + hL (2(1 + 2)δIUS ).
(C5)

Hence,′′
0 = R(ρE ) − R(σE
) + R(σE) − R(ρE )
(0)

(0)

(1)

(1)

≥ R(US ρS US† ) − R(ρS ) − R(US ρS US† ) + R(ρS )√√− 2 fL (2(1 + 2)δIUS )gL (dE dS ) + hL (2(1 + 2)δIUS )= GUS + LUS√√− 2 fL (2(1 + 2)δIUS )gL (dE dS ) + hL (2(1 + 2)δIUS ) ,(C6)≥ log (Tr[ρXρ ] − kρ − σk1 kXρ k∞ )kρ − σk1 kXρ k∞= Dmax (ρ) + log 1 −Tr[ρXρ ]≥ Dmax (ρ) + log (1 − kρ − σk1 dS )≥ Dmax (ρ) − 2kρ − σk1 dS(D2)
The first inequality is because Xρ is a suboptimal solution for σ. The second inequality is because of thesame argument in (17). The third inequality is because
it holds that kXρ k∞ ≤ dS from the second constraintin (D1) together with the fact that the maximally mixed
state I/dS is a stabilizer state, and that Tr[ρXρ ] ≥ 1 because I serves as a suboptimal solution for X that gives
Tr[ρI] = 1. The fourth inequality is because it holds thatlog(1 − x) ≥ −2x for 0 ≤ x ≤ 1/2 (note that we take thebase 2 for the logarithm), where we used the assumptionthat kρ − σk1 < 1/(2dS ). Note also that the logarithmin (D2) is always well-defined because Tr[ρXρ ] ≥ 1 andkρ − σk1 kXk∞ ≤ 1/2. The statement is reached by combining the assumption that Dmax (ρ) ≥ Dmax (σ).

11[1] Åberg, J. (2014), Phys. Rev. Lett. 113, 150402.
[2] Åberg, J. (2006), arXiv:quant-ph/0612146.
[3] Åberg, J. (2013), Nature Communications 4, 1925.
[4] Albarelli, F., M. G. Genoni, M. G. A. Paris, and A. Ferraro (2018), Phys. Rev. A 98, 052350.
[5] Alicki, R., and M. Fannes (2004), Journal of Physics A: Mathematical and General 37 (5), L55.
[6] Anderson, J. T., G. Duclos-Cianci, and D. Poulin (2014), Phys. Rev. Lett. 113, 080501.
[7] Anshu, A., M.-H. Hsieh, and R. Jain (2018), Phys. Rev. Lett. 121, 190504.
[8] Baumgratz, T., M. Cramer, and M. B. Plenio (2014), Phys. Rev. Lett. 113, 140401.
[9] Bombı́n, H. (2015), New. J. Phys. 17 (8), 083002.
[10] Bombin, H., and M. A. Martin-Delgado (2006), Phys. Rev. Lett. 97, 180501.
[11] Boyd, S., and L. Vandenberghe (2004), Convex Optimization (Cambridge University Press, New York).
[12] Brandão, F. G. S. L., M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens (2013),
Phys. Rev. Lett. 111, 250404.
[13] Brandão,F.,
M.
Horodecki,
N.
Ng,
J.
Oppenheim,
andS.
Wehner(2015),
Proceedings of the National Academy of Sciences 112 (11), 3275.
[14] Brandão, F. G. S. L., and G. Gour (2015), Phys. Rev. Lett. 115, 070503.
[15] Bravyi, S., D. Browne, P. Calpin, E. Campbell, D. Gosset, and M. Howard (2019), Quantum 3, 181.
[16] Bravyi, S., and J. Haah (2012), Phys. Rev. A 86, 052329.
[17] Bravyi, S., and A. Kitaev (2005), Phys. Rev. A 71, 022316.
[18] Campbell, E. T., and M. Howard (2017), Phys. Rev. A 95, 022316.
[19] Campbell, E. T., and M. Howard (2018), Quantum 2, 56.
[20] Chamberland, C., T. Jochym-O’Connor, and R. Laflamme (2016), Phys. Rev. Lett. 117, 010501.
[21] Chen, L., and L. Yu (2014), Phys. Rev. A 89, 062326.
[22] Chen, L., and L. Yu (2016), Phys. Rev. A 93, 042331.
[23] Chiribella, G., Y. Yang, and R. Renner (2019), arXiv e-prints arXiv:1908.10884 [quant-ph].
[24] Chitambar, E., and G. Gour (2019), Rev. Mod. Phys. 91, 025001.
[25] Christandl, M., and A. Winter (2004), Journal of Mathematical Physics 45 (3), 829.
[26] Coecke, B., T. Fritz, and R. W. Spekkens (2016), Inf. Comput. 250, 59.
[27] Datta, N. (2009), International Journal of Quantum Information 07 (02), 475.
[28] Del Rio, L., L. Kraemer, and R. Renner (2015), arXiv:1511.08818.
[29] Dı́az, M. G., K. Fang, X. Wang, M. Rosati, M. Skotiniotis, J. Calsamiglia, and A. Winter (2018), Quantum 2, 100.
[30] Duan, L.-M., and C. Monroe (2010), Rev. Mod. Phys. 82, 1209.
[31] Duclos-Cianci, G., and D. Poulin (2015), Phys. Rev. A 91, 042315.
[32] Duclos-Cianci, G., and K. M. Svore (2013), Phys. Rev. A 88, 042325.
[33] Eisert, J., K. Jacobs, P. Papadopoulos, and M. B. Plenio (2000), Phys. Rev. A 62, 052317.
[34] Fowler, A. G., S. J. Devitt, and C. Jones (2013), Scientific Reports 3, 1939.
[35] Fowler, A. G., and C. Gidney (2018), arXiv e-prints arXiv:1808.06709 [quant-ph].
[36] Fowler, A. G., M. Mariantoni, J. M. Martinis, and A. N. Cleland (2012), Phys. Rev. A 86, 032324.
[37] Fuchs, C. A., and J. van de Graaf (1999), IEEE Transactions on Information Theory 45 (4), 1216.
[38] Gallego, R., and L. Aolita (2015), Phys. Rev. X 5, 041008.
[39] Genoni, M. G., M. G. A. Paris, and K. Banaszek (2008), Phys. Rev. A 78, 060303.
[40] Gidney, C., and A. G. Fowler (2019), Quantum 3, 135.
[41] Gottesman, D., and I. L. Chuang (1999), Nature 402 (6760), 390.
[42] Gour, G. (2017), Phys. Rev. A 95, 062314.
[43] Gour, G., M. P. Müller, V. Narasimhachar, R. W. Spekkens, and N. Y. Halpern (2015), Physics Reports 583, 1 .
[44] Gour, G., and R. W. Spekkens (2008), New J. Phys. 10 (3), 033023.
[45] Gross, D. (2006), Journal of Mathematical Physics 47 (12), 122107.
[46] Haah, J., and M. B. Hastings (2018), Quantum 2, 71.
[47] Hansen, F. (2008), Proceedings of the National Academy of Sciences 105 (29), 9909.
[48] Hayashi, M., and H. Tajima (2017), Phys. Rev. A 95, 032132.
[49] Horodecki, M., and J. Oppenheim (2013), Nature Communications 4, 2059.
[50] Horodecki, M., and J. Oppenheim (2013), Int. J. Mod. Phys. B 27, 1345019.
[51] Horodecki, R., P. Horodecki, M. Horodecki, and K. Horodecki (2009), Rev. Mod. Phys. 81, 865.
[52] Howard, M., and E. Campbell (2017), Phys. Rev. Lett. 118, 090501.
[53] Jochym-O’Connor, T., and R. Laflamme (2014), Phys. Rev. Lett. 112, 010505.

12[54] Jones, C. (2013), Phys. Rev. A 87, 042305.
[55] Karasawa, T., J. Gea-Banacloche, and M. Ozawa (2009), J. Phys. A: Math. Theor. 42 (22), 225303.
[56] Karasawa, T., and M. Ozawa (2007), Phys. Rev. A 75, 032324.
[57] Krishnan Vijayan, M., E. Chitambar, and M.-H. Hsieh (2019), arXiv e-prints arXiv:1906.04959 [quant-ph].
[58] Lami, L., B. Regula, X. Wang, R. Nichols, A. Winter, and G. Adesso (2018), Phys. Rev. A 98, 022335.
[59] Li, L., K. Bu, and Z.-W. Liu (2018), arXiv:1812.02572.
[60] Litinski, D. (2019), arXiv e-prints arXiv:1905.06903 [quant-ph].
[61] Liu, Z.-W., K. Bu, and R. Takagi (2019), Phys. Rev. Lett. 123, 020401.
[62] Liu, Z.-W., X. Hu, and S. Lloyd (2017), Phys. Rev. Lett. 118, 060502.
[63] Marvian, I., and R. W. Spekkens (2014), Nature Communications 5, 3821.
[64] Marvian, I., and R. W. Spekkens (2016), Phys. Rev. A 94, 052324.
[65] Morikuni, Y., H. Tajima, and N. Hatano (2017), Phys. Rev. E 95 (3), 032147.
[66] Nikahd, E., M. Sedighi, and M. Saheb Zamani (2017), Phys. Rev. A 96, 032337.
[67] One could alternatively argue the impossibility of implementing entangling gates only using local operations and sharedentanglement by noting that no signalling is allowed under this setup.
[68] O’Gorman, J., and E. T. Campbell (2017), Phys. Rev. A 95, 032338.
[69] Ozawa, M. (2002), Phys. Rev. Lett. 89, 057902.
[70] Ozawa, M. (2003), International Journal of Quantum Information 01 (04), 569.
[71] Paetznick, A., and B. W. Reichardt (2013), Phys. Rev. Lett. 111, 090505.
[72] Pirker, A., J. Wallnöfer, and W. Dür (2018), New. J. Phys. 20 (5), 053054.
[73] Plenio, M. B., and S. Virmani (2007), Quant. Inf. Comput. 7 (1), 001.
[74] Preskill, J. (1998), “Fault-tolerant quantum computation,” in Introduction to Quantum Computation and Information,
pp. 213–269.
[75] Regula, B. (2018), J. Phys. A: Math. Theor. 51, 045303.
[76] Rivas, A., S. F. Huelga, and M. B. Plenio (2010), Phys. Rev. Lett. 105, 050403.
[77] Seddon, J. R., and E. T. Campbell (2019), Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475[78] Shiraishi, N., and H. Tajima (2017), Phys. Rev. E 96, 022138.
[79] Shor, P. W. (1995), Phys. Rev. A 52, R2493.
[80] Shor, P. W. (1996), in Proceedings of 37th Conference on Foundations of Computer Science, pp. 56–65.
[81] Skrzypczyk, P., A. J. Short, and S. Popescu (2014), Nature Communications 5, 4185.
[82] Soeda, A., P. S. Turner, and M. Murao (2011), Phys. Rev. Lett. 107, 180501.
[83] Steane, A. M. (1996), Phys. Rev. Lett. 77, 793.
[84] Steane, A. M. (1997), Phys. Rev. Lett. 78, 2252.
[85] Stinespring, W. F. (1955), Proceedings of the American Mathematical Society 6 (2), 211.
[86] Streltsov, A., G. Adesso, and M. B. Plenio (2017), Rev. Mod. Phys. 89, 041003.
[87] Tajima, H., N. Shiraishi, and K. Saito (2018), Phys. Rev. Lett. 121, 110403.
[88] Tajima, H., N. Shiraishi, and K. Saito (2019), arXiv e-prints arXiv:1906.04076 [quant-ph].
[89] Tajima, H., E. Wakakuwa, and T. Ogawa (2016), arXiv e-prints arXiv:1611.06614 [quant-ph].
[90] Takagi, R. (2019), Sci. Rep. 9 (1), 14562.
[91] Takagi, R., and B. Regula (2019), Phys. Rev. X 9, 031053.
[92] Takagi, R., B. Regula, K. Bu, Z.-W. Liu, and G. Adesso (2019), Phys. Rev. Lett. 122, 140402.
[93] Takagi, R., T. J. Yoder, and I. L. Chuang (2017), Phys. Rev. A 96, 042302.
[94] Takagi, R., and Q. Zhuang (2018), Phys. Rev. A 97, 062337.
[95] Tasaki, H. (2016), Phys. Rev. Lett. 116, 170402.
[96] Theurer, T., D. Egloff, L. Zhang, and M. B. Plenio (2019), Phys. Rev. Lett. 122, 190405.
[97] Uola, R., T. Kraft, J. Shang, X.-D. Yu, and O. Gühne (2019), Phys. Rev. Lett. 122, 130404.
[98] Vedral, V., M. B. Plenio, M. A. Rippin, and P. L. Knight (1997), Phys. Rev. Lett. 78, 2275.
[99] Veitch, V., C. Ferrie, D. Gross, and J. Emerson (2012), New. J. Phys. 14 (11), 113011.
[100] Veitch, V., S. A. H. Mousavian, D. Gottesman, and J. Emerson (2014), New. J. Phys. 16 (1), 013009.
[101] Vidal, G., and R. Tarrach (1999), Phys. Rev. A 59, 141.
[102] Vollbrecht, K. G. H., and R. F. Werner (2001), Phys. Rev. A 64, 062307.
[103] Wakakuwa, E. (2017), arXiv:1709.07248.
[104] Wakakuwa, E., A. Soeda, and M. Murao (2017), IEEE Transactions on Information Theory 63 (8), 5372.
[105] Wakakuwa, E., A. Soeda, and M. Murao (2019), Phys. Rev. Lett. 122, 190502.
[106] Wang, X., M. M. Wilde, and Y. Su (2019), New J. Phys. 21, 103002.
[107] Wigner, E. P., and M. M. Yanase (1963), Proceedings of the National Academy of Sciences of the United States of America
49, 910.
[108] Winter, A. (2016), Communications in Mathematical Physics 347 (1), 291.

13[109] Yang, D., M. Horodecki, and Z. D. Wang (2008), Phys. Rev. Lett. 101, 140501.
[110] Yoder, T. J., R. Takagi, and I. L. Chuang (2016), Phys. Rev. X 6, 031039.
[111] Zanardi, P., C. Zalka, and L. Faoro (2000), Phys. Rev. A 62, 030301.
[112] Zhang, C., B. Yadin, Z.-B. Hou, H. Cao, B.-H. Liu, Y.-F. Huang, R. Maity, V. Vedral, C.-F. Li, G.-C. Guo, and D. Girolami(2017), Phys. Rev. A 96, 042327.