Universal limitations on implementing resourceful unitary evolutions
aa r X i v : . [ qu a n t - ph ] F e b Universal limitations on implementing resourceful unitary evolutions
Ryuji Takagi ∗ and Hiroyasu Tajima † Center for Theoretical Physics and Department of Physics,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Yukawa Institute for Theoretical Physics, Kyoto University,Oibuncho Kitashirakawa Sakyo-ku, Kyoto, 606-8502, Japan
We derive a trade-off relation between the accuracy of implementing a desired unitary evolutionusing a restricted set of free unitaries and the size of the assisting system, in terms of the resourcegenerating/losing capacity of the target unitary. In particular, this relation implies that, for anytheory 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 includingenergy, asymmetry, coherence, entanglement, and magic, imposing ultimate limitations inherent insuch important physical settings, as well as providing insights into operational restrictions in generalresource theories.
I. INTRODUCTION
One of the ultimate goals in quantum information sci-ence is to understand the operational enhancement madepossible by quantum phenomena as well as limitations onthe enhancement imposed by laws of quantum mechan-ics. This is not only an important theoretical questionbut also of practical relevance, as recent years have wit-nessed the burgeoning development in manipulation ofsystems on small scales, in which quantum effects playcentral roles.Any quantum information processing involves time evo-lution of quantum states, and the most fundamentalbuilding block for the quantum dynamics is unitary evo-lution. Even though general quantum dynamics is de-scribed by completely positive trace preserving (CPTP)maps, also called quantum channels, any channel actingon a system can be simulated by an appropriate unitaryoperation applied over the system and an environment[1], and thus any quantum evolution can be realized ifone has access to an arbitrary unitary. However, due totechnological limitations as well as restrictions imposedby laws of physics, physical systems usually do not allowone to apply an arbitrary unitary. This makes it essen-tial to consider to what extent a desired unitary dynamicscan be realized only using a limited set of accessible uni-taries. This question has been specifically addressed forthe systems with additive conserved quantities, in whichonly unitaries that respect the conservation laws can beapplied [2–8]. In particular, Ref. [7] has derived a lowerbound for the necessary amount of quantum fluctuationthat the ancillary state must possess to implement a de- ∗ [email protected] † [email protected] sired unitary in terms of its implementation accuracy andthe amount of energy that the target unitary can create,and they further derived lower and upper bounds thatalways match asymptotically in the region where the im-plementation error is small [8]. The presented boundslead to a fundamental no-go theorem that prohibits theperfect implementation of any unitary that can create en-ergy using an energy conserving unitary and finite-sizedancillary state.However, there are various settings where other typesof quantities can play the main role, and one can askwhether this type of trade-off relation is a general prop-erty shared by generic physical situations. This line ofthought naturally leads to the idea of resource theories,which are general frameworks that deal with quantifica-tion and manipulation of precious quantities considered“resource” under a given setting [9]. The resource the-oretic framework allows for systematic investigation onspecific physical settings [10–27] and has turned out tobe especially useful for providing a unifying operationalview to general class of quantities [28–42]. In this context,it can be seen that the previous works [7, 8] dealt witha specific theory (i.e. theory of asymmetry with U(1)group [16, 17]), and it has remained elusive whether onecan extend the relevant consideration to more generalresources.Here, we address the above question for the settingwhere 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 trade-off relations between the implementation accuracy, theamount of resources that the target unitary can change,and the size of the ancillary system, which are applicableto a wide class of physical settings that satisfy severallenient conditions. These relations immediately lead tono-go theorems that prohibit us from implementing anyresourceful unitary with perfect accuracy only using freeunitaries and aiding states defined in a system with finitesize, which qualitatively reproduces the results in [7, 8]as a special case. We also apply our results to severalimportant settings and discuss significance of the results.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 Sec-tion III, our first main result on the trade-off relationbetween accuracy, the amount of resources the target uni-tary 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 SectionV, we apply our results to various resources such as en-ergy, 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 RESOURCEMEASURES
Let H d denote the Hilbert space with dimension d and D ( H d ) be the set of density operators acting on H d . Also,let U F ( d ) ⊆ U ( d ) be some set of unitaries acting on H d and define U F := S d U F ( d ), which we call a set of freeunitaries . The set of free unitaries is usually determinedby the system of interest, and it can be most naturallyunderstood as free operations in the context of resourcetheories. A resource theory is specified by its set of freestates and free operations, which are considered givenfor free under the interested physical setting, and an im-portant requirement for free operations is that they arenot capable of creating any resources out of free states.For instance, for the setting where two parties are physi-cally separated apart, a reasonable theory comes with theset of separable states as free states and the set of localoperations and classical communication (LOCC) as freeoperations. Motivated by the resource theoretic consider-ations, we also define resource measures as the maps fromstates to non-negative real numbers. If one assumes someunderlying resource theory of quantum states, one natu-ral choice is to take resource monotones (which evaluatezero for free states and do not increase under applica-tion of free operations) defined in the theory as resourcemeasures.Once some resource theory is provided, one can nat-urally consider U F as the set of unitaries that are alsofree operations (e.g. the set of local unitaries for the caseof entanglement.) However, although considering the un- derlying resource theory is conceptually useful, for ourpurpose as long as the set of free unitaries is given, onedoes not necessarily need to assume an underlying struc-ture of the resource theory. Indeed, as we shall see laterit is sometimes convenient to only consider the set of freeunitaries, not explicitly taking into account the underly-ing set of free states. In the same vein, we do not imposethe monotonicity property for resource measures in gen-eral. Instead, we consider the following properties for aresource measure R determined by the given set of freeunitaries, which play major roles in later discussions. Property 1: (Invariance under free unitaries) R ( ρ ) = R ( V ρV † ) , ∀ V ∈ U F . Property 2: (Continuity) There exist non-negative in-creasing functions f , g with lim x → f ( x ) = 0, g ( x ) < ∞ , ∀ x < ∞ , and a real function h withlim x → h ( x ) = 0 such that | R ( ρ ) − R ( σ ) | ≤ f ( D ( ρ, σ )) g ( d ) + h ( D ( ρ, σ )) (1)for ρ, σ ∈ D ( H d ) 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 the fact that free unitaries donot change the resource contents attributed to quantumstates, and it is especially a natural property when appli-cation of a free unitary can be reversed by another freeunitary. Property 2 states that if two states are close toeach other, the amount of resources possessed by thesestates should be also close. Property 3 is the propertythat if a state is prepared independently of another state,the resource contents attributed to the two states are eval-uated as the sum of the amount of resources possessedby each state. As we see in Section V, these propertiesare shared by a number of known resource measures, andwe shall obtain ultimate bounds on implementation accu-racy of desired unitary in terms of the resource measuressatisfying these conditions.We also define the resource generating power and re-source losing power for unitary U [38, 43–46]: G U := max ρ (cid:8) R ( U ρU † ) − R ( ρ ) (cid:9) , (2) L U := − min ρ (cid:8) R ( U ρU † ) − R ( ρ ) (cid:9) . (3)Note that G U , L U ≥ U because there alwaysexists a state ρ that is invariant under U , for which onecan for instance take ρ = | u ih u | where | u i is an eigenstateof the unitary. III. IMPLEMENTATION OF RESOURCEFULUNITARIES
Once the concept of free unitaries is introduced, onecan ask what can be done with them and what are ulti-mate limitations imposed on the tasks accomplished bythe given free unitaries. One of the fundamental ques-tions that is both practically and theoretically importantis whether we can implement (or simulate) non-free uni-taries, which we call resourceful unitaries , only using freeunitaries with the aid of the ancillary system.More specifically, our aim is to simulate the given uni-tary U S on the Hilbert space H S by a channel Λ S imple-mented by a free unitary V SE ∈ U F acting on the Hilbertspace H S ⊗ H E and some ancillary state ρ E ∈ D ( H E ),i.e. Λ S ( · ) := Tr E [ V SE ( · ⊗ ρ E ) V † SE ] . (4)The tuple I := ( H E , V SE , ρ E ) defines a specific imple-mentation of the channel. A standard way of evaluatingthe closeness of two quantum channels is to see how closethe output states from these channels are when the chan-nels are allowed to act on only part of the input space.In order to take into account the worst-case input, wedefine the error for the given implementation I , which isa type of gate fidelity, as δ U S I := max ρ S δ U S I ( ρ S ) (5)where δ U S I ( ρ S ) := L e ( ρ S , Λ U † S ◦ Λ S ) , (6)Λ U ( · ) := U · U † (7)and L e ( ρ S , Λ) := p − F e ( ρ S , Λ)) , (8) F e ( ρ S , Λ) := q h ψ | SR [Λ ⊗ id R ]( ψ SR ) | ψ i SR . (9)where | ψ i SR is a purification of ρ S . A related distancemeasure is the Bures distance for two quantum states: L ( ρ, σ ) := p − F ( ρ, σ )) (10)where F ( ρ, σ ) := k√ ρ √ σ k is the Uhlman fidelity. Thechoice of this distance measure is primarily due to themathematical convenience in later discussions, but be-cause 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 distin-guishability. Then, we obtain the following trade-off relation be-tween resourcefulness of desired unitary, implementationaccuracy, and dimension of the ancillary system with re-spect to any resource measure satisfying the three prop-erties above. Theorem 1.
Let R be a resource measure satisfyingProperties 1, 2, 3 and f L , g L , h L , G U S , L U S be the func-tions defined in (1) , (2) , (3) with respect to R and theBures distance: D ( ρ, σ ) := L ( ρ, σ ) . Then, for any imple-mentation I , it holds that G U S + L U S ≤ α L ( δ U S I , d E ) + β L ( δ U S I ) . (11) where α L ( x, y ) := f L (2 √ x ) g L ( y ) + 2 f L (2 x ) g L ( d S · y ) , β L ( x ) := h L (2 √ x )+ 2 h L (2 x ) with d E := dim H E , d S :=dim H S . The proof of Theorem 1 can be concisely stated byutilizing the “no-correlation lemma” shown in [8], whichquantitatively clarifies the fact that in order to implementa unitary on the target system approximately, the corre-lation between the target system and the external devicemust become weak. We defer a detailed proof to the Ap-pendix. Note that α L and β L are increasing functionsthat 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 theancillary system and the implementation accuracy, andin particular the result indicates that in order to imple-ment a resourceful unitary the dimension of the ancillarysystem must grow as the implementation becomes bet-ter, and at the limit of perfect implementation the sizeof the ancillary system must diverge. Notably, Theorem1 holds for any resource measure that satisfies Properties1, 2, 3, which ensures a wide applicability of the trade-off relation. This observation immediately leads to thefollowing fundamental no-go theorem.
Corollary 2.
Given the set of free unitaries U F and afinite dimensional ancillary system H E with dim H E < ∞ , it is impossible to perfectly implement any unitarythat can generate (or lose) nonzero resources in terms ofat least one resource measure satisfying Properties 1, 2,3 by means of Eq. (4) . Theorem 1 and Corollary 2 suggest an important im-plication — one might think that if a target operationcan only create a certain amount of resource, supplyinga state defined in a finite-dimensional space with roughlythe same amount of resource would be enough to accom-plish the desired implementation. The above results statethat it is not the case when it comes to the unitary im-plementation, and Theorem 1 in particular provides aquantitative estimation of the necessary dimension evenwhen a non-zero error is allowed.It is also convenient to rewrite Theorem 1 in terms ofthe trace norm and diamond norm.
Corollary 3.
Suppose the implementation I =( H E , ρ E , V SE ) implements channel Λ S with the errormeasured by the diamond norm: δ U S I , ⋄ := k Λ U S − Λ S k ⋄ .Let R be a resource measure satisfying Properties 1, 2,3 and f , g , h , G U S , L U S be the functions definedin (1) , (2) , (3) with respect to R and the trace norm: D ( ρ, σ ) := k ρ − σ k . Then, it holds that G U S + L U S ≤ α ( δ U S I , ⋄ , d E ) + β ( δ U S I , ⋄ ) (12) where α ( x, y ) := f (cid:0) √ x (cid:1) g ( y ) + 2 f (4 √ x ) g ( d S · y ) and β ( x ) := h (cid:0) √ x (cid:1) + 2 h (4 √ x ) . This is a direct consequence from Theorem 1, but weinclude a proof in the Appendixes for completeness.
IV. RELAXATION OF ADDITIVITYCONDITION
Although a large class of resource theories possessgeneric resource measures that satisfy Property 1 andProperty 2, the additivity condition (Property 3) israther a peculiar one. In fact, classes of resource mea-sures that can be defined for any convex resource theory(e.g. relative entropy measure, robustness measure, con-vex roof measure etc.) are often only subadditive forproduct states. Thus, relaxing the additivity conditionis highly desired in order for the results to be applicableto more generic scenarios.Here, we relax the additivity condition into that for pure product states. It gives us much more freedom tochoose resource measures because some important mea-sures are additive only for pure product states. Exam-ples for such measures include relative entropy of entan-glement [47] and (logarithm of) stabilizer extent for thetheory of magic [48], which we discuss later in detail.To this end, we introduce a relaxed version of Prop-erty 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/losingpower for pure input states: G pU := max | ψ i (cid:8) R ( U | ψ ih ψ | U † ) − R ( | ψ ih ψ | ) (cid:9) (13) L pU := − min | ψ i (cid:8) R ( U | ψ ih ψ | U † ) − R ( | ψ ih ψ | ) (cid:9) . (14)For the same reason that G U , L U ≥
0, it also holds that G pU , L pU ≥ U .Then, we obtain the following trade-off relation. Theorem 4.
Let R be a resource measure satisfyingProperties 1, 2, 3’ and f L , g L , h L , G pU S , L pU S be the func-tions defined in (1) , (13) , (14) with respect to R and theBures distance: D ( ρ, σ ) := L ( ρ, σ ) . Then, for any im-plementation I = ( H E , V SE , ρ E ) with a pure state ρ E , itholds that G pU S + L pU S ≤ (cid:16) f L (2(1 + √ δ U S I ) g L ( d E d S ) + h L (2(1 + √ δ U S I ) (cid:17) . (15)A proof can be found in the Appendixes. It is worthnoting that R does not have to be defined for generalmixed states; as long as it is well-defined for pure states,the statement holds and the continuity (Property 2) canbe relaxed to that for pure states.This Theorem leads to a variant of the aforementionedno-go theorem on perfect implementability of a resource-ful unitary. Corollary 5.
Given the set of free unitaries U F and afinite dimensional ancillary system H E with dim H E < ∞ , 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 satisfyingProperties 1, 2, 3’ by means of Eq. (4) with ρ E being apure state. These results encompass a standard setup where someunit resource state (e.g. Bell state for entanglement, uni-form superposition state for coherence), which is usuallypure, is prepared in the ancillary system. Although usingthe unit state as a resource supply appears to be moreeffective than using a mixed state, interestingly the re-quirement for Theorem 4 to hold is more lenient thanthat for Theorem 1, imposing more severe restriction onthe achievable accuracy for the implementation with apure ancillary state.
V. APPLICATIONS
Here, we examine the validity of our results by apply-ing them to specific physical settings. Although thereis no systematic way of constructing a resource measuresatisfying the three properties to our knowledge, it turnsout 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 { S i } Mi =1 with an observable H tot = H ⊗ I ⊗ M − + I ⊗ H ⊗ I ⊗ M − + . . . where H i are local observables associ-ated with subsystem S i . For these observables, we choosethe set of free unitaries as the ones that conserve the ex-pectation values for any states, or equivalently, commutewith the observable. Namely, we choose U F = n U S ...S M (cid:12)(cid:12)(cid:12) [ H tot , U S ...S M ] = 0 o . (16)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 freeunitaries can be considered time evolutions that respectthe energy conservation law, which in particular play keyroles in thermodynamics on small scales [18, 19, 49–57].For this theory, natural resource measures one can takewill be the expectation value of the observable: R ( ρ S ) :=Tr[ ρ S H S ]. It is clear that this measure satisfies Property1 and 3. Regarding Property 2, let us take the observableof the form H S = P d S − j =0 j | j ih j | . Then, we get | R ( ρ ) − R ( σ ) | = | Tr[( ρ − σ ) H S ] | = | X j ( ρ jj − σ jj ) H S,j |≤ X j | ( ρ jj − σ jj ) || H S,j |≤ X j | ( ρ jj − σ jj ) |k H k ∞ = k ∆( ρ − σ ) k ( d S − ≤ k ρ − σ k ( d S −
1) (17)where ρ jj = h j | ρ | j i , σ jj = h j | σ | j i , H S,j = h j | H S | j i , ∆is the dephasing with respect to the eigenbasis of H S , andwe used the contractivity of the trace norm under CPTPmaps in the last inequality. Thus, for this case one cantake f ( x ) = x , g ( x ) = x , and h ( x ) = − x in Corollary 3,and we conclude that the finite dimensional environmentdoes not allow for perfect implementation of unitary thatchanges the energy by any energy-conserving unitary andan energy “battery” state, which qualitatively reproducesthe results in [7, 8]. Although we considered the observ-able with uniform spectrum, a similar argument can beapplied to other observables with more general form.It will be worth pointing out that this is a situationwhere our approach in which one does not necessarilyneed to assume any underlying resource theory becomesuseful, since the concept of free states and free operationsfor this setting can be ambiguous — from the perspectivethat the energy is resource, one could say that the groundstate | i is free, but in that case the set of free unitariesdefined 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 | i and | i ) also becomes free in 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 resourcetheory can be naturally identified as the resource theoryof asymmetry [16, 17]. In particular, the resource theoryof asymmetry with U(1) group with unitary representa-tion U t = e iH S t is equipped with a family of resourcemonotones that are additive for product states known asmetric-adjusted skew informations [58–60]. One of the ex-amples in this family is the well-known Wigner-Yanaseskew information [61, 62] defined as I W Y ( ρ, H S ) = −
12 Tr([ √ ρ, H S ] )= Tr( ρH S ) − Tr( √ ρH S √ ρH S ) . (18)Since this satisfies Property 1 and 3, Theorem 1 andCorollary 2 can be applied with respect to this measureas well, providing another way of looking at the trade-offrelation.Finally, when the observable of interest is the Hamilto-nian, the free unitaries in (16) preserve the Gibbs state τ = exp( − H S /T ) /Z where T is the temperature and Z is the partition function of the system. This motivates usto consider the “athermality”, a measure indicating thedistance from the Gibbs state to the given state, and es-pecially the free energy is recovered by taking the relativeentropy as a distance measure: A R ( ρ ) := S ( ρ || τ ) = 1 T ( F ( ρ ) − F ( τ )) (19)where F ( ρ ) := Tr[ ρH S ] − T S ( ρ ) is the free energy. Itis then easy to see that this also satisfies all the threeproperties. B. Coherence
Consider the theory of coherence [12–14] where oneis interested in the degree of superposition with re-spect to the given preferred basis {| i i} . For this the-ory, the set of incoherent states I := conv( {| i ih i |} )is a reasonable choice for the free states, and one cannaturally choose the relevant free unitaries U F ( d ) = n U (cid:12)(cid:12)(cid:12) U = P d − j =0 e iθ j | π ( j ) ih j | o where π is the permuta-tion on { , . . . , d − } , which is often called the set ofincoherent unitaries.As a resource measure, let us consider a standard co-herence measure, the relative entropy of coherence: C R ( ρ ) := min σ ∈ I S ( ρ || σ ) = S (∆( ρ )) − S ( ρ ) . (20)For this measure, it is easy to see that Property 1 is sat-isfied. The explicit form of C R in (20) ensures Property 3as well because of the additivity of the von Neumann en-tropy for product states. As for Property 2, recall thefollowing asymptotic continuity property that holds forrelative entropy measure M R ( ρ ) := inf σ ∈F S ( ρ || σ ) with F being any convex and closed set of positive semidefi-nite operators that contains at least one full-rank opera-tor [63]: | M R ( ρ ) − M R ( σ ) | ≤ κǫ + (1 + ǫ ) b (cid:18) ǫ ǫ (cid:19) (21)for any two states k ρ − σ k ≤ ǫ where κ :=sup τ,τ ′ { M R ( τ ) − M R ( 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: | C R ( ρ ) − C R ( σ ) | ≤ ǫ log d + (1 + ǫ ) b (cid:18) ǫ ǫ (cid:19) , (22)for which we find f ( x ) = x , g ( x ) = log x , and h ( x ) =(1 + x ) b ( x/ (1 + x )). Since this measure is also faithful,i.e. C R ( ρ ) = 0 iff ρ ∈ I , Corollary 3 implies that anycoherence generating unitary that can create a coherentstate out of an incoherent state cannot be implementedwith zero-error with the aid of any coherent state actingon a finite-dimensional ancillary system. C. Entanglement
Arguably, entanglement is one of the most importantresources to consider, which has a strong connectionto operational tasks in quantum information processing.In particular, using only local operations and classicalcommunication to implement desired global operationswith the help of preshared entanglement is a key idea ofquantum network and distributed quantum computing[64, 65], and methodology as well as necessary entangle-ment cost for implementing global gates with local oper-ations and classical communication have been consideredfor various settings [66–72]. Our formalism addresses amore restricted scenario where the parties only have ac-cess to local gates in order to implement a desired globalgate with the aid of preshared entanglement. Our resultsinduce necessary size of the shared entangled state andimply the impossibility of perfectly implementing any en-tangling gate with finite-sized aiding system. Since it isclearly possible to perfectly implement any global uni-tary if classical communication is allowed (via quantumteleportation), our results clarify the significance of classi-cal communication for the situations such as distributedquantum computing [73]. In order to apply our results, we need to find an en-tanglement measure satisfying the three properties. Inparticular, one needs to be careful about the additivityproperty since some well-known entanglement measures(e.g. such as the (max-)relative entropy of entanglement[74, 75], robustness of entanglement [76]) are only sub-additive even for product states, and it had been indeedan important program to find an additive measure ofentanglement. As a result, the squashed entanglementwas introduced as an additive entanglement measure [77],and its continuity was also shown [78]. In addition, theconditional entanglement of mutual information [79] wasintroduced as another additive and continuous measureof entanglement. Remarkably, this measure can be easilyextended to multipartite entanglement, which allows ourresults to be applied to the multipartite scenarios.On the other hand, Theorem 4 allows us to avoid thissubtlety and take an even simpler entanglement measure.For instance, the relative entropy of entanglement is ad-ditive for pure product states, as can be seen by not-ing 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-resilientfashion, which is so called fault-tolerant quantum compu-tation [80, 81], encoding quantum states into quantumerror correcting codes and carrying out logical compu-tation inside the code space is essential. Since manypromising error correcting codes allow for relatively ef-ficient implementation of the logical Clifford gates in afault-tolerant manner [82–86], for the situations wherethose codes are in use, Clifford gates can be naturallyconsidered “free”. However, since Clifford gates do notform a universal gate set, some non-Clifford gate needsto be implemented fault-tolerantly, and a popular way ofrealizing it is via the gate teleportation [87], in which“magic states” [88] are injected as resources of “non-Cliffordness”. Since good logical magic states are hardto prepare in general, a magic-state distillation protocol[88] should be run beforehand to increase the quality ofthe noisy magic states. However, a large overhead costcomes with the distillation protocols and how to reducethe overhead has been under active research [89–100] (er-ror correcting codes that avoid using the magic-state dis-tillation have also been investigated [101–108]), and thiscostly nature of magic states motivates us to consider theresource 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 con-vex combinations of pure states produced by Cliffordgates [23]. By definition, non-Clifford gates are able tocreate non-stabilizer sates out of stabilizer states, and asdescribed above it is an essential building block for univer-sal quantum computation. This operationally motivatedframework leads us to a natural question on how well anon-Clifford gate could be implemented by Clifford gateswith the aid of magic states as resources. Our results ad-dress this question by considering appropriate resourcemeasures for magicness. We consider the cases of qubits(dimension 2) and quopits (qudits with odd-prime dimen-sions) separately.
1. Qubits
Although one can consider valid magic monotones de-fined for multiqubit states (e.g. relative entropy of magic[23], robustness of magic [24]), they are not additive forproduct states in general, which prevents us from apply-ing Theorem 1. However, Theorem 4 turns out to beuseful in this case since there indeed exists a measure de-fined for pure states and additive for pure product mul-tiqubit states. To this end, consider the stabilizer extent introduced in [48]: ξ ( | ψ i ) := min X i | c i | ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ψ i = X i c i | φ i i (23)where | φ i i are pure stabilizer states. The stabilizer ex-tent was originally introduced for investigating the over-head cost for classically simulating quantum circuits, butwe find that it is also useful for our purpose, providinganother perspective to this measure. Let us take our re-source measure as R ( | ψ ih ψ | ) = log ξ ( | ψ i ). It was shownthat the stabilizer extent is multiplicative for tensor prod-ucts between states supported on up to three qubits [48],and thus R satisfies Property 3’. Property 1 is also sat-isfied because of the monotonicity of ξ under Cliffordgates and reversibility of Clifford unitary under anotherClifford unitary (since Clifford gates constitute a group).As for Property 2, we first remark that our measure co-incides with the max-relative entropy of magic for purestates as shown in Ref. [35], where the max-relative en-tropy measure is defined as D max ( ρ ) := min n r (cid:12)(cid:12)(cid:12) ρ (cid:22) r σ, σ ∈ STAB o (24)where STAB refers to the set of stabilizer states, and (cid:22) denotes the inequality with respect to the positivesemidefiniteness. Then, we prove the following continu-ity 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 stabilizerextent is derived as a special case of this result. It wouldbe also worth noting that the following result holds forthe max-relative entropy measure defined for any convexresource theory that includes the maximally mixed stateas a free state. (One can also easily extend the relationto the theories with at least one full-rank free state.) Proposition 6.
Let ρ, σ ∈ D ( H d S ) and suppose that k ρ − σ k < / (2 d S ) . Then, it holds that | D max ( ρ ) − D max ( σ ) | ≤ k ρ − σ k d S . (25)The proof is presented in the Appendixes. Our resultsprovide an interesting implication for implementation ofnon-Clifford gates. Suppose we are given qubits actingon system A and try to implement some non-Cliffordgate U NC on the subsystem A ⊂ A by applying Cliffordgates on A . Let N be the number of qubits supportedon the subsystem A \ A . Then, our results imply thatin order to realize the implementation accuracy ǫ withrespect to the diamond norm, the required number ofqubits N must scale as Ω (cid:18) log (cid:18) G pU NC + L pU NC √ ǫ (cid:19)(cid:19) . Thisobservation explicitly tells us the importance of measure-ment + feedforward (adaptive) operations for quantumcircuits to gain their power.
2. Quopits
For the case when the dimension of the system thateach qudit acts on is odd-prime, “mana” was introducedas a magic monotone [23]: M ( ρ ) := log X u | W ρ ( u ) | ! (26)where W ρ ( u ) is the discrete Wigner function for state ρ [109]. The mana essentially measures the total negativ-ity of the discrete Wigner function, which is motivatedby the fact that stabilizer states only take non-negativevalue for the discrete Wigner function. An importantproperty 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 con-tinuous (although it is not asymptotically continuous asshown in [23]), and Property 1 can be also easily seen bythe monotonicity of mana under Clifford gates and thefact that the application of Clifford gate can be reversedby another Clifford gate. Thus, Theorem 1 and Corollary2 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 [110]. However, the discrete Hud-son’s theorem [109] ensures that it is faithful for purestates, which is enough to show that any non-Cliffordunitary cannot be implemented with zero-error with afinite number of magic states. VI. TOWARD FULL GENERALITY
Although Theorem 4 covers most of the known impor-tant settings, one could still argue that some theory ofinterest may not come with a resource measure that satis-fies all the three properties, especially the additivity con-dition. Here, we focus on the qualitative no-go statementand see that it is quite unlikely for the perfect implemen-tation of resourceful unitary to be possible even in moregeneral settings.Suppose that free unitary V SE and pure state | φ i allow for an exact implementation of U S , i.e.Tr E h V SE ( ρ S ⊗ | φ ih φ | E ) V † SE i = U S ρ S U † S for any ρ S . Bytaking δ U S I = 0 in (C1), we getTr S h V SE ( ρ S ⊗ | φ ih φ | E ) V † SE i = σ ′ E (27)where σ ′ E is a pure state. Since states with pure reducedstates are only product states, we know that the totalstate must look like V SE ( ρ S ⊗ | φ ih φ | E ) V † SE = U S ρ S U † S ⊗ σ ′ E . (28)Then, we get for any ρ S and any measure R that is in-variant under free unitaries that R ( ρ S ⊗ | φ ih φ | ) = R ( V SE ( ρ S ⊗ | φ ih φ | ) V † SE )= R ( U S ρ S U † S ⊗ σ ′ E ) (29)Thus, for the given theory, unless any resource measurewith Property 1 (but not necessarily Property 2, 3, 3’)satisfies (29) for any ρ S , it is impossible to implementthe target U S exactly. Note that this is a very strongrestriction, and when R is additive for product states,Corollaries 2 and 5 are reproduced.Let us impose another natural condition on R that itbe a subadditive monotone for some resource theory inwhich composition of free states and partial trace are freeoperations. For such cases, one can show that R ( | φ ih φ | ) = R ( σ ′ E ) as follows. Take a free state τ S and η S = U † S τ S U S .Then, we get R ( | φ ih φ | ) ≥ R ( τ S ⊗ | φ ih φ | )= R ( U S τ S U † S ⊗ σ ′ E ) ≥ R ( σ ′ E ) (30)and R ( σ ′ E ) ≥ R ( U S η S U † S ⊗ σ ′ E )= R ( η S ⊗ | φ ih φ | ) ≥ R ( | φ ih φ | ) . (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 unitariesand (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) holdsfor any ρ S for resourceful unitary U S since it would indi-cate that attaching ancillary states with the same amountof resources to two states with different amount of re-sources would necessarily produce the states with thesame amount of resources. We leave the thorough analy-sis on how general the no-go statement can be made forfuture work. VII. CONCLUSIONS
We considered a general setting where one aims to im-plement a target unitary with access to a restricted set ofunitaries as well as ancillary system. We derived a trade-off 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 proper-ties: invariance under free unitaries, continuity, and ad-ditivity for product states. Using this relation, we pre-sented a fundamental no-go theorem on the perfect imple-mentation of resourceful unitaries with finite-dimensionalancillary systems. We further relaxed the subtle condi-tion in the above three properties, additivity for productstates, and showed an analogous trade-off relation thatonly requires the resource measures to be additive forpure product states, in addition to the other two prop-erties. We exemplified the wide validity of our resultsby applying them to various important settings and dis-cussed the physical significance implied by the results forspecific settings. We finally discussed the feasibility of ex-tending our no-go results to even more general settingsthat do not assume all the properties for the resourcemeasures we considered.For future work, it will be intriguing to clarify whethersome of the required properties for resource measures con-sidered in this work can be dropped to obtain a similartrade-off relation. It will also be interesting to investigatehow good our lower bounds are in general by constructingupper bounds with explicit protocols that approximatelyimplement desired unitaries.
Note added . — Recently, we became aware of the in-dependent related work by Chiribella, Yang, and Renner[111].
ACKNOWLEDGMENTS
We thank Tomoyuki Morimae for fruitful discussions.R. T. acknowledges the support of NSF, ARO, IARPA,and the Takenaka Scholarship Foundation. H. T. ac-knowledges the support of JSPS (Grants-in-Aid for Sci-entific Research No. JP19K14610).
Appendix A: Proof of Theorem 1
We first retrieve the main lemma we use for the readers’convenience.
Lemma 7 (No-correlation lemma [8]) . Let Λ AB be achannel on the composite system AB and U A be a unitaryoperation on A . We consider three possible initial statesof A : ρ (0) A , ρ (1) A , and ρ (0+1) A := ( ρ (0) A + ρ (1) A ) / and writethe initial state of B as ρ B . We refer to the final statesof AB and B with the initial state ρ ( i ) A ( i = 0 , , ) as σ ( i ) AB := Λ AB ( ρ ( i ) A ⊗ ρ B ) , (A1) σ ( i ) B := Tr A [ σ ( i ) AB ] . (A2) Let Λ A be the channel implemented by the implementa-tion I = ( H E , Λ AB , ρ B ) , i.e. Λ A ( · ) := Tr B [Λ AB ( · ⊗ ρ B )] and write the accuracy of implementation of U A with im-plementation I for input state ρ ( i ) A as δ U, ( i ) I := δ U I ( ρ ( i ) A ) as in (7) . Then, for any U A and I , we have the followingrelations:1. It holds that L ( σ ( i ) AB , U A ρ ( i ) A U † A ⊗ σ ( i ) B ) ≤ δ U A , ( i ) I . (A3)
2. There exists a state σ ′ (0+1) B of B such that L ( σ (0) B , σ ′ (0+1) B ) + L ( σ ′ (0+1) B , σ (1) B ) ≤ √ δ U A , (0+1) I . (A4) Moreover, if ρ B is a pure state and Λ AB is a unitaryoperation, one can take a pure state for σ ′ (0+1) B . We are now in a position to prove Theorem 1.
Proof.
Define ρ ( i ) S , i = 0 , ρ (0) S := argmax( R ( U S ρ S U † S ) − R ( ρ S )) (A5) ρ (1) S := argmin( R ( U S ρ S U † S ) − R ( ρ S )) (A6)and corresponding final states on SE and E as σ ( i ) SE := V SE ( ρ ( i ) S ⊗ ρ E ) V † SE , (A7) σ ( i ) E := Tr S [ σ ( i ) SE ] . (A8) Due to Property 1 and 3 of the resource measure R , wehave R ( ρ ( i ) S ) + R ( ρ E ) = R ( σ ( i ) SE ) . (A9)Using (A3), we get L ( σ ( i ) SE , U S ρ ( i ) S U † S ⊗ σ ( i ) E ) ≤ δ U S I . (A10)Due to Property 2 of R and (A9), (A10), we obtain | R ( ρ ( i ) S ) + R ( ρ E ) − R ( U S ρ ( i ) S U † S ) − R ( σ ( i ) E ) |≤ f L (2 δ U S I ) g L ( d E d S ) + h L (2 δ U S I ) . (A11)Using the triangle inequality and (A11), we get | R ( ρ (0) S ) − R ( U S ρ (0) S U † S ) − R ( σ (0) E ) − R ( ρ (1) S ) + R ( U S ρ (1) S U † S ) + R ( σ (1) E ) |≤ (cid:16) f L (2 δ U S I ) g L ( d E d S ) + h L (2 δ U S I ) (cid:17) . (A12)Another use of the triangle inequality leads to | R ( σ (0) E ) − R ( σ (1) E ) |≥ | R ( U S ρ (0) S U † S ) − R ( ρ (0) S ) − R ( U S ρ (1) S U † S ) + R ( ρ (1) S ) |− (cid:16) f L (2 δ U S I ) g L ( d E d S ) + h L (2 δ U S I ) (cid:17) = G U S + L U S − (cid:16) f L (2 δ U S I ) g L ( d E d S ) + h L (2 δ U S I ) (cid:17) (A13)where we used G U S , L U S ≥ R , we get | R ( σ (0) E ) − R ( σ (1) E ) |≤ f L (2 √ δ U S I ) g L ( d E ) + h L (2 √ δ U S I ) . (A14)Combining (A13) and (A14), we finally obtain G U S + L U S ≤ f L (2 √ δ U S I ) g L ( d E ) + h L (2 √ δ U S I )+ 2 (cid:16) f L (2 δ U S I ) g L ( d E d S ) + h L (2 δ U S I ) (cid:17) = α L ( δ U S I , d E ) + β L ( δ U S I ) . (A15) Appendix B: Proof of Corollary 3
Proof.
Recall the relation between the Bures distance andthe trace distance [112]12 ( L ( ρ, σ )) ≤ k ρ − σ k ≤ L ( ρ, σ ) , (B1)0which also implies δ U S I ≤ q δ U S I , ⋄ . Then, (A3) and (A4)imply 12 k σ ( i ) SE − U S ρ ( i ) S U † S ⊗ σ ( i ) E k ≤ q δ U S I , ⋄ (B2)and 12 k σ (0) B − σ (1) B k ≤ q δ U S I , ⋄ (B3)Then, 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 ρ E is pure ensures that there exists a pure state σ ′ E thatsatisfies (A4), namely L ( σ ( i ) E , σ ′ E ) ≤ L ( σ (0) E , σ ′ E ) + L ( σ (1) E , σ ′ E ) ≤ √ δ U S I . (C1)Then, we obtain L ( σ ( i ) SE , U S ρ ( i ) S U † S ⊗ σ ′ E ) ≤ L ( σ ( i ) SE , U S ρ ( i ) S U † S ⊗ σ ( i ) E )+ L ( U S ρ ( i ) S U † S ⊗ σ ( i ) E , U S ρ ( i ) S U † S ⊗ σ ′ E ) ≤ δ U S I + L ( σ ( i ) E , σ ′ E ) ≤ √ δ U S I (C2)where in the first inequality we used the triangle inequal-ity, in the second inequality we used (A3) and the factthat L ( ρ ⊗ σ, ρ ⊗ τ ) = L ( σ, τ ), and in the third inequalitywe used (C1).Let ρ (0) S and ρ (1) S be pure states that achieve (13) and(14) respectively. Then, Property 1 and 3’ of R lead to R ( σ ( i ) SE ) = R ( ρ ( i ) S ) + R ( ρ E ) . (C3)and R ( U S ρ ( i ) S U † S ⊗ σ ′ E ) = R ( U S ρ ( i ) S U † S ) + R ( σ ′ E ) . (C4)Combining Property 2, (C2), (C3), (C4), we get | R ( ρ ( i ) S ) + R ( ρ E ) − R ( U S ρ ( i ) S U † S ) − R ( σ ′ E ) |≤ f L (2(1 + √ δ U S I ) g L ( d E d S ) + h L (2(1 + √ δ U S I ) . (C5)Hence,0 = R ( ρ E ) − R ( σ ′ E ) + R ( σ ′ E ) − R ( ρ E ) ≥ R ( U S ρ (0) S U † S ) − R ( ρ (0) S ) − R ( U S ρ (1) S U † S ) + R ( ρ (1) S ) − (cid:16) f L (2(1 + √ δ U S I ) g L ( d E d S ) + h L (2(1 + √ δ U S I ) (cid:17) = G U S + L U S − (cid:16) f L (2(1 + √ δ U S I ) g L ( d E d S ) + h L (2(1 + √ δ U S I ) (cid:17) , (C6) which proves the statement. Appendix D: Proof of Proposition 6
Proof.
We assume D max ( ρ ) ≥ D max ( σ ) without loss ofgenerality. The definition of max-relative entropy mea-sure (24) admits the following dual form [113]:maximize log Tr[ ρX ]subject to X (cid:23) τ X ] ≤ , ∀ τ ∈ STAB . (D1)Let X ρ be an optimal solution that achieves (D1) forstate ρ . Then, we obtain D max ( σ ) ≥ log Tr[ σX ρ ] ≥ log (Tr[ ρX ρ ] − k ρ − σ k k X ρ k ∞ )= D max ( ρ ) + log (cid:18) − k ρ − σ k k X ρ k ∞ Tr[ ρX ρ ] (cid:19) ≥ D max ( ρ ) + log (1 − k ρ − σ k d S ) ≥ D max ( ρ ) − k ρ − σ k d S (D2)The first inequality is because X ρ is a suboptimal so-lution for σ . The second inequality is because of thesame argument in (17). The third inequality is becauseit holds that k X ρ k ∞ ≤ d S from the second constraintin (D1) together with the fact that the maximally mixedstate I /d S is a stabilizer state, and that Tr[ ρX ρ ] ≥ I serves as a suboptimal solution for X that givesTr[ ρ I ] = 1. The fourth inequality is because it holds thatlog(1 − x ) ≥ − x for 0 ≤ x ≤ / k ρ − σ k < / (2 d S ). Note also that the logarithmin (D2) is always well-defined because Tr[ ρX ρ ] ≥ k ρ − σ k k X k ∞ ≤ /
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