Quantum Superreplication of States and Gates

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Front. Phys. 11(3), 110304 (2016) DOI 10.1007/s11467-016-0556-7 REVIEW ARTICLE Quantum superreplication of states and gates

Giulio Chiribella†, Yuxiang Yang

Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China Corresponding author. E-mail: †[email protected] Received December 18, 2015; accepted January 8, 2016

Although the no-cloning theorem forbids perfect replication of quantum information, it is sometimes possible to produce large numbers of replicas with vanishingly small error. This phenomenon, known as quantum superreplication, can occur for both quantum states and quantum gates. The aim of this paper is to review the central features of quantum superreplication and provide a uniﬁed view of existing results. The paper also includes new results. In particular, we show that when quantum superreplication can be achieved, it can be achieved through estimation up to an error of size O(M/N2), where N and M are the number of input and output copies, respectively. Quantum strategies still oﬀer an advantage for superreplication in that they allow for exponentially faster reduction of the error. Using the relation with estimation, we provide i) an alternative proof of the optimality of Heisenberg scaling in quantum metrology, ii) a strategy for estimating arbitrary unitary gates with a mean square error scaling as log N/N2,andiii) a protocol that generates O(N 2) nearly perfect copies of a generic pure state U|0 while using the corresponding gate U only N times. Finally, we point out that superreplication can be achieved using interactions among k systems, provided that k is large compared to M 2/N 2. Keywords quantum cloning, quantum metrology, quantum superreplication, Heisenberg limit, quantum networks PACS numb ers 03.67.-a, 03.67.Ac, 03.65.Wj, 03.67.Bg

Contents provides a working principle for quantum cryptography with applications to key distribution [5, 6], unforgeable 1 Introduction 1 banknotes [7], and secret sharing [8]. The no-cloning the- 2 Quantum state superreplication 3 orem forbids perfect cloning. A natural question, origi- 3 The relation between state superreplication and nally asked by Buˇzek and Hillery [9], is how well cloning state estimation 8 can be approximated by the processes allowed by quan- 4 Superreplication with reduced interaction size 8 tum mechanics. The question is relevant both to crypto- 5 Quantum gate superreplication 11 graphic applications and to the foundations of quantum 6 Applications of gate superreplication 13 theory, shedding light on the relationship between quan- 7 Conclusion and outlook 17 tum and classical copy machines [10–15] and providing Acknowledgements 17 benchmarks that certify the advantages of quantum in- References 17 formation processing over classical information processing [16–20]. Due to the broad spectrum of applications, 1 Introduction the research into optimal cloning machines is still an ac- tive and fruitful line of investigation [21]. The no-cloning theorem [1, 2] is one of the cornerstones Among the cloning machines allowed by quantum me- of quantum information theory, with implications per- chanics, one can distinguish two types: deterministic and meating the entire ﬁeld [3, 4]. Most famously, the im- probabilistic machines. Deterministic machines produce possibility of copying non-orthogonal quantum states approximate copies with certainty, whereas probabilistic machines sometimes produce a failure message indicat- ing that the copying process has gone wrong. In general, ∗Special Topic: Quantum Communication, Measurement, and Computing (Eds. G. M. D’Ariano, Youjin Deng, Lu-Ming Duan probabilistic machines can produce more accurate copies &Jian-WeiPan). at the price of a reduced probability of success. For ex-

c The Author(s) 2016. This article is published with open access at www.springer.com/11467 and journal.hep.com.cn/fop REVIEW ARTICLE ample, Duan and Guo [22] showed that a set of non- When δ is zero, this means that the number of extra orthogonal states can be cloned without error as long as copies scales as N, evading the limitation posed by the they are linearly independent. More recently, Fiuráˇsek asymptotic no-cloning theorem. When δ is larger than [23] showed that probabilistic cloners can offer an advan- zero, the number of extra copies grows faster than N,and tage even for linearly dependent states, including, e.g., one can produce up to M =Θ(N 2) copies. In both cases, coherent states of harmonic oscillators with known am- we refer to this phenomenon as superreplication, empha- plitude. Ralph and Lund [24] proposed a concrete optical sizing that the number of extra copies grows beyond the setup achieving noiseless probabilistic amplification and limits imposed by the asymptotic no-cloning theorem. cloning of coherent states. The possibility of noiseless Note, however, that the number of output copies allowed probabilistic amplification was later extended to the case byEq.(2)cangrowatmostatarateM =Θ N 2− , of a Gaussian-distributed coherent state amplitude [19]. where is an arbitrarily small constant. In finite dimen- Although the probability of success vanishes as the ac- sions, the quadratic replication rate is the ultimate limit curacy increases, nearly perfect amplification of coherent for superreplication of clock states, in close connection states has been observed experimentally for small values with the Heisenberg limit in quantum metrology [29, 30]. of the amplitude [25–28]. In addition to producing a large number of extra The recent developments in probabilistic amplification copies, probabilistic cloners exhibit better scaling of the and cloning of coherent states motivate the search for error, which vanishes as exp[−cN 2/M (N)] for a suitable new scenarios where non-orthogonal states can be cloned constant c>0 depending on the particular set of clock with vanishingly small error. In [29], we considered the states under consideration. In contrast, the error for task of cloning clock states, that is, quantum states gen- deterministic machines can scale at most as 1/N 4 [29]. erated by time evolution under a known Hamiltonian. The benefits of probabilistic cloners, however, are not Here, the cloner is given N identical copies of the same without cost. The price to pay is a very small proba- clock state and attempts to produce M = M(N)copies. bility of success; precisely, the probability of superrepli- Inspired by the asymptotic framework of information cation has to be small compared to exp [−M/N]. This theory, we considered the scenario where N is large. Here, fact severely limits the ability to observe superreplica- themainquestionishowfastM(N) can grow under the tion, especially when M grows much faster than N.Nev- condition that the copying process is reliable, meaning ertheless, the cloner can be devised in such a way that, that the global error of all the output copies vanishes in when superreplication fails, the original input state is the large-N limit. For deterministic cloners, we showed almost undisturbed. Precisely, Winter’s gentle measure- that the number of reliable copies has to scale as ment lemma [31] implies that the error introduced√ by the failure of superreplication scales as O p ,where M(N)=N [1 + f(N)] with lim f(N)=0. (1) succ N→∞ psucc is the probability of success. For superreplication That is, the number of extra copies produced by the processes with δ>0, this means that the error intro- cloner must be negligible compared to the number of in- duced by the failure of superreplication vanishes faster δ/ put copies. This result can be seen as an asymptotic no- than exp −N 2 . As a consequence, the state resulting cloning theorem, which extends the original no-cloning from failed superreplication can still be used to achieve theorem from perfect cloners to cloners that become per- standard, deterministic cloning with asymptotically op- δ/ fect in the asymptotic limit. The asymptotic no-cloning timal performance; for large N, the error exp −N 2 is theorem originates from the requirement that the error covered by the error introduced by deterministic cloning. vanishes globally on all the clones. By using tomography, In summary, superreplication is a rare event, but trying it is generally easy to produce copies that individually to observe it does not prevent the application of standard have an error vanishing on the order of 1/N . However, deterministic cloning techniques. this does not guarantee that the error vanishes at the Interestingly, the limitation on the probability of suc- global level. Indeed, the errors accumulate when mul- cess is lifted if we consider the problem of cloning gates tiple copies are examined jointly. Consequently, only a instead of states. Cloning a quantum gate means simu- cloner satisfying Eq. (1) can have a vanishing error at lating M uses of it while actually using it only N

110304-2 Giulio Chiribella and Yuxiang Yang, Front. Phys. 11(3), 110304 (2016) REVIEW ARTICLE nian. They devised a quantum network that simulates up tocol reduces the size of the interactions by a factor to N 2 uses of an unknown phase gate by using it only N M/N2−, which is asymptotically large for all the times. The network works deterministically and has van- replication rates allowed by Eq. (2). ishing error on average over all input states. We refer to 4) We construct a simple protocol that estimates an this phenomenon as gate superreplication. The discovery unknown unitary gate with a mean square error by Dür et al. opened the question of whether superrepli- of log N/N2. The protocol uses gate superreplica- cation can be achieved not only for phase gates, but also tion to produce M =Θ(N 2/ log N) copies and gate for arbitrary quantum gates. In Ref. [34], we answered tomography to estimate the gate within a mean the question in the affirmative, constructing a univer- square error of size 1/M . The resulting scaling of sal quantum network that replicates completely unknown the error is close to the optimal scaling 1/N 2 [35– unitary gates. The network has vanishing error on almost 38]. all input states, except for a vanishingly small fraction 5) We analyze the task of generating M copies of a of the Hilbert space. These “bad states” can be charac- quantum state U|0 given N uses of a completely terized explicitly, making it easy to identify applications unknown unitary gate U [34]. In this setting we where gate superreplication can be safely employed. show that M =Θ(N 2−) copies of the state can be In this paper, we review the key facts about supergenerated with fidelity going to one in the large-N replication of states and gates, unifying the ideas pre- limit for every >0. sented in the literature and emphasizing the connections 6) We examine replication of quantum gates in the between superreplication and other tasks, such as esti- worst case over all possible input states. In the mating quantum gates and generating states using gates worst-case scenario, we establish an asymptotic no- as oracles. In addition to this review, the paper contains cloning theorem for quantum gates,whichstates a number of new results: that the number of reliable copies of the gate scales as M(N)=N[1 + f(N)], where f(N)vanishesin the large-N limit. 1) We show that whenever quantum superreplication is achievable, it can also be achieved through es- The paper is organized as follows. Section 2 introduces timation. However, the error for estimation-based the task of asymptotic cloning and precisely analyses the strategies will vanish according to a power law, phenomenon of quantum state superreplication. Section whereas the error for genuine quantum strategies 3 analyses the relationship between state superreplica- vanishes faster than any polynomial. tion and quantum metrology. Section 4 discusses the 2) We establish an equivalence between the optimal- scale of the interactions needed for state superreplica- ity of Heisenberg scaling in quantum metrology and tion. In Section 5, we review the existing results for su- the ultimate limit on the rate of superreplication, perreplication of quantum gates. Several applications of which is encapsulated in Eq. (2). On the one hand, gate superreplication are presented in Section 6. Finally, Heisenberg scaling implies the impossibility of pro- conclusions are drawn in Section 7. ducing more than M = O(N 2)copiesofaclock state with vanishing error [29]. On the other hand, here we prove the converse result, showing that the 2 Quantum state superreplication limit on the replication rate set by Eq. (2) implies the optimality of Heisenberg scaling. Deterministic cloners. Although perfect cloning of 3) We explore the possibility of achieving superrepli- non-orthogonal quantum states is forbidden by the no- cation without acting globally on all N input copies cloning theorem [1, 2], approximate cloning can be re- and on M − N additional blank copies. Specifically, alized to various degrees of accuracy, depending on the we consider strategies where the N input copies set of states to be cloned [10–15]. Consider a scenario in are divided into subgroups and each subgroup gen- which one is given N identical systems, each prepared erates new copies, mimicking a scenario in which in the same state |ψx, and the goal is to generate M groups of cells fuse together and then split into ap- systems in a state close to M perfect copies of the state proximate clones. For replication strategies of this |ψx. For the moment, we assume the cloning process form, we show that interactions among groups of to be deterministic, meaning that it produces approxi- O(M 2/N 2−) particles are necessary and sufficient mate clones with unit probability. Mathematically, a de- to achieve superreplication. With respect to the terministic cloner is described by a completely positive original superreplication protocol, the modified pro- trace-preserving linear map C, a.k.a. a quantum channel,

Giulio Chiribella and Yuxiang Yang, Front. Phys. 11(3), 110304 (2016) 110304-3 REVIEW ARTICLE sending states on the Hilbert space H⊗N to states on the The asymptotic no-cloning theorem. Inspired by Hilbert space H⊗M ,whereH is the Hilbert space of a the asymptotic framework of information theory, we now single copy. The cloning accuracy can be quantified as focus on quantum cloners in the limit N →∞.Wemodel the average fidelity between the ideal state of M perfect the cloning process as a sequence of cloners (CN )N∈N, copies and the actual output state of the cloner. Explic- where CN transforms N copies into M(N) approximate itly, the average fidelity is given by copies for a given function M(N). We call the cloning process reliable if the cloning fidelity goes to one in the F N →M p |ψ ψ |⊗M C |ψ ψ |⊗N , det[ ]= xTr x x x x asymptotic limit, namely, x lim Fdet[N → M(N)] = 1 . (3) N→∞ where px is the prior probability of the state |ψx.The The key question here is how many extra copies can be quantum channel that maximizes the fidelity is the opti- produced reliably. Can we produce N extra copies or mal N-to-M cloner for the ensemble {|ψx ,px}.Ingen- more? To provide a rigorous answer, we define the rate eral, the form of the optimal channel varies for different of a cloning process as ensembles, depending on both the states and their prior log[M(N) − N] α := lim inf . probabilities. N→∞ log N That is, if the rate is α, the number of output copies Example: the equatorial qubit cloner. The gen- grows as M(N)=N +Θ(N α). We say that a cloning eral settings of optimal cloning can be nicely illus- rate α is achievable if and only if there exists a reliable trated in the example of equatorial qubit states [39– cloning process that has a rate equal to α.Foragivenset 41]. Here, the ensemble√ consists of states of the form −it of states, the task is to find the maximum achievable rate |ψt =(|0 +e |1)/ 2, where θ is drawn uniformly at over all cloning processes. For deterministic processes, random from the interval [0, 2π). The input state can be the following theorem tightly limits the number of copies expanded as that can be produced reliably. N N ⊗N 1 −int |ψt = e |N,n, Theorem 1 (Asymptotic no-cloning theorem for quan- 2N/2 n n=0 tum states [29]). No deterministic process can reliably α where {|N,n|n =0,...,N} is the orthonormal basis clone a continuous set of quantum states at a rate 1. of the Dicke states, which are defined as The intuition for the proof comes from the standard 1 ⊗(N−n) ⊗n quantum limit of metrology [30], which limits the preci- |N,n := Uπ |0 |1 , N! n!(N − n)! sion of deterministic estimation of a parameter t encoded π∈SN into N product copies of a state |ψt. The standard quan- where π is a permutation of N qubits, Uπ is the unitary tum limit states that the mean square error vanishes as operator that implements the permutation π,andthe c/N,wherec is a constant that depends on the encoding S sum running over the symmetric group is N . t →|ψt. Intuitively, a deterministic cloner that violates For convenience of discussion, we focus on the case Theorem 1 would contradict the standard quantum limit, where both N and M are even. In this case, the optimal because one could increase the precision by first cloning † cloning channel has the simple form C(ρ)=VρV ,where the probe states and then measuring them. V is the isometry, which is defined as [41] The argument based on the standard quantum limit of N M N N metrology is partly heuristic. A complete argument can V |N, + m = |M, + m ,m∈ − , . 2 2 2 2 be made for clock states, i.e., quantum states of the form −itH † Plugging the above relation into the definition of the av- |ψt =e |ψ ,t∈ R ,H = H. erage fidelity (3), one obtains the optimal value The argument is conceptually independent of the stan-

Fdet[N → M] dard quantum limit and allows one to prove a strong ⎡ ⎤ 2 converse of the asymptotic no-cloning theorem. N/2 1 ⎣ N M ⎦ = N M Theorem 2 (Strong converse of the asymptotic no- 2N+M + m + m m=−N/2 2 2 cloning theorem [29]). Every deterministic process that √ α> 2 MN clones clock states at a rate 1 will necessarily have ≈ N,M 1 . (4) N M + N a vanishing ﬁdelity in the large- limit.

110304-4 Giulio Chiribella and Yuxiang Yang, Front. Phys. 11(3), 110304 (2016) REVIEW ARTICLE

The asymptotic no-cloning theorem implies that a de- where p(x|yes) is the conditional probability given by terministic cloning process cannot produce more than Bayes’ rule. Inserting Eqs. (5) and (6) into the above a negligible number of extra replicas in the asymptotic expression, we obtain the explicit formula limit. Let us check a few examples. The first example is F [N → M] the cloning of equatorial qubit states introduced earlier prob ⊗M ⊗N N pxTr |ψxψx| M (|ψxψx| ) in the paper. In√ the large- limit, the cloning fidelity is x yes . = ⊗N (8) given by F ≈ 2 MN/(M +N); cf. Eq. (4). It is straight- y py Tr [Myes(|ψyψy| )] forward to see that any achievable cloning rate must The maximum fidelity over all possible quantum instru- be smaller than one. Another example is the universal ments represents the ultimate performance allowed by cloning of pure states. For d-dimensional quantum sys- d N− d M− quantum mechanics, even if the probability of success is tems, the optimal fidelity is F = + 1 / + 1 [42]. N M arbitrarily small. In the following, we will focus on the For large N and M, the fidelity scales as (N/M)d−1 and maximum fidelity in the asymptotic limit of large N and converges to one if and only if the cloning rate satisfies M. the condition α<1. The example of universal cloning shows that one can always find a deterministic process that reliably produces M = N +Θ(N δ) copies for every Superreplication of equatorial qubit states. Let desired exponent δ>0. us start from the simple example of equatorial qubit It is important to stress that the asymptotic no-cloning states. In this case, the optimal probabilistic cloner is theorem holds for continuous sets of states but does not given by a quantum instrument with successful opera- M ρ QρQ† place any restriction on the cloning rate of discrete sets tion yes( )= given by of states. For example, every finite set of quantum states M N M m M can be cloned with vanishing error in the large-N limit, Q |N, m 2+ |M, m , + = M N + M 2 M+N N 2 no matter how large is [15]. Hence, for finite sets of +m 2 2 states, every rate is achievable. N N m ∈ − , (9) Probabilistic cloners. The asymptotic no-cloning the- 2 2 orem poses a stringent limit on the ability to replicate (recall that we are assuming that N and M are even information. In the following we will see that the limit for simplicity). Inserting the expression for the optimal can be broken by allowing the cloner to be probabilistic. cloner into the fidelity (8), we obtain the optimal value To analyze probabilistic cloners, it is useful to intro- N/2 duce the notion of a quantum instrument [43, 44]. A 1 M F [N → M]= quantum instrument consists of an indexed set of com- prob 2M M + m m=−N/2 2 pletely positive, trace non-increasing maps (a.k.a. quantum operations), {M , M ,...}. When an input state ρ N 2 1 2 1 − 2exp − , (10) is fed into the quantum instrument, the quantum oper- 2M M p M ρ ation i occurs with probability i =Tr[ i( )], and where we have used Hoeffding’s inequality. One can im- the instrument outputs a quantum system in the state mediately see that any cloning rate α<2isachievable; ρ M ρ /p = i( ) i. In the context of our cloning problem, we in this case, the cloning error vanishes faster than the consider a quantum instrument consisting of two quan- inverse of any polynomial in N. Interestingly, such rapid M M M tum operations, yes and no,where yes describes decay of the error is impossible for deterministic cloners, the realization of a successful cloning process. When act- whose error can vanish at most as 1/N 4 [29]. In sum- N |ψ ⊗N ing on the -copy input state x , the probabilistic mary, allowing the cloner to claim failure yields two ad- cloner succeeds with probability vantages: i) the cloning rate can be boosted beyond the ⊗N p(yes|x)=Tr Myes(|ψxψx| ) , (5) limits of the asymptotic no-cloning theorem, and ii) the cloning error vanishes as N at a speed that could not be M in which case it produces the -copy output state achieved by deterministic cloners. The contrast between ⊗N deterministic and probabilistic cloners can also be seen in ρx = Myes(|ψxψx| )/p(yes|x) . (6) the non-asymptotic setting; for example, when N =20 Conditional on the success of the cloning process, the and M = 120, the probabilistic fidelity maintains the average fidelity is equal to high value of 94.52%, whereas the deterministic fidelity ⊗M drops to 69.39%. Fprob[N → M]= p(x|yes) Tr |ψxψx| ρx ,(7) x The optimal probabilistic cloner dramatically outper-

Giulio Chiribella and Yuxiang Yang, Front. Phys. 11(3), 110304 (2016) 110304-5 REVIEW ARTICLE forms all deterministic cloners in terms of rate and ac- replicationcan be achieved with a success probability 1− curacy. However, these advantages have a cost; the price pyes > exp −M/N for every >0. to pay is that the probability of success vanishes in the Informally, we can think of superreplication as a lot- large-N limit. For example, the probability of success for tery where one invests the initial copies of the state in the the qubit cloner of Eq. (9) is given by hope of obtaining a much larger number of approximate ⊗M ⊗N 2 copies. Despite the low probability of success, the super- ψt| Q|ψt p [N → M]= replication lottery has almost no risk of loss; indeed, it yes ⊗N † ⊗N 2 |ψt| Q Q|ψt | is possible to guarantee that when replication fails, the N/2 cloner returns the N input copies, up to an error that 1 M , ∀t ∈ , π . is vanishingly small compared to exp [−M/(2N)]. This = N M M [0 2 ) 2 N+M + m 2 m=−N/2 2 result follows from the gentle measurement lemma [31]. Because the error is small, the input copies can still be Using Stirling’s approximation, one can see that the used for deterministic cloning. Recall that the cloning above probability satisfies the asymptotic equality error of deterministic cloners vanishes at most as 1/N 4. −N − N (ln 2 M ) pyes[N → M] ≈ e ,N M. Hence, the error introduced by the failed attempt at superreplication is negligible compared to the cloning error. This example indicates a trade-off between the cloning In summary, one can achieve the same asymptotic per- rate and the success probability. In the following, we will formance as the optimal deterministic cloning while still see that such a trade-off is a generic feature of state su- allowing for the chance to observe the exotic superrepli- perreplication. cation phenomenon. General case: superreplication of quantum Building on this observation, we can construct a cloner clocks. Superreplication of equatorial qubit states can that makes a series of attempts to copy clock states be generalized to the broad family of clock states of the probabilistically, where every failed attempt results in −itH † form |ψt =e |ψ with t ∈ R and H = H.Forev- slight deterioration of the input copies [45]. This cloner ery family of clock states, one can find a probabilistic increases the probability of success, at the price of a per- N-to-M cloner that achieves a fidelity [29] formance that decreases with the number of attempts. In Fig. 1, we show a plot of the fidelity as a function p2 N 2 N F N → M − K −2 min 4 of the total probability of success for equatorial qubit prob[ ] 1 2 exp M + MK states with N =50andM = 1000. The plot shows a independently of t. Here, K is the number of distinct en- high fidelity/low probability region corresponding to the ergy levels of the Hamiltonian H,andpmin is the smallest first attempts and a rapid decrease in the fidelity with non-zero probability in the probability distribution of the increasing number of attempts. energy for the state |ψ. It is immediately obvious that every cloning rate α<2isachievable,andthusthat The Heisenberg limit for superreplication. We clock states can be superreplicated. Again, the cloning error vanishes with N faster than the inverse of every polynomial—a rapid scaling that could not be achieved by deterministic cloners. In other words, the probabilistic cloner produces not only more but also better replicas. The exponential decay of the success probability is a necessary condition for superreplication of clock states; indeed, every reliable superreplication process must have a success probability that is vanishingly small compared to exp [−M/N]. In other words, as soon as a cloning process violates the asymptotic no-cloning theorem, the probability of the process must vanish in the large-N Fig. 1 Replication of equatorial qubit states through successive attempts of probabilistic cloning. The figure shows the tradeoff be- limit. A strong converse result can also be proven [29]: tween fidelity and success probability for N =50andM = 1000. Theorem 3 Every cloning process with rate α 1 and The points on the blue line represent successive attempts to generate M copies through the optimal probabilistic process. The black success probability of order exp [−M(N)/N ] or larger dashed line represents the fidelity of the optimal deterministic will necessarily have vanishing fidelity. cloner. In this example, the errors introduced by failed attempts cause the fidelity to fall below the optimal deterministic fidelity On the other hand, it is possible to show that super- when the probability of success becomes larger than 20%.

110304-6 Giulio Chiribella and Yuxiang Yang, Front. Phys. 11(3), 110304 (2016) REVIEW ARTICLE have seen that clock states can be superreplicated for ev- reliable and can have every desired rate α,thusbreach- ery cloning rate smaller than two. Higher cloning rates ing the Heisenberg limit α<2. The catch is, of course, are forbidden by the following theorem. that the probability of success will decay faster for processes with a higher rate. Pandey et al. provide a general Theorem 4 (Heisenberg limit for superreplication [29]) bound [46], implying that a replication process with rate In finite dimensions, no physical process can achieve a α must satisfy the relation cloning rate larger than two for a set of states containing 2 α clock states. pyes[N → M] exp −(α − 1)r N ln N . The limit on the cloning rate originates in the Heisen- State superreplication beyond clock states. Super- berg limit in quantum metrology [30], which can be ex- replication of quantum clocks implies superreplication of tended to probabilistic strategies in the case of finite- other relevant families of states, including the multiphase dimensional systems [29]. The intuitive idea is the follow- covariant states ing: the Heisenberg limit implies that the probabilistic d− ⊗N √ 1 √ estimation of t from the state |ψt has a minimum iθj |ψθ = p |0 + pje |j,θj ∈ [0, 2π) , N −2 0 mean square error scaling as . If one could produce j=1 M =Θ(N α) clones with sufficiently small error, these ∀j =1,...,d− 1 . clones could be used to estimate t. Performing individual measurements of the clones and collecting the statis- To see that these states can be superreplicated, it tics, one could make the mean square error as small as is enough to consider the family of√ clock states α −itH d−1 O(1/N ). Clearly, the Heisenberg limit implies α 2. |ψt =e |ψ,where|ψ = pj |j, H = √ j=0 The argument based on the Heisenberg limit is par- d−1 n |jj| n j j=0 j ,and j is the -th prime number. With tially heuristic because is relies on the assumption that this choice, the closure of the set {|ψt|t ∈ R} co- the replication error vanishes sufficiently rapidly. How- incides with the full set of multiphase covariant states, ×d−1 ever, it is possible to make a complete argument based {|ψθ|θ ∈ [0, 2π) }. On the other hand, an N-to-M on direct optimization of the probabilistic cloner. This cloner for the clock states will achieve a fidelity of argument is conceptually independent of the Heisenberg 2p2 N 2 4N limit and makes it possible to prove a strong converse F N → M − d − min , prob[ ] 1 2 exp M + Md [29]. pmin ≡ min pj Theorem 5 (Strong converse of the Heisenberg limit for j superreplication) Every physical process that replicates independently of t. By taking limits, we then obtain that clock states with replication rate α>2 will necessarily every multiphase covariant state can be cloned with the have vanishing fidelity, no matter how small its probabil- same fidelity. ity of success is. Another multiparameter family of states that can be Note, however, that the restriction to finite- superreplicated is the family of all maximally entan- dimensional systems is essential for the above conclu- gled states. The superreplication of maximally entangled sions. For infinite-dimensional systems, the probabilistic states will be discussed in detail in Section 5. Heisenberg bound can easily become trivial. For exam- It is important to note that not all quantum states ple, consider the set of all coherent states with fixed can be superreplicated. First, the symmetry of the input amplitude r>0, which can be seen as an infinite- ensemble can sometimes inhibit superreplication. For in- dimensional example of clock states: stance, if one tries to clone an arbitrary unknown finite- dimensional state, the probabilistic cloner will yield the ∞ n † 2 r −ita a −r /2 same fidelity as the optimal deterministic cloner [29]. The |ψt =e |ψ , |ψ =e |n , 0 0 n! n=0 lack of probabilistic advantages for universal cloning is an √ appealing feature from the viewpoint of the many-worlds where a |n = n |n − 1, ∀n ∈ N.Now,coherentstates interpretation of quantum mechanics [47] and can even with known amplitude can be probabilistically cloned be viewed as an axiom: with a fidelity arbitrarily close to 1 [23] by using Ralph and Lund’s noiseless probabilistic amplifier [24]. This Axiom 1 (Many-Worlds Fairness) The maximum rate means that we can pick every function M(N) and build a at which a completely unknown state can be cloned is the probabilistic cloner that transforms N input copies into same in all possible worlds. M(N) approximate copies with an error smaller than, say, 1/N . The cloning process constructed in this way is Interestingly, Many-Worlds Fairness rules out the vari-

Giulio Chiribella and Yuxiang Yang, Front. Phys. 11(3), 110304 (2016) 110304-7 REVIEW ARTICLE ant of quantum mechanics on real Hilbert spaces originally considered by Stueckelberg [48] and recently an- 3 The relation between state superreplication alyzed by Hardy and Wootters [49, 50]. The argument and state estimation for excluding quantum mechanics on real Hilbert spaces is the following: if the wavefunction had only real am- Superreplication via state estimation. A funda- plitudes, then one branch of the wavefunction would al- mental fact about quantum cloning is its asymptotic low one to superreplicate every state of a quantum bit, equivalence with state estimation [10–15]; when the num- whereas the other branches would not, in violation of ber of output copies becomes large, the optimal cloning Many-Worlds Fairness. The fact that ordinary quantum performance can be achieved by a classical cloning strat- mechanics allows for complex amplitudes results in the egy that consists of measuring the input copies and us- Many-World Fairness property and in the impossibility ing the outcome of the measurement as an instruction to of superreplicating completely unknown pure states. prepare the output copies. In the standard setting, one N An interesting open question is what are the proba- fixes the number of input copies and lets the num- M bility distributions on the Bloch sphere that allow for ber of output copies go to infinity. For probabilistic superreplication, interpolating between the uniform dis- cloning, this scenario has been analyzed by Gendra et tribution (for which superreplication is impossible) and al. [53], who proved the asymptotic equivalence between a distribution concentrated on the equator (for which probabilistic cloning and probabilistic state estimation. f superreplication is possible). Similarly, it is interesting For clock states that are dense in an -dimensional to wonder whether there exists a family of optimal clon- manifold, Gendra et al. showed that the probabilis- F N → M ∝ N 2/M f/2 ers that interpolate between the optimal universal cloner tic fidelity decays as prob[ ] ( ) for M N 2 and the optimal probabilistic cloner that achieves su- , whereas the deterministic fidelity decays as F N → M ∝ N/M f/2 perreplication. Such interpolation between universal and det[ ] ( ) .Anopenquestioniswhether non-universal cloning was known in the deterministic probabilistic estimation also makes it possible to achieve case [51], while the probabilistic case is still open. superreplication. We now show that this is indeed the Another limitation on superreplication can be ob- case. served for coherent states of a harmonic oscillator Consider a classical cloning strategy for the states {|ψx|x ∈ X} based on probabilistic estimation of the parametrized as ⊗M parameter x and on preparation of the state |ψxˆ , ∞ n 2 z wherex ˆ is the estimate. This is not the most general |z =e−|z| /2 |n ,z∈ C . n! classical strategy, but it will be sufficient to establish the n=0 possibility of superreplication. The successful quantum When z is chosen at random from a Gaussian distribu- operation is given by −λ |z|2 tion p(z)=λ/π e , λ>0, the optimal probabilistic ⊗M M (ρ)= |ψxψx| Tr[Px ρ] , (12) fidelity is given by [19] yes ˆ ˆ ˆ xˆ∈X (1+λ)N M ,λ<− 1, where {Px}x∈X are positive operators satisfying the nor- F [N → M]= M N (11) prob ,λ M − . malization condition 1 N 1 Px + P = I From the above expression, we see that the fidelity van- ? x∈X ishes for all replication rates α>1. This fact is in stark P contrast to the situation for coherent states with known for some operator ? 0 associated with the unsuccess- amplitude [23, 24], for which every rate is achievable. ful outcome of the estimation strategy. The fidelity of the Eq. (11) implies that superreplication with a rate strategy can be computed using Eq. (7), which gives α = 1 is still possible, provided that λ is sufficiently 2M F [N → M]= p(x|yes) p(ˆx|x) |ψx|ψx| large. For example, for λ 1, one can probabilistically prob ˆ x,xˆ transform N copies into M =2N copies, thus violat- E |ψ |ψ |2M , ing the asymptotic no-cloning theorem. Here it is impor- = ( x xˆ ) tant to stress the role of prior information; in its absence where E denotes the expectation with respect to the (λ = 0), the fidelity of the optimal probabilistic cloner is probability distribution p(x, xˆ|yes) := p(x|yes) p(ˆx|x). equal to the fidelity of the optimal deterministic cloner Using the convexity of the function f(y)=yM ,wethen originally proposed by Braunstein et al. [52]. obtain the inequality

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2M 2 M N |ψ E(|ψx|ψxˆ| ) E(|ψx|ψxˆ| ) , tum limit using copies of the clock state t to estimate the parameter t deterministically with or equivalently an error scaling as 1/N 1+, >0. Then, Eq. (13) M M Fprob[N → M] (Fprob[N → 1]) , would imply that one can produce clones with M =Θ(N 1+/2), thus violating Theorem 2. M which means that the -copy fidelity of our measure- 2) Suppose that one could violate the Heisenberg limit M and-prepare strategy is lower-bounded by the -th of probabilistic metrology using N copies of the power of the single-copy fidelity. Now, suppose that the clock state |ψt to estimate the parameter t proba- /N β probabilistic single-copy fidelity approaches one as 1 bilistically with an error scaling as 1/N 2+ , >0. β> for some 0, namely, Then, Eq. (13) would imply that one can produce M clones with M =Θ(N 2+/2), thus violating The- F N → − O 1 . prob[ 1] 1 N β orem 5. Then, Bernoulli’s inequality yields the bound In conclusion, we have shown a complete equivalence M between the limits on the scaling of the error in quantum Fprob[N → M] 1 − O . (13) N β metrology and the limits on the replication rate set by From the bound, it is clear that every replication rate Theorems 2 and 5. α<βcan be achieved; the fidelity converges to one M N /N β N as long as ( ) vanishes in the large- limit. In 4 Superreplication with reduced interaction summary, we have proven the following. size Theorem 6 Let S = {Ux | x ∈ X} be a continuous set of gates. A probabilistic strategy that estimates the gates The divide-and-clone approach. To realize the op- N M in S with fidelity larger than 1 − O(1/N β) can be used timal -to- probabilistic cloner, a global interaction M to replicate the states in S at every rate α<βvia a involving at least systems is required. This can be measure-and-prepare strategy. seen in the example of the equatorial qubit cloner, which is defined as an evolution acting coherently on the ba- Theorem 6 applies to arbitrary sets of states and to sis of the Dicke states; cf. Eq. (9). Of course, the need probabilistic strategies with arbitrary constraints on the for interactions among large numbers of systems makes probability of success. For clock states, assuming no con- it challenging to implement cloning. Here we investigate straint on the probability of success, the single-copy fi- the possibility of reducing the scale of the interactions. delity of phase estimation exhibits the Heisenberg scal- A simple strategy to reduce the interaction scale is a F N → − O /N 2 ing prob[ 1] = 1 1 [54]. Hence, the clas- “divide-and-clone” strategy in which one divides the in- sical strategy described above achieves superreplication put copies into groups and performs optimal cloning on for every replication rate α<2. Note, however, that each group. Suppose that we divide the N input copies the classical superreplication strategy has an error that into groups of N := Θ(N β) copies. Applying the optimal 2−α vanishes with the power law 1/N , whereas the quan- N -to-M probabilistic cloner to each individual group, tum superreplication strategy has an error that vanishes we then achieve a fidelity faster than every polynomial. p2 N 2 N F N → M − K −2 min( ) 4 Deriving precision limits from superreplication. [ ] 1 2 exp M + M K We have seen that the replication rate for clock states is M determined by the Heisenberg limit in the probabilistic for each group. The value of canbechoseninor- M case and by the standard quantum limit in the determin- der to reach the desired replication rate; to obtain M istic case. On the other hand, we have also seen strong copies overall, one needs to satisfy the condition M M N/N M converse results (Theorems 2 and 5) that prove the opti- = . For a replication process producing = N 1+δ M N δ+β mality of the replication rates without invoking the pre- Θ( ) copies, the condition yields =Θ( ). N/N cision limits of quantum metrology. In fact, the quantum Because there are groups, the overall fidelity of metrology limits can be derived from our bounds on the this strategy is replication rates. The argument proceeds by contradic- F [N → M]=(F [N → M ])N/N tion: 1 − Θ(N 1−β)exp −Θ N β−δ .

1) Suppose that one could violate the standard quan- From the above bound, it is immediately clear that the ﬁ-

Giulio Chiribella and Yuxiang Yang, Front. Phys. 11(3), 110304 (2016) 110304-9 REVIEW ARTICLE delity converges to one whenever β>δ. Hence, quantum cloner after K − 1 steps, which result in the generation clocks can be superreplicated with interactions among of KN approximate copies. At the first step, the input M = O N 2δ+ = O M 2/N 2− particles for every state is the N-copy state desired >0. Now, recall that the original superrepli- N N cation strategy [29] requires interactions among O(M) 1 −int |Ψ t, = e |N,n , particles. This means that the modified strategy reduces 0 2N/2 n 1 n=0 the interaction scale by a factor of N 2−/M . N On the other hand, the modified strategy does not where the subscript 1 indicates the first group of | make it possible to achieve superreplication with inter- qubits. The first cloner transforms Ψt,0 into a state on actions among less than Θ(N δ) systems. Indeed, super- the first and second group, yielding 2N qubits in the replication can be achieved only if the fidelity of the in- state dividual processes approaches one. By the Heisenberg N N N 2 −int limit, this condition is satisfied only if N and M sat- | t, N | N, n , Ψ 1 = 1 N/ n e 2 + 1 2 M N 2 n 2+ 2 isfy the asymptotic relation ( ) . Recalling =0 β δ that M has to scale as N + and that N scales as N N N β N −int N , we then obtain that β>δis a necessary con- = 1 e n − x1 N/2+x1 dition. In summary, the strategy of dividing the input n=0 x1∈Sn,1 copies into non-interacting groups achieves superreplica- N ·|N,n − x11|N, + x12, tion with M =Θ(N 1+δ) output copies if and only if the 2 N δ M/N size of each group grows faster than = . where N1 is a normalization constant, and the set Sn,1 is defined via the relation A sequential cloning approach. Another approach to N N Sn,K = x := (x1,...,xk) xj ∈ − , , reducing the scale of interactions is to generate clones by 2 2 local mechanisms. For example, we can imagine a cloning j process that involves repeated interactions among O(N) xl ∈ [n − N,n] , ∀ j ∈{1,...,K} . (14) systems. Let us analyze this idea for equatorial qubit l=1 states; here we consider a sequence of cloners that take Iterating the cloning process for K − 1 steps, we finally N to 2N copies arranged in the following circuit: obtain K groups of qubits in the joint state N −int |Ψt,K−1 = NK−1 e cn,x

n=0 x∈Sn,K−1 K−1 N N ·|N,n− xi |N, + x ···|N, + xK− K , 1 2 1 2 2 1 i=1 Each wire in the circuit represents a composite sys- c tem of N qubits. At each step, the quantum opera- where n,x is a coefficient given by tion M(N) performs the optimal N-to-2N probabilistic N K−1 N cloning given by the map cn,x := K−1 . n − xi N/2+xj i=1 j=1 N 2N |N, m−→ N+m | N,N m , NK + 2N N 2 + The overall fidelity of the cloner with the state of 2 3N N m 2 2 + perfect copies does not depend on t and is given by N N N ∀m ∈ − , , N F N → KN 1 2 2 seq[ ]= NK K−1 2 n − i xi n=0 x∈Sn,K−1 =1 M N ⎡ ⎤ as one can see by replacing with 2 inEq.(9).Inthe K−1 N following, it will be convenient to express the map as · ⎣ ⎦ . (15) N/2+xj 2N j=1 N/2+n N |N,n−→ |2N, + n , ∀n ∈ [0,N] . 2N N For K>2, the cloning fidelity (15) is strictly 3N n 2 2 smaller than the fidelity (8) of the optimal probabilis- We now analyze the performance of the sequential tic cloner. Fig. 2 compares the sequential cloner, optimal

110304-10 Giulio Chiribella and Yuxiang Yang, Front. Phys. 11(3), 110304 (2016) REVIEW ARTICLE

given access to a black box implementing an unknown quantum gate. In these applications, the uses of the gate are a resource; indeed, a no-cloning theorem for gates as- serts that it is impossible to perfectly simulate two uses of an unknown gate by using it only once [32]. Still, a natural question is: how well can we simulate M uses of the unknown gate with N

Fig. 2 Comparison between the three cloners. In this figure we that superreplication of quantum gates is possible if and compare the performance of the network cloner to the one of the only if one can find a sequence of networks with the prop- probabilistic cloner as well as the one of the deterministic cloner. erty that the simulation error vanishes in the asymptotic The fidelity is plotted as a function of the output number, fix- N limit and the number of extra copies of the input gate ing the input number to = 8. The green line (with numerics N represented by red dots) represents the fidelity-output curve for grows as or faster. In the following, we will see that the network cloner. The blue line (with numerics represented by gate superreplication can be achieved deterministically red dots) stands for the fidelity-output curve for the probabilistic for almost all input states. The first result of this type cloner, and the black line (with numerics represented by black dots) was discovered by Dür et al. [33] for phase gates, that stands for the fidelity-output curve for the deterministic cloner. is, gates of the form probabilistic cloner, and optimal deterministic cloner for d−1 −i nt N = 8. Note that the fidelity of the sequential cloner Ut = e |nn| , ∀t ∈ [0, 2π) . decays quickly with the number of cloning steps. In our n=0 example, the fidelity of the sequential cloner approaches The performance of the simulation was quantified by the that of the deterministic cloner as the number of out- fidelity between the output state of the simulation and M N 2 put copies approaches = = 64. This numerical the output state of M perfect uses of the gate Ut,aver- result suggests that it may not be possible to achieve su- aged over t and over all possible input states. Using a de- perreplication sequentially. However, it is an open ques- terministic network, the authors showed how to simulate tion whether superreplication can be achieved by the the M N 2 uses with asymptotically unit fidelity. This re- sequential cloner or by similar mechanisms. For exam- sult can be extended from phase gates to arbitrary gates ple, one could construct a quantum cellular automaton [34], as discussed in the following paragraphs. for cloning, as proposed by D’Ariano et al. [55]. They considered a tree-shaped network of deterministic 1-to- Universal gate superreplication. Given N uses of a 2 cloners. To achieve superreplication, one would have unitary gate, the goal is to simulate M parallel uses of to extend the model, allowing for probabilistic cloners. (N) the gate. Let us denote by CU the quantum channel Moreover, symmetry arguments imply that probabilistic that results from inserting N uses of the unknown gate 1-to-2 cloners do not offer any advantage over their de- in the cloning network. With this notation, the average terministic counterparts [29]. Hence, one has to consider fidelity is given by local cloners with more input copies. An interesting possibility is to construct a probabilistic cellular automaton Fgate[N → M] where the building blocks are the optimal k-to-2k prob- U | U † ⊗M C(N) | | U ⊗M | , abilistic cloners with k>1. = d dΨ Ψ ( ) U ( Ψ Ψ ) Ψ (16) 5 Quantum gate superreplication

We have seen that state superreplication necessarily has a vanishing success probability in the asymptotic limit. Here we analyze the task of replicating quantum gates, where superreplication can be achieved with unit probability of success on average over all input states. Fig. 3 Quantum gate cloning. A quantum network for gate cloning. Given N uses of an unknown unitary gate U (green boxes Gate superreplication. In many applications, includ- on the left), the network (blue boxes on the left) simulates M>N ing quantum metrology and quantum algorithms, one is uses of U (green boxes on the right).

Giulio Chiribella and Yuxiang Yang, Front. Phys. 11(3), 110304 (2016) 110304-11 REVIEW ARTICLE where dU is the normalized Haar measure over the group ity spaces can be eliminated without losing any informa- of all unitary gates, and dΨ is the normalized Haar tion about U. Likewise, they can be inflated by adding measure over the manifold of pure M-partite states. ancillas. The working principle for simulation of U ⊗M By Markov’s inequality, a simulation with gate fidelity given U ⊗N will be to conveniently adjust the size of the Fgate[N → M] 1 − works with fidelity F 1 − δ on multiplicity spaces, as in the following protocol. all pure states except for a small fraction, whose proba- Protocol 1 (Gate superreplication network [34]) bility is smaller than /δ. The replication network is constructed as follows: A replication process is described as a sequence of networks in which the N-th network transforms N copies of 1) Choose an ancilla A such that the rank of IjN ⊗ IA the unknown gate into M(N) approximate copies. We is no smaller than the rank of IjM for every j ∈ say that the replication process is reliable if and only if [0,N/2]; 2) Compose the gate U ⊗N with the identity on A,thus lim Fgate[N → M(N)] = 1 . N→∞ U ⊗N ⊗ I N/2 U ⊗ I ⊗ obtaining the gate A = j=0 ( j jN In other words, for a reliable replication process, one IA); F N → M N − ⊗N has [ ( )] √1 N for some N converging to 3) Project the resulting gate U ⊗ IA into the sub- zero. Choosing δN = N , we then have that the replica- space tion process simulates the desired gate with fidelity larger N/2 than 1−N on all pure states except for a small fraction, ⊗N √ HN = (Rj ⊗MjM ) ⊂H ⊗HA, whose probability is smaller than N /δN = N .Asin j=0 the case of state replication, we say that a process has a replication rate where the gate acts as log[M(N) − N] N/2 α = lim inf , U U ⊗ I . N→∞ log N = ( j jM ) j=0 and we say that the rate α is achievable if and only if H there exists a reliable replication process with that rate. 4) Embed the subspace N into the Hilbert space H⊗M H In the following, we will see that every rate smaller . With a slight abuse of notation, we use N H⊗N ⊗H than 2 is achievable. For simplicity, we discuss the case both for the subspace of A and for the cor- H⊗M of qubit gates, although the protocol can be extended responding subspace of . M in a straightforward way to unitary gates in arbitrary 5) Given an input state of qubits, perform a pro- finite-dimensional quantum systems. jective measurement that projects the qubits either H The key to construction of the universal gate cloning into the subspace N or into its orthogonal com- H⊥ H network is decomposition of the Hilbert space of K plement N . If the qubits are projected into N , U qubits into orthogonal subspaces corresponding to dif- then apply the gate . If the qubits are projected H⊥ ferent values of the total angular momentum. In the for- into N , then perform the identity. mula, one has The network described by Protocol 1 perfectly imi- ⊗M ⊗M K/2 tates U for any state in the subspace HN ⊂H . ⊗K H (Rj ⊗MjK) , Now, it is possible to show that a random pure state in j ⊗M =0 H has a large overlap with the subspace HN when- 2 where j is the quantum number of the total angular mo- ever M N . Indeed, we have the following lemma. mentum, Rj is a representation space, and Mj is a mul- Lemma 1 Let FN be the fidelity between the pure state tiplicity space [56]. With respect to this decomposition, ⊗M |Ψ∈H and its projection |ΨN = PN |Ψ/PN |Ψ. the action of K parallel uses of a generic qubit unitary The expectation value of the fidelity over the uniform U has the block-diagonal form measure is given by K/2 ⊗K E F |P | U (Uj ⊗ IjK) , ( N )= dΨ Ψ N Ψ j=0 Tr[PN ] = where IjK is the identity on the multiplicity space MjK. 2M ⊗K Note that U acts nontrivially only in the represen- N/2 djmjM tation spaces. This means that essentially, the multiplic- = 2M j=0

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N 2 O M/N N 1 − (M +1)exp − . qubits to ( ) qubits by dividing the input gates 2M into groups of O(N 2/M ) gates. This fact can be proven by the same argument as that used for quantum states. In the above inequality, dj and mjM are the dimensions of the representation space Rj and the multiplicity space MjM , respectively. The detailed proof of the last 6 Applications of gate superreplication inequality can be found in the supplemental material of Superreplication of maximally entangled states. Ref. [34]. Using the above lemma, it is easy to show that Inspired by gate superreplication, one can construct a the fidelity is close to one whenever M is small compared protocol for superreplicating maximally entangled states to N 2. Indeed, the cloning fidelity (16) is lower-bounded that achieves any cloning rate α<2 [34]. For qubits, as we can cast an arbitrary maximally entangled state in + + F [N → M] the form |ΦU =(U ⊗ I)|Φ ,where|Φ is the Bell gate √ state, |Φ+ =(|00 + |11)/ 2. The replication protocol † ⊗M † ⊗M dU dΨΨ|(U ) U PN |ΨΨ|PN U U |Ψ works as follows: First, the unitary gate U can be extracted from the state via a gate teleportation protocol 2 = dΨ (Ψ|PN |Ψ) [62]. The success probability of a single gate extraction is 1/4, which implies that U ⊗N can be extracted from N 2 copies of the maximally entangled state with probabil- dΨ Ψ|PN |Ψ ity (1/4)N . The extracted gates are then used in the gate 2 U ⊗M =[E(FN )] superreplication network, which simulates the gate M N α α< N 2 with vanishing error whenever grows as , 2. 1 − 2(M +1)exp − . Finally, the approximation of U ⊗M is applied locally to 2M M copies of the Bell state |Φ+, thus providing M ap- In conclusion, the gate fidelity converges to one for all proximate copies of the state |ΦU . The fidelity of the replication processes with replication rate α<2, estab- replicas can be bounded as lishing the possibility of gate superreplication. Similar d + ⊗M † ⊗M bounds can be found for arbitrary -dimensional sys- Fent[N → M]= dU Φ | (U ⊗ I) tems, in which case we have · C(N) ⊗I | + +|⊗M U ⊗ I ⊗M | +⊗M d(d−1) N 2 U Φ Φ ( ) Φ 2 Fgate[N → M] 1 − 2(M +1) exp − , 2M (2M +1)F [N → M] − 1 = gate ∀d<∞. (17) 2M The inequality follows from the concentration of the where the relationship between the entanglement fidelity Schur–Weyl measure [57, 58], which has applications and the gate fidelity derived by Horodecki et al. [63] was in quantum estimation [59] and information compres- used. The above equality implies that the fidelity of the sion [60, 61]. The bound (17) allows one to produce state cloning protocol goes to one if and only if the fi- M =Θ(N 2−) copies with an error that vanishes su- delity of gate superreplication goes to one. Therefore, the perpolynomially fast. In addition, one can produce M = above protocol can replicate maximally entangled states Θ[N 2/ log N] copies with an error vanishing faster than at every rate α<2. 1/N k for every desired k>0. For this purpose, it is The idea of gate teleportation followed by gate super- enough to choose replication can also be applied to achieve superreplication of other families of states. For instance, gate su- N 2 M = perreplication yields an alternative superreplication pro- 2(d2 − d + k)logN tocol for equatorial qubit states. Given√ N copies of an so that Eq. (17) becomes equatorial qubit state (|0 +e−it|1)/ 2, one can first generate N copies of the maximally entangled qubit state 1 √ F [N → M] 1 − O , (|00 +e−it|11)/ 2byapplyingaCNOTgatetoeach gate N k log N d(d−1)/2 input copy. Then, the maximally entangled qubit state N d. (18) can be superreplicated. Finally, applying a CNOT gate Finally, we note that the size of the interactions used in to each output copy makes it possible to transform the the superreplication network can be reduced from O(M) (approximate) copies of the maximally entangled qubit

Giulio Chiribella and Yuxiang Yang, Front. Phys. 11(3), 110304 (2016) 110304-13 REVIEW ARTICLE state into (approximate) copies of the equatorial qubit rate α 2 must have vanishing fidelity. Hence, the pos- state. The net result of this protocol is superreplication sibility of achieving gate superreplication at rates larger of equatorial qubit states. than quadratic is excluded. Note that the above argu- Similar arguments apply to superreplication of max- ment applies not only to deterministic gate replication imally entangled states in arbitrary finite dimensions. networks, but also to probabilistic networks. Using the Every maximally entangled state can be√ parametrized as correspondence between entanglement fidelity and gate | U ⊗ I | + | + / d d−1 |n|n ΦU =( ) Φ ,where Φ =1 n=0 . fidelity, we obtain the following. Again, the unitary gate U can be probabilistically ex- 2 Theorem 7 Every physical process that replicates phase tracted from the state |ΦU with probability p =1/d . gates with a replication rate α>2 will necessarily have Hence, maximally entangled states can be superrepli- vanishing gate fidelity (no matter how small the proba- cated by i) simulating M copies of the gate U and ii) bility of success). applying the simulated gates locally to M copies of the state |Φ+. The fidelity is then given by An alternative optimality proof for the quadratic replication rate was presented by Sekatski et al. [64], who de- F N → M U +|⊗M U † ⊗ I ⊗M ent[ ]= d Φ ( ) vised an argument to bound the replication fidelity based on the no-signaling principle. (N) + + ⊗M ⊗M + ⊗M · CU ⊗I |Φ Φ | (U ⊗ I) |Φ Supergeneration of maximally entangled states. dM F N → M − ( +1) gate[ ] 1 , Gate superreplication can be achieved deterministically, = M (19) d whereas state superreplication can be achieved only with where the relationship between the entanglement fidelity a vanishing probability. This sharp difference originates and the gate fidelity [63] is again used. In conclusion, in the fact that states and gates are inequivalent re- the cloning fidelity for maximally entangled states ap- sources; whereas gates can be used to deterministically proaches one if and only if the cloning fidelity for uni- generate states as |ψU = U|0, the converse process tary gates approaches one. Hence, superreplication of is forbidden by the no-programming theorem [65]. It quantum gates implies (probabilistic) superreplication of is then useful to distinguish between the task of state maximally entangled states. In turn, superreplication of cloning, where the input consists of N copies of the state maximally entangled states implies superreplication of |ψU = U|0,andthetaskofstate generation,wherethe the uniform-weight multiphase states input consists of N copies of the gate U. Determinis- ⎛ ⎞ tic gate superreplication cannot be used to achieve de- d−1 1 ⎝ iθj ⎠ terministic state superreplication, but it can be used to |ψθ = √ |0 + e |j , d j=1 achieve deterministic state supergeneration,thatis,the generation of up to N 2 almost perfect copies of the quan- θj ∈ [0, 2π) , ∀j =1,...,d− 1 , tum state |ψU from N copies of the gate U.Ageneral which are unitarily equivalent to the maximally entan- supergeneration protocol works as follows: d− | √1 | | 1 iθj |j|j gled states Φθ = d 0 0 + j=1 e . 1) Use a gate superreplication protocol to simulate M N 2 uses of the gate U. Upperboundontherateofgatereplication.The 2) Apply the simulated gates to the state |0⊗N . relationship between gate replication and replication of maximally entangled states can be used to derive the ultimate limit on the rates of gate replication in finite di- In general, the protocol can be tailored to the specific mensions. The argument proceeds by contradiction: Sup- set of states that one wants to generate. For example, U pose that it is possible to devise a network that simulates one could have i) clock states, where is of the form α U e−itH U Θ(N )usesofU for α 2 with vanishing error. Then we = , ii) maximally entangled states, where is U V ⊗ I could use it to superreplicate maximally entangled states of the product form = A B with respect to some at a rate α 2. Now, the family of maximally entangled bipartition of the Hilbert space, and iii) arbitrary pure U states contains the family of clock states states, where is a generic unitary gate. The fidelity of supergeneration depends on the proto- −int |Φt =(Ut ⊗ I) |Φ+ ,Ut = e |nn| . col used to replicate the gates. For example, the phase n gate replication by Dür et al. makes it possible to su- Replication of these states at a rate α 2wouldcontra- pergenerate clock states [33], whereas our universal gate dict Theorem 5, which implies that every cloner with a replication makes it possible to supergenerate maximally

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entangled states of bipartite systems [34]. Interestingly, the maximally entangled state |ΦU with M copies as a universal gate replication does not work for the set of all particular strategy for estimation of the unitary gate U pure states parametrized as with N copies, meaning that we have ˆ 2 {|ψU = U|0|U ∈ SU(d) , |0∈H}. pN (U|U) ≡ pM (ΦUˆ |ΦU ),M=Θ(N / log N) . In this case, the approach of simulating M uses of the Then, the estimation fidelity for the maximally entan- gate U and applying it to the state |0⊗N does not work gled state can be easily converted into the gate fidelity ⊗N ⊥ because the state |0 lies in the subspace HN ,where for the corresponding gate using the relation [63] our gate superreplication network fails. However, we will d F M − F N ( +1) est,ent[ ] 1 . see below that arbitrary pure states can be supergener- est,gate[ ]= d ated via a suitable protocol based on gate estimation. Because the state estimation fidelity converges to one as 1/M , the gate estimation fidelity will also converge Gate estimation with quasi-Heisenberg scaling. to one as 1/M . Recalling that M scales as N 2/ log N, Supergeneration of maximally entangled states has an this proves Eq. (21). The error scaling e =logN/N2 elegant application to quantum metrology. Specifically, beats the central limit scaling of classical statistics and it allows for an easy proof of the fact that an un- is close to the optimal quantum scaling 1/N 2,whichwas known quantum gate can be estimated with an error 2 derived in Refs. [35–37] for qubits and in Ref. [38] for scaling as log N/N . Here the error is defined as eN = general d-dimensional systems. The usefulness of our new 1 − F , [N], where F , [N] is the fidelity of gate est gate est gate derivation is that the proof is much simpler than full op- estimation with N copies, namely, timization of the estimation strategy. Interestingly, for d = 2, an alternative estimation Fest,gate[N]= dU dUpˆ N (Uˆ|U) Fgate(U,Uˆ ) , (20) strategy achieving scaling by log N 2/N 2 was proposed by Rudolph and Grover [66]. Their protocol is sequential where pN (Uˆ|U) is the probability distribution resulting and uses unentangled states to reach a quasi-Heisenberg from the estimation, and Fgate(U,Uˆ ) is the gate fidelity, which is defined as scaling. It is an open question whether a similar protocol exists for d>2. 2 F (U,Uˆ ):= dψ ψ|U †Uˆ|ψ . gate Universal supergeneration of pure states. We now show that all pure states can be supergenerated. To this We now show that gate superreplication can be used purpose, we parametrize the manifold of pure states as to achieve estimation fidelity scaling as {|ψU = U |0|U ∈ SU(d)},where|0 is a fixed pure log N state. To achieve supergeneration, we use a classical Fest,gate[N] 1 − O . (21) N 2 strategy based on estimation of the unknown gate U and |ψ ⊗M The proof goes as follows: given N copies of the gate on preparation of the state Uˆ conditional on the U M N 2/ N estimate Uˆ. The fidelity of this strategy is given by , we can produce =Θ( log )copiesofthe maximally entangled state |ΦU ,withanerrorvanish- 2M F [N → M]= dU dUpˆ N (Uˆ|U) |ψ |ψU | , ing faster than 1/N k for every desired k>0; cf. Eq. pure Uˆ (18). In particular, we set k = 2. With this choice, the where pN (Uˆ|U) is the probability distribution resulting error vanishes faster than 1/M ,andtheM approximate from gate estimation. The above choice of strategy im- copies can be used for state estimation, resulting in an plies that we have the bound estimate of the maximally entangled state with fidelity M Fpure[N → M] (Fpure[N → 1]) . (22) F M − O 1 est,ent[ ] 1 M Now, note that the single-copy fidelity satisfies the relation according to the central limit theorem. Here, the estima- F N → tion fidelity is defined as pure[ 1] 2 = dV dU dUpˆ N (UVˆ |UV) |ψ |ψUV | F M U Upˆ | | 2 , Uˆ V est,ent[ ]= d d M (ΦUˆ ΦU ) ΦUˆ ΦU ˆ ˆ 2 = dV dU dUpN (U|U) |ψUˆ V |ψUV | where pM (Φ |ΦU ) is the probability distribution result- Uˆ ing from estimation of a maximally entangled state with V U Upˆ Uˆ|U |ψ | Uˆ †U |ψ |2 M nearly perfect copies. Now, we regard estimation of = d d d N ( ) V V

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that works on all input states. The problem can be ad- = dU dUpˆ N (Uˆ|U) Fgate(U,Uˆ ) dressed by evaluating the worst-case fidelity = Fest,gate[N] , (23) Fgate,worst[N → M]= min where F , [N] is the gate estimation fidelity defined |Ψ ∈H⊗M , |Ψ =1 est gate in Eq. (20), and in the second equality we used the U | U † ⊗M C(N) | | U ⊗M | , fact that the optimal gate estimation strategy is covari- d Ψ ( ) U ( Ψ Ψ ) Ψ (25) ant [67], i.e., it satisfies the condition pN (UV,UVˆ )= (N) pN (U,Uˆ ) for every U,U,Vˆ ∈ SU(d). where CU is the channel implemented by the replication Combining Eq. (23) with the bounds (22) and (21), we network. In this setting, we say that a gate replication finally obtain process is reliable in the worst case if and only if M log N F [N → M] 1 − O , lim Fgate,worst[N → M(N)] = 1, pure N 2 N→∞ which means that arbitrary pure states can be super- and we say that the replication rate α is achievable in the generated at every rate smaller than quadratic. Because worst case if and only if there exists a replication proto- the above protocol is based on estimation, the error van- col that has that rate and is reliable in the worst case. ishes only as a power law. It is an open question whether The supremum over all achievable rates is determined as there exists a universal quantum supergeneration proto- follows. col whose error vanishes faster than any inverse polyno- Theorem 8 (Asymptotic no-cloning theorem for quan- mial. tum gates) For finite-dimensional quantum systems, no Equivalence between gate superreplication and physical process can reliably replicate phase gates at a gate estimation. As in the case of states, superreplica- rate α 1 in the worst-case scenario. tion can be achieved by gate estimation. The argument is thesameasthatusedtoderiveEq.(13);inshort,onecan Here we give a heuristic proof based on optimal phase show that every estimation strategy with fidelity scaling estimation. When an unknown phase gate is used N β as Fest,gate[N]=1− O(1/N ) can be used to achieve times, the optimal estimation strategy has a mean square gate replication with fidelity error c/N 2 for some suitable constant c>0 [68, 69]. The bound holds for both deterministic and probabilistic F N → M − O M/Nβ , gate[ ] 1 ( ) (24) strategies [70], and the fact that the optimal estimation which means that every replication rate smaller than β strategy can be achieved deterministically plays a crucial can be achieved. In particular, we know from Eq. (21) role in our argument. Now, suppose that one can produce M /M β that we can choose β =2− . Hence, every replication reliable replicas up to an error scaling as 1 for β> rate smaller than quadratic can be achieved via state es- some 2. In this case, one could use the replicas for timation. Note, however, that the replication error goes phase estimation, reducing the error to to zero only as a power law, whereas the quantum repli- c 1 e = + O . cation network has much better scaling of the error. M 2 M β The connection between gate superreplication and estimation can be used to prove the Heisenberg limit [30]. This result follows from the fact that the error of a de- The proof is by contradiction: Suppose that one could terministic estimation strategy is a continuous function estimate the parameters of a unitary gate with a fidelity of the input state. Because the above strategy cannot be F N − O /N 2+ scaling as est,gate[ ]=1 (1 ). Then, Eq. (24) better than the optimal strategy, we obtain the inequal- would imply that one can simulate M = O N 2+/2 ity copies of the gate, in contradiction to Theorem 7. In c c 1 summary, the quadratic scaling of the Heisenberg limit + O , (26) N 2 M 2 M β can be derived solely from considerations regarding the optimal rates of gate superreplication. which implies that M can grow at most as M = N + Asymptotic no-cloning theorem for quantum Θ(N α)withα<1. In addition, by Taylor-expanding gates in the worst-case scenario. We have seen that the r.h.s. of Eq. (26), we obtain the inequality β<3−α, gate superreplication can be achieved on all input states so the error can vanish at most as 1/N 3. In summary, except for a vanishingly small fraction. A natural ques- superreplication and exponentially vanishing errors are tion is whether one can find a superreplication protocol forbidden in the worst-case scenario.

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esting to examine how the replication rates are affected 7 Conclusion and outlook by the presence of limited resources or, conversely, what resources are needed to achieve a desired replication rate. In this paper, we reviewed the phenomenon of superreplication of quantum states and gates, emphasizing the ap- Acknowledgements This work was supported by the Founda- plications and connections with other tasks in quantum tional Questions Institute (FQXi-RFP3-1325), the National Natu- information processing. In particular, we clarified the re- ral Science Foundation of China (11450110096, 11350110207), and the 1000 Youth Fellowship Program of China. lation between superreplication and the precision limits of quantum metrology, showing that i) estimation can be used to achieve superreplication with an error vanish- References ing as O(M/N2), probabilistically in the case of states and deterministically in the case of gates, ii) gate super- 1. W. Wootters and W. Zurek, A single quantum cannot be replication allows for a simpler proof of the quadratic cloned, Nature 299(5886), 802 (1982) precision enhancement in the estimation of an unknown 2. D. Dieks, Communication by EPR devices, Phys. Lett. A gate, and iii) the optimality of Heisenberg scaling for 92(6), 271 (1982) estimation of unitary gates can be derived from the ul- 3. V. Scarani, S. Iblisdir, N. Gisin, and A. Acin, Quantum timate limit on the rate of superreplication. In addition, cloning, Rev. Mod. Phys. 77(4), 1225 (2005) we showed that N uses of a completely unknown gate 4. N. J. Cerf and J. Fiuráˇsek, Optical quantum cloning, are sufficient to generate O(N 2) approximate copies of Progress in Optics 49, 455 (2006) the corresponding pure state with an error that vanishes 5. C. Bennett and G. Brassard, Quantum cryptography: Public in the large-N limit. key distribution and coin tossing, in: Conference on Com- Among the future research directions, an important puters, Systems and Signal Processing (Bangalore, India), pp 175–179, 1984 one is the study of replication processes for mixed states 6. A. Ekert, Quantum cryptography based on Bells theorem, and noisy channels. We expect that in this case there Phys. Rev. Lett. 67(6), 661 (1991) is also an equivalence between replication and estima- 7. S. Wiesner, Conjugate coding, ACM Sigact News 15(1), 78 tion. It is well known that in the presence of noise, (1983) Heisenberg scaling is often inhibited, leading to different 8. M. Hillery, V. Buˇzek, and A. Berthiaume, Quantum secret sub-Heisenberg scalings [71–75]. We then expect that the sharing, Phys. Rev. A 59(3), 1829 (1999) replication rates will also have intermediate values rang- 9. V. Buˇzek and M. Hillery, Quantum copying: Beyond the α α ing between =1and = 2, depending on the type of no-cloning theorem, Phys. Rev. A 54(3), 1844 (1996) noise. The study of superreplication in the noisy case is 10. N. Gisin and S. Massar, Optimal quantum cloning machines, also expected to shed light on the optimal strategies for Phys. Rev. Lett. 79(11), 2153 (1997) estimation of noisy channels, the performance of which 11. D. Bruss, A. Ekert, and C. Macchiavello, Optimal univer- is sometimes hard to characterize analytically. Another sal quantum cloning and state estimation, Phys. Rev. Lett. interesting research direction is the study of replication 81(12), 2598 (1998) processes that are subject to constraints, e.g., on the 12. J. Bae and A. Ac´ın, Asymptotic quantum cloning is state ability to perform joint operations on composite sys- estimation, Phys. Rev. Lett. 97(3), 030402 (2006) tems. Recently, Kumagai and Hayashi [76] investigated 13. G. Chiribella and G. M. D’Ariano, Quantum information be- the problem of cloning bipartite quantum states using comes classical when distributed to many users, Phys. Rev. only local operations and classical communication. When Lett. 97(25), 250503 (2006) the state to be cloned is perfectly known,√ they showed 14. G. Chiribella, On quantum estimation, quantum cloning and that the number of extra copies scales as N.Notethe finite quantum de Finetti theorems, in: Theory of Quantum contrast with the situation where the limitations of local Computation, Communication, and Cryptography, Lecture operations and classical communication are not imposed, Notes in Computer Science, Volume 6519, pp 9–25, Springer, in which case one can produce Θ(N δ) extra copies for 2011 every δ<1. In addition to the constraints due to the 15. G. Chiribella and Y. Yang, Optimal asymptotic cloning ma- locality of the operations, other physical constraints can chines, New J. Phys. 16(6), 063005 (2014) arise from conservation laws, such as energy and an- 16. S. L. Braunstein, C. A. Fuchs, and H. J. Kimble, Criteria gular momentum conservation [77–81]. The search for for continuous-variable quantum teleportation, J. Mod. Opt. 47(2-3), 267 (2000) energy-preserving cloners was considered in our earlier 17. K. Hammerer, M. M. Wolf, E. S. Polzik, and J. I. Cirac, work [45], where we identified the optimal operations for Quantum benchmark for storage and transmission of coher- transformation of pure states. In this scenario, it is inter- ent states, Phys. Rev. Lett. 94(15), 150503 (2005)

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18. G. Adesso and G. Chiribella, Quantum benchmark for tele- 35. G. Chiribella, G. M. D’Ariano, P. Perinotti, and M. F. Sac- portation and storage of squeezed states, Phys. Rev. Lett. chi, Efficient use of quantum resources for the transmission 100(17), 170503 (2008) of a reference frame, Phys. Rev. Lett. 93(18), 180503 (2004) 19. G. Chiribella and J. Xie, Optimal design and quantum 36. E. Bagan, M. Baig, and R. Munoz-Tapia, Quantum reverse benchmarks for coherent state amplifiers, Phys. Rev. Lett. engineering and reference-frame alignment without nonlocal 110(21), 213602 (2013) correlations, Phys. Rev. A 70(3), 030301 (2004) 20. G. Chiribella and G. Adesso, Quantum benchmarks for 37. M. Hayashi, Parallel treatment of estimation of SU(2) and pure single-mode Gaussian states, Phys. Rev. Lett. 112(1), phase estimation, Phys. Lett. A 354(3), 183 (2006) 010501 (2014) 38. J. Kahn, Fast rate estimation of a unitary operation in 21. H. Fan, Y. N. Wang, L. Jing, J. D. Yue, H. D. Shi, Y. L. SU(d), Phys. Rev. A 75(2), 022326 (2007) Zhang, and L. Z. Mu, Quantum cloning machines and the 39. H. Fan, K. Matsumoto, X. B. Wang, and M. Wadati, Quan- applications, Phys. Rep. 544(3), 241 (2014) tum cloning machines for equatorial qubits, Phys. Rev. A 22. L. M. Duan and G. C. Guo, Probabilistic cloning and identi- 65(1), 012304 (2001) fication of linearly independent quantum states, Phys. Rev. 40. D. Bruß, M. Cinchetti, G. Mauro D’Ariano, and C. Mac- Lett. 80(22), 4999 (1998) chiavello, Phase-covariant quantum cloning, Phys. Rev. A 23. J. Fiuráˇsek, Optimal probabilistic cloning and purification 62(1), 012302 (2000) of quantum states, Phys. Rev. A 70(3), 032308 (2004) 41. G. M. D’Ariano and C. Macchiavello, Optimal phase- 24. T. Ralph and A. Lund, Nondeterministic noiseless linear covariant cloning for qubits and qutrits, Phys. Rev. A 67(4), amplification of quantum systems, in: Ninth International 042306 (2003) Conference on Quantum Communication, Measurement and 42. R. F. Reinhard, Optimal cloning of pure states, Phys. Rev. Computing (QCMC), Volume 1110, pp 155–160, AIP Pub- A 58(3), 1827 (1998) lishing, 2009 43. E. B. Davies and J. T. Lewis, An operational approach to 25. G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. quantum probability, Commun. Math. Phys. 17(3), 239 Pryde, Heralded noiseless linear amplification and distilla- (1970) tion of entanglement, Nat. Photonics 4(5), 316 (2010) 44. M. Ozawa, Quantum measuring processes of continuous ob- 26. F. Ferreyrol, M. Barbieri, R. Blandino, S. Fossier, R. Tualle- servables, J. Math. Phys. 25(1), 79 (1984) Brouri, and P. Grangier, Implementation of a nondetermin- 45. Y. Yang and G. Chiribella, Optimal energy-preserving con- istic optical noiseless amplifier, Phys. Rev. Lett. 104(12), versions of quantum coherence, arXiv: 1502.00259, 2015 123603 (2010) 46. S. Pandey, Z. Jiang, J. Combes, and C. Caves, Quantum lim- 27. M. A. Usuga, C. R. Müller, C. Wittmann, P. Marek, R. its on probabilistic amplifiers, Phys. Rev. A 88(3), 033852 Filip, C. Marquardt, G. Leuchs, and U. L. Andersen, Noise- (2013) powered probabilistic concentration of phase information, 47. H. Everett, “Relative state” formulation of quantum me- Nat. Phys. 6(10), 767 (2010) chanics, Rev. Mod. Phys. 29(3), 454 (1957) 28. A. Zavatta, J. Fiuráˇsek, and M. Bellini, A high-fidelity noise- 48. E. Stueckelberg, Quantum theory in real Hilbert space, Hel- less amplifier for quantum light states, Nat. Photonics 5(1), vetica Physica Acta 33, 727 (1960) 52 (2011) 49. L. Hardy and W. K. Wootters, Limited holism and real- 29. G. Chiribella, Y. Yang, and A. C.C. Yao, Quantum replica- vector-space quantum theory, Found. Phys. 42(3), 454 tion at the Heisenberg limit, Nat. Commun. 4(2915) (2013) (2012) 30. V. Giovannetti, S. Lloyd, and L. Maccone, Quantum- 50. W. Wootters, Optimal information transfer and real-vector- enhanced measurements: Beating the standard quantum space quantum theory, arXiv: 1301.2018, 2013 limit, Science 306(5700), 1330 (2004) 51. Y. N. Wang, H. D. Shi, Z. X. Xiong, L. Jing, X. J. Ren, L. Z. 31. A. Winter, Coding theorem and strong converse for quantum Mu, and H. Fan, Unified universal quantum cloning machine channels,IEEE Transactions on Information Theory 45(7), and fidelities, Phys. Rev. A 84(3), 034302 (2011) 2481 (1999) 52. S. Braunstein, N. Cerf, S. Iblisdir, P. van Loock, and S. Mas- 32. G. Chiribella, G. M. D’Ariano, and P. Perinotti, Optimal sar, Optimal cloning of coherent states with a linear amplifier cloning of unitary transformation, Phys. Rev. Lett. 101(18), and beam splitters, Phys. Rev. Lett. 86(21), 4938 (2001) 180504 (2008) 53. B. Gendra, J. Calsamiglia, R. Muñoz-Tapia, E. Bagan, and 33. W. Dür, P. Sekatski, and M. Skotiniotis, Deterministic G. Chiribella, Probabilistic metrology attains macroscopic superreplication of one-parameter unitary transformations, cloning of quantum clocks, Phys. Rev. Lett. 113(26), 260402 Phys. Rev. Lett. 114(12), 120503 (2015) (2014) 34. G. Chiribella, Y. Yang, and C. Huang, Universal superrepli- 54. B. Gendra, E. Ronco-Bonvehi, J. Calsamiglia, R. Munoz- cation of unitary gates, Phys. Rev. Lett. 114(12), 120504 Tapia, and E. Bagan, Quantum metrology assisted by ab- (2015) stention, Phys. Rev. Lett. 110(10), 100501 (2013)

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55. G. M. D’Ariano, C. Macchiavello, and M. Rossi, Quantum optimal adaptive measurements for quantum interferometry, cloning by cellular automata, Phys. Rev. A 87(3), 032337 Phys. Rev. Lett. 85(24), 5098 (2000) (2013) 70. G. Chiribella, G. M. D’Ariano, and P. Perinotti, Memory 56. W. Fulton and J. Harris, Representation Theory: A First effects in quantum channel discrimination, Phys. Rev. Lett. Course, Volume 129, Springer Science & Business Media, 101(18), 180501 (2008) 1991 71. U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. 57. R. Alicki, S. Rudnicki, and S. Sadowski, Symmetry prop- Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walms- erties of product states for the system of Nn-level atoms, J. ley, Optimal quantum phase estimation, Phys. Rev. Lett. Math. Phys. 29(5), 1158 (1988) 102(4), 040403 (2009) 58. P.-L. Méliot, Kerov’s central limit theorem for Schur–Weyl 72. B. M. Escher, R. L. de Matos Filho, and L. Davidovich, Gen- measures of parameter 1/2, arXiv: 1009.4034, 2010 eral framework for estimating the ultimate precision limit in 59. I. Marvian and R. Spekkens, A generalization of Schur–Weyl noisy quantum-enhanced metrology, Nat. Phys. 7(5), 406 duality with applications in quantum estimation, Commun. (2011) Math. Phys. 331(2), 431 (2014) 73. R. Demkowicz-Dobrzański, J. Kolodyński, and M. Gu, The 60. A. W. Harrow, Applications of coherent classical commu- elusive Heisenberg limit in quantum-enhanced metrology, nication and the Schur transform to quantum information Nat. Commun. 3, 1063 (2012) theory, PhD thesis, Massachusetts Institute of Technology, 74. R. Chaves, J. B. Brask, M. Markiewicz, J. Kolodyński, and 2005 A. Ac´ın, Noisy metrology beyond the standard quantum 61. Y. Yang, D. Ebler, and G. Chiribella, Efficient quantum limit, Phys. Rev. Lett. 111(12), 120401 (2013) compression for ensembles of identically prepared mixed 75. R. Demkowicz-Dobrzański and L. Maccone, Using entangle- states, arXiv: 1506.03542, 2015 ment against noise in quantum metrology, Phys. Rev. Lett. 62. I. L. Chuang and D. Gottesman, Demonstrating the viability 113(25), 250801 (2014) of universal quantum computation using teleportation and 76. W. Kumagai and M. Hayashi, A new family of probability single-qubit operations, Nature 402(6760), 390 (1999) distributions and asymptotics of classical and locc conver- 63. M. Horodecki, P. Horodecki, and R. Horodecki, General sions, arXiv: 1306.4166, 2013 teleportation channel, singlet fraction, and quasidistillation, 77. M. Ozawa, Conservative quantum computing, Phys. Rev. Phys. Rev. A 60(3), 1888 (1999) Lett. 89(5), 057902 (2002) 64. P. Sekatski, M. Skotiniotis, and W. Dür, No-signaling 78. J. Gea-Banacloche and M. Ozawa, Constraints for quantum bounds for quantum cloning and metrology, Phys. Rev. A logic arising from conservation laws and field uctuations, J. 92(2), 022355 (2015) Opt. B 7(10), S326 (2005) 65. M. A. Nielsen and I. L. Chuang, Programmable quantum 79. M. Ahmadi, D. Jennings, and T. Rudolph, The Wigner- Phys. Rev. Lett. gate arrays, 79(2), 321 (1997) Araki-Yanase theorem and the quantum resource theory of 66. T. Rudolph and L. Grover, Quantum communication com- asymmetry, New J. Phys. 15(1), 013057 (2013) plexity of establishing a shared reference frame, Phys. Rev. 80. I. Marvian and R. Spekkens, The theory of manipulations Lett. 91(21), 217905 (2003) of pure state asymmetry (I): Basic tools, equivalence classes 67. A. Holevo, Probabilistic and Statistical Aspects of Quantum and single copy transformations, New J. Phys. 15(3), 033001 Theory, North-Holland, 1982 (2013) 68. V. Buˇzek, R. Derka, and S. Massar, Optimal quantum 81. I. Marvian and R. W. Spekkens, Extending Noethers theo- clocks, Phys. Rev. Lett. 82(10), 2207 (1999) rem by quantifying the asymmetry of quantum states, Nat. 69. D. Berry and H. M. Wiseman, Optimal states and almost Commun. 5(3821) (2014)

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