CContextual advantage for state-dependent cloning
Matteo Lostaglio and Gabriel Senno ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona), 08860, Spain QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands
A number of noncontextual models existwhich reproduce different subsets of quan-tum theory and admit a no-cloning theorem.Therefore, if one chooses noncontextuality asone’s notion of classicality, no-cloning cannotbe regarded as a nonclassical phenomenon.In this work, however, we show that thereare aspects of the phenomenology of quan-tum state cloning which are indeed nonclassi-cal according to this principle. Specifically, wefocus on the task of state-dependent cloningand prove that the optimal cloning fidelitypredicted by quantum theory cannot be ex-plained by any noncontextual model. We de-rive a noise-robust noncontextuality inequalitywhose violation by quantum theory not onlyimplies a quantum advantage for the task ofstate-dependent cloning relative to noncontex-tual models, but also provides an experimentalwitness of noncontextuality.
An important guiding principle for quantum theo-rists is the identification of genuine nonclassical effectscertified by rigorous theorems. Given a quantum phe-nomenon, the relevant question is: Are there classi-cal models able to reproduce the observed operationaldata? Here we investigate this question in the contextof a cloning experiment.The no-cloning theorem [1–3] is widely regardedas a central result in quantum theory. Informally,the theorem states the impossibility of copying quan-tum information, and is contrasted with the fact thatclassical information, on the other hand, can be per-fectly copied. More precisely, there is no machine(formally, a quantum channel) that can take two dis-tinct and nonorthogonal states {| ψ i , | ψ i} sent atrandom as inputs and output the corresponding copies {| ψ i ⊗ | ψ i , | ψ i ⊗ | ψ i} [4].While no-cloning is often regarded as an intrinsi-cally quantum feature, one would like to back thatclaim by a precise theorem stating what operationalfeatures cannot be explained within classical models.The theorem should hence define a precise notion of‘classicality’ and show that such notion leads to op-erational predictions incompatible with the relevantquantum statistics [5]. At the operational level, wecan schematically think of an experiment as a setof black-boxes each corresponding to certain sets ofoperational instructions. At the ontological level we look for theoretical explanations of the empirical datawithin the framework of ontological models . This is avery broad class of models involving an arbitrary setof physical states evolving according to some laws andultimately determining (the probabilities of) the mea-surement outcomes. This analysis forces us to look forany plausible alternative explanation of the empiricaldata collected in a quantum experiment before we cer-tify it as “nonclassical”. But, which ontological modelsshould be deemed “classical"?Clearly, the broader the chosen notion of classical-ity is, the stronger the resulting no-go theorem is.Since the scenario of quantum cloning does not featurespace-like separated measurements, we need a differ-ent notion of ‘classicality’ than the ubiquitous Bell’slocality. Hence, in this work we identify nonclassicalfeatures as those that cannot be explained within anynoncontextual model, in the generalized sense intro-duced in Ref. [6]. It is a known fact that, with respectto this broad notion, no-cloning by itself should notbe regarded as a nonclassical phenomenon. There are,in fact, several examples of noncontextual models forsubsets of quantum theory with a no-cloning theorem[7, 8]. The mechanism behind no-cloning in noncon-textual theories is simple: non-orthogonal quantumstates | ψ i , | ψ i correspond to overlapping probabilitydistributions µ ( λ ) , µ ( λ ) over the posited set of phys-ical states λ and there is no deterministic nor stochas-tic process mapping { µ , µ } to { µ ⊗ µ , µ ⊗ µ } [9].The existence of these models proves that no-cloningcannot be interpreted as a nonclassical phenomenonwhen the notion of classicality is taken to be thatof noncontextuality. Hence, we need to look moreclosely at the phenomenology of quantum cloning ifwe are to identify aspects of it that are nonclassicalaccording to the principle of noncontextuality.In this work, we identify a strongly nonclassical as-pect in the ultimate limits of imperfect cloning. Thequestion of what is the best fidelity with which a givenset of quantum states can be cloned has been widelystudied since the pivotal work of Bužek and Hillary in1996 [10] (for a review on quantum cloning, see, e.g. ,Ref. [11]). We find that the optimal fidelity predictedby quantum theory for the cloning of two distinct, non Crucially, cloning should be distinguished from the notionof broadcasting. Broadcasting only requires the creation of ajoint distribution with marginals µ i and can be done perfectlyby a generalized CNOT. Accepted in
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Quantum2020-04-20, click title to verify. Published under CC-BY 4.0. a r X i v : . [ qu a n t - ph ] A p r rthogonal pure states cannot be reproduced by anynoncontextual model which complies with the opera-tional phenomenology featured in a quantum cloningexperiment. Specifically, contextuality provides anadvantage to the maximum copying fidelity. Our re-sult directly links contextuality to a quantum advan-tage [5, 12, 13]. At the operational level, we can schematically thinkof an experiment as a set of black-boxes each corre-sponding to certain sets of operational instructions. We can distinguish three kinds of black-boxes:1. A preparation black-box P s initialises the system;2. A transformation black-box T takes in a systemprepared according to P s and transforms it intosome new preparation, denoted T ( P s ) .3. A measurement black-box M s takes a prepara-tion P s as input and returns an outcome x withprobability p ( x | P s , M s ) .4. An experiment consists of collecting the statistics p ( x | T ( P s ) , M s ) for various choices of the blackboxes P s , T and M s .The set of P s , T , M s and corresponding observedstatistics p ( x | T ( P s ) , M s ) are the defining elements ofan operational theory . Noncontextuality is a restric-tion on the ontological models that try to explain thestatistics of some operational theory. An ontologicalmodel for an operational theory is one which [14]:1. Makes every preparation P s correspond to sam-pling from a probability distribution µ s ( λ ) oversome set of ontic variables λ . λ s are referred toas ‘hidden variables’ in the context of Bell non-locality and they form a (measurable) set Λ .2. Represents transformations by matrices T ( λ | λ ) of transition probabilities ( T ( λ | λ ) ≥ , R dλ T ( λ | λ ) = 1 ∀ λ ) acting on the correspondingprobability density.3. Represents a measurement M s by a responsefunction ξ s ( x | λ ) giving the probability of out-come x given that the hidden variable takes thevalue λ ( ξ s ( x | λ ) ≥ , P x ξ s ( x | λ ) = 1 ∀ λ ). While empirical data is always to some degree theory-laden ,the word “operational” here signifies that we are striving to-wards the ideal of the most low-level instructions we can imag-ine (e.g. press this button, write down an outcome when acorresponding light flashes etc.). This is to be opposed tohigh-level instructions that refer to theoretical entities, such as“lower the potential barrier in which the electron is trapped”.
Figure 1:
Cloning experiment.
Top: black-box of the cloningprotocol; one of two preparation procedures P x , x = a, b isperformed with equal probability, the resultant state is sentthrough a cloning machine (independent of x ), which respec-tively prepares P γ , γ = α, β ; a test measurement M xx for thetarget preparation P xx is performed and passed with prob-ability P ( M aa | P α ) (or P ( M bb | P β ) ). Bottom: ontologicaldescription of the same experiment, where preparing P x cor-responds to sampling λ with probability µ x ( λ ) , the cloningmachine maps λ λ with probability T ( λ | λ ) and M xx gives a ‘pass’ outcome with probability ξ xx (1 | λ ) . An ontological model then defines its predictions as p ( x | T ( P s ) , M s ) = Z dλdλ µ s ( λ ) T ( λ | λ ) ξ s ( x | λ ) . (1) Two operational procedures (be them preparations,measurements or transformations) are said to be op-erationally equivalent if they cannot be distinguishedby any experiment. Noncontextuality, in the gener-alized form introduced in [6], is a restriction to on-tological models requiring that if two procedures areoperationally equivalent, they must be represented bythe same object in the ontological model . This notioncan be seen as an extension of the traditional one ofKochen-Specker [6, 15].In this work we will be concerned with operationalequivalences only at the level of preparations. Twopreparations P s and P s are operationally equivalentif they cannot be distinguished by any measurements: p ( x | P s , M ) = p ( x | P s , M ) , ∀ M, which, for short, we will denote by P s ’ P s . Theassumption of (preparation) noncontextuality is then P s ’ P s ⇒ µ s ( λ ) = µ s ( λ ) . (2) This principle can be understood as an ‘identity ofthe indiscernibles’ and, together with locality, it canbe seen as a successful methodological principle fortheory construction [16]. Examples of noncontex-tual ontological models include classical Hamiltonianmechanics, Hamiltonian mechanics with a resolutionlimit on phase space [7] and Spekken’s toy model [8].
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Quantum2020-04-20, click title to verify. Published under CC-BY 4.0. Operational features of quantumcloning - ideal scenario
We now describe the operational features of optimalstate-dependent quantum cloning which, as we willshow, are impossible to explain with noncontextualmodels (see also Fig. 1). We will make the assump-tion that certain perfect correlations are observed, butwe will later remove these idealizations. For all two-outcome measurements M s we will use the shortcuts p ( x = 1 | P, M s ) ≡ p ( M s | P ) and ξ s (1 | λ ) ≡ ξ s ( λ ) .Let P a and P b denote the experimental proceduresfollowed to prepare the states | a i and | b i to be cloned.As an operational signature of the fact that | a i and | b i are two pure and, in general, nonorthogonal states,we consider the ‘test measurements’ M a , M b , withoutcomes x ∈ { , } , giving the operational statis-tics p ( M a | P a ) = p ( M b | P b ) = 1 . In the quantumformalism, this statistics is reproduced by perform-ing the projective measurements {| a ih a | , − | a ih a |} and {| b ih b | , − | b ih b |} (with x = 1 corresponding tothe first outcome). We will use the notation c ab := p ( M b | P a ) , which is called ‘confusability’ in Ref. [5],for the probability of observing the first outcome ofthe M b measurement when the system is initializedaccording to P a . Clearly, in the ideal quantum exper-iment one observes c ab = |h a | b i| .The two preparations P a , P b go through a cloningmachine T , which outputs new preparations P α = T ( P a ) , P β = T ( P b ) . In quantum theory, the optimal-state dependent cloning operation is a unitary U and, hence, the preparations P α and P β correspondto pure states | α i := U | a i , | β i := U | b i respec-tively, with | i the initial state of some ancillaryregister. Operationally, and similarly to the discus-sion above, the purity of the outputs implies that wecan perform test measurements M α , M β satisfying p ( M α | P α ) = 1 , p ( M β | P β ) = 1 (again, by performingthe measurements described in the quantum formal-ism as {| α ih α | , − | α ih α |} and {| β ih β | , − | β ih β |} ).The experiment ends by testing what the fidelitybetween the output and the ideal clone is. To doso, given the ideal clones P aa , P bb we introducetest-measurements M aa , M bb and assume one ob-serves the statistics p ( M aa | P aa ) = p ( M bb | P bb ) =1 , p ( M bb | P aa ) = |h aa | bb i| = |h a | b i| . In a quan-tum experiment this is realized by preparing states | aa i , | bb i and performing the projective measurements {| aa ih aa | , − | aa ih aa |} , {| bb ih bb | , − | bb ih bb |} .Then, denoting by c αaa := P ( M aa | P α ) , c βbb := P ( M bb | P β ) , the (global) cloning fidelity is opera-tionally defined to be F g := 12 c αaa + 12 c βbb , i.e., the average probability that the imperfect clones P α and P β pass the corresponding test measure-ments for the ideal clones, M aa and M bb respec- tively. In quantum theory, the optimal cloning uni-tary achieves [17] F Q , opt g := 14 (cid:20)q (1 + c ab )(1 + √ c ab )+ q (1 − c ab )(1 − √ c ab ) (cid:21) , (3) with c ab = |h a | b i| .This brief summary captures the main operationalfeatures of the traditional ‘optimal state-dependentcloning’ and highlights the main issue with thisapproach: it leaves no room to leverage opera-tional equivalences to further study its potential non-classical aspects. To fix that, we follow Ref. [5] andexploit another operational consequence of the purityof | a i , | b i : the existence of preparations P a ⊥ , P b ⊥ sat-isfying p ( M a | P a ⊥ ) = p ( M b | P b ⊥ ) = 0 and such that themixture P a / P a ⊥ / (tossing a fair coin and follow-ing either P a or P a ⊥ ) is operationally equivalent to themixture P b / P b ⊥ / : P a / P a ⊥ / ’ P b / P b ⊥ / .In the idealized quantum experiment one observesthis operational statistics by preparing pure states | a ⊥ i , | b ⊥ i in the span of { | a i , | b i } and satisfying (cid:10) a (cid:12)(cid:12) a ⊥ (cid:11) = (cid:10) b (cid:12)(cid:12) b ⊥ (cid:11) = 0 as well as | a ih a | + (cid:12)(cid:12) a ⊥ (cid:11)(cid:10) a ⊥ (cid:12)(cid:12) = | b ih b | + (cid:12)(cid:12) b ⊥ (cid:11)(cid:10) b ⊥ (cid:12)(cid:12) . The same discussion can be re-peated for each of the pairs { ( a, b ) , ( α, aa ) , ( β, bb ) } .To conclude, here is an operational account (with-out any reference to quantum theory) of the featuresthat we demand are observed in the idealized scenarioof the cloning experiment: there exists P s , P s ⊥ , M s such thatO1 p ( M s | P s ) = 1 , p ( M s | P s ⊥ ) = 0 for s = a, b, α, β, aa, bb .O2 P s + P s ⊥ ’ P s + P s , for all ( s, s ) in { ( a, b ) , ( α, aa ) , ( β, bb ) } . In any ontological model, a cloning experiment is de-scribed as follows (see Fig. 1). A preparation devicerandomly prepares either P a or P b , i.e., it samplesa λ from either the distribution µ a ( λ ) or µ b ( λ ) . Thisstate is sent into the cloning machine that maps λ intosome new λ with probability T ( λ | λ ) . For example, if λ = ( x , p ) one could have λ = ( x , p , x , p ) . This λ is sent into a testing device doing the measurement M aa if P a was prepared, or M bb if P b was prepared.Upon receiving λ , the device gives an outcome x withprobability ξ aa ( λ ) or ξ bb ( λ ) .The assumption of noncontextuality (more pre-cisely, preparation noncontextuality [6]) and linearityapplied to the operational equivalences in O2 requires Accepted in
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Quantum2020-04-20, click title to verify. Published under CC-BY 4.0. hat any noncontextual ontological model must sat-isfy (see Eq. (2) ) µ s ( λ ) + 12 µ s ⊥ ( λ ) = 12 µ s ( λ ) + 12 µ s ( λ ) , (4) for all ( s, s ) in { ( a, b ) , ( α, aa ) , ( β, bb ) } and λ ∈ Λ .Our main result is that no noncontextual ontologicalmodel can reproduce the operational features listedO1-O2 and match the optimal cloning fidelity pre-dicted by quantum theory. More precisely: Theorem 1 (Optimal cloning fidelity in noncontex-tual models) . Let P α = T ( P a ) , P β = T ( P a ) be theachieved outputs of a cloning process with inputs P a , P b and target outputs P aa , P bb . Suppose one observesthe operational features O1-O2. Then, for any non-contextual model we have that F g ≤ F NC g = 1 − c ab c aa,bb . (5) Proof.
The first part of the proof essentially followsthe argument given in Ref. [6] Sec. VIIIA and repro-duced in Ref. [5] Sec. IVA, slightly adapted to use thefewer assumptions of the statement. We have that1 = p ( M k | P k ) = Z S k dλµ k ( λ ) ξ k ( λ ) , k = s, s , where S k denotes the support of µ k . From this equa-tion, it follows that ξ k ( λ ) = 1 almost everywhere on S k (that is, modulo sets of measure zero). Further-more,0 = p ( M k | P k ⊥ ) = Z S k ⊥ µ k ⊥ ( λ ) ξ k ( λ ) , k = s, s , from which it follows that ξ k ( λ ) = 0 almost every-where on S k ⊥ . Hence, S k ∩ S k ⊥ = ∅ modulo sets ofzero measure.The operational equivalence of assumption 1 im-plies that in a noncontextual model µ s ( λ ) + µ s ⊥ ( λ ) = µ s ( λ ) + µ s ( λ ) , ∀ λ ∈ Λ . (6)Since S s ∩ S s ⊥ = S s ∩ S s = ∅ modulo a set ofzero measure, this implies µ s ( λ ) = µ s ( λ ) for almostall λ ∈ S s ∩ S s . Hence, using the facts above, the ‘ norm distance between µ s and µ s reads ( k µ s − µ s k := R dλ | µ s ( λ ) − µ s ( λ ) | ). k µ s − µ s k = Z Λ \ S s dλµ s ( λ ) + Z Λ \ S s dλµ s ( λ )= 2 − Z S s ∩ S s dλµ s ( λ )= 2 − Z S s ∩ S s dλµ s ( λ ) ξ s ( λ ) . Note that the last integral can be extended to Λ.In fact, by contradiction suppose that ξ s ( λ ) = 0for some nonzero measure set X ⊆ S s \ S s . Then, from Eq. (6), it follows that, for almost all λ ∈ X ,0 < µ s ( λ ) = µ s ( λ ). However, as we discussed ξ s ( λ ) = 0 almost everywhere on S s , which givesthe desired contradiction. Hence the integral can beextended to S s ∪ S s and, trivially, to all Λ. In con-clusion, k µ s − µ s k = 2 − Z Λ dλµ s ( λ ) ξ s ( λ ) = 2(1 − c ss ) , (7)where c ss = p ( M s | P s ). Using the triangle inequality, k µ aa − µ bb k ≤ k µ aa − µ α k + k µ α − µ β k + k µ β − µ bb k . By definition, µ α ( λ ) = R dλ T ( λ | λ ) µ a ( λ ), for astochastic matrix T ( λ | λ ). Similarly, µ β ( λ ) = R dλ T ( λ | λ ) µ b ( λ ), with the same stochastic matrix.Since R dλT ( λ | λ ) = 1 and T ( λ | λ ) ≥
0, one can read-ily verify from the convexity of the absolute value that k µ α − µ β k ≤ k µ a − µ b k (data processing inequality),which implies k µ aa − µ bb k ≤ k µ α − µ aa k + k µ a − µ b k + k µ β − µ bb k . (8)We can apply Eq. (7) to each of the couples ( s, s ) onthe right hand side of Eq. (8), obtaining k µ aa − µ bb k ≤ − c αaa )+2(1 − c ab )+2(1 − c βbb ) . (9)Let us now show k µ aa − µ bb k ≥ − c aa,bb ). First,notice that k µ aa − µ bb k = Z S aa \ S bb dλµ aa ( λ ) + Z S bb \ S aa dλµ bb ( λ ) + Z R dλ ( µ aa ( λ ) − µ bb ( λ )) + Z R dλ ( µ bb ( λ ) − µ aa ( λ )) , with R := { λ ∈ S aa ∩ S bb : µ aa ( λ ) ≥ µ bb ( λ ) } and R := ( S aa ∩ S bb ) \ R . Next, k µ aa − µ bb k = 2 − Z R dλµ bb ( λ ) − Z R dλµ aa ( λ ) ≥ − Z R ∪ R = S aa ∩ S bb dλµ aa ( λ )= 2 − Z S aa ∩ S bb dλµ aa ( λ ) ξ bb ( λ ) ≥ − c aa,bb )where the first inequality follows from µ aa ( λ ) ≥ µ bb ( λ ) ∀ λ ∈ R and the second equality follows from ξ bb ( λ ) = 1 almost everywhere in S bb . Finally, substi-tuting this in Eq. (9) and rearranging the terms gives12 c αaa + 12 c βbb ≤ − c ab c aa,bb F g = c αaa + c βbb the global cloningachieved by non-contextual ontological models thatcomply with the operational features O1-O2 is upperbounded as in Eq. (5). Accepted in
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Quantum2020-04-20, click title to verify. Published under CC-BY 4.0. igure 2: Maximum tradeoff between cloning fidelity F g andconfusability c ab allowed for noncontextual models (blue line,Eq. (5) ) versus optimal tradeoff achievable in quantum theory(red line, Eq. (3) ). In Fig. 2 we compare the optimal quantum cloning(global) fidelity of Eq. (3) with the maximum non-contextual cloning fidelity of Eq. (5) , taking into ac-count that, in quantum experiments, one observes c aa,bb = c ab . One can see, for any < c ab < , thatquantum mechanics achieves higher copying fidelitiesthan what is allowed by the principle of noncontextu-ality. Hence, the phenomenology of optimal cloningcannot be reproduced within noncontextual ontologi-cal models. Contextuality provides an advantage forthe maximum copying fidelity. Interestingly, the above derivation also gives an al-ternative, simple proof of the main result of Ref. [5].In fact, an intermediate technical result in the proof ofTheorem 1 is that in the presence of the operationalfeatures O1-O2, noncontextual models must have adirect relation between the experimentally accessibleconfusabilities c ss = p ( M s | P s ) and the ‘ distancebetween the corresponding probability distributions: k µ s − µ s k = 2(1 − c ss ) . (10) (This was implicitly shown in Ref. [5] Sec. IVA, butusing infinitely many extra operational assumptions.That is, they assume O2 for all pairs of orthogonalstates).Since the maximum probability s ab of distinguish-ing two preparations P a and P b is at most / k µ a − µ b k / , it immediately follows s ab ≤ − c ab / ,which is the optimal state discrimination probabilityin noncontextual models, as given in Ref. [5]. Con-versely, it is not immediately obvious how the tech-niques of Ref. [5] could be adapted to obtain our re-sult on cloning, due to our use of the data processinginequality in Theorem 1.We also note that the noncontextual bound oncloning is tight. Denote by S s the support of µ s . Con-sider a model in which µ ss = µ s µ s and ξ s ( λ ) = 1 if Of course, when c ab = 0 - as it is for classical, i.e., orthogo-nal, states - both the quantum and the noncontextual fidelitiesare 1. λ ∈ S s and zero otherwise. A cloning strategy thatsaturates the bound is as follows: if the input λ isin S a \ S b , output ( λ, λ ) , with λ sampled accordingto µ a ; otherwise, output ( λ, λ ) with λ sampled ac-cording to µ b . Notice that this sets µ β = µ b µ b and,hence, c βbb = 1 ( µ b is copied perfectly). On the otherhand, µ α ( λ, λ ) = µ a ( λ ) µ a ( λ ) for λ ∈ S a \ S b and µ α ( λ, λ ) = µ a ( λ ) µ b ( λ ) for λ ∈ S a ∩ S b and, hence, c αaa = Z dλdλ µ α ( λ, λ ) ξ aa ( λ, λ )= Z S a × S a dλdλ µ α ( λ, λ )= Z ( S a \ S b ) × S a dλdλ µ a ( λ ) µ a ( λ ) + Z ( S a ∩ S b ) × S a dλdλ µ a ( λ ) µ b ( λ )= (1 − c ab ) + c ab · c ba = 1 − c ab + c ab , where, in the last equality, we use the opera-tional fact that c ab = c ba . Finally, this gives F g = (1 − c ab + c ab ) + = F NC g . In Appendix A wecomplete this strategy with a concrete choice of µ a , µ b , µ aa ⊥ , µ bb ⊥ , µ α ⊥ and µ β ⊥ complying with O1 andsatisfying Eq. (4) for all the operational equivalencesin O2.This optimal strategy seems to suggest the follow-ing intuition behind the theorem: our assumption ofpreparation noncontextuality on the input prepara-tions P a , P b imply that the distributions µ a ( λ ) and µ b ( λ ) overlap “too much” (formally, it implies maxi-mal ψ -epistemicity, c ab = R S b dλµ a ( λ ) [18]), hence thecloning performance turns out worse than in quan-tum mechanics. Furthermore, noncontextuality im-plies that µ a and µ b coincide on their overlap, whichimplies a direct relation between c ab and the ‘ norm k µ a − µ b k . Crucially the latter cannot be increasedby the cloning machine, since k · k decreases underpost-processing.However, this mechanism can only be part of thestory. First, the cloning performance is not mono-tonically decreasing with increasing overlap, since for c ab = 1 one can clone perfectly. Second, cloning isdefined as the creation of two independent copies ofthe preparations P a or P b , but these do not neces-sarily correspond to two independent copies µ a µ a , µ b µ b (this assumption, which we do not make, iscalled preparation independence [19]). Nevertheless,we showed that a no-go theorem results from the ob-served overlaps c ab , c aa,bb and noncontextuality as-sumptions only as a consequence of information pro-cessing inequalities and the triangle inequality.We note in passing that our proof technique can beabstracted and applied to other tasks as follows:1. First, given a set of observed overlaps { c ss } , non-contextuality applied to the operational equiva-lences P s + P s ⊥ ’ P s + P s gives the equa-tions (10) . Accepted in
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Quantum2020-04-20, click title to verify. Published under CC-BY 4.0. . Second, verify if the equations (10) are compati-ble with triangle and data processing inequalitiesand the performance of quantum protocol un-der consideration (in this case, state-dependentcloning).In fact, the same proof technique can be extendedto nonideal scenarios (with Eq. (10) replaced byEq. (12) ), as we now see. Theorem 1 is a no-go result for noncontextual ontolog-ical models aimed at explaining the phenomenologyof state-dependent quantum cloning. However, theinequality derived in Eq. (5) is not a proper noncon-textuality inequality because the operational featuresconsidered refer to an idealized experiment. In anyreal experiment, on the other hand, one will need toconfront the following nonidealities: • The correlations in O1 will only approximativelyhold in data collected in a real experiment. • O2 will only be approximatively realized.Theorem 2 below extends Theorem 1 beyond the ideallimit, allowing for the observation of nonperfect corre-lations in O1, such as those generated by a cloning ex-periment carried out with nonideal preparations andtest measurements. As we will discuss later, thereare general techniques to deal with the idealizationin O2, so that the problem of deriving an experimen-tally testable statement reduces to the elimitation ofthe idealization in O1. Specifically, we want to weakenit toO1ni p ( M s | P s ) ≥ − (cid:15) s , p ( M s | P s ⊥ ) ≤ (cid:15) s for s = a, b, α, β, aa, bb ,where ‘ni’ stands for ‘non-ideal’. Theorem 2 (Optimal cloning fidelity in noncontex-tual models – noise-robust version) . With the nota-tion of Thm. 1, suppose that one observes the opera-tional features O1ni and O2. Then, for any noncon-textual model we have that F g ≤ F NC , ni g = 1 − c ab c aa,bb . (11) where Err = ( (cid:15) b + 2 (cid:15) bb + (cid:15) aa ) . Note that, while we gave an independent and sim-pler proof of Theorem 1, we can now see it as a corol-lary of the result above once all error terms are setof zero. Another interesting case is when all errorterms are equal, (cid:15) b = (cid:15) bb = (cid:15) aa := (cid:15) , which gives F NC , ni g = 1 − c ab + c aa,bb + 2 (cid:15) . In fact, we can givea slightly stronger and symmetric bound than theabove. For the specific form, see Appendix B. The proof of Theorem 2 follows the same lines asthat of Theorem 1. The key addition is to extendEq. (10) to the noisy setting. Specifically, we showthat in the presence of the operational features O1ni-O2, noncontextual models must satisfy |k µ s − µ s k − − c ss ) | ≤ (cid:15) s , (12) and similarly if we exchange s and s . In other words,the relation of Eq. (10) holds approximatively, andwe can bound its violation with the experimentallyaccessible noise level. The proof of this result is moreinvolved than in the ideal scenario, so we postponethe derivation to Appendix B.Eq. (12) imposes a strict relation, in any noncon-textual model and beyond the ideal scenario, betweenthe ‘ distance of two epistemic states and their opera-tionally accessible confusability. Hence, we anticipatethat these relations will be of broader use to identifyquantum advantages beyond state-dependent cloning.For instance, following the same reasoning given afterTheorem 1, these inequalities provide an alternativeand intuitive derivation of the tight noise-robust non-contextual bound on state discrimination of Ref. [5], s ab ≤ + k µ a − µ b k ≤ − c ab − (cid:15) b . Having derived a noise-robust version of our noncon-textual bound, the next step is to investigate whetherquantum mechanics violates it. We consider a stan-dard noise model in which the ideal quantum prepara-tions, measurements and unitary transformation areall thwarted by a depolarizing channel N v with noiselevel v ∈ [0 , : N v ( ρ ) = (1 − v ) ρ + v I / . A direct calculation (see Appendix C) shows thatthis sets (cid:15) = v (31 − v + 9 v ) / in Eq. (11) . If oneuses the unitary transformation that is optimal forstate-dependent cloning in the noiseless setting, onegets a quantum strategy whose global average fidelityreads F Q , noisy g ( v ) := (1 − v ) F Q , opt g + 14 v (3 − v + v ) (13) which coincides with the optimal for v = 0 . For v > ,however, and unlike in the ideal case, the tradeoff be-tween c ab and F g is not necessarily above the non-contextual bound. For example, for v = 0 . a vi-olation can be observed only for c ab ∈ [0 . , . ,see Fig. 3. Nevertheless, a preliminary comparisonwith the experimental results of Ref. [20] suggeststhat the required low level of noise is not beyond cur-rent experiments. In fact, in terms of the parame-ter C s = 1 / p ( M s | P s ) + 1 / p ( M s ⊥ | P s ⊥ ) defined inRef. [20] ( C s = 1 in the ideal scenario), v = 0 . cor-responds to C s ≈ . for s = a, b and C s ≈ . for s = aa, bb , and Ref. [20] experimentally realized C s = 0 . . Accepted in
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Quantum2020-04-20, click title to verify. Published under CC-BY 4.0. .0 0.2 0.4 0.6 0.8 1.00.0000.0050.0100.0150.020 c ab v Figure 3:
Noise-resistance of the quantum advantage incloning.
This plot shows the maximum value of the noise pa-rameter v of a depolarizing channel (affecting preparations,measurements and transformation) for which the quantumvalue of the cloning fidelity (Eq. (13) ) is above the noncon-textual bound, as a function of the confusability between theinputs c ab . As we mentioned, the only remaining idealiza-tion is the operational verification of O2. Letus suppose that, in an experiment, after do-ing tomography, one determines that the actualexperimental realizations of the ideal preparations are P (1) a , P (1) a ⊥ , P (1) b , P (1) b ⊥ , P (1) aa , P (1) aa ⊥ , P (1) bb , P (1) bb ⊥ , P (1) α , P (1) α ⊥ , P (1) β and P (1) β ⊥ . These ’primary’ preparations will,in general, not respect the required operationalequivalences in O2, due to unavoidable imperfectionsin the experimental realisation. Luckily, there aregeneral considerations to tackle this idealization [20].The first thing to notice is that if one can experi-mentally achieve a set of preparations P (1) s , then onecan also prepare any convex combination of them, i.e.any preparation in the convex hull C of the prepara-tions P (1) s . By the linearity of Eq. (1) , one can thencompute the measurement statistics of all the prepa-rations in C . Therefore, as put forward in Sec. IV ofRef. [20], to go ahead with the experimental verifica-tion of Theorem 2 one only needs to find ‘secondary’preparations P (2) s in C whose measurement statisticssatisfy the operational equivalences in O2; that is, weonly needO2ni O2 is satisfied for some preparations P (2) s in theconvex hull C of the experimental preparations P (1) s .This post-processing hence allows one to apply The-orem 2 even if the collected data does not satisfy O2.One can think of the secondary preparations as noisyversions of the primary preparations. Hence, the price We will later discuss the assumption that can one access atomographically complete set of measurements. one pays in this construction is that the correspond-ing noise parameters (cid:15) s = p ( M s | P (2) s ) in O1ni will ingeneral be larger. Note that, even if (cid:15) s is too largecompared to (cid:15) s to see any violation in Theorem 2,one can get around this issue by adding extra experi-mental preparations P (1)extra to enlarge C , as explicitlydone in Ref. [20]. To summarize, there are good gen-eral tools to deal with imperfections in the operationalequivalences O2.As a final remark, it is useful to briefly talk aboutloopholes. These are all those assumptions that can-not be conclusively tested by any experimental means.In a nonlocality experiment, for example, these in-clude the assumption that the two sides cannot com-municate and the ability to choose the measurementfreely, i.e., independently of any other variable rele-vant to the experiment. In a contextuality experi-ment the notion of operational equivalence relies onthe knowledge of a tomographically complete set ofmeasurements. However, if quantum theory is notcorrect, the tomographically complete set of a post-quantum theory may contain extra unknown measure-ments (just like a future theory may allow signalling).Recent work has shown that the problem can be miti-gated by the addition of extra (known) measurementsand preparations (see Ref. [21]), but this goes beyondthe scope of the present work. We have shown that the operational statistics ob-served in the optimal state-dependent quantumcloning is incompatible with the predictions of ev-ery noncontextual ontological model. In particular,for given overlap, the noncontextual global cloningfidelity is strictly smaller than the quantum predic-tion. A similar result continues to hold in more re-alistic experiments which are unavoidably affected bynoise (while the effect can be ‘washed out’ by exces-sive experimental imperfections). This identifies con-textuality as the resource for optimal state-dependentquantum cloning.From a foundational point of view, it would be rel-evant to explore whether the relation between contex-tuality and cloning fidelity, that we proved for optimalstate-dependent cloning, extends to the other typesof imperfect cloning studied in the literature, mainlyphase-covariant and/or universal cloning, as well asto probabilistic cloning [11]. From an applications’point of view, one important open question is if ournoncontextual bound can be used to prove a contex-tual advantage for quantum information processingtasks which rely on optimal quantum state-dependentcloning ( e.g. , [22, 23]).Finally, it may be possible to use the connectionbetween ‘ norm and confusability developed here tounderstand what aspects of other quantum informa- Accepted in
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Quantum2020-04-20, click title to verify. Published under CC-BY 4.0. ion primitives, such as quantum teleportation, aretruly nonclassical. Acknowledgements.
We are grateful to JosephBowles for useful comments on a draft of thismanuscript. We acknowledge financial supportfrom the the European Union’s Marie Sklodowska-Curie individual Fellowships (H2020-MSCA-IF-2017,GA794842), Spanish MINECO (Severo Ochoa SEV-2015-0522 and project QIBEQI FIS2016-80773-P),Fundacio Cellex and Generalitat de Catalunya(CERCA Programme and SGR 875).
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The logico-algebraic approach to quantum me-chanics , pages 293–328. Springer, 1975. DOI:10.1007/978-94-010-1795-4_17.[16] Robert W Spekkens. The ontological identity ofempirical indiscernibles: Leibniz’s methodologi-cal principle and its significance in the work ofEinstein. 2019. URL https://arxiv.org/abs/1909.04628v1 .[17] Dagmar Bruß, David P DiVincenzo, Artur Ek-ert, Christopher A Fuchs, Chiara Macchiavello,and John A Smolin. Optimal universal andstate-dependent quantum cloning.
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Accepted in
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Quantum2020-04-20, click title to verify. Published under CC-BY 4.0. ion. Physical Review A , 62(4):042304, 2000.DOI: 10.1103/PhysRevA.62.042304.
Accepted in
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Quantum2020-04-20, click title to verify. Published under CC-BY 4.0. Noncontextual model saturating the bound in Theorem 1
To complement the cloning strategy given in the main text, in this section we give a concrete choice of distribu-tions µ aa , µ bb , µ aa ⊥ , µ bb ⊥ and µ α ⊥ satisfying the operational features targeted by Theorem 1. The supports ofall these distributions, which we set to be subsets of [0 , × [0 , , are plotted in Figure 4. All the distributionsare constantly on their support. Notice that since the cloning map given in the main text makes µ β ≡ µ bb ,it follows that to satisfy the operational equivalence for ( µ β , µ bb ) we must have µ β ⊥ ≡ µ bb ⊥ . We let the readerverify, by inspecting the plots, that the remaining requirements implied by the operational features O1 and O2are satisfied by these distributions (and the choice of response functions made in the main text). (a) S aa (b) S aa ⊥ (c) S bb (d) S bb ⊥ (e) S α ⊥ Figure 4: Supports of distributions µ aa , µ bb , µ aa ⊥ , µ bb ⊥ and µ α ⊥ satisfying the restrictions imposed on noncotextual modelsby the requirements of Theorem 1. The distributions are -valued in the filled regions (i.e. in their support). Accepted in
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Quantum2020-04-20, click title to verify. Published under CC-BY 4.0. Generalization and proof of Theorem 2
In this section we will prove a slightly stronger and more symmetric bound on the noncontextual cloning fidelity F NC g from which the bound in Thm. 2 in the main text follows straightforwardly as a corollary. Theorem 3.
With the notation of Thm. 1, suppose that one observes the operational features O1ni and O2:O1ni p ( M s | P s ) ≥ − (cid:15) s , p ( M s | P s ⊥ ) ≤ (cid:15) s for s = a, b, α, β, aa, bb, O2 P s + P s ⊥ ’ P s + P s , for all ( s, s ) in { ( a, b ) , ( α, aa ) , ( β, bb ) } .Then, for any noncontextual model we have that F NC g ≤ { (cid:15) b − c ab , (cid:15) a − c ba } { c aa,bb + (cid:15) bb , c bb,aa + (cid:15) aa } (cid:15) aa + (cid:15) bb . (14) For the proof of Theorem 3, we make use of the following lemma relating the ‘ distance of two epistemicstates in any ontological model satisfying the hypothesis of the theorem and their operationally accessibleconfusability. Lemma 4.
Let P s , P s be preparations. Suppose there exists preparations P s ⊥ , P s and a two outcome mea-surement M s such that1. P s + P s ⊥ ’ P s + P s , p ( M k | P k ) ≥ − (cid:15) k , p ( M k | P k ⊥ ) ≤ (cid:15) k , k = s, s . Then, in a noncontextual ontological model, { − c ss − (cid:15) s , − c s s − (cid:15) s } ≤ k µ s − µ s k ≤ { − c ss + (cid:15) s , − c s s + (cid:15) s } , (15) Proof.
We denote by S s the support of µ s . Define a partition S s ∪ S s = t i =1 R i , as summarized in Figure 5: • R = S s \ ( S s ∩ S s ), R = S s \ ( S s ∩ S s ). • R = { λ ∈ S s ∩ S s | µ s ( λ ) ≥ µ s ( λ ) } , R = { λ ∈ S s ∩ S s | µ s ( λ ) < µ s ( λ ) } . Figure 5: Sketch of the relevant regions in the proof of Lemma 4.
Then, k µ s − µ s k = Z dλ | µ s ( λ ) − µ s ( λ ) | = Z R dλµ s ( λ ) + Z R dλµ s ( λ ) + Z R dλ [ µ s ( λ ) − µ s ( λ )] + Z R dλ [ µ s ( λ ) − µ s ( λ )]= 2 − Z R ∪ R dλ [ µ s ( λ ) + µ s ( λ )] + Z R dλ [ µ s ( λ ) − µ s ( λ )] + Z R dλ [ µ s ( λ ) − µ s ( λ )]= 2 − Z R dλµ s ( λ ) − Z R dλµ s ( λ ) . (16) Accepted in
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Quantum2020-04-20, click title to verify. Published under CC-BY 4.0. onsider, c ss − Z R dλµ s ( λ ) − Z R dλµ s ( λ ) = Z R ∪ R ∪ R dλµ s ( λ ) ξ s ( λ ) − Z R dλµ s ( λ ) − Z R dλµ s ( λ ) ≤ Z R ∪ R dλµ s ( λ ) ξ s ( λ ) − Z R dλµ s ( λ )= Z R ∪ R dλ [ µ s ( λ ) + µ s ( λ ) − µ s ⊥ ( λ )] ξ s ( λ ) − Z R dλµ s ( λ )= − Z R dλµ s ( λ ) ξ s ( λ ) + Z R ∪ R dλ [ µ s ( λ ) − µ s ⊥ ( λ )] ξ s ( λ ) ≤ Z R ∪ R dλµ s ( λ ) ξ s ( λ ) ≤ Z dλµ s ( λ ) ξ s ( λ ) = p ( M s | P s ) ≤ (cid:15) s , where we used ξ s ≤ R R dλµ s ( λ ) ξ s ( λ ) = 0 and in the final inequality we used assumption 2. Then,using Eq. (16), k µ s − µ s k ≤ − c ss + (cid:15) s ) . Furthermore, recalling that ξ s = 1 − ξ s , Z R dλµ s ( λ ) + Z R dλµ s ( λ ) − c ss = Z R dλµ s ( λ ) + Z R dλµ s ( λ ) − Z R ∪ R ∪ R dλµ s ( λ ) ξ s ( λ ) ≤ Z R dλµ s ( λ ) ξ s ( λ ) + Z R dλµ s ( λ ) ξ s ( λ ) − Z R dλµ s ξ s ( λ ) ≤ Z R dλµ s ( λ ) ξ s ( λ ) + Z R dλµ s ( λ ) ξ s ( λ ) ≤ Z dλµ s ( λ ) ξ s ( λ ) = p ( M s | P s ) ≤ (cid:15) s . where in the first inequality we used that µ s ( λ ) ≤ µ s ( λ ) in R and µ s ( λ ) ≥ µ s ( λ ) in R . In the final inequality,we used assumption 2. Hence, we have that c ss − Z R dλµ s ( λ ) − Z R dλµ s ( λ ) ≥ − (cid:15) s and, using Eq. (16), that k µ s − µ s k ≥ − c ss + (cid:15) s ) . (17)Finally, noting that k µ s − µ s k = k µ s − µ s k and that the above derivation is symmetric under the exchangeof s with s we arrive to the desired result2 max { − c ss − (cid:15) s , − c s s − (cid:15) s } ≤ k µ s − µ s k ≤ { − c ss + (cid:15) s , − c s s + (cid:15) s } , (18)Notice that for the lower bound in Eq. (17) (and, hence, the left hand side of Eq. (18)) we did not use assumption1 of operational equivalence. Given the above we can now prove Theorem 3:
Proof of Theorem 3.
In the first part we proceed as in the ideal case. From the triangle inequality and thecontractivity of the ‘ norm under stochastic processes (which gives k µ α − µ β k ≤ k µ a − µ b k ), one can show thatthe following equation holds (see Eq. (8)): k µ aa − µ bb k ≤ k µ α − µ aa k + k µ a − µ b k + k µ β − µ bb k . (19)Using both upper and lower bounds for the ‘ distance derived in Lemma 4, this implies2 max { − c aa,bb − (cid:15) bb , − c bb,aa − (cid:15) aa }≤ − c αaa ) + 2 (cid:15) aa + 2 min { − c ab + (cid:15) b , − c ba + (cid:15) a } +2(1 − c βbb ) + 2 (cid:15) bb , which can be rearranged to give the claimed bound on F NC g . Accepted in
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Quantum2020-04-20, click title to verify. Published under CC-BY 4.0. Quantum violation of noise-contextual bound under depolarizing noise
C.1 Introducing noise
We will assume that all experimental procedures in the ideal quantum cloning experiment (that is, preparations,measurements and transformations) are affected by a depolarizing channel N v with noise level v ∈ [0 , : N v ( ρ ) = (1 − v ) ρ + v I . Therefore, for x ∈ { a, b } , the ideal input preparations transform as | x i 7→ N v ( | x ih x | ) = (1 − v ) | x ih x | + v I , so that the actual input preparations become ρ x := Tr [ N v ( | x ih x | )] = (1 − v ) | x ih x | + v I ab , with I ab the projector over the span( {| a i , | b i} ) . The ideal cloning transformation U becomes N v ◦ U = (1 − v ) U + v D , where D ( ρ ) = I / for all ρ . Hence, the actual outcomes χ ∈ { α, β } correspondent to input x ∈ { a, b } become ρ χ := N v ◦ U ( ρ x ) = [(1 − v ) U + v D ] (cid:18) (1 − v ) | x ih x | + v I (cid:19) = (1 − v ) | χ ih χ | + (1 − (1 − v ) ) I . The actual target copies would become N v ( | xx ih xx | ) . While this is the minimal amount of noise in thispreparation required by our model, not all operational equivalences are satisfied under it. A simple (albeitlikely not optimal) way to fix this issue is to let the noise act for a second step; hence, define ρ xx := N v ◦ N v ( | xx ih xx | ) = (1 − v ) | xx ih xx | + (1 − (1 − v ) ) I . Finally, for the ideal measurements, they transform as (for x ∈ { a, b } , χ ∈ { α, β } ), {| x ih x | , I ab − | x ih x |} 7→ M x := (cid:26) (1 − v ) | x ih x | + v I ab , (1 − v )( I ab − | x ih x | ) + v I ab (cid:27) , {| xx ih xx | , I − | xx ih xx |} 7→ M xx := (cid:26) (1 − v ) | xx ih xx | + v I , (1 − v )( I − | xx ih xx | ) + v I (cid:27) , {| χ ih χ | , I − | χ ih χ |} 7→ M χ := (cid:26) (1 − v ) | χ ih χ | + v I , (1 − v )( I − | χ ih χ | ) + v I (cid:27) . C.2 Orthogonal preparations and operational equivalences
We now introduce the orthogonal preparations, necessary for the satisfaction of the operational equivalences.We start with the ones pertaining to the pair of input preparations ( a, b ) . For, x ∈ { a, b } , let ρ x ⊥ := (1 − v ) (cid:12)(cid:12) x ⊥ (cid:11)(cid:10) x ⊥ (cid:12)(cid:12) + v I ab , with | x ⊥ i ∈ span( {| a i , | b i} ) and (cid:10) x (cid:12)(cid:12) x ⊥ (cid:11) = 0 . Note that these are naturally thought as the noisy version of theperfect orthogonal preparations, ρ x ⊥ = N v ( (cid:12)(cid:12) x ⊥ (cid:11)(cid:10) x ⊥ (cid:12)(cid:12) ) . Now, let us check that the operational equivalence issatisfied, ρ a + 12 ρ a ⊥ = (1 − v ) | a ih a | + (cid:12)(cid:12) a ⊥ (cid:11)(cid:10) a ⊥ (cid:12)(cid:12) v I ab I ab − v ) | b ih b | + (cid:12)(cid:12) b ⊥ (cid:11)(cid:10) b ⊥ (cid:12)(cid:12) v I ab ρ b + 12 ρ b ⊥ . Next, we consider the pair of preparations s ∈ ( α, aa ) . Let ρ s ⊥ := (1 − v ) | s ih s | + (1 − (1 − v ) ) I , with | s ⊥ i ∈ span( {| α i , | aa i} ) , and (cid:10) s (cid:12)(cid:12) s ⊥ (cid:11) = 0 . ρ s ⊥ can be seen as the state resulting from preparing | s ⊥ i andletting the noise channel act for two steps, i.e., ρ s ⊥ = N v ◦ N v ( (cid:12)(cid:12) s ⊥ (cid:11)(cid:10) s ⊥ (cid:12)(cid:12) ) . Accepted in
Quantum2020-04-20, click title to verify. Published under CC-BY 4.0.
Quantum2020-04-20, click title to verify. Published under CC-BY 4.0. e can now see that with this choice of states the operational equivalences are satisfied: ρ α + 12 ρ α ⊥ = (1 − v ) | α ih α | + (cid:12)(cid:12) α ⊥ (cid:11)(cid:10) α ⊥ (cid:12)(cid:12) − (1 − v ) ) I − v ) I αaa − (1 − v ) ) I
4= (1 − v ) | aa ih aa | + (cid:12)(cid:12) aa ⊥ (cid:11)(cid:10) aa ⊥ (cid:12)(cid:12) − (1 − v ) ) I ρ aa + 12 ρ aa ⊥ , with I αaa the projector over the span( {| α i , | aa i} ) . Following the same argumentation, one can see that for theremaining pairs of preparations ( β, bb ) and ( aa, bb ) , if we let ρ β ⊥ := (1 − v ) (cid:12)(cid:12) β ⊥ (cid:11)(cid:10) β ⊥ (cid:12)(cid:12) + (1 − (1 − v ) ) I ,ρ bb ⊥ := (1 − v ) (cid:12)(cid:12) bb ⊥ (cid:11)(cid:10) bb ⊥ (cid:12)(cid:12) + (1 − (1 − v ) ) I , with | bb ⊥ i , | β ⊥ i ∈ span( {| β i , | bb i} ) and (cid:10) bb (cid:12)(cid:12) bb ⊥ (cid:11) = (cid:10) β (cid:12)(cid:12) β ⊥ (cid:11) = 0 , the operational equivalence for ( β, bb ) is satisfied: ρ β + 12 ρ β ⊥ = 12 ρ bb + 12 ρ bb ⊥ . And letting ρ aa ⊥ := (1 − v ) (cid:12)(cid:12) aa ⊥ (cid:11)(cid:10) aa ⊥ (cid:12)(cid:12) + (1 − (1 − v ) ) I ,ρ bb ⊥ := (1 − v ) (cid:12)(cid:12)(cid:12) bb ⊥ ED bb ⊥ (cid:12)(cid:12)(cid:12) + (1 − (1 − v ) ) I , with | aa ⊥ i , | bb ⊥ i ∈ span( {| aa i , | bb i} ) , and (cid:10) aa (cid:12)(cid:12) aa ⊥ (cid:11) = D bb (cid:12)(cid:12)(cid:12) bb ⊥ E = 0 , the operational equivalence for ( aa, bb ) issatisfied: f ρ aa + 12 ρ aa ⊥ = 12 f ρ bb + 12 ρ bb ⊥ . Notice that ρ aa ⊥ , ρ aa ⊥ and ρ bb ⊥ , ρ bb ⊥ are alternative choices of orthogonal preparations, tailored to each pair ofpreparations appearing in the operational equivalences. C.3 Noise parameter and Error term in the NC bound
In this subsection, we find the expression for each of the measurement error probabilities appearing in the errorterm in Eq. (11) as a function of the noise parameter v of the depolarizing channel.For x ∈ { a, b } , − (cid:15) x = p ( M x | P x ) = Tr (cid:20)(cid:18) (1 − v ) | x ih x | + v I ab (cid:19) (cid:18) (1 − v ) | x ih x | + v I ab (cid:19)(cid:21) = (1 − v ) + 2 v (1 − v ) Tr (cid:20) | x ih x | (cid:21) + v Tr (cid:20) I ab (cid:21) = (1 − v ) + v (1 − v ) + v ⇒ (cid:15) x = v − v . For χ ∈ { α, aa } , − (cid:15) α = p ( M α | P α ) = Tr[ ρ α M α ] = 1 − (cid:15) aa = p ( M aa | P aa ) = Tr[ ρ aa M aa ]= Tr (cid:20)(cid:18) (1 − v ) | χ ih χ | + (cid:0) − (1 − v ) (cid:1) I (cid:19) (cid:18) (1 − v ) | χ ih χ | + v I (cid:19)(cid:21) = (1 − v ) + (1 − v ) v − (1 − v ) )(1 − v )4 + (1 − (1 − v ) ) v
4= 14 (4 − v + 9 v − v )= ⇒ (cid:15) aa = (cid:15) α = 34 v (3 − v + v ) , Accepted in
Quantum2020-04-20, click title to verify. Published under CC-BY 4.0.
Quantum2020-04-20, click title to verify. Published under CC-BY 4.0. nd analogously for χ in { β, bb } and { aa, bb } . Hence, Err = (cid:15) α + (cid:15) β + (cid:15) a + (cid:15) b + 2( (cid:15) aa + (cid:15) bb ) = 2( v − v · v (3 − v + v ) = 12 v (31 − v + 9 v ) . Following the same arguments, it is easy to see that, for the case of symmetric confusabilities, the error term inEq. (11) becomes, as a function of v , Err = 18 v (31 − v + 9 v ) C.4 Quantum performance
In this last subsection, we compute the global average fidelity F Q g in the noisy setting of the optimal quantumcloner for the ideal setting as a function of the noise channel’s parameter v . F Q g = 12 Tr[ M aa ρ α ] + 12 Tr[ M bb ρ β ]= Tr (cid:20)(cid:18) (1 − v ) | xx ih xx | + (cid:0) − (1 − v ) (cid:1) I (cid:19) (cid:18) (1 − v ) | χ ih χ | + v I (cid:19)(cid:21) = (1 − v ) |h xx | χ i| + (1 − (1 − v ) )(1 − v )4 + (1 − v ) v − (1 − v ) ) v
4= (1 − v ) |h xx | χ i| + 14 v (3 − v + v ) . Hence, F Q , noisy g ( v ) := (1 − v ) F Q , opt g + 14 v (3 − v + v ) . Accepted in
Quantum2020-04-20, click title to verify. Published under CC-BY 4.0.