Self-testing mutually unbiased bases in the prepare-and-measure scenario
SSelf-testing mutually unbiased bases in the prepare-and-measure scenario
Máté Farkas ∗ and Jędrzej Kaniewski Institute of Theoretical Physics and Astrophysics,National Quantum Information Centre, Faculty of Mathematics,Physics and Informatics, University of Gdansk, 80-952 Gdansk, Poland Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
Mutually unbiased bases (MUBs) constitute the canonical example of incompatible quantummeasurements. One standard application of MUBs is the task known as quantum random accesscode (QRAC), in which classical information is encoded in a quantum system, and later part of it isrecovered by performing a quantum measurement. We analyse a specific class of QRACs, known asthe d → QRAC, in which two classical dits are encoded in a d -dimensional quantum system. Itis known that among rank-1 projective measurements MUBs give the best performance. We show(for every d ) that this cannot be improved by employing non-projective measurements. Moreover,we show that the optimal performance can only be achieved by measurements which are rank-1projective and mutually unbiased. In other words, the d → QRAC is a self-test for a pair ofMUBs in the prepare-and-measure scenario. To make the self-testing statement robust we proposemeasures which characterise how well a pair of (not necessarily projective) measurements satisfiesthe MUB conditions and show how to estimate these measures from the observed performance.Similarly, we derive explicit bounds on operational quantities like the incompatibility robustnessor the amount of uncertainty generated by the uncharacterised measurements. For low dimensionsthe robustness of our bounds is comparable to that of currently available technology, which makesthem relevant for existing experiments. Lastly, our results provide essential support for a recentlyproposed method for solving the long-standing existence problem of MUBs.
I. INTRODUCTION
Mutually unbiased bases (MUBs) play an importantrole in many quantum information processing tasks.They are optimal for quantum state determination [1, 2],information locking [3, 4], and the mean king’s problem[5, 6]. Moreover, they give rise to the strongest entropicuncertainty relations (among projective measurements)[7–9]. One intuitive way to look at them is the follow-ing: imagine that we encode a classical message in a purestate corresponding to an element of a basis. Then, if wemeasure this state in a basis unbiased to the initial one,each measurement outcome occurs with the same proba-bility. That is, we do not learn anything about the orig-inally encoded message. Formally, two bases {| a i (cid:105)} di =1 and {| b j (cid:105)} dj =1 in C d are mutually unbiased if |(cid:104) a i | b j (cid:105)| = 1 d ∀ i, j ∈ [ d ] := { , , . . . , d } . (1)Due to their importance, significant effort has beendedicated to investigating their structure (see [10] for asurvey and [11] for a classification in dimensions 2–5). Itis known that in dimension d , there are at least 3 and atmost d + 1 MUBs and the upper bound is saturated inprime power dimensions. The maximal number of MUBsin composite dimensions is a long-standing open problem(see [12–17] for the case of dimension 6).Another scenario in which MUBs perform well is theso-called d → quantum random access code (QRAC) ∗ [email protected] [18, 19]. In this setup, two classical dits are encoded intoa qudit, and the aim is to recover one of them chosenuniformly at random. It is well-known that sending aquantum system gives an advantage over sending a clas-sical system (of the same dimension) [20] and this factis used in many quantum information protocols [21–25].It is commonly believed that the optimal performance ofthe d → QRAC is achieved when the measurementscorrespond to a pair of MUBs in dimension d , but thisclaim has only been proven for a restricted class of mea-surements [26].The observation that quantum systems can give riseto stronger-than-classical correlations was first made byBell [27] (although in a slightly different setup). More-over, it turns out that some of these strongly non-classicalcorrelations can be achieved in an essentially unique man-ner. That is, the observed statistics allow us to iden-tify the employed states and measurements (up to lo-cal isometries and extra degrees of freedom). The mostprominent example of this kind is the well-known CHSHinequality [28], which is maximally violated by a pairof MUBs in dimension 2 on both sides [29–32]. When-ever such an inference — characterising the state and/ormeasurements based solely on the observed statistics —can be made, it is referred to as self-testing [33–35].Self-testing is closely related to the concept of device-independent (DI) quantum information processing, inwhich the devices used in the protocol are a priori un-trusted [36–40]. It is clear that what makes DI cryptog-raphy possible is precisely the self-testing character ofthe correlations observed during the protocol. By nowself-testing is a well-developed field [41–48] and includesresults which are robust to noise [49–55]. Such state- a r X i v : . [ qu a n t - ph ] M a r ments are of particular interest, as they can be directlyapplied to experiments [56].Recently the notion of self-testing has been extendedto prepare-and-measure scenarios [57]. In this setup, apreparation device creates one of many possible quantumstates and then sends it to a measurement device. Thelatter performs one of many possible measurements onthe state, and then produces a classical output. Thisscenario encompasses many important quantum com-munication protocols, e.g. the BB84 and B92 quantumcryptography protocols [58, 59], and the aforementionedQRACs.In the prepare-and-measure scenario one cannot distin-guish between classical and quantum systems, unless ad-ditional restrictions are imposed. The standard choice isto place an upper bound on the dimension of the systemtransmitted between the devices [60–62]. This is oftenreferred to as the semi-device-independent (SDI) modelfor which several cryptographic protocols have been pro-posed [63–65]. In analogy to the DI model, it is clear thatthe security of SDI protocols is related to self-testing re-sults in the prepare-and-measure scenario.In this paper, we investigate the self-testing propertiesof the d → QRAC. In [57], the authors derive robustself-testing results for d = 2 and ask whether similarstatements hold for larger d . We resolve this questionby deriving a robust self-testing statement for arbitrary d . We show that the optimal performance in the d → QRAC certifies that the two measurements correspond toMUBs. To make the statement robust we propose newmeasures which characterise how close a pair of POVMsis to the MUB arrangement and derive explicit boundson those in terms of the QRAC performance. Finally,we use this characterisation to obtain explicit bounds onoperationally relevant quantities like the incompatibilityrobustness [66] or the amount of uncertainty produced.
II. SETUP
In the d → QRAC scenario (see Fig. 1), on thepreparation side Alice gets two uniformly random in-puts, i, j ∈ [ d ] . Based on these inputs she prepares a d -dimensional state ρ ij , and sends it to Bob who is onthe measurement side. He gets a uniformly random input y ∈ { , } , which tells him which of Alice’s inputs he issupposed to guess. If y = 1 , he aims to guess i , otherwise j . This is performed by a measurement on ρ ij , which wedescribe by the operators { A i } i for y = 1 , and { B j } j for y = 2 , where A i , B j ≥ , (cid:80) i A i = (cid:80) j B j = I and i, j ∈ [ d ] . The outcome of the measurement determinesBob’s guess and the figure of merit is the average successprobability (ASP), which can be written, using the abovenotation, as ¯ p = 12 d (cid:88) ij tr (cid:2) ρ ij ( A i + B j ) (cid:3) . (2) i j ybρ ij Alice Bob
Fig. 1: Schematic representation of the d → QRACprotocol.
III. IDEAL SELF-TEST
To obtain the ideal self-testing statement we derive anachievable upper bound on the ASP and identify situa-tions in which all the steps in the derivation are tight.Note that tr (cid:2) ρ ij ( A i + B j ) (cid:3) ≤ || A i + B j || , where || . || is theoperator norm, and since ( A i + B j ) ≥ , one can alwaysfind a state ρ ij such that this inequality is saturated. Letus from now on assume that the preparations are alwayschosen optimally, which allows us to focus solely on themeasurements. Finding the maximal ASP can be per-formed using operator norm inequalities and other toolsfrom matrix analysis, and yields the following theorem. Theorem 1.
The average success probability of the d → QRAC is upper bounded by ¯ p ≤ (cid:18) √ d (cid:19) =: ¯ p Q , (3) and this bound can only be attained if Bob’s mea-surements are rank-1 projective and mutually unbiased.Moreover, in the optimal case the prepared states are theunique eigenstates of A i + B j , corresponding to the high-est eigenvalue. It was previously known that this upper bound holdsif we restrict ourselves to rank-1 projective measure-ments and that among these measurements only MUBscan actually achieve it [26]. What we show is that theQRAC performance cannot be improved by employingnon-projective measurements and that the optimal per-formance indeed requires MUBs, even if we allow forgeneric measurements. Note that this does not followfrom any extremality argument, as in general projectivemeasurements are not the only extremal d -outcome mea-surements [67].For a complete proof, we refer the reader to AppendixA. Here, we state that the crucial step is to use operatornorm inequalities to show that the ASP is bounded by ¯ p ≤
12 + 12 d (cid:88) ij (cid:112) t ij , (4)where t ij := tr( A i B j ) ≥ , and therefore (cid:80) ij t ij = d .The right-hand side is strictly Schur-concave in { t ij } ij ,and hence is uniquely maximised by the uniform distri-bution, t ij = d [68], which yields ¯ p Q . A separate argu-ment implies that to reach ¯ p Q both measurements mustbe rank-1 projective and combining these two facts leadsto the conclusion that the two measurements correspondto MUBs.Theorem 1 implies that the d → QRAC is an SDIself-test for a pair of MUBs in dimension d : observing theoptimal ASP implies that the two measurements consti-tute a pair of MUBs. One might wonder whether theself-testing statement can be made even stronger, in thesense of providing more details about the measurements,but this is not possible. It is easy to check that every pairof MUBs is capable of producing the optimal ASP. Thisideal self-test is crucial for the success of the methodsdescribed in [26], as there it is essential that the optimalQRAC ASP can only be obtained with an arbitrary pairof MUBs. IV. ROBUST SELF-TEST
Since in a real experiment one never observes the op-timal performance, the ideal self-testing result is not suf-ficient. Instead, we need a robust self-testing statement,which tells us what can be certified in the case of sub-optimal performance.Inequality (4) implies that observing the optimal ASPforces the distribution { t ij } ij to be uniform. For sub-optimal performance we immediately get a bound onthe -Rényi entropy ( H ( { q i } ) = 2 log (cid:2)(cid:80) i √ q i (cid:3) ) of thedistribution { t ij /d } ij , which we call the overlap entropy H S ( A, B ) := H (cid:0) { t ij /d } ij (cid:1) . More concretely, from (4)we deduce that H S ( A, B ) ≥ (cid:2) d √ d (2¯ p − (cid:3) . (5)This bound is non-trivial as long as ¯ p > + d √ d andobserving ¯ p = ¯ p Q implies H S ( A, B ) = log ( d ) , which isthe maximal value of the overlap entropy for a pair ofPOVMs. For d = 4 the lower bound is plotted in Fig. 2.Looking at the overlap entropy is not sufficient, be-cause the maximal value can be achieved by measure-ments which are not MUBs, for instance the trivialmeasurements corresponding to A i = B j = I /d . Themissing part is an argument showing that the measure-ments are close to being rank-1 projective. For a d -outcome measurement { A i } i acting on C d this propertycan be assessed by looking at the sum of the norms, N ( A ) := (cid:80) i || A i || , since for all measurements N ( A ) ≤ d and the maximal value is attained if and only if themeasurement is rank-1 projective. Therefore, saturat-ing N ( A ) = N ( B ) = d and H S ( A, B ) = log ( d ) certifiesthe MUB arrangement.To obtain a bound on N ( A ) we need a stronger version p H S (cid:72) A , B (cid:76) d (cid:61) Fig. 2: Lower bound on the overlap entropy for ¯ p ∈ [ + d √ d , ¯ p Q ] in dimension 4.of Eq. (4). In the Appendix B we show that ¯ p ≤
12 + 12 d (cid:88) ij (cid:2) s ij − (2 − √ s ij n ij (cid:3) , (6)where n ij := 1 − (cid:0) || A i || + || B j || (cid:1) and s ij := (cid:12)(cid:12)(cid:12)(cid:12) √ A i (cid:112) B j (cid:12)(cid:12)(cid:12)(cid:12) .This bound reduces to Eq. (4) if we omit the negativeterm and bound s ij by √ t ij , which constitutes an alter-native derivation of Theorem 1 (as n ij = 0 for all i, j implies that both measurements are rank-1 projective).The important feature of Eq. (6) is that it allows us tolower bound the sum of the norms. In Appendix B weshow that for ¯ p > ¯ p := + d (cid:112) ( d − d we have N ( A ) ≥ d − √ d (cid:16) − (cid:112) d (2¯ p − − ( d − (cid:17) (7)and by symmetry the same bound holds for N ( B ) . It iseasy to check that for ¯ p = ¯ p Q , the right-hand side evalu-ates to d , i.e. the optimal performance certifies that bothmeasurements are rank-1 projective. The lower boundgiven in Eq. (7) is plotted for d = 4 in Fig. 3. p N (cid:72) A (cid:76) d (cid:61) Fig. 3: Lower bound on the sum of the norms for ¯ p ∈ (¯ p , ¯ p Q ] in dimension 4.Since Eqs. (7) and (5) allow us to robustly certifythe two defining properties of MUBs (rank-1 projectiv-ity and uniformity of overlaps, respectively), combiningthem yields a robust self-test for MUBs. Note that the ro-bustness is limited by Eq. (7) which requires that ¯ p > ¯ p . V. OPERATIONAL BOUNDS
In the previous paragraph we have focused on quanti-ties tailored to certify closeness to the MUB arrangement.Let us now show that a similar approach can be used toderive bounds on quantities which have an immediateoperational meaning.We begin with the incompatibility robustness. Wesay that two POVMs { A i } i and { B j } j are compatible(or jointly measurable) if there exists a parent POVM { M ij } ij , such that (cid:80) j M ij = A i and (cid:80) i M ij = B j forall i, j . Otherwise they are incompatible, which is of-ten taken as the definition of non-classicality. In orderto quantify incompatibility beyond this binary charac-terisation, the notion of incompatibility robustness hasbeen introduced [66]. Consider the noisy POVMs, A ηi = ηA i + (1 − η ) tr A i I /d , and similarly B ηj . The incompat-ibility robustness η ∗ of A and B is defined as the largest η such that { A ηi } i and { B ηj } j are compatible. Accord-ing to this measure MUBs are highly incompatible, but,perhaps surprisingly, they are not the most incompati-ble among rank-1 projective measurements in dimension d [69].Recently an analytic upper bound on η ∗ has been de-rived for an arbitrary set of POVMs [70]. For a pair ofPOVMs the bound reads η ∗ ≤ d max ij || A i + B j || − (cid:80) i (tr A i ) − (cid:80) j (tr B j ) d (cid:80) i tr A i + d (cid:80) j tr B j − (cid:80) i (tr A i ) − (cid:80) j (tr B j ) . (8)All the quantities appearing in this expression can bebounded using the previously developed methods, whichleads to a bound which depends only on the observedperformance ¯ p . Since the final bound is rather complex,we do not present it here and refer the interested readerto Appendix C. The important feature of the bound isthat for the optimal performance ¯ p = ¯ p Q we recover thecorrect value of the incompatibility robustness for a pairof MUBs, i.e. η ∗ = √ d/ √ d +1 . In Fig. 4 we plot the boundfor d = 4 over the region where it is non-trivial.We note here that similar bounds can be derived forother measures of incompatibility robustness using thesame techniques. Among these is a measure that usesarbitrary POVMs as noise [71], for which MUBs are themost incompatible pair of POVMs (of any number ofoutcomes) in dimension d [72]. This can also be certifiedby observing ¯ p = ¯ p Q .The second operational quantity we consider is theamount of randomness produced by the uncharac-terised measurements. For a POVM A , let H ( A ) ρ := p Η (cid:42) d (cid:61) Fig. 4: Upper bound on the incompatibility robustnessover the non-trivial region in dimension 4. H (cid:0)(cid:8) tr( A i ρ ) (cid:9) i (cid:1) be the Shannon entropy of the outcomestatistics of A on the state ρ . Maassen and Uffink deriveda state-independent lower bound on H ( A ) ρ + H ( B ) ρ forrank-1 projective measurements [7]. For our purposeswe need a more general statement which covers non-projective measurements. Such a bound has been derivedin [73] and reads H ( A ) ρ + H ( B ) ρ ≥ − log c, (9)where c := max ij (cid:12)(cid:12)(cid:12)(cid:12) √ A i (cid:112) B j (cid:12)(cid:12)(cid:12)(cid:12) . Therefore, we need anupper bound on s ij and such a bound has already beenderived in Appendix B. The final statement reads H ( A ) ρ + H ( B ) ρ ≥− (cid:18) p − d (cid:112) d ( d − − d (2¯ p − ] (cid:19) . (10)The optimal performance certifies log d bits of random-ness, which is the maximal value for a pair of projectivemeasurements. We plot the above bound for d = 4 overthe region where it is non-trivial in Fig. 5.We note that a similar bound can be derived for theone-shot analogue of the Shannon entropy, the min-entropy H min (which coincides with the ∞ -Rényi en-tropy), which is often preferred in cryptographic scenar-ios. It was shown in [74] that for a pair of POVMs, H min ( A ) ρ + H min ( B ) ρ ≥ − log (cid:16) √ c (cid:17) , for which wecan derive a similar bound to that of (10). VI. SUMMARY AND OUTLOOK
We have shown that the d → QRAC constitutesa robust self-test for MUBs in arbitrary dimension. Ob-serving sufficiently high ASP allows us to deduce that theemployed measurements are close to being rank-1 projec-tive and that their overlaps are close to being uniform. p d (cid:61) (cid:72) A (cid:76) Ρ (cid:43) H (cid:72) B (cid:76) Ρ Fig. 5: Lower bound on the entropic uncertainty overthe non-trivial region in dimension 4.The same approach can be used to bound operationallyrelevant quantities like the incompatibility robustness orthe amount of randomness produced. For low dimen-sions the robustness of our bounds makes them interest-ing from the experimental point of view.The most obvious direction for further research isto use our self-testing results to prove SDI securityof prepare-and-measure quantum key distribution usinghigh-dimensional systems. One of the main componentsof the SDI security proof given in [63] is the relation be-tween the observed QRAC performance and the random-ness produced for d = 2 (qubits). In this work we deriveprecisely such relations for arbitrary d and we believethat one can use them directly in security proofs.There is an important difference between SDI self-testing and DI self-testing. In the usual DI self-testing wecertify systems up to local isometries and extra degrees offreedom. Since the second equivalence is not relevant inthe SDI setup (the dimension of the system is fixed), one might expect that SDI self-testing should characterise themeasurements up to a unitary transformation. However,this is generally not the case: while in some dimensionsall pairs of MUBs are equivalent up to unitaries (andpossibly complex conjugation), e.g. d = 2 , , , there aredimensions where this is not the case, e.g. d = 4 [11].It is natural to ask whether these inequivalent classes ofMUBs can be distinguished by considering more com-plex QRACs. In fact, a related version of this ques-tion appears readily if we consider n d → QRACs with n > . In this case it is known that different classesof n -tuples of MUBs perform differently [26, 75]. Nu-merical evidence for n = 3 and low d suggests that theoptimal performance is achieved by one of these classes,so one might conjecture that such QRACs self-test thisparticular class. Again, it is not clear how to certify theremaining classes.The d → QRAC analysed in this paper is closelyrelated, at least in spirit, to the family of Bell inequalitiesproposed by Bechmann-Pasquinucci and Gisin [76]. Wehope that the understanding gained in this work will helpus to prove self-testing statements for those inequalities.It would be particularly interesting to see whether theneed for “more-than-unitary” freedom can also appear inthe standard nonlocality-based self-testing.
ACKNOWLEDGEMENTS
We would like to thank Michał Oszmaniec for fruitfuldiscussions. MF acknowledges support from the PolishNCN grant Sonata UMO-2014/14/E/ST2/00020. JK ac-knowledges support under POLONEZ programme whichhas received funding from the European Union’s Hori-zon 2020 research and innovation programme under theMarie Skłodowska-Curie grant agreement no. 665778.
Appendix A: Ideal self-test
In the main text, we establish that the QRAC ASP can be upper bounded by ¯ p ≤ d (cid:88) ij || A i + B j || , (A1)and this can always be saturated by suitable states ρ ij on the preparation side. In order to bound the above quantity,we use a special case of a matrix norm inequality derived by Kittaneh [77], applied to the square-root function andthe operator norm. For further purposes, we briefly reproduce the proof here as well. We will make use of the factthat for operators A, B on a Hilbert space, || A ⊕ B || = max {|| A || , || B ||} [78]. Theorem 2.
Let
A, B ≥ be operators on a Hilbert space. Then || A + B || ≤ max {|| A || , || B ||} + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ A √ B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) .Proof. Consider the block-operator X = (cid:18) √ A √ B (cid:19) , and thus X † X = A + B. (A2)Therefore || A + B || = (cid:12)(cid:12)(cid:12)(cid:12) X † X (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) XX † (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) A √ A √ B √ B √ A B (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) A B (cid:19) + (cid:18) √ A √ B √ B √ A (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) A B (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) √ A √ B √ B √ A (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = max {|| A || , || B ||} + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ A √ B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (A3)where we used some basic properties of the operator norm (see e.g. [78]; or [77] for a more detailed and general versionof the proof).Using the above theorem, we get ¯ p ≤ d (cid:88) ij (cid:16) max {|| A i || , || B j ||} + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:112) A i (cid:112) B j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:17) . (A4)From (cid:80) i A i = (cid:80) j B j = I it follows that A i , B j ≤ I , and thus || A i || , || B j || ≤ . Then ¯ p ≤ d (cid:88) ij (cid:16) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:112) A i (cid:112) B j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:17) = 12 + 12 d (cid:88) ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:112) A i (cid:112) B j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A5)Now we use the fact that for any operator O , || O || ≤ || O || F , where || O || F := (cid:112) tr( O † O ) is the Frobenius norm [78].Therefore ¯ p ≤
12 + 12 d (cid:88) ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:112) A i (cid:112) B j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F = 12 + 12 d (cid:88) ij (cid:113) tr( A i B j ) . (A6)Recall that t ij := tr( A i B j ) and, therefore, t ij ≥ and (cid:80) ij t ij = d . The right-hand side of Eq. (A6) is a symmetric andstrictly concave function of the t ij , and as such, it is strictly Schur-concave (see e.g. [68]). Therefore, it is maximised uniquely by setting all the t ij uniform, t ij = d for all i, j ∈ [ d ] . The upper bound on the ASP set by such t ij is then ¯ p ≤
12 + 12 d (cid:88) ij √ d = 12 (cid:16) √ d (cid:17) . (A7)Note that this bound is saturated by measuring in MUBs (see also [26]).Now, let us turn our attention to necessary conditions for saturating the above bound. We first show that at leastone of the measurements must be rank-1 projective in order to reach the optimal ASP. Saturating the upper boundrequires tr( A i B j ) = d for all i, j ∈ [ d ] and by summing over one of the indices, we see that tr A i = tr B j = 1 for all i, j . Investigating the chain of inequalities obtained above, it is necessary for optimality that max {|| A i || , || B j ||} = 1 forall i, j ∈ [ d ] , otherwise ¯ p < d (cid:80) ij (1 + (cid:12)(cid:12)(cid:12)(cid:12) √ A i (cid:112) B j (cid:12)(cid:12)(cid:12)(cid:12) ) ≤ (cid:16) √ d (cid:17) . Assume that there exists a j ∗ such that || B j ∗ || < .Then in order to fulfil max {|| A i || , || B j ∗ ||} = 1 for all i ∈ [ d ] , it is necessary that || A i || = 1 for all i ∈ [ d ] . Since theseoperators must all be trace-1 and positive semi-definite, it follows that A i = | a i (cid:105)(cid:104) a i | for all i ∈ [ d ] . If there is no such j ∗ , then || B j || = 1 for all j ∈ [ d ] , and we arrive at an analogous condition for B j . Thus, without loss of generality wecan assume that A i = | a i (cid:105)(cid:104) a i | for all i ∈ [ d ] .The rest of this appendix is dedicated to showing that the other measurement must also be rank-1 projective. Letus analyse the inequality derived by Kittaneh and in order to do so, we first recall a few definitions from matrixanalysis. We denote by L ( H ) the algebra of linear operators on the Hilbert space H , and by || . || H the Hilbert spacenorm. The numerical range of an operator O is W ( O ) := {(cid:104) x | Ox (cid:105) | || x || H = 1 } , while the numerical radius is w ( O ) := sup || x || H =1 |(cid:104) x | Ox (cid:105)| . By construction every complex number c ∈ W ( O ) satisfies | c | ≤ w ( O ) and we alwayshave w ( O ) ≤ || O || [78].In Theorem 2, the inequality comes from the triangle inequality and to investigate when this holds as an equalitywe use a result by Barraa and Boumazgour [79]. Theorem 3.
Let
S, T ∈ L ( H ) be non-zero. Then the equation || S + T || = || S || + || T || holds if and only if || S || || T || ∈ W ( S † T ) . For a finite-dimensional Hilbert space the numerical range is always closed [78], thus in our case the closure in thetheorem is redundant. It is immediate to see that a necessary condition for the operators S and T to saturate thetriangle inequality is that || S || || T || ≤ w ( S † T ) . On the other hand, from the submultiplicativity of the operator norm,we know that w ( S † T ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) S † T (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) S † (cid:12)(cid:12)(cid:12)(cid:12) || T || = || S || || T || , and hence this condition is equivalent to w ( S † T ) = || S || || T || .We will also use the following bound on the numerical radius, obtained by Kittaneh [80]. Theorem 4. If O ∈ L ( H ) , then (cid:0) w ( O ) (cid:1) ≤ (cid:12)(cid:12)(cid:12)(cid:12) O † O + OO † (cid:12)(cid:12)(cid:12)(cid:12) . (A8)We are now ready to derive a necessary condition to saturate Kittaneh’s inequality in Theorem 2. Lemma 5.
Let
A, B ≥ be operators on a Hilbert space. Then, the equality || A + B || = max {|| A || , || B ||} + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ A √ B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) holds only if || A || = || B || .Proof. Let us denote the block-operators appearing in the proof of Theorem 2 by: S = (cid:18) A B (cid:19) = S † , T = (cid:18) √ A √ B √ B √ A (cid:19) = T † . (A9)Then, following from Theorem 3 and the discussion below it, a necessary condition for A, B ≥ to saturate Kittaneh’sinequality is that w ( ST ) = || S || || T || = max {|| A || , || B ||} (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ A √ B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) .Applying Theorem 4 to ST , we get that ( ST ) † ST = (cid:18) √ AB √ A √ BA √ B (cid:19) ,ST ( ST ) † = (cid:18) A BA B AB (cid:19) , (A10)and hence (cid:0) w ( ST ) (cid:1) ≤
12 max (cid:110) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ AB √ A + A BA (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ BA √ B + B AB (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:111) ≤
12 max (cid:110) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ AB √ A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A BA (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ BA √ B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B AB (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:111) = 12 max (cid:110) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ AB (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A √ B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A √ B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ AB (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:111) = 12 (cid:16) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A √ B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ AB (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:17) ≤ (cid:16) || A || + || B || (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ A √ B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max (cid:8) || A || , || B || (cid:9) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ A √ B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A11)Here, in the second line, we used the triangle inequality, in the third line the identity || O || = (cid:12)(cid:12)(cid:12)(cid:12) O † O (cid:12)(cid:12)(cid:12)(cid:12) and inthe fourth line submultiplicativity. The last inequality is trivial, and is only saturated if || A || = || B || . Therefore, || A + B || = max {|| A || , || B ||} + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ A √ B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) only if || A || = || B || .This lemma shows that saturating the upper bound on the ASP implies that || B j || = || A i || = 1 for all i, j ∈ [ d ] .It was also necessary that tr B j = 1 , and therefore (similarly to the A i ), B j = | b j (cid:105)(cid:104) b j | for all j ∈ [ d ] , and bothmeasurements must be rank-1 projective. From here, it follows immediately from the condition tr( A i B j ) = d , thatthe bases defining the measurements must be mutually unbiased. Appendix B: Robust self-test
While it is clear what it means for two measurements to be exactly mutually unbiased, there are multiple waysof turning this definition into an approximate statement (particularly if we allow for non-projective measurements).For our purposes it is natural to split the definition of MUBs into two stand-alone conditions and consider themseparately.The first condition, which is usually implicit in the definition of MUBs, is that both measurements are projectiveand that the measurement operators are rank-1. Let { A i } i be a d -outcome measurement on a d -dimensional systemand let us consider the sum of the norms, N ( A ) := (cid:80) i || A i || . This is a suitable quantity, because N ( A ) = (cid:88) i || A i || ≤ (cid:88) i tr A i = d and since || A i || ≤ , the maximum is achieved iff every measurement operator is a rank-1 projector. Therefore, thedifference between (cid:80) i || A i || and the maximal value d tells us how much { A i } i deviates from being rank-1 projective.The second condition, often referred to as the MUB condition, requires that the overlap between every pair ofmeasurement operators is the same. The question here is how to generalise the overlap to non-projective measurements.The quantity (cid:112) tr( A i B j ) discussed in the main text is a valid generalisation of the overlap in the sense that it reducesto the overlap for rank-1 projective measurements. However, the argument given below naturally leads to a differentquantity, namely (cid:12)(cid:12)(cid:12)(cid:12) √ A i (cid:112) B j (cid:12)(cid:12)(cid:12)(cid:12) . Note that this is a commonly used definition of the overlap, e.g. in the context ofuncertainty relations.The main purpose of this appendix is to derive a lower bound on N ( A ) as a function of the observed performance.However, in order to do that, we must first derive explicit bounds on the range of (cid:12)(cid:12)(cid:12)(cid:12) √ A i (cid:112) B j (cid:12)(cid:12)(cid:12)(cid:12) .In our argument we use the following technical lemma. Lemma 6.
The function h ( x, y ) := x + y − αxy − (cid:112) x + y for α := 2 − √ satisfies h ( x, y ) ≥ for x, y ∈ [0 , .Proof. If we express x and y in terms of the polar coordinates x = r cos( θ − π/ ,y = r sin( θ − π/ , the function becomes h ( r, θ ) = r (cid:2) cos( θ − π/
4) + sin( θ − π/ − (cid:3) − αr (cid:2) θ − π/ (cid:3) = r (cid:0) √ θ − (cid:1) + αr θ. To cover the square x, y ∈ [0 , we prove the statement for r ∈ [0 , √ and θ ∈ [ π/ , π/ . For fixed θ the function h ( r, θ ) is a quadratic function of r and the coefficient of the quadratic term is non-positive. This means that in orderto determine the minimum value, it suffices to consider the extreme points, i.e. r = 0 and r = √ . Since h (0 , θ ) = 0 ,we only have to look at the latter. We have h ( √ , θ ) = 2 sin θ − √ α cos 2 θ = − α sin θ + 2 sin θ + 2 − √ α (1 − sin θ ) (cid:18) sin θ − √ (cid:19) and it is easy to see that for θ ∈ [ π/ , π/ each term is non-negative.Moreover, we use the following operator norm inequality derived by Kittaneh [81]. Theorem 7.
For positive semidefinite operators A and B acting on a finite-dimensional Hilbert space we have || A + B || ≤ (cid:32) || A || + || B || + (cid:114) ( || A || − || B || ) + 4 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ A √ B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:33) . (B1)In our argument A and B will be particular measurement operators from the two measurements. We define the generalised overlap between A i and B j as s ij := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:112) A i (cid:112) B j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∈ [0 , . Another relevant quantity of a pair of measurement operators is the norm deficiency defined as n ij := 1 − ( || A i || + || B j || ) / ∈ [0 , . It is easy to see that if n ij = 0 for all i, j , we have (cid:88) i || A i || = (cid:88) j || B j || = d, i.e. both measurements are rank-1 projective. Our goal now is to relate the right-hand side of Eq. (B1) to s ij and n ij .First, note that || A i || − || B j || = 2 || A i || − ( || A i || + || B j || ) ≤ − − n ij ) = 2 n ij and similarly || B j || − || A i || ≤ n ij . These two inequalities imply that (cid:0) || A i || − || B j || (cid:1) ≤ n ij and plugging this back into Eq. (B1) gives || A i + B j || ≤ − n ij + (cid:113) n ij + s ij . Applying the inequality derived in Lemma 6 to s ij and n ij gives || A i + B j || ≤ s ij − αs ij n ij , where α = 2 − √ . Applying this upper bound to Eq. (A1) immediately yields ¯ p ≤ d (cid:88) ij (cid:0) s ij − αs ij n ij (cid:1) = 12 + 12 d (cid:88) ij s ij − α d (cid:88) ij s ij n ij . (B2)Let us first bound the range of s ij , i.e. find explicit functions of ¯ p denoted by s min and s max such that s ij ∈ [ s min , s max ] for all i, j . To do this we drop the last term in Eq. (B2) to obtain ¯ p ≤
12 + 12 d (cid:88) ij s ij . To bound the sum of s ij we bound the operator norm by the Frobenius norm: s ij = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:112) A i (cid:112) B j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:112) A i (cid:112) B j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F = (cid:113) tr( A i B j ) = (cid:112) t ij and finally use the normalisation condition (cid:80) ij t ij = d . Let us now separate one term from the rest of the sum. Forsimplicity we choose the first term, i.e. s , but by symmetry the same argument applies to every s ij . We obtain ¯ p ≤
12 + 12 d (cid:16) s + (cid:88) ij (cid:54) =11 s ij (cid:17) ≤
12 + 12 d (cid:16) s + (cid:88) ij (cid:54) =11 (cid:112) t ij (cid:17) . (B3)Since the remaining sum contains d − terms, concavity of the square root implies that (cid:88) ij (cid:54) =11 d − (cid:112) t ij ≤ (cid:115) (cid:80) ij (cid:54) =11 t ij d − (cid:114) d − t d − ≤ (cid:115) d − s d − , where in the last step we used the fact that s ≤ √ t . Plugging this bound into Eq. (B3) gives ¯ p ≤
12 + 12 d (cid:16) s + (cid:113) ( d − d − s ) (cid:17) =: f ( s ) . Computing the derivative of f shows that f is increasing for s < / √ d and decreasing for s > / √ d . Themaximum achieved for s = 1 / √ d corresponds to the optimal ASP. This implies that the lowest and highest valuesof s compatible with the observed ¯ p can be determined by computing the two solutions of the equality ¯ p = 12 + 12 d (cid:16) s + (cid:113) ( d − d − s ) (cid:17) . s ∈ [ s min , s max ] , where s min := 2¯ p − − d (cid:112) d ( d − − d (2¯ p − ] , (B4) s max := 2¯ p − d (cid:112) d ( d − − d (2¯ p − ] . (B5)The optimal performance, i.e. ¯ p = + √ d , implies that s min = s max = √ d . Moreover, since both functions arecontinuous in ¯ p , for sufficiently good performance we obtain bounds stronger than the trivial s ≥ and s ≤ .This concludes the first part of the argument, i.e. providing explicit bounds on the range of the generalised overlaps.For the second part of the argument, in which we show that the measurements are close to being rank-1 projective,we need all the overlaps to be bounded away from , i.e. s min > . According to Eq. (B4) this is guaranteed as longas ¯ p > ¯ p for ¯ p := 12 + 12 d (cid:112) ( d − d. Using the concavity result while keeping the negative term in Eq. (B2) leads to ¯ p ≤
12 + 12 d (cid:16) s + (cid:113) ( d − d − s ) (cid:17) − α d (cid:88) ij s ij n ij . Without loss of generality we can assume that s is the smallest overlap and then ¯ p ≤
12 + 12 d (cid:16) s + (cid:113) ( d − d − s ) (cid:17) − αs d (cid:88) ij n ij , which is equivalent to (cid:88) ij n ij ≤ αs (cid:18) s + (cid:113) ( d − d − s ) − d (2¯ p − (cid:19) . (B6)To analyse the right-hand side, we define g ( x ) := 1 + (cid:115) ( d − (cid:18) dx − (cid:19) − d (2¯ p − x , and now our goal is to maximise g ( x ) over x ∈ [0 , / √ d ] , as s min ≤ / √ d . Recall that we work under the assumptionthat ¯ p > ¯ p and therefore p − > . We can analytically compute the derivative dg/dx and set it to to concludethat the only stationary point corresponds to x ∗ := (cid:112) d (2¯ p − − ( d − d (2¯ p −
1) = (cid:115) d − d − d (2¯ p − . Evaluating the second derivative d g/dx at x ∗ tells us that this is a maximum and since this is the only stationarypoint, it must be the unique maximiser in the interval [0 , / √ d ] . Therefore, in Eq. (B6) we can set s = x ∗ to obtain (cid:88) ij n ij ≤ α (cid:16) − (cid:112) d (2¯ p − − ( d − (cid:17) . Finally, we can use this bound to obtain lower bounds on the sums of the norms (cid:80) i || A i || and (cid:80) j || B j || for the individualmeasurements. Since (cid:88) ij n ij = d − d (cid:16) (cid:88) i || A i || + (cid:88) j || B j || (cid:17) , we can use the trivial bound N ( B ) = (cid:80) j || B j || ≤ d to obtain N ( A ) = (cid:88) i || A i || ≥ d − d (cid:88) ij n ij ≥ d − αd (cid:16) − (cid:112) d (2¯ p − − ( d − (cid:17) . (B7)Clearly, the same lower bound holds for N ( B ) .1 Appendix C: Incompatibility robustness
In this appendix we derive an analytic upper bound on the incompatibility robustness as a function of the observedASP. We start with a bound derived recently in [70]: η ∗ ≤ d max ij || A i + B j || − (cid:80) i (tr A i ) − (cid:80) j (tr B j ) d (cid:80) i tr A i + d (cid:80) j tr B j − (cid:80) i (tr A i ) − (cid:80) j (tr B j ) . (C1)The aim is to bound all the terms appearing in this formula by quantities which we have already bounded in Ap-pendix B.Let us start with the numerator. The first term is easy to bound since || A i + B j || ≤ s ij , and max ij s ij ≤ s max given in Eq. (B5).To bound the second term we use the fact that for positive semidefinite operators (tr A ) ≥ tr A and then boundthe Frobenius norm by the operator norm: (tr A i ) ≥ tr A i = || A i || F ≥ || A i || . To bound the sum of the squares (cid:80) i || A i || we use a standard inequality for vector p -norms which for d -dimensionalvectors reads || x || ≥ √ d || x || . Applying this to the real vector whose components are given by x i = || A i || yields (cid:88) i || A i || ≥ d (cid:16) (cid:88) i || A i || (cid:17) . Putting the two inequalities together gives (cid:88) i (tr A i ) ≥ d (cid:16) (cid:88) i || A i || (cid:17) , which can be bounded using Eq. (B7).The first term in the denominator we have already bounded: from the previous argument we see that (cid:88) i tr A i ≥ d (cid:16) (cid:88) i || A i || (cid:17) . Bounding the last term turns out to be slightly more involved, so we state it as a separate lemma.
Lemma 8.
Let { A i } i be a d -outcome measurement acting on C d . If (cid:88) i || A i || ≥ q, then (cid:88) i (tr A i ) ≤ d + ( d − q )( d − q + 1) . Proof.
Before proceeding to the technical details, let us briefly explain the idea behind the proof. Suppose we aregiven a partition of the d measurement outcomes into two disjoint sets. Moreover, we are promised that the traceof the measurement operators corresponding to the outcomes in the first (second) set belongs to the interval [0 , ( [1 , d ] ). It turns out that an upper bound on the desired quantity can be derived in terms of simple properties of thispartition. Maximising this bound over all valid partitions leads to the main result of the lemma.Formally, we are given two sets X and Y such that X ∪ Y = [ d ] and X ∩ Y = ∅ . Moreover, we have i ∈ X = ⇒ tr A i ∈ [0 , ,i ∈ Y = ⇒ tr A i ∈ [1 , d ] . n := | X | , γ := (cid:80) i ∈ X tr A i and clearly n − γ ≥ . (C2)Moreover, the assumption of the lemma implies q ≤ (cid:88) i || A i || = (cid:88) i ∈ X || A i || + (cid:88) i ∈ Y || A i || ≤ (cid:88) i ∈ X tr A i + | Y | = γ + d − n and therefore n − γ ≤ d − q. (C3)For the rest of the argument let us think of n and γ as some fixed values. Once we derive the final upper bound interms of these two variables, we will maximise it over the allowed pairs of n and γ .For i ∈ X we have (tr A i ) ≤ tr A i and therefore (cid:88) i ∈ X (tr A i ) ≤ (cid:88) i ∈ X tr A i = γ. To bound the second term we must explicitly determine the allowed combinations of { tr A i } i ∈ Y . Since { tr A i } i ∈ Y ∈ [1 , d ] | Y | and (cid:88) i ∈ Y tr A i = d − γ, the valid choices of { tr A i } i ∈ Y form a polytope. It is easy to see that all the vertices of this polytope correspond tosetting | Y | − values to and the last value to [ d − γ − ( | Y | − . Since (cid:80) i ∈ Y (tr A i ) is a convex function of thetraces, the maximal value is achieved at a vertex and therefore (cid:88) i ∈ Y (tr A i ) ≤ ( | Y | −
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