Experimentally efficient methods for estimating the performance of quantum measurements
EExperimentally efficient methods for estimating the performance of quantummeasurements
Easwar Magesan and Paola Cappellaro
Research Laboratory of Electronics and Nuclear Science and Engineering Department, MIT, Cambridge MA, 02139, U.S.A.
Efficient methods for characterizing the performance of quantum measurements are important inthe experimental quantum sciences. Ideally, one requires both a physically relevant distinguishabilitymeasure between measurement operations and a well-defined experimental procedure for estimatingthe distinguishability measure. Here, we propose the average measurement fidelity and error betweenquantum measurements as distinguishability measures. We present protocols for obtaining boundson these quantities that are both estimable using experimentally accessible quantities and scalablein the size of the quantum system. We explain why the bounds should be valid in large generalityand illustrate the method via numerical examples.
I. INTRODUCTION
Measurement plays a fundamental role in the quantumsciences as it allows one to evolve and extract informa-tion from a quantum system. From a practical perspec-tive, measurements are imperative for the successful im-plementation of a wide variety of protocols in quantuminformation processing (QIP) and communication [1], aswell as high-precision metrology [2]. If large-scale QIPis achieved experimentally, certain tasks that are conjec-tured to be classically inefficient will become realizableusing a scalable number of resources [3, 4].Various models of computation have been proposed forQIP, such as the standard circuit model [5], measurementbased quantum computation (MBQC) [6], and topologi-cal quantum computation [7]. Each of these models re-lies greatly on the ability to perform accurate measure-ments. A large number of physical systems have beenproposed as candidates for implementing QIP. A short,and certainly non-exhaustive, list of systems includes su-perconducting circuits [8, 9], Nitrogen-vacancy (NV) cen-ters [10], trapped ions [11], NMR [12], quantum dots [13],and optical implementations [14]. Measurement schemesvary greatly across these systems and so it is importantto have straightforward protocols for comparing measure-ments that are both independent of the particular typeof implementation and scalable in the size of the system.Providing such protocols is the main goal of this paper.There has been a significant amount of research to-wards completely characterizing the error on the opera-tions used to process and measure quantum informationvia quantum process [15–17] and measurement tomog-raphy [18, 19]. In principle, these tomographic meth-ods can be used to characterize the error affecting anyquantum operation or measurement realized in an exper-imental setting. Unfortunately, there are various signif-icant drawbacks to complete process and measurementtomography. Process tomography requires an exponen-tial number of resources in the number n of quantumbits (qubits) that comprise the system (the number ofparameters required to just describe the process scalesas 16 n ), is not robust against state-preparation and mea-surement errors [20], and requires intensive classical post- processing of measurement data [21]. Not surprisingly,since measurement tomography is realized by essentiallyinverting process tomography [18, 19], it suffers from sim-ilar drawbacks to quantum process tomography. In par-ticular, complete measurement tomography requires theability to prepare a complicated set of pure input stateswith extremely high precision (possibly by performingcomplex unitary gates), scales badly in the size of thesystem, and can require lengthy post-processing of thetomographic data.In many cases, one may only be interested in a sub-set of parameters characterizing the noise, or determin-ing the strength of the noise process rather than the en-tire process itself. For instance, in fault-tolerance [22–25], as long as the strength of the noise affecting state-preparation, gates, and measurements is bounded bysome value, large-scale computation is possible. As a re-sult, various methods for partially characterizing the er-ror of quantum gates [26–35] have recently been proposedwhich attempt to circumvent many of the problems asso-ciated with full process tomography. The error is definedvia the “quantum average gate fidelity”, which is derivedfrom the quantum channel fidelity between two quantumprocesses. Since quantum gates are unitary operations,the average gate fidelity, and higher order moments, takea simple form and can be calculated analytically [36–38].Similarly, partial estimation of measurement errors bymore efficient methods than full measurement tomogra-phy would be extremely valuable, but have yet to beconsidered in significant detail. There are a variety ofquestions to consider in this endeavor, such as how todeal with abundance of different measurement schemesacross the various implementations, and the ambiguityregarding what the single parameter characterizing theerror should be (see Ref. [39] for two proposals of suchdistance measures). Since measurements are non-unitaryprocesses, the quantum channel fidelity between the mea-surement and noise process is a very complex quantityand cannot be calculated analytically as is the case forquantum gates.One of the main goals of this paper is to initiate re-search into scalable verification and characterization ofmeasurement devices. Here, we discuss methods for a r X i v : . [ qu a n t - ph ] A ug characterizing the error of a quantum measurement andpresent protocols for estimating the performance of ameasurement with respect to these error measures. Theprotocols are scalable in the number of qubits comprisingthe system. In addition, the protocols are very generaland should be straightforward to implement in most sys-tems. They require the ability to prepare a random statefrom the basis that constitutes the ideal measurement,and perform the noisy measurement (perhaps consecu-tively). As quantum systems continue to scale to largersizes, and full tomography of the measurement becomesinfeasible, the protocols provided here can be used asfast and straightforward methods to verify that the per-formance of a measuring device achieves some desiredthreshold.Specifically, we propose the average measurement fi-delity and error rate of a quantum measurement as nat-ural measures of the error, and provide scalable, exper-imentally implementable protocols for estimating thesequantities. The output of the protocol is a lower (upper)bound on the average fidelity (error rate) of the mea-surement. Computing these bounds from the measure-ment data is straightforward, and generic estimates ofhow many trials need to be performed can be obtainedusing statistical methods. We provide a direct compar-ison of the time-complexity between our protocol andthat of completely reconstructing the noisy measurementPOVM elements via tomography. Our method scales in-dependently of d (where d is the dimension of the system,e.g. d = 2 n for an n -qubit system) while a complete re-construction scales as O ( d ). Thus, while our methodgives much less information than full tomography, it isscalable, straightforward to implement, and involves lesspost-processing of measurement data. Ideally, the pro-tocols we provide here will be useful as a starting pointfor further analysis and research into more efficient to-mographic methods of quantum measurements.We focus mainly on the case where we attempt toimplement a finite-dimensional projective measurement(also known as a projective-valued measure (PVM)), butinstead implement a general quantum measurement inthe form of a positive operator-valued measure (POVM).In particular, we analyze the following two scenarios:1. The ideal measurement is a rank-1 PVM and weare only concerned with characterizing measure-ment probabilities.2. The ideal measurement is a rank-1 PVM and weare concerned with simultaneously characterizingmeasurement probabilities and output states.Rank-1 PVM’s are significant in QIP since each com-putational model mentioned above can achieve universalquantum computation using rank-1 PVM’s. We leaveextending our results to arbitrary-rank PVM’s as a di-rection of future research, however we anticipate the re-sults presented here can be carried over to these mea-surements. Moreover, while the most general measure-ment model is a POVM, any POVM can be realized as a PVM on an extended Hilbert space by attaching an-cilla systems to the Hilbert space of interest [40]. Thus,characterizing the quality of PVM’s can potentially pro-vide direct information about one’s ability to implementPOVM’s.The structure of the presentation is as follows. First,in Sec. (II) we set notation, discuss the framework ofPVM’s and POVM’s, and define the noisy measurementmodels. Next, in Sec.(III), we define the quantum averagemeasurement fidelity and quantum average measurementerror , which are the operationally relevant metrics weuse to compare measurement operations. In Sec. (IV) wepresent the main result for the first of the two scenar-ios listed above. We provide an expression for a lower(upper) bound on the fidelity (error rate), and presentan experimental protocol for obtaining this bound. InSec. (V) we give a derivation of the bound and showit satisfies certain necessary conditions to be useful inpractice. Sec. (VI) provides a discussion of the generalvalidity of the bound and numerical examples for thecase of a single-qubit system are analyzed. Our derivedbounds are shown to be valid for every example ana-lyzed, confirming the discussion regarding the validity ofthe method. We discuss the extension of the method torank-1 PVM’s with output states in Sec. (VII) and pro-vide the experimental protocol for this case in Sec. (VIII).The resource analysis for the protocols are provided inSec. (IX) and concluding remarks are made in Sec. (X).The reader interested in the bounds and protocols with-out most of the technical details is referred to the textsurrounding Eq. (5.1) for rank-1 PVM’s and the text sur-rounding Eq. (7.14) for rank-1 PVM’s with output states.By definition, the average fidelity and error rate ofa quantum measurement are equivalent (see Sec. III).Hence, the majority of the presentation is in terms of theaverage fidelity, however it is important to keep in mindthat analogous results hold for the error rate. II. MEASUREMENT OF QUANTUM SYSTEMSAND THE ERROR MODEL
Denote the quantum system by S and suppose it isrepresented by a Hilbert space H of dimension d < ∞ ( d = 2 n for an n -qubit system). The most general typeof measurement one can make is a “positive operator-valued measure” (POVM). A POVM consists of a set { E k } of linear operators on H that satisfy E k ≥ , (cid:88) k E k = . (2.1)If the input state to the measurement is σ , the probabilityof obtaining outcome “ k ” is given by r k = tr ( E k σ ) . (2.2)An important subset of POVM measurements are“projection-valued measures” (PVM’s), which corre-spond to the case of each E k being equal to a projectionoperator Π k . Hence, in addition to the conditions listedin Eq. (2.1), Π k = Π k for each k .PVM’s are a very general class of quantum measure-ments and, as previously mentioned, are sufficient forperforming universal quantum computation in variouscomputational models [5, 6]. For instance, computationalbasis measurements allow for universality in the standardcircuit model and single-qubit projective measurementsprovide universality in MBQC. Other important exam-ples of PVM’s in quantum information theory are paritymeasurements, which are used extensively in quantumerror-correction and fault tolerance [41–44]. It is impor-tant to note that, by Naimark’s theorem [40], any POVMcan be implemented via a PVM on a larger Hilbert space.Hence, methods for determining error rates of PVM’s canalso give direct information regarding the error associatedwith performing general POVM measurements.PVM’s are in a 1-1 correspondence with observables(Hermitian operators) O , where O is non-degenerate ifand only if the PVM consists of rank-1 projective ele-ments . We write the observable O as O = b (cid:88) k =1 λ k Π k , (2.3)where the eigenvalues λ k correspond to the measurementvalues and Π k := (cid:80) d k j =1 | ψ kj (cid:105)(cid:104) ψ kj | is the rank- d k projectoronto the eigenspace spanned by the eigenvectors | ψ kj (cid:105) .Hence, (cid:80) bk =1 d k = d and the PVM is rank-1 if and onlyif d k = 1 for every k . If M represents the ideal PVMwithout post-selection then M has the following actionon each input state σ , M ( σ ) = b (cid:88) k =1 Π k σ Π k = b (cid:88) k =1 p k (cid:20) Π k σ Π k tr (Π k σ ) (cid:21) , (2.4)where we have defined p k = p k ( σ ) = tr (Π k σ ) . (2.5)In the case of analyzing only measurement probabil-ities, the PVM is a mapping from the set of quantumstates to the probability vectors ( p , ..., p b ), M ( σ ) = ( p ( σ ) , ..., p b ( σ )) = (tr (Π σ ) , ..., tr (Π b σ )) . (2.6)This case is especially interesting since in many scenar-ios one is mainly interested in the output value of themeasurement rather than the output state itself. Forinstance, the output of a quantum algorithm is usuallythe value of a measurement in the computational basis.Hence, as long as the measurement values are obtainedwith correct probabilities, the measurement is deemedsuccessful. In addition, in many current implementations of mea-surements, the action of performing the measurement op-eration to obtain the output value destroys or drasticallyalters the state of the system. For instance in photo-detection, which forms the basis of various measurementschemes in optical, atomic, and superconducting systems,the measurement consists of recording the occurrence ofa photon. If the state of the system is encoded into adegree of freedom (mode) of the photon, such as polar-ization or frequency, detection of the photon can recordthe output of a measurement of this degree of freedom. Incommon photodetectors, such as avalanche photodiodes,the photon is lost in the process of creating a currentthrough the photoelectric effect. Hence, the system en-coding the information to be measured is destroyed, butthe measurement output can be accessed.An example where the system is not destroyed yetthe state is not preserved from the measurement is theNitrogen-Vacancy (NV) center in diamond [10]. In thiscase, the processes of measurement and ground-state po-larization are identical. Hence, while one can obtainthe relevant measurement outcomes and statistics, theoutput state of the measurement is always the groundstate. There are situations where preserving the mea-surement output state is useful. For instance, in anyparadigm where quantum information is evolved by per-forming measurements, such as in measurement-basedquantum computation [6], one requires the state of thesystem to be well-preserved under measurements. Inmeasurement-based quantum computing, quantum infor-mation is evolved by performing single-qubit measure-ments on a highly entangled initial state. Any degrada-tion of the state or system caused by the measurementprocess can be highly detrimental to obtaining a highfidelity output.In the general case of Eq. (2.4), the noisy measurement E is modeled by E ( σ ) = b (cid:88) k =1 r k ( σ ) ρ k ( σ ) , (2.7)where we allow both the noisy measurement probabilities r k and output states ρ k to be functions of σ . By Eq. (2.4), r k ( σ ) and ρ k ( σ ) are ideally given by p k ( σ ) = tr(Π k σ )and Π k σ Π k tr(Π k σ ) respectively. If we analyze only measurementprobabilities, we have E ( σ ) = ( r ( σ ) , ..., r b ( σ )) = (tr ( E σ ) , ..., tr ( E b σ )) , (2.8)where E is allowed to be of a completely general formby assuming it is modeled by a POVM { E k } bk =1 . Thefirst scenario we are interested in is to compare the prob-ability distributions in Eq. (2.6) and (2.8). We analyzethe more general case of measurement probabilities andoutput states (ie. comparing Eq.’s (2.4) and (2.7)) inSec. (VII). We first discuss the figure of merits we willuse to compare the ideal measurement process M andnoisy measurement process E . III. QUANTUM AVERAGE MEASUREMENTFIDELITY AND ERROR
A completely question is, how should we compare theideal and actual measurements M and E ? A set of crite-ria that a distance measure, ∆, for comparing ideal andreal quantum processes should satisfy has been given pre-viously [45]. Currently, no known ∆ satisfies all of thesecriteria simultaneously. Thus, one must settle for ∆ tosatisfy a subset of these criteria, in addition to other cri-teria that may be useful for the particular task at hand.As mentioned previously, the average gate fidelity F E , U is a useful method for comparing an intended unitaryoperation U and actual quantum process E . There arevarious reasons for the utility of the average gate fidelity,for instance, it satisfies the following properties:1. There is a straightforward method for evaluating F E , U (given a description of U and E ),2. F E , U has a well-motivated physical interpretation,3. All states are taken into account in an unbiasedmanner when calculating F E , U ,4. F E , U is experimentally accessible via efficient pro-tocols.An important drawback of the average gate fidelity isthat it is not a metric. We would like similar propertiesto hold for our method of comparing ideal and real mea-surements. Let us briefly outline how the average gatefidelity is derived from more general quantities, whichwill provide intuition for how to define our method forcomparing measurements.The average gate fidelity is derived from the state-dependent quantum channel fidelity, which is a standardmethod for comparing quantum operations. If E and E are quantum operations and σ is a quantum state, thequantum channel fidelity between E and E , denoted F E , E , is given by the standard state fidelity between E ( σ ) and E ( σ ), F E , E ( σ ) = F ( E ( σ ) , E ( σ ))= (cid:18) tr (cid:113)(cid:112) E ( σ ) E ( σ ) (cid:112) E ( σ ) (cid:19) . (3.1)When one of the operations in Eq. (3.1) is unitary (say E = U ), the channel fidelity is called the quantum gatefidelity, and when σ is pure ( σ = | ψ (cid:105)(cid:104) ψ | ), the gate fidelitytakes the extremely simple form F E , U ( | ψ (cid:105)(cid:104) ψ | ) = tr ( E ( | ψ (cid:105)(cid:104) ψ | ) U ( | ψ (cid:105)(cid:104) ψ | ))= (cid:104) ψ | Λ( | ψ (cid:105)(cid:104) ψ | ) | ψ (cid:105) (3.2)where Λ = U † ◦ E . (3.3) The average quantum gate fidelity , denoted F E , U , is ob-tained by integrating over all pure input states. Theintegral is taken over the unitarily invariant Haar mea-sure (also known as the Fubini-Study measure) on the setof pure states [46]. In this paper we denote the Fubini-Study measure by µ . This gives F E , U = (cid:90) tr ( E ( | ψ (cid:105)(cid:104) ψ | ) U ( | ψ (cid:105)(cid:104) ψ | )) dψ = (cid:90) (cid:104) ψ | Λ( | ψ (cid:105)(cid:104) ψ | ) | ψ (cid:105) dψ = (cid:80) j tr( A j )tr( A j ) + dd + d (3.4)where { A j } is any set of Kraus operators for Λ [36]. Thus,the average gate fidelity reduces to an extremely simpleform because one of the operations is unitary.Following this intuition, from Eq.’s (2.4), (2.7),and (3.1), we have that for the ideal ( M ) and noisy ( E )measurements F ( E ( σ ) , M ( σ ))= tr (cid:118)(cid:117)(cid:117)(cid:117)(cid:116)(cid:118)(cid:117)(cid:117)(cid:116) b (cid:88) k =1 r k ρ k (cid:32) b (cid:88) k =1 p k (cid:20) Π k σ Π k tr (Π k σ ) (cid:21)(cid:33) (cid:118)(cid:117)(cid:117)(cid:116) b (cid:88) k =1 r k ρ k , (3.5)where the state-dependence in p k , r k , ρ k is omitted fornotational convenience. The operational significance ofthis quantity comes from the fact that it is just the stan-dard fidelity between the two quantum states E ( σ ) and M ( σ ). More precisely, Eq. (3.5) is related to the maxi-mum distinguishability between probability distributionsone could obtain using any POVM measurement on thestates E ( σ ) and M ( σ ) [47].Unlike the unitary case in Eq. (3.2), such an expres-sion does not reduce to a simple form in general. How-ever, ideally, the figure of merit we use to distinguish M and E will contain evenly weighted information from all possible input states and also have direct operationalsignificance. As a result, we define the quantum aver-age measurement fidelity , denoted F E , M , to be the Haarintegral of Eq. (3.5) over pure input states F E , M = (cid:90) F ( E ( | ψ (cid:105) ) , M ( (cid:104) ψ | )) dψ. (3.6)We also define r E , M = 1 − F E , M (3.7)to be the quantum average measurement error . In thecase of only analyzing measurement probabilities, that is M and E are given by Eq.’s (2.6) and (2.8) respectively,the states associated to p k and r k can be taken to be Π k ,which gives F E , M = (cid:90) (cid:32) b (cid:88) k =1 √ p k r k (cid:33) dψ = (cid:90) (cid:32) b (cid:88) k =1 (cid:112) tr(Π k | ψ (cid:105)(cid:104) ψ | )tr( E k | ψ (cid:105)(cid:104) ψ | ) (cid:33) dψ, (3.8) r E , M = 1 − (cid:90) (cid:32) b (cid:88) k =1 (cid:112) tr(Π k | ψ (cid:105)(cid:104) ψ | )tr( E k | ψ (cid:105)(cid:104) ψ | ) (cid:33) dψ. (3.9)The direct relationship between r E , M and F E , M im-plies statements and bounds proven about one directlyapplies to the other. Thus, as mentioned in the intro-duction, we will phrase the majority of the discussion interms of F E , M . Since F E , M does not have a simple formlike the average gate fidelity in Eq. (3.4), our goal is toprovide efficient methods for estimating F E , M that canbe experimentally implemented in a simple manner.An important reason for using the average measure-ment fidelity is that it provides a state-independent dis-tance measure which can be connected to the diamondnorm distance [48] between quantum operations [49].Computing the diamond norm is an exponentially hardtask since one needs a complete description of the quan-tum operations. Thus, the diamond norm is neitherstraightforward to calculate nor experimentally acces-sible. However, it is commonly used in fault-tolerantanalyses of threshold error rates of physical operations.Thus, information about the diamond norm provided bythe average measurement error defined above can poten-tially provide information regarding the ability to per-form fault-tolerant computation. IV. EXPERIMENTAL PROTOCOL: RANK-1PVM’S
In this section, we look at obtaining a lower boundon F E , M for measurement probabilities of rank-1 PVM’s.Again, we note that upper bounds on r E , M are equivalentto lower bounds on F E , M and so, for both clarity andconsistency, we phrase the following discussion in termsof only F E , M .There are d PVM elements, { Π , ..., Π d } , each of whichis a rank-1 projection operator. From Eq. (2.8), the out-come of the measurement of a state | ψ (cid:105)(cid:104) ψ | can be asso-ciated to a member from { , ..., d } with frequency distri-bution( r , ..., r d ) = (tr( E | ψ (cid:105)(cid:104) ψ | ) , ..., tr( E d | ψ (cid:105)(cid:104) ψ | )) , (4.1)where the E j represent the noisy POVM elements. Our expression for the lower bound is given by lb = 1 + dX d , (4.2)where X = 1 d (cid:88) ( l,m ) ∈D √ u l u m , (4.3) u l = tr(Π l E l ) , (4.4)and D = { , ..., d − } × { , ..., d − } . (4.5)The u l measure the overlap between the l ’th ideal PVMelement and l ’th noisy POVM element. Thus, lb canintuitively be thought of as a parameter that measureshow well ideal PVM elements are preserved when inputto the noisy measurements (with the inclusion of dimen-sional factors).Goal: Obtain a lower bound, lb , for F E , M as definedin Eq. (3.9).The experimental protocol to obtain lb is as follows:Protocol:Step 1: Choose a pair of indices ( l, m ) uniformly at ran-dom from D .Step 2: Independently, for both j = l and j = m ,a): Prepare the quantum state | ψ j (cid:105) , perform the noisymeasurement E on Π j = | ψ j (cid:105)(cid:104) ψ j | , and record whetheroutcome “ j ” is obtained,b): Repeat a) many times and denote the frequencyof obtaining “ j ” by ˆ u j , that is, ˆ u j is an estimator of u j = tr(Π j E j ),(see Sec. IX A for a discussion of the number of repeti-tions required to estimate u j to a desired accuracy andconfidence).Step 3: Repeat Steps 1 and 2 K times, where K is dic-tated by the desired accuracy and confidence in estimat-ing lb (see Sec. IX B for a discussion of the size of K ).Step 4: Compute an estimator ˆ lb for the lower bound lb of F E , M defined in Eq. (5.1) via the formulaˆ lb = 1 + d ˆ X d , (4.6)whereˆ X = 1 K (cid:88) ( k ,k ) (cid:112) ˆ u k ˆ u k (4.7)is an estimator of X defined in Eq. (4.7) and the { (1 , ) , ..., ( K , K ) } are the K trials dictated by Step3. This concludes the protocol.There are various important points about the protocolthat should be emphasized. First, the number of trialsrequired in Steps 2b) and 3 are independent of d . Thus,the time-complexity of the entire protocol is independentof d , and depends only on the desired accuracy and con-fidence of the estimate ˆ lb of lb (see Sec. IX). Second, lb can be estimated from the above protocol using only:1. Applications of the noisy measurement and2. The ability to prepare the d pure input states | ψ j (cid:105) .Lastly, it is straightforward to show the following twoproperties of lb hold: (see Sec. V D)1. In the limit of F E , M ↑ lb ↑ . (4.8)2. lb scales well in d , that is, if each u k = tr(Π k E k ) ison the order of 1 − δ then, as d → ∞ , lb → − δ (and does not converge to 0 or some other smallconstant).These are clearly necessary conditions for lb to be a goodlower bound on F E , M . Property 1 implies that, looselyspeaking, in the small error limit lb can be taken as anestimate of F E , M . Property 2 implies that as d → ∞ ,the lower bound remains on the order of 1 − δ . Thisis important because if the lower bound generically con-verges to 0 (or some other constant) as d → ∞ , then thelower bound is ineffective at providing useful informationabout the noise. V. DERIVATION OF THE LOWER BOUND
We need to show lb := 1 + Xd d = d + (cid:80) dk =1 u k + (cid:80) l (cid:54) = m √ u l u m d ( d + 1) (5.1)is a lower bound for F E , M . First, we have from Eq. (3.8), F ( E ( σ ) , M ( σ )) = (cid:32)(cid:88) k √ r k p k (cid:33) , (5.2) where the state-dependence is implicit in the p k and r k .Taking the integral over all pure states gives (cid:90) F ( E ( | ψ (cid:105)(cid:104) ψ | ) , M ( | ψ (cid:105)(cid:104) ψ | )) dψ = d (cid:88) k =1 (cid:20)(cid:90) r k p k dψ (cid:21) + (cid:88) l (cid:54) = m (cid:20)(cid:90) √ r l r m p l p m dψ (cid:21) . (5.3)With these tools in hand, let us look at the sums inEq. (5.3) separately. A. d (cid:88) k =1 (cid:90) r k p k dψ We have (cid:90) r k p k dψ = (cid:90) tr( E k | ψ (cid:105)(cid:104) ψ | )tr(Π k | ψ (cid:105)(cid:104) ψ | ) dψ = (cid:90) tr ([ E k ⊗ Π k ] | ψ (cid:105)(cid:104) ψ | ⊗ | ψ (cid:105)(cid:104) ψ | ) dψ = tr (cid:18) [ E k ⊗ Π k ] (cid:90) | ψ (cid:105)(cid:104) ψ | ⊗ dψ (cid:19) . (5.4)To compute this integral we use Schur’s Lemma whichstates that for any positive integer t , [38, 50] (cid:90) | ψ (cid:105)(cid:104) ψ | ⊗ t dψ = Π sym ( t, d )tr [Π sym ( t, d )] , (5.5)where Π sym ( t, d ) is the projector onto the symmetric sub-space of the t -partite Hilbert space (whose t factor spaceseach have dimension d ). The symmetric subspace of a t -partite Hilbert space consists of states that are left un-changed under permutations of the factor spaces. Rea-soning for why Eq. (5.5) holds is as follows. First, bysymmetry of | ψ (cid:105)(cid:104) ψ | ⊗ t , (cid:82) | ψ (cid:105)(cid:104) ψ | ⊗ t dψ must only have sup-port on the symmetric subspace. In addition, since thisoperator is invariant under multiplication by any unitaryoperator of the form U ⊗ t , and has unit trace, Schur’sLemma implies that it must be equal to the normalizedprojector onto the symmetric subspace, which is just theright hand side of Eq. (5.5). This implies (cid:90) r k p k dψ = 2tr ([ E k ⊗ Π k ] Π sym (2 , d )) d ( d + 1) . (5.6)Now, one can show that Π sym ( t, d ) is equal to the nor-malized sum of the t ! elements in the group of permuta-tion operators on the t -partite Hilbert spaceΠ sym ( t, d ) = 1 t ! (cid:88) σ P σ . (5.7)This can be shown by first noting that the set of all per-mutations is a subgroup of the unitary group. Thus thesquare of the normalized sum is just equal to the normal-ized sum itself (cid:32) t ! (cid:88) σ P σ (cid:33) = 1 t ! (cid:88) σ P σ . (5.8)Hence, since t ! (cid:80) σ P σ commutes with any permutation P τ , it must be equal to the projection operator onto thesymmetric subspace. As a simple example, in the caseof t = 2, there are two permutation operators, , andthe SWAP operation which swaps the two factor spaces.Thus Π sym (2 , d ) = ⊗ + SWAP2 . (5.9)Eq.’s (5.6) and (5.9) imply (cid:90) r k p k dψ = tr ([ E k ⊗ Π k ] [ ⊗ + SWAP]) d ( d + 1)= tr( E k Π k ) + tr( E k )tr(Π k ) d ( d + 1)= tr( E k Π k ) + d tr( E k d ) d ( d + 1) , (5.10) and so d (cid:88) k =1 (cid:90) r k p k dψ = d (cid:88) k =1 tr( E k Π k ) + d tr( E k d ) d ( d + 1)= (cid:80) dk =1 u k + dd ( d + 1) , (5.11)since (cid:80) dk =1 E k = . This gives the first two terms in thenumerator in Eq. (5.1). B. (cid:88) l (cid:54) = m (cid:90) √ r l r m p l p m dψ We have (cid:90) √ r l r m p l p m dψ = (cid:90) (cid:112) tr(Π l | ψ (cid:105)(cid:104) ψ | )tr( E l | ψ (cid:105)(cid:104) ψ | ) (cid:112) tr(Π m | ψ (cid:105)(cid:104) ψ | )tr( E m | ψ (cid:105)(cid:104) ψ | ) dψ. (5.12)Computing the above integral analytically is difficult because of the square root in the argument. Now we assumethe following inequality holds (see Sec. VI) (cid:90) (cid:112) tr(Π l | ψ (cid:105)(cid:104) ψ | )tr( E l | ψ (cid:105)(cid:104) ψ | ) (cid:112) tr(Π m | ψ (cid:105)(cid:104) ψ | )tr( E m | ψ (cid:105)(cid:104) ψ | ) dψ ≥ (cid:90) (cid:112) tr(Π l | ψ (cid:105)(cid:104) ψ | )tr(Π l E l Π l ( | ψ (cid:105)(cid:104) ψ | )) (cid:112) tr(Π m | ψ (cid:105)(cid:104) ψ | )tr(Π m E m Π m ( | ψ (cid:105)(cid:104) ψ | )) dψ = (cid:90) (cid:112) (cid:104) ψ l | E l | ψ l (cid:105) tr(Π l ( | ψ (cid:105)(cid:104) ψ | )) (cid:112) (cid:104) ψ m | E m | ψ m (cid:105) tr(Π m ( | ψ (cid:105)(cid:104) ψ | )) dψ = √ u l u m (cid:90) tr(Π l ( | ψ (cid:105)(cid:104) ψ | ))tr(Π m ( | ψ (cid:105)(cid:104) ψ | )) dψ. (5.13)Note that for each j it is generally not true that E j − u j Π j is positive semidefinite (if this were the case then theabove inequality would always hold). From Eq. (5.5) we have since Π l and Π m are projectorsonto orthogonal subspaces, (cid:90) tr(Π l ( | ψ (cid:105)(cid:104) ψ | ))tr(Π m ( | ψ (cid:105)(cid:104) ψ | )) dψ = (cid:90) tr [(Π l ⊗ Π m ) | ψ (cid:105)(cid:104) ψ | ⊗ | ψ (cid:105)(cid:104) ψ | ] dψ = tr (Π l Π m ) + tr (Π l ) tr (Π m ) d ( d + 1)= 1 d ( d + 1) . (5.14)Thus (cid:90) √ r l r m p l p m dψ ≥ (cid:112) tr ( E l Π l ) tr ( E m Π m ) d ( d + 1)= √ u l u m d ( d + 1) , (5.15)which gives the last term in the numerator of Eq. (5.1).In total, assuming the inequality in Eq. (5.13) holds,we have that1 + dX d = d + (cid:80) dk =1 u k + (cid:80) l (cid:54) = m √ u l u m d ( d + 1) (5.16)is a lower bound for F E , M . Before analyzing the validityof the lower bound, we briefly compare its expression withthat of the average gate fidelity and show that it satisfiescertain necessary conditions to be useful in practice. C. Comparison With Quantum Gate Fidelity
There is a nice parallel between the lower bound onthe average measurement fidelity and the exact expres-sion for the average quantum gate fidelity. As previouslymentioned, the quantum average gate fidelity takes a par-ticularly nice form because one of the operations is uni-tary [36]. Indeed, if one compares the unitary U andquantum operation E , the average gate fidelity between U and E , F E , U , is given by F E , U = Ad + 1 d + 1 , (5.17)where A = 1 d (cid:88) k tr( A k )tr( A † k ) , (5.18)and { A k } is any set of Kraus operators for the quantumoperation Λ = U † ◦ E .Our expression for a lower bound on the average mea-surement fidelity F E , M takes a similar form: lb = Xd + 1 d + 1 (5.19)where X = 1 d (cid:88) l,m (cid:112) tr(Π l E l )tr(Π m E m ) . (5.20)In some sense this is not surprising since each quantity isa Haar integral over functions of two copies of a quantumstate | ψ (cid:105)(cid:104) ψ | . A direction of further research is to under-stand properties of the average measurement fidelity inmore detail, and draw more parallels with well-knowndistinguishability measures such as the average gate fi-delity. D. Necessary Conditions For lb To Be a UsefulLower Bound on F E , M In this subsection we discuss two necessary conditions lb must satisfy in order to be a useful lower bound on F E , M ;1. Limit of No Error: As the measurement error goesto 0, lb ↑ d : lb scales well in the dimension d of thesystem.Here, we show lb satisfies both of these criteria.
1. Limit of No Error
Suppose the measurement error goes to 0 in that thePOVM elements converge to the ideal PVM elements E j → Π j . (5.21)This implies for each j ∈ { , ..., d } , u j = tr(Π j E j ) ↑ . (5.22)Hence, from Eq.’s (5.1) and (4.7), lb ↑ . (5.23)Thus lb satisfies the necessary condition of converging to1 in the limit of no errors.
2. Scaling in d Let us now explicitly show that lb scales well in d . Moreprecisely, if each u k = tr(Π k E k ) is on the order of 1 − δ for some δ > d → ∞ , lb → − δ . Hence, thelower bound does not generically converge to a constantvalue that is independent of the E k . Such an effect wouldrender the lower bound useless as the system size growslarge.Again, by Eq.’s (5.1) and (4.7) we see that if each u k satisfies u k ∼ − δ (5.24)for some δ > lb ∼ Xd d = 1 + (1 − δ ) d d . (5.25)Hence as d → ∞ , lb → − δ, (5.26)which is what we wanted to show. VI. VALIDITY OF THE LOWER BOUND
The lower bound in Eq. (5.1) is valid provided the in-equality in Eq. (5.13) holds. The goal of this section is to show that this inequality holds in very general situations.To set notation, we define f l,m ( | ψ (cid:105) ) = (cid:112) tr(Π l | ψ (cid:105)(cid:104) ψ | )tr( E l | ψ (cid:105)(cid:104) ψ | ) (cid:112) tr(Π m | ψ (cid:105)(cid:104) ψ | )tr( E m | ψ (cid:105)(cid:104) ψ | ) ,g l,m ( | ψ (cid:105) ) = (cid:112) tr(Π l | ψ (cid:105)(cid:104) ψ | )tr(Π l E l Π l | ψ (cid:105)(cid:104) ψ | ) (cid:112) tr(Π m | ψ (cid:105)(cid:104) ψ | )tr(Π m E m Π m | ψ (cid:105)(cid:104) ψ | )= √ u l u m tr(Π l | ψ (cid:105)(cid:104) ψ | )tr(Π m | ψ (cid:105)(cid:104) ψ | ) (6.1)so that we want to show (cid:90) [ f l,m ( | ψ (cid:105) ) − g l,m ( | ψ (cid:105) )] dψ ≥ . (6.2)We demonstrate that Eq. (6.2) should hold by showingthat the set of states for which f l,m ( | ψ (cid:105) ) − g l,m ( | ψ (cid:105) ) < | ψ (cid:105) satisfies f l,m ( | ψ (cid:105) ) − g l,m ( | ψ (cid:105) ) < | f l,m ( | ψ (cid:105) ) − g l,m ( | ψ (cid:105) ) | will be small. So, let us look atthe measure of states | ψ (cid:105) that could satisfy f l,m ( | ψ (cid:105) ) − g l,m ( | ψ (cid:105) ) < | f l,m ( | ψ (cid:105) ) − g l,m ( | ψ (cid:105) ) | will look like for such astate.First note that (cid:90) [ f l,m ( | ψ (cid:105) ) − g l,m ( | ψ (cid:105) )] dψ = (cid:90) k l,m ( | ψ (cid:105) ) h l,m ( | ψ (cid:105) ) dψ, (6.4)where k l,m ( | ψ (cid:105) ) := (cid:112) tr(Π l | ψ (cid:105)(cid:104) ψ | )tr(Π m | ψ (cid:105)(cid:104) ψ | ) , (6.5) h l,m ( | ψ (cid:105) ) := (cid:112) tr( E l | ψ (cid:105)(cid:104) ψ | )tr( E m | ψ (cid:105)(cid:104) ψ | ) (6.6) − √ u l u m (cid:112) tr(Π l | ψ (cid:105)(cid:104) ψ | )tr(Π m | ψ (cid:105)(cid:104) ψ | ) . (6.7)Since k l,m ( | ψ (cid:105) ) ≥ | ψ (cid:105) , we have f l,m ( | ψ (cid:105) ) − g l,m ( | ψ (cid:105) ) < ⇔ h l,m ( | ψ (cid:105) ) < . (6.8)However, if h l,m ( | ψ (cid:105) ) < (cid:104) ψ | E l − u l Π l | ψ (cid:105) < (cid:104) ψ | E m − u m Π m | ψ (cid:105) < . (6.9)Without loss of generality, suppose (cid:104) ψ | E l − u l Π l | ψ (cid:105) <
0. Thus, if | ψ (cid:105) satisfies f l,m ( | ψ (cid:105) ) − g l,m ( | ψ (cid:105) ) < (cid:104) ψ | E l − u l Π l | ψ (cid:105) < {| ψ k (cid:105)} basis, we have E l − u l Π l is equal to E l except in the ( l, l )’th entry which is0. For instance if l = 0 then E − u Π is given by E , . . . . . . E ,d − E ,d − E , E , . . . . . . ...... . . . . . . . . . . . . ...... . . . . . . . . . . . . ... E d − , . . . . . . . . . ... E d − , . . . . . . . . . . . . E d − ,d − . (6.10)Hence, E l − u l Π l is not positive semidefinite unless theoff-diagonals in the l ’th row (column) are equal to 0. Thisis because, since E l is positive semidefinite, the followingbound on the off-diagonal elements holds [51] (cid:12)(cid:12)(cid:12) E i,jl (cid:12)(cid:12)(cid:12) ≤ (cid:113) E i,il (cid:113) E j,jl . (6.11)Now, note that if | ψ (cid:105) = | ψ l (cid:105) then (cid:104) ψ | E l − u l Π l | ψ (cid:105) = 0,and if | ψ (cid:105) is orthogonal to | ψ l (cid:105) , (cid:104) ψ | E l − u l Π l | ψ (cid:105) ≥ | ψ l (cid:105) for which one couldhave (cid:104) ψ | E l − u l Π l | ψ (cid:105) < . (6.12)That is, the function on pure states defined by (cid:104) ψ | E l − u l Π l | ψ (cid:105) has a minimum (which can be less than 0) thatis achieved for some state close to | ψ l (cid:105) . As | ψ (cid:105) movesfarther away from | ψ l (cid:105) , and accumulates more amplitudein the subspace orthogonal to | ψ l (cid:105) , we have (cid:104) ψ | E l − u l Π l | ψ (cid:105) ≥ . (6.13)Thus, Eq. (6.9) can only be satisfied if | ψ (cid:105) has largeenough amplitude in the subspace defined by | ψ l (cid:105) . Theamount of amplitude that is required depends on the sizeof the off-diagonal elements (coherence) in the l ’th col-umn (row) of E l . If the coherence is not too large then | ψ (cid:105) will need to have large amplitude in | ψ l (cid:105) to satisfyEq. (6.9), and as the coherence grows larger, | ψ (cid:105) canpotentially satisfy Eq. (6.9) without being very close to | ψ l (cid:105) .0The key point is that the measure of the set of statesfor which | ψ (cid:105) is close to | ψ l (cid:105) (or | ψ m (cid:105) ) will be small (andby Levy’s Lemma [52] decreases exponentially in the di-mension d ). Moreover, if | ψ (cid:105) has large amplitude in | ψ l (cid:105) ,and so (cid:104) ψ | E l − u l Π l | ψ (cid:105) < m | ψ (cid:105)(cid:104) ψ | )is small since | ψ l (cid:105) and | ψ m (cid:105) are orthogonal. Thus, by def-inition, if | ψ (cid:105) is close to either | ψ l (cid:105) or | ψ m (cid:105) , k l,m ( | ψ (cid:105) ) willbe small. Hence, k l,m ( | ψ (cid:105) ) acts like a modulating fac-tor in the expression for f l,m ( | ψ (cid:105) ) − g l,m ( | ψ (cid:105) ) to ensure f l,m ( | ψ (cid:105) ) − g l,m ( | ψ (cid:105) ) is close to 0 when | ψ (cid:105) is close toeither of | ψ l (cid:105) or | ψ m (cid:105) .In total then, the measure of states for which | ψ (cid:105) isclose to | ψ l (cid:105) or | ψ m (cid:105) is small and, for any such state, f l,m ( | ψ (cid:105) ) − g l,m ( | ψ (cid:105) ) will be close to 0. Equivalently, theset of states for which f l,m ( | ψ (cid:105) ) − g l,m ( | ψ (cid:105) ) < | f l,m ( | ψ (cid:105) ) − g l,m ( | ψ (cid:105) ) | will beclose to 0. This is what we wanted to show and so weexpect Eq. (6.2) to hold for most practical cases.Note that the bounds are guaranteed to be valid ifthere is no coherence at all, that is, if the POVM elementsare diagonal in the {| ψ k (cid:105)} basis. In the next section weobtain a sufficient condition for Eq. (6.2) to hold in thecase of a single qubit. In Sec. VI B we perform a detailednumerical investigation of the single-qubit case and showour lower bound always holds, that is, for all values of co-herence magnitude, the bounds are valid. This providesevidence that the bounds will be valid in large generality.Clearly, a more in-depth investigation of sufficient con-ditions for the bounds to be valid is desirable, howeverthe above argument and numerical results of Sec. (VI B)indicate that the bounds and protocol will be valid inmost practical cases. A. Sufficient Condition For the Single-Qubit Case
Using the notation of the previous section, for a singlequbit, the lower bound in Eq. (5.1) is valid if (cid:90) [ f , ( | ψ (cid:105) ) − g , ( | ψ (cid:105) )] dψ ≥ . (6.14)Our goal in this section is to obtain a sufficient conditionfor when Eq. (6.14) holds.We use the Bloch sphere representation of a singlequbit: | ψ (cid:105) = cos (cid:18) θ (cid:19) | (cid:105) + e ı φ sin (cid:18) θ (cid:19) | (cid:105) (6.15)where θ ∈ [0 , π ), φ ∈ [0 , π ). Hence, the POVM elementsare given by E = (cid:18) u γγ tr( E Π ) (cid:19) , (6.16) E = (cid:18) tr( E Π ) − γ − γ u (cid:19) , (6.17)where, since the results will depend on the magnitude ofthe coherence in the POVM elements, we assume withoutloss of generality that γ ∈ R and γ > (cid:82) [ f , ( | ψ (cid:105) ) − g , ( | ψ (cid:105) )] dψ , we need to understand whenone of (cid:104) ψ | E − u Π | ψ (cid:105) < (cid:104) ψ | E − u Π | ψ (cid:105) < f , ( | ψ (cid:105) ) − g , ( | ψ (cid:105) ) < (cid:104) ψ | E − u Π | ψ (cid:105) < . (6.18)We have (cid:104) ψ | E − u Π | ψ (cid:105) = 2Re (cid:18) γ cos (cid:18) θ (cid:19) e iφ sin (cid:18) θ (cid:19)(cid:19) + tr( E Π ) sin (cid:18) θ (cid:19) (6.19)and it is straightforward to show (cid:104) ψ | E − u Π | ψ (cid:105) < φ ∈ (cid:18) π , π (cid:19) ,θ ∈ (cid:20) , (cid:18) − tr( E Π )2 γ cos( φ ) (cid:19)(cid:21) . (6.20)Hence, denoting the set of states that satisfy (cid:104) ψ | E − u Π | ψ (cid:105) < A ,γ , we have that A ,γ is contained inthe set of all states with φ and θ given by Eq. (6.20).As expected, the measure of A ,γ , µ ( A ,γ ), is extremelysmall for weak coherence and grows larger as the coher-ence increases in magnitude.The uniform measure on the Bloch sphere has densityfunction 14 π sin( θ ) . (6.21)Hence, from Eq.’s (6.20), µ ( A ,γ ) is at most14 π (cid:90) π π (cid:90) (cid:16) − tr( E γ cos( φ ) (cid:17) sin( θ ) dθdφ = 14 − π (cid:90) π π cos (cid:18) (cid:18) − tr( E Π )2 γ cos( φ ) (cid:19)(cid:19) dφ. (6.22)Now, if | ψ (cid:105) ∈ A ,γ θ < (cid:18) − tr( E Π )2 γ cos( φ ) (cid:19) , (6.23)1and sotr(Π | ψ (cid:105)(cid:104) ψ | ) ≤ sin (cid:18) arccot (cid:18) − tr( E Π )2 γ cos( φ ) (cid:19)(cid:19) = cos ( φ )cos ( φ ) + (cid:16) tr( E Π )2 γ (cid:17) . (6.24) Thus, we have that for weak coherence γ , the set of stateswhich satisfy Eq. (6.18) will have small measure boundedby Eq. (6.22). Using a symmetric argument for the state | (cid:105) gives the following set of equations, µ ( A ,γ ) ≤ − π (cid:90) π π cos (cid:18) (cid:18) − tr( E Π )2 γ cos( φ ) (cid:19)(cid:19) ,µ ( A ,γ ) ≤ − π (cid:90) π π cos (cid:18) (cid:18) − tr( E Π )2 γ cos( φ ) (cid:19)(cid:19) , If | ψ (cid:105) ∈ A ,γ , tr(Π | ψ (cid:105)(cid:104) ψ | ) ≤ cos ( φ )cos ( φ ) + (cid:16) tr( E Π )2 γ (cid:17) with φ ∈ (cid:18) π , π (cid:19) , If | ψ (cid:105) ∈ A ,γ , tr(Π | ψ (cid:105)(cid:104) ψ | ) ≤ cos ( φ )cos ( φ ) + (cid:16) tr( E Π )2 γ (cid:17) with φ ∈ (cid:18) π , π (cid:19) . (6.25)We can now obtain a sufficient condition for (cid:90) [ f , ( | ψ (cid:105) ) − g , ( | ψ (cid:105) )] dψ = (cid:88) j =0 (cid:90) A j,γ [ f , ( | ψ (cid:105) ) − g , ( | ψ (cid:105) )] dψ + (cid:90) ( A ,γ ∪ A ,γ ) c [ f , ( | ψ (cid:105) ) − g , ( | ψ (cid:105) )] dψ ≥ . (6.26)By the set of equations in Eq. (6.25) one can show (cid:90) A ,γ [ f , ( | ψ (cid:105) ) − g , ( | ψ (cid:105) )] dψ ≥ (cid:90) A ,γ − √ u u tr(Π | ψ (cid:105)(cid:104) ψ | )tr(Π | ψ (cid:105)(cid:104) ψ | ) dψ = − √ u u π (cid:90) π π sin (cid:18) arccot (cid:18) − tr( E Π )2 γ cos( φ ) (cid:19)(cid:19) dφ. (6.27)By Eq. (6.24) (cid:90) A ,γ [ f , ( | ψ (cid:105) ) − g , ( | ψ (cid:105) )] dψ ≥ − √ u u π (cid:90) π π cos ( φ )cos ( φ ) + (cid:16) tr( E Π )2 γ (cid:17) dφ, =: − δ ,γ (6.28) and similarly (cid:90) A ,γ [ f , ( | ψ (cid:105) ) − g , ( | ψ (cid:105) )] dψ ≥ − √ u u π (cid:90) π π cos ( φ )cos ( φ ) + (cid:16) tr( E Π )2 γ (cid:17) dφ =: − δ ,γ . (6.29)Hence (cid:90) [ f , ( | ψ (cid:105) ) − g , ( | ψ (cid:105) )] dψ ≥ − ( δ ,γ + δ ,γ ) + (cid:90) ( A ,γ ∪ A ,γ ) c [ f , ( | ψ (cid:105) ) − g , ( | ψ (cid:105) )] dψ, (6.30)where µ (( A ,γ ∪ A ,γ ) c ) ≥
12 + 14 π (cid:90) π π cos (cid:18) (cid:18) − tr( E Π )2 γ cos( φ ) (cid:19)(cid:19) + 14 π (cid:90) π π cos (cid:18) (cid:18) − tr( E Π )2 γ cos( φ ) (cid:19)(cid:19) (6.31)2and, if | ψ (cid:105) ∈ ( A ,γ ∪ A ,γ ) c , f , ( | ψ (cid:105) ) − g , ( | ψ (cid:105) ) ≥ . (6.32)This gives the desired sufficient condition for (cid:82) [ f , ( | ψ (cid:105) ) − g , ( | ψ (cid:105) )] dψ ≥
0, and so a sufficient con-dition for the lower bound to be valid: (cid:90) ( A ,γ ∪ A ,γ ) c [ f , ( | ψ (cid:105) ) − g , ( | ψ (cid:105) )] dψ ≥ δ ,γ + δ ,γ (6.33)Note that the right hand side of Eq. (6.33) is typicallysmall and the left hand is non-negative with the integraltaken over a set with large measure.We have shown here that, for a single qubit, an analyt-ical sufficient condition for the lower bound to be validdoes indeed exist. This sufficient condition explicitly de-fines a regime where our bounds are provably valid. How-ever, clearly it will also be useful to numerically analyzethe exact relationship between the average measurementfidelity and our derived lower bound lb for the single-qubit case. This analysis will further aid in shaping ourunderstanding of the validity of the bounds and is per-formed in the next section. We find that the lower boundis always valid, independent of the magnitude of the co-herence. B. Numerical Examples For the Single-Qubit Case
In the previous section we obtained a sufficient condi-tion for lb to be valid in the case of a single qubit. Wenow numerically analyze various single-qubit examplesto observe the exact behavior of lb relative to the exactvalue of F E , M , which will also help clarify many of thetechnical details presented thus far.In order to perform integration over the Fubini-Studymeasure on single-qubit states, we again use the Blochsphere representation of a single qubit state | ψ (cid:105) = cos (cid:18) θ (cid:19) | (cid:105) + sin (cid:18) θ (cid:19) e iφ | (cid:105) , (6.34)where θ ∈ [0 , π ], φ ∈ [0 , π ]. Thus any function f onpure states can be written as a function f ( θ, φ ) on theunit sphere S ⊂ R . In addition, the integral of f ( θ, φ )with respect to the Fubini-Study measure is just the usualdouble integral14 π (cid:90) πθ =0 (cid:90) πφ =0 f ( θ, φ ) sin( θ ) dθdφ, (6.35)where the normalization π ensures we are integratingover a probability measure.Our goal is to provide a direct comparison between F E , M and lb for the case of a single-qubit projective mea- surement in the computational basis. As before, we writethe noisy POVM elements as { E , E } . From Eq. (5.3),we have that the exact value of the average measurementfidelity, F E , M , is given by F E , M = (cid:90) F ( E ( | ψ (cid:105)(cid:104) ψ | ) , M ( | ψ (cid:105)(cid:104) ψ | )) dψ = (cid:88) k =0 (cid:20)(cid:90) r k p k dψ (cid:21) + (cid:88) l (cid:54) = m (cid:20)(cid:90) √ r l r m p l p m dψ (cid:21) = (cid:88) k =0 (cid:20)(cid:90) r k p k dψ (cid:21) + 2 (cid:90) √ r r p p dψ (6.36)where, using the Bloch sphere representation and the el-ements of the POVM { E , E } , we can write p = cos (cid:18) θ (cid:19) ,p = 1 − p = sin (cid:18) θ (cid:19) ,r = E , cos (cid:18) θ (cid:19) + E , sin (cid:18) θ (cid:19) + Re (cid:16) E , e iφ (cid:17) sin( θ ) ,r = 1 − r . (6.37)For each k = 0 , (cid:82) r k p k dψ exactly bythe techniques introduced in Sec. V A (specifically seeEq. (5.11)). However as noted in Sec. V B, calculat-ing (cid:82) √ r r p p dψ analytically is generally not possi-ble. One can however directly input the expressions inEq.’s (6.37) into (cid:82) √ r r p p dψ so that a numerical anal-ysis can be performed.The POVM elements { E , E } that model the noisemust each be positive semidefinite and E + E = .Hence, we write E = (cid:18) u γγ ∗ − u (cid:19) , (6.38) E = (cid:18) − u − γ − γ ∗ u (cid:19) , (6.39)where γ ∈ C is given by γ = re iφ (6.40)and φ ∈ [0 , π ). We have that the linear operator E ( E )is positive semidefinite if and only if its leading principalminors are non-negative. Since d = 2, this reduces to thecondition | γ | = r ≤ (cid:112) u (1 − u ) . (6.41)For various cases, we observe how the exact measure-ment fidelity compares to lb as R approaches its maxi-3mum value R max := (cid:112) u (1 − u ) . (6.42)Since the results will only depend on R and not φ , with-out loss of generality, one can assume γ ∈ R so that γ = R . As a verification of this, the numerics given belowwere performed for various φ ∈ [0 , π ) and, as expected,the results were independent of φ .Three of the cases we analyzed are u = 0 .
99, 0 . . lb and ub for each case arecontained in Table I. Plots of the average measurementerror F E , M as a function of | γ | for each case are given inFig. 1. The horizontal axis is cut off at R max for eachcase. Thus, all admissible values of the coherence areprovided. TABLE I. Values of lb and ub for 3 Cases of u . u .
99 0 .
995 0 . lb . . . ub . . . lb is always a lower bound for F E , M (the curvesin the plot do not intersect). This agrees with the dis-cussion in Sec. (V) where we noted that lb should bea valid lower bound in large generality (even for largecoherence) due to both the anti-correlation of the ran-dom variables tr(Π | ψ (cid:105)(cid:104) ψ | ) and tr(Π | ψ (cid:105)(cid:104) ψ | ), and themeasure of the set of states for which g , > f , be-ing small. Second, as expected, the difference between lb and F E , M grows smaller as the magnitude of the coher-ence increases. Thus, the bounds become more tight asthe coherence in the POVM elements increases.These results suggest that the bounds are always validregardless of how large | γ | is relative to u . We testedwhether there were any violations when u was variedfrom 0.5 to 0.9999 in increments of 10 − , and the coher-ence was varied from its minimum to maximum in eachcase. We found no violations of the lower bound. VII. CHARACTERIZING MEASUREMENTPROBABILITIES AND OUTPUT STATES FORRANK-1 PVM’S
Ideally, a quantum measurement produces an outputstate that can be utilized for further purposes. In thiscase one would like to characterize the error on both themeasurement statistics and output states of the measure-ment. In this section we briefly outline such a methodfor rank-1 PVM’s and discuss conditions for the lowerbound on the measurement fidelity F E , M to be valid. Measurement Fidelity and Coherence Strength
Exact Measurement FidelityLower Bound on Measurement Fidelity M ea s u r e m en t F i de li t y Magnitude of Coherence
FIG. 1. Plot of exact average measurement fidelity F E , M (blue-dashed) against coherence strength | γ | for three differ-ent values of u : 0 .
99 (upper left), 0 .
995 (bottom left), and0 .
999 (bottom right). The lower bound lb for F E , M in eachcase is also given (green). The magnitude of the coherencewas varied in each case up to its maximum value R max . From Eq. (2.4) we have that M is given by M ( σ ) = d (cid:88) k =1 Π k σ Π k = b (cid:88) k =1 p k Π k , (7.1)where p k = tr (Π k σ ) , (7.2)and the output state Π k is independent of the input state σ . Note that this assumption is only valid because theideal measurement is a rank-1 PVM. We assume the noisymeasurement process E is of the form E ( σ ) = b (cid:88) k =1 r k ρ k (7.3)where, as before, r k = tr( E k σ ) (7.4)4is modeled via a POVM { E k } dk =1 . We assume the ρ k areindependent of σ since, ideally, the output states Π k areindependent of σ . However, it may be the case that thereis some dependence of ρ k on the input σ . When the de-pendence is weak we expect the results presented here togenerally still be valid, however the extent to which this isthe case is left as an area for further research. When thereis large dependence, clearly new techniques will have tobe applied. There are simple tests one could performto help detect such state-dependence of output states.For instance, suppose we take two input states | ψ (cid:105) and | ψ (cid:105) . We input | ψ (cid:105) into the measurement and observewhether outcome “ k ” is obtained. Whenever it is, wecan make a successive measurement on the output state ρ k ( | ψ (cid:105) ) and observe whether outcome “ k ” is obtainedagain. Repeating this many times produces the statis-tic tr( ρ k ( ψ ) E k ). Repeating this procedure for | ψ (cid:105) givestr( ρ k ( ψ ) E k ). If these quantities differ then there mustbe some input state-dependence in the output states ofthe measurement.The average measurement fidelity is still given by F E , M = (cid:90) F E , M ( | ψ (cid:105)(cid:104) ψ | ) dψ = (cid:90) F (cid:32)(cid:88) k r k ρ k , (cid:88) k p k Π k (cid:33) dψ (7.5)and, as before, r E , M = 1 − (cid:90) F E , M ( | ψ (cid:105)(cid:104) ψ | ) dψ. (7.6)In this case where we are also interested in outputstates, the expression for F E , M is much more compli-cated. However, we can use strong concavity of the fi-delity [53] to obtain F E , M = (cid:90) F ( E ( | ψ (cid:105) ) , M ( (cid:104) ψ | )) ≥ (cid:90) (cid:32)(cid:88) k √ r k p k (cid:112) tr ( ρ k Π k ) (cid:33) dψ. (7.7)Since tr ( ρ k Π k ) is just the fidelity between the k ’th idealand noisy output states of the measurement, we define F k := F ( ρ k , Π k ) = tr ( ρ k Π k ) . (7.8)Moreover, since there is only state-dependence in p k and r k , we have F E , M ≥ (cid:88) ( l,m ) ∈D (cid:20)(cid:90) √ r l r m p l p m dψ (cid:21) (cid:112) F l F m . (7.9)From Sec. V, and specifically Eq. (5.15), this gives F E , M ≥ dY d , (7.10) where Y := 1 d (cid:88) ( l,m ) ∈D √ u l u m (cid:112) F l F m . (7.11)At this point, we could define lb = dY d (and thus ub = 1 − dY d ), except the F k cannot be obtained di-rectly from the noisy measurement E . The reason forthis is that F k describes the overlap between the outputstate from the ideal measurement, Π k , and the outputstate from the real measurement, ρ k . Equivalently, thisis the same as the result obtained by making an idealmeasurement in the Π k basis on the real measurementoutput state ρ k . By definition however, we cannot makean ideal measurement M . All we can do is make an-other real measurement E and look at resulting outcomeprobabilities, namely tr( ρ k E k ).Let us denote Q k to be this probability Q k = P [obtaining outcome k from E if the input is ρ k ]= tr( ρ k E k ) (7.12)where the notation “ P ( · )” means “probability of”. Thus, if it is true that for each kF k = tr ( ρ k Π k ) ≥ Q k = tr ( ρ k E k ) , (7.13)then we have F E , M ≥ lb := 1 + dZ d , (7.14)where Z := 1 d (cid:88) ( l,m ) ∈D √ u l u m (cid:112) Q l Q m . (7.15)Motivation for why the assumption in Eq. (7.13) willoften be true is that ideally, for each k , E k = Π k . Hence,tr ( ρ k Π k ) is just the k ’th diagonal element of ρ k , ρ k,kk .Since E k ≤ , if E k is diagonal in the {| ψ k (cid:105)} basis, thenEq. (7.13) is guaranteed to hold. This is because theprobability amplitude of Π k is spread across the diag-onal elements of E k . Even when there is coherence inthe E k (non-zero off-diagonal elements) we expect the in-equality should still hold in realistic situations, howevera more detailed analysis of conditions for this to occur isrequired. Note also that, realistically, one does not needEq. (7.13) to hold for every k , just a large enough num-ber to ensure Y ≥ Z . We now provide the experimentalprotocol for determining lb . VIII. EXPERIMENTAL PROTOCOL: RANK-1PVM’S WITH OUTPUT STATES
Goal: Obtain lb in the case of imperfect measurementprobabilities and output states.5Protocol:Step 1: Choose a pair of indices ( l, m ) uniformly at ran-dom from D = { , ..., d − } × { , ..., d − } .Step 2: For each j = l and j = m ,a) Prepare the quantum state Π j , perform the noisymeasurement E on Π j , and record whether outcome “ j ”is obtained,b) If outcome “ j ” is obtained in a), repeat the mea-surement on the current state of the system, and recordwhether outcome “ j ” is obtained again,c) Repeat Steps a) and b) many times and denote thefrequency of obtaining “ j ” in each step by ˆ u j and ˆ Q j respectively.(see below for a discussion of the number of repetitionsrequired to estimate u j and Q j to desired accuracies).Step 3: Repeat Steps 1 and 2 K times, where K is dic-tated by the desired accuracy and confidence in estimat-ing lb (see below for a discussion of the size of K ).Step 4: Compute the estimator, ˆ lb , of lb (where lb is givenin Eq. (7.14)) to accuracy and confidence dictated by the K trials { (1 , ) , ..., ( K , K ) } from Step 3ˆ lb = 1 + d K (cid:88) ( k ,k ) √ u k u k (cid:112) Q k Q k d . (8.1)This concludes the protocol.We should emphasize various points about this proto-col. First, similar to the case of no output states (seeSec IV), the number of trials required to implement theabove protocol is independent of d , and only depends onthe desired accuracy and confidence of the estimates ineach of Steps 2 c) and 3. See Sec.’s IX A and IX B forrespective discussions about the time-complexity of eachof these steps.Second, lb can be estimated using only:1. Sequential applications of the noisy measurementand,2. The ability to prepare each of the d pure inputstates Π j .Lastly, it is straightforward to show that the two neces-sary conditions given previously in Sec. V D for lb to bea useful lower bound also hold here. Indeed, 1. In the limit of F E , M ↑ lb ↑ , (8.2)and2. lb scales well in d .The first condition holds since, as F E , M ↑
1, it mustbe the case that E k → Π k and ρ k → Π k . Therefore, forevery k , u k → Q k →
1. The second condition holdsusing an analogous argument as that given in Sec. V D 2
IX. RESOURCE ANALYSIS
In this section we discuss the time-complexity and re-sources required for the protocols and compare the time-complexity of the protocol with a full reconstruction ofthe noisy measurement (ie. a full reconstruction of thePOVM elements E k ). First, we analyze the number oftrials required in Step 2b) of the protocol in Sec. IV andStep 2c) of the protocol in Sec. VIII. Afterwards, we an-alyze the number of trials required in Step 3 of bothprotocols. A. Number of Trials Required in Step 2b) ofSec. IV and Step 2c) of Sec. VIII
We first explicitly analyze the number of trials for Step2b) of Sec. IV. The discussion carries over in a straight-forward manner to Step 2c) of Sec. VIII.
1. Step 2b) of Sec.IV
For each k ∈ { , , ..., d − } , we would like to under-stand how many samples are required to obtain an esti-mate of u k . Let us fix k and define z k = tr(Π k ( − E k ))where, since { E k } dk =1 is a POVM, − E k = d (cid:88) j (cid:54) = k E j . (9.1)Then z k ≥ ,u k + z k = tr(Π k ) = 1 , (9.2)so { u k , z k } = { u k , − u k } forms a probability dis-tribution on the binary measurement outcome space { “ k ” , “not k ” } . Let us define the random variable V which takes the values { “ k ” , “not k ” } with associatedprobabilities { u k , − u k } . Then, V is an asymmetricBernoulli random variable [54] (it is asymmetric since ingeneral u k (cid:54) = ), and we can call outcome “ k ” a successand outcome “not k ” a failure.6Let ˆ u k be the estimator of u k obtained in 2b of theprotocol. Since u k is just the probability of success in aBernoulli random variable, we can use maximum likeli-hood estimation (MLE) techniques [54] to determine thenumber of trials required to estimate u k to some desiredaccuracy and confidence. As the trials performed in theprotocol are independent, standard MLE gives that ˆ u k is just equal to the frequency of obtaining outcome “ k ”.Thus, if N is the number of trials and n k is the numberof times outcome “ k ” is observed, standard MLE givesthe estimate ˆ u k = n k N (9.3)which agrees with intuition.There can be a problem with this procedure if u k isvery close to 1. In particular, if “ k ” is observed in all N trials then a probability of 1 is assigned to “ k ” anda probability of 0 is assigned to “not k ”, which is nota physically realistic scenario. This is an example of amuch more general problem that can arise when usingMLE to assign probabilities to rare events that are notobserved because the data set is not large enough. Onesolution to this problem is to utilize additive (Laplace)smoothing methods to augment the MLE procedure [55].The main idea behind such a smoothing technique is toassign higher (non-zero) weight to low probability out-comes. In our case, the low probability outcome is “not k ” and the weight can be controlled by a parameter λ according to the formulaˆ u k = n k + λN + 2 λ . (9.4)The factor of 2 multiplying λ in the denominator arisesbecause there are two possible outcomes of the experi-ment, and is required to ensure { ˆ u k , − ˆ u k } is a proba-bility distribution. Note that when this smoothing tech-nique is used, it will tend to fairly bias the estimate of u k to be smaller than the actual value of u k , so the esti-mation will be honest.The key point in terms of time complexity, which wenow prove, is that the number of trials N required to esti-mate u k to accuracy (cid:15) with confidence 1 − δ is independent of the size of the system. Since the set of possible prob-ability distributions { u k , z k } satisfies certain consistencyconditions, the estimator ˆ u k converges in distribution to u k , ˆ u k D −→ u k . (9.5)Moreover, the mean of ˆ u k is equal to u k and the varianceof ˆ u k scales as Var (ˆ u k ) ∼ N I ( { u k , z k } ) , (9.6)where I ( { u k , z k } ) is the Fisher information [54] of oneobservation of the true probability distribution. The Fisher information contained in one observation is a mea-sure of how much information about u k is gained on av-erage from observing { “ k ” , “not k ” } (with distribution { u k , z k } ). If u k ∼ { “ k ” , “not k ” } is a Bernoulli random variable, itis possible to explicitly compute the Fisher informationof one observation [54], I ( { u k , z k } ) = 1 u k z k = 1 u k (1 − u k ) . (9.7)Hence, from Eq.’s (9.6) and (9.7), we have thatVar (ˆ u k ) ∼ u k (1 − u k ) N . (9.8)Now, suppose we want to estimate u k to accuracy (cid:15) withconfidence 1 − δ , ie. we want P ( | ˆ u k − u k | ≥ (cid:15) ) ≤ δ. (9.9)By Chebyshev’s theorem we have for any integer j ≥ P ( | ˆ u k − u k | ≥ jσ (ˆ u k )) ≤ j . (9.10)Choose j δ to be the smallest j such that1 j δ ≤ δ. (9.11)Then we have P (cid:32) | ˆ u k − u k | ≥ j δ (cid:112) u k (1 − u k ) √ N (cid:33) ≤ j δ , (9.12)and so we set j δ (cid:112) u k (1 − u k ) √ N ≤ (cid:15). (9.13)This gives N ≥ j δ u k (1 − u k ) (cid:15) . (9.14)In total, if N ≥ j δ u k (1 − u k ) (cid:15) with j δ ≥ δ , then P ( | ˆ u k − u k | ≥ (cid:15) ) ≤ δ. (9.15)Under the assumption that the u k are independent of d ,we have that N is independent of d . Hence, the num-ber of trials required to estimate u k to accuracy (cid:15) andconfidence 1 − δ is independent of d . If the probabilities u k do depend on d then the number of trials will be afunction of d . Determining the extent to which the u k can have dependence on d is an interesting question andlikely depends on the particular scenario at hand. Thisis left as a topic for further investigation.7
2. Step 2c) of Sec. VIII
In the protocol of Sec. VIII we have to estimate both u k and Q k where, u k is the probability of obtaining outcome“ k ” on the first measurement and Q k is the probabil-ity of obtaining outcome “ k ” on a second measurement given the result of the first measurement is “ k ”. Notethat observing “ k ” in the second measurement is also aBernoulli random variable, so the discussion from abovecarries over in an analogous manner to estimating Q k .Let N and N be the number of trials required to esti-mate u k and Q k to each of their respective accuracies andconfidences. Suppose we perform the first measurement N times (so we have estimated u k to its desired accuracyand confidence). If the number of times “ k ” is observedover these trials is greater than N then we have esti-mated both u k and Q k to their desired accuracies. If thenumber of times “ k ” is observed in the first measurementis less than N then we keep repeating until N outcomesof “ k ” are recorded in the first measurement. Let M bethe number of times the first measurement has to be per-formed before N values of “ k ” are recorded. The totalnumber of trials is no more than N = min { N , M } . (9.16)As N only depends on the accuracies, confidences, andvalues of u k and Q k , it is independent of d . B. Number of Trials Required in Step 3 ofSec. (IV) and Step 3 of Sec. (VIII)
Let us now discuss how many trials are required in Step3 of each of the protocols. The argument is the samefor each protocol so, without loss of generality, we usethe notation from Sec. IV (the discussion for Sec. (VIII)follows by replacing X with Z and making appropriatechanges). The random variable X : D → [0 ,
1] is definedby X ( l, m ) = √ u l u m (9.17)(for Sec. VIII this will be Z ( l, m ) = √ u l u m √ Q l Q m ) andthe expectation value of X is denoted X . Suppose onewants to estimate X to accuracy (cid:15) and confidence 1 − δ .Let ˆ X denote the estimator of X obtained from takingthe average of K independent samples, { X , ..., X K } , of X ˆ X = 1 K (cid:88) ( k ,k ) √ u k u k . (9.18)By Hoeffding’s inequality P (cid:16)(cid:12)(cid:12)(cid:12) ˆ X − X (cid:12)(cid:12)(cid:12) ≥ (cid:15) (cid:17) ≤ e − k(cid:15) )2 K ( b − a )2 = 2 e − K(cid:15) b − a )2 (9.19) where [ a, b ] is the range of X (here [ a, b ] ⊆ [0 , δ = 2 e − K(cid:15) b − a )2 , (9.20)gives K = ln (cid:0) δ (cid:1) ( b − a ) (cid:15) ≤ ln (cid:0) δ (cid:1) (cid:15) (9.21)which is independent of d . In practice, b − a (cid:28) K . C. Complete Reconstruction of POVM Elements
For completeness, we provide an explicit protocol anddetermination of the time-complexity required to per-form complete tomography of the noisy POVM elements { E k } dk =1 . The idea is to input various pure states intothe noisy measurement and analyzing the frequency ofobtaining particular outcomes. If the matrix representa-tion of E k in the | ψ m (cid:105) basis is written as E i,jk , then thereare d + d elements on the main diagonal and upper trian-gular section of E k that need to be estimated (since each E k ≥
0, the lower triangular part of E k is completelydetermined by the upper triangular part). One can esti-mate the elements of E k by first defining the pure states | ψ + i,j (cid:105) = | ψ i (cid:105) + | ψ j (cid:105)√ | ψ − i,j (cid:105) = | ψ i (cid:105) + i | ψ j (cid:105)√ . Then, since | ψ i (cid:105)(cid:104) ψ j | = | ψ + i,j (cid:105)(cid:104) ψ + i,j | + i | ψ − i,j (cid:105)(cid:104) ψ − i,j |− (cid:18) i (cid:19) | ψ i (cid:105)(cid:104) ψ i | − (cid:18) i (cid:19) | ψ j (cid:105)(cid:104) ψ j | , (9.22)we have E i,jk = tr (cid:0) E k Π + i,j (cid:1) + i tr (cid:0) E k Π − i,j (cid:1) − (cid:18) i (cid:19) tr( E k Π i ) − (cid:18) i (cid:19) tr( E k Π j ) . (9.23)The algorithm to determine the set { E k } dk =1 is as follows.Step 1: For each state | ψ j,j (cid:105) ∈ {| ψ k,k (cid:105)} dk =1 , input | ψ j,j (cid:105) into the noisy measurement E many times and record thefrequency of obtaining each of the d different possibleoutcomes “ k ”. For each k this gives tr( E k | ψ j,j (cid:105)(cid:104) ψ j,j | ).Step 2: For each state | ψ + i,j (cid:105) ∈ {| ψ + k,l (cid:105)} dk,l =1 , input | ψ + i,j (cid:105) into the noisy measurement E many times andrecord the frequency of obtaining each of the d dif-ferent possible outcomes “ k ”. For each k this givestr( E k | ψ + i,j (cid:105)(cid:104) ψ + i,j | ) = tr (cid:0) E k Π + i,j (cid:1) .8Step 3: For each state | ψ − i,j (cid:105) ∈ {| ψ − k,l (cid:105)} dk,l =1 , input | ψ − i,j (cid:105) into the noisy measurement E many times andrecord the frequency of obtaining each of the d dif-ferent possible outcomes “ k ”. For each k this givestr( E k | ψ − i,j (cid:105)(cid:104) ψ − i,j | ) = tr (cid:0) E k Π − i,j (cid:1) .Step 4: Combine all of the elements estimated in Steps1 through 3 to construct the E k . Step 1 gives the diagonalelements of the E k since E j,jk = tr( E k | ψ j,j (cid:105)(cid:104) ψ j,j | ) . (9.24)Eq. (9.23) and Steps 1 through 3 give all of the off-diagonal elements E i,jk . This concludes the protocol.The number of trials one will have to perform is againdictated by MLE. It is important to note that the MLEprocedure in this case is more involved than the sim-ple Bernoulli procedure for our protocol (described inSec. IX A). This is because, when the noisy measurementis performed, one must keep track of which value of j isobtained (not just whether the outcome was j or not).Thus the number of trials in each step will be greaterthan that required to estimate the u k because eventswith small probability may rarely be seen (if at all).Smoothing techniques will likely have to be employedto ensure rare events are not assigned zero probability.Thus, the determination of each tr( E k | φ (cid:105)(cid:104) φ | ), where | φ (cid:105) is one of | ψ j,j (cid:105) , | ψ + i,j (cid:105) , or | ψ − i,j (cid:105) , requires greater time-complexity than that of estimating each u k . Since thereare d +2 d ( d − = d such | φ (cid:105) , one will have to estimate d different probabilities over Steps 1 through 3 using MLE.Hence, the full reconstruction requires the estimation of d probabilities with more complicated post-processingof the measurement data (as well as a larger number oftrials to estimate each probability). In addition, thereis an added complexity in preparing d different inputstates since more complex rotations may be required. X. DISCUSSION
We have provided a straightforward, efficient, and ex-perimentally implementable method for obtaining esti-mates for lower bounds on the average fidelity of projec-tive (rank-1) quantum measurements. As realizations ofquantum protocols scale to larger sizes, and full measure-ment tomography becomes impossible to implement, ourprotocol can potentially be used as a simple method tobenchmark the performance of a measuring device.We have discussed conditions for the validity of thebounds and explained why they should hold in extremelygeneral situations. The bounds could also potentially beuseful as estimates of the average measurement fidelityin the small-error regime. We have presented a set ofnumerical examples for a single-qubit system. In everyinstance analyzed, that is, for all possible values of thecoherence in the noisy POVM operators, the bounds were found to be valid. This provides further evidence thatthe bounds should hold in extremely general situationsand should be useful in practice. In addition, the boundsbecame better approximations of the actual measurementfidelity as the magnitude of the coherence increased.The protocols are scalable and only require the abilityto prepare states from a basis set and perform the noisymeasurement (sequentially when there are output states)to estimate d probabilities. In addition, post-processingof the data is completely straightforward and avoids thedifficulties in associating large sets of tomographic datato valid mathematical objects. This can be comparedwith a full reconstruction of the POVM elements of thenoisy measurement, which requires the preparation of d input states, the estimation of d probabilities, and moreinvolved post-processing. In addition, the d input statesrequired for tomography can be highly complex. In manysituations, these input states will have to be preparedusing complicated unitary rotations. Minimizing the re-quired set of input states, as well as their complexity, isimportant for obtaining a more faithful characterizationof the measurement that is less prone to state-preparationerrors.It is important to note that the experimental proto-cols provide an estimate for a lower bound on the aver-age measurement fidelity. More precisely, one chooses anumber of trials K to obtain the estimate, where K de-pends on the desired accuracy (cid:15) and confidence 1 − δ ofthe estimate. Thus, under the assumption that the lowerbound is valid, the true value of the average measure-ment fidelity is no more than (cid:15) units of distance below the estimated lower bound with confidence 1 − δ .There are a number of different questions and avenuesfor future research. First, we have focused on the caseof rank-1 PVM’s, however we expect that our results canbe extended to higher-rank PVM’s, especially low-rankPVM’s in large Hilbert spaces. As well, since any POVMcan be implemented via a PVM on an extended Hilbertspace, the protocol can potentially give information re-garding the quality of implementations of POVM mea-surements. It will also be useful to analyze the extent towhich the ideas presented here can be used to character-ize non-ideal POVM measurements that are not imple-mented as PVM’s on a larger space.While we have shown the bounds derived here shouldhold in large generality, a deeper understanding of thevalidity of the bounds will clearly be useful. In mostphysically relevant cases, where the coherence is notoverwhelmingly large, the protocol should produce validbounds. In any noise estimation or characterizationscheme, there is a trade-off between the amount of in-formation one is able to extract and the amount of re-sources required to implement the scheme. Here, we ob-tain a single parameter which serves as an upper (lower)bound on the error (fidelity) of the measurement in ascalable amount of time. Ideally, one would like as muchinformation about the physical measurement as possible.Further analysis of schemes that give more information9about an imperfect measurement than average fidelitiesor errors, while retaining properties such as scalability,will clearly be useful.As previously mentioned, the algorithms given here re-quire the ability to prepare the basis states {| ψ k (cid:105)} whichconstitute the measurement. In practice, these statestypically have errors and may actually be created by ameasurement procedure, which is exactly what we wantto characterize. There are various systems however wherestate preparation is very different from the measurementprocedure. For instance, in superconducting qubit sys-tems [8, 9], the system is initialized to the ground stateby cooling the system to extremely low temperatures.Coupling the system to a superconducting resonator in acircuit-QED set-up [9, 56] allows one to perform bothstate preparation (by applying unitary rotations) andmeasurements. Typically, unitary rotations have muchhigher fidelities than measurements and so one expectsstate preparation to be much more accurate than mea-surements, which is ideal for the protocols presented inthis paper.We note that running the protocol with noisy statescan still provide valid bounds. For instance, if the noisy input states are given by a convex combination of ele-ments of {| ψ k (cid:105)} then the bounds still hold. The same isalso true for noisy input states with small coherence inthe {| ψ k (cid:105)} basis. The performance of the bounds undermore general noise models on input states is a topic forfuture research.The scalable protocols presented here can be useful fordetermining the quality of experimental quantum mea-surements. There is still much to investigate with re-gard to useful metrics for comparing measurements andproposing experimentally efficient methods for character-izing measurement devices. As experimental quantumsystems scale to larger sizes, such methods will be usefulfor characterizing and controlling multi-qubit systems. ACKNOWLEDGMENTS
E.M. acknowledges financial support from the NationalScience Foundation through grant NSF PHY-1125846.The authors are grateful for helpful discussions withAlexandre Cooper, Joseph Emerson, Jay Gambetta, Ma-soud Mohseni, and Marcus Silva. [1] R. Feynman, Internat. J. Theoret. Phys , 6 (1982).[2] V. Giovannetti, S. Lloyd, and L. Maccone, Science ,1330 (2004).[3] P. Shor, in Proceedings of the 35’th Annual Symposium onFoundations of Computer Science (FOCS) (IEEE Press,Los Alamitos, CA, 1994), pp. 124–134.[4] S. Lloyd, Science , 1073 (1996).[5] D. Deutsch, Proc. Roy. Soc. Lond. A , 97 (1985).[6] R. Raussendorf and H. Briegel, Phys. Rev. Lett. , 5188(2001).[7] A. Kitaev, Annals of Physics , 2 (1997).[8] V. Bouchiat et al., Phys. Scr. A T76 , 165 (1998).[9] J. Koch et al., Phys. Rev. A , 042319 (2007).[10] J. Wrachtrup and F. Jelezko, Journal of Physics: Con-densed Matter , S807 (2006).[11] I. Cirac and P. Zoller, Phys. Rev. Lett. , 4091 (1995).[12] D. Cory, A. Fahmy, and T. Havel, in Proceedings ofthe 4th Workshop on Physics and Computation (Boston,MA, 1996).[13] D. Loss and D. Divincenzo, Phys. Rev. A , 120 (1998).[14] E. Knill, R. Laflamme, and G. J. Milburn, Nature ,46 (2001).[15] I. Chuang and M. Nielsen, J. Mod. Opt. , 2455 (1997).[16] J. B. Altepeter et al., Phys. Rev. Lett. , 193601 (2003).[17] M. Mohseni and D. A. Lidar, Phys. Rev. Lett. , 170501(2006).[18] J. S. Lundeen et al., Nature Physics , 27 (2009).[19] L. Zhang et al., Nature Photonics , 364 (2012).[20] S. Merkel et al., Self-consistent quantum process tomog-raphy (2012), arXiv:quant-ph/1211.0322.[21] T. Monz et al., Phys. Rev. Lett. , 130506 (2011).[22] P. Shor, in
Proceedings of the 37’th Annual Symposium onFoundations of Computer Science (FOCS) (IEEE Press,Burlington, VT, 1996). [23] D. Aharonov and M. Ben-Or, in
Proceedings of the29th Annual ACM Symposium on Theory of Computing(STOC) (1997).[24] E. Knill, R. Laflamme, and W. Zurek, Proc. R. Soc.Lond. A , 365 (1997).[25] J. Preskill,
Fault tolerant quantum computation (1997),arXiv:quant-ph/9712048.[26] J. Emerson et al., Science , 1893 (2007).[27] M. Silva et al., Phys. Rev. A , 012347 (2008).[28] E. Knill et al., Phys. Rev. A , 012307 (2008).[29] E. Magesan, J. M. Gambetta, and J. Emerson, Phys.Rev. Lett. , 180504 (2011).[30] E. Magesan et al., Phys. Rev. Lett. , 080505 (2012).[31] O. Moussa et al., Phys. Rev. Lett. , 070504 (2012).[32] A. Bendersky, F. Pastawski, and J. Paz, Phys. Rev. Lett. , 190403 (2008).[33] M. P. da Silva, O. Landon-Cardinal, and D. Poulin, Phys.Rev. Lett. , 210404 (2011).[34] C. Schmiegelow et al., Phys. Rev. Lett. , 100502(2011).[35] S. Flammia and Y.-K. Liu, Phys. Rev. Lett. , 230501(2011).[36] M. Nielsen, Phys. Lett. A , 249 (2002).[37] J. Emerson, R. Alicki, and K. Zyczkowski, J. Opt. B:Quantum and Semiclassical Optics , S347 (2005).[38] E. Magesan, R. Blume-Kohout, and J. Emerson, Phys.Rev. A , 012309 (2011).[39] O. Oreshkov and J. Calsamiglia, Phys. Rev. A , 032336(2009).[40] V. Paulsen, Completely Bounded Maps and Operator Al-gebras , vol. 78 (Cambridge University Press, UK, 2002).[41] D. Gottesman,
Stabilizer codes and quantum error cor-rection (1997), ph.D. Thesis, arXiv:quant-ph/9705052.[42] Chaos, Solitons and Fractals , 1749 (1999). [43] M. D. Reed et al., Nature , 382 (2012).[44] D. DiVincenzo and F. Solgun (2012), arXiv:1205.1910.[45] A. Gilchrist, N. Langford, and M. Nielsen, Phys. Rev. A , 062310 (2005).[46] I. Bengtsson and K. Zyczkowski, Geometry of Quan-tum States: An Introduction to Quantum Entanglement (Cambridge University Press, Cambridge, UK, 2006).[47] C. Fuchs and J. van de Graaf, IEEE Trans. Inf. Th. ,1216 (1999).[48] A. Kitaev, Russian Mathematical Surveys , 1191(1997).[49] S. Beigi and R. Koenig, New J. Phys. , 093036 (2011). [50] J. Renes et al., J. Math. Phys. , 2171 (2004).[51] R. Horn and C. Johnson, Matrix Analysis (CambridgeUniversity Press, Cambridge, UK, 1990).[52] M. Ledoux,
The Concentration of Measure Phenomenon (American Mathematical Society, 2001).[53] C. A. Fuchs, Ph.D. thesis, University of New Mexico(2006), preprint quant-ph 9601020.[54] M. J. Schervish,
Theory of Statistics (Springer, NewYork, USA, 1997), 1st ed.[55] C. Manning, P. Raghavan, and H. Sch¨utze,
Introductionto Information Retrieval (Cambridge University Press,New York, NY, 2008).[56] A. Blais et al., Phys. Rev. A69