Table Of Content

Quantum non-demolition measurement saturates fidelity trade-off Ladislav Miˇsta Jr. and Radim Filip Department of Optics, Palacký University, 17. listopadu 50, 772 07 Olomouc, Czech Republic (Dated: February 1, 2008) A general quantum measurement on an unknown quantum state enables us to estimate what the state originally was. Simultaneously, the measurement has a destructive effect on a measured quantum state which is reflected by the decrease of the output fidelity. We show for any d-level systemthatquantumnon-demolition(QND)measurementcontrolledbyasuitablypreparedancilla is a measurement in which the decrease of the output fidelity is minimal. The ratio between the estimation fidelity and theoutput fidelity can be continuously controlled by the preparation of the ancilla. Different measurement strategies on the ancilla are also discussed. Finally, we propose 5 a feasible scheme of such a measurement for atomic and optical 2-level systems based on basic 0 controlled-NOTgate. 0 2 PACSnumbers: 03.67.-a n a J Measurement in quantum mechanics changes drasti- and based on the outcomes of the measurement she can 6 cally measured quantum state. Moreover, this change guess the state. Alternatively, Eve can guess the state cannot be done arbitrarily small. This main feature of frommeasurementonanancillarysystemthatpreviously 1 quantummeasurementcanbesimplyprovedbyperform- interacted with the original state. Both the strategies v ingtheestimationofthestateafterthemeasurement. At producetwostates,anestimateofthe originalstatethat 1 first sight this is a negative effect which does not allow is hold by Eve ρest and an output state ρout after the 3 many operations well known from classicalphysics. For- measurementthat continues towardBob. The quality of 0 tunately, there is also a positive aspect of this property. Eve’sguessescanbecharacterizedbythemeanfidelityG 1 0 In principle, it can be exploited to make communication (estimationfidelity)definedasG= ψ ρest ψ dψ where h | | i 5 between two distant stations secure against eavesdrop- dψ is the integral over the space of pure states and dψ R 0 ping attacks. Namely, secret information can be sent by is the measure invariant with respect to unitary trans- / quantum states in such a way that any measurement on Rformations. The perturbation introduced by Eve to the h p the transmitted states canbe detected andconsequently originalstatecanbecharacterizedbythemeanfidelityF - any attack on the link can be revealed [1]. This prop- (output fidelity) of the output state F = ψ ρout ψ dψ. t h | | i n erty represents a fundamental distinction between quan- Accordingtothelawsofquantummechanicsthefidelities R a tummeasurementandclassicalmeasurementthatcanbe F and G must satisfy the following inequality [3]: u made in principle state non-destructive. Such an ideal q classical measurement has a quantum analogue called : 1 1 2 v quantum non-demolition (QND) measurement [2]. The F G + (d 1) G . i QND measurement is non-destructive in the sense that r − d+1 ≤r − d+1 s − (cid:18)d+1 − (cid:19) X is preserves probabilistic distribution of so called non- (1) r a demolition variable of the measuredsystem andsimulta- neously, the measurement results give a perfect copy of The inequality sets a tightest bound between the mean the non-demolition variable statistics. From this point fidelity G of estimation of an unknownstate from a gen- of view, they can be used as a perfect distributor of in- eral deterministic quantum operation on a single qudit formation encoded in the non-demolition variable of a and the mean fidelity F of the state after the operation. quantumstate. Allnoisearisinginthemeasurementpro- Particularlyimportantarequantumoperationsthatsat- cess is transfered to the complementary variables. The urate the inequality (1). Namely, these operations intro- present work is devoted to (1) the analysis of the fun- duce the least possible disturbance to the original state damental property of the QND measurement and (2) to in the sense that for a given estimation fidelity G they the feasibleapplicationoftheQNDmeasurementforop- provide the highest possible output fidelity F. timaldistributionofinformationencodedinanunknown Inthisarticleweshowgenerallyforaquditthataper- system variable. fectQNDmeasurementrandomlyperformedalongallthe basis in the Hilbert space which is controlledby a quan- Suppose Alice is given a d-level quantum system S tum state of ancilla saturates the inequality (1). In par- (qudit S) in an unknown pure state ψ and she sends ticular,suchtheQNDmeasurementforasinglequbitcan S | i thestatetoBob. SupposethereisEvebetweenAliceand be implemented by the basic controlled-NOT (CNOT) Bob that wants to guess this state whereas disturbing it operation. The perfect QND measurement means that to the least possible extent. For this purpose Eve can Evehasaperfectcopyofstatisticsofthe non-demolition measure the state directly by a projective measurement variable. Further,wediscussindetailanimperfectQND 2 M where α and β are positive real numbers satisfying the A G normalization condition α2 + β2 + 2αβ/√d = 1. The interaction U transforms the state as QND S S d d IN T T−1 OUT ψ τ U c a αµ + β µ . (5) S A i i S i A j A F | i | i → | i  | i √d | i  i=1 j=1 X X   Then, the ancilla is measured in the basis µ d . If {| iiA}i=1 the ancilla is found in the state µ then Eve prepares r A RND | i her qudit E in the state a . The proposed scheme r E | i realizes a general quantum operation on qudit S that FIG.1: Theschemeof optimal measurement with aminimal canbe describedby asetofoperatorsAr, (r =1,...,d). disturbance: QND–QNDinteraction,T–twirlingoperation, In the basis {|aiiS}di=1 the operators are represented by RND– random-numbergenerator, M – state discriminator. diagonal matrices with elements β (A ) = αδ + δ , (6) measurement and we compare influence of different dis- r ij ir √d ij (cid:18) (cid:19) criminations ofancillary states. Finally, we shortly sum- where i,j = 1,...,d and δ is the Kronecker sym- marizefeasibleexperimentalimplementationsofthepro- ij bol. Since the operators satisfy the resolution of unity posed measurement scheme and discuss open problems d stimulating future research. i=1A†rAr = 11 the operation is deterministic and one Our protocol depicted in Fig. 1 consists of two steps. can use the following formulas [3]: P At the outset we consider the protocol without twirling d operationsT andT−1. InthefirststepthequditS inan F = 1 d+ TrA 2 , (7) unknown state ψ S is coupled by the two-qudit unitary d(d+1) | r| ! | i r=1 interactionU toanotherancillaryquditA. Inthesecond X d step the informationabout the state ψ is gained from 1 asuitableprojectivemeasurementon|thieSancillaanditis G = d(d+1) d+ Ehar|A†rAr|ariE!. (8) r=1 convertedinto the state ofanotherqudit E. Inorderthe X operationtosaturatetheinequality(1)theinteractionU Hence, one obtains using Eq. (6) that must satisfy two following conditions. First, there must 2 2 exist a normalizedvector µ A of the ancilla A such that 1+ α+√dβ 1+ α+ β | i √d F = , G= . (9) d (cid:16)d+1 (cid:17) (cid:16)d+1 (cid:17) U ψ µ c a µ , (2) S A i i S i A | i | i → | i | i Substituting finally these mean fidelities back into the i=1 X inequality (1) we find that they saturate the inequal- where a d and µ d are the orthonormal ity. This means that the fidelities (9) lie for any state {| iiS}i=1 {| iiA}i=1 bases in the state spaces of the qudit S and A, respec- (4) of ancilla on the very boundary of the quantum me- tively, and c = a ψ . The interaction (2) represents chanically allowed region defined by the inequality (1). i S i S h | i aperfect QNDcoupling. Second, theremustexista nor- Moreover, by changing continuously the parameter α malized vector κ that “switches off” the interaction in this state from 1 to 0 one can continuously move A | i U, i. e. along the whole boundary from its one extreme point (Gmax,Fmin)=(2/(d+1),2/(d+1))toitsotherextreme U 1 d point (Gmin,Fmax) = (1/d,1). Up to now we have con- ψ κ ψ µ . (3) S A S i A sidered the device in Fig. 1 without twirling operations | i | i →| i √d | i Xi=1 T andT−1. Suchaschemeisnotuniversalastheoutput Obviously, performing the measurement on the ancilla state fidelity f = ψ ρout ψ is dependent on the input h | | i afterthe transformation(3)inthe basis µ d gives state ψ . Inordertoobtainauniversaldevicewherethe no information on the input state. Now{|wiieAs}hi=ow1 that fidelit|y if is state independent and therefore f = F it is if an unitary interaction U satisfies the two conditions sufficient to place the QND interaction in between two then it can be used to construct a quantum operation twirling operations as is depicted in Fig. 1 [4]. that saturates the inequality (1). Moreover, the flow of Interestingly, the controllable optimal quantum oper- information between the qudit E and the original qudit ation (6) can be implemented using the qudit CNOT S can be controlled by the preparation of the ancilla A. gate UCNOT defined as UCNOT i S j A = i S i j A Let us assume that the ancilla is prepared in the super- where ⊕ denotes addition modu|loid|aind {|ii|Si,A|}di=⊕1 aire position chosen sets of basis states of qudits S and A [5]. For the CNOT gate the relevant states of the ancilla satis- τ =αµ +β κ , (4) fying the conditions (2) and (3) are µ = 0 and A A A A A | i | i | i | i | i 3 κ = (1/√d) d i , respectively. Eve then mea- |suirAes the ancilla ini=t1h|eibAasis i d and prepares the P {| iA}i=1 1,0 state r if she finds the ancilla in the state r . No- E A | i | i F tice, that optimal quantum operation saturating the in- 0,9 equality(1)canbealternativelyrealizedviateleportation scheme with two entangled ancillas [3]. 0,8 Aspecific featureofthe quantumoperation(6)isthat it is diagonal in the basis a d and thus it pre- G serves these basis states. {T|hieirSe}foi=re1, our scheme can F, 0,7 be interpreted as the QND measurement on the qudit S of some observable (non-demolition variable) = 0,6 G A d a a a withnon-degenerateeigenvaluesa . In facit=,1thie| QiiNSSDhmi|easurement preserving the basis stiates 0,5 0,0 0,2 0,4 0,6 0,8 1,0 Pa d canbeimplementedwithamoregeneralclass {| iiS}i=1 O of two-quditunitary interactions. To illustrate this, sup- poseEveisgivenanunitaryinteractionU satisfyingonly FIG. 2: The dependence of the fidelities F and G on the the condition(2) where,in addition, the states ofancilla overlap O=|hµ1|µ2i|2. µ d are in generalnonorthogonal. Clearly, in this {| iiA}i=1 case Eve’s best strategy is to discriminate among these states and since she has no a priori information about Pe,min = 1 1 µ1 µ2 2 /2 [6].(In this formula the occurrence of the states (the complex amplitudes c − −|h | i| i are unknown) the states have equal a priori probabili- and in wh(cid:16)at folplows we drop t(cid:17)he subscript A in states µ d ). Hence, using the formula and Eq. (10) one ties. In order to preserve the deterministic character of {| iiA}i=1 obtains the output fidelity and the estimation fidelity in her operationwhen in each run of the protocol the mea- the form surementontheancillauniquelydetermineswhichofthe banasaismsbtiagtueosu{s|adiiisEcr}idim=i1naistitoonb[6e]porfetphaersetdatsehse hµas to.uIsne F = 2+|hµ1|µ2i|cosφ, (11) {| iiA} 3 this approachshe applies a generalizedmeasurementΠ , d i 3+ 1 µ1 µ2 2 i = 1,...,d (Πi ≥ 0, i=1Πi = 11) on the ancilla A dis- G = −6|h | i| , (12) criminating among these states and prepares the state p P a if she detected Π . Making use of the Eqs.(7) and | riE r where φ = arg µ1 µ2 . Substituting now the fidelities (8) one then finds that h | i into the inequality (1) for d = 2 one finds that they are saturated only if φ = 2kπ, where k is an integer. The F = 2+ d1 di6=j=1Ahµi|µjiA, G= 2−Pe, (10) undesirable phase φ can be removed and thus this opti- d+1 d+1 malityconditioncanbeestablishedbyapplyingthephase P shift a1 S a1 S and a2 S e−iφ a2 S onBob’squbit where P = 1 (1/d) d µ Π µ is Eve’s error after|theiin→ter|actiionU.|Asiac→onsequ|encie,anytwo-qubit e − r=1Ah r| r| riA rate. The present quantum operation apparently pre- unitaryinteractionU satisfyingthecondition(2)supple- serves the basis statesPa d and therefore it can be mentedbythesuitableadditionalphaseshiftonaqubitS {| iiS}i=1 again interpreted as a QND measurement. The ques- canbeusedfortheQNDmeasurementthatsaturatesthe tionthatcanbe riseninthis contextiswhether alsothis inequality (1) for d=2. If, however,the states µ 2 {| ii}i=1 QND measurement allows to gain maximum possible in- are nonorthogonal,it is not possible to achieve the max- formationontheinputstatesimilarlyasinthepreviously imumpossible estimationfidelity Gmax =2/3andthere- discussed scheme. The Eq. (10) reveals that this is not fore one cannot move along the whole boundary of the thecaseassoonasthestates µ arenonorthogonal. regiondefinedbytheinequality(1)ford=2bychanging i A {| i } Namely, Eve’s error rate is always greater than zero for the state µ of the ancilla.The dependences of the fi- A | i nonorthogonal states, i. e. P > 0, and therefore Eve’s delitiesF andGgiveninEqs.(11)and(12)whereφ=0 e estimation fidelity will be always less that the highest on the overlap O= µ1 µ2 2 is depicted in Fig. 2. |h | i| possible value Gmax =2/(d+1) [7]. Another interesting However,themaximumpossibleestimationfidelitycan question is whether and when the fidelities (10) satu- beachieved,atleastonasuitablesub-ensemble,ifEvere- ratethe inequality(1). Since the fidelityF inEq.(10)is placestheambiguousdiscriminationbytheunambiguous independentofthemeasurementontheancillathecorre- discrimination [9] of the states µ 2 . This strategy {| ii}i=1 sponding estimation fidelity G will be maximized if Eve can be used if the states are linearly independent and performs optimal discrimination of the ancillary states it allows to discriminate them perfectly with a certain µ d that minimizes the error rate P . For d > 2 probability P of inconclusive result. The correspond- {| iiA}i=1 e I theminimalerrorratecanbefoundonlynumerically[8]. inggeneralizedmeasurementhasthreecomponentsΣ1 = However, for qubits (d = 2) the minimal error rate can µ⊥2 µ⊥2 /(1+ µ1 µ2 ), Σ2 = µ⊥1 µ⊥1 /(1+ µ1 µ2 ) | ih | |h | i| | ih | |h | i| be calculated analytically and it is given by the formula (|µ⊥i i is orthogonal state to the state |µii) and Σ0 = 4 2 11− i=1Σi (Σi ≥ 0). The component Σ0 corresponds optimal fidelity measurement can be proposed, for ex- to the inconclusive result and this measurement is opti- ample, for qudit with d = 4 as it was suggested in our P malinthesensethattheprobabilityP attainsminimum previous work [18]. Second, the effect of ambiguous dis- I possible value PI,min = µ1 µ2 . Apparently, if Eve de- criminationofancillarystatesoutgoingQNDinteraction |h | i| tects the conclusive result Σ , i =1,2 then she prepares has been discussed. As we know there is still no bound i the state a . Therefore, on the sub-ensemble corre- onmaximalsuccessrateforthiskindofmeasurementon i E | i sponding to the conclusive (C) results Eve prepares the a quantum system. It is an open question if such the stateρ = 2 c 2 a a forwhichthemeanes- optimal measurement can be also based on the QND in- timatioEn,Cfidelityi=a1c|hiie|v|esiimEaExhimi|um possible value, i. e. teraction. Third, in fact we decomposed optimal fidelity GC = GmaxP= 2/3. On the same sub-ensemble, Bob measurementwith minimaldisturbance for asingle copy receives the same mixed state as prepares Eve whence into two steps: programmable QND coupling and dis- F = 2/3. The obtained result clearly illustrates that crimination of ancillary states. Thus QND interaction C if Eve uses unambiguous discrimination of the states is not only optimal for accessing information encoded in µ 2 than in cases when she detects the conclusive single preferred basis but also it is optimal for universal r{e|suiilt}i=sh1e is able to obtain the best possible estimate measurementwithoutapreferredbasis. Thedifferenceis of the state ψ even if the QND interaction is imper- only in the twirling operation which effectively changes S fect and enc|odies the information on the state into the the preferred basis. For many identical copies of input nonorthogonalstates of ancilla. state it is an open question if optimal fidelity measure- mentisbasedonthe samemethod. Canbe this strategy To experimentally test this peculiar property of the used to approachoptimal fidelity measurementon many QND interaction, the experimentalists can use a recent identical copies [19] ? At the end, the problem of the progress in the implementations of the CNOT gates be- quantumcomplementarityanderasurefortheQNDcou- tween the atoms and photons. Due to a possible long pling is closely related [20, 21]. It is known that perfect time for manipulations of qubits (represented by long- two-qubit QND coupling with arbitrary pure ancillary lived electronic states) and high efficiency of state de- state is perfectly reversible if Eve implements an appro- tection, trapped and cooled ions are ideally suited for priate measurement and Bob performs according to the implementations of quantum operations. A single-ion measurement results an appropriate unitary operation. CNOT gate hasbeen realizedsome time agoin[10]. Re- cently, a two-ion CNOT gate based on 40Ca+ ions in a The remaining problem to be discussed is also how the imperfectQNDcouplingcanbereversedusingonlylocal linear Paul trap which were individually addressed us- operations and classical communication. ingfocusedlaserbeamshasbeenimplemented[11]. Also two-ion π-phase gate demonstrated with 9Be+ ions in a In this article, a fundamental property of the QND harmonic trap [12] can be used for the same purpose. measurement came to light: performing QND measure- Further, probabilisticCNOT gates,where the qubits are ment randomly along all basis in Hilbert space of the destroyed upon failure, have been experimentally tested system the tightest bound (1) between the estimation fi- in optical systems. Despite of the fact that these CNOT delityGandtheoutputfidelityF ofthemeasuredsystem gates for polarization qubits [13] and path qubits [14] canbesaturated. Tochangeoptimalratiobetweenthese arenotdeterministictheyaresufficienttoexperimentally fidelities it is sufficient to controlthe ancilla of the QND prove the fundamental trade-off between the estimation measurement. Even when the used QND measurement fidelity G and the output fidelity F. To achieve univer- is not perfect we can still optimally control the fidelity salcharacterofdisturbanceintroducedintothemeasured trade-off but only in a restricted range if the output an- statemutuallyinversetwirlingoperations[15]havetobe cillary states are optimally ambiguously discriminated. implemented on qubit S before and after the QND in- These results are not only important from the funda- teraction (see Fig. 1). In summary, in all the mentioned mental point of view but they can be used to distribute experimentalimplementations ofthe CNOT gatesthe fi- information carried by quantum state without any pre- delity trade-offcan be thus directly experimentally mea- ferred basis. sured. There are few important and interesting consequences that have to be noticed. They can stimulate broad dis- Acknowledgments cussion and future work. First, our result shows that it is allowed to achieve any optimal fidelity measurement withaminimaldisturbanceby”programming”theQND We would like to thank J. Fiura´ˇsek and M. Jeˇzek for interactionbyasingleprogramancillarystate. Thisisan stimulatingdiscussions. Theresearchhasbeensupported interesting result in the context of a previous proof that by Research Project ”Measurement and information in it is not possible to programme any single-qubit unitary Optics”ofthe CzechMinistry ofEducation. We alsoac- operationandmeasurementusingonlyasinglequbitpro- knowledge partial support by the Czech Ministry of Ed- gram [16, 17]. More generally, any optimal fidelity mea- ucation under the project OCP11.003 in the framework surement of a qudit can be programmed by an ancilla of the EU grant COST P11. R. F. thanks support by withthed-dimensionalHilbertspace. Anetworkforthis Project 202/03/D239 of the Grant Agency of the Czech 5 Republic. [1] C. H. Bennett et al., in Proceedings of the IEEE Inter- [11] F. Schmidt-Kaler et al., Nature422, 408 (2003) national Conference on Computers, Systems, and Sig- [12] D. Leibfried et al., Nature422, 412 (2003). nalProcessing,Bangalore,India(IEEE,NewYork,1984), [13] K. Sanaka et al., Phys. Rev. A 66, 040301 (2002); T. pp.175-179. B. Pittman et al., Phys. Rev. A 68, 032316 (2003); S. [2] V.B. Braginsky et al., Science 209, 547 (1980). Gasparoni et al., Phys. Rev. Lett. 93, 020504 (2004); Z. [3] K.Banaszek, Phys. Rev.Lett. 86, 1366 (2001). Zhao et al., arXiv:quant-ph/0404129. [4] R.F.Werner,Phys.Rev.A58,1827(1998);J.Fiuráˇsek, [14] J.L.O’Brienetal.,Nature426,264(2003);G.J.Pryde Phys.Rev.A 70, 032308 (2004). et al., Phys.Rev.Lett. 92, 190402 (2004). [5] G. Alber et al., arXiv:quantum-ph/0102035. [15] C. H.Bennett et al., Phys. Rev.Lett. 76, 722 (1996). [6] C. W. Helstrom, Quantum Detection and Estimation [16] M.A.NielsenandI.L.Chuang,Phys.Rev.Lett.79,321 Theory (AcademicPress, New York,1976). (1997). [7] D. Bruß and C. Macchiavello, Phys. Lett. A 253, 249 [17] M.DuˇsekandV.Buˇzek,Phys.Rev.A66,022112(2002). (1999). [18] L. Miˇsta Jr. and R.Filip, arXiv:quant-ph/0407017. [8] M. Jeˇzek et al., Phys. Rev.A 65, 060301 (2002). [19] K. Banaszek et al., Phys.Rev. A 64, 052307 (2001). [9] I.D.Ivanovich,Phys.Lett.A123,257(1987);D.Dieks, [20] B. G. Englert, Phys.Rev.Lett. 77, 2154 (1996). Phys. Lett. A 126, 303 (1988); A. Peres, Phys. Lett. A [21] T. C. Ralph et al., arXiv:quant-ph/0412149. 128, 19 (1988). [10] C. Monroe et al.,Phys. Rev.Lett. 75, 4714 (1995).