Table Of Content

Optimal minimum-cost quantum measurements for imperfect detection Erika Andersson SUPA, Department of Physics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom (Dated: January 4, 2012) Knowledge of optimal quantum measurements is important for a wide range of situations, including quantum communication and quantum metrology. Quantum measurements are usually optimised with an ideal experimental realisation in mind. Real devices and detectors are, however, imperfect. This has to be taken into account when optimising quantum measurements. In this paper, we derivethe optimal minimum-cost and minimum-error measurements for a general model of imperfect detection. PACSnumbers: 03.65.Ta,03.67.-a,42.50.Dv 2 1 0 I. INTRODUCTION of the detector [18]. This is not self-evident, since an 2 amplifier will add noise, which could degrade the perfor- n Quantum measurements may be optimized with re- mance of a measurement. a spect to a range of criteria. For example, when distin- In this paper, we will derive the optimal minimum- J 1 wguitishhpinrgiorbeptrwoebeanbialitsieestpof,qwueanmtauymwsatnattetsoρmj,inoimcciuserrtinhge cporsotcemsseaissunroetmpeenrftecfot.rItthteurcnasseouwthtehnatthwehefinnadlisdtientgecutiisohn- j average error in the result, or, somewhat more gener- ing between two quantum states, the optimal measure- ] h ally, the average cost [1–3]. Other possibilities are to ment strategy we should aim for remains the same as in p maximise the information contained in the result, or to the ideal case, although the cost changes. For three or t- askthatmeasurementresultsshouldbeunambiguous[2]. more states, the optimal measurement strategy in gen- n The optimal measurement is often a generalized quan- eral changes, and we give an example of this. We finish a tum measurement. Thesegobeyondstandardprojective with a discussion. u q quantum measurements, and are referred to as proba- [ bility operator measures (POMs) or positive operator- valued measures (POVMs) [1–3]. How to optimally per- II. GENERALIZED QUANTUM 1 formaquantummeasurementisrelevantforawiderange MEASUREMENTS v of applications, from quantum communication including 7 8 quantum key distribution [4], to parameter estimation Generalized quantum measurements (POMs) can be 3 and quantum metrology, such as for optimal quantum realised in terms of a projective measurement in an ex- 0 estimation of the Unruh-Hawking effect [5]. tended Hilbert space [1, 3]. This is used in both exist- 1. Derivations of optimal quantum measurement strate- ingandsuggestedexperimentalrealisations. TheHilbert 0 gies have, however, aimed at finding the optimal ideal space of the quantum system ρS to be measured can 2 quantummeasurement,thatis,thebestthatcanbedone either be extended through a direct sum, e.g. by us- 1 providedthatanactualexperimentalrealisationis ideal. ing extra atomic levels or adding more optical paths, or : v Anadvantageisthatthetheoreticalderivationoftheop- through a tensor product by coupling ρS to an auxiliary Xi timal ideal measurement can be separated from working quantumsystemρA. Broadlyspeaking,thenumberofdi- out how to do the experimental realization. The same mensionsinthetotalextendedHilbertspacecorresponds r theoretically optimal measurement may be realised in tothenumberofoutcomesofthemeasurement. Lesswell a very different ways, depending on the character of the knownis thatbyrealisingthe measurementsequentially, quantum system to be measured. Generalized quantum itisalsopossibletolimitthetotalnumberofdimensions measurementshavebeenrealizedonphotonpolarisation, needed at any one time to d+1, where d is the dimen- see e.g. [6–13], and very recently on NV-centres in dia- sion of ρ [19]. We can also realise the measurement by S mond [14]. They could also be realized on ions or atoms usingatmost2ddimensions,bycouplingρ toanauxil- S with existing experimental tools [15–17]. iaryqubit,measuringthequbit,andrepeating[20]. This Real experiments are of course not ideal. One might realisation is the most efficient in the sense that fewer askwhetherimperfectionsindevicecomponentschanges operations are needed on average. thestrategythatweshouldaimtoimplement. Thisques- Any ideal experimental realisation of a generalized tion has largely been overlooked. It turns out that the quantum measurement will thus employ a final projec- measurementweshouldaimtoimplementdoesindeedin tive measurement in some basis on some quantum sys- generaldependonthe particularpropertiesoftheexper- tem. The final projective measurement is preceded by imental devices used. An indication that this might be a unitary transform in the extended Hilbert space. For thecaseisgivenbyunambiguouscomparisonofquantum example, for a measurement on an atom or ion, we may states, where for detectors with less than unit efficiency, couple the levels of the atom or ion to some additional it is sometimes advantageousto use an amplifier in front atomiclevelse.g. usinglaserpulsesorpassagethrougha 2 cavity [15, 17], followedby a final measurementofwhich able to implement might be described by the measure- energyleveltheatomorionoccupies. Duetoexperimen- ment operators tal constraints, the final measurement is restricted to be ∞ a projection in the energy eigenbasis. If making measurements on a photon, the final measurement might be Πno click = Xq(0|n)|nihn| adetectionofwhichpaththephotonexitsfrom,andwith n=0 ∞ whichpolarisation,inasuitablepolarisationbasis[7–13]. Πclick = Xq(1n)n n, (2) Thisfinalmeasurementisprecededbyanopticalnetwork | | ih | n=0 which includes wave plates and beam splitters, effecting a unitary transform in the extended space. where n is a photon number state. This measurement Formally,anygeneralizedmeasurementstrategyis de- thus ha|sionly two outcomes, and q(0n) and q(1n) are scribed by a set of measurement operators Πi, acting in theprobabilitiestoobtain“noclick”an|da“click”r|espec- the space of ρS, the system to be measured. If the sys- tively,whentherewherenphotonspresent. Asimilarde- tem is prepared in a state ρj, then the probability to scription results for photon-number-resolving detectors, obtain outcome i is given by p(ij) = Tr(Πiρj). The and also when a number of photodetectors are used to | fact that the sum of the probabilities for all outcomes is detect whatpathor with whatpolarisationa photonex- equal to 1 corresponds to the condition PiΠi = 1, and its. Yetanotherexampleisthedetectionofthestateofa the fact that probabilities for all outcomes are nonnega- Rydberg atomby field ionization. By detecting for what tive correspondsto Πi 0 (and consequently Πi have to field strength the atom is ionized one can infer what en- ≥ be Hermitian). Note that we may have more outcomes ergylevelitmostlikely wouldhavebeen in[17,21]. The thanwehavedimensionsinthespaceofthesystemtobe distribution functions for when the electron is released measured, and that Πi need not be orthonormal. Each may however overlap for different energy levels. This measurement operator Πi is obtained by taking the cor- also results in a measurement of the type in Eq. (1). responding projective measurement operators in the ex- Thereisaminortechnicalpointwhichshouldbemen- tended Hilbert space and projecting them onto the sub- tioned in order to further justify the generality of our space of ρS. error model. If the measurement operators Πj are pro- portionaltopurestateprojectors,thenthefinalmeasure- mentintheextendedspacecanbechosenasaprojection III. IMPERFECT DETECTION in a complete basis,with eachoutcome correspondingto onepurebasisstate j . Ourmodelforerrorsinthefinal Sources of error in the experimental realisation can measurement is then| niatural. The measurement opera- now be divided in two categories, errors in the unitary tors Π may however also be mixed. This is of course j transform preceding the final measurement, and errors possible to realise using a final measurement comprising in the final projective measurement. In this paper, we projectors onto more than one orthonormal state. How- will consider the latter type of errors. Errors in the uni- ever, the index j in q(ij) most naturally refers to pure- tary transform are of course also likely to occur in any state projectors j j i|n the final measurement, rather realization, and this will be the subject of future work. thantoprojector|soihnt|omorethanoneorthonormalstate. In particular, suppose that when a perfect projective Nevertheless,alsoformixedΠ ,wemayarrangeforthe j measurement would have given result j, then our mea- corresponding final measurement operator to be a pure surementwillinsteadgiveresultiwithprobabilityq(ij), state projector in an extended space. This means that | wtichemreatPrixi.q(Ti|hji)s=is1a;vqe(riy|jg)eanreerathlleyealpepmliecnatbsleofdaessctroipchtiaosn- pthreoadbeislictriiepstiinonthuesfiinngalqm(ie|ja)sutroemdeesnctriibseagmaiisnidneanttuirfiacla.tFioonr of errors in the final measurement. We may be able to example, suppose that the measurementin the extended vary the unitary transform that precedes the final mea- Hilbert space is described by the projectors 1 1 + EE surement, but the final measurement and the q(ij) are 2 2 and 3 3. We may then further|coiuplhe t|he oftenfixedbythenatureofthechosendetectionpr|ocess, |syisEteEmh |to a q|ubiEitE,hin|itially in the state 0 , using e.g. q including the efficiency of the detectors, and what basis the unitary operation | i states we must project on. Any non-ideal measurement weareabletoimplementwillthenbedescribedbymixed U = 1 1 0 0 + 2 2 0 0 EE qq EE qq | i h |⊗| i h | | i h |⊗| i h | measurement operators + 1 1 1 1 + 2 2 1 1 EE qq EE qq | i h |⊗| i h | | i h |⊗| i h | + 3 3 1 0 + 3 3 0 1. (3) Πi =Xq(ij)Πj, Xq(ij)=1, (1) | iEEh |⊗| iqqh | | iEEh |⊗| iqqh | e | | j i Thefinalmeasurementmaythenbe realizedasa projec- where Π are the measurement operators for an uncon- tivemeasurementonthequbit,with 0 0 correspond- j qq | i h | strainedgeneralizedmeasurementstrategy,acting in the ing to 1 1 + 2 2 and 1 1 corresponding to EE EE qq | i h | | i h | | i h | space of ρ . 3 3. This procedure is easily generalized so that S EE | i h | Forexample,whentryingtoimplementaphotonnum- any mixed measurementoperatorsΠ will correspondto j ber measurement, the measurement that we are actually a pure state projector in some extended space. Related 3 to this, one realizes that it may well be advantageous to costs C and modified risk operators W . It is also jk j arrange to use as few final measurement states as possi- clear theat if the states ρ are orthogonalf, corresponding j ble. Any extra unitary transforms are of course likely to to “perfectly distinguishable” classical states, then the introduce experimental errors. But in this initial work measurement in the presence of misidentification prob- we wantto optimize with respectto imperfections in the abilities does not change; it remains a projective mea- final projective measurement only. surementon(thesubspacesof)thedifferentρ . Thefact j that the measurement strategy changesis in this sense a quantum feature. IV. MINIMUM-COST MEASUREMENTS FOR We can freely choose which final measurement states IMPERFECT DETECTION are assigned to which initial state ρ , and should choose i sothattheobtainedcostisoptimal. Thiscanbedoneby Wewillnowderivetheoptimalminimum-coststrategy checking what the optimal measurement is for each pos- when the final measurement we can realize is restricted, sible assignment and picking the best one; there are m! so that the measurementoperatorsaregivenby Eq. (1). assignments if there are m basis states. Roughly speak- The measurement we are actually realizing is described ing,the mostprobableinitialstatesshouldbe associated by the measurementoperatorsΠ , whereasthe measure- with those basis states which we can identify most ac- i ment we are aiming to realize eis described by the mea- curately. Related to this, if the final detection process surement operators Π . is defective enough, then for some assignments it may j To briefly review results related to optimal minimum- happen that when a final outcome i is obtained, this is cost measurements [1], suppose that quantum state ρ morelikely tohaveoccurredasaresultofanotherinitial j occurs with prior probability pj. We choose a measure- state ρj than the initial state ρi itself. As a particu- ment with measurement operators Π , and obtaining re- larly simple example, consider distinguishing with min- i sult i when the state prepared was actually ρ carries a imum error between the orthogonal states 0 and 1 , j | i | i cost of Cij. The average cost will then be given by occurring with probabilities p0 and p1. We make an imperfect projection in the 0 , 1 basis, with Π0 = C¯ =TrXCijpjΠiρj =TrXWiΠi =TrΓ, (4) q(00)0 0 +q(01)1 1,Π1 ={|qi(1| 0i)}0 0 +q(11)e1 1. ij i If fo|r|exiahm|ple q(|01|)iph1 |>eq(00)p0|, th|einh t|he fin|al|reishul|t | | “0”ismorelikelytohaveoccurredbecausethe statewas where 1 rather than 0 , and we should guess “1 ” even if we | i | i | i obtain result “0”. We should therefore also check what Wi =XCijpjρj and Γ=XWiΠi =XΠiWi (5) the optimal cost is for different reassignments of final j i i outcomes to other states. are called the risk operator corresponding to result i Nevertheless, checking all possible different assign- and the Lagrange operator, respectively. Γ takes care mentsofoutcomes,inordertoobtaintheoveralloptimal of the constraint PiΠi = 1, and is Hermitian. For a detection strategy for imperfect detection, is straightfor- minimum-error measurement we may choose Cij = δij ward if there is a finite number of outcomes. We will − and Wi = piρi. The minimum-cost measurementoper- proceed to look at examples. − ators satisfy the conditions (W Γ)Π =Π (W Γ) = 0 i (6) i i i i − − ∀ A. Distinguishing between two non-orthogonal W Γ 0 i. (7) i states − ≥ ∀ Consider now a mixed measurement strategy with measurement operators given by Eq. (1). The average Suppose that we want to distinguish between ρ0 cost for this measurement strategy will be and ρ1, occurring with prior probabilities p0 and p1, with minimum cost, with misidentification probabilities Ce = TrXWiΠei =TrXCikpkρkq(i|j)Πj iq(w0h|0e)n,qt(h0e|1p)r,eqp(a1r|e0d),sqt(a1t|e1)w.aTshjeiscCost,ofofroib,tjai=ni0n,g1r.eTsuhlet i ijk ij ideal measurement strategy we should try to implement = TrXCjkpkρkΠj =TrXWjΠj, (8) is optimal for the modified risk operators e f jk j where W0 = C00p0ρ0+C01p1ρ1, f e e W1 = C10p0ρ0+C11p1ρ1. (10) Cjk =XCikq(ij) and Wj =XCjkpkρk. (9) f e e e | f e i k Theoptimalmeasurementfordistinguishingbetweentwo It immediately follows that the ideal strategy we should non-orthogonal states with minimum cost was given by aimtoperform,ifthefinalmeasurementhasthemisiden- Helstrom [1]. It is a projection in the eigenbasis of the tification probabilities q(ij), is optimal for the modified operator O = W0 W1. Using the first equation in Eq. | e f −f 4 (9) and the second equation in Eq. (1), we find that measurementstrategyhas threemeasurementoperators. When q increases, it pays less and less try to identify O =[p(11)+p(22) 1](W0 W1). (11) ψ1 , and the trace a of Π1 = aρ1 decreases, starting e | | − − |fromi 2/3. At the same time, Tr Π2 = Tr Π3 increases. This means that unless p(1|1) + p(2|2) = 1, the mea- Π2,3 are unnormalised projectors onto pure states that surement strategy we should aim to implement does not becomecloserandcloserto =1/√2(0 1 ). When changewhendetectionisimperfect. Ifp(11)+p(22)>1, |±i | i±| i | | q qc = (1+1/√3)/2 0.211, a = 0 and the optimal thentheoptimalΠ0 isaprojectorontotheeigenstatesof m≥easurement has only t≈wo non-zero measurement oper- tOeheweiitghennsetgaatteisvewietihgepnovsaitluivees,eaignednvΠa1luiess.aIpfrtohjeercetaorreoanntyo ators. Then Π1 = 0, and Π2,3 are projectors onto the states . This remains optimal for all q q 1/2. c zeroeigenvalues,thenthecorrespondingeigenstatesmay |±i ≤ ≤ Thus, for any non-zero value of q, the measurement we beassignedtoeitherresultwithoutchangingtheaverage should aim for is different from the optimal minimum- cost. Iftheresultsaresufficiently“scrambled”,morepre- error measurement for q =0. cisely, when q(11)+q(22) < 1, we should still perform | | The case we have considered, distinguishing between the samemeasurement,but with Π0 and Π1 swapped. If three or more symmetric states, is relevant for quan- q(11)+q(22)=1, then this implies q(22)=q(21) and | | | | tum key distribution (QKD), see e.g. [24] and references q(11) = q(12), i.e. that the measurement results are | | therein. As argued above, our detection error model di- completely random. There is then no point in making rectly applies to photodetection. In a realisation of sim- a measurement at all. We should just pick the ρ which i ilar QKD protocols, it is therefore likely that optimal gives the least averagecost basedon the prior probabili- operationwouldrequiresimilarmodificationsofthemea- ties and costs C . ij surements performed. The minimum cost, given in Eq. (8), does of course increase. These results also hold for the special case of distinguishing between ρ0 and ρ1 with minimum error. The strategy we should try to implement stays the same,buttheerrorprobabilityincreases. Thatthisholds V. CONCLUSIONS even when the misidentification probabilities q(ij) are | not symmetric is not entirely intuitive. Knowledgeofoptimalquantummeasurementsforreal- isticexperimentalcomponentsisimportantinordertobe B. Distinguishing between three symmetric states able to selectthe best possible measurements for a given situation. Inthiswork,wederivedtheoptimalminimum- We will now see that when distinguishing between cost measurement one should aim to implement for a threepurequantumstates,themeasurementstrategywe general model of imperfect detection. When a perfect should aim for may change when the detection is imper- measurement would have given result j, then the detec- fect. Consider the three equiprobable states tiongivesresultiwithprobabilityq(ij). Thisleadsto a | modificationofthecostsofdifferentoutcomes,andhence 1 1 the measurement strategy we should aim to implement ψ1 = 0 , ψ2 = (0 +√3)1 , ψ3 = (0 √3)1 . | i −| i | i 2 | i | i | i 2 | i− | i in general changes. In the special case of distinguish- (12) ing between two quantum states,the measurementstays The ideal measurement that distinguishes between these the same, and only the cost changes. For three states, states with minimum error has the measurement opera- we gave a simple example, relevant e.g. for quantum tors Πi =2/3ψi ψi =2/3ρi for i=1,2,3 [1, 2, 8]. key distribution, where the optimal measurement strat- | ih | Suppose now that in the final detection, outcome 1 egy changes significantly if the detection is imperfect. is sometimes misidentified as outcome 2 or 3 with equal Itwouldbeinterestingtoinvestigatehowimperfectde- probabilityq,butthatotherwisethedetectionisperfect. tection affects other types of measurements, such as un- That is, we have q(11) = 1 2q and q(21) = q(31) = | − | | ambiguousor error-freeeasurements[2]. If the detection q, with 0 q 1/2. Also, q(22) = q(33) = 1 and ≤ ≤ | | is imperfect, then it may not be possible to distinguish q(12) = q(32) = q(13) = q(23) = 0. We find C11 = | | | | e somestatesunambiguouslyanymore. Weshouldthenin- 2q 1,C12 = C13 = q,C22 = C33 = 1, and all other steadconsidera maximumconfidence measurement[25]. − e e − e e − Cij =0. Furthermore, Also, errors in an experimental realisation will not only e comefromimperfectionsinthefinaldetection,butneces- 1 W1 = [(2q 1)ρ1 q(ρ2+ρ3)], sarily also from errors in operations on the system to be f 3 − − measured. Finding optimal measuements for these cases 1 1 W2 = ρ2, W3 = ρ3. (13) will be the subject of further work. f −3 f −3 The author gratefully acknowledges useful discussions The optimal measurements for such mirror-symmetric with Mark Hillery and Mario Ziman, and financial sup- situations are known [22, 23]. For small q, the optimal port from EPSRC EP/G009821/1. 5 [1] C.W.Helstrom, Quantum detection and estimation the- Lett. 93, 200403 (2004). ory (Academic, New York,1976). [13] P. J. Mosley et al.,Phys.Rev.A 77, 012113 (2008). [2] J.A.Bergou,U.Herzog,andM.Hillery,Ch.11inQuan- [14] G. Waldherr, A. Dada, F. Jelezko, E. Andersson, and J. tumstateestimation(Springer,Berlin,2004);A.Chefles, Wrachtrup,in preparation. ibid.,Ch. 12. [15] S.Franke-Arnoldetal.,Phys.Rev.A63,052301(2001). [3] S.M.Barnett,Quantum information(OxfordUniversity [16] E. Andersson,Phys. Rev.A 64, 032303 (2001). Press, Oxford,2009). [17] A. Dada et al.,Phys. Rev.A 83, 042339 (2011). [4] S.J. van Enk,Pys. Rev.A 66, 042313 (2002). [18] C. S.Hamilton et al.,Phys. Rev.A 79, 023808 (2009). [5] M. Aspachs, G. Adesso and I. Fuentes, Phys. Rev. Lett. [19] G. Wangang M. Ying,arXiv:0608235. 105, 151301 (2010). [20] E. Andersson and D.K.L.Oi, Phys.Rev.A 77, 052104 [6] B. Huttneret al.,Phys.Rev. A 54, 3783 (1996). (2008). [7] R.B. M. Clarke et al., Phys.Rev.A 63, 040305 (2001) [21] H. Walther et al.,Rep.Prog. Phys. 69, 1325 (2006). [8] R.B. M. Clarke et al., Phys.Rev.A 64, 012303 (2001). [22] E. Andersson et al.,Phys. Rev.A 65, 052308 (2002). [9] J. Mizuno et al., Phys.Rev.A 65, 012315 (2001). [23] C.-L. Chou, Phys.Rev. A 70, 062316 (2004). [10] M. Fujiwara et al.,Phys. Rev.Lett. 90, 167906 (2003). [24] D.SychandG.Leuchs,New.J.Phys.12,053019(2010). [11] Y.Mitsumori et al.,Phys.Rev.Lett.91, 217902 (2003). [25] S. Croke et al., Phys.Rev.A 72, 052116 (2005). [12] M.Mohseni,A.M.Steinberg,andJ.Bergou,Phys.Rev.