Experimental implementation of unambiguous quantum reading Michele Dall’Arno,1,2 Alessandro Bisio,2,3 Giacomo Mauro D’Ariano,2,3 Martina Mikov´a,4 Miroslav Jeˇzek,4 and Miloslav Duˇsek4 1ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, E-08860 Castelldefels (Barcelona), Spain 2Quit group, Dipartimento di Fisica “A. Volta”, via Bassi 6, I-27100 Pavia, Italy 3Istituto Nazionale di Fisica Nucleare, Gruppo IV, via Bassi 6, I-27100 Pavia, Italy 4Department of Optics, Faculty of Science, Palacky University, 17. listopadu 12, CZ-77146 Olomouc, Czech Republic (Dated: January 10, 2012) We providethe optimal strategy for unambiguous quantumreading of optical memories, namely when perfect retrieving of information is achieved probabilistically, for the case where noise and 2 loss are negligible. We describe the experimental quantum optical implementations, and provide 1 experimental results for the single photon case. 0 2 I. INTRODUCTION talrealization. Inthishypothesisquantumreadingofop- n tical devices can be recasted to a discrimination among a J In the engineering of optical memories (such as CDs optical devices with low energy and high precision. 9 or DVDs) and readers, a tradeoff among several param- In the ideal reading of a classical bit of information eters must be taken into account. High precision in the from an optical memory, namely in the discrimination h] retrievingofinformationissurelyanindefeasibleassump- of a quantum optical device from a set of two, different p tion, but also energy requirements, size and weight can scenarios can be distinguished. A possibility is the on- - play a very relevant role for applications. Clearly, size the-flyretrievingofinformation(e.g. multimediastream- nt and weight of the device increase with the energy, and ing), where the requirement is that the reading opera- a using a low energetic radiation to read information re- tion is performed fast - namely, only once, but a modest u duces the heating of the physical bit, thus allowing for amountoferrorsintheretrievedinformationistolerable. q smallerimplementationofthebititself. Moreover,many Thisscenariocorrespondstotheproblemofminimumen- [ physical media (e.g., superconducting devices) dramat- ergyambiguousdiscriminationofopticaldevices[10–12], 2 ically change their optical property if the energy flow whereone guessesthe unknowndevice andthe taskis to v overcomes a critical threshold. minimize the probability of making an error. 5 Inthe problemofquantumreading[1–5]ofopticalde- Ontheotherhand,inasituationofcriticalityoferrors 3 9 vices one’s task is to exploit the quantum properties of and very reliable technology,the perfect retrieving of in- 5 light in order to retrieve some classical digital informa- formationisanissue. Then,unambiguousdiscrimination . tion stored in the optical properties of a given media, of optical devices [13], where one allows for an inconclu- 1 1 making use of as few energy as possible. The quantum sive outcome (while, in case of conclusive outcome, the 1 readingofopticalmemorieswasfirstintroducedin[1]. A probability of error is zero) becomes interesting. 1 realistic model of digital memory was considered, where In [2] an optimal strategy for the first scenario - : eachcelliscomposedofabeamsplitterwithtwopossible namely, the minimum energy ambiguous discrimination v i reflectivities. A single optical input is available to the of optical devices - has been provided for the ideal case. X reading device, while the other one introduces thermal Thisstrategy,thatexploitsfundamentalpropertiesofthe r noise in the readingprocess,so that the problemconsid- quantumtheorysuchasentanglement,allowsfortheam- a eredisthediscriminationoftwolossyandthermalGaus- biguous discrimination of beamsplitters with probability sianchannels. Itwasshownthat,for fixedmeannumber of errorunder any giventhreshold, while minimizing the of photons irradiated over each memory cell, even in the energy requirement. The proposed optimal strategy has presence ofnoise and loss,a quantum sourceoflight can been compared with a coherent strategy, reminiscent of retrieve more information than any classical source - in the one implemented in common CD readers, showing particular in the regime of few photons and high reflec- that the former saves orders of magnitude of energy if tivities. This provided the first evidence that the use of compared with the latter, and moreover allows for per- quantum light can provide great improvements in appli- fect discrimination with finite energy. cationsinthetechnologyofdigitalmemoriessuchasCDs In this paper we first extend the results of [2] to the or DVDs. case of unambiguous ideal quantum reading - namely, In practical implementations noise can sometimes be the minimum energy unambiguous discrimination of op- noticeablyreduced[6]. Ontheotherhand,ingeneralloss tical devices. We provide the optimal strategy for un- inherently affects quantum optical setups. Nevertheless, ambiguous discrimination of beamsplitters with proba- a theoretical analysis of the ideal, i.e. lossless and noise- bility of failure under a given threshold, while minimiz- less,quantumreadingprovidesatheoreticalinsightofthe ing the energy requirement. We show that the optimal problemandameaningfulbenchmarkforanyexperimen- strategy does not require any ancillary mode - while in 2 the presence of noise and loss ancillary states improve number operator on . In the following, for any pure H the performance of the quantum reading setup [1, 14]. state ψ , we denote with ψ := ψ ψ the corresponding | i | ih | Both strategies for ambiguous and unambiguous quan- projector. For any Fock space , we denote with n a H | i tumreadingreducetothesameoptimalstrategyforper- Fock basis in (0 denotes the state of the vacuum). H | i fect discrimination if the probability of error (in the for- Suppose we want to discriminate between two linear mercase)orthe probabilityoffailure(inthe lattercase) optical passive devices U and U . If a single use of the 1 2 issettozero. Then,wepresentsomeexperimentalsetups unknown device is available, the most general strategy implementing such optimal strategies which are feasible consistsofpreparingabipartiteinputstateρ ( ) ∈B H⊗K withpresentdayquantumopticaltechnology,intermsof ( is an ancillary Fock space with mode operators b ), i K preparations of single-photon input states, linear optics applying locally the unknown device and performing a andphotodetectors. Finally,wewillnoticethatintheex- bipartite POVM Π on the output state ( )ρ = x K perimental implementation of perfect discrimination the (U I )ρ(U† I ) (x can be either 1 orU2).⊗ I x⊗ K x ⊗ K noise is negligible. H U There are only a few papers reporting on the experi- ?>ρ x Π=<. (2) mentalimplementationofdiscriminationofquantumde- 89 K :; vices. Ref. [15]dealswithperfectdiscriminationbetween single-bit unitary operations using a sequential scheme. The choice of Π in Eq. (2) depends on the figure of In Ref. [16] the authors demonstrate unambiguous dis- merit taken into account. For example, for ambiguous crimination of non-orthogonal processes employing en- discrimination Π = Π1,Π2 and one’s task is to mini- { } tanglement. Inthe presentpaper wereportonanexper- mize the probability of error imental realization of the perfect quantum-process dis- P (ρ,U ,U ):=Tr[( )(ρ)Π +( )(ρ)Π ], crimination optimized with respect to the minimal en- E 1 2 U1⊗IH 2 U2⊗IH 1 ergy flux through the unknown device. with 0 P (ρ,U ,U ) 1/2. When p = p = 1/2 the E 1 2 1 2 ≤ ≤ The paper is organized as follows. First, in Section II minimalprobability oferrorhas beenprovento be given we provide general results for the unambiguous discrim- by the following function [18] of ρ, ination of optical devices (Section IIA) and the optimal 1 strategy for the unambiguous discrimination of beam- PE(ρ∗,U1,U2)= (1 (( 1 2) K)ρ 1), (3) 2 −|| U −U ⊗I || splitters (Section IIB). Then, in Section III we describe some experimental setups for the optimal ambiguous where X 1 =Tr[√X†X] denotes the trace norm. || || (SectionIIIA),unambiguous(SectionIIIB),andperfect For unambiguous discrimination Π = Π1,Π2,ΠI , { } (Section IIIC) discrimination of beamsplitters. Then, in Tr[( 1 H)(ρ)Π2]=Tr[( 2 H)(ρ)Π1]=0 and one’s U ⊗I U ⊗I Section IV, we provide the results of the experimental task is to minimize the probability of inconclusive out- implementation of the setup for perfect discrimination. come (failure) Finally, Section V is devoted to conclusions and future P (ρ,U ,U ):=Tr[( + )(ρ)Π ], (4) perspectives. F 1 2 U1⊗IH U2⊗IH I with 0 P (ρ,U ,U ) 1. F 1 2 ≤ ≤ UpondenotingwithE (ρ):=Tr[ρ(N I )]theenergy D K ⊗ II. UNAMBIGUOUS QUANTUM READING that flows through the unknown device, the total energy of the input state is E(ρ):=E +Tr[ρ(I N )]. D H K ⊗ We introduce now the ambiguous (unambiguous) A M-modes quantum optical device [17] is described quantum reading problem. For any set of two optical by a unitary operator U relating M input optical modes devices U ,U and any threshold q in the probability withannihilationoperatorsaion i,toM outputoptical { 1 2} modes with annihilation operatoHrs a′i on Hi′, where Hi oρf∗etrhraotra(fllaoiwlusreu)s, fitondamthbeigmuoinuismlyum(uneanmerbgiyguionupsulty)stdaitse- denotes the Fockspace ofthe opticalmode i. We denote criminate between U and U with probability of error the total Fock space as = . 1 2 H iHi (failure) not greater than q, namely An optical device is calledNlinear if the operators of theoutputmodesarerelatedtotheoperatoroftheinput ρ∗ =arg min E(ρ). (5) modes by a linear transformation, namely ρs.t. P(ρ,U)≤q whereP(ρ,U)=P (ρ,U)forthe ambiguousdiscrimina- a′ a A B tion problem and PE(ρ,U)=P (ρ,U) for the unambigu- (cid:18)a′† (cid:19)=S(cid:18)a† (cid:19), S :=(cid:18)B¯ A¯ (cid:19) (1) ous discrimination problem. F where S is called scattering matrix, X¯ denotes the com- plex conjugate of X, a=(a ,...a ) is the vector of an- A. Unambiguous quantum reading of optical 1 N nihilation operators of the input mode, and analogously devices a′ fortheoutputmodes. IfB =0inEq. (1)thedeviceis calledpassive andconservesthetotalnumberofphotons, The problem in Eq. (5) has been already solved in that is ψ N ψ = ψ U†NU ψ with N := a†a the [2] for the case of ambiguous discrimination. Here we h | | i h | | i i i i P 3 generalizethe resultsobtainedin[2]tothe unambiguous of the probability of errorin Eq. (3) (for ambiguous dis- discrimination problem. crimination)andtheprobabilityoffailureinEq. (4)(for First, notice that for any POVM Π we have P (( unambiguous discrimination), namely the linearity in ρ, F 1 U ⊗ EIK(ρ)ρ),,Iso,Uw2eUc1†a)n=resPtrFic(tρ,oUur1,aUn2a)lyasinsdtoEt(h(eUc1a⊗seIinK)wρh)ic=h tthheeemqounaolittoiensicPitEy(|i0nih0ψ|)U=PψF(|0(Eihq0.|)(7=))0. [Eq. (4)],and |h | | i| U =I andU =U,andidentifyP (ρ,I,U)=P (ρ,U). 1 2 F F Then, notice that without loss of generality the con- B. Unambiguous quantum reading of beamsplitters straint in Eq. (5) can be restated as P (ρ,U) = q. F Indeed, for any POVM Π we have that P (ρ,U) is a F continuous function, and that P (0 0 ,U) = 1. So A beamsplitter is a two-mode linear passive quan- F for any ρ with P (ρ,U) < q there| eixhis|ts a 0 < α tum optical device such that A SU(2) in Eq. (1). 1 such that P ((F1 α)ρ + α 0 0 ,U) = q. Sinc≤e In the following we will use the∈basis n,m with E((1 α)ρ+αF0 0−)<E(ρ), t|heihco|nstraint in Eq. (5) respect to which A is diagonal with eige{n|valuei}s e±iδ, becom−es P (ρ,U|)i=h |q. 0 δ π. With this choice, for any ψ = F ∞≤ α≤ n,m , we have U n,m =eiδ(n−m)|ni,m , Proposition 1 (Optimal state is pure). For any optical Pson,mth=a0t n,ψm|U ψi = ∞ | αi 2eiδ(n−m)| anid device U and any threshold q in the probability of failure h | | i n,m=0| n,m| ψ N ψ = ∞ α P2(n+m). Wenoticethatboth sPuFc(hρ,thUa)t,ρth∗eirsepeuxriset.sastateρ∗ which minimizes Eq. (5) hthe|se e|xipresPsionn,ms=on0l|ynd,emp|ends on the squared modulus ofthecoefficientsα ,sowecanassumeα tobereal n,m n,m Proof. Notice that Eq. (5) is equivalent to C(ρ,U) := and positive. pP (ρ,U)+(1 p)E(ρ), for any fixed value of p. If ρ∗ Here x ( x ) denotes the maximum (minimum) in- F ⌊ ⌋ ⌈ ⌉ is the state tha−t minimizes C(ρ,U), for q := P(ρ∗,U) teger number smaller (greater) than x. wehavethatE(ρ∗)givestheminimumpossiblevaluefor Proposition 3 (Unambiguous quantum reading of the energy. Since PF(ρ,U)andE(ρ) are linearfunctions beamsplitters). For any beamsplitter U and for any of ρ, it follows that C(ρ,U) is a linear function of ρ and thresholdqintheprobabilityoffailure,thereexistsastate its minimum is attained on the boundary of its domain, ψ∗ which minimizes Eq. (6) such that namely for a pure state ψ∗ . | i 1 ψ∗ = α(0,n∗ + n∗,0 )+β 00 , (8) As a consequence of Proposition 1, Eq. (5) can be | i √2 | i | i | i restated as where α = 1−q , β = 1 α2, n∗ = | | 1−cos(δn1) | | −| | ψ∗ =arg min E(ψ). (6) argmin⌊x∗⌋,⌈x∗⌉qE(ψ∗), and x∗ = mpin(x R+ δx = ψ s.t. P(ψ,U)=q tan(δx/2)). ∈ | For pure states, the probability of failure in the unam- Proof. First we prove that the optimal state in Eq. 6 is biguous discriminationwhen p =p =1/2 givenby Eq. a superposition of NOON states. For any state ψ = 1 2 | i (4) has been proved to be given by [13] α n,m , the state ψ′ = 1/2 α′(l,0 + n,m n,m| i | i l l | i P0,l ) with α′ 2 = α 2 ips suchPthat P (ψ∗,U)= ψ U ψ . (7) | i | l| |n−m|=l| nm| F |h | | i| ∞P Proposition 2 (No ancillary modes are required). For hψ′|N|ψ′i= α2nm|n−m|≤hψ|N|ψi, (9) any optical device U and any threshold q in the proba- n,Xm=0 bility of failure P (ρ,U), there exists a state ρ∗ which ∞ F minimizes Eq. (5) such that ρ∗ ∈H. |hψ′|U|ψ′i|=(cid:12)(cid:12)(cid:12)n,Xm=0α2nmcos(δ|n−m|)(cid:12)(cid:12)(cid:12)≤|hψ|U|ψi|. Proof. Any pure input state can be written as ψ = (cid:12) (cid:12) | i So we have ψ U(cid:12) ψ R and the constr(cid:12)aint in Eq. (6) |Pχiiicia|riei|χnoiirmwahleizreed|iistiasteasnionrtKho.noIrfmwael bdaesfiisnein|ψH′ian:=d beTcohmeneswhψe|hUpr||oψvie|=tihq∈.at the optimal state is the su- c i 0 , it follows that P (ψ,U) = P (ψ′,U) while EP(iψ)i| iE| (iψ′), which proofs oFf the statemFent. perposition of two NOON states. Let |ψ∗i = ≥ 1/2 α∗(n,0 + 0,n ) be the optimal state and let Since no ancillary modes are required, the energy tphe setP{nα∗n}n h|aveiN |≥ 3inot-null elements. Then there exist n and n such that α ,α = 0 and cos(δn ) ED(ψ) that flows through the unknown device is equal 1 2 n1 n2 6 1 ≤ to the total energy of the input state E(ρ), so minimiz- q ≤ cos(δn2). Define |χi := 1/√2 i=1,2βni(|ni,0i + ingtheformerinsteadthanthelatter-namely,replacing 0,n ) such that χ U χ = q, anPd ξ := 1/√2(1 i | i h | | i | i − E(ψ) with ED(ψ) in Eq. (6) - does not change the opti- ǫ)−1/2 nγn(|n,0i+|0,ni), where mal state. P α if n=n ,n Notice that the generalizationof the results in [2] pro- γ = n 6 1 2 , videdherebasicallydependsonsomecommonproperties n (cid:26) α2n−ǫβn2 if n=n1,n2 p 4 and ǫ min(α /β ,α /β ). Notice that ξ U ξ = the proposed setups for quantum reading make use of q, and≤ψ∗ N nψ1∗ n=1 ǫn2χ Nn2χ + (1 ǫ) ξhN| ξ|.i If three-modes input states - namely, an ancillary mode is h | | i h | | i − h | | i χ N χ = ψ∗ N ψ∗ the statement follows with ψ = employed. This choice is due to the requirement to have hχ|. |Ifi χhN|χ |=i ψ∗ N ψ∗ , either χ N χ| i< an input state with fixed number of photons in order | i h | | i 6 h | | i h | | i ψ∗ N ψ∗ or ξ N ξ < ψ∗ N ψ∗ , that contradicts to be able to take into account loss. For this reason, our hthe|hyp|othiesis hth|at |ψ∗i ishthe|opt|imail state. setupminimizestheenergyE (ρ)thatflowsthroughthe D | i Finally we prove that the optimal state is the su- unknowndevice,whilethetotalenergyoftheinputstate perposition of a NOON state and the vacuum. Let is fixed. |ψ∗i=1/√2 i=1,2αni(|ni,0i+|0,nii). Then In the following, for any beamsplitter X we denote P with A the A matrix of X in Eq. (1), so we write X n cos(δn ) n cos(δn )+q(n n ) ψ∗ N ψ∗ = 2 1 − 1 2 1− 2 . h | | i cos(δn ) cos(δn ) It is easy to verify (in [2] a p1roo−f of this2 fact is pro- AX =(cid:18)rtXX −rtXX (cid:19), A†X =(cid:18) rtXX rtXX (cid:19). − vided) that it is not restrictive to set n =0, so one has 2 ψ∗ N ψ∗ = (1 q)n(1 cos(δn1))−1. Then one can We define the reflectivity RX and the transmittivity TX hsee|tha|t itiis not−restricti−ve to choose π/2 ≤ δn1 ≤ π, of X as RX := |rX|2 and TX = |tX|2, respectively, with where ψ∗ N ψ∗ can be proven to be a convex func- RX +TX =1. tion thhat a|ttai|nsiits minimum for n1 = x∗ , x∗ , with The general setup is given by a Mach-Zender interfer- x∗ = min(x R+ δx = tan(δx/2)). The⌊ st⌋at⌈eme⌉nt im- ometer with beamsplitters B and B†, acting on modes ∈ | mediately follows. 2 and 3. In one of the harms of the interferometer (cor- responding to mode 2), the following beamsplitters are Notice that from Proposition 3 it immediately fol- inserted lows that unambiguous discrimination between beam- 1 splitters U and I can be achieved only if the threshold '&!0 q in the probability of failure P (ρ,U) satisfies the in- =< F 2 N D I,U D† N† equality q ≥ cos(δn∗) (an analogous inequality, namely '&!0 Π , K(q) ≥ cos(δn∗) with K(q) = 4q(1−q), holds in the 3 B B† case of ambiguous quantum reapding addressed in [2]). '&!1 :; In [2], the optimalcoherentstrategy - namely, a strat- egy making use of coherent input states and homodyne where N is a 50/50 beamsplitter, I,U is the unknown measurements - for ambiguous quantum reading is pro- beamsplitter,andD isthebeamsplitterdiagonalizingU. vided. A comparison between the optimal strategy and The POVM Π is different for ambiguous and unambigu- theoptimalcoherentstrategyshowedthattheformerre- ous quantum reading. quires much less energy than the latter when the same It is easy to verify that the composition of beamsplit- threshold in the probability of erroris allowed. Here, we ters DN reduces to a phase shifter on mode 2, namely notice that, since the support of coherent states is the entire Fock space, no measurement exists projecting on 1 1 1 1 0 its orthogonal complement. For this reason, no coher- A = , A A = . (10) ent strategy exists for the unambiguous discrimination D √2(cid:18) i −i(cid:19) D N (cid:18)0 i (cid:19) of optical devices. It is easy to check that this phase shifter is irrelevant,so in the following we will disregard it. III. EXPERIMENTAL SETUP FOR QUANTUM READING A. Experimental setup for ambiguous quantum InthisSectionweprovideexperimentalsetupsforam- reading of beamsplitters biguous, unambiguous, and perfect quantum reading, which are feasible with present quantum optical tech- nology. The input is a single-photon state, that can be The optimal strategy for ambiguous quantum reading realized e.g. through spontaneous parametric down con- of beamsplitters has been provided in [2]. Here we de- version or through the attenuation of a laser beam. The scribe an experimental setup implementing such strat- evolution is given by a circuit of beamsplitters, one of egy, namely the ambiguous discrimination of a beam- which is the unknown one, and the final measurement is splitter randomly chosen from the set I,U with equal { } implemented through photodetectors. prior probabilities, with probability of error PE(ρ,U) In Proposition 2 we proved that, for the unambiguous under a given threshold q and minimal energy flow quantum reading of optical devices, no ancillary modes trough the unknown device. In the following we set are required. The same result has been proved for the K(q) := 4q(1 q). According to [2], in order to have − case of ambiguous quantum reading in [2]. Nevertheless, P (ρ,U)pq, we must have K(q) √R . E U ≤ ≥ 5 The experimental setup is then given by wherethereflectivitiesandtransmittivitiesofbeamsplit- ters B, M and N are given by 1 ❴✤ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴✤ '&!0 ✤ I"%#$ ✤ q rU 1 q R = − , T = − , 2 I,U ✤ M ✤ B 1 rU B 1 rU '&!0 3 B B† ✤✤✤ N† ΠU*-+, ,✤✤✤ RM = (√−1+rU(1−+pqq)(2q−r−U))2, '&!1 ✤ ΠI*-+, ✤ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ( q(1+r )+√q r )2 U U T = − , wherethereflectivitiesandtransmittivitiesofbeamsplit- M p (1+q)2 ters B, M and N† are given by R = 1 q, T =√q. N N − p K(q) r 1 K(q) The optimal measurement for unambiguous discrimi- U RB = − , TB = − , nation [13] is implemented by the two beamsplitters M 1 r 1 r U U − − and N and by the three photocounters Π , Π , and Π (1 K(q))(K(q) r ) U I F R = − − U , surrounded by the dashed line. The conditional proba- M (1 2q)2 bilities p of detecting a photon in photodetector Π − X|Y X (1 K(q))(1+rU) given that the unknown device is Y are given by T = − , M (1 2q)2 − pU|U =pI|I =1 q, pI|U =pU|I =0, − R = 1 q, T =√q. N p − N pF|U =pF|I =q. The optimal measurement for ambiguous discrimina- DetectingaphotoninΠU orΠI impliesthattheunknown tion[18]isimplementedbythetwobeamsplittersM and beamsplitter is certainly U or I, respectively, while de- N† andbythetwophotocountersΠU andΠI surrounded tectingaphotoninΠF declaresafailurewithprobability bythedashedline(nomeasurementisperformedonout- q. put mode 1). The conditional probabilities p of de- X|Y tecting aphotoninphotodetectorΠ giventhatthe un- X known device is Y are given by C. Experimental setup for perfect quantum reading of beamsplitters p =p =1 q, p =p =q. U|U I|I I|U U|I − The optimal strategies for ambiguous and unambigu- DetectingaphotoninΠ orΠ impliesthattheunknown ous quantum reading of beamsplitters of [2] and Propo- U I beamsplitter is U or I, respectively, with probability of sition 3 reduce to the same optimal strategy for perfect error q. quantum reading of beamsplitters when the threshold q in the probability of error (for the ambiguous case) and failure (for the ambiguous case) is set to zero. Inthiscase,theconditionr 0givenbyProposition U B. Experimental setup for unambiguous quantum ≤ 3 can be satisfied upon defining U as the composition of reading of beamsplitters a beamsplitter V with r 0 with π/2 phase shifters V ≥ ± on its input and output modes, according to We provided the optimal strategy for unambiguous 1 1 quantum reading of beamsplitters in Proposition 3. π/2 π/2 − − Here we describe an experimental setup implementing 2 U := 2 V , (11) such strategy, namely the unambiguous discrimination π/2 π/2 of a beamsplitter randomly chosen from the set I,U { } with equal prior probabilities, with probability of failure and it is easy to verify that P (ρ,U) under a given threshold q and minimal energy flsioFtwiontr3o,uginhotrhdeeruntoknhoawvne dPeFv(iρce,.U)Accoqr,dwinegmtousPtrhoapvoe- AU =(cid:18)e−0iπ2 ei0π2 (cid:19)AV (cid:18)e−0iπ2 ei0π2 (cid:19)=(cid:18)−trVV −−rtVV (cid:19), ≤ q √R . U with 0 r ,t 1, so that r 0 and perfect discrim- ≥ V V U The experimental setup is given by ination≤is indeed≤possible. ≤ The experimental setup is then given by 1 ❴✤ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴✤ 1 '&!0 I,U ✤✤ M ΠU*-+, ✤✤ '&!0 ΠU*-+, 2 I,U '&!0 ✤✤ ΠF*-+, ,✤✤ '&!0 2 ΠU′*-+,, (12) 3 B B† ✤ N ✤ 3 B B† '&!1 ✤ ΠI*-+, ✤ '&!1 ΠI*-+, ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ 6 where the reflectivity and transmittivity of beamsplitter we employ a collinear frequency-degenerate SPDC pro- B are given by cess with type-II phase matching in a 2-mm-long BBO crystal pumped by a cw laser diode (Coherent Cube) at r 1 R = V , T = . 405nm. In this process pairs of photons at 810nm are B B 1+rV 1+rV created. Photonsfromeachpairareseparatedbyapolar- izing beam splitter and coupled into single-mode optical Notice that the two π/2 phase shifters on mode 1 in − fibers. One of them is led directly to trigger detector the Equation (11) are irrelevant and can be discarded, DetT which heralds the creationof a pair (Perkin-Elmer since the one onthe input mode acts onthe vacuumand SPCM AQR-14FC, dark counts 180s−1, total efficiency the one on the output mode is immediately followed by cca 50%). The other one enters MZI through variable a photodetector, so we can redefine ratio coupler VRC-in. 1 1 An additional variable ratio coupler, VRC-mid, repre- 2 U := 2 V . (13) sentsanunknowndevice. Whenitsreflectivityequalsone π/2 π/2 it correspondsto device I.To switchto device U one has tosetrequiredsplittingratioandapplyadditionalphase Theoptimalmeasurementforperfectdiscriminationis shift, see scheme (13). For practical reasons we apply— implemented by the three photocounters ΠU, ΠU′, and without the loss of generality—a cumulative phase shift ΠI. The conditional probabilities pX|Y of detecting a infrontofthebeamsplitter. Intheexperimentthephase photoninphotodetectorΠX giventhatthe unknownde- shiftsareintroducedbyelectro-opticalphasemodulators vice is Y are given by (PM) manufactured by EO Space. Their half-wave volt- ages are about 1.5V. These phase modulators exhibit pU|U =1−rV, pU′|U =rV, pI|I =1, relativelyhighdispersion. ThereforeonePMisplacedin pI|U =pU|I =pU′|I =0 each interferometer arm in order to compensate disper- sion effects. In case of device U we use the PM in the Detecting a photon in ΠU or ΠU′ implies that the un- upper interferometer arm to apply the additional phase knownbeamsplitteriscertainlyU,whiledetectingapho- shift of π. ton in Π implies that the unknown device is certainly I The output fibers from the unknown device and from I. the interferometerare ledto detectors DetU, DetU’, and DetI. These detectors are parts of Perkin-Elmer quad module SPCM-AQ4C (dark counts 370-440s−1, total ef- IV. EXPERIMENTAL IMPLEMENTATION OF ficiencies about 50%). QUANTUM READING To reduce the effect of the phase drift caused by fluc- tuations of temperature and temperature gradients we To demonstrate experimental feasibility of quantum apply both passive and active stabilization. The exper- reading we have built a laboratory setup for perfect dis- imental setup is covered by a shield minimizing air flux crimination of two beam splitters according to scheme around the components. Besides, after each three sec- (12) [see Fig. (1)]. It consists of a Mach-Zehnder inter- ondsofmeasurementanactivestabilizationisperformed. ferometer (MZI) with an additional beam splitter in its It measures intensities for phase shifts 0 and π/2 and if upper arm. This additional beam splitter has a variable necessary it calculates phase compensation and applies splitting ratio and it serves as an unknown device to be corrective voltage to the phase modulator in the lower discriminated. interferometer arm. This results in the precision of the phasesettingduringthemeasurementperiodbetterthan π/200. For each pair of devices U and I the proper splitting ratio of fiber couplers VRC-in and VRC-out must be set inordertodiscriminatethesedevicesoptimally. Wehave mademeasurementsfor11differentdevicesUwithinten- sityreflectances0,0.1,0.2,...,1. Foreachpairofdevices U and I the counts at detectors DetU, DetU’, and DetI were cumulated during 30 three-secondmeasurementin- tervals interlaced by stabilization procedures. All mea- surements were done in coincidence with the trigger de- FIG.1. (Coloronline)Schemeoftheexperiment. VRC–vari- tector DetT. It means we measured coincidence counts ableratiocouplers,PM–phasemodulators, Det–detectors, CU,CU′,CI betweendetectorsDetT-DetU, DetT-DetU’, C – coincidence electronics. and DetT-DetI, respectively, using 3ns coincidence time window. These results were normalized to obtain rela- Weuseaheraldedsinglephotonsourcebasedonspon- tive frequencies, fj = Cj/(CU +CU′ +CI),j = U,U′,I, taneous parametric down conversion (SPDC). Namely, which can be compared with theoretical probabilities of 7 detection. Rv pU|U pU′|U pI|U fU|U fU′|U fI|U Measured relative frequencies and theoretical proba- 0.0 1.000 0.000 0 0.986 0.000 0.014 bilities arelistedinTables I andII andshowninFigures 0.1 0.684 0.316 0 0.680 0.295 0.025 2 and 3, respectively. Table I and Fig. 2 show the re- 0.2 0.553 0.447 0 0.551 0.440 0.009 sultsobtainedwithdeviceIinserted,TableII andFig. 3 0.3 0.452 0.548 0 0.455 0.542 0.003 summarize the results for devices U. Eachrow in the ta- 0.4 0.368 0.633 0 0.369 0.623 0.008 blescorrespondstoonepairofIandUwithR beingthe v 0.5 0.293 0.707 0 0.288 0.691 0.021 reflectivityofdeviceU.Onecanobserveverygoodagree- ment between theory and experiment. Small discrepan- 0.6 0.225 0.775 0 0.219 0.758 0.022 ciesappearmainlyduetoimperfectionsinsplitting-ratio 0.7 0.163 0.837 0 0.160 0.830 0.010 settings, phase fluctuations, and polarization misalign- 0.8 0.106 0.894 0 0.100 0.891 0.009 ment. In coincidence measurements the contribution of 0.9 0.051 0.949 0 0.046 0.946 0.007 detector noise is completely negligible. 1.0 0.000 1.000 0 0.000 0.980 0.020 TABLE II. Results for devices U. Rv – reflectivity of de- Rv pU|I pU′|I pI|I fU|I fU′|I fI|I vice U; pU|U,pU′|U,pI|U – theoretical probabilities of pho- ton detection at detectors DetU, DetU’, DetI, respectively, 0.0 0 0 1 0.000 0.002 0.998 0.1 0 0 1 0.000 0.012 0.988 fU|U,fU′|U,fI|U – relative frequencies measured at detectors DetU, DetU’, DetI, respectively (measured in coincidence 0.2 0 0 1 0.000 0.018 0.982 with DetT). 0.3 0 0 1 0.000 0.012 0.988 0.4 0 0 1 0.000 0.023 0.977 0.5 0 0 1 0.000 0.022 0.978 0.6 0 0 1 0.000 0.014 0.986 0.7 0 0 1 0.000 0.011 0.989 0.8 0 0 1 0.000 0.013 0.987 0.9 0 0 1 0.000 0.018 0.982 1.0 0 0 1 0.000 0.021 0.979 TABLEI.ResultsfordeviceI.Rv –reflectivitiesofdevicesU; pU|I,pU′|I,pI|I –theoreticalprobabilitiesofphotondetection atdetectorsDetU,DetU’,DetI,respectively,fU|I,fU′|I,fI|I – relativefrequenciesmeasuredatdetectorsDetU,DetU’,DetI, respectively (measured in coincidence with DetT). FIG.3. (Coloronline) ResultsfordevicesU:Detectionprob- abilities andmeasured relativefrequencies as functionsof re- flectivity Rv. Different reflectivities correspond to different devices U (VRC-mid). V. CONCLUSION In this paper we considered unambiguous quantum readingofopticalmemories,ontheassumptionthatnoise and loss are negligible. In Section II we showedthat the optimal strategy for the unambiguous discrimination of optical devices can be derived by extending the results proved for the ambiguous case. In Section III we presented some experimental imple- mentation of quantum reading for both the ambiguous FIG. 2. (Color online) Results for device I: Detection prob- and the unambiguous case. In the proposed setups the abilities and measured relative frequenciesas functions of re- input state is fixed to be a single photon state. By mak- flectivity Rv. Different reflectivities correspond to different ing use of an ancillary mode it was possible to tune the devices U (VRC-mid). amount of energy flowing through the device. Finally, in Section IV we provide experimental results 8 for the perfect quantum reading. The advantage of the ACKNOWLEDGMENTS implemented setup is that in ideal case there is exactly one photon at the output ports. It makes detection rel- We thank Michal Sedla´k for very useful suggestions, atively easy. Nevertheless, it is still a superposition of a Helena Fikerov´a for her help with software for ac- single photonandvacuumwhatis enteringthe unknown tive stabilization of the interferometer, and Ivo Straka device. So the unknown device is exposed just to a frac- for the construction of the two-photon source. This tion of energy of a single photon in average. 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