Experimental quantum reading with photon counting
Giuseppe Ortolano, Elena Losero, Ivano Ruo Berchera, Stefano Pirandola, Marco Genovese
EExperimental quantum reading with photon counting
Giuseppe Ortolano,
1, 2, ∗ Elena Losero, Ivano Ruo Berchera, Stefano Pirandola, and Marco Genovese Quantum metrology and nano technologies division,INRiM, Strada delle Cacce 91, 10153 Torino, Italy DISAT, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy Department of Computer Science, University of York, York YO10 5GH, United Kingdom
The final goal of quantum hypothesis testing is to achieve quantum advantage over all possibleclassical strategies. In the protocol of quantum reading this advantage is achieved for informationretrieval from an optical memory, whose generic cell stores a bit of information in two possible lossychannels. For this protocol, we show, theoretically and experimentally, that quantum advantage isobtained by practical photon-counting measurements combined with a simple maximum-likelihooddecision. In particular, we show that this receiver combined with an entangled two-mode squeezedvacuum source is able to outperform any strategy based on statistical mixtures of coherent statesfor the same mean number of input photons. Our experimental findings demonstrate that quantumentanglement and simple optics are able to enhance the readout of digital data, paving the way toreal applications of quantum reading and with potential applications for any other model that isbased on the binary discrimination of bosonic loss.
Introduction. – In the vast panorama of quantumtechnologies [1, 2], the most practical area is arguablythat of quantum sensing, well developed with both dis-crete [3] and continuous variable systems [4–6]. In thisarea, quantum metrology [7] deals with the estimationof unknown parameters encoded in a state or a physi-cal transformation, while quantum hypothesis testing [8]deals with the discrimination of a discrete set of states [9–12] or quantum channels [13, 14]. In particular, the prob-lem of quantum channel discrimination [4, 15] is knownto have a very rich theoretical structure due to its inher-ent double optimization nature, which involves findingboth the best input states and the optimal output mea-surements.In 2011, Ref. [16] modeled the information retrievalfrom an optical memory as a problem of bosonic chan-nel discrimination. In fact, a memory cell can be rep-resented as a reflector (e.g., a beam splitter) with twopossible values of the reflectivity, which is equivalent toconsidering two possible lossy channels acting on the in-coming photons. In this scenario, one can show that theuse of a quantum source of light (and, in particular, en-tangled) can sensibly boost the retrieval of informationfrom the cell with respect to classical input states, i.e.,having positive-P representations [17, 18].The idea of quantum reading has been further exploredin a series of papers (e.g., see Refs. [19–24] among oth-ers). A preliminary experiment [25] was performed for aperfect fully-unitary variant of the protocol, where zerodiscrimination error was achieved by analyzing the coin-cidences at the two outputs of the beam-splitter cell. Forsuch an ideal unitary discrimination no entanglement isneeded. However, in a realistic scenario, only one outputof the cell is available for detection, so that the processis clearly non-unitary and must be described by a lossyquantum channel (as in the original proposal). For thisreason, a truly quantum reading experiment has yet to be performed.In this work, we experimentally demonstrate the origi-nal protocol of quantum reading [16] showing that a two-mode squeezed vacuum state (TMSV) [26, 27] is able tooutperform any classical state in retrieving informationfrom an absorbing layer in a coated glass-slide, mimickingthe memory cell. Remarkably, this advantage is achievedwithout resorting to any complicated Helstrom-like mea-surement [8, 28, 29], but just resorting to photon countingof the output followed by a maximum likelihood decision.Quantum advantage is proven notwithstanding the pres-ence of more than 20% experimental loss. This robust-ness to losses and the simplicity of detection scheme pavethe way to possible real applications of quantum-readingin a next future.
Theoretical model .– Let us store a bit u = { , } ina memory cell by means of two equiprobable lossy chan-nels, E and E , with transmissivities τ and τ . Recall idlers RxTx 𝜏 𝑢 𝑢𝑁 𝐿 ⊗ cell FIG. 1:
Quantum reading of a memory cell . A memory cellencodes a bit u in a lossy channel with transmissivity τ u . Thecell is read by a transmitter (Tx) which irradiates M signalmodes and N mean total photons over the cell, plus extra L idler modes sent to the output. The receiver (Rx) performsa generally-joint measurement of signals and idlers, decodingthe bit u up to some error probability p err . Quantum readingcorresponds to using a quantum source of light for the Tx,so that we outperform any classical source in the readoutof the bit. The scheme can be realized in reflection or intransmission, as done in our experiment. a r X i v : . [ qu a n t - ph ] A p r that a lossy channel with transmissivity τ correspondsto the following input-output transformation of the fieldoperator ˆ a → √ τ ˆ a + i (cid:112) (1 − τ )ˆ v , where ˆ v describes anenvironmental vacuum mode [30]. To retrieve the bit,consider a transmitter and a receiver. The transmitterirradiates M signal modes over the cell, for a total of N mean photons, and also sends additional L idler modesdirectly to the output. The receiver measures the trans-mitted signals and the idlers, guessing the classical bit u up to an error probability p err (see Fig. 1).Assuming an optimal measurement at the receiver, theminimization of p err over all transmitters with fixed sig-nal energy N is difficult to solve. If we restrict the anal-ysis to classical transmitters, described by a state withpositive P-representation (mixture of coherent states),then the minimum error probability is given by [16] p claerr ≥ C ( N, τ , τ ) := 1 − (cid:112) − e − N ( √ τ −√ τ ) . (1)Equivalently, the maximum information accessible toclassical transmitters cannot exceed the bound 1 − H ( C ),where H ( · ) denotes the binary Shannon entropy [31].Consider now a multi-mode quantum transmitter in atensor product of M TMSV states | TMSV (cid:105) ⊗ MS,I . EachTMSV state irradiates ¯ n mean photons per mode anddescribes an entangled pair of signal (S) and idler (I)modes, so that we have a total of M signals and corre-sponding L = M idlers. Let us assume that ¯ n is chosensuch that M ¯ n = N mean photons are globally irradiatedover the cell. Then, for sufficiently large M , it is possibleto show that the error probability p err goes below theclassical bound C . In terms of the gain G = 1 − H ( p err ) − [1 − H ( C )] , (2)one can show that G may approach 1, meaning that thequantum transmitter retrieves all the information whilethe bit cannot be read by any classical strategy [16].In the following we show that a similar result can beachieved performing a photon counting measurement atthe output and a maximum likelihood decision, in theplace of the unspecified optimal receiver. Note that quan-tum advantage has been demonstrated by photo countingmeasurement strategies for parameter estimation [32–38],where the goal is to estimate the value of a continuousparameter τ . In that case, in fact, it can be proven thatsuitable quantum resources and photon counting mea-surements allow one to reach the ultimate (non-adaptive)quantum limits in precision [38–41]. However, for thediscrete-case considered here, i.e., for a problem of bi-nary channel discrimination, such a proof has not beengiven and the effective performance of photon countinghas not been investigated yet. Photon counting strategy.–
When photon countingmeasurements are performed over the signal and idlermodes of a bipartite state ρ , the output is a classical random variable n = ( n S , n I ), distributed as p ( n ) = (cid:104) n S , n I | ρ | n S , n I (cid:105) , where | n k (cid:105) is the eigenstate with eigen-value n k of the number operator ˆ n k = ˆ a † k ˆ a k of the fieldand k = S, I . The effect of a lossy channel E τ on thesignal mode of a bipartite state is to combine its initialphoton distribution p ( n ) with a binomial distribution B ( n (cid:48) S | n S , τ ) with n S trials and success probability τ , sothat the outcome n will be distributed according to p ( n | τ ) = ∞ (cid:88) m = n S p ( m, n I ) B ( n S | m, τ ) . (3)Let us suppose that n is the outcome of photon-counting measurements after a lossy channel with un-known transmissivity τ u (for u = 0 , τ u is given by p ( τ u | n ) = p ( n | τ u ) p ( τ u ) p ( n ) = p ( n | τ u ) p ( n | τ ) + p ( n | τ ) , (4)where the last equality follows from the condition of equi-probable channels, p ( τ u ) = 1 /
2. To assign a value to therecovered bit, the optimal strategy is to choose the value u = 0 , u = arg max u p ( τ u | n ). Because p ( τ u )is uniform, this is equivalent to a maximum likelihooddecision, i.e., to choose u = arg max u p ( n | τ u ).The corresponding error probability will be given by p err ( τ , τ | n ) = min u p ( τ u | n ). Therefore, by averagingover the distribution of the outcomes p ( n ), we may writethe following expression for the mean error probability p err ( τ , τ ) = (cid:88) n min u p ( τ u | n ) p ( n )= 12 (cid:88) n min u p ( n | τ u ) . (5)The error probability above describes the performanceachievable by a photon-counting receiver in the read-ing scenario of Fig. 1 where the transmitter irradiatesa generic bipartite state. In general, the formula canbe applied to a transmitter with arbitrary M and L byconsidering an M + L vectorial variable n . Let us nowapply this analysis to evaluate the corresponding perfor-mances with classical and quantum states. Without lossof generality, in the following we assume that τ < τ .The photon counting performance with a classicaltransmitter, i.e., described by a state with positive P-representation, is optimized by the use of a single sig-nal mode ( M = 1 and L = 0) with N mean photons,whose photon number statistics is a Poisson distribution P N ( n ). It is easy to show that there is a threshold value n th := N ( τ − τ ) / log( τ /τ ) such that, for every n ≤ n th ,one has P N ( τ | n ) > P N ( τ | n ), thus the value τ is cho-sen. The error probability will be given by p cla,phcerr ( τ , τ ) = 12 (cid:20) − γ ( τ ) − γ ( τ ) (cid:98) n th (cid:99) ! (cid:21) , (6)where (cid:98) x (cid:99) is the floor of x , γ ( τ u ) := Γ( (cid:98) n th + 1 (cid:99) , N τ u ),and Γ( x, y ) is the incomplete gamma function. In otherwords, Eq. (6) establishes a lower bound on the errorprobability that can be achieved by using classical trans-mitters and photon counting.Let us now study the photon-counting performancethat is achievable by a quantum transmitter based oncopies of TMSV states. We consider the transmitter’sstate | TMSV (cid:105) ⊗ M S,I , where each signal-idler TMSV state | TMSV (cid:105)
S,I ∝ (cid:80) n (cid:112) P ¯ n ( n ) | n (cid:105) S | n (cid:105) I is maximally corre-lated in the number of photons and locally characterizedby a single-mode thermal distribution P ¯ n ( n ) = ¯ n n / (¯ n +1) n +1 . The product state | TMSV (cid:105) ⊗ M S,I preserves the per-fect correlation between the total photon numbers, rede-fined as (cid:80) Mm =1 n ( m ) S/I → n S/I , while the marginal distribu-tion becomes multi-thermal P N,M ( n S/I ), with mean pho-ton number N. Fixing N and increasing M , this distribu-tion becomes narrower and tends to a Poisson distribu-tion P N ( n S/I ) with mean occupation number
N/M → τ u on the signal path transforms the input joint prob-ability P N,M ( n S , n I ) into the output probability distri-bution P N,M ( n S , n I | τ u ) = P N,M ( n I ) B ( n S | n I , τ u ). Pho-ton counting is then performed on both the signal andidler modes, and a maximum likelihood decision is finallytaken. In fact, we can identify a threshold value n thS = (cid:26) log( τ /τ )log[(1 − τ ) / (1 − τ )] + 1 (cid:27) − n I , (7)and choose τ if n S < n thS , corresponding to the con-dition P N,M ( n S , n I | τ ) > P N,M ( n S , n I | τ ). Otherwisewe choose τ . This strategy provides an error proba-bility p qua,phcerr for the TMSV-based transmitter and thephoton-counting receiver. We explicitly evaluate the per-formance of this strategy in the numerical study below. Theoretical predictions.–
Numerical investigationshows a quantum advantage even with a single TMSVstate. However, the described narrowing of the marginaldistributions, resulting from the spread of the energy overan high number of copies M , makes the discriminationmore effective, so this is the regime that we will con-sider and exploit in our experiment. We have studiedthe following information gain G = 1 − H ( p qua,phcerr ) − [1 − H ( p claerr )], where we have assumed, for p claerr , eitherthe optimal classical bound in Eq. (1) or the classicalphoton counting bound of Eq. (6).As we can see from Fig. 2, there is an evident informa-tion gain, which may approach the maximum value of 1,meaning that, in certain regions the use of quantum re-sources allows the full recovery of the stored information,whereas no information could be retrieved by classicalmeans.In Fig. 2(A,B) we see that, increasing the mean photonnumber, the maximum of the advantage shifts towards higher reflectivity τ . Intuitively, this is explained by thefact that the gain becomes larger when classical strategiesstart to fail. For example, although non-optimal, anotherclassical discrimination strategy can be to measure themean photon number, that is either N or τ N (assuming τ = 1). This approach of mean-energy-discrimination(MED) fails when the difference in the average photoncounts becomes smaller than the noise associated withthe Poisson fluctuations, i.e., when τ > − N − . Thesaturation of this inequality defines the red line in Fig. 2.In fact, in Fig. 2(A), this curve follows the contourlines of the plot, denoting the start of the maximum gainregion. Of course, when τ is approaching τ = 1 there isno way to distinguish among the channels, neither clas-sical nor quantum, and the information gain drops tozero. The competition between these two tendencies de-termines the maximum of the gain. When comparingwith the optimal classical bound in Fig. 2(B), the regionsare in general narrower, and the maximum deviates fromthe MED curve. However, note that Eq. (1) represents atheoretical lower bound which may be non-tight.The biggest limitation in an experimental realizationof this procedure is given by photon losses of different na-ture, interaction with the environment and optical com-ponents, as well as the intrinsic quantum efficiency of thedetectors. Their combined effect can be accounted witha unique coefficient, the detection efficiency 0 ≤ η ≤ E η and E τ , commute and their total effect is given the compos-ite pure-loss channel E ητ . Due to this indistinguishabil-ity, the classical limits, in this scenario can be computedperforming the substitution τ u → ητ u in Eq. (1) andEq. (6), resulting in a decreased accuracy for discrimi-nation. An equivalent way to obtain these classical lim-its is to consider the signal energy reduction caused by η , yielding the same result. When quantum-correlatedsystems are considered, however, aside from the energyreduction, an additional effect induced by losses is theworsening of the correlations, therefore decreasing theadvantage that can be obtained. This drop in the gaincan be seen from Figs. 2(C-D), where the scenario withan efficiency η = 0 .
76 is reported. The maximum gainis reduced to (cid:39) / (cid:39) /
6, depending on the clas-sical benchmark considered. Still, this is a macroscopicamount of information due to the fact that it refers togain per cell . DCA B
FIG. 2: Information gain G of quantum reading as a function of the lower transmissivity τ and total mean number ofphotons N (higher transmissivity is set to τ = 1). The information gain is computed assuming a TMSV-state transmitterwith large number of copies ( M (cid:39) ) and a receiver based on photon counting. In panel A , the classical benchmark isthe photon-counting performance with classical states of Eq. (6). In panel B , the benchmark is the optimal classical limit inEq. (1). In both panels, the red curve represent the MED strategy described in the text, marking the limit after which thechannels are classically indistinguishable. In panels C and D , we consider the case of imperfect quantum efficiency η = 0 .
76 forboth the signal and idler systems (so that τ u → ητ u for u = 0 , C , and the gain over the optimal classical limit in panel D . In these panels, the dashed lines indicate the regions whereexperimental data were collected. These data points are those reported in Fig. 4. Experimental results.–
A scheme of the experi-mental set-up is reported in Fig. 3(a). The multi-modestate | TMSV (cid:105) ⊗ M S,I is experimentally produced exploit-ing the spontaneous parametric down conversion pro-cess in a non linear crystal. We pump a (1cm) type-II-Beta-Barium-Borate (BBO) crystal with a CW laserof λ p = 405nm and power of 100mW. An interferentialfilter (IF) at (800 ± λ d = 2 λ p = 810nm). The correlation in mo-mentum of two down-converted photons is mapped intospatial correlations at the back focal plane of a lens with f F F = 1cm focal length. This plane is then imaged tothe detection plane by a second lens.The detector is a charge-coupled-device (CCD) camera(Princeton Instrument Pixis 400BR Excelon), workingin linear mode, with high quantum efficiency (nominally >
95% at 810nm) and few e − / (Pixel · Frame) of electronicnoise. The physical pixels of the camera measure 13 µ m.A 12 ×
12 hardware binning is performed on them, inorder to lower the acquisition time and increase the read-out signal-to-noise ratio. The total photon counts n S and n I are obtained integrating the signal over the twospatially correlated detection areas S S and S I , for signaland idler respectively. The total number of spatial modescollected is M s ∼ and the temporal modes can beestimated to be M t ∼ (for a deeper discussion onthese estimates see [42]). Since N I ∼ , the meanoccupation number is N I / ( M s · M t ) ∼ − (cid:28)
1, meaningthat the marginal distributions are well approximated byPoissonian ones.The memory cell is implemented inserting in the focalplane of the first lens a coated glass-slide with a deposi-tion of variable transmission 0 . < τ <
1. The bit of
Pump SIIF Far-field lens 𝑓 𝐹𝐹 Imaging lensBBO 𝒒 = 0 𝒒−𝒒 −𝒙𝒙 𝜏 𝑆 𝐼 𝑆 𝑆 𝑛 𝐼 𝑛 𝑆 Glass slide CCD camera 𝜏 (a) 𝝉 𝝉 (c) 𝝉 𝝉 𝑛 𝑆 (⋅ 10 ) (b) 𝑛 𝐼 ⋅ 10 𝑛 𝑆 (⋅ 10 ) FIG. 3: (a) Simplified schematic of the experimental set-up.In the BBO crystal the multi-mode TMSV source is gener-ated. The signal beam passes through the memory cell in-vestigated, whose transmissivity can be either τ or τ andis then detected in the S S region of the CCD camera. Theidler beam goes directly to the S I region of the CCD. n S and n I are the total photon counts over S S and S I . BBO: Type-II-Beta-Barium-Borate non linear crystal. IF: interferentialfilter (800 ± n S in function of n I , for 1000 frames. Blue dots correspond to τ ∼ . τ = 1. (c) n S rel-ative frequency distribution for τ ∼ .
996 (blue histogram)and τ = 1 (red histogram). information is stored in the presence ( τ = τ ) or absence( τ = τ = 1) of the deposition.The effect at the base of the quantum enhancementcan be visualized comparing Fig. 3(b) and Fig. 3(c). Thejoint distributions of n S and n I for τ and τ = 1, dueto their squeezed shape, are less overlapped with respectto the marginal distributions of n S only, increasing theirdistinguishability. Note that the squeezed shape thejoint distributions of Fig. 3(b) is purely due to quantumcorrelations and cannot be achieved by any classicalsource.The parameters necessary for the subsequent analysis( N , τ , η , η , electronic noise ν e ) are estimated in acalibration phase. In particular, the channels efficienciesare estimated using the absolute calibration method pre-sented in Refs. [43–45]. The error probability in the dis-crimination between τ and τ is evaluated on two sets Theoretical gain over optimal boundExperimetal Gain over optimal boundTheoretical gain over PC boundExperimental gain over PC bound
𝑵 = 𝟏.𝟏𝟓 × 𝟏𝟎 𝑵 = 𝟑.𝟏 × 𝟏𝟎 𝑵 = 𝟓.𝟐 × 𝟏𝟎 (a)(b)(c) GGG 𝝉 𝝉 𝝉 FIG. 4: Experimental gain G of quantum reading (bits) as afunction of the lower transmissivity τ . The three panels re-fer to different mean photon number in the signal beam: (a) N = 1 . · , (b) 3 . · , and (c) 5 . · . Blue data refersto the gain with respect to the classical optimal bound inEq. (1). Red data refers to the gain with respect to the clas-sical photon-counting bound given in Eq. (6), obtained fromthe marginal distribution of the signal. The experimental pa-rameters, estimated independently in a calibration step, arethe mean signal energy N , the detection efficiency of signaland idler channel η S and η I and the electronic noise ν e . Apartthe value of N , which is intentionally different in the threepanels, the other parameters are kept fixed to: η S = 0 . η I = 0 .
77, and ν e ∼ . of frames (10000 frames per set are acquired), one foreach known value of the transmittance. For each framewe compute P N,M ( n S , n I | τ u ), using the values of the pa-rameters estimated in the calibration, and we assign tothe frame the value of τ u that makes this probabilityhigher. The comparison of the true known value of τ u over each set with the guessed ones, allows estimatingthe error frequency p experr for each set.The experimental gain G evaluated from p experr is re-ported in Fig. 4, both with respect to the optimal clas-sical bound (blue curves) and to the classical photon-counting bound (red curves). The three panels are ob-tained for a different number of photons in the signalbeam, i.e., N ∼ . · , 3 . · and 5 . · respec-tively, corresponding to the sections lines in the theoreti-cal Figs. 2(C-D). In Fig. 4, the error bands on the theoret-ical curves have been obtained via numerical simulation.Experimental data show a good accordance with the the-oretical model, with the majority of the data falling inthe confidence region at 1 standard deviation. In all threecases, we find a clear quantum advantage. In perfect ac-cordance with theory, we find that the maximum gain in-creases with the mean signal energy but at the expensesof a narrowing of the region in which the quantum en-hancement can be found. Conclusion.–
In this work we have provided an ex-perimental demonstration of the quantum reading pro-tocol, showing how entanglement is able to boost theretrieval of classical information from an optical memorycell, outperforming any classical strategy for the samenumber of input photons. We have shown, theoreti-cally and experimentally, that quantum advantage canbe achieved by means of a simple receiver strategy basedon photon counting measurements followed by a maxi-mum likelihood decision test. In this way, we were ableto demonstrate values, for the quantum advantage, whichare close to the performance originally foreseen by usingoptimal, but highly-theoretical, joint quantum measure-ments.In our experiment, we considered the realistic scenariowhere only a single output from the cell is accessiblefor detection and we were able to show quantum ad-vantage despite the presence of extra optical losses onboth the signal and idler paths. Because of all theseaspects, our results pave the way for a realistic andpractical implementation of quantum reading techniques,whose implications go beyond the memory model andmay involve spectroscopic applications. For instance,our results implicitly show the feasibility of a quantum-enhanced detection of absorbance at some frequency of aspectrum. Thus, this work represents a significant step inthe progress of quantum technology, demonstrating thefeasibility with easily accessible resources of a quantumscheme of huge practical interest.
Acknowledgments.–
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