Filtering of the absolute value of photon-number difference for two-mode macroscopic quantum superpositions
Magdalena Stobińska, Falk Töppel, Pavel Sekatski, Adam Buraczewski, Marek Żukowski, Maria V. Chekhova, Gerd Leuchs, Nicolas Gisin
aa r X i v : . [ qu a n t - ph ] J a n Filtering of the absolute value of photon-number differencefor two-mode macroscopic quantum superpositions
M. Stobi´nska,
1, 2
F. T¨oppel,
3, 4
P. Sekatski, A. Buraczewski, M. ˙Zukowski,
1, 7
M. V. Chekhova,
3, 8
G. Leuchs,
3, 4 and N. Gisin Institute of Theoretical Physics and Astrophysics,University of Gda´nsk, ul. Wita Stwosza 57, 80-952 Gda´nsk, Poland Institute of Physics, Polish Academy of Sciences, Al. Lotnik´ow 32/46, 02-668 Warsaw, Poland Max Planck Institute for the Science of Light, Guenther-Scharowsky-Str. 1/Bldg. 24, 91058 Erlangen, Germany Institute for Optics, Information and Photonics,University of Erlangen-N¨urnberg, Staudtstr. 7/B2, 91058 Erlangen, Germany Group of Applied Physics, University of Geneva,Chemin de Pinchat 22, CH-1211 Geneva, Switzerland Faculty of Electronics and Information Technology,Warsaw University of Technology, ul. Nowowiejska 15/19, 00-665 Warsaw, Poland University of Science and Technology of China, Hefei, Anhui, China Department of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory, 119991 Moscow, Russia (Dated: September 2, 2018)We discuss a device capable of filtering out two-mode states of light with mode populationsdiffering by more than a certain threshold, while not revealing which mode is more populated. Itwould allow engineering of macroscopic quantum states of light in a way which is preserving specificsuperpositions. As a result, it would enhance optical phase estimation with these states as wellas distinguishability of “macroscopic” qubits. We propose an optical scheme, which is a relativelysimple, albeit non-ideal, operational implementation of such a filter. It uses tapping of the originalpolarization two-mode field, with a polarization neutral beam splitter of low reflectivity. Next, thereflected beams are suitably interfered on a polarizing beam splitter. It is oriented such that itselects unbiased polarization modes with respect to the original ones. The more an incoming two-mode Fock state is unequally populated, the more the polarizing beam splitter output modes areequally populated. This effect is especially pronounced for highly populated states. Additionally,for such states we expect strong population correlations between the original fields and the tappedone. Thus, after a photon-number measurement of the polarizing beam splitter outputs, a feed-forward loop can be used to let through a shutter the field, which was transmitted by the tappingbeam splitter. This happens only if the counts at the outputs are roughly equal. In such a case,the transmitted field differs strongly in occupation number of the two modes, while information onwhich mode is more populated is non-existent (a necessary condition for preserving superpositions).
I. INTRODUCTION
The set of efficiently produced quantum states oflight is limited. It is especially difficult to producenon-classical non-Gaussian superpositions. Nevertheless,with quantum state engineering certain properties of ac-cessible states can be modified or enhanced. In particu-lar, measurement induced state operations which facili-tate preparing a quantum state for some further tasks, al-low filtering out states of required features and may leadto non-Gaussian characteristics of the resulting states.Often, they involve intensity measurements, for whichcrucial are threshold detectors, selecting Fock states ortheir superpositions with sufficiently high population.Examples of low-threshold detectors are realized with sin-gle photon on-off detectors or human eyes [1, 2]. Theycan be applied in setups that perform POVM measure-ments [3] leading to quantum operations. As a result, itis possible to block light of unwanted properties (too lowor too high intensity). More complicated filters for Fockstates utilize interference effects [4, 5]. A more challeng-ing task is to construct a filter selecting states of certainproperties (on request), while preserving quantum super- positions. This is very important for superpositions ofthe Schr¨odinger-cat type.Recently, macroscopic quantum superpositions becameexperimentally accessible for light in the form of themicro-macro singlet state [6] and the entangled brightsqueezed vacuum [7]. In the former state, producedby optimal quantum cloning, a single photon is entan-gled with a “macroscopic” qubit in a polarization singletstate. The latter is a macroscopic analog of two-photonpolarization Bell states [8]. Since these states combinequantum properties with macroscopic population andcould enable efficient light-matter coupling, they are in-teresting for quantum information technology: quantummemory [9–11], quantum key distribution [12], quantummetrology [13, 14] and macroscopic Bell tests [15, 16].However, their distinguishability is low in analog detec-tion and they are easily destroyed by losses [17–20]. Spe-cial quantum state filtering applied to these states giveshope to solve the problem of detection and to enhancetheir properties useful for quantum technology tasks.We present a theory of a device capable of filtering outtwo-mode states of light with mode populations differingby more than a certain threshold. We call it modulusof intensity difference filter (MDF). It performs a non-Gaussian operation and works as quantum scissors [21]for general two-mode Fock state superpositions. We showthat, effectively, MDF filters out superpositions of N00N-like components, allowing an enhanced optical phase es-timation with macroscopic quantum states of light. Wealso show that it improves distinguishability of “macro-scopic” qubits in realistic scenarios.We propose a simple optical scheme, which gives anapproximate operational implementation of such a filterfor two orthogonal (linear) polarization modes. The fieldis fed into a polarization neutral (tapping) beam splitterof low reflectivity. The weak reflected modes are suitablyinterfered on a polarizing beam splitter oriented such thatit selects diagonal and anti-diagonal polarization modeswith respect to the original ones. The more an incomingtwo-mode Fock state is unequally populated, the morethe output modes are roughly equally populated. Sincethe reflected and transmitted beams are correlated, esti-mating the modulus of population difference for the for-mer gives an estimate for the latter. This effect is es-pecially pronounced for highly populated states. After aphoton-number measurement of the outputs of the polar-izing beam splitter, a feed-forward loop can be used to letthrough a shutter the field, which was transmitted by thetapping beam splitter, only in the case of roughly equalcounts at the outputs. Such a field differs strongly in oc-cupation number of the two modes, while information onwhich mode is more populated is non-existent. Thus, anecessary condition for preserving superpositions is sat-isfied.The paper is organized as follows. In Section II we dis-cuss the theoretical description and properties of modu-lus of intensity difference filter. In Section III we ana-lyze the action of the theoretical MDF on “macroscopic”qubits, a part of micro-macro polarization singlet state.Section IV is devoted to the operational scheme givingeffectively an MDF.
II. THEORY AND PROPERTIES OF MDF
We define an MDF as a device which performs thefollowing projection operation P δ th = ∞ X k,l =0; | k − l |≥ δ th | k, l ih k, l | , (1)where | k, l i is a two-mode Fock state. For simplicity, letus consider polarization modes. If δ th > | k − l | < δ th ), and preserves the oneswith the modulus of difference above it ( | k − l | ≥ δ th ).We would like to comment on two key features of thefilter. First of all, it estimates the absolute value of thedifference instead of the difference. This procedure is ex-perimentally more demanding, but it has an advantage. (a) YESYESNO ( S ) ( S ) δ th δ th kl − − − − − − − − − + + + + + + + + + ( S ) ( S ) δ th δ th kl (b)FIG. 1: Comparison of two filtering techniques: absolute dif-ference (MDF) (a) and orthogonality filter (OF) (b). Thedots in (b) symbolize specific possible measurement results ofphoton numbers. The state of the field filtered by an OF isrepresented by one of the dots. In the case of MDF, the stateis projected onto the whole YES region, which preserves quan-tum coherence of components occupying both regions. k and l denote numbers of photons in two orthogonal polarizationmodes. Since all non-zero eigenvalues of the operator P δ th areequal to 1, the filter does not provide any information onwhich polarization mode was more populated. Thus, if aqubit is encoded in highly populated polarization states,like e.g. in Eq. (2), it does not discriminate these statesand filters them fairly. This property is important for allquantum protocols requiring state preparation withoutthe state readout. The other main feature is that thefiltering is performed in a “yes”-“no” manner: the exactvalue of the modulus is never measured. This is a keyproperty for quantum protocols which require engineer-ing preserving the superposition. For these reasons wecall this device a filter.These features are the main difference between theMDF and the orthogonality filter (OF) executing directintensity difference measurements [22]. The OF is thebasic element in setups performing measurement inducedoperations on macroscopic polarization states [15]. Con-trary to the MDF which performs a non-destructive mea-surement, the OF destroys superpositions and allows onlyfor efficient state discrimination in detection, not filter-ing, and is not suitable for preselection strategies in Belltests [15]. In the case of a micro-macro singlet, it identi-fies the state and breaks entanglement. The action of theMDF and OF is compared in Fig. 1. MDF projects onto S and S area. Superpositions of components belongingto S and S are preserved. OF, combined with photo-multipliers, projects the state on a Fock state either in S or S , illustrated as a red or blue dot in Fig. 1. III. FILTERING OF “MACROSCOPIC” QUBITS
Let us analyze the action of the operator P δ th on spe-cific “macroscopic” qubits (macro-qubits), which are themacroscopic part of micro-macro polarization singlets.They are produced by optimal phase covariant quan-tum cloning via phase sensitive parametric amplifica-tion [2, 22, 23] of single photons of a defined polarization( ϕ or ϕ ⊥ , respectively) | Φ i = ∞ X i,j =0 γ ij (cid:12)(cid:12) i + 1 , j i , (2) | Φ ⊥ i = ∞ X i,j =0 γ ij (cid:12)(cid:12) j, i + 1 i , where e.g. states | k, l i represent k photons in po-larization state | ϕ i , and l in | ϕ ⊥ i , which in turnare defined as | ϕ i = ( e iϕ | H i + e − iϕ | V i ) / √ | ϕ ⊥ i = i ( e iϕ | H i − e − iϕ | V i ) / √ H and V represent linear horizontal and vertical polariza-tions. The probability amplitudes equal γ ij =cosh g − ((tanh g ) / i + j p (1 + 2 i )!(2 j )! /i ! /j !, where g isthe parametric gain. Due to a different parity of occupa-tion numbers of the two polarizations, the states | Φ i and | Φ ⊥ i are orthogonal.In a recent experiment [22], realizations of such statescontained up to 4 sinh g ≃ photons on average.However, in high photon number regime the detectors arenot single photon resolving, but distinguish counts vary-ing by at least ±
150 photons [23]. Thus, macro-qubitsare hardly distinguishable with direct detection [22].To overcome this problem, an MDF could be used toenhance the distinguishability. Two important traits ofthe states are crucial. The average number of photonsin polarization ϕ in | Φ i is three times higher than thenumber of photons in polarization ϕ ⊥ , and vice versa for | Φ ⊥ i . Further, if one excludes superposition compo-nents with approximately identical numbers of photonsin the two polarizations, this ratio increases. Thus, anMDF would definitely increase the distinguishability ofthe states.Imagine a scheme which uses an MDF, and behindit we place detection station which measures number ofphotons in the two polarization modes. In such a case thedistinguishability may be quantified in terms of photondistributions p Φ ( k, l ) = |h k, l | Φ i| and p Φ ⊥ ( k, l ) givingthe probabilities of finding simultaneously k photons inpolarization ϕ and l in ϕ ⊥ . For the filtered macro-qubitswith the operator P δ th they equal (see Appendix A) p Φ ( k, l ) = ∞ X i,j =0; | i +1 − j |≥ δ th ˜ γ ij δ k, i +1 δ l, j , p Φ ⊥ ( k, l ) = p Φ ( l, k ) , (3)where ˜ γ ij are renormalized γ ij , and δ a,b is the Kroneckerdelta. Since the distribution p Φ ⊥ is mirror reflected withrespect to p Φ along the k = l line, we divide the space( k, l ) into two triangular areas S for k ≥ l and S for k < l . The distinguishability reads v = P ( S )Φ − P ( S )Φ ⊥ = P ( S )Φ − P ( S )Φ , (4)where P ( S i )Φ = P k,l ∈ S i p Φ ( k, l ) is the probability of find-ing | Φ i in S i and P ( S )Φ + P ( S )Φ = 1. It increases if | Φ i ( | Φ ⊥ i ) starts to occupy mostly one of S i regions, e.g. S (a) − − − − k l − − − − − − . × − . × − S S (b) − − − − − − k l − − − − − − . × − . × − S S FIG. 2: Photon distribution p Φ for the macroscopic state | Φ i computed for g = 1 .
87 and filtering threshold δ th = 0 (a) and δ th = 200 (b). k and l denote numbers of photons in twoorthogonal polarization modes. The one-dimensional plotsshow values of p Φ for k = 0 (the left one) and l = 0 (thebottom one), respectively. ( S ), with increasing δ th . Fully distinguishable (indistin-guishable) states have v = 1 ( v = 0).Originally, the photon-number distribution p Φ ( k, l ) oc-cupies both S and S and is almost equally distributedbetween them giving v = 0 .
64, independently of the gain g , see Fig. 2a. Fig. 2 is plotted for g = 1 .
87. The filter-ing cuts out a stripe, √ δ th wide, located symmetricallyalong the k = l line. In Fig. 2b we took δ th = 200.The state | Φ i occupies two disjoint regions of space: thebottom ( S ) and top ( S ) triangles, but increasing thethreshold from δ th = 0 to δ th = 200 reduces the contri-bution of p Φ in S : the peak value goes down originallyfrom 8 . · − to 1 . · − . Simultaneously, the distri-bution peak in S increases from 1 . · − to 3 . · − .Similar behavior is observed for higher gains. The be-havior of p Φ ⊥ is identical but mirror reflected. Thus,distinguishability increases.The effect of increased distinguishability remains evenin the presence of losses. The losses can be modeled bya beam splitter (BS) with a reflectivity R (see AppendixA) put in front of an ideal detector. The p Φ distributionsevaluated for g = 1 . δ th = 200 and 50% and 90% oflosses are depicted in Fig. 3. The loss results in shiftingthe distribution towards the origin of the coordinates,i.e. the vacuum state. The distribution peaks become (a) − − − − − − k l − − − − − − . × − . × − S S (b) − − − − − − k l − − − − − − . × − . × − S S FIG. 3: Photon distribution p Φ for the macroscopic state | Φ i computed for g = 1 .
87, filtering threshold δ th = 200 and 50%(a) and 90% (b) of losses. k and l denote numbers of photonsin two orthogonal polarization modes. The one-dimensionalplots show values of p Φ for k = 0 (the left one) and l = 0 (thebottom one), respectively. . . . . . . R . . . . . . v δ th = 0 δ th = 40 δ th = 200 δ th = 400 FIG. 4: Distinguishability v of macro-qubits (Eq. (4)) eval-uated for gain g = 1 .
87 and several threshold values δ th asfunction of losses R . smooth and symmetric. The edges along the thresholdlines are blurred and the bigger the losses, the smallerthe width of the gap. It disappears completely for 90%of losses. With increasing losses the height of the upperand left peak first drops, and next increases, because thetotal probability over the whole space ( k, l ) has to be 1.For states (2) we have numerically computed their dis- FIG. 5: An approximate operational scheme of an MDF. Thebox MDF in (b) is the setup given in (a). The details are inthe main text. tinguishability v for gain g = 1 .
87 and several filteringthresholds δ th as a function of losses, see Fig. 4. If nofiltering is applied, then v = 0 .
64, but drops quickly to 0if
R > .
9. If δ th increases, v increases as well and ap-proaches unity with a reasonable probability of success,e.g. v = 0 .
96 with p s = 10 − . Obviously, for R = 1 thestates become vacuum and we get v = 0 independentlyof δ th (this is indicated by an open circle in the uppercurves and a full circle in the solid line in Fig. 4). IV. SIMPLE OPERATIONAL SCHEME FORAPPROXIMATE MDF
Our scheme for an approximate realization of an MDFfor polarization modes is shown in Fig. 5a. The setupin Fig. 5b shows its application for the measurement in-duced operations on quantum states. It uses tapping ofthe original field, with a polarization neutral BS of a lowreflectivity (Fig. 5b). The reflected beams, a r , a r ⊥ aresuitably interfered on a polarizing beam splitter (PBS)oriented such that it selects unbiased polarization modeswith respect to the original ones (Fig. 5a). The more anincoming two-mode Fock state is unequally populated,the more the output modes are roughly equally pop-ulated. This effect is especially pronounced for highlypopulated states, and additionally for such states we ex-pect strong population correlations between the originalfields and the tapped one. Thus, after a photon-numbermeasurement of PBS outputs, a feed-forward loop can beused to let through a shutter the field, that was trans-mitted by the tapping BS. This happens only in the caseof roughly equal counts at the outputs. Such a field dif-fers strongly in occupation number of the two modes,while information on which mode is more populated isnon-existent (a necessary condition for preserving super-positions).Let us move to the details of operation of the part ofthe device shown in Fig. 5a. A two-mode r , r ⊥ polar-ization light beam enters PBS which works in a basis d , d ⊥ unbiased with respect to the basis in which wewrite the original superposition. For example, the beamcould be defined in diagonal/ anti-diagonal basis, whilePBS may select in left-handed/right-handed polarizationbasis. Let us denote the annihilation operators of thepolarization modes entering PBS by a r , a r ⊥ . PBS trans-forms them according to the unitary operation such thatits output mode operators equal a d = 1 / √ a r + a r ⊥ ), a d ⊥ = 1 / √ a r ⊥ − a r ). The two orthogonally polar-ized exit beams d and d ⊥ , propagate to a pair of detec-tors, which measure their photon numbers I d = K and I d ⊥ = L .We will examine the work of the setup (Fig. 5a)by its action on a general two-mode polarization in-put state which is a Fock state | n, m i r . Detection be-hind PBS projects this state onto a two-mode Fock state | K, L i d = √ K ! L ! a † dK a † d ⊥ L | i . The states | n, m i r forma basis in the considered subspace of photon states.Note, that one can introduce a different indexation ofthe basis, namely | ( S r + ∆ r ) , ( S r − ∆ r ) i r , where S r = n + m and ∆ r = n − m , which is one-to-one. Let us denote such basis states | Ψ S r , ∆ r i r . Thestates | K, L i d also form such a basis, which is relatedto the previous one via the unitary transformation ofBS. The probability of obtaining | K, L i d from | Ψ S r , ∆ r i r input is p ( K, L | S r , ∆ r ) = |h Ψ S r , ∆ r | K, L i d | . However, p ( K, L | S r , ∆ r ) = p ( S r , ∆ r | K, L ) due to the bi-stochasticnature of such quantum probabilities [24]. Note that themeasured total number of photons S = K + L , if the ini-tial state is | Ψ S r , ∆ r i r , must be S = S r . Let us change thevariables L and K , so that they would correspond to thequantities useful for the further analysis of the filtering:the total sum S and the population difference ∆ = L − K of the registered photons. The probability distributionof the occupation difference ∆ r in the incoming modes r and r ⊥ given that S and ∆ were measured p S, ∆ (∆ r ) = p ( S r , ∆ r | ( S − ∆) , ( S + ∆)), due to the fact that underBS transformation p ( S r , ∆ r | ( S − ∆) , ( S + ∆)) is pro-portional to the Kronecker delta δ S r ,S , simplifies to thefollowing p S, ∆ (∆ r ) = 12 S ( S − ∆2 )!( S +∆2 )! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S − ∆2 X q =0 S +∆2 X p =0 δ p + q, S − δ (5) (cid:18) S − ∆2 q (cid:19)(cid:18) S +∆2 p (cid:19) ( − p q ( S − ∆ r )!( S +∆ r )! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The calculations that lead one to the formula closely re-semble the ones presented in Appendix B, for a slightlymore general process.The analysis of Eq. (5) shows that for a Fock stateinput with | ∆ r | ≈ | ∆ | ≈ S with higher prob-ability than | ∆ | ≈ Vice versa , when | ∆ r | ≈ S theresult | ∆ | ≈ | ∆ | ≈ S [25]. Thus,the filter works probabilistically and for any outcome S (a) p ( | ∆ r | ≥
30) = 0 . p ( | ∆ r | <
30) = 0 . − − −
30 0 30 100 200 ∆ r . . . . . . . p , (∆ r ) (b) p ( | ∆ r | ≥
30) = 0 . p ( | ∆ r | <
30) = 0 . − − −
30 0 30 100 200 ∆ r . . . . . . . p , (∆ r ) (c) p ( | ∆ r | ≥
30) = 0 . p ( | ∆ r | <
30) = 0 . − − −
30 0 30 100 200 ∆ r . . . . . . . p , (∆ r ) FIG. 6: Distribution of the population difference p S, ∆ (∆ r ) ina superposition Fock input state | Ψ in i = | n, m i r conditionedon the measurement of S = 200 photons and ∆ = 0 (a),∆ = 80 (b), ∆ = 200 (c) at the PBS output. The verticaldashed lines show the threshold δ th = 30. The probabilitythat | ∆ r | ≥
30 is given by p ( | ∆ r | ≥ and ∆ obtained all values of ∆ r are possible, but notequally probable. So we argue if K and L differ little,∆ ≈ | ∆ r | ≈ S is the most probable case, which meansthat a large initial population difference is anticipated.If K and L differ a lot, ∆ ≈ S , we obtain that | ∆ r | ≈ p S, ∆ (∆ r ) plotted for exemplary values of S = 200,∆ = 0, ∆ = 80 and ∆ = 200. The erratic shape of dis-tributions in Figs. 6 reveals the interference between twonon-zero Fock states entering a beamsplitter.Imposing a filtering threshold in Eq. (1) correspondsto fixing two independent threshold values. We choosea threshold value δ th for which we check if | ∆ r | ≥ δ th .Next, since the process is probabilistic (is governed bythe probability distribution p S, ∆ (∆ r )), we fix the levelof trust for it, i.e. the minimum probability, e.g. equal90%, with which the condition | ∆ r | ≥ δ th is fulfilled. Theprobability that the condition holds true is denoted by p ( | ∆ r | ≥ δ th ). It is evaluated by summing all probabili-ties p S, ∆ (∆ r ) of these possibilities where | ∆ r | ≥ δ th , i.e.for ∆ r ∈ [ − S, − δ th ] ∪ [ δ th , S ]. Thus, if for a fixed value of δ th S increases, the probability p ( | ∆ r | > = δ th ) increasesas well. In Fig. 6 we set δ th = 30. For ∆ = 200, the prob-ability of | ∆ r | ≥
30 equals p ( | ∆ r | ≥
30) = 0 . < . p ( | ∆ r | ≥
30) = 0 . t and reflectivity r ≪ t . This, in case of high photon num-bers, means splitting with highest probability of photonnumbers (of incoming two-mode Fock basis states) alsoin this proportion and that the initial ratio of occupa-tions of the two polarization modes in a Fock componentis preserved in the reflected and transmitted beams.We will illustrate the action of the tapping and thefeed-forward loop from Fig. 5b using a Fock state | Ψ in i = | n, m i with an unknown initial population difference∆ = n − m . After the tapping BS, v photons of n arereflected from the first and w photons of m are reflectedfrom the second input mode. The possible mode popula-tion differences equal ∆ r = v − w in the reflected beamand ∆ t = n − v − m + w in the transmitted beam, where v ∈ [0 , n ], w ∈ [0 , m ]. The mode occupation differenceregistered at the detectors reads again ∆ = L − K . Ifthe reflectivity of the tapping BS is r = 10%, the analy-sis of the probability distribution for the BS shows thatfor highly populated input ∆ r ≃ . and ∆ t ≃ . .Now, the problem is reduced to the previously discussed:from the analysis below Eq. (5) we know, that if the mea-sured in MDF ∆ ≃
0, than entering MDF difference ∆ r and thus ∆ t are large; vice versa , if ∆ is large, ∆ r ≃ t ≃
0. In this setup, we directly setthe threshold δ th from Eq. (1) for the transmitted beam,i.e. we require that | ∆ t | ≥ δ th , and the analysis of thereflected beam by MDF tells us the probability distri-bution of the population difference for the transmittedbeam p S, ∆ (∆ t ) and thus, the probability p ( | ∆ t | ≥ δ th )with which this condition is fulfilled. Only if it is highenough, the MDF opens the shutter.The above discussion applies also for Fock superposi-tion states. See Appendix B for the complete calculusof the state evolution through the setup from Fig. 5b foran arbitrary superposition state and the derivation of theprobability distribution of the population difference forthe transmitted beam p S, ∆ (∆ t ) (Eq. 17).Finally, we would like to mention that the assumptionof the accurate measurement of K and L numbers is jus- tified: a setup involving losses after the tapping BS isequivalent to a setup with losses introduced in the re-flected beam before the detectors. In the latter case,losses account for the imperfect detection. Thus, con-sidering losses only in the transmitted part and perfectdetection in the reflected part gives the full view. In ex-periments, a measurement accuracy of 150 photons, to-gether with mean photon numbers per mode 10 , wouldgive a very good relative accuracy.The discussion concerning weak invasibility of theMDF measurement on the beam leaving the shutter ismoved to the Appendix C. V. CONCLUSIONS
Thus, we have shown that the MDF is feasible andallows one to perform a threshold measurement whilemaintaining quantum superpositions. It works for anyhighly populated two-mode polarization states contain-ing a single frequency and wavevector mode. Realizationof such a device is demanding, but the properties of theMDF are worth the effort. The filter would be useful inthe engineering of macroscopic quantum states of light.In the case of macro-qubits it circumvents the problemof inefficient detection, and improves distinguishability.Thus, it makes them useful in quantum information andmetrology protocols.
Acknowledgments
This work was supported by the EU FP7 Marie CurieCareer Integration Grant No. 322150 ”QCAT”, MNiSWgrant No. 2012/04/M/ST2/00789 and FNP Homing Plusproject. MZ acknowledges EU Q-ESSENCE project,MNiSW (NCN) grant No. N202 208538. MC acknowl-edges EU FP7 BRISQ2 project (grant No. 308803). Cal-culations were performed at CI TASK in Gda´nsk andCyfronet in Krak´ow.
Appendix A: Action of theoretical MDF onmacro-qubits taking into account losses
After filtering with the operator P δ th the macro-qubitsin Eq. (2) take the form | Φ i = ∞ X i,j =0; | i +1 − j |≥ δ th ˜ γ ij (cid:12)(cid:12) i + 1 , j i , (6) | Φ ⊥ i = ∞ X i,j =0; | i +1 − j |≥ δ th ˜ γ ij (cid:12)(cid:12) j, i + 1 i , where the new probability amplitudes ˜ γ ij ensure the cor-rect normalization. Next, the filtered macro-qubits aresubjected to losses, modeled by a BS with the reflectivity R , which transforms them into mixed states ρ Φ = ∞ X i,j =0; | i +1 − j |≥ δ th ˜ γ ij ∞ X i ′ ,j ′ =0; | i ′ +1 − j ′ |≥ δ th ˜ γ i ′ j ′ min(2 i +1 , i ′ +1) X n =0 min(2 j, j ′ ) X m =0 c (2 i +1) n c (2 j ) d c (2 i ′ +1) n c (2 j ′ ) d | i + 1 − n, j − m ih i ′ + 1 − n, j ′ − m | , (7) ρ Φ ⊥ = ∞ X i,j =0; | i +1 − j |≥ δ th ˜ γ ij ∞ X i ′ ,j ′ =0; | i ′ +1 − j ′ |≥ δ th ˜ γ i ′ j ′ min(2 i +1 , i ′ +1) X n =0 min(2 j, j ′ ) X m =0 c (2 i +1) n c (2 j ) d c (2 i ′ +1) n c (2 j ′ ) d | j − m, i + 1 − n ih j ′ − m, i ′ + 1 − n | , where c ( x ) n = q(cid:0) xn (cid:1) R n (1 − R ) x − n is the BS probabilityamplitude for the BS reflecting of n from x photons.The photon number distribution for these states is p Φ ( k, l ) = Tr { ρ Φ | k, l ih k, l |} , (8) p Φ ⊥ ( k, l ) = Tr { ρ Φ ⊥ | k, l ih k, l |} ,p Φ ( k, l ) = ∞ X i,j =0; | i +1 − j |≥ δ th ˜ γ ij (cid:16) c (2 i +1)2 i +1 − k (cid:17) (cid:16) c (2 j )2 j − l (cid:17) Θ(2 i + 1 − k )Θ(2 j − l ) , (9) p Φ ⊥ ( k, l ) = ∞ X i,j =0; | i +1 − j |≥ δ th ˜ γ ij (cid:16) c (2 i +1)2 i +1 − l (cid:17) (cid:16) c (2 j )2 j − k (cid:17) Θ(2 i + 1 − l )Θ(2 j − k ) , where Θ( x ) = 1(0) for x ≥ x < Appendix B: MDF measurement and the stateevolution in tapping and feed-forward loop
In this appendix we will present the evolution of aninput state | Ψ in i = P n,m ξ nm | n, m i entering the setupdepicted in Fig. 5b.Fig. 7 illustrates each stage of the experiment per-formed by this setup. At stage 1 this state impingeson a tapping BS, with the reflectivity coefficient r , whichacts independently on both polarization modes. This re-sults in transformation U BS | n, m i . Its action on a singlepolarization Fock state reads U BS | , n i = n X v =0 c ( n ) v | v i r | n − v i t , (10) c ( n ) v = s(cid:18) nv (cid:19) r v (1 − r ) n − v . FIG. 7: Physical implementation of the MDF with the no-tation indicating the state evolution in different parts of thesetup.
The index r ( t ) corresponds to the reflected (transmitted)part. The input state is transformed to | Ψ i = U BS | Ψ in i where | Ψ i = X n,m ξ nm n X v =0 m X w =0 c ( n ) v c ( m ) w | v, w i r | n − v, m − w i t . (11)Next, in stage 2, the reflected beam impinges on the PBS.It transforms the operators a r and a r ⊥ according to thetransformation a d = 1 / √ a r + a r ⊥ ), a d ⊥ = 1 / √ a r ⊥ − a r ). The reflected part | v, w i r = √ v ! w ! ( a † r ) v ( a † r ⊥ ) w looksas follows U PBS | v, w i r = √ v ! w ! 1 √ v + w ( a † d − a † d ⊥ ) v ( a † d + a † d ⊥ ) w | i = √ v ! w ! 1 √ v + w v X p =0 w X q =0 (cid:18) vp (cid:19)(cid:18) wq (cid:19) ( a † d ) p ( − a † d ⊥ ) v − p ( a † d ) q ( a † d ⊥ ) w − q | i = √ v ! w ! 1 √ v + w v X p =0 w X q =0 (cid:18) vp (cid:19)(cid:18) wq (cid:19) ( − v − p ( a † d ) p + q ( a † d ⊥ ) v + w − p − q | i . (12)After the PBS the state equals | Ψ i = U PBS U BS | Ψ in i| Ψ i = X n,m ξ nm n X v =0 m X w =0 c ( n ) v c ( m ) w √ v ! w ! √ v + w v X p =0 w X q =0 (cid:18) vp (cid:19)(cid:18) wq (cid:19) ( − v − p p ( p + q )! ( v + w − p − q )! | p + q, v + w − p − q i d | n − v, m − w i t . (13)In stage 3 the detectors detect two Fock states | K, L i d and project the state | Ψ i to | Ψ i = d h K, L |U PBS U BS | Ψ in i | Ψ i = X n,m ˜ ξ nm n X v =0 m X w =0 c ( n ) v c ( m ) w √ v ! w ! √ v + w v X p =0 w X q =0 (cid:18) vp (cid:19)(cid:18) wq (cid:19) ( − v − p p ( p + q )! ( v + w − p − q )! δ K,p + q δ L,v + w − p − q | n − v, m − w i t = X n,m ˜ ξ nm n X v =0 m X w =0 c ( n ) v c ( m ) w √ v ! w ! √ v + w v X p =0 w X q =0 (cid:18) vp (cid:19)(cid:18) wq (cid:19) ( − v − p √ K ! L ! δ K,p + q δ L,v + w − K | n − v, m − w i t . (14)The coefficients ˜ ξ nm are renormalized to ensure normal-ization of | Ψ i .For the further discussion of the filtering process it isuseful to compute the conditional photon number distri-bution for the transmitted beam p K,L ( k, l ) = |h k, l | Ψ i| p K,L ( k, l ) = K ! L ! (cid:16) X n,m ˜ ξ nm n X v =0 m X w =0 c ( n ) v c ( m ) w √ v ! w ! √ v + w δ L,v + w − K δ k,n − v δ l,m − wv X p =0 w X q =0 (cid:18) vp (cid:19)(cid:18) wq (cid:19) ( − v − p δ K,p + q (cid:17) . (15)We change the variables L and K so that they were cor-responding to the quantities useful for the filtering: thetotal sum of the registered photons S = L + K and thedifference in the occupation of the polarization modes∆ = L − K . We obtain p S, ∆ ( S t , ∆ t ) with S t = k + l ,∆ t = k − lp S, ∆ ( S t , ∆ t ) = (cid:0) S +∆2 (cid:1) ! (cid:0) S − ∆2 (cid:1) ! (cid:16) X n,m ˜ ξ nm n X v =0 m X w =0 c ( n ) v c ( m ) w √ v ! w ! √ v + w δ S +∆2 ,v + w − S − ∆2 δ S t +∆ t ,n − v δ S t − ∆ t ,m − wv X p =0 w X q =0 (cid:18) vp (cid:19)(cid:18) wq (cid:19) ( − v − p δ S − ∆2 ,p + q (cid:17) . (16)The probability distribution for the occupation differencein the transmitted beam ∆ t is given by p S, ∆ (∆ t ) = ∞ X S t =0 p S, ∆ ( S t , ∆ t ) . (17)The filtering is performed in stage 4 of the experiment.Here, the detectors’ readings are analyzed and only thoseevents and realizations of | Ψ i are accepted where ∆ ≃ (a) p ( | ∆ t | ≥ . p ( | ∆ t | < . − − − − −
40 0 40 80 120 160 200 ∆ t . . . . . . . . . . . p , (∆ t ) (b) p ( | ∆ t | ≥ . p ( | ∆ t | < . − − − − −
40 0 40 80 120 160 200 ∆ t . . . . . . . . p , (∆ t ) (c) p ( | ∆ t | ≥ . p ( | ∆ t | < . − − − − −
40 0 40 80 120 160 200 ∆ t . . . . . . . . . . . p , (∆ t ) FIG. 8: Distribution of the population difference p S, ∆ (∆ t )in the transmitted beam t after the shutter for the state inEq. (18) with S = 200 assuming that S = 20 photons wereregistered in the reflected beam and the difference measuredby detectors was ∆ = 0 (a), ∆ = 10 (b), ∆ = 20 (c). Thevertical dashed lines show the threshold δ th = 150. The prob-ability that | ∆ t | ≥
150 is given by p ( | ∆ t | ≥ Example
We consider a simple superposition of Fock states withfixed total photon number S (it allows avoiding the sum-mation over S t in Eq. (17)) and with a uniform distribu-tion of the occupation difference ∆ | Ψ in i = 1 / p S + 1 S X n =0 | n, S − n i . (18)In Fig. 8 we have depicted the probability distributions p S, ∆ (∆ t ) for this state with S = 200 for three cases:∆ = 0, ∆ = 10 and ∆ = 20 for S = 20. These plotsreveal that for small ∆ ≈ t in the transmitted beam are large. The higher ∆is, the more probable are the superposition componentswith ∆ t = 0 to be present in the output beam. Wetook δ th = 150 and the probabilities that | ∆ t | ≥ . . .
001 for ∆ = 0, ∆ = 10, ∆ = 20,respectively.
Appendix C: Small disturbance by MDFmeasurement of ”macroscopic” qubits
In reality, one would aim at applying the MDF to morecomplex quantum states, the superpositions like the onegiven in Eq. (2), which constitute a “macroscopic” qubit.The goal of the MDF apart form filtering of those statesand increasing their distinguishability in classical detec-tion, is to avoid discriminating between them. Moreover,usually the experimental conditions are not perfect andin the analysis of the action of the filter one has to takeinto account the multi-mode character of the input stateand the losses. We will discuss these issues in this section.Imagine a source producing a micro-macro polariza-tion singlet state of the form | Ψ − i = ( | i A | Φ ⊥ i B −| ⊥ i A | Φ i B ) / √
2. The macroscopic part B of the singlet (a) p ( | ∆ t | ≥
40) = 0 . p ( | ∆ t | <
40) = 0 . − − −
40 0 40 80 120 ∆ t . . . . . . . . . . p , (∆ t ) (b) p ( | ∆ t | ≥
40) = 0 . p ( | ∆ t | <
40) = 0 . − − −
40 0 40 80 120 ∆ t . . . . . . . . . . . p , (∆ t ) (c) p ( | ∆ t | ≥
40) = 0 . p ( | ∆ t | <
40) = 0 . − − − −
30 0 30 60 90 120 ∆ t . . . . . . . . . . p , (∆ t ) FIG. 9: Distribution of the population difference p S, ∆ (∆ t )(Eq. 17) in the transmitted beam t after the shutter for ρ in = 1 / | Φ ih Φ | + | Φ ⊥ ih Φ ⊥ | ) for g = 1 .
87 assuming that S = 20 photons were registered in the reflected beam andthe difference measured by detectors was ∆ = 0 (a), ∆ = 10(b), ∆ = 20 (c). The vertical dashed lines show the thresh-old δ th = 40. The probability that | ∆ t | ≥
40 is given by p ( | ∆ t | ≥ is fed to the setup in Fig. 5b. The initial state reads ρ in = 1 / | Φ ih Φ | + | Φ ⊥ ih Φ ⊥ | ) . (19)The state passes through the whole setup in Fig. 5b. InFig. 9 we depicted the probability distributions p S, ∆ (∆ t )(Eq. (17) with ˜ ξ nm = ˜ γ nm ) for this state as a functionof the population difference ∆ t in the transmitted beam t after the shutter. In our computation we assumed the v = 0 . − − − − − k l S S − − − − − − v = 0 . − − − − − k l S S − − − − − − v = 0 . − − − − − k l S S − − − − − − v = 0 . − − − − − k l S S − − − − − − FIG. 10: Photon number distribution p Φ (Eq. (20)) and dis-tinguishability v (Eq. (4)) of the macroscopic state | Φ i pro-cessed by the setup from Fig. 5b, computed for g = 1 .
87, thelevel of trust 90% and δ th = 0 (a), δ th = 5 (b), δ th = 10(c), δ th = 15 (d). k and l denote numbers of photons in twoorthogonal polarization modes. g = 1 . S = 20 photons registered in the reflectedbeam and chose δ th = 40. The probabilities p ( | ∆ t | ≥ | ∆ t | ≥
40 are: 0 .
87, 0 .
77, 0 .
01 for ∆ = 0, ∆ = 10,∆ = 20, respectively.We also computed the photon number distributions(useful for the distinguishability estimation) for ρ in pro-cessed by the setup in Fig. 5b and compared them withthe distributions obtained in theoretical filtering per-formed by P δ th which are displayed in Fig. 2. The photonnumber distribution for ρ in reads p Φ ( k, l ) = X S ∈ S p S, ∆=0 ( k, l ) , (20)where p S, ∆=0 ( k, l ) is given by Eq. (15) and S is a set of S for which the filter shutter is open, i.e. the probabilityof | ∆ t | ≥ δ th evaluated for ρ in is greater than a givenlevel of trust. We chose δ th = 0, 5, 10, 15 and the level oftrust 90%. The distribution p Φ ( k, l ) and the correspond-ing distinguishabilities are depicted in Fig. 10. Althoughthere is no clear separation between the regions S and (a) p ( | ∆ t | ≥
35) = 0 . p ( | ∆ t | <
35) = 0 . − − − −
30 0 30 60 90 120 ∆ t . . . . . . . . . p , (∆ t ) (b) p ( | ∆ t | ≥
35) = 0 . p ( | ∆ t | <
35) = 0 . − − − −
30 0 30 60 90 120 ∆ t . . . . . . . . . p , (∆ t ) (c) p ( | ∆ t | ≥
35) = 0 . p ( | ∆ t | <
35) = 0 . − − − −
30 0 30 60 90 120 ∆ t . . . . . . . . . . . p , (∆ t ) FIG. 11: Distribution of the population difference p S, ∆2 (∆ t )(Eq. (21)) in the transmitted beam t after the shutter forthe two-mode state in Eq. (19) for g = 1 .
87 assuming that S = 20 photons were registered in the reflected beam and thedifference measured by detectors was ∆ = 0 (a), ∆ = 10 (b),∆ = 20 (c). S here, still, some low-probability gap appears which re-sults in the increase of the distinguishability. For δ th = 0,5, 10 and 15, the distinguishabilities are 0 .
72, 0 .
93, 0 . .
97, respectively.
Multi-mode case and Losses
Let us consider two spatial or frequency modes in theinput state in Eq. (19). Since the two modes are inde-pendent, the probability distribution p K,L ( k, l ) resultingfrom detecting K = n + n and L = m + m photonsin the detectors, where n ( n ) and m ( m ) are the con-tributions which come from the first (second) mode, isgiven by the convolution p K,L ( k, l ) = K X n =0 L X m =0 k X k =0 l X l =0 p n ,m ( k , l ) p K − n ,L − m ( k − k , l − l ) . (21) (a) p ( | ∆ t | ≥
35) = 0 . p ( | ∆ t | <
35) = 0 . − − − −
20 0 20 40 60 80 ∆ t . . . . . . . . . . p , R (∆ t ) (b) p ( | ∆ t | ≥
35) = 0 . p ( | ∆ t | <
35) = 0 . − − − −
20 0 20 40 60 80 ∆ t . . . . . . . . . . p , R (∆ t ) (c) p ( | ∆ t | ≥
35) = 0 . p ( | ∆ t | <
35) = 0 . − − − −
20 0 20 40 60 80 ∆ t . . . . . . . . . . . p , R (∆ t ) FIG. 12: Distribution of the population difference p S, ∆ R (∆ t )(Eq. (22)) in the transmitted beam t after the shutter forthe state in Eq. (19) subjected to 20% of losses for g = 1 . S = 20 photons were registered in the reflectedbeam and the difference measured by detectors was ∆ = 0 (a),∆ = 10 (b), ∆ = 20 (c). g = 1 . S = K + L = 20, ∆ = L − K = 0,10, 20), but for lower threshold δ th = 35 we achievedsimilar values of probabilities for a successful filtering p ( | ∆ t | ≥
35) equal to: 0 . . .
061 for ∆ = 0, 10,20, respectively.Next, we computed the probability distribution p S, ∆ R (∆ t ) (Eq. (22) with ˜ ξ nm = ˜ γ nm in Appendix D) forthe state in Eq. (19) subjected to R = 20% of losses, seeFig. 12. Clearly, the filtering effect is preserved even forhigh losses. The higher gain and thus, the state popu-lation, the higher losses are tolerable. Effectively, lossesdiminish the available threshold values in comparison tothe ideal case. Appendix D: Losses
The probability distribution p S, ∆ R (∆ t ) for the state inEq. (19) subjected to losses R reads p S, ∆ R ( S t , ∆ t ) = X n,m ˜ ξ nm n X v =0 m X w =0 f ( v, w ) X n ′ ,m ′ ˜ ξ n ′ m ′ n ′ X v ′ =0 m ′ X w ′ =0 f ( v ′ , w ′ ) min( n − v,n ′ − v ′ ) X x =0 ˜ c ( n − v ) x ˜ c ( n ′ − v ′ ) x δ n − v − x,n ′ − v ′ − x δ n ′ − v ′ − x, S t +∆ t m − w,m ′ − w ′ ) X y =0 ˜ c ( m − w ) y ˜ c ( m ′ − w ′ ) y δ m − v − y,m ′ − v ′ − y δ m ′ − v ′ − y, S t − ∆ t , (22)where f ( v, w ) = c ( n ) v c ( m ) w √ v ! w ! 2 w + v v X p =0 w X q =0 (cid:18) vp (cid:19)(cid:18) wq (cid:19) ( − v − p δ S r +∆ r ,v + w − S − ∆2 δ S − ∆2 ,p + q , (23)˜ c ( n ) k = s(cid:18) nk (cid:19) R k (1 − R ) n − k . (24) [1] P. Sekatski, N. Brunner, C. Branciard, N. Gisin and C.Simon, Phys. Rev. Lett. , 113601 (2009).[2] P. Sekatski, B. Sanguinetti, E. Pomarico, N. Gisin, andC. Simon, Phys. Rev. A , 053814 (2010).[3] M. Nielsen and I. Chuang, Quantum Computation andQuantum Information (CUP, 2000).[4] K. Sanaka, K. J. Resch, and A. Zeilinger, Phys. Rev.Lett. , 083601 (2006).[5] K. J. Resch, J. L. O’Brien, T. J. Weinhold, K. Sanaka,B. P. Lanyon, N. K. Langford and A. G. White, Phys.Rev. Lett. , 203602 (2007).[6] F. De Martini, F. Sciarrino, and C. Vitelli, Phys. Rev.Lett. , 253601 (2008).[7] T. Sh. Iskhakov, M. V. Chekhova, G. O. Rytikov, andG. Leuchs, Phys. Rev. Lett. , 113602 (2011).[8] M. Stobi´nska, F. T¨oppel, P. Sekatski andM. V. Chekhova, Phys. Rev. A , 022323 (2012).[9] J. Appel, E. Figueroa, D. Korystov, M. Lobino and A. I.Lvovsky, Phys. Rev. Lett. , 093602 (2008).[10] S. Burks, J. Ortalo, A. Chiummo, X. Jia, F. Villa, A.Bramati, J. Laurat and E. Giacobino, Opt. Express ,3777 (2009).[11] L. V. Gerasimov, I. M. Sokolov, D. V. Kupriyanov, andM. D. Havey, arXiv:1111.6669.[12] N. Gisin, G. Ribordy, W. Tittel and H. Zbinden, Rev.Mod. Phys. , 145 (2002).[13] C. Vitelli, N. Spagnolo, L. Toffoli, F. Sciarrino, and F.De Martini, Phys. Rev. Lett. , 113602 (2010).[14] N. Spagnolo, C. Vitelli, V. G. Lucivero, V. Giovannetti,L. Maccone, and F. Sciarrino, arXiv:1107.3726.[15] C. Vitelli, N. Spagnolo, F. Sciarrino, and F. De Martini,Phys. Rev. A , 062319 (2010).[16] E. Pomarico, B. Sanguinetti, P. Sekatski, H. Zbinden, and N. Gisin, arXiv:1104.2212.[17] M. Stobi´nska, P. Horodecki, A. Buraczewski, R.W. Chhajlany, R. Horodecki, and G. Leuchs,arXiv:0909.1545.[18] M. Stobi´nska, P. Sekatski, A. Buraczewski, N. Gisin, andG. Leuchs, Phys. Rev. A , 034104 (2011).[19] A. Buraczewski and M. Stobi´nska, Comp. Phys. Com-mun. , 2245 (2012).[20] C. Vitelli, N. Spagnolo, L. Toffoli, F. Sciarrino and F. DeMartini, Phys. Rev. A , 032123 (2010).[21] D. T. Pegg, L. S. Phillips, and S. M. Barnett, Phys. Rev.Lett. , 1604 (1998).[22] F. De Martini, F. Sciarrino, and Ch. Vitelli, Phys. Rev.Lett. , 253601 (2008).[23] T. Iskhakov, M. V. Chekhova, and G. Leuchs, Phys. Rev.Lett. , 183602 (2009).[24] A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic Publishers, Dordrecht, The Nether-lands, 1993).[25] R.A. Campos, B. E. A. Saleh and M. C. Teich, Phys.Rev. A40