Optimal implementation of two-qubit linear optical quantum filters
OOptimal implementation of two-qubit linear optical quantum filters
Jarom´ır Fiur´aˇsek, Robert St´arek, and Michal Miˇcuda
Department of Optics, Palack´y University, 17. listopadu 1192/12, 771 46 Olomouc, Czech Republic
We design optimal interferometric schemes for implementation of two-qubit linear optical quantumfilters diagonal in the computational basis. The filtering is realized by interference of the two photonsencoding the qubits in a multiport linear optical interferometer, followed by conditioning on presenceof a single photon in each output port of the filter. The filter thus operates in the coincidence basis,similarly to many linear optical unitary quantum gates. Implementation of the filter with linearoptics may require an additional overhead in terms of reduced overall success probability of thefiltering and the optimal filters are those that maximize the overall success probability. We discussin detail the case of symmetric real filters and extend our analysis also to asymmetric and complexfilters.
I. INTRODUCTION
Quantum information processing with linear optics [1–3] relies on encoding of qubits into states of single pho-tons and implementation of various quantum operationsby multiphoton interference, followed by photon countingmeasurements and postselection based on the measure-ment outcomes. Scalable linear optical quantum gatescan be in principle realized with the use of auxiliarysingle photons and feedforward operations controlled bythe outcomes of measurements on auxiliary modes [4, 5].During the past two decades, quantum information pro-cessing with linear optics has evolved rapidly, driven inrecent years by important advances in design of inte-grated quantum optics circuits on a chip [6], highly effi-cient superconducting single-photon detectors [7, 8] andsingle-photon sources [9, 10]. Although full-scale quan-tum computing with linear optics still appears to be tech-nologically very demanding, the linear optics platformproved to be very useful for proof-of-principle tests ofvarious concepts and protocols in quantum informationprocessing, and small-scale linear optical quantum pro-cessors may find their applications in advanced quantumcommunication networks, where the role of light as theinformation carrier is indispensable.A central topic in quantum computing with linear op-tics is to design and realize various two-qubit [5] and mul-tiqubit [11–15] linear optical quantum gates. Besides uni-tary gates, non-unitary quantum operations, commonlyreferred to as quantum filters, also play an essential rolein quantum information processing. A quantum filtercan be defined as a trace-decreasing completely posi-tive map with a single Kraus operator M that satisfies M † M ≤ I and transforms a general input state ρ in as ρ out = M ρ in M † . This output state is not normalizedand P S = Tr[ ρ out ] is the probability of successful filter-ing. Quantum filters find their applications for instancein optimal quantum state discrimination [16, 17], entan-glement concentration and distillation [18–21], or in en-gineering of highly nonclassical states of light by condi-tional photon addition or subtraction [22–26].In the present work we investigate optimal linear opti-cal implementation of a two-qubit quantum filter diago- nal in the computational basis, M = m | (cid:105)(cid:104) | + m | (cid:105)(cid:104) | + m | (cid:105)(cid:104) | + | (cid:105)(cid:104) | , (1)where | m jk | ≤
1, and without loss of generality we set m = 1. We concentrate on the resource-effective imple-mentation that does not require any auxiliary photons.The filter is realized by interference of the two photonsencoding the qubits in a suitably designed multiport opti-cal interferometer, and successful filtering is heralded bypresence of a single photon in each output of the filter.The filter thus operates in the coincidence basis, simi-larly to a number of linear optical unitary quantum gatesdesigned and realized to date. In practice, the verifica-tion of presence of a single photon in each output of thefilter would require destructive coincidence two-photondetection. The quantum filters M can be considered asgeneralization of two-qubit controlled-phase gates, wherephase modulation is replaced by amplitude modulation.Specifically, for m = 1 and m = m = 0 the filter(1) becomes the quantum parity check [27–29] that is use-ful for implementation of the linear optical CNOT gate[28, 30, 31] and for generation of entangled multiphotoncluster states [32].It turns out that, depending on the filter parameters,it may not be possible to implement the filter without ad-ditional reduction of probability of success. This meansthat instead of filter M we implement an equivalent butless efficient filter √ P L M , where P L is the probability re-duction factor imposed by the linear optical setup. Ourgoal is to design optimal interferometric schemes for thetwo-qubit quantum filters (1), that maximize the prob-ability P L . This task is similar to the design of optimaltwo-qubit linear optical phase gates operating in the co-incidence basis [33, 34]. However, in contrast to the op-timal controlled-phase gate, we find that different mode-coupling configurations are optimal depending on the fil-ter parameters. Importantly, fully analytical results canbe obtained for the optimal interferometer parametersand the resulting maximum success probability P L .The rest of the paper is organized as follows. In Sec-tion II we present a general description of the linear opti-cal interferometric scheme that implements the two-qubitquantum filters. In Section III we discuss in detail real- a r X i v : . [ qu a n t - ph ] F e b ization of a symmetric filter with real coefficients and inSection IV we extend our analysis to asymmetric andcomplex filters. Finally, Section V contains a brief sum-mary and conclusions. The Appendix contains technicalproof of the allowed structure of the interferometer thatimplements the quantum filter. II. LINEAR OPTICAL TWO-QUBIT QUANTUMFILTERS
A conceptual scheme of linear optical setup implement-ing the two-qubit quantum filter (1) is depicted in Fig. 1.Each qubit is encoded into state of a single photon thatcan propagate in two modes denoted as A , A , and B , B for the qubit A and B, respectively. Presence of aphoton in mode A , B represents logical state | (cid:105) whilephoton in mode A , B encodes logical state | (cid:105) . Thequantum filter is implemented by interference of the twophotons in a multiport optical interferometer followed byverification of presence of a single photon in each pair ofoutput modes A , A and B , B . In practice, this ver-ification can be performed destructively by conditioningon suitable two-photon coincidence detection. The linearoptical quantum filter thus operates in the coincidencebasis, similarly to certain linear optical two-qubit CNOTand controlled-phase gates [5, 35–37].A multiport optical interferometer can be described bya unitary matrix U that specifies the coupling betweenthe input and output modes. Note that in addition tothe four modes that encode the qubits the interferometermay contain also additional auxiliary modes. In terms ofcreation operators c † j associated with each mode we have c † j, in = (cid:88) k U j,k c † k, out . (2)Let | A j , B k (cid:105) , where j, k ∈ { , } , denote the input two-photon Fock state corresponding to the two-qubit prod-uct state | j (cid:105) A | k (cid:105) B . Conditional on observation of a singlephoton in each pair of output modes A , A and B , B ,the input state transforms according to | A j , B k (cid:105) → (cid:88) m,n =0 W A m ,B n | A j ,B k | A m , B n (cid:105) , (3)where W A m ,B n | A j ,B k = U A j ,A m U B k ,B n + U A j ,B n U B k ,A m . (4)Correct implementation of the quantum filter (1) requiresthat W A m ,B n | A j ,B k = (cid:112) P L m jk δ jm δ kn , (5)where P L ≤ P L . FIG. 1. Two-qubit linear optical quantum filter operating inthe coincidence basis. Qubits are encoded into states of singlephotons in pairs of modes A , A and B , B . The scheme in-volves also auxiliary input modes C j that are all prepared ina vacuum state. All modes are coupled in a multiport opticalinterferometer described by a unitary matrix U . Successfulimplementation of the filter is indicated by coincidence detec-tion of two photons, one in modes A , A , the other in modes B , B . Design of the optimal linear optical quantum filters issimilar to the construction of optimal linear optical two-qubit quantum controlled-phase gate operating in the co-incidence basis [33, 34]. In particular, one can show thatonly one pair of the information-encoding modes can beinterferometrically coupled, see the Appendix for a proof.Consequently, the 4 × U AB = ( U j,k ), where j, k ∈ { A , B , A , B } , has a block-diagonal structure,consisting of a general 2 × × V = ( V j,k ) isa submatrix of a unitary matrix. Define two row vecors v j = ( V j, , V j, ). Matrix V is a submatrix of a unitarymatrix if and only if the vector norms and scalar productsatisfy the inequalities, | (cid:126)v | ≤ , | (cid:126)v | ≤ , | (cid:126)v · (cid:126)v | ≤ (1 − | (cid:126)v | )(1 − | (cid:126)v | ) . (6)Here the last inequality guarantees that the vectors (cid:126)v and (cid:126)v can be completed to orthogonal vectors of unitlength. III. SYMMETRIC REAL FILTER
In this section we investigate optimal interferometricschemes for implementation of a two-qubit real symmet-ric quantum filter specified by Kraus operator M = a | (cid:105) | + b ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) + | (cid:105)(cid:104) | , (7)where 0 ≤ a ≤ ≤ b ≤
1. Note that if a = b then the filter factorizes and becomes a product of twosingle-qubit filters that each attenuate the amplitude ofbasis state | (cid:105) by factor b . Otherwise, the quantum fil-ter is an entangling operation that can create entangledstates from input separable states. When optimizing thesuccess probability P L , it is necessary to consider threedifferent configurations: coupling of modes A and B ,coupling of modes A and B , and finally also coupling ofmodes B and A . Note that due to the symmetry of thefilter, the fourth possible configuration, where modes B and A are coupled, is fully equivalent to the configura-tion where modes A and B are coupled, and thereforeneed not be considered separately. In what follows wediscuss each of the above listed configurations in detail. A. Coupling of modes A and B Assuming coupling of modes A and B and the order-ing of modes A , B , A , B we can write the correspond-ing 4 × U as follows (see the Appendix), U AB = τ A b τ A x τ B y τ B b τ A
00 0 0 τ B . (8)Here τ A and τ B represent the amplitude attenuation ofmodes A and B , respectively, and the parameters x and y specify the strength of the interferometric couplingbetween modes A and B . Without loss of generality,we can assume that all matrix elements of U AB are real.The parameters x and y are related by the condition xy = a − b . (9)We are thus left with three free parameters τ A , τ B and x that shall be optimized to maximize the probability P L = τ A τ B . (10)Since U AB is a submatrix of a unitary matrix, the follow-ing constraints must be satisfied (c.f. also Eq. (6)), τ A ≤ τ B ≤ . (11) τ A ( b + x ) ≤ , τ B ( b + y ) ≤ , (12)and τ A τ B b ( x + y ) ≤ [1 − τ A ( b + x )][1 − τ B ( b + y )] . (13)Taking into account the constraint (9), and introducingnew parameters z , τ B = zτ A , and γ = | y/x | z , this lastinequality can be rewritten as (cid:112) P L b ( z − + z )+ (cid:112) P L | a − b | ( γ − + γ ) − P L (2 b − a ) ≤ . (14)Since x + x − ≥ , ∀ x > , (15) we get2 (cid:112) P L ( b + | a − b | ) − P L (2 b − a ) ≤ . (16)This inequality yields a nontrivial upper bound on P L if a > b . Assuming equality in Eq. (16), and carefully an-alyzing the two roots of the resulting quadratic equationfor √ P L , (cid:112) P L = a ± b √ a − b (2 b − a ) = 1( b ∓ √ a − b ) , (17)we find that P L is upper bounded by the smaller root,and P L ≤ b + √ a − b ) , a > b . (18)Another useful inequality can be obtained by taking theproduct of the two inequalities (12). We get P L [ b + b | b − a | ( µ + µ − ) + ( b − a ) ] ≤ , (19)where µ = | x/y | . With the use of inequality (15) thisyields P L ≤ b + | b − a | ) . (20)We now explicitly present the optimal interferometricconfigurations that are all symmetric, τ A = τ B = τ and x = ± y . We have to distinguish four different cases ac-cording to the values of the filter parameters a and b .(i) a ≤ b , 2 b − a <
1. As shown in Fig. 2(a), in thiscase it is optimal to couple the modes A and B on abeam splitter with amplitude transmittance t = b √ b − a . (21)Subsequently, each of the modes A and B is attenuatedwith amplitude factor ν = (cid:112) b − a (22)by sending it through a beam splitter with amplitudetransmittance ν whose auxiliary mode is prepared in vac-uum state. In this case P L = 1 and the linear opticalimplementation does not impose any extra reduction ofthe overall success probability of the quantum filtering.(ii) a ≤ b , 2 b − a >
1. The optimal scheme is drawnin Fig. 2(b) and is similar to that in case (i). However,instead of attenuating modes A and B we have to at-tenuate modes A and B with amplitude transmittance τ A = τ B = 1 √ b − a . (23)Subsequently, the probability P L drops below 1 and weget P L = (2 b − a ) − . The scheme is optimal because P L saturates the inequality (20). FIG. 2. Optimal optical interferometers implementing thetwo-qubit quantum filters (7) by interferometric coupling ofmodes A and B . The labels of beam splitters indicate theiramplitude transmittances. Mode attenuation is realized bypropagation through a beam splitter with suitable transmit-tance, whose auxiliary input mode is prepared in vacuumstate. The four schemes (a)-(d) represent optimal setups fordifferent values of filter parameters a and b . For details, seetext. (iii) a > b , b + √ a − b ≤
1. The optimal interfero-metric scheme is shown in Fig. 2(c). Modes A and B are injected into a Mach-Zehnder interferometer formedby two balanced beam splitters. One arm of the inter-ferometer is attenuated with amplitude transmittance t and the other with amplitude transmittance t (cid:48) , where t = b − (cid:112) a − b , t (cid:48) = b + (cid:112) a − b . (24)In this case we achieve P L = 1.(iv) a > b , b + √ a − b >
1. The optimal scheme isshown in Fig. 2(d) and is similar to the scheme for case(iii). However, only one of the interferometer arms isattenuated, with amplitude transmittance t = b − √ a − b b + √ a − b . (25)Furthermore, modes A and B are each attenuated byfactor τ A = τ B = 1 b + √ a − b . (26) Consequently, we have P L = 1( a + 2 b √ a − b ) . (27)The scheme is optimal because the achieved P L saturatesthe bound (18). B. Coupling of modes A and B Let us now investigate the configuration where modes A and B are interferometrically coupled instead of themodes A an B . Keeping the same ordering of modes A , B , A , B , the relevant 4 × U can bewritten as U AB = τ A τ B ba τ A τ A x τ B y ba τ B , (28)where xy = a − b a (29)and the probability P L can be expressed as P L = τ A τ B a . (30)The conditions following from the requirement that (28)is a submatrix of a unitary matrix yield τ A (cid:18) b a + x (cid:19) ≤ , τ B (cid:18) b a + y (cid:19) ≤ , (31)and 2 √ P L a ( b + | a − b | ) − P L a (2 b − a ) ≤ . (32)This last inequality was obtained by the same procedureas the inequality (16) and it implies the following upperbound on P L , P L ≤ a ( b + √ a − b ) , a > b . (33)By taking the product of the two inequalities (31) andutilizing the constraint (29) we find that P L ≤ a ( b + | b − a | ) . (34)For a < b this bound is stricter than the bound (20).Similarly, for a > b the inequality (33) is stricter thanthe inequality (18). It follows from the inequalities (33)and (34) that with the coupling of modes A and B we can achieve P L = 1 only if a = b . Physically, for a (cid:54) = b there will always be a nonzero probability thatfor the input state | (cid:105) the two photons will bunch andwill both end up either in mode A or B , resulting inthe failure of the filter. We can therefore conclude thatthe interferometric coupling of modes A and B cannotyield higher P L than coupling of modes A and B . . . . . . . b . . . . . . a . . . . . . . . FIG. 3. Maximum probability P L for the symmetric two-qubitquantum filters (7) achievable with linear optical interfero-metric schemes is plotted as a function of filter parameters a and b . The large yellow area represents filters for which P L = 1. The gray line indicates the points a = b that cor-respond to filters which factor into products of single-qubitfilters. C. Coupling of modes B and A Finally, we consider an asymmetric configurationwhere modes B and A are coupled. The correspondingmatrix U AB can be expressed as U AB = bτ A τ B ab τ B x τ A y τ A
00 0 0 τ B , (35)where xy = b − ab . (36)The requirement that U AB is a submatrix of a unitarymatrix yields the constraints τ A (1 + y ) ≤ , τ B (cid:18) x + a b (cid:19) ≤ , (37)and τ A τ B (cid:16) x + y ab (cid:17) ≤ (cid:2) − τ A (1 + y ) (cid:3) (cid:20) − τ B (cid:18) x + a b (cid:19)(cid:21) , (38)together with τ A ≤ τ B ≤ τ A , hence also P L = τ A τ B , until at least oneinequality is saturated.Let us first assume that one of the inequalities (37)is saturated. It immediately follows from Eq. (38) that x + ya/b = 0 must hold, which together with (36) yields x = √ ab (cid:112) a − b , y = − (cid:114) a − b a . (39) This solution exists in the parameter region a > b . Itfollows from the inequalities (37) that the maximum pos-sible values of τ A,B are given by τ A = a a − b , τ B = min (cid:18) , b a (2 a − b ) (cid:19) . (40)Consequently, the maximum achievable P L for this casecan be expressed as P L = min (cid:18) a a − b , b (2 a − b ) (cid:19) , a > b . (41)Let us now assume that only the inequality (38) is satu-rated. Since the saturation means that equality holds in(38), we can use it to express τ A in terms of x and τ B , τ A = x [ b − τ B ( b x + a )] x b + ( b − a ) − x τ B (2 a − b ) . (42)The optimal values of τ B and x can be determined bysolving the extremal equations ∂∂τ B ( τ A τ B ) = 0 , ∂∂x ( τ A τ B ) = 0 , (43)where τ A is given by Eq. (42). In the region a > b we recover the optimality condition (39). In the region a < b we obtain additional potentially optimal solution x = √ ab (cid:112) b − a, y = (cid:114) b − aa , (44)and τ A = a ( √ b − a + √ a ) , τ B = b a √ b − a + √ a ) . (45)Note that this solution is acceptable only if all the in-equalities (11) and (37) are satisfied. Additionally, wehave to consider also the extremal point τ B = 1. Oninserting this into Eq. (42), we have P L = τ A = x ( b − b x − a )] x b + ( b − a ) − x (2 a − b ) . (46)The optimal x maximizing P L can be found from theextremal equation ∂P L ∂x = 0 . (47)After some algebra, this yields two roots x = ( a + b )( a − b ) b (2 a + b − b ) , P L = ( a + b ) (2 a + b − b ) , (48)and x = ( b − a )( b − a ) b ( b + b − a ) , P L = ( a − b ) ( b + b − a ) . (49) FIG. 4. The black area indicates the range of parameters a and b of a symmetric two-qubit quantum filter (7) for whichthe coupling of modes B and A leads to maximum successprobability P L . We emphasize that the formulas (48) or (49) representvalid potential optimal points only if x ≥ P L must be performed overall the above considered configurations and all the iden-tified potentially optimal solutions. The maximal P L ,optimized over all the coupling configurations, is plottedin Fig. 3. Remarkably, we find that for a certain rangeof parameters a and b satisfying a > b the asymmetricscheme where modes B and A are coupled outperformsthe symmetric scheme where modes A and B are cou-pled, and achieves higher P L . This area of parameterswhere the coupling of modes B and A is optimal isdepicted in Fig. 4. We note that the interferometric cou-pling described by matrix (35) can be realized by interfer-ence of modes B and A in a Mach-Zehneder interferom-eter formed by two generally unbalanced beam splitters,and the signal in each interferometer arm should be suit-ably attenuated, c.f. Fig. 2(c). The splitting ratios ofthe beam splitters and the attenuation factors can be de-termined by singular value decomposition of the matrix U AB [34]. IV. ASYMMETRIC AND COMPLEX FILTERS
The optimization procedure discussed in the previoussection can be extended to asymmetric and complex two-qubit filters. Here we illustrate it on the examples oftwo-qubit asymmetric filter with real coefficients and atwo-qubit symmetric complex filter. We shall focus onthe configuration where modes A and B are coupled.Configurations where other pairs of modes are coupledcan be treated in a similar manner. For an asymmet-ric filter one has to consider separately both coupling ofmodes A , B and A , B because the symmetry is bro-ken. A. Asymmetric real filter
Let us consider linear optical implementation of anasymmetric real filter M = a | (cid:105)(cid:104) | + b A | (cid:105)(cid:104) | + b B | (cid:105)(cid:104) | + | (cid:105)(cid:104) | , (50)where 0 ≤ a ≤
1, and 0 ≤ b A ≤ b B ≤ A and B , the matrix U AB can be conveniently parameterizedas U AB = b A τ A b A τ A x b B τ B y b B τ B τ A
00 0 0 τ B , (51)where xy = ab − , (52)and we have defined the parameter b = √ b A b B . Since U AB is a submatrix of a unitary matrix, the followinginequalities must hold, similarly to the previously studiedcase of symmetric filter: τ A ≤ , τ B ≤ , (53) b A τ A (1 + x ) ≤ , b B τ B (1 + y ) ≤ , (54)and η A η B ( x + y ) ≤ [1 − η A (1 + x )][1 − η B (1 + y )] , (55)where η A = b A τ A and η B = b B τ B . With the use ofcondition (52), the last inequality can be rewritten as η A (1 + x ) + η B (1 + y ) − η A η B (cid:16) − ab (cid:17) ≤ . (56)For any filter (50), the optimal interferometer maximiz-ing P L = τ A τ B can always be designed such that theinequality (56) is saturated and equality holds. Fist notethat if one of the inequalities (54) is saturated, then alsoinequality (55) is saturated and equality must hold, be-cause both the left and righ-hand sides of Eq. (55) mustbe equal to 0. Assume now a configuration where noneof the inequalities (54) and (55) is saturated. If τ A or τ B is smaller than 1, then we can increase their valueuntil either equality holds in (55) or τ A = τ B = 1. Foran optimal configuration with τ A = τ B = 1 we can in-crease or decrease the free parameter x while keeping theconstraint (52) until equality holds in Eq. (55).We now discuss the various options that have to beconsidered. Let us first consider the option τ A = τ B = 1,i.e. P L = 1. In this case, x and y can be determined bysolving Eqs. (52) and (56), where equality is assumed tohold. We obtain x = 12 b A (cid:104) q + (cid:112) q − a − b ) (cid:105) ,y = 12 b B (cid:104) q − (cid:112) q − a − b ) (cid:105) , (57)where q = 1 − b A − b B + (2 b − a ) . (58)Since x and y must be real and non-negative, the solu-tion (57) exists only if q ≥ | a − b | . (59)Additionally, the inequalities (54) must also hold, whichreduces to x ≤ b A − , y ≤ b B − , (60)where x and y are given by Eq. (57). To sum up, P L = 1 is achievable with coupling of modes A and B if and only if the inequalities (59) and (60) are satisfied.Let us now assume that τ A = 1 but τ B can be smallerthan 1. Assuming equality in Eq. (56) we get, τ B = b A x [1 − b A (1 + x )] b x + ( a − b ) − b A x (2 b − a ) . (61)The optimal x that maximizes τ B can be determined bysolving the extremal equation ∂τ B ∂x = 0 . (62)This leads to quadratic equation for x with roots x = ( a − b )(1 − b A ) b A ( b + ab A − b b A ) ,x = ( b − a )(1 + b A ) b A ( b − ab A + 2 b b A ) . (63)These roots represent valid solutions provided that x ≥ y and τ B are determined by Eqs. (52) and (61), re-spectively. For an asymmetric filter we must also inde-pendently consider the configuration τ B = 1 because thesymmetry is broken. Following a similar procedure asbefore, we obtain τ A = b B y [1 − b B (1 + y )] b y + ( a − b ) − b B y (2 b − a ) , (64)and the potentially optimal y read y = ( a − b )(1 − b B ) b B ( b + ab B − b b B ) ,y = ( b − a )(1 + b B ) b B ( b − ab B + 2 b b B ) . (65)Once again these solutions are valid only if y ≥ τ A and τ B can be smaller than 1. Assuming equality in (56),we can express τ A as a function of τ B and x , τ A = b B y [1 − b B τ B (1 + y )] b y + ( a − b ) − b B τ B y (2 b − a ) . (66) On inserting this formula into the extremal equations ∂∂τ B ( τ A τ B ) = 0 , ∂∂y ( τ A τ B ) = 0 , (67)we obtain after some algebra the following expressionsfor x and y , x = (cid:112) | a − b | b , y = sgn( a − b ) (cid:112) | a − b | b . (68)If a ≤ b , then x = − y and at least one of the inequalities(54) is saturated. However, the inequalities (53) mayrepresent an additional bound. We can succinctly express τ A and τ B as follows, τ A = min (cid:18) , b b A (2 b − a ) (cid:19) ,τ B = min (cid:18) , b b B (2 b − a ) (cid:19) . (69)If a > b , then the extremal equations (67) lead to thefollowing expressions for τ A and τ B , τ A = b B b A √ a − b + b ) , τ B = b A b B √ a − b + b ) . (70)These formulas represent valid solutions only if the in-equalities (53) and (54) are satisfied. B. Symmetric complex filter
Let us finally investigate realization of symmetric two-qubit filters with complex coefficients. Without loss ofgenerality, we can restrict ourselves to the filters M = ae iϕ | (cid:105)(cid:104) | + b | (cid:105)(cid:104) | + b | (cid:105)(cid:104) | + | (cid:105)(cid:104) | , (71)where a and b are real and positive, because the relativephase shifts of states | (cid:105) and | (cid:105) can be set to zeroby suitable phase shifts applied to modes A and B ,respectively. We shall again focus on the configurationwhere modes A and B are coupled. The matrix U AB has the same structure as for real symmetric filters, U AB = bτ A τ A x τ B y bτ B τ A
00 0 0 τ B , (72)only the condition on parameters x and y changes to, xy = ae iϕ − b . (73)Since x and y are generally complex, the conditions im-plied by U AB being a submatrix of a unitary matrix mustbe written as follows, τ A ( b + | x | ) ≤ , b τ A ( b + | y | ) ≤ , (74) FIG. 5. Two examples of a possible implementation of the two-qubit linear optical quantum filters. (a) On-chip implementationwith path encoding of the qubit states. Lines represent the optical waveguides and their crossings balanced directional couplers.The empty circles indicate variable phase shifters. Sub-blocks in colored boxes act as variable beam splitters. Blue boxes servefor mode swapping that ensures coupling of the desired pair of modes. The two red boxes realize the required interferometriccoupling of the selected pair of signal carrying modes, and the four green boxes serve for tunable signal attenuation in eachmode. (b) Bulk-optics implementation with polarization encoding and interferometers formed by calcite beam displacers thatintroduce lateral shift between the vertically and horizontally polarized beams. The polarization states are transformed withhalf-wave plates (green elements) that play the role of beam splitters, and the interferometric phase shifts can be set andcontrolled by tilting thin glass plates (orange elements). and b τ A τ B | x + y ∗ | ≤ [1 − τ A ( b + | x | )][1 − τ B ( b + | y | )] . (75)Taking into account the symmetry of the filter, one canshow that P L = 1 can be achieved provided that theinequalities (74) and (75) are satisfied for a symmetricconfiguration with | x | = | y | and τ A = τ B = 1. Aftersome algebra, this yields the following condition, b + b (cid:112) s + a cos ϕ − b ) + s ≤ . (76)where s = | xy | = (cid:112) a + b − ab cos ϕ. (77)If the inequality (76) does not hold, then the optimalconfiguration is symmetric, with x = y = (cid:112) ae iϕ − b , (78)and τ A = τ B = 1 b + b (cid:112) s + a cos ϕ − b ) + s . (79)This yields P L = (cid:104) b + b (cid:112) s + a cos ϕ − b ) + s (cid:105) − . (80)For ϕ = 0 we recover the results for symmetric real filterderived in Section III A. Also, for a = b = 1 we recoverfrom Eq. (80) the maximum probability of implementa-tion of a two-qubit linear optical controlled-phase gate[33, 34], P CP ( ϕ ) = (cid:20) (cid:12)(cid:12)(cid:12) sin ϕ (cid:12)(cid:12)(cid:12) + 2 (cid:114)(cid:12)(cid:12)(cid:12) sin ϕ (cid:12)(cid:12)(cid:12) − sin ϕ (cid:21) − . (81) V. CONCLUSIONS
We have designed optimal interferometric schemes forimplementation of two-qubit linear optical quantum fil-ters operating in the coincidence basis. The consideredlinear optical realization of the quantum filters may im-pose an extra cost in terms of reduced success probabil-ity of successful filtering and the designed schemes maxi-mize the success probability of the filter. The symmetricreal filters were analyzed in particular detail and, inter-estingly, we have found that for a certain range of pa-rameters the optimal scheme is asymmetric in the sensethat it couples a pair of modes corresponding to logicalstate | (cid:105) of one qubit and logical state | (cid:105) of the otherqubit, which contrasts the symmetry of the consideredfilter. Our investigation of the optimal implementationof optical quantum filters complements the earlier studieson optimal realization of linear optical unitary quantumgates. The required interferometric setup can be imple-mented on-chip with integrated optics where a tunablebeam splitter can be realized using a Mach-Zehnder inter-ferometer with tunable phase shift [6, 38, 39]. A universalintegrated optics circuit that can realize all of the opti-mal interferometric schemes is drawn in Fig. 5(a). As asecond example, in Fig. 5(b) we show a possible bulk op-tics realization based on polarization qubit encoding andutilization of inherently stable interferometers formed bya sequence of calcite beam displacers [11, 40–42]. The in-vestigated two-qubit linear optical quantum filters mayfind applications in linear optics quantum informationprocessing and quantum state engineering. ACKNOWLEDGMENTS
We acknowledge support by the Czech Science Foun-dation under Grant No. 19-19189S.
Appendix: Derivation of structure of matrix U AB In this Appendix we determine the most general formof the 4 × U j,k , j, k ∈ A , B , A , B , that de- scribes interferometric coupling which enables implemen-tation of the diagonal two-qubit quantum filter (7). Re-call that input two-photon Fock state | A j , B k (cid:105) trans-forms as follows, | A j , B k (cid:105) → (cid:88) m,n =0 ( U A j ,A m U B k ,B n + U A j ,B n U B k ,A m ) | A m , B n (cid:105) , (A.1)where we assume operation in the coincidence basis andrestrict ourselves to the outputs where a single photonis present in each pair of modes A , A and B , B . Re-call also that the implementation of a diagonal two-qubitquantum filter M = (cid:80) j,k =0 m jk | jk (cid:105)(cid:104) jk | requires that U A j ,A m U B k ,B n + U A j ,B n U B k ,A m = (cid:112) P L m jk δ jm δ kn . (A.2)Throughout the following discussion we assume that allfour coefficients m jk are nonzero. Let us first provethat all four diagonal matrix elements U A j ,A j and U B k ,B k must be nonzero. Assume that U A ,A = 0. In order toobtain nonzero m A ,B and m A ,B the matrix elements U B ,A , U A ,B , U A ,B , and U B ,A must be all nonzero.However, this implies that U A ,A U B ,B + U A ,B U B ,A (A.3)is nonzero, which is in contradiction with the requiredstructure (A.2). Specifically, nonzero term (A.3) impliesthat the input state | A , B (cid:105) is transformed to a statethat contains non-vanishing contribution of | A , B (cid:105) ,which is not compatible with the diagonal form of thetargeted quantum filter. We have thus proved by con-tradiction that U A ,A must be nonzero. The same proofapplies also the the other three matrix elements U A ,A , U B ,B , and U B ,B .We next show that the four matrix elements U A ,A , U A ,A , U B ,B , and U B ,B must be zero. We again provethis by contradiction. We provide the proof for U A ,A .Equation (A.2) implies that U A ,A U B ,B = − U A ,B U B ,A ,U A ,A U B ,B = − U A ,B U B ,A , (A.4) U A ,A U B ,B = − U A ,B U B ,A . If we take the product of the first two equations (A.4)and make use of the third equality (A.4), we obtain U A ,A U B ,B ( U A ,A U B ,B + U A ,B U B ,A ) = 0 . (A.5) Since the term in the parentheses is equal to √ P L m and thus nonzero, we have U A ,A U B ,B = 0 . (A.6)This implies that also U A ,B U B ,A = 0 . (A.7)If U A ,A (cid:54) = 0, then also U B ,A (cid:54) = 0, and U A ,B = 0,which follows from Eqs. (A.4) and (A.6) and from theabove proved condition U B ,B (cid:54) = 0. It follows that theamplitude U A ,A U B ,B + U A ,B U B ,A (A.8)is nonzero, although it should vanish. Therefore, U A ,A = 0 must hold and similarly we can show thatalso U A ,A = U B ,B = U B ,B = 0.Let us now assume that modes A and B are interfero-metrically coupled and U A ,B (cid:54) = 0, as well as U B ,A (cid:54) = 0.We show that the other pairs of modes A j and B k cannotbe coupled. It follows immediately from Eq. (A.4) that U A ,B = U B ,A = 0 . (A.9)We next consider the following amplitudes that shouldalso vanish, U B ,B U A ,A + U A ,B U B ,A = 0 ,U A ,A U B ,B + U A ,B U B ,A = 0 , (A.10)Since U B ,B = U A ,A = 0 and U A ,B (cid:54) = 0, U B ,A (cid:54) = 0,we get U A ,B = U B ,A = 0 . (A.11)Finally, from the requirement that the following two am-plitudes should vanish, U A ,B U B ,A + U A ,A U B ,B = 0 ,U B ,A U A ,B + U A ,A U B ,B = 0 , (A.12)0we can deduce that U A ,B = U B ,A = 0 . (A.13)To summarize our findings: out of the 16 matrix elements U j,k , where j, k ∈ { A , A , B , B } , only 6 elements arenonzero: the four diagonal elements U j,j and two ele-ments representing interferometric coupling of a singlepair of modes A j and B k , e.g. U A ,B and U B ,A . Thematrix (8) considered in Section IIIA of the manuscript(or its variants obtained by swapping the modes A and A and/or B and B ) therefore represents the most gen-eral permissible interferometric coupling for the imple-mentation of two-qubit diagonal quantum filters.We note that, strictly speaking, this result holds only if all four m jk are nonzero. If two or three filter parame-ters m jk vanish, then it can be shown that the filter canbe implemented with P L = 1 and coupling of one pair ofmodes is sufficient to achieve this. In fact, the only non-trivial configuration is m = m = 0 while m (cid:54) = 0,and this is covered by the optimal symmetric quantumfilters discussed in Section III. 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