Efficient Optimal Minimum Error Discrimination of Symmetric Quantum States Antonio Assalini, Gianfranco Cariolaro, and Gianfranco Pierobon Department of Information Engineering (DEI), University of Padua, Via Gradenigo 6/B, 35131, Padova, Italy (Dated: January,2010) Thispaperdealswiththequantumoptimaldiscriminationamongmixedquantumstatesenjoying geometricaluniformsymmetrywithrespecttoareferencedensityoperatorρ0. Itiswell-knownthat the minimal error probability is given by the positive operator-valued measure (POVM) obtained as a solution of a convex optimization problem, namely a set of operators satisfying geometrical symmetry, with respect to a reference operator Π0, and maximizing Tr(ρ0Π0). In this paper, by resolving the dual problem, we show that the same result is obtained by minimizing the trace of a 0 1 semidefinite positive operator X commuting with the symmetry operator and such that X ≥ ρ0. The new formulation gives a deeper insight into the optimization problem and allows to obtain 0 2 closed-form analyticalsolutions,asshownbyasimplebutnottrivialexplanatoryexample. Besides thetheoretical interest,theresult leadstosemidefiniteprogramming solutionsof reducedcomplex- n ity, allowing to extend the numerical performance evaluation to quantum communication systems a modeled in Hilbert spaces of large dimension. J 9 I. INTRODUCTION set of smaller dimension. The simplified formulation of ] h the dual problem is illustrated with an example where p InaquantumsystemAlicepreparesthequantumchan- a closed-form solution is easily found. For problems of - large dimension, that cannot be solved analytically, the nelintooneofseveralquantumstates. Bobmeasuresthe t n quantumchannelbyasetofmeasurementoperatorsand, reduced number of variables in the simplified dual state- a ment becomes useful to the numericalsolutionby means on the basis of the result, it guesses the choice made by u of semidefinite programming (SDP) tools. the transmitter. These actions lead to a classical chan- q [ nel, and the problem arises of finding the measurement operatorsthatprovideoptimalperformanceaccordingto 1 a predefined criterion, in this paper the minimum error II. GENERAL FORMULATION OF MINIMUM v probability. However, to solve the problem, except for ERROR DISCRIMINATION 5 8 some particular cases, has appeared to be a very diffi- 3 cult task since the pioneeringcontributions in the seven- The quantum decision problem is formalized in a N- 1 ties [1–3]. dimensional complex Hilbert space H [1], where an en- 1. In recent years, particular attention has been paid to semble of quantum states ρ , i = 0,1,...,M 1, with i 0 quantumstatessatisfyinggeometricalsymmetry[4–7],in prior probabilities q 0, M−1q = 1, is gi−ven. The 0 view of applications to optical communication systems. quantum states ρ arie≥densityi=o0peraitorson H, i.e., (self- 1 In some specific cases, including symmetric pure quan- i P adjoint) positive semidefinite (PSD) operators (ρ 0), v: tum states [4, 5] and symmetric mixed quantum states withunittrace,Tr(ρ )=1. Fornotationconvenienic≥e,we i Xi with a characteristic structure [6], the solution named denote by P the class of the PSD operators on H. The square rootmeasurement(SRM) provesoptimal. Never- measurementoperatorsΠ ,i=0,1,...,M 1,constitute ar theless, in general, SRM represents a suboptimal strat- a POVM having the propierties Π P and− M−1Π = egy, although it provides pretty good performance in I, where I is the identity operatori o∈n H. The it=ra0nsitiion many scenarios. P probabilities of the resulting quantum channel become In this paper, we are concerned with the construction p(j i)=Tr(ρ Π ), so that the probability of correct de- of optimal POVM for the discrimination of symmetric | i j tection is given by P = M−1q Tr(ρ Π ). mixed quantum states. We present some results that c i=0 i i i Hence, the problem of finding the maximum proba- provideintuition into the problemandoffer perspectives P bility of correct state discrimination can be concisely on its solution. stated as follows. Optimal quantum state discrimination represents a convex optimization problem, and, as such, it can be Primal problem (PP1). Find the maximum of the formulated in a primal and in a dual form, with the lat- probability of correct detection P = M−1q Tr(ρ Π ) terhavingareducednumberofvariablesandconstraints over the class of the POVM on H.c i=0 i i i [3, 6]. In this paper we investigate how primal and dual P problems simplify with symmetric quantum states. For The analytical solution of PP1 is in general difficult the primal problem this study was already considered in since P has to be maximized over the whole M-tuple of c [6]. Herein, we extend the analysis to the dual prob- measure operators Π . As a matter of fact, closed-form i lem, where the optimal solution can be searched in a resultsto the primalproblemare availableonly for some 2 particular quantum mechanical systems, e.g., the binary Primal problem for symmetric quantum states case [1]. Nevertheless, since the objective is to search a (PP2). Find the maximum of P = Tr(ρ Π ), with c 0 0 global maximum of a linear function into a convex set, Π P and such that M−1SiΠ S−i =I. the problem can be faced by means of numerical tools P0ro∈of: This formulatioi=n0was fi0rst given in [6]. It such as SDP. Besides, according to classical results in can straightforwardlyPbe proved by using (1) and convex optimization theory [8], in place of the primal (2) in PP1, so that P = M−1q Tr(ρ Π ) = problem, it is in general more convenient to consider its M−1 1 Tr(Siρ S−iSiΠcS−i) = Tri=(0ρ Πi ). Miorieover, corresponding dual problem, since it presents a smaller i=0 M 0 0 P 0 0 if Π 0 then Π =SiΠ S−i 0. (cid:4) number of variables and constraints [3, 6]. P 0 ≥ i 0 ≥ Dual problem (DP1). Minimize the trace of the optimization operator X over the class P, subject to the B. Dual Problem for Symmetric Quantum States constraints X q ρ , i = 0,1,...,M 1. Once found a i i ≥ − minimum trace operator Xopt, its trace gives the maxi- The optimization problem PP2 is comparatively sim- mum probability of correct detection, Pc =Tr(Xopt). ple, and, perhaps, this is the reason why no particular attention has been paid in the literature to the study of The equalities (X q ρ )Π = Π (X q ρ ) = opt i i i i opt i i − − the dual theorem to obtain an alternative formulation. 0, i = 0,1,...,M 1, are necessary conditions on the − In the following we investigate this point. optimalPOVM.Theseconditionsbecomesufficient,once thesearchedmeasureoperatorsareconstrainedtobelong Dual problem for symmetric quantum states to P and to solve the identity on H. (DP2). Minimize the trace of the optimization operator X over the class P, subject to the constraints X 1 ρ ≥ M 0 and XS = SX. Once found a minimum trace operator III. DISCRIMINATION OF SYMMETRIC X , its trace gives the maximum probability of correct opt QUANTUM STATES detection, P =Tr(X ). c opt Proof: Define ρ′ = Si 1 ρ S−i. Let V be the fea- In quantum detection an important role is played by i M 0 geometrically uniform symmetry [4–6]. Among the sev- sible set according to th(cid:0)e gen(cid:17)eral dual problem in Sec- emreatlrgicenmeirxaelidzaqtuioannst,umwestcaotnessidgeernetrhaetebdasfricomcaasereofferseynmce- tii=on0I,I1,,i..e..,.,tMheset1o,faPnSdDleotpVe′rabteortshXe sseutcohftPhaStDXop≥erρa′i-, density operator ρ0 as torsX′ suchtha−t X′ ≥ρ′0 andX′S =SX′. The proofis organized in two steps. In the first step it is shown that ρi =Siρ0S−i , i=0,1,...,M 1 , (1) V′ V, while in the secondstepit is provedthatfor any − X ⊂V there exist X′ V′ such that Tr(X′) = Tr(X). wherethesymmetryoperator S isunitary(SS† =S†S = Th∈en, the search of X∈can be confined into V′. I) and such that SM = I. The geometry implicitly - Step 1: If X′ V′, for the commutativity between requires that the mixed states are equiprobable, i.e., X′ and S we get∈X′ = SX′S−1 and recursively X′ = q = 1/M, i = 0,1,...,M 1. In [6] it was shown that i − SiX′S−i. Hence, for any i optimalPOVMhavingthesamesymmetrycanalwaysbe found. Hence, without loss of generality, we can assume X′ ρ′ =X′ Siρ′S−i that − i − 0 =SiX′S−i Siρ′S−i (3) Π =SiΠ S−i , i=0,1,...,M 1 , (2) − 0 i 0 − =Si(X′ ρ′)S−i 0, − 0 ≥ where Π is the reference measure operator. Conse- 0 quently, for a fixed M, the knowledge of ρ , Π and S since X′ ρ′ for assumption. is sufficient to fully describe the state ensem0ble0and the - Step 2:≥For0each X V we consider ∈ POVM. M−1 The density operators (1) have all the same rank, and 1 X′ = S−iXSi . the same holds for the measure operators(2). As proved M in [9], the optimal measure operators can be assumed Xi=0 to have rank no higher than that of the corresponding Being X ρ′ for each i, it follows that density operators, namely rank(Π ) rank(ρ ). ≥ i 0 0 ≤ M−1 1 X′ S−iρ′Si =ρ′ . ≥ M i 0 A. Primal Problem for Symmetric Quantum States i=0 X Moreover,recalling that SM =I The specific geometry of the state ensemble can be exploited to get insight into how solving the state dis- M−1 1 crimination problem. In particular, the primal problem SX′S−1 = S−(i−1)XSi−1 =X′ PP1 can be rewritten in a simpler form as follows. M i=0 X 3 and then X′ commutes with S. Finally, Table I: Number of real decision variables d, equality 1 M−1 M−1 constraints Ce and inequality constraints Ci for the opti- Tr(X′)= Tr(S−iXSi)= Tr(XSiS−i) mization problems. M i=0 i=0 X X M−1 d Ce Ci 1 = Tr(X)=Tr(X), PP1 MN2 1 M M Xi=0 PP2 N2 1 1 and the proof is complete. (cid:4) DP1 N2 0 M Therefore,thesearchoftheunknownoptimizationop- DP2 N2 0 2 eratorX canberestrictedtothesubclassofPcomposed DP3 PNi¯=0Ni2 0 1 by the PSD operators that commute with the symmetry operator S. The optimalΠ canbe found by the relations(X 1 ρ )Π = Π (X0 1 ρ ) = 0 subject to Π Popatn−d D. Problem dimension and number of constraints M 0 0 0 opt− M 0 0 ∈ M−1SiΠ S−i =I. i=0 0 Thecomplexityofalinearprogram,togetthesolution In the next section, we develop a formulation, where of a convex optimization problem, is hard to evaluate in P the commutation condition XS = SX is replaced by an terms of arithmetic operations (see [8] for details). Nev- alternative constraint. ertheless, since we are seeking a global optimum, the di- mension of the feasible region for the objective function C. Alternative Formulation of the Dual Problem gives an order of the complexity of the problem [8]. The set of self-adjoint operators on H forms an N2– Since the symmetry operator S is known a-priori, dimensional realvector space. Therefore, for a given op- it is possible to exploit its spectral characterization to timization problem we can find the dimension d of the rewrite the dual problem DP2 as follows. correspondent real space on which the considered objec- tivefunctionisdefined. Inotherterms,wefindthenum- Other form for the dual problem for symmetric ber of realvariables d in a given objective function. The Nq¯uantNumdistsitnactteesig(eDnPv3a)lu.esLoeftSλ,0,aλn1d,.N..,bλeN¯−th1ebmeutlthie- resultsaresummarizedinTableIwhereCe andCi repre- i sent, respectively, the number of equality and inequality ≤ plicity of λ . Let U be a basis of eigenvectors of S, with i constraintsfor a givenproblem. Note that the PSD con- the first N eigenvectors corresponding to λ , the next 0 0 dition on self-adjoint operators is counted as an inequal- N1 eigenvectors corresponding to λ1, and so on. Then, ity, e.g., the relation Π P is counted as the inequality minimize the trace of the optimization operator X˜ over 0 ∈ Π 0. the subclass of P consisting of block-diagonal operators 0T≥he dual problem DP3 presents the smaller number of with blocks of dimension N , under the constraint i variablesandconstraintsamongtheconsideredoptimiza- X˜ ≥ M1 U†ρ0U. Once found a minimum trace operator tionproblemsand,inparticular,d= N¯−1N2issmaller X˜opt, its trace gives the maximum probability of correct thanN2,dependingonthespectrumofti=h0esymi metryop- detection, Pc =Tr(X˜opt). erator S. If S has all distinct eigenvPalues, then N¯ = N, Proof: By DP2 the optimization operator X can N = 1 for each i, and the optimization operator X˜ in i be chosen to commute with S. Therefore, given an DP3 becomes diagonal, giving d=N <N2. eigenbasis U for S, we can write X = UX˜U† where, by well-known results on simultaneous diagonalization of commuting self-adjointoperators[10], X˜ turns outto be IV. EXAMPLE OF APPLICATION block diagonal with the size of the blocks given by the multiplicity of the eigenvalues of S. Moreover, we find In this section, the previous results are applied to the that Tr(X)=Tr(UX˜U†)=Tr(X˜) and the constraint discrimination of an ensemble of M symmetric mixed X ≥ M1 ρ0 becomes X˜ ≥ M1 U†ρ0U. (cid:4) quantum states on a 2-dimensional (N = 2) complex Hilbert space. The symmetry operator is The commutative requirement XS = SX is then replaced by fixing the block-diagonal structure on X˜. cos π sin π GΠiviesnsoalnutoiopntimofa(lUX˜X˜opt,Uth†e re1feρre)nΠce=opΠtim(UalX˜opeUra†tor S =" sin(cid:0)MMπ (cid:1) −cos (cid:0)MπM(cid:1)# , (4) 0 opt − M 0 0 0 opt − 1 ρ )=0, subject to Π P and M−1SiΠ S−i =I. anditrepresentsaline(cid:0)art(cid:1)ransform(cid:0)atio(cid:1)ngivenbyacoun- M 0 0 ∈ i=0 0 The new formulationof the dual problem is givenin a terclockwise rotation through angle π/M. We assume formthatis particularlysuitable foPrSDP computational that the reference density operator has the general form tools[11]and,moreover,itisanalyticallymoretractable thatthepreviousversion. Wenowquantifythecomplex- ρ = α β , (5) 0 ity of the different approaches. " β 1 α # − 4 where α and β are real numbers. Since ρ is PSD, the The optimal reference measure operator Π can 0 0 feasible values of α and β are constrained as 0 α be found from the conditions given in Section IIIC 1 and β α(1 α), respectively. Without≤loss o≤f (UX˜ U† 1 ρ )Π = Π (UX˜ U† 1 ρ ) = 0, that | | ≤ − opt − M 0 0 0 opt − M 0 generality we assume α 1/2. in this example simplify as (x˜ I 1 ρ )Π =Π (x˜ I The operatpor S has ≥two non-degenerate eigenvalues 1 ρ ) = 0, being X˜ = x˜ I1. −NoMte 0that0 these0 co1nd−i- equal to λ (S) = eiπ/M and λ (S) = e−iπ/M. There- M 0 opt 1 1 2 tions also imply that ρ Π =Π ρ . The optimal Π can 0 0 0 0 0 fore, the corresponding eigenvectors define the basis then be numerically found by solving a linear system of equations including the additional requirements Π P U = 1 1 1 . (6) and M−1SiΠ S−i =I. 0 ∈ √2" i i # It iis=0 usefu0l to observe that the condition − Consequently,forthedualproblemDP3the optimization Mi=−P01SiΠ0S−i =I implicitly fixes the value of operator X˜ has to be diagonal Pthe trace of Π0. In fact, Tr( Mi=−01SiΠ0S−i) = M−1Tr(SiΠ S−i) = M−1Tr(Π S−iSi) = i=0 0 i=0P 0 X˜ = x˜1 0 , (7) MTr(Π0) and being Tr(I)=N it follows that P P " 0 x˜2 # Tr(Π0) = N/M. By simple algebra it can be found a closed-formexpression for Π in the following two cases. 0 where, both x˜ and x˜ are real and non-negative, being 1 2 X˜ PSD. The constraint X˜ 1 U†ρ U reads 1. β =0. The optimal Π0 is given by ≥ M 0 2 1 0 X˜ 1 1 (2α−1)+i2β , (8) Π0 = M "0 0# . (13) ≥ 2M "(2α 1) i2β 1 # − − In particular, setting α = 1/3 the same numerical and after some simple algebra,1 we find that it can be results obtained in [12] are found. rewritten as 2. β = α(1 α). The measure operator results (2Mx˜ 1)(2Mx˜ 1) [(2α 1)2+(2β)2] 0 , (9) − 1 2 − − − − ≥ p 2 Π = ρ . (14) with x˜ 1/(2M) and x˜ 1/(2M). Hence, the min- 0 0 1 2 M ≥ ≥ imum of Tr(X˜) = x˜ +x˜ is obtained for x˜ = x˜ = 1 2 1 2 The constraint M−1SiΠ S−i =I becomes 1/(2M)(1+ (2α 1)2+(2β)2. Inconclusion,themin- i=0 0 imum error pprobab−ility Pe =1−Pc =1−Tr(X˜) is eM2nsemMi=b−0le1hρais=aIpasrhPtoicwuilnagr tshtrautctthuereq.uIanndteuemd,sstuacthe M 1 1 a sPpecific geometry has been considered by Yuen P = − (2α 1)2+(2β)2 . (10) e M − M − et al. in[3,(IV.4)]andtheresultsthereinreported p are in agreement with (11) and (14). We note that the first term on the right-hand side of (10) corresponds to a blind guessing on the equiproba- Theproposedformulationofthedualproblemhasalso ble elements belonging to the quantum state ensemble, provedusefultonumericallysolvesystemsoflargedimen- while the secondterm represents the gaindue to the op- sions, where the computational complexity sets a severe timal quantum discrimination. It is interesting to ob- limit to the possibility of finding an optimal solution. serve that the minimum errorprobability is obtained for This is the case of pulse position modulation (PPM), a β = α(1 α) and it results modulationformatcandidatefordeepspacecommunica- − tions [13]. In quantum PPM the states are defined in a p 2 P =1 . (11) Hilbert space given by the tensorial product of M sub- e − M spaces, each of dimension n, and, therefore, the overall Forinstance,(11)holdsforα=1andβ =0whichisthe space dimension N grows exponentially with the PPM case of study considered by Helstrom [1] and Ban et al. order M, being N = nM [14]. In [15], DP3 is applied to [5], which models linearly dependent spin–1/2 quantum quantumPPM usingthe softwareCVXforSDP [11]. The states, where the generating density operator has rank- solution of DP3 results considerably faster than DP1. As one ρ = (˛ψ )(ψ ) with pure state (˛ψ )= 1 . When a limit case, for values of N about 1000 the discrimina- 0 0 ¯ 0 0 0 tion problem was successfully solved with DP3 while, on ρ0 is diagonal, i.e., β =0, (10) simplifies ash i the same processing unit, CVX fails with DP1, because of computer memory limits. 2 P =1 α . (12) e − M V. CONCLUSIONS 1 Itis recalled that an Hermitianmatrix isPSD ifand onlyifits We have studied the dual problem for minimum error principalminorsareallnon-negative. probability discriminationof symmetric quantum states. 5 It hasbeen shownthat the optimizationoperator,inthe quantum states on Hilbert spaces of large dimensions, objectivefunction, canbe assumedto commute withthe such as modulated coherent states in optical communi- symmetry operator. This result leads to an alternative cations. formulationofthedualproblem,thatpresentsareduced number of variables and constraints. The obtained dual statement is convenient to find analytical solutions to Acknowledgments the discrimination problem, as we showed with an illus- trativeexample. Ontheotherhand,thenewformulation also permits a computationally efficient numerical solu- The authorswouldlike to thank M.Sasakiforencour- tion by means of SDP methods. This property is partic- agingcomments. 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