Distinguishability of the symmetric states
aa r X i v : . [ qu a n t - ph ] M a r Distinguishability of the symmetric states
M. A. Jafarizadeh a ∗ , P.Sadeghi b † , d.Akhgar a ‡ , P.Mahmoudi c § a Department of Theoretical Physics and Astrophysics, University of Tabriz, Tabriz 51664, Iran. , b Marand Faculty of Engineering, University of Tabriz, Tabriz, Iran. , c Department of Physics, Azarbaijan Shahid Madani University. 53714-161. Tabriz. Iran
August 3, 2018
Abstract
In this paper, the distinguishability of multipartite geometrically uniform quantumstates obtained from a single reference state is studied in the symmetric subspace. Wespecially focus our attention on the unitary transformation in a way that the producedstates remain in the symmetric subspace, so rotation group with J y as the generatorof rotation is applied. The optimal probability and measurements are obtained for thepure and some special mixed separable states and the results are compared with thoseobtained at the previous articles for the special cases. The results are valid for lin-early dependent states. The discrimination of these states is also investigated using the ∗ E-mail:[email protected] † E-mail:[email protected] ‡ E-mail:[email protected] § E-mail:[email protected] iscrimination separable measurement. We introduce appropriate transformation to gain the optimalseparable measurements equivalent to the optimal global measurements with the sameoptimal probability. iscrimination Discrimination of nonorthogonal quantum states is a fundamental and important problem inquantum information theory. In distinguishing a quantum state that belongs to the set ofknown quantum states with given prior probabilities, one possibility is to find a set of positiveoperator valued measure(POVM) that maximizes the probability of correct detection, whichcalled minimum error discrimination[1, 2, 3]. In the 1970s, necessary and sufficient conditionsfor an optimum measurement have been derived by Holevo, Helstrom, and Yuen et al. [4, 5, 6].However, solving problems by means of them, except for some particular cases, is a difficulttask. Jafarizadeh et al. [1] presented optimality conditions, by using Helstrom family ofensembles, which is not only powerful in solving problems but also easy to apply. Accordingly,using this technique, we obtain the optimal measurements.In many discrimination problems, considerable attention is paid to use local quantumoperations and classical communication between the components (LOCC)[2, 3, 7]. However,using LOCC does not have a simple mathematical structure to give analytical optimization, andfor some cases can’t achieve the optimal probability which obtained by global measurements;therefore, obtained information reduce [7, 3]. In order to partially overcome the defects, someresearchers began to solve alternative problem by separable operation, to investigate the localdistinguishability [8, 9, 10, 11]. These operators are free of entanglement and made strictsuperset of LOCC [7].Dimension of Hilbert space for n -qubit systems grows exponentially by n . In order todecrease the complexity of discrimination problem for these particles, we restrict the problemto a set of states that possesses sufficient symmetry. Symmetric subspace contains the stateswhich are invariant under the permutations of particles. This symmetric subspace is spannedby the n + 1 Dicke states. Dicke states are produced and detected experimentally [12, 13, 14].In addition, Dicke states are proposed for certain tasks in quantum information theory [15, 15]. iscrimination / { ρ k = U k ρ ( U k ) † , k =0 , , ..., m } , where U is unitary matrices [17]. GU states have well-known examples such asQuadrature amplitude modulation (QAM), pulse-position modulated ( PPM) and phase-shift-keyed( PSK) that discrimination of them investigated extensively [18, 19]. We select ρ as pureor special mixed separable state in symmetric subspace and as U k = exp( − i kJ y π/m ) thatrotates a spin- j state by 2 kπ/m with respect to the J y -axis. We obtain optimal probabilityof correct detection and optimal global measurement, while, results are valid for arbitrary k ,even for linearly dependent states. The set of pure GU states in the symmetric subspace whichare perfectly discriminated by the obtained measurements, are identified. Also, separableform of optimal global measurements is obtained. By Mapping the optimal measurementfrom the symmetric space to entire space of n -qubit, we succeed in obtain optimal separablemeasurements equivalent to optimal global measurements with the same error probability.In Sec. II a brief review of the minimum error discrimination is presented. In Sec. III andIV optimal detection of GU pure and mixed states in the Symmetric subspace are investigatedand the optimal probability and the optimal global measurements are obtained, respectively. InSec. V appropriate transformation is introduced to gain the optimal separable measurementsequivalent to the optimal global measurements with the same optimal probability, and Sec.VII is devoted to the conclusions. We assume a quantum system is prepared from a collection of given states which represented bym density operators { ρ i , ρ i ≥ , T r ( ρ i ) = 1 , i = 0 , ...m } , and transmission probability to thereceiver for each of them is p i is P mi p i = 1. The aim is to obtain the set of positive semidefinite iscrimination { Π i , P Π i = I } , in the way that the output state of operator Π i , represent state ρ i .Therefore, the probability of correct discrimination for each ρ i is T r ( ρ i Π i ). In the minimumerror approach, the set of measurement operators are looked for which provide maximumprobability of correct discrimination as follows p opt = 1 − p error = m X p i T r ( ρ i Π i ) . (2-1)The necessary and sufficient conditions of discrimination with the maximum-success probabilityis m X p i Π i ρ i − p j ρ j ≥ , ∀ j = 1 , ..., m. (2-2)In the Ref. [1] has been demonstrated that the necessary and sufficient conditions are equivalentto a Helstrom family of ensembles; then a more suitable form of the conditions of the minimumerror discrimination is presented as M = p j ρ j + ( p − p j ) τ j , ∀ j, (2-3)where M = P mi =1 p i ρ i Π i and { τ i , τ i ≥ } is the conjugate state of ρ i . Also, eigenvector of τ i with zero eigenvalue is proportional to Π i [1],Π i τ i = 0 . (2-4)In the following two sections, the new technique is applied for optimal detection of GU pureand mixed states in the Symmetric subspace. In this section we derive the maximum attainable value of the success probability in the methodof the minimum error discrimination probability for GU Symmetric states of n -qubit with equalthe priori probabilities. iscrimination { P | ψ i = | ψ i , P ∈ S n } , are calledSymmetric subspace states. For the n -qubit in the Hilbert space, ⊗ ni H , common eigenvectors J z , J are standard orthogonal bases and the symmetric subspace H s is indicated with j = n/ | ψ i = n X q = − n c q | j, q i z . (3-5)where c q is the probability amplitude. We distinguish the set of states { ρ k = U k ρ ( U k ) † } ,which ρ is in the symmetric subspace and U k is a unitary operator. This set is well-known asGU states. We specially are interest in the unitary transformation which the produced statesremain in the symmetric subspace, in a way that, σ y and J y are selected as generator of rotationfor each qubit and generator for n -qubit, respectively. Therefore, unitary transformation iswritten as U = exp( − i πm σ y ) ⇒ U = n z }| { U ⊗ U... ⊗ U = e − i πm J y , (3-6)the number of states, m , may be equal or greater than the dimension of H s , in other words, it isnot necessary to states be linear independent .We consider states with equal initial probability,1 /m , hence, from Eq. (2-3) for all k M = U k [ 1 m ρ + ( p − m ) τ ]( U k ) † = U k M ( U k ) † k = 0 , , ..., m − . (3-7)Thus, M and U k commute, in addition, by Cayley-Hamilton theorem in the subspace j = n/ M is written as M = P n − i =0 a i J iy and from Eq. (2-3) one obtains p = n X i =0 a i T r ( J iy ) , (3-8)Which p is Helestrom ratio and p opt < p [1]. Then optimization problem are given by min p = n X i =0 a i T r ( J iy ) , (3-9) subject to − τ = − ( n X i =0 a i J iy − m | ψ ih ψ | ) ≤ , (3-10) iscrimination max g ( Z ) = 1 m h ψ | Z | ψ i , (3-11) subject to Z ≥ T r ( J iy ) − T r ( Z J iy ) = 0 i = 0 , , ..., n − , (3-12)From slackness conditions τ Z = 0 and Eq. (2-4) Π i is concluded,Π = Z = | z ih z | , (3-13)where | z i is expressed by eigenvector of J y , | z i = P n q = − n α q | j, q i y . Using Eq. (3-12), P q q i (1 −| α | q ) = 0. So, for all i , . . . n ) ( n −
1) ( n − . . . ( − n )( n ) ( n − ( n − . . . ( − n ) ... . . . ( n ) n . . . ( − n ) n − | α | n ...1 − | α | − n = 0 , (3-14)matrix of coefficients is the same of the vandermonde matrix, since there is no two equal rows,determinant of the matrix of coefficients is non-zero, thus, we conclude that | α | q = 1 and | z i = P q e iθ q | j, q i y . Inserting the above result into the equation τ | z i = 0 , one obtains[ n − X i =0 a i J iy − m | ψ ih ψ | ] n X q = − n e iθ q | j, q i y = 0 n X q = − n [ e iθ q n − X i =0 a i q i − λm y h j, q | ψ i ] | j, q i y = 0 e iθ q n − X i =0 a i q i = λm y h j, q | ψ i , (3-15)and | n − X i =0 a i q i | = | λ | m | y h j, q | ψ i| , (3-16) iscrimination λ = h ψ | z i .After this, all coefficients in the initial state are real. Equation J y | J, q i ∗ y = − q | J, q i ∗ y , yields | y h j, q | ψ i| = | y h j, − q | ψ i| , (3-17)and | n X i =0 a i q i | = | n X i =0 a i ( − q ) i | . (3-18)Hence, Helestrom ratio, p = P ni =0 a i P q = n q = − n q i , is zero for the odd numbers of i , in the Eq. (3-9),and only even numbers of i have non-zero terms. For λ = | λ | e iθ λ , and y h j, q | ψ i = | y h j, q | ψ i| e iθ y h j,q | ψ i , last term of Eq. (3-15) yields θ q = θ λ + θ y h j,q | ψ i , and implies: λ = n X q = − n h ψ | j, q i y e iθ q , (3-19)so | λ | = n X q = − n | y h j, q | ψ i| . (3-20)Strong duality for the optimal value of dual problem, p opt , yield p opt = p ,thus p opt = n X i =0 a i T r ( J iy ) = n X q = − n e − iθ q | λ | e iθ λ m | y h j, q | ψ i| e iθ h j,q | ψ i = | λ | m . (3-21)Eq. (3-21) is valid for all of the number of states, m . The unnormalized vector | z i in theoptimal measurement operator, Π = | z ih z | , is | z i = n X q = − n e iθ q | j, q i y = 1 √ m n X q = − n e iθ y h j,q | ψ i | j, q i y . (3-22)Therefore, one obtains the set of the optimal measurements as { Z i = U k Z ( U k ) † , k =0 , , ..., m − } . In the case that the set of GU states are linearly independent, the opti-mal measurements, | z k i = U k | z i , are projective and discrete Fourier transformation of | z i iscrimination h z k | z h i = 1 m X q, q ′ e − iθ y h j,q | ψ i + iθ y h j,q ′| ψ i (cid:18) y h j, q | (cid:16) U † (cid:17) k (cid:19) (cid:16) U h | j, q ′ i y (cid:17) = 1 m n X − n e i πm q ( k − h ) = 1 m e − i π ( n /2 +1) m ( k − h ) n +1 X p =1 e i πm p ( k − h ) = δ k,h ∀ n + 1 = m. (3-23)This result is in agreement with pretty good measurement.If we suppose reference state as | ψ i = | j, q i z , it is always possible to make θ y h j,q | ψ i ,constant, therefore, | z i has a simple form as | z i = 1 √ m n X q = − n | j, q i y (3-24)This result is consistent with Ref [2].For the perfect discrimination, Eq. (3-21) is written as n X q = − n e iθ q | y h j, q | ψ i| = √ m. (3-25)For example, for generating state | ψ i = √ n +1 P n q = − n e iθ q | j, q i y , the set of GU states, which arelinearly independent, are discrete Fourier transformation of | ψ i and orthogonal, so perfectlyis discriminated. For discrimination of GU mixed states in the symmetric subspace, we select separable mixedstate, ρ = r | n , n ih n , n | + (1 − r ) | n , − n ih n , − n | , (4-26) iscrimination p − m ) τ = n − X i =0 a i J iy − m ( r | n , n ih n , n | + (1 − r ) | n , − n ih n , − n | ) , (4-27)Thus, the dual problem is written as follows: max g ( Z ) = 1 m ( r h n | Z | n i + (1 − r ) h − n | Z | − n i ) subject to Z ≥ T r ( J iy ) − T r ( Z J iy ) = 0 i = 0 , , ..., n − , (4-28)From τ | z i = 0, and | z i = P n q = − n α q | j, q i y , Eq. (3-16) for each q is written as e iθ q n − X i =0 a i q i = rm h j, q | n i X ´ q h n | j, ´ q i e iθ ´ q + 1 − rm h j, q | − n i X ´ q h − n | j, ´ q i e iθ ´ q , (4-29) e − iπJ y | j, n i z = | j, − n i z and y h j, q | j, − n i = e iπq y h j, q | j, n i are concluded From z h j, ´ m | e − iπJ y | j, m i z = δ m + ´ m, . Thus, Eq. (4-29) is given by e iθ q n − X i =0 a i q i = h j, q | n i m Ω q , (4-30)where Ω q = P ´ q h n | j, ´ q i e iθ ´ q [ r + (1 − r ) e iπ ( q − ´ q ) ]. It is always possible to make θ y h j,q | n , n i = 0, for all q s, and from Ω q = Ω q +2 , is concluded which exp ( iθ q ) takes only two different values. Withoutloss the generality, phase of | j, − j + 2 s i y and | j, − j + (2 s + 1) i y for s = 0 , , ... is given 1 andexp( iθ ), respectively. Therefore,Ω q = X s h n | j, − j + 2 s i [ r + (1 − r ) e iπ ( q + n ) e − iπ s ]+ e iθ X s h n | j, − j + (2 s + 1) i [ r + (1 − r ) e iπ ( q + n ) e − iπ (2 s +1) ]= ( r + (1 − r ) e iπ ( q + n ) ) A + e iθ ( r + (1 − r ) e iπ ( q + n ) ) B, (4-31)where P s h n | j, − j + 2 s i = A , P s h n | j, − j + (2 s + 1) i = B , and Ω q = − j +2 s = A + e iθ (2 r − B , e − iθ Ω q = − j +(2 s +1) = (2 r − e − iθ A + B . iscrimination p opt = q = n X q = − n e − iθ q h j, q | n i m Ω q = Ω q = − j +2 s m A + e − iθ Ω q = − j +(2 s +1) m B = 1 m [( X q h j, q | n i ) + 2 AB (cos( θ )(2 r − − , (4-32)From Eq. (4-32), cos( θ ) = 1 and cos( θ ) = − r > and r < , respectively. For thesystems with odd number of qubit, A = B . Therefore, p opt is simplified to the following form p opt = cos( θ )(2 r − − m ( X q h j, q | n i ) . (4-33)As a special case, for mixed three-qubit states, p opt becomes p opt = 13 (1 + | r − | ) . (4-34)This is in agreement with the results in the Ref. [20].Like the pure states for the linearly independent states, m = n + 1, the optimal measure-ments are discrete Fourier transformations of | z i and projective. In the Majorana representation any symmetric state of n -qubit, | φ S i , which is invariant underthe permutation, is uniquely made from the sum of all permutations of n single qubit state as | φ s i = 1 √ k X g = S n g | ϕ i| ϕ i ... | ϕ n i , (5-35)which k is the normalization factor. The vector of | ϕ i i is made from the roots of the followingfunction Φ( t ) = n X k =0 ( − k nk a k t k , (5-36)where t = e iϕ tan( θ ), | ϕ i i = cos θ | i + e iϕ sin θ | i and a k is expansion coefficient of | φ S i in theDick basis. iscrimination | z i is in the symmetric subspace thus it is always possible to write | z i in theMajorana representation as | z i = 1 √ k X g = S n g | ϕ i| ϕ i ... | ϕ n i = 1 √ k X g ∈ S n g | ϕ i , (5-37)the term of | ϕ i| ϕ i ... | ϕ n i is inserted by | ϕ i .Similar to the general furrier transformation, we introduce the following map on quantumstate | ϕ i | τ ρij i = s d ρ n ! X g ∈ S n ρ ij ( g ) g | ϕ i , (5-38)where S n is Symmetric group and ρ ( g ) is an irreducible representation of S n with the dimensionof d ρ . From Eq. (3-21), Majorana representation is equivalent, up to a constant, to transformedform of | ϕ i , by the trivial representation, | τ i = √ n ! P g ∈ S n ( g | ϕ i ).Here, we prove one of the important properties of this map which | τ i and each symmet-ric state, | ψ i , is orthogonal to the transformed form of non-trivial representation of S n . Bymultiplying of h ψ | , into the Eq. (5-38) one obtains h τ | τ ρij i = h τ | s d ρ n ! X g ∈ S n ρ ij ( g ) g | ϕ i = s d ρ n ! X g ∈ S n ρ ij ( g ) h τ | g | ϕ i = h τ | ϕ i s d ρ n ! X g ∈ S n ρ ij ( g ) . (5-39)In this step, we show that P g ∈ S n ρ ( g ) equal to zero. If P g ∈ S n ρ ( g ) = A then for all ´ g ∈ G isobtained ρ (´ g ) A = Aρ (´ g ) , (5-40)and from the shor lemma for an irreducible representation X g ∈ S n ρ ( g ) = λI d ρ . (5-41) iscrimination X g ∈ S n χ ρ ( g ) = λd ρ , (5-42)where χ ρ ( g ) is the character of group elements. For the finite group P g ∈ S n χ ρ ( g ) χ ∗ ´ ρ ( g ) = 0,and for the trivial representation χ ∗ ´ ρ ( g ) = 1. Therefore, one obtains λ = 0, and consequently P g ∈ S n ρ ( g ) = 0.This indicates that P g ∈ S n ρ ij ( g ) = 0, thus, h τ | τ ρi j i = 0 ∀ ρ = 1 . (5-43)So, (cid:12)(cid:12)(cid:12) τ ρi j E by non-trivial representations, is not in the symmetric space. In fact from the Schur-Weyl duality theorem [21] the Hilbert space of n -qubit, H ⊗ n , expressed in term of subspaceswhich are invariant under irreducible representation (irrep) of S n , U , i.e, H ⊗ n = X M H irrep S n ,U . (5-44)Then, | τ i , Belongs to the Symmetric Hilbert subspace and from (5-43) all (cid:12)(cid:12)(cid:12) τ ρi j E , which ρ = 1,are the states in the other subspaces.In following, in order to find optimum separable operation, the orthogonal terms are addedto the trivial representation. X ij X ρ | τ ρij ih τ ρij | = X ´ g,g ∈ S n X ij,ρ d ρ n ! ρ ij ( g ) ρ ∗ ij (´ g ) g | ϕ ih ϕ | ´ g = X ´ g,g ∈ S n X ρ d ρ n ! T r ( ρ ( g ) ρ † (´ g )) g | ϕ ih ϕ | ´ g = X ´ g,g ∈ S n X ρ d ρ n ! T r ( ρ ( g ´ g − )) g | ϕ ih ϕ | ´ g = X ´ g,g ∈ S n [ X ρ d ρ n ! χ ρ ( g ´ g − )] g | ϕ ih ϕ | ´ g = X ´ g,g ∈ S n δ g, ´ g g | ϕ ih ϕ | ´ g = X g ∈ S n g | ϕ ih ϕ | g = X g ∈ S n g [ | ϕ i| ϕ i ... | ϕ n ih ϕ |h ϕ | ... h ϕ n | ] g iscrimination
14= ( | ϕ i h ϕ | ⊗ | ϕ i h ϕ | ⊗ ... | ϕ n i h ϕ n | ) + ( | ϕ i h ϕ | ⊗ | ϕ i h ϕ | ⊗ ... | ϕ n i h ϕ n | ) + ... + ( | ϕ n i h ϕ n | ⊗ | ϕ i h ϕ | ⊗ ... ) + ..... (5-45)According to the last term in the Eq. (5-45) is yielded each term of last summation is separable,and the probability of correct discrimination pure state, in the symmetric subspace, by thislocal operator is p localopt = 1 m T r ( c X ij X σ | τ σij ih τ σij | ρ ) = 1 m T r ( √ c | τ ih τ |√ cρ ) + cm X ij X σ =1 T r ( | τ σij ih τ σij | ρ )= p globalopt + c X ij X σ =1 |h τ σij | ψ i| = p globalopt . (5-46)where, | z i = √ c | τ i .Therefore, the optimum separable operation equivalent to the optimumglobal operation is achieved. The discrimination of GU states in the symmetric subspace of n -qubit particles is investigated.In this subspace for the general pure states, the optimal probability and the optimal globalmeasurements are obtained by the presented method in the Ref. [1]. Also, as a special case,the mixed reference states which Included the convex combination of up and down states arestudied and the results are consistent with the results in the Ref. [2] and [20]. The following,we introduce a mapping to gain the optimal separable measurements equivalent to the optimalglobal measurements with the same optimal probability. The achieved results are always validfor make of the separable operations which are in the symmetric subspace. We expect tothe same results are expressed in the other subspaces and this paper offers a good startingpoint. To make the states which are orthogonal to symmetric subspace, we use Majoranarepresentation,however, this representation is not valid for general case of qudit state, so,finding of the appropriate map to make the separable measurement from each symmetricmeasurement, is difficult and under investigation. iscrimination References [1] M. A. Jafarizadeh, R. Sufiani, and Y. M. Khiavi, Phys. Rev. A , 337-394 (1973).[5] C. W. Helstrom, Quantum Detection and Estimation theory , New York: Academic, (1976).[6] H. p. Yuen, R. S. Kennedy and M. Lax, IEEE T. Inform. Theory
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