5 HAO San-Ru, HOU o-yu, XI Xiao-Qiang and YUE Rui-Hong Vol. 37 states form a basis in the Hilbert space of two qubits. In general, the two parties hav

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1 ommun. Theor. Phys. (eijing, hina) 37 () pp 49{54 c International Academic Publishers Vol. 37, No., February 5, Accessible Information for Equally-Distant Partially-Entangled Alphabet State Resource HAO San-Ru, HOU o-yu, XI Xiao-Qiang and YUE Rui-Hong alculating Physics Division, Department of omputer Teaching, Hunan Normal University, hangsha 48, hina Institute of Modern Physics, Northwest University, Xi'an 769, hina (Received April 9, Revised June, ) Abstract We have proposed a quantum system with equally-distant partially-entangled alphabet states which has the minimal mutual overlap and the highly distinguishability, these quantum states are used as the \signal states" of the quantum communication. We have also constructed the positive operator-valued measure for these \signal states" and discussed their entanglement properties and measurement of entanglement. We calculate the accessible information for these alphabet states and show that the accessible information is closely related to the entanglement of the \signal states": the higher the entanglement of the \signal states", the better the accessible information of the quantum system, and the accessible information reaches its maximal value when the alphabet states have their maximal entanglement. PAS numbers: 3.65.z, a, c Key words: accessible information, alphabet states, POVM, entanglement Introduction The mutual information I(X : Y ) [;6] was one of the rst information-theoretic quantities investigated with respect to quantum systems. [7 8] It quanties how correlated two messages are, i.e., how much we know about a message drawn from X n when we have read a message drawn from Y n. The maximum mutual information of an ensemble of quantum states obtained between the states of the ensemble and the outcomes of a positive operator valued measurement (POVM) [9;] on these states is called as accessible information. [3;5] Information theory is the theory of messages composed from letters. In a general communication setting, let fx i Xg be input letters and let f i g be their prior probabilities. Let us denote output letters by fy i Y g. The Shannon mutual information is dened in terms of the conditional probability P (jji) of obtaining output y j provided that the X letter sent was x i, which is written as X P (jji) I(X : Y ) = i P (jji)log i Pk kp (jjk) : () j In classical information theory, the channel matrix [P (jji)] is given and xed, characterizing the noise in the channel. In contrast, in a quantum information theoretic context where signal carriers are to be quantum states transmitted without noise, the channel matrix generally becomes a variable. ecause the act of quantum detection itself generally has a probabilistic output, therefore the channel matrix depends on the choice of quantum detection strategy. A quantum detection strategy is described by a POVM on a Hilbert space H. A POVM is any set f ^Fj g of Hermitian positive operators forming a resolution of the identity ^F + j = ^Fj ^Fj 8 j X j ^F j = ^I : () The detection operator ^Fj corresponds to the output letter y j and the conditional probabilities are given by P (jji) = Tr(^Fj ^ i ) : (3) Thus in the quantum context the optimization of I(X : Y ) is carried out with respect to the choice of POVM f ^Fj g for xed ensemble " = f^ i i g. The accessible information of the ensemble " is the maximum value of I(X : Y ). In this paper, our task is to study the accessible information for the ensemble of the equally-distant partiallyentangled alphabet states. In Sec. we start our discussion with the construction of quantum systems of equallydistant partially-entangled alphabet states needed to investigate our problem. Then in Sec. 3 we construct the POVM for this alphabet quantum state system. In Sec. 4 we discuss the accessible information and entanglement. Finally, in Sec. 5 we summarize our results and give concluding remarks. Equally-Distant Partially-Entangled Alphabet States oth sides of communication channel (Alice and ob) who share a pure two-qubit state j i A can generate three other states j ji A (j = 3 4) such that the four The project supported in part by Foundation of the Science and Technology ommittee of hina, and Foundation of the Science and Technology ommittee of Hunan Province of hina under the contract FSTH-5

2 5 HAO San-Ru, HOU o-yu, XI Xiao-Qiang and YUE Rui-Hong Vol. 37 states form a basis in the Hilbert space of two qubits. In general, the two parties have to perform operations on both qubits to generate the orthogonal states j ji A. Nevertheless, there is an exception if the original state j i A is one of the four ell states, [6 7] then by performing unitary transformations on just one of the two qubits (let us assume Alice is the operations), the other three ell states that form the ell basis of the two-qubit system can be generated. Specically, let us assume that the system is initially in the ell state j i A = p (ji A ji + ji A ji ) (4) where ji X and ji X (X = A ) are basis vectors in the Hilbert space H X of the qubit X (in what follows we will use the shorthand notation ji = jiji where we will clearly omit subscripts indicating the subsystem). Now we introduce four local (single-qubit) operations ^S = ^ = (jihj + jihi) ^S = ^ x = (jihj + jihj) ^S 3 = ^ y = i(jihj ; jihj) ^S 4 = ^ z = (jihj ; jihj) (5) where ^ ( = x y z) are three Pauli operators. When the operators ^Sk act on the rst (Alice's) qubit of the ell state (Eq. (4)), e.g., j i k = ^Sk ^j i (k = 3 4), we nd the four ell states. [6] This means that by performing just local operations the two-qubit states are changed globally in such a way that the four outcomes are perfectly distinguishable (i.e., the four ell states are mutually orthogonal). In fact, we can say that the four outcomes are mutually equally (and maximally) distant, which can be expressed by their mutual overlap R kl = jh kj lij = kl. Recently, arenco and Ekert, [8] Hausladen et al., [9] and ose et al. [] have discussed how the channel capacity depends on the degree of entanglement between the two qubits. Specically, these authors have analyzed the situation when initially Alice and ob share a two-qubit system in a state j i = ji + ji : (6) Then Alice is performing locally one of the four unitary operations ^Sk given by Eq. (5). These operations on Eq. (6) generate four possible states j k i called as \alphabet" states which are used in the communication channel. Very recently, Ziman and uzek [7] discussed another equally-distant partially-entangled alphabet states j ki (k = 3 4), which are generated by performing a set of local unitary operations ^Uk (k = 3 4) on one of the two qubits of the alphabet j i given by Eq. (6). In order to let the elements of the set fj ki j ki = ^Uk j i k = 3 4g be equally distant (i.e., the mutual overlaps of these four states are equal), and simultaneously be mutually distinguishable as possible as they can, these transformations ^Uk must obey the following conditions ( + for k = l R kl = jh j ^U ^U k l j ij = (7) R for k 6= l with R being as small as possible. In addition, the transformations under consideration have to fulll the ell limit, i.e., when!, or j i! ell state, [6] they must generate four maximally entangled mutually orthogonal two-qubit states. The explicit construction of the set of the local unitary transformation ^Uk is given in Ref. [7], and the equally-distant partially-entangled alphabet states in the basis fji ji ji jig read j i = ( ) p3 j i = j 3i = j 4i = r 3 r 3 ; p 3 p3 ; ; ip 3 p ; + ip 3 p 6 6 ; p ; ip 3 p + ip 3 p ; p 3 p : (8) 3 3 onstruction of the POVM for Equally- Distant Partially-Entangled Alphabet States ecause we want to discuss the accessible information for the ensemble (Eq. (8)), and the accessible information depends on the choice of quantum detection strategy which is described by a POVM on H, we rst discuss the POVM for the equally-distant partially-entangled alphabet states (Eq. (8)). Evidently, these \signal states" (8) are nonorthogonal, jh j ij = jh j 3ij = jh j 4ij = jh j 3ij = jh j 4ij = jh 3j 4ij = 3 4 (9) where 4 = jj ; jj. In fact, the mutual overlaps between these \signal states" aforementioned are minimal because we have asked that the \signal states" obtained by performing the transformation U k on alphabet state (6) have the minimal mutual overlaps in Sec., therefore these \signal states" (8) are the maximal distinguishable states. Let Alice prepare one of the four possible pure states (Eq. (8)), each occurring with a priori probability i = =4. ob's task is to nd out as much as he can about what Alice prepared by making a suitable measurement POVM. Now we begin to construct the POVM operators and invoke a general procedure that is usually at least pretty

3 No. Accessible Information for Equally-Distant Partially-Entangled Alphabet State Resource 5 good. The POVM constructed by this procedure is usually called as \pretty good measurement" (PGM). onsider the vectors fj iig that are not orthogonal. We want to devise a POVM that can distinguish these vectors reasonably well. Let us rst construct G = 4X i= j iih ij () and substitute Eq. (8) into Eq. (), then one can see G = jj jj jj jj A : () It is obvious that this operator G is a positive operator on the space spanned by the j ii. Therefore, on that subspace, G has an inverse, G ;, and that inverse has a positive square root G ;=, G ;= = p =jj =jj =jj =jj Now we dene the operator set ff i g as and we see that A : () F i = G ;= j iih ijg ;= (3) F ( ) = F ( ) = F 3 ( ) = F 4 ( ) = jjjj 3 p jjjj 6 p jjjj 6 p jjjj jjjj A (4) jjjj p jjjj jjjj ; p p p jjj jjjj ; p p jjjj ;jjjj A (5) ; p ;jjjj jjjj p p p jjjj (; + 3 i)jjjj ;( + 3 i) ; p ; p ;( + p 3 i)jjjj p jjjj (; + p 3 i) p ( + p 3 i) (; + p 3 i) ;( + p 3 i) p p jjjj ( ; p 3 i)jjjja ; p ( ; p 3 i) ( + p p (6) 3 i)jjjj jjjj p p p jjjj ;( + 3 i)jjjj (; + 3 i) ; p (; + p 3 i)jjjj p jjjj ;( + p 3 i) p ( ; p 3 i) ;( + p 3 i) (; + p 3 i) p p jjjj ( + p 3 i)jjjj ; p ( + p 3 i) ( ; p p 3 i)jjjj jjjj A : (7) It is straightforward to calculate and we will see that 4X i= F i = (8) on the span of the j ii. From Eqs (4) (7), we can see that all these four operators ff i g are Hermitian matrices, i.e., F + i = F i. In addition, by calculating directly, one can show that all these four operator matrices ff i g have the same eigenvalues and they are one-rank matrix operators, so they are positive operators, i.e., F j. Therefore, these operators ff i g dened by Eq. (3) satisfy the condition (), then we have constructed a POVM for the ensemble Eq. (8). 4 Accessible Information and Entanglement In this section, we nd the accessible information for the ensemble of the equally-distant partially-entangled alphabet states (Eq. (8)). The accessible information is dened as the maximal mutual information between the equally-distant partially-entangled alphabet states (with probabilities /4 each) and the elements of the POVM measuring these states. In quantum context, the accessible information of the ensemble " = fj ii i g of the pure state (Eq. (8)) is carried out with respect to the choice of POVM for the xed ensemble ". For the pure state ensemble, the conditional probabilities related to the detection operators F i which correspond to the output states are given by P (jji) = h ijf j ( )j ii (9) where F j ( ) are the detection operators given by Eqs (4) (7) which are matrix functions of parameters of and.

4 5 HAO San-Ru, HOU o-yu, XI Xiao-Qiang and YUE Rui-Hong Vol. 37 In what follows, we will calculate Eq. (9). efore performing our calculations, we rst introduce the polar coordinates. For = j j e i and = j j e i', the detection operator F i ( ) can be reduced as a matrix function of the parameters of the complex angles and ', i.e. F i ( ' ), and we will show that P ' (iji) = h ijf i ( ' )j ii = (jj + jj + e ;i( ;' ) + e i( ;' ) ) (i = 3 4) () P ' (jji) = h ijf j ( ' )j ii = 6 (jj + jj ; e ;i( ;' ) ; e i( ;' ) ) for i 6= j : () Now suppose that Alice and ob start to communicate with each other. Alice prepares one of four two-qubit states (Eq. (8)) each occurring with a priori probability i = =4. It follows that the mutual information of the input pure states ensemble " = fj ii i g is n (jj + jj ; e ;i( ;' ) ; e i( ;' ) ) log 3 ; e ;i( ;' ) + e i( ;' ) I ' (X : Y ) = jj + jj + (jj + jj + e ;i( ;' ) + e i( ;' ) ) log + e ;i( ;' ) + e i( ;' ) jj + jj The accessible information is dened as the maximal mutual information, i.e., o : () I(X : Y ) = max I ' (X : Y ) : (3) ' In order to get the maximal value of above equation, we rst note that I ' (X : Y ) is a function of ; ', and we can obtain the extreme value of the function I ' (X : Y ) by dierentiating it with respect to ; ', i.e., If one does as this procedure, one can get If let = jj e i and = jj e i', we have I ' (X : Y )=( ; ' ) = : e i( ;' ) = : (4) e i( ;' ) = e i(;') : (5) It is obvious that the parameters ' fullling the above equation enable I ' to get the extreme value, and this value is the maximal value (because of I ' =( ; ' ) < at the extreme point). Substituting Eqs (5) and () into Eq. (3), we obtain the accessible information as follows: I(X : Y ) = n (jj + jj ; jjjj)log 3 ; 4jjjj + (jj + jj + jjjj) log + 4jjjj o : (6) 3(jj + jj ) (jj + jj ) Let us assume that the \signal states" (Eq. (8)) are normalized states, so the and should satisfy the following relationship jj + jj = (7) therefore the accessible information (Eq. (6)) can be rewritten as follows: n p ( ; jj ; jj ) log 3 ; 4 jjp ; jj 3 p p o + ( + jj ; jj )log(+ 4jj ; jj ) : (8) I(X : Y ) = In order to discuss the relationship between the entanglement and the accessible information of the \signal states" (Eq. (8)), now we briey discuss the measures of entanglement. Every measure of entanglement E(^) should satisfy the following necessary conditions for a given density matrix ^: [ ] (i) E(^) = if and only if ^ is separable (ii) A local unitary transformation leaves E(^) invariant (iii) E(^) cannot increase under local general measurements (LGM), classical communications () and post selection of subensemble (PSS). For a pure state ^ of a bipartite system, one can prove that the entropy S satises all aforementioned conditions and we can choose it as a measure of entanglement. The entropy S expression is S(^) = ;Tr ^A log ^A = ;Tr ^ log ^ (9) where ^A = Tr A^ are the reduced density matrices for the subsystems A and. For the pure state j 3i of Eq. (8), the ^ density matrix can be written as ^ = j 3ih 3j (3)

5 No. Accessible Information for Equally-Distant Partially-Entangled Alphabet State Resource 53 and the reduced density matrix for the subsystem A follows ^A = Tr ^ = 3 jj jihj + ; + p 3 i 3 p jj jihj ; + p 3 i 3 p jj jihj + 3 jj jihj + 3 jj jihj + ; p 3 i 3 p jj jihj + + p 3 i 3 p jj jihj + 3 jj jihj : (3) In order to nd the eigenvalues of the reduced density matrix ^A, we rewrite the above expression in a matrix form, 3 ^A = (jj + jj ) ; + p 3 i 3 p (jj ; jj ) ; + p A : (3) 3 i 3 p (jj ; jj 3 (jj + jj ) From the expression about the reduced density ^A, one can easily nd the eigenvalues E = [ (jj ; jj )] : (33) The measure of entanglement S(^) (9) is then written as S(^) = ;(jj log jj + jj log jj ) : (34) ecause and satisfy the relation (7), the measure of entanglement S can be expressed by jj in the following form S(^) = ;(jj log jj + ( ; jj ) log ( ; jj )) : (35) Following the same procedure, one can nd that all these four \signal states" j ii i = 3 4, have the same entanglement as the \signal state" j 3i. How the accessible information depends on the entanglement of the \signal states" can be shown by Fig.. Fig. The gure shows how the accessible information I(x : y) and the measure of entanglement S(^) of the equally-distant partially-entangled alphabet \signal states" depend on the entanglement alphabet jj. The solid line shows the accessible information and the dashed line shows the measure of entanglement of the equallydistant partially-entangled alphabet \signal states". First, we can see from Fig. that the accessible information I(x : y) rstly increases and then decreases as the alphabet jj varies in the range jj and reaches the maximal value at jj = = p Secondly, the measure of entanglement of the \signal states" (Eq. (8)) reaches the maximal value at the point jj = = p. So the \signal states" (Eq. (8)) are the maximal entanglement states when jj = = p and the partially-entangled alphabet \signal states" when jj 6= = p. We can also see directly from Fig. that the larger the entanglement of the \signal states" (8), the larger the accessible information (8) and the maximally entangled \signal states" give the unit accessible information per qubit at jj = = p. 5 oncluding Remarks In general, i.e., jj 6= p =, the mutual overlap of the equally-distant partially-entangled alphabet \signal states" (Eq. (8)) is nonzero (See Eq. (9)), so they are not orthogonal states. And in this case, the \signal states" are the partially-entangled quantum states, the best measure method used by ob is the POVM so that the accessible information attains the optimal value. As jj ;! p =, the accessible information I(x : y) trends to the maximal value. When jj = p =, the mutual overlaps of the \signal states" are zero (because of 4 = ), the \signal states" (Eq. (8)) are orthogonal states and maximally-entangled states (like-ell states). In that case, the POVM used by ob changes into the Von Neumann measurement (VNM) which is the optional measurements, the communication outcomes are best measured and the accessible information reaches the maximal value. As we have seen and discussed [3] that the more distinguishable the \signal states", the better the mutual information of the communication system. It can be seen from Eq. (9) that the distinguishability of the \signal states" varies as the entanglement parameter jj changes and it reaches the complete distinguishable one if jj = p =. The distinguishability of the \signal states" means the extent of mutual overlap between the dierent \signal states", the smaller the mutual overlap between the dierent \signal states", the higher the distinguishability of the \signal states". Therefore, improving the distinguishability of the \signal states" can enhance the accessible information, and one can gain the maximal accessible infor-

6 54 HAO San-Ru, HOU o-yu, XI Xiao-Qiang and YUE Rui-Hong Vol. 37 mation if the \signal states" are complete distinguishable. The \pretty good measurement" with the highly distinguishable measurement states can improve the mutual information. First, Alice is able to convey more information to ob by \improving" her \signal state" distinguishability. She is better choosing \signal states" from higher distinguishable signal states than choosing from lower distinguishable ones. Second, ob is able to read more information if he performs a POVM measurement on the \signal state" qubits instead of measuring them in other methods. Non-orthogonal \signal states" may provide a means for encrypting information. [4] In practice, Alice may send the alphabet \signal states" with jj being close to = p which have higher distinguishability and higher entanglement, they can reach better accessible information provided that ob chooses the POVM measurement and they simultaneously have a good encryption. The average mutual information of the random block messages [3] for the equally-distant partially-entangled alphabet \signal state" systems is a very interesting problem, moreover the quantum channels of the random block messages for the equally-distant partially-entangled alphabet \signal state" ensembles are also interesting as well as useful, we will discuss them in our future works. Acknowledgments We are grateful to P. Wang and K.J. Shi for helpful comments and one of the author (S.R. Hao) wishes to thank L.Y. Wang for his encouragement and support. References [] A.S. Holevo and J. Multivar, Anal. 3 (973) 337. [].W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, New York (976). [3] A. Peres, Quantum Theory: oncepts and Methods, Kluwer Academic Publishers, Dortrecht (993) pp 79{ 89. [4] A. Peres and W.K. Wootters, Phys. Rev. Lett. 66 (99) 9. [5] K. Kato, M. Osaki, M. Sasaki and O. Hirota, IEEE Trans. ommun. 47 (998) 48. [6].A. Fuchs, LANL quant-ph/96. [7] K.J. ostrom, LANL quant-ph/95. [8] S.R. HAO and L.Y. WANG, Acta Physica Sinica 49 () 6. [9] Y.. Eldar and G.D. Forney, LANL quant-ph/53. [] A. Peres, Quantum Theory: oncepts and Methods, oston, Kluwer (995). [] H.P. Yuen, R.S. Kennedy and M. Lax, IEEE Trans. Inform. Theory IT- (975) 5. [] M. an, K. Kurukowa, R. Momose and O. Hirota, Int. J. Theor. Phys. 36 (997) 69. [3] L.. Levitin, \Optimal Quantum Measurements for Two Pure and Mixed States" in Quantum ommunications and Measurement, eds V.P. elavkin, O. Hirota and R.L. Hudson, Plenum Press, New York (995) pp 439{448. [4] M. Sasaki, S.M. arnett, R. Jozsa, M. Osaki and O. Hirota, Phys. Rev. A59 (999) 335 LANL quantph/986. [5] P.W. Shor, LANL quant-ph/977. [6] J. Preskill, Quantum Theory of Information and omputation, [7] M. Ziman and V. uzek, LANL quant-ph/975. [8] A. arenco and A.K. Ekert, J. Mod. Opt. 4 (995) 53. [9] P. Hausladen, R. Jozsa,. Schumacher, M. Westmoreland and W.K. Wootters, Phys. Rev. A54 (996) 869. [] S. ose, M.. Plenio and V. Vedral, J. Mod. Opt. 47 () 9. [] V. Vedral, M.. Plenio, M.A. Rippin and P.L. Knight, Phys. Rev. Lett. 78 (997) 75. [] V. Vedral and M.. Plenio, Phys. Rev. A57 (998) 69. [3] S.R. HAO,.Y. HOU, X.Q. XI and R.H. YUE, to appear in hinese Phys. [4] G. Gilbert and M. Hamrick, LANL quant-ph/97.