Quantum and Classical Information Theory with Disentropy

Transcription

1 Quantum and Classcal Informaton Theory wth Dsentropy R V Ramos rubensramos@ufcbr Lab of Quantum Informaton Technology, Department of Telenformatc Engneerng Federal Unversty of Ceara - DETI/UFC, CP 6007 Campus do Pc Fortaleza-Ce, Brazl Abstract In the present work, some concepts of uantum and classcal nformaton theory usng the dsentropy are ntroduced The dsentropy s a new concept based on the Lambert-Tsalls W functon It s shown that theorems smlar to Shannon s theorems can be establshed usng the dsentropy Furthermore, the dsentanglement of bpartte states can be calculated usng the dsentropy At last, snce the Lambert- Tsalls functon can assume real values for negatve values of the argument, the dsentropy of some uasprobablty dstrbuton can be calculated Key words Lambert-Tsalls W functon; uantum dsentropy; entanglement 1 Introducton In a recent work the Lambert-Tsalls W functon was ntroduced [1] Bascally, t s the functon the solves the euaton W z W z e z (1) In (1) s the Tsalls nonextensvty parameter and e (x) s the -exponental [2] When = 1, one has e (x) = e x and, hence, W =1 s the famous Lambert W functon [3-5] An mportant property of the Lambert-Tsalls W functon s x W x W x ln ln, (2) where + s the -addton operaton and ln (x) s the -logarthm functon [6] Usng (2), the Tsalls entropy of a dscrete varable can be rewrtten as S p ln p p W p p ln W p 1 p W p ln W p n n n n n n n n n n n n n (3) The term D p W p n n n (4) n (3) s postve and t has been called dsentropy snce t s maxmal for a delta dstrbuton and mnmal for a unform dstrbuton [1] In ths drecton, the goal of ths

2 work s to show new defntons and applcatons of the dsentropy n uantum and classcal nformaton theory The present work s outlned as follows: n Secton 2, some basc concepts of nformaton theory wth dsentropy are provded; In Secton 3 the uantum dsentropy s ntroduced; Secton 4 brngs some applcatons of the dsentropy n classcal nformaton theory whle Secton 5 shows the applcaton of the dsentropy n the calculaton of the dsentanglement In Secton 6 the man dfference between entropy and dsentropy s dscussed At last, the conclusons are drawn n Secton 7 2 Basc concepts of nformaton theory usng dsentropy Let the random varable X = {x k k = 1,2,,K} to represent the possble outcomes of an event (lke a measurement) The result x k appears wth probablty p k where p k 0 and The amount of dsnformaton assocated to p k s gven by d = W (p k ) One may note that W (0) = 0 and W (x) > 0 for x > 0 [1] The average amount of dsnformaton s the dsentropy K D X p W p k k k1 (5) For a delta dstrbuton one has D = W (1), ts maxmal value, whle for a unform dstrbuton ( ), ts mnmal value Hence, the dsentropy measures the certanty of X Followng e (5), the jont dsentropy of the random varables X and Y s gven by KJ, D X, Y p x, y W p x, y, k j k j k1, j1 (6) where p(x k,y j ) s the jont probablty of X = x k and Y = y j The mutual dsentropy, by ts turn, s defned as M D X, Y D X D Y D X, Y (7) Usng the mutual dsentropy, the condtonal dsentropy can be defned n the tradtonal way as beng D X Y D X, Y D Y (8) In (8) the certanty about X after observaton of Y s eual to the dfference of certantes of the par (X,Y) and Y The relatve dsentropy can be defned as

3 D X Y p W p W t R k k k k (9) In (9) the dstrbuton p n (t n ) s assocated to the possble values of the random varable X (Y) At last, the dsentropc uncertanty s wrtten as D X D Y 2W 1 (10) Obvously, the value 2W (1) s not the tght bound value when X and Y are correlated 3 Quantum dsentropy The uantum verson of the dsentropy s D W, Q n n n (11) where n s are the egenvalues of the densty matrx The uantum mutual dsentropy s M D, D D D Q A B Q A Q B Q AB B A A B Tr AB (12) (13) The uantum mutual dsentropy s non-negatve The uantum condtonal dsentropy can be defned n the tradtonal way as beng D A B D D Q Q AB Q B (14) The uantum condtonal dsentropy can be negatve or postve for entangled states but t s negatve for dsentangled states The uantum relatve dsentropy, by ts turn, s gven by, D W W Q n n n n (15) where n and n are, respectvely, the egenvalues of the densty matrces and 4 Applcatons of classcal dsentropy The Shannon s source codng theorem states that H(u) L H(u)+1 bts, where H(u) s the Shannon entropy of an alphabet u wth r symbols Each symbol s selected

4 by the source wth probablty p, = 1,,r Furthermore,, where l s the number of bts of the -th codeword Smlarly, one can defne a source codng theorem usng the dsentropy: D u 1 D u, (16) where s an average value For = 1, one has r L p log W p p l log W p 1 1 r In order to guarantee a postve value for regardless the probablty dstrbuton {p }, one has the followng upper bound for l : (17) l log 2 W p (18) Now, consder a nosy channel modelled by the condtonal probablty p(y x) Fg 1 Nosy channel modelled by the condtonal probablty p(y x) The nput of the nosy channel s the random varable X wth probablty dstrbuton p X (x) The output random varable s Y The Shannon-Hartley s theorem states that, n order to avod errors, the transmsson rate R (bts/s) must be smaller than the channel s capacty gven by ( ), where I(X,Y) s the mutual ( ) nformaton Smlarly, one can defne a channel capacty based on dsentropy as beng D M C Inf D X, Y px x (19) For a gven channel, the probablty dstrbuton p X (x) that maxmzes C should be the same that mnmzes 5 Applcatons of uantum dsentropy It was shown that the uantum dsentropy can be used to measure the dsentanglement of a pure bpartte of ubt state: The dsentanglement measure of the two-ubt state (here 0 and 1 are the Schmdt coeffcents and = 1) s gven by the uantum dsentropy of the partal state A = Tr B ( ) ( B could also be used) [1]

5 D D W W E Q A (20) From (20) t s easy to note the dsentanglement s maxmal when A s a pure state ( 0 1 = 0) and mnmal when A s the maxmally mxed state ( 0 = 1 = 1/2) The dsentanglement of a mxed state can be calculated usng mn D p D p E Q A Tr A B (21) (22) The dsentanglement can also be calculated by the dstance between uantum states, usng the relatve uantum dsentropy as dstance measure: D R D mn E Q E (23) In (23) E s the set contanng all entangled states The uantum dsentropy can also be used to analyse the securty of uantum protocols For example, a uantum key dstrbuton protocol (QKD) s consdered secure f the mutual nformaton between Alce and Bob s larger than the mutual nformaton between Alce and Eve: I AB > I AE Smlarly, under the same condtons, a QKD protocol can be consdered secure f ( ) ( ) 6 Dsentropy versus Entropy At a frst glance, t seems that dsentropy does not brng somethng really new It seems to be just an opposte pont of vew: order nstead of dsorder or dsentanglement nstead of entanglement However, ths s not true Frstly, one can note the Lambert- Tsalls functon can assume real values for a range of negatve values of the argument In fact W (z) assume real values for z 1 for z 1 for 1, 2, 4, 6,8 2 (24) (25) One may note that, as expected, z tends to -1/e when tends to 1 Snce the Lambert-Tsalls functon can assume real values for a range of negatve values of the argument, the dsentropy of some seuences wth negatve values can be calculated For example, the separablty crteron for 2x2 and 2x3 systems proposed by Peres and the Horodeck famly [7,8] states that f the partal transpose of the densty matrx does not have negatves egenvalues then the state s separable otherwse t s entangled The

6 dsentropy of the partal transpose of the densty matrx can be calculated f the negatve egenvalue wth maxmal ampltude s larger than the lmts n (24)-(25) In ths case, the partal transpose of a hghly entangled bpartte state can have negatve dsentropy Consderng the contnuous case, the dsentropy of some uas-probablty dstrbutons can be calculated f the maxmal negatve value s larger than the lmts n (24)-(25) If w(x,p) s the Wgner functon of a gven uantum state, then ts contnuous dsentropy s gven by D w x, p W w x, p dxdp px (26) If one consders a dynamcal system n whch the Wgner functon of the uantum state vares at tme, w(x,p;t), one can calculate rate of dsentropy decay, for Gaussan and non-gaussan uantum states, gven by D r w x, p; tw wx, p; t dxdp t t px (27) In [9] the entropy producton was calculated but only for Gaussan states Snce the calculaton of the dsentropy decay can be done for Gaussan and non-gaussan states, one can check the relevance of the uantumness n the dynamc of the dsentropy 7 Conclusons The dsentropy based on the W functon can brng new results n uantum and classcal nformaton theory In partcular, the uantum dsentropy s useful n the calculaton of some dsentanglement measures In problems where the entropy should be maxmzed (mnmzed), the dsentropy should be mnmzed (maxmzed) One may note that t s hard to get analytcal results once the Lambert-Tsalls functon does not have the sum-product relaton found n the logarthmc functon Fnally, the operatonal meanng of (16) and (19) reures further nvestgaton On the other hand, the calculaton of the dsentropy of uas-probablty functons can brng new nsghts n the uantum-classcal transton problem Acknowledgements Ths study was fnanced n part by the Coordenação de Aperfeçoamento de Pessoal de Nível Superor - Brasl (CAPES) - Fnance Code 001, and CNP va Grant no / Also, ths work was performed as part of the Brazlan Natonal Insttute of Scence and Technology for Quantum Informaton

7 References 1 G B da Slva and R V Ramos, The Lambert-Tsalls W functon, arxv: , C Tsalls, Possble generalzaton of Boltzmann-Gbbs statstcs, J Stat Phys 52, 479, R M Corless, G H Gonnet, D E G Hare, D J Jeffrey and D E Knuth, On the Lambert W functon, Advances n Computatonal Mathematcs, vol 5, , S R Vallur, D J Jeffrey, R M Corless, Some applcatons of the Lambert W functon to Physcs, Canadan Journal of Physcs, vol 78 n 9, , Ken Roberts, SR Vallur, Tutoral: The uantum fnte suare well and the Lambert W functon, Canadan Journal of Physcs, vol 95, no 2, , T Yamano, Some propertes of -logarthm and -exponental functons n Tsalls statstcs, Physca A, 305, , Asher Peres, Separablty Crteron for Densty Matrces, Phys Rev Lett, 77, 8, , Mchal Horodeck, Pawel Horodeck and Ryszard Horodeck, Separablty of mxed states: necessary and suffcent condtons, Phys Lett A, 223, 1-8, J P Santos, G T Land, and M Paternostro, The Wgner entropy producton rate, Phys Rev Lett, 118, [220601], 2017