Reexamination of Quantum Data Compression and Relative Entropy

Transcription

1 Wlfrd Laurer Uversty Scholars Laurer Physcs ad Computer Scece Faculty Publcatos Physcs ad Computer Scece 2008 Reexamato of Quatum Data Compresso ad Relatve Etropy Alexe Kaltcheko Wlfrd Laurer Uversty, akaltcheko@wlu.ca Follow ths ad addtoal works at: Recommeded Ctato Kaltcheko, Alexe, "Reexamato of Quatum Data Compresso ad Relatve Etropy" (2008). Physcs ad Computer Scece Faculty Publcatos Ths Artcle s brought to you for free ad ope access by the Physcs ad Computer Scece at Scholars Laurer. It has bee accepted for cluso Physcs ad Computer Scece Faculty Publcatos by a authorzed admstrator of Scholars Laurer. For more formato, please cotact scholarscommos@wlu.ca.

2 Reexamato of quatum data compresso ad relatve etropy Alexe Kaltcheko* Departmet of Physcs ad Computer Scece, Wlfrd Laurer Uversty, Waterloo, Otaro, Caada N2L3C5 Receved 5 May 2008; publshed 7 August 2008 Schumacher ad Westmorelad Phys. Rev. A 64, have establshed a quatum aalog of a well-kow classcal formato theory result o a role of relatve etropy as a measure of ooptmalty classcal data compresso. I ths paper, we provde a alteratve smple ad costructve proof of ths result by costructg quatum compresso codes schemes from classcal data compresso codes. Moreover, as the quatum data compresso or codg task ca be effectvely reduced to a quas classcal oe, we show that relevat results from classcal formato theory ad data compresso become applcable ad therefore ca be exteded to the quatum doma. DOI: /PhysRevA PACS umber s : Hk, Lx, Ta I. INTRODUCTION Quatum relatve etropy, defed as S tr log 2 tr log 2, s always o-egatve ad s equal to zero f ad oly f =, where ad are desty operators defed o the same fte-dmesoal Hlbert space. For a revew of the propertes of quatum relatve etropy as well as the propertes of quatum vo Neuma etropy, defed as S =tr log 2, see, for stace, Ref. 1. Both quatum vo Neuma etropy ad quatum relatve etropy ca be vewed as quatum extesos of ther respectve classcal couterparts ad hert may of ther propertes ad applcatos. Such hertace however s ofte ether automatc or obvous. For example, classcal formato theory Ref. 2 Chap. 5 provdes the followg atural terpretatos of Shao etropy ad relatve etropy. A compresso code scheme, for example, Huffma code, whch s made optmal for a formato source wth a gve probablty dstrbuto p of characters, would requre H p bts per put letter to ecode a sequece emtted by the source, where H p stads for the Shao etropy of the probablty dstrbuto p ad s defed by H p p x log 2 p x. Thus, f usg a compresso code 1 scheme made optmal for a formato source wth probablty dstrbuto q, also called a q-source, oe could compress such a source wth H q bts per put character. If, to compress the same q-source, oe uses a dfferet compresso code scheme optmal for a source wth a dfferet probablty dstrbuto p.e., p-source, t wll requre H q +D q p bts per put character, where D q p stads for the relatve etropy fucto for the probablty dstrbuto q ad p ad s defed by D q p q x log 2 q x p x. Accordgly, the quatty D q p represets a addtoal ecodg cost whch arses from ecodg a q-source wth a *akaltcheko@wlu.ca 1 See Sec. II A for the formal deftos of formato sources, compresso codes, etc. 1 2 code optmal for a p-source. Ths classcal data compresso terpretato of classcal Shao ad relatve etropes exteds 3,4 to the quatum doma f oe replaces classcal formato sources havg probablty dstrbutos p ad q wth quatum-formato sources havg desty matrces ad, respectvely, as well as classcal Shao ad relatve etropes wth quatum vo Neuma ad relatve etropes, respectvely. I ths paper, we provde a alteratve smple ad costructve proof of ths result. We buld quatum compresso codes schemes from classcal compresso codes. As the orgal source desty matrx ca smply be replaced wth, a so-called effectve source matrx 5 wth respect to our computatoal bass, our quatum compresso or codg task s effectvely reduced to a quas classcal oe, wth classcal formato sources havg probablty dstrbutos p ad p, whch are formed by the egevalues of ad, respectvely. Thus, relevat results from classcal formato theory ad data compresso become applcable ad therefore ca be exteded to the quatum doma. Our paper s orgazed as follows. I Sec. II, we brefly revew some classcal data compresso deftos ad results, cludg formato sources ad source codes also kow as data compresso codes. I Sec. III, we troduce quatum-formato sources, quatum data compresso, ad dscuss how classcal data compresso codes ca be tured to quatum data compresso codes. I Sec. IV, we troduce a effectve desty matrx for a quatum-formato source. I Sec. V, we state our ma results: Lemma V.1 ad Theorem 3 followed by a dscusso. We coclude the paper wth a bref cocluso secto. II. CLASSICAL DATA COMPRESSION A. Classcal formato sources, compresso codes, ad optmal data compresso Let A= a 1,a 2,...,a A be a data sequece alphabet, where the otato A stads for the cardalty of A. Bary alphabet 0, 1 ad Lat alphabet a,b,c,d,...,x,y,z are wdely-used alphabet examples. We wll also use the otato sequece-ame to deote the sequece legth. Let A*, A +, ad A be, respectvely, the set of all fte sequeces, /2008/78 2 / The Amerca Physcal Socety

3 ALEXEI KALTCHENKO cludg the empty sequece, the set of all fte sequeces of postve legth, ad the set of all fte sequeces, where all the sequeces are draw from A. For ay postve teger, A deotes the set of all sequeces of legth from A.A sequece from A s sometmes called a A sequece. To save space, we deote a A sequece x 1,x 2,...,x by x for every teger 0. Defto II.1. A alphabet A depedetly ad detcally dstrbuted IID formato source s a IID dscreet stochastc process, whch s defed by a sgle-letter margal probablty dstrbuto of characters from A. We exted the sgle-letter probablty dstrbuto to a probablty measure o the probablty space costructed for the set A + of all A sequeces. We say A-source or p-source to emphasze that the source has a certa alphabet A or a probablty dstrbuto p. Thus, we assume that every A sequece s geerated by a IID formato source. Defto II.2. A alphabet A source code C s a mappg from A + to 0,1 +, where, accordg to our otato, A + s the set of all A strgs ad 0,1 + s the set of all bary strgs. We restrct our atteto to uquely decdable codes, that s, for every ad every x 1,x 2,...,x, we have C 1 C x 1,x 2,...,x = x 1,x 2,...,x. For every ad every x 1,x 2,...,x, we call a sequece C x 1,x 2,...,x a codeword. We ofte call a source code a compresso code ad use the terms source code ad compresso code terchageably. Defto II.3. For ay gve source ad a alphabet A source code C, we defe the compresso rate of C bts per put character by 1 C x 1,x 2,...,x p, where the legth of a compressed sequece codeword C x 1,x 2,...,x s dvded by ad averaged wth respect to formato source s probablty measure p. Theorem 1 (optmal data compresso). For a IID formato source wth probablty measure p ad ay compresso code, the best achevable compresso rate s gve by Shao etropy H p, whch s defed by H p p x log 2 p x. 3 x A Defto II.4. For ay probablty dstrbuto p, we call a source code optmal for a IID source wth probablty dstrbuto p f the compresso rate for ths code s equal to Shao etropy H p. We deote such a code by C p. Remark 1. Optmal compresso code s exst for ay ad every source s probablty dstrbuto. B. Classcal data compresso ad relatve etropy Suppose we have a compresso code C p, that s, C p s optmal for a A-source wth a probablty dstrbuto p. What happes to the compresso rate f we use C p to compress aother A-source wth a dfferet dstrbuto q? Its ot dffcult to fer that the compresso rate wll geerally crease, but by how much? The crease the compresso rate wll be equal to relatve etropy fucto D q p, whch s defed by q x D q p q x log 2 x A p x, where relatve etropy fucto D q p s always oegatve 4 ad equal to zero f ad oly f q p. More precsely, f usg a code C q.e., optmal for a dstrbuto q, oe could ecode a q-source wth H q bts per put character. If, stead, oe uses a code C p.e., optmal for p to ecode a q-source, t wll requre H q +D q p bts per put character. III. TURNING CLASSICAL SOURCE CODES INTO QUANTUM TRANSFORMATIONS I ths secto we revew how a classcal oe-to-oe mappg code ca be aturally exteded 2 to a utary quatum trasformato. But frst, we eed to defe quatumformato sources. A. Quatum-formato sources Iformally, a quatum detcally ad depedetly dstrbuted IID formato source s a fte sequece of detcally ad depedetly geerated quatum states say, photos, where each photo ca have oe of two fxed polarzatos. Each such state lves ts ow copy of the same Hlbert space ad s characterzed by ts ow copy of the same desty operator actg o the Hlbert space. We deote the Hlbert space ad the desty operator by H ad, respectvely. The, the sequece of states lves the space H ad s characterzed by the desty operator actg o H. For example, case of a photo sequece, we have dm H=2. Thus, to defe a quatum source, we ust eed to specfy ad H. I ths paper, we restrct our atteto to quatum sources defed o two-dmesoal Hlbert spaces. Ofte, ust s gve ad H s ot explctly specfed as all Hlbert spaces of the same dmesoalty are somorphc. B. Turg classcal source codes to quatum trasformatos Now we are ready to expla how a classcal source code ca be tured to a quatum trasformato. For the rest of the paper, we wll oly be cosderg bary classcal sources wth alphabet A= 0,1 ad quatum-formato sources defed o a two-dmesoal Hlbert space. Let B 0, 1 be a arbtrary but fxed orthoormal bass the quatum-formato source space H. We call B a computatoal bass. The, for a sequece of quatum states, the orthoormal bass wll have 2 vectors of the form e 1 e 2 e 3 e, where, for every teger 1, e B 0, 1. To emphasze the symbolc correspodece betwee bary strgs of legth ad the bass vectors H, we ca rewrte Eq. 5 as 2 For more detals, see Refs. 5 or

4 REEXAMINATION OF QUANTUM DATA COMPRESSION AND e 1 e 2 e 3...e, where for every teger 1, e 0,1. I other words, f we add the Drac s ket otato at the begg ad the ed of ay bary strg of legth, the we wll obta a bass vector H. Thus, for ay computatoal bass B H, postve teger, ad a classcal fully reversble fucto mappg for -bt bary strgs, we ca defe a utary trasformato o H as follows: Let : 0,1 0,1 be a classcal fully reversble fucto for -bt bary strgs. For every ad, we defe a utary operator U B, :H =H H out by the bases vectors mappg U B, e 1...e = e 1...e, where for every teger 1, e 0,1 ad e B 0, 1. We pot out that H out deotes a output space whch s a somorphc copy of the put space H =H, ad H out may sometmes cocde wth H. A elegat way of desgg quatum crcuts to mplemet U B, was gve by Lagford 6, where U B, was computed by the followg two cosecutve operatos defed o H H out : e 1...e e 1 out...e out e 1...e e 1 out...e out e1...e, e 1...e e 1 out...e out e 1...e 1 e 1 out...e out e 1 out...e out, where e,e out 0,1, e 1 out...e out H out, ad the otato stads for btwse modulo-2 addto. I the above, relato 7 maps a bass vector e 1...e to the bass vector e 1...e e 1...e. Relato 8 maps a bass vector e 1...e e 1...e to the bass vector e 1...e.. Aother terestg approach for computg U B, was suggested 5, where Jozsa ad Presell used a so-called dgtalzato procedure. Remark 2. We reterate that, gve a quatum source s Hlbert space H, the quatum trasformato U B, etrely depeds o, ad s determed by the selecto of the computatoal bass B H ad a classcal fully reversble fucto mappg for -bt bary strgs. We also pot out that as log as ad 1 ca be classcally computed lear tme ad space, the U B, ca be computed 7 9 log- lear tme ad space, too. C. Quatum data compresso overvew We wsh to compress a sequece of quatum states so that at a later stage the compressed sequece ca be decompressed. We also wat that the decompressed sequece wll be the same as, or close to the orgal sequece. The measure of closeess s called quatum fdelty. There are a few dfferet deftos of fdelty. Oe wdely used s the fdelty betwee ay pure state ad mxed state, defed by F, =. We ow troduce otato U B,C for a utary trasformato defed as U B,C ª U B, =C. Thus, the utary trasfor- mato U B,C acts o a product space H H ad ca be vewed as a quatum exteso of the classcal compresso code C. So to call U B,C, we wll use terms utary trasformato ad quatum compresso code terchageably. Defto III.1. Gve a quatum IID source wth desty operator, we defe a compresso rate of U B,C by 1 U B,C z 1 z 2 z 3...z, 9 where z 1 z 2 z 3...z H s a sequece emtted by the quatum source, ad the legth of compressed quatum sequece U B,C z 1 z 2 z 3...z s dvded by ad averaged the sese of a quatum-mechacal observable. 3 If we cut the compressed sequece U B,C z 1 z 2 z 3...z at the legth slghtly more tha U B,C z 1 z 2 z 3...z, the t ca be decompressed wth hgh fdelty. Thus, U B,C z 1 z 2 z 3...z s the average umber of qubts eeded to fathfully represet a ucompressed sequece of legth. See 4 for a detaled treatmet. Theorem 2 (Schumacher s optmal quatum data compresso [3]). For a quatum IID source wth a desty operator ad ay compresso scheme, the best achevable compresso rate wth a hgh expected fdelty s gve by the vo Neuma etropy, whch s defed by S =tr log Defto III.2. A compresso code, for whch the above rate s acheved, s called optmal. Lemma III.1. For a desty operator, we choose the egebass of to be our computatoal bass B. Let C p be a classcal compresso code, whch s optmal for a probablty dstrbuto formed my the set of the egevalues of. The, a utary trasformato U B,Cp s a optmal quatum compresso code for a quatum-formato source wth the desty operator. IV. EFFECTIVE SOURCE DENSITY MATRIX We recall a quatum bary detcally ad depedetly dstrbuted IID formato source s defed by a desty operator actg o a two-dmesoal Hlbert space. Let B= b be our computatoal bass ad suppose we have a quatum IID source wth desty operator actg o our two-dmesoal computatoal Hlbert space. Let be a orthoormal decompostos of. For every ad b, Jozsa ad Presell defed 5 a so-called effectve source desty matrx wth respect to 3 Of course, the legth a quatum sequece s a quatummechacal varable. See 4 for a detaled dscusso o the subect

5 ALEXEI KALTCHENKO a computatoal orthoormal bass b as follows: b b, where P ad P s a doubly stochastc matrx defed by P = b b 0. We observe the followg obvous fact. If the source egebass cocdes wth the computato bass,.e., = b, the effectve source desty matrx cocdes wth the actual source desty matrx,.e., =. V. MAIN RESULT Defto V.1. Wth ay desty matrx, we assocate a probablty dstrbuto formed by the desty matrx s egevalues ad deote t by puttg the desty matrx otato as a subscrpt: p. That s, p = 0, 1, where 0, 1 are the egevalues of. Lemma V.1. Let ad be desty matrces defed o the same Hlbert space, wth orthoormal decompostos = ad =. Let be the effectve desty matrx of wth respect to the bass, ad let be the egevalues of. The, we have the followg relato betwee quatum ad classcal etropes: S + S = D p p + H p, 11 where D s classcal relatve etropy, H s classcal Shao etropy, p ad p are the probablty dstrbutos formed by egevalues sets of ad, respectvely. That s, we have p = ad p =. Proof 1. By the defto of effectve desty matrx, we have P, where P s a doubly stochastc matrx, defed by P =. By the defto of quatum relatve etropy, we have S tr log 2 tr log 2. Frst, we look at the quatty tr log 2, tr log 2 =tr log 2 = log 2. = tr log 2 Substtutg the detty = to the above equato, we obta tr log 2 = log O the other had, we have the followg detty for the log fucto: log 2 = log 2, 13 ad, therefore, substtutg the rght-had-sde of Eq. 13 to log 2, we obta log 2 = = = log 2 log 2 log 2 P. Combg Eqs. 12 ad 14, we obta tr log 2 = log 2 P = log 2 = log 2. P Substtutg the rght-had sde of the quatum etropy detty S = log 2 ad that of Eq. 15 to the defto of quatum relatve etropy, we have S = Thus, we obta log 2 log 2 = S log 2 = S log 2. log 2 + log 2 S = S + D + H, where D s the classcal relatve etropy of the probablty dstrbutos ad ; H s the classcal etropy of the dstrbuto. We ca rewrte Eq. 17 as Eq. 11, whch completes the proof. I Sec. II A, we have troduced a otato C p to deote a classcal data compresso code mappg optmal for a source probablty dstrbuto p. The, accordg to our otato, the code C p wll be optmal for a classcal IID source wth probablty dstrbuto formed by the egevalues of a desty matrx. Lemma V.2 (classcal data compresso ad relatve etropy). Let z z 1,...,z be a sequece emtted by a classcal IID source wth a margal probablty dstrbuto formed by the egevalues of. The, from classcal data compresso theory, we have 1 C p z = D p p + H p, 18 p where the otato the left-had sde stads for the legth of compressed sequece codeword dvded by ad aver

6 REEXAMINATION OF QUANTUM DATA COMPRESSION AND aged wth respect to the probablty measure p. Combg Lemma V.1, Lemma V.2, ad the results of Sec. III, we obta the followg theorem: Theorem 3 (quatum data compresso ad relatve etropy). Let z 1 z 2 z 3...z H be emtted by a quatum IID source wth desty operator defed o a Hlbert space H. Let be a arbtrary desty operator defed o the same Hlbert space H. We choose the egebass of to be our computatoal bass B, ad let be the effectve matrx of wth respect to B. The, the followg relatos hold: S + S = D p p + H p = 1 U B,C z p 1 z 2 z 3...z, 19 where the otato the rght-had sde stads for the legth of compressed quatum sequece dvded by ad averaged the sese of a quatum-mechacal observable. Remark 3. Quatum relatve etropy S has the followg operatoal meag. If a quatum-compresso code, optmal for oe quatum-formato source wth a desty operator, s appled to aother source wth a desty operator, the the crease of the compresso rate s equal to S. Remark 4. By fxg ad the computatoal bass B.e., the egebass of, we wll also fx. The, for a fxed, D p p s mmzed ad s therefore equal to zero whe =. The, as t follows from Theorem 3, for a arbtrary computatoal bass B, the best possble compresso rate s equal to H p =S ad s acheved wth trasformato U B,Cp.I 5, such setup s called msmatched bases compresso. If we choose the egebass of to be our computatoal bass B, the we have =, ad optmal quatum compresso s acheved wth U B,Cp, wth compresso rate H p =S. Dscusso 1. As log as U B,Cp s computed a computatoal bass B, we ca smply replace the orgal source desty matrx wth, the effectve oe wth respect to the computatoal bass. The, our quatum compresso or codg task s effectvely reduced to a classcal oe, wth classcal formato sources havg probablty dstrbutos p ad p. Therefore, all relevat results from classcal formato theory ad data compresso become applcable. We have already see oe such result, the statemet of data compresso o optmalty D p p +H p. To gve aother example, we cosder the followg task of quatum relatve etropy estmato. Suppose we have two quatum IID sources wth ukow desty matrces ad, where ad defed o the same Hlbert space. We wat to estmate quatum relatve etropy S. Let z 1 z 2 z 3...z ad x 1 x 2 x 3...x be emtted by the sources wth desty matrces ad, respectvely. From Theorem 3, oe could hope to estmate the quatty S +S by smply measurg the average legth of codeword U B,Cp z 1 z 2 z 3...z, the estmate S by measurg the average legth of codeword U B,Cp z 1 z 2 z 3...z, ad the subtract the latter from the former. For such straghtforward approach to work, oe should have quatum compresso codes U B,Cp ad U B,Cp, whch are optmal for sources wth desty matrces ad, respectvely. But as ad are ukow, so are U B,Cp ad U B,Cp. Fortuately, classcal data compresso, there exst so-called uversal compresso codes 10, whch wll let estmate classcal quattes D p p +H p ad H p eve wthout kowg probablty dstrbutos p ad p. So, based o these classcal codes, we ca costruct quatum codes 11 utary trasformatos to estmate S. VI. CONCLUSION We have provded a smple ad costructve proof of a quatum aalog of a well-kow classcal formato theory result o a role of relatve etropy as a measure of ooptmalty data compresso. We have costructed quatum compresso codes schemes from classcal data compresso codes. Moreover, as the quatum data compresso or codg task ca be effectvely reduced to a quas classcal oe, we show that relevat results from classcal formato theory ad data compresso become applcable ad therefore ca be exteded to the quatum doma. 1 V. Vedral, Rev. Mod. Phys. 74, T. Cover ad J. Thomas, Elemets of Iformato Theory Wley-Iterscece, New York, B. Schumacher, Phys. Rev. A 51, B. Schumacher ad M. D. Westmorelad, Phys. Rev. A 64, R. Jozsa ad S. Presell, Proc. R. Soc. Lodo, Ser. A 459, J. Lagford, Phys. Rev. A 65, C. Beett, IBM J. Res. Dev. 17, E. Berste ad U. Vazra, Quatum complexty theory, SIAM J. Comput H. Buhrma, J. Tromp, ad P. Vtay, Proceedgs of the Iteratoal Coferece o Automata, Laguages ad Programmg, Lecture Notes Computer Scece Sprger- Verlag, Berl J. Zv ad N. Merhav, IEEE Tras. If. Theory 39 4, A. Kaltcheko, Uversal No-Destructve Estmato of Quatum Relatve Etropy, Proceedgs of the 2004 IEEE Iteratoal Symposum o Iformato Theory USA, 2004 p