at iona l Co n f er ence \ Qu an tu m O p t i cs V " , Ko ƒci el isko, Po la nd , 200 1 O n M i x ed St ate s Entan gle m ent an d Q u antu m C om m u n ication : As p ects of Qu antu m C h an n els Th eor y P .H oro d eck i Fa cul t y of A ppl i ed Phy sics and Ma the mati cs, T echnical Uni versity of Gda ¥sk Naruto wi cza 11/ 12, 80 -952 G da¥sk, Po l and Basi c aspects of qua ntu m communication based on quantum channe ls theo ry are re view ed to gether w ith suitable examples .T he key notion of quantum channel capacity is discussed.T he role of mixed states entanglement as a practical to ol in quantum channels theory is explain ed in detail.A pplication of recent results on non-positi ve partial transp ose bound entanglement con j ecture to binding entanglement channels show s that validity of the conj ecture for some states implies n o n a dd i t i v it y of tw o-w ay quantum channel capacity .Some asp ects of robustness of entanglement in context of positive maps theory are also discussed .
at iona l Co n f er ence \ Qu an tu m O p t i cs V " , Ko ƒci el isko, Po la nd , 200 1 O n M i x ed St ate s Entan gle m ent an d Q u antu m C om m u n ication : As p ects of Qu antu m C h an n els Th eor y P .H oro d eck i Fa cul t y of A ppl i ed Phy sics and Ma the mati cs, T echnical Uni versity of Gda ¥sk Naruto wi cza 11/ 12, 80 -952 G da¥sk, Po l and Basi c aspects of qua ntu m communication based on quantum channe ls theo ry are re view ed to gether w ith suitable examples .T he key notion of quantum channel capacity is discussed.T he role of mixed states entanglement as a practical to ol in quantum channels theory is explain ed in detail.A pplication of recent results on non-positi ve partial transp ose bound entanglement con j ecture to binding entanglement channels show s that validity of the conj ecture for some states implies n o n a dd i t i v it y of tw o-w ay quantum channel capacity .Some asp ects of robustness of entanglement in context of positive maps theory are also discussed .
R em ar k s on En tan gle m en t , its Sou r ces an d A p p lic ation s M .`uk ow sk i I n st yt ut F izy ki T eorety cznej i Astro Ùzyki , Uni wersytet G da¥ski W i ta Stw osza 57, 80-952 G da ¥sk, Po l and and Insti tut f ur Ex peri m enta l physi k, Uni versi t at Inns bruck, 6020 Innsbruck, Austri a W e shall discuss the cent ral r ole pla yed in the quantu m information theory by entan glement .A generali zed version of Bell ' s theorem w ill be presented , as well as metho ds in multiphoton interf erometry to obtain correlations violatin g the Bell inequali tie s.Finally , w e shall present some quantum inf ormationa l applicati on s of entanglement.PAC S numb ers: 03.65.U d, 03.67.{a, 42.50.A r 1.I n t r o d u ct io n Current adv ances i n the op erati ona l contro l of qua ntum pheno m ena enabl e us to use the para do xi cal aspects of the qua ntum wo rl d as sources of pra cti cal appl i cati ons.The mai n areas i n whi ch qua ntum pheno m ena can b e harnessed to p erform ta sks im p ossibl e for classical theo ry of i nf orm ati on are qua ntum crypto gra phy (a l ready a rel i abl e techno l ogy), qua ntum com muni cati on (a t the sta ge of bi rth, e.g. the tel eporta ti on exp eri ments) and qua ntum com puta ti on (stunni ng theo reti cal resul ts, brea ki ng the l i m its of the T uri ng -t yp e machines, but sti l l far away fro m op erati onal real i zati ons).The basic no ti on i n the qua ntum i nform a ti on theo ry i s tha t of the qua ntum bi t or the qubi t.Cl a ssical bi ts can ta ke one of the two p ossibl e values 0 or 1 ( \ on " or \ o˜" , etc.).The qubi ts (the simpl est quantum system s wi th pure sta tes descri bed by a two -dim ensional Hi l b ert space, e.g.spins 1/ 2), since they are qua ntum enti ti es, they can b e in an arbi tra ry sup erp ositi on of two ba sis states j 0 i and j 1 i : j q u bi t i = ˜j 0 i + Ù j 1 i :(1 ) (2 1)
of Qu antu m State s M .K uƒ Cent rum F i zyki T eorety cznej , Pol ska Ak adem ia Na uk al .Lo tni k §w 32/ 46, 02-668 Warsza wa , Po l and ( Recei ved N ovember 5, 2001)T h e ad vanta ges of quantu m inf ormation pro cessing and transmissio n over corresp ondi ng classical operations are connected to the existence in comp osite quantum -mechanical systems states without classical counterparts | the so-called entangled states .T he pap er show s and examines v arious applica tion s of entangled states in the quantum inf ormation theory .Recent results connected with characteri zation and geometry of entangled states are examined with the emphasis on some unsolved mathematical pro blems.
Although the concept of quantum entanglement has been known for about seventy years, it only recently quit the realms of meta-theoretical discussions when it was discovered how entanglement can be exploited to compute and communicate with an unprecedented power. The primary motivation of the work presented in this thesis has been to contribute to the big e#ort that has been done during the last decade to understand and quantify quantum entanglement. We have developed advanced techniques of linear and multilinear algebra to investigate and classify entangled pure and mixed quantum states, and discussed some novel applications in the field of quantum information theory. The results presented in this thesis are mainly of interest from a fundamental point a view: entanglement is the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought [186]. It is however a real privilege that fundamental research in quantum information theory bears the tools of tomorrow's electrical engineers: the ongoing miniaturization of electronic components will soon reach a scale where quantum mechanical e#ects play a major role.
Quantum entangled states play very important roles in quantum information processing,such as quantum teleportation,dense coding,error correction,cryptography,and computation.The theory of entanglement mainly concerns the characterization of quantum entanglement and the classification of quantum states,as well as its applications in quantum information processing.We present some basic concepts and results,including:the relationship between locality and reality in quantum mechanics and Bell inequalities,and that between Bell inequalities and the separability of quantum states;the definition of separability for pure and mixed quantum states,and some separability criteria such as the positive map approach,positive partial transpose,reduction criterion,realignment,entanglement witnesses,and the covariance matrix approach and local uncertainty relations;some entanglement measures such as the entanglement of formation,concurrence,relative entropy,negativity,tangle,entanglement of assistance,and their estimation;the change in entanglement when the physical system evolves or undergoes interaction with the environment.
g B oson s T ra p p ed in H ar m on ic P ote ntial : Qu alit y of B ogoliu b o v Ap p ro x im ation M .A .Z a¤u sk a-K otu r , M. Ga j da and J. Mo sto w sk i In st yt ut Fi zy ki , Po l i sh A cadem y of Sciences and Col l ege of Science Al .Lo tni k §w 32/ 46, 02-668 W arszawa, Po l and ( Recei ved Ju ly 26, 2 001)W e st udy a syste m of tra pp ed bosoni c partic les interactin g by mo del harmonic forces.O ur mo del allo w s for a detailed examination of the notion of an order parameter (a condensate w ave function ).By decomp osing a single particle density matrix into coherent eigenmo des w e study an e˜ect of interaction on the condensate.W e show that su£ciently strong interactions cause that the condensate disapp ears even if the whole system is in its lowest energy state.I n the second part of our pap er we discuss the v alidity of the Bogoliub ov approximation by comparing its prediction s w ith results inferred from the exactly soluble mo del.In particular we examine an energy spectrum , occupation , and Ûuctuation s of the condensate.W e conclude that Bogoliu bov approach gives a quite accurate descripti on of the system in the limit of w eak interactions.PAC S numb ers: 03