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.
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.
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)
Quantum information theory is at the frontiers of physics, mathematics and information science, offering a variety of solutions that are impossible using classical theory. This book provides an introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. After a gentle introduction to the necessary mathematics the authors describe the geometry of quantum state spaces. Focusing on finite dimensional Hilbert spaces, they discuss the statistical distance measures and entropies used in quantum theory. The final part of the book is devoted to quantum entanglement - a non-intuitive phenomenon discovered by Schrödinger, which has become a key resource for quantum computation. This richly-illustrated book is useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.
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 .
Information and computation theory have undergone a spurt of new growth, and a renewal of their historic connection to basic physics, as they have expanded to treat the intact transmission and processing of quantum states, and the interaction of such 'quantum information' with traditional forms of information. We may wonder why this did not happen earlier, as quantum principles have long been accepted as fundamental to all of physics. Perhaps the founders of information and computation theory, such as Shannon, Turing and von Neumann, were too accustomed to thinking of information processing in macroscopic terms, not yet having before them the powerful examples of the genetic code and ever-shrinking microelectronics. Be that as it may, information until recently has largely been thought of in classical terms, with quantum mechanics playing a supporting role in the design of the equipment to process it, and setting limits on the rate at which it could be sent through certain channels. Now we know that a fully quantum theory of information and information processing offers, among other benefits, a brand of cryptography whose security rests on fundamental physics, and a reasonable hope of constructing quan- tum computers that could dramatically speed up the solution of certain mathematical problems. These benefits depend on distinc- tively quantum properties such as uncertainty, interference and entanglement. At a more fundamental level, it has become clear that an information theory based on quantum principles extends and completes classical information theory, just as complex numbers extend and complete the reals. Besides quantum generalizations of classical notions such as sources, channels and codes, the new theory includes two complementary, quantifiable kinds of information— classical information and quantum entanglement. Classical infor- mation can be copied at will, but can only be transmitted forward in time, to a receiver in the sender's forward light cone. Entanglement in contrast, cannot be copied, but can connect any two points in space-time. Conventional data processing operations destroy entanglement, but quantum operations can create it and use it for various purposes, such as speeding up certain classical computa- tions and assisting in the transmission of classical information or intact quantum states. Part of the new quantum information theory is the qualitative and quantitative study of entanglement, and its interactions with classical information. Any means, such as an optical fibre, of delivering quantum systems more or less intact from one place to another, may be viewed as a quantum channel. Unlike classical channels, which are well characterized by a single capacity, quantum channels have several distinct capacities, depending on what one is trying to use them for, and what auxiliary resources are brought into play. New effects involving quantum information continue to be discovered, not only in the traditional areas of computation, channel capacity, and cryptography, but in areas such as commu- nication complexity and game theory.
This thesis addresses a number of problems within the emerging field of quantum information science. Quantum information science can be said to encompass the more-established disciplines of quantum computation and quantum information, as well as rather more recent attempts to apply concepts, tools and techniques from these disciplines to gain greater understanding of quantum systems in general. The role of entanglement — non-classical correlation — has been of particular interest to date. Part I contributes to this later goal. In particular, we establish a connection between the energy of a many-body quantum system and the idea of an entanglement witness from the theory of mixed-state entanglement. This connection allows mathematical results about entanglement witnesses to be translated into physical results about many-body quantum systems, specifically energy and temperature thresholds for entanglement. For the case of two qubits we are able to establish fairly detailed results about the behaviour of entanglement with temperature. We also study entanglement in systems of indistinguishable particles, where even the question of which quantum states should be regarded as entangled has been the subject of much controversy. We aim to clarify this issue by applying Wiseman and Vaccaro’s notion of entanglement of particles to a number of wellunderstood model systems. We discuss the advantages of the entanglement of particles approach compared with other methods in common use. Finally, we study the operational meaning of superselection rules in quantum physics, in particular the connection to the existence or not of an appropriate reference frame. We propose an experiment that aims to exhibit a coherent superposition of an atom and a molecule, apparently in violation of the commonly-accepted particle-number superselection rule. This result sheds light on the entanglement of particles approach to entanglement of indistinguishable particles. Part II returns to a fundamental question at the heart of quantum computation and quantum information, namely: how many quantum gates are required to perform a particular quantum computation? In other words, how efficiently can a quantum computer solve a particular computational problem? We establish a connection between this question and the field of Riemannian geometry. Intuitively, optimal quantum circuits correspond to “free-falling” along the shortest path between two points in a curved space. This opens up the possibility of using Riemannian geometry to study quantum computation, a possibility that was previously unknown. We provide explicit calculations of all the basic geometric quantities associated with the space, and give some preliminary results of applying geometric ideas to quantum computing. Finally, we explore more generally the connection between optimal control and quantum circuit complexity, of which the Riemannian metric described above can be viewed as a special case.