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# jan28 - It 142—55 l'l“>-——A|ll’> FIGURE 2...

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Unformatted text preview: It?) 142—55 l'l“>-——A|ll’> FIGURE 2 Deﬁnition of the adjoint opera- tor A‘ of an operator A using the correspondence between kets and ("l’l AT <¢ll=<¢lﬁll bras. l A' is therefore a linear operator, deﬁned by the formula : l|¢’>=Alc>«=~<c’|=<e|A* (13-47) From (B-4?), it is easy to deduce another important relationship satisﬁed . by the operator A‘. Using the properties of the scalar product, one can always write: <chp>=<ch>* (3-43} I where | no) is an arbitrary ket of 6”. Using expressions (BA?) for |Ilﬂ> and (M, we obtain: l<tﬁ|A'|w>=<w|Allﬁ>*l (13-49} a relation which is valid for all |cp> and H; ). The Bloch Sphere Page 1 of 2 The Bloch Sphere The most general state of a quantum two—level system can be written in the form ht} = or ID} +a'll1} where e and ,l'i' are complex numbers, so it might seem like one has four real parameteTs to work with. However, the state has to be normalized, so In]1 + [,3]2 = l, and an overall phase makes no difference, so either a or ,8 can be chosen to be real. This leads to the “canonical“ parametrization |1,i‘}= cor-i E ID} + em all] E I1} {1] in terms or" only two real numbers 6' and o, with natural ranges [I 5 El 5 a: and [I S p S 2111'. These are the same as the polar angles in 3-dimensional spherical coordinates, and this leads to the representation of the state {1] as a point on a unit sphere called the Bloc}: sphere. In the notation of (1}, the state Ill} is represented by the North pole, and the state II} by the South pole, whereas the states on the equator correspond to superpositions of |o} and ll} with equal weights (5' = 792]. and different phases, parametrized by the azimuthal angle {a (the “longitude"). Pin alternative notation is |l]} E Jute} and I1} E has}. Given any state on the sphere, the diametrically opposite point always represents an orthogonal state. All the pictures of Bloch spheres generated by the applets on these pages can be freely rotated with the mouse or using the sliders provided. We have tried to indicate depth by graying out the parts of the sphere or the axes that are farthest from the viewer. http:ffeomp.uarlt.eduf~j geabanafbloe happsl'bloehhtml lﬂﬁf 21308 Quantum Information Processing analyze if they are probabilistic. Examples include n‘tan},r optimizatim] and physics simu— lation algorithms. In some cases. the best available probabilistic algorithm is more effi— cient than any known deterministic algorill'lrn. All example is an algorithm for determinv ing whether a number is prime or not. IL is not known whether ever}.-r probabilistic algo- rithm can he derandomixed efﬁciently. There are important communication problems that can be solved probabilisticslly but not detorrninisticully. For a survey of these algo- rithms, see Rajiv Gupta {19943}. What is the djﬁerenee between bits, pbits, and quhits'! Cine way to visualize the difference and see the enrichment provided by pbits and qubits is shown in ﬁgure 1. Bit Plait D D ‘l 1 lt} Stale: :1 or 1 his. {1 - Pin} 010} + iii} In2 + in2 = I Figure 1. Bomparlttg State Spaces of Different Information Units The states of a hit correspond to mo pointc.The states at e pbtt can be thought of as comet: combinations of e hit's states and therefore can he trlsualized as lying on the line cmneﬁlng the two bit states. A quhlt’e pure states correspond to the surface at the unit sphere in three dimensions, where the loglcel states correspond to the poles.Thls representation of quhit states ls called the Bloch sphere.The explicit correspondence ls discussed at the end of the section “Mixtures and Density Operm' Aim referto the deﬁnition and use of the Bloch sphere In the article “NI-1R and armature Intonnetlm Processing" on page 225.There, the cor- respondence between the pure states and the sphere to physically motivated and comes from a way of I.I'ie'uting e spin-1r: system as a shall quantum magnet. lntulthrolyr, a state is determined by the direction of the north pole of the magnet. Processing ﬂue Quiz-it The quantum section of the not gate for hits exchanges the two logical states; that is. using leet notation. nottatci+ so = 6:11} + out = 1310} + oh} . {B} In vector notation. this equation becomes 3 £411 Mamas Science Numb-:12? zoos ...
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