Question

Question 1 (m) The probability of observing a particular vnlue or eigenetate of a state vector is given by Prob.la = a.) (P.) where P. 4) dal is a projector onto the set of stntess which correspond to a given mesure ment vlue. Consider the following state in a three-dimension Hilbert space 1 1) 10) + V3 Using the projector approach, cakulate the probability of finding the system in the states: 10), 11), and 2), respectively. Of course you can quickly pull these off from the amplitudes but for now we want to put emphasis on formulating the projectors (operntons) and wulking your way through to the probabilities using them 1 [ 12 marks] (b) At times it helps to be able to change basis and perform qubit measurements using a completely different computational basis. This can allow us to acces parts of the system we can't access in some "traditional" basis. Recall at some point we did mention the I+) = (0) + |19), It us formally introduce the full plus-minus bussis is =) = (0) + 1)) Take for instance the state: 1.) = V0 + 1). Suppose we have a detector which measures or rather can only distinguish between states in the plus-minns basis. What are the probabilities of mensuring (0) and (1) in the plus-minus busis? 8 marks]

Answer 1

Q1 we have given 147= Js 1°) + to 117+ e 127 J3 to vector land prabablity get any P (a=an) = <41 Pan 147 <4147. Pana land Land (+147= (tj kul tys <11-ete) [ t, 103 til 13 tot ang <4147 =) - t - 33 + 1 / 3 = 1 so for 107. P (107) <410) col4) <410) Lol47 (4147 2 V K014>2

P(10) = I 201471 I fgtutol? P(107) = 3 simillest 2 P(17) = 1<1/47) lotte to 12 PU17= 1/2 & 도 P(128) =<214712 loto tf31² P (<11) w/- (Ь New bassis. Now given that 1 > = tl107 117) and The state 117 (47= 57 107+ probablity to get lo7 & 117 in the 4+2 basis.

5 ( tz 1+3+ / 2 1*>) + 7 (tz 1+7 / 117) * £3 1774 1 1 3 1 -> + to 1+² toll? (+3+ tu)! It) + (foto) I-> 147 P(H₂) = (41+Lt 147 L4147 L4147- (Nomlized.) 2 Pll+7)= 12+1+)/ -> I te tjt ! = 0.971 1 к Plu) - K41-712- 1 ts to 12 = 0.028 Now probality to lo7 8 112 in basly get 12 80 1+ p(10)) = <+10/2014) (se) the = Lt1t7 1 P( 117= <+117 211 +) 24ity fox J2 P (107)- 107 2017 +2x+2 1 -N 1 P (117) - L-117411-7 (+1-7 fax tz 1 2

8 Now The probablity (tinal.) To get ) 07 & 117 p (107) = 0.971 x 1 + oi 098x1 2 P' (107) 0.4995 ? 0,5 & for 117 pl (112) 0.028x + 0-07) x 1 P' (11>) =) 015