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Unformatted text preview: escribed by a qubit. Very roughly speaking, the spin is a quantum
description of the magnetic moment of an electron which behaves like a spinning charge. The two allowed states can roughly be thought of as clockwise
rotations (“spinup”) and counter clockwise rotations (“spindown”). We will
say much more about the spin of an elementary particle later in the course. 1.7. QUBITS 15 Measurement Example I: Phase Estimation
Now that we have discussed qubits in some detail, we can are prepared to
look more closesly at the measurement principle. Consider the quantum
state,
eiθ
1
 ψ = √  0 + √  1 .
2
2
If we were to measure this qubit in the standard basis, the outcome would
be 0 with probability 1/2 and 1 with probability 1/2. This measurement
tells us only about the norms of the state amplitudes. Is there any measurement that yields information about the phase, θ?
To see if we can gather any phase information, let us consider a measurement in a basis other than the standard basis, namely
1
+ ≡ √ (0 + 1)
2 and 1
− ≡ √ (0 − 1).
2 What does φ look like in this new basis? This can be expressed by ﬁrst
writing,
1
0 = √ (+ + −)
2 and 1
1 = √ (+ − −).
2 Now we are equipped to rewrite ψ in the {+ , −}basis,
1
eiθ
ψ = √ 0 + √ 1)
2
2
1
e iθ
= (+ + −) +
(+ − −)
2
2
1 + eiθ
1 − e iθ
=
+ +
− .
2
2
Recalling the Euler relation, eiθ = cos θ + i sin θ, we see that the probability
of measuring + is 1 ((1 + cos θ)2 + sin2 θ) = cos2 (θ/2). A similar calcula4
tion reveals that the probability of measuring − is sin2 (θ/2). Measuring
in the (+ , −)basis therefore reveals some information about the phase
θ.
Later we shall show how to analyze the measurement of a qubit in a
general basis. 16 CHAPTER 1. INTRODUCTION Measurement example II: General Qubit Bases
What is the result of measuring a general qubit state ψ = α 0 + β 1, in
a general orthonormal basis v , v ⊥ , where v =a0 + b1 and v ⊥ =
b∗ 0 − a∗ 1? You should also check that v and v ⊥ are orthogonal by
showing that v ⊥ v = 0.
To answer this question, let us make use of our recently acquired bra
ket notation. We ﬁrst show that the states v and v ⊥ are orthogonal,
that is, that their inner product is zero:
v ⊥ v = (b∗ 0 − a∗ 1)† (a 0 + b 1)
= (b 0 − a 1)† (a 0 + b 1) = ba 00 − a2 10 + b2 01 − ab 11 = ba − 0 + 0 − ab
=0 Here we have used the fact that ij = δij .
Now, the probability of measuring the state ψ and getting v as a
result is,
Pψ ( v ) =  v  ψ  2 =  ( a ∗ 0 + b ∗ 1 ) ( α  0 + β  1 )  2
=  a ∗ α + b∗ β  2 Similarly,
2
Pψ ( v ⊥ ) = v ⊥  ψ =  ( b 0 − a 1 ) ( α  0 + β  1 )  2
=  bα − a β  2...
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This document was uploaded on 09/22/2013.
 Fall '13

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