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Unformatted text preview: vector spaces, is also endowed with an inner product. And an inner
product is a way of taking two states (vectors in the Hilbert space) and getting
a number out. For instance, deﬁne
ak | k ,
k where the kets |k form a basis, so are orthogonal. If we instead write this
state as a column vector, a0 a1 |ψ = . .
a N −1 Then the inner product of |ψ with itself is a0 a1
ψ , ψ = a∗ a∗ · · · a∗ 1 · .
. a N −1 N −1 N −1
ak ak =
| ak | 2
= k=0 k=0 The complex conjugation step is important so that when we take the inner
product of a vector with itself we get a real number which we can associate 10 CHAPTER 1. INTRODUCTION with a length. Dirac noticed that there could be an easier way to write this
by deﬁning an object, called a “bra,” that is the conjugate-transpose of a ket,
ψ | = |ψ † =
k a∗ k | .
k This object acts on a ket to give a number, as long as we remember the rule,
j | |k ≡ j |k = δjk
Now we can write the inner product of |ψ with itself as ψ |ψ = =
j,k = a∗ j | j a∗ ak j | k
k ak | k a∗ ak δjk
j j,k =
k | ak | 2 Now we can use the same tools to write the inner product of any two states,
|ψ and |φ, where
bk | k .
k Their inner product is,
ψ |φ =
j,k a ∗ bk j | k =
j a ∗ bk
k k Notice that there is no reason for the inner product of two states to be real
(unless they are the same state), and that
ψ |φ = φ|ψ ∗ ∈ C
In this way, a bra vector may be considered as a “functional.” We feed it a
ket, and it spits out a complex number. 1.6. THE MEASUREMENT PRINCIPLE 11 The Dual Space
We mentioned above that a bra vector is a functional on the Hilbert space.
In fact, the set of all bra vectors forms what is known as the dual space. This
space is the set of all linear functionals that can act on the Hilbert space. 1.6 The Measurement Principle −1
This linear superposition |ψ = k=0 αj |j is part of the private world of the
electron. Access to the information describing this state is severely limited —
in particular, we cannot actually measure the complex amplitudes αj . This is
not just a practical limitation; it is enshrined in the measurement postulate
of quantum physics.
A measurement on this k state system yields one of at most k possible
outcomes: i.e. an integer between 0 and k − 1. Measuring |ψ in the standard
basis yields j with probability |αj | 2 .
One important aspect of the measurement process is that it alters the
state of the quantum system: the eﬀect of the measurement is that the new
state is exactly the outcome of the measurement. I.e., if the outcome of the
measurement is j , then following the measurement, the qubit is in state |j .
This implies that you cannot collect any additional information about the
amplitudes αj by repeating...
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This document was uploaded on 09/22/2013.
- Fall '13