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Unformatted text preview: the measurement.
Intuitively, a measurement provides the only way of reaching into the
Hilbert space to probe the quantum state vector. In general this is done by
selecting an orthonormal basis e0 , . . . , ek−1 . The outcome of the measurement is ej with probability equal to the square of the length of the projection
of the state vector ψ on ej . A consequence of performing the measurement
is that the new state vector is ej . Thus measurement may be regarded as a
probabilistic rule for projecting the state vector onto one of the vectors of the
orthonormal measurement basis.
Some of you might be puzzled about how a measurement is carried out
physically? We will get to that soon when we give more explicit examples of
quantum systems. 1.7 Qubits Qubits (pronounced “cuebit”) or quantum bits are basic building blocks that
encompass all fundamental quantum phenomena. They provide a mathemat 12 CHAPTER 1. INTRODUCTION ically simple framework in which to introduce the basic concepts of quantum
physics. Qubits are 2state quantum systems. For example, if we set k = 2,
the electron in the Hydrogen atom can be in the ground state or the ﬁrst
excited state, or any superposition of the two. We shall see more examples of
qubits soon.
The state of a qubit can be written as a unit (column) vector ( α ) ∈ C2 .
β
In Dirac notation, this may be written as:
 ψ = α  0 + β  1 with α, β ∈ C and α2 + β 2 = 1. This linear superposition ψ = α 0 + β 1 is part of the private world of
the electron. For us to know the electron’s state, we must make a measurement. Making a measurement gives us a single classical bit of information —
0 or 1. The simplest measurement is in the standard basis, and measuring ψ
in this {0 , 1} basis yields 0 with probability α2 , and 1 with probability
β 2 .
One important aspect of the measurement process is that it alters the state
of the qubit: 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
of ψ = α 0 + β 1 yields 0, then following the measurement, the qubit is
in state 0. This implies that you cannot collect any additional information
about α, β by repeating the measurement.
More generally, we may choose any orthogonal basis {v , w} and measure the qubit in that basis. To do this, we rewrite our state in that basis:
ψ = α v + β w. The outcome is v with probability α  2 , and w with
probability β  2 . If the outcome of the measurement on ψ yields v , then
as before, the the qubit is then in state v . Examples of Qubits Atomic Orbitals
The electrons within an atom exist in quantized energy levels. Qualitatively
these electronic orbits (or “orbitals” as we like to call them) can be thought
of as resonating standing waves, in close analogy to the vibrating waves one
observe...
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This document was uploaded on 09/22/2013.
 Fall '13

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