Quantum Information Processing

is, |α| 2 + |β| 2 =1. Such a superposition or vector is said to be normalized. (For a complex
number given by γ = x + iy, one can evaluate |γ| 2 = x 2 + y 2 . Here, x and y are the
real and imaginary part of γ , and the symbol i is a square root of –1, that is, i 2 = –1. The
conjugate of γ is γ = x – iy. Thus, |γ | 2 = γγ). Here are a few examples of states given in
both the ket and vector notation:
ψ 1
ψ 2
ψ 3
( + ) ⎛1
2⎞
= ↔ ⎜ ⎟
2
⎜
⎝1
2
⎟
,
⎠
3 4 ⎛ 35⎞
= − ↔
5 5
⎜
⎝−45
⎟ ,
⎠
i3i4⎛i35⎞ = − ↔
5 5
⎜
⎝−i45
⎟ .
⎠
and
The state |ψ 3 〉 is obtained from |ψ 2 〉 by multiplication with i. It turns out that two
states cannot be distinguished if one of them is obtained by multiplying the other by a
phase e iθ . Note how we have generalized the ket notation by introducing expressions
such as |ψ〉 for arbitrary states.
The superposition principle for quantum information means that we can have states
that are sums of logical states with complex coefficients. There is another, more familiar
type of information, whose states are combinations of logical states. The basic unit of
this type of information is the probabilistic bit (pbit). Intuitively, a pbit can be thought of
as representing the as-yet-undetermined outcome of a coin flip. Since we need the idea
of probability to understand how quantum information converts to classical information,
we briefly introduce pbits.
A pbit’s state space is a probability distribution over the states of a bit. One very
explicit way to symbolize such a state is by using the expression {p:, (1 – p):}, which
means that the pbit has probability p of being and 1 – p of being . Thus, a state of a
pbit is a probabilistic combination of the two logical states, where the coefficients are
nonnegative real numbers summing to 1. A typical example is the unbiased coin in the
process of being flipped. If tail and head represent and , respectively, the coin’s state
is {1/2:, 1/2:}. After the outcome of the flip is known, the state collapses to one of the
logical states and . In this way, a pbit is converted to a classical bit. If the pbit is
probabilistically correlated with other pbits, the collapse associated with learning the
pbit’s logical state changes the overall probability distribution by a process called
conditioning on the outcome.
A consequence of the conditioning process is that we never actually “see” a
probability distribution. We only see classical deterministic bit states. According to the
frequency interpretation of probabilities, the original probability distribution can only
be inferred after one looks at many independent pbits in the same state {p:, (1 – p):}:
In the limit of infinitely many pbits, p is given by the fraction of pbits seen to be in the
state o. As we will explain, we can never see a general qubit state either. For qubits,
there is a process analogous to conditioning. It is called measurement and converts qubit
states to classical information.
Information processing with pbits has many advantages over deterministic information
processing with bits. One advantage is that algorithms are often much easier to design and
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Quantum Information Processing