t b a b a

in states corresponding to a blend or superposition of these states. In other words, a qubit
can exist as a zero, a one, or simultaneously as both 0 and 1, with a coefficient representing
the probability amplitude for each state. Actually, a qubit is a 2-dimensional, normalized
vector in a Hilbert space with base vectors | 0〉 and | 1〉. An arbitrary single-qubit state can
be represented as:
| ψ〉 = α | 0〉 + β | 1〉
This is a normalized state where the probability amplitudes α and β are complex numbers,
with |α| 2 + |β| 2 = 1. A classical computer with n bits has 2 n possible states and this is an
n-dimensional vector space over Z2.
The inner product of two state vectors represents the generalized ”angle” between the
states and gives an estimate of the overlap between the states | ψa〉 and | ψb〉. The inner
product 〈ψa | ψb〉 = 0 defines orthogonal states; the implication of 〈ψa | ψb〉 = 1 is that | ψa〉
and | ψb〉 are the same state.
A quantum computer with n qubits is a state in a 2 n -dimensional complex vector space.
It can be represented as the tensor product ”⊗” of the respective state spaces of all the
individual qubits. We abbreviate: | 0〉⊗ | 1〉 =| 01〉, etc. The tensor product of two qubits
over the bases {| 0〉, | 1〉} can be expressed with the four basic vectors | 00〉, | 01〉, | 10〉,
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