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2. NANOMEMS PHYSICS: Quantum Wave-Particle Phenomena 67

time, i.e., to the degree that a and b may adopt an infinity of values, the qubit
has the potential to be in any of these. A quantum system possessing n qubits
n
is said to have 2 accessible mutually orthogonal quantum states. For
example, a system containing two noninteracting qubits will have the four
states: 00 , 01 , 10 , 11 . States such as these, which represent the
juxtaposition of independent or noninteracting systems (qubits), are called
tensor product states.

2.4.1 Quantum Entanglement

In general, a tensor product provides the mathematical description of the
state of a system that is constituted by bringing together noninteracting
quantum systems, assuming that they remain without interacting [60].
Comprehending this concept is useful to get a clear understanding of the
definition of an entangled state [107-111].
In particular, if associated with two quantum systems there are vector
spaces V of dimension N , in which resides a vector φ , and V of
1 1 2
dimension N , in which resides a vector χ , and where N and N may
2 1 2
be finite or infinite, then the tensor product of V and V is denoted by the
1
2
vector space V [60],
V = V ⊗ V , (62)
1
2
of dimension N N , where the vector,
1 2

φ ⊗ χ = φ χ , (63)

associated with the overall space V , is called the tensor product of φ and
χ .

If the vectors φ and χ can be expressed in terms of the respective
u
v
bases { } and { }, so that,
i i
φ = ¦ a i u , (64)
i
i

and