Page 81 - Principles and Applications of NanoMEMS Physics

68 Chapter 2

χ = ¦ b j v j , (65)
j

then, the tensor product may be written as,

φ ⊗ χ = ¦ a i b j u i ⊗ v j , (66)
i, j

from where it is seen that the components of a tensor product vector are the
products of the components of the two vectors of the product. An example
will help appreciate the meaning of a tensor product immediately. Let V
x
y
x
and V be two vector spaces in which the bases {} and {}, reside.
y
Then the tensor product of the spaces is given by,
V = V ⊗ V , (67)
xy x y

and the tensor product of the bases is given by,

xy = x y . (68)

Consequently, if X and Y are operators in V , then we have,
xy
X xy = X x ( ) = x x ( ) = x x y = x xy , (69)
y
y
Y xy = ( ) yYx = ( ) yyx = y x y = y xy . (70)

Essentially, then, the operators acting over a tensor product of spaces operate
only on the vector space to which they belong.
Now, assume that the global state of the system is embodied by the
wavefunction ψ ∈ V = V ⊗ V . Then, according to the above,
1 2
ψ = ψ ⊗ ψ , where ψ ∈ V and ψ ∈ V . A quantum system is
1 2 1 1 2 2
said to be entangled if it is impossible to express its global state as the tensor
product, i.e., ψ ≠ ψ 1 ⊗ ψ 2 . Thus, in an entangled system, it is not
possible to act on one of its vector states independently without perturbing
the others. It is said then, that the states in an entangled system are
correlated.