Page 188 - Principles and Applications of NanoMEMS Physics

176 Chapter 4

Constructing a pseudo-pure state makes use of the concept of deviation
density matrix. This is the arbitrary matrix δ for which −δ ρ = I λ for some
constant λ . From this definition, and inspection of Eq. (29), it is clear that
the matrix 0ε 0 is in fact a deviation matrix from the equilibrium state of

one nuclear spin. An interesting property of the deviation matrix is that, if
m ˆ is a traceless operator, then [193],
m
tr ( ) ˆ =δ m ( ( tr ρ + I λ ) ) ˆ
= tr ( ) trm ˆ +ρ ( ) ˆ . (33)
m
= tr ρ m
( ) ˆ

Thus, the expectation value (the measurement) of a traceless observable may
be obtained either from the density matrix or from the deviation matrix, as
prescribed in Eq. (33). A pseudo-pure state, in fact, is defined as one whose
equilibrium state has the deviation δ = ε 0 0 . Its significance is as

follows. If we are interested in the probability of p of measuring state 1 ,
1
given that the initial state was 0 , then this is given by [193],

p = 1 U 0 0 U + 1
1
= tr ( 0U 0 U + 1 1 )

= tr ( 0U 0 U + (I σ− z )) 2/ , (34)

= tr ( ( U 0 0 U + ) ( 0Utr− 0 U σ z )) 2/
+
+
= ( 1− tr ( 0U 0 U σ )) 2/
z

Where U is the total unitary operator associated with a computation.
Therefore, Eq. (34) indicates that by measuring the initial and final
expectation values of ı , a = tr (įı ) İ= , and
z z
+
′ a = tr ( ′ıį ) = İtr ( 0U 0 U σ ), respectively, one can determine p . In
z z 1
fact, p = (1− ( aa ′ )) 2 , independent of the scale İ .
1
Most importantly, the technique may be extended to the case in which one
desires to determine the probability p of measuring the state 1 , in the
1 1
case in which this state refers to the first qubic resulting from applying a