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Note the difference in signs of the two exponents. For example, an X-rotation
applied to the density matrix would be:
ρ(t + ∆t) = e −iπ∆t(X σx) ρ e iπ∆t(X σx) (3.71)
The density matrix of an n-qubit state is a hermitian 2 n × 2 n matrix. Its diagonal
elements give the probabilities of the possible measurement outcomes, e.g.:
ρ =
⎡
⎢
⎣
⎤
P | 00 〉 a b c
a ∗ P | 01 〉 d e
⎥
b ∗ d ∗ P | 10 〉 f ⎦ (3.72)
c ∗ e ∗ f ∗ P | 11 〉
The complex off-diagonal elements capture coherences between the states. Since
the probabilities are real and need to sum to unity, i.e. Tr(ρ) = 1, the number of
degrees of freedom of the density matrix for an n-qubit system is given by:
D(n) = n 2 − 1 (3.73)
3.4.5 Decoherence – The Kraus Operators
One way to simulate the decoherence of a qubit state expressed in the density
matrix formalism is with the use of the Kraus Operators [Kraus, 1983]:
[ ]
1 0
K 1a (∆t) =
0 e −∆t/2T 1
[ √
0 1 − e
−∆t/T 1
K 1b (∆t) =
0 0
]
(3.74)
(3.75)
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