Introduction to Quantum Physics and Information Processing

92 Introduction to Quantum Physics and Information Processing

Can we write a differential equation for time evolution, like the Schr¨odinger

equation 3.23 for the state vector? If we differentiate Equation 5.31 with re-

spect to time:

d

dt

ρ(t) =

i

~



−

ˆ

H

ˆ

Uρ(0)

ˆ

U

†

+

ˆ

Uρ(0)

ˆ

H

ˆ

U

†



=

i

~



−

ˆ

Hρ(t) + ρ(t)

ˆ

H



=

i

~

h

ρ(t),

ˆ

H

i

. (5.33)

For an open system, the evolution of the density matrix can no longer be

expressed as a unitary transformation. The description of such evolution is

beyond the scope of this text.

5.3 Composite Systems

There is another sense in which density operators are a useful way to

describe nature. In general, it is impossible to isolate the system of in