Michael A. Nielsen

1
Density Matrices and Quantum Noise
Michael A. Nielsen University of Queensland

• Goals
• To review a tool the density matrix that is
  used to describe noise in quantum systems.
• To give more practical examples.

2
Density matrices
Generalization of the quantum state used to
describe noisy quantum systems.
Terminology Density matrix Density operator
Quantum subsystem
Ensemble
Fundamental point of view
3
What were going to do in this lecture, and why
were doing it
Most of the lecture will be spent mastering the
density matrix.
Weve got to master arather complex formalism.
It might seem a little strange, since the density
matrix is never essential for calculations its
a mathematical tool, introduced for convenience.
Why bother with it?
The density matrix seems to be a very deep
abstraction once youve mastered the formalism,
it becomes far easier to understand many other
things, including quantum noise, quantum
error-correction, quantum entanglement, and
quantum communication.
4
Review Outer product notation
As we remember, this is a matrix, we showed how
to calculate it
5
Outer product notation
6
Outer product notation
One of the advantages of the outer product
notation is that it provides a convenient tool
with which to describe projectors, and thus
quantum measurements.
7
REMINDER Ensemble point of view
Probability of outcome k being in state ?j
Probability of being in state ?j
8
Qubit example REMINDER calculate the density
matrix
Conjugate and change kets to bras
Density matrix
Density matrix is a generalization of state
9
Qubit example a measurement using density matrix
Pr(0)
Pr(1)
10
Why work with density matrices?
Answer Simplicity!
We know the probabilities of states and we want
to find or check the density matrix
The quantum (mixed) state is
?
Sum of these probabilities must be equal one
11
Two-qubit example calculating the density matrix
knowing probabilties of states
Entangled states
As we see, this formalism is also good for some
states being entangled
Sum of probabilities on diagonal (trace) is one
12
Dynamics and the density matrix
Initial density matrix
13
Dynamics and the density matrix
This way, we can calculate a new density matrix
from old density matrix and unitary evolution
matrix U This is analogous to calculate a new
state from old state and unitary evolution matrix
U. The new formalism is more powerful since it
refers also to mixed states.
S1 U S0
14
Single-qubit example calculating new density
matrix by operating with an inverter on old
density matrix
Completely mixed state
15
How the density matrix changes during a
measurement
16
Characterizing the density matrix
What class of matrices correspond to possible
density matrices?
Trace of a density matrix is one
17
(No Transcript)
18
Summary of the ensemble point of view
19
Problems to Solve
Illustrate on matrices
Exercise Prove that tr(aihb) hbai.
Illustrate on matrices