1. University of Queensland
Quantum Noise
Michael A. Nielsen
University of Queensland
Goals:
To introduce a tool – the density matrix – that is used to describe noise in quantum systems, and to give some examples.
The first two lectures today are going to introduce the basic background in quantum mechanics that you need to know in order to do quantum information science.
The goal of this and the next lecture is to introduce _all_ the basic elements of quantum mechanics, using examples drawn from quantum information science. I’m going to do this assuming only elementary linear algebra, and mathematical maturity about that I would expect for a good third or fourth year undergraduate. Notice, by the way, that over the next few days I’m going to use the umbrella term “quantum information science” to encompass quantum information and computation.
There are two groups of people that these lectures are for.
The first is for people with little or no background in quantum mechanics, who’d like to learn the subject, or at least brush up on it.
Thus, the approach today is going to be fairly elementary, and some experts may wish to go and spend some time enjoying the sights of Brisbane. However, they may also like to say, and participate in and observe an experimental approach to the teaching of quantum mechanics, an approach that I think has some very considerable advantages over the standard approach.

2. Fundamental point of view
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 we’re going to do in this lecture, and why we’re doing it
Most of the lecture will be spent understanding the density matrix.
Unfortunately, that means we’ve got to master a rather complex formalism.
It might seem a little strange, since the density matrix is never essential for calculations – it’s a mathematical tool, introduced for convenience.
Why bother with it?
The density matrix seems to be a very deep abstraction – once you’ve mastered the formalism, it becomes far easier to understand many other things, including quantum noise, quantum error-correction, quantum entanglement, and quantum communication.

4. Outer product notation

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. The trace operation
Exercise: Prove that tr(|aihb|) = hb|ai.

8. I. Ensemble point of view
Probability of outcome k being in state j
Probability of being in state j

9. Qubit example: calculate the density matrix
conjugate
Density matrix
Density matrix is a generalization of state

10. Qubit example: a measurement using density matrix

11. Why work with density matrices?
Answer: Simplicity!
The quantum state is: ?
We know the probabilities of states and we want to find or check the density matrix

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.

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

18. Summary of the ensemble point of view