Michael A. Nielsen

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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.

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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
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What were going to do in this lecture, and why
were doing it
Most of the lecture will be spent understanding
the density matrix.
Unfortunately, that means 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.
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Outer product notation
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Outer product notation
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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.
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The trace operation
Exercise Prove that tr(aihb) hbai.
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I. Ensemble point of view
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Qubit examples
conjugate
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Qubit example
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Why work with density matrices?
Answer Simplicity!
The quantum state is
?
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Two-qubit example
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Dynamics and the density matrix
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Single-qubit examples
Completely mixed state
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How the density matrix changes during a
measurement
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Characterizing the density matrix
What class of matrices correspond to possible
density matrices?
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Summary of the ensemble point of view