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## Michael A. Nielsen University of Queensland

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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.**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**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 arather 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.**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.**The trace operation**Exercise: Prove that tr(|aihb|) = hb|ai.**I. Ensemble point of view**Probability of outcome k being in state j Probability of being in state j**Qubit example: calculate the density matrix**conjugate Density matrix Density matrix is a generalization of state**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**Dynamics and the density matrix**Initial density matrix**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.**Single-qubit example: calculating new density matrix by**operating with an inverter on old density matrix “Completely mixed state”**Characterizing the density matrix**What class of matrices correspond to possible density matrices? Trace of a density matrix is one