Presentation on theme: "Dirac Notation and Spectral decomposition"— Presentation transcript:
1 Dirac Notation and Spectral decomposition Michele Mosca
2 review”: Dirac notation For any vector , we let denote , the complex conjugate of .We denote by the inner product between two vectors anddefines a linear function that maps(I.e … it maps any state to the coefficient of its component)
3 More Dirac notation defines a linear operator that maps This is a scalar so I can move it to front(I.e. projects a state to its componentRecall: this projection operator also corresponds to the “density matrix” for )
4 More Dirac notationMore generally, we can also have operators like
5 Example of this Dirac notation For example, the one qubit NOT gate corresponds to the operatore.g.(sum of matrices applied to ket vector)This is one more notation to calculate state from state and operatorThe NOT gate is a 1-qubit unitary operation.
6 Special unitaries: Pauli Matrices in new notation The NOT operation, is often called the X or σX operation.
7 Recall: Special unitaries: Pauli Matrices in new representation Representation of unitary operator
8 What is ??It helps to start with the spectral decomposition theorem.
9 Spectral decomposition Definition: an operator (or matrix) M is “normal” if MMt=MtME.g. Unitary matrices U satisfy UUt=UtU=IE.g. Density matrices (since they satisfy =t; i.e. “Hermitian”) are also normalRemember: Unitary matrix operators and density matrices are normal so can be decomposed
10 Spectral decomposition Theorem Theorem: For any normal matrix M, there is a unitary matrix P so thatM=PPt where is a diagonal matrix.The diagonal entries of are the eigenvalues.The columns of P encode the eigenvectors.
11 Example: Spectral decomposition of the NOT gate This is the middle matrix in above decomposition
12 Spectral decomposition: matrix from column vectors
13 Spectral decomposition: eigenvalues on diagonal Eigenvalues on the diagonal
24 “Von Neumann measurement in the computational basis” Suppose we have a universal set of quantum gates, and the ability to measure each qubit in the basisIf we measure we get with probabilityWe knew it from beginning but now we can generalize
25 Using new notation this can be described like this: We have the projection operatorsand satisfyingWe consider the projection operator or “observable”Note that 0 and 1 are the eigenvaluesWhen we measure this observable M, the probability of getting the eigenvalue b is and we are in that case left with the state
26 Polar Decomposition Left polar decomposition Right polar decomposition This is for square matrices