# 2. Introduction to quantum mechanics

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1 2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian conj/ adjoint of matrix Inner product

2 Basis, vector representation For a set of vectors The set spanning constitutes a basis for Linear operators A linear operator Notation means

3 Matrix representation For spanning a matrix representation of Numbers, spanning means form matrix Linear operator (basis given) matrix representation (to be used interchangeably...) Pauli matrices,

4 Inner vector product A inner product on is We use notation Hilbert space = inner product space The vectors are orthogonal if The norm of a vector is An orthonormal set of vectors obey

5 Vector representation With respect to an orthonormal basis for the inner product is We thus have (an orthonormal basis will be used unless otherwise stated)

6 Outer vector product The outer product is a linear operator Cauchy-Schwartz inequality For two vectors Completeness relation For vectors forming an orthonormal basis for

7 Eigenvectors and eigenvalues The eigenvector with to obeys the eigenvalue. The diagonal representation of in terms of eigenvalues is (for diagonalizable and orthonormal eigenvectors Hermitian operators The Hermitian conjugate/adjoint of We have An Hermitian operator obeys and is, ) of

8 Projection operator The operator is a projection operator Properties Orthogonal complement Normal operator An operator is normal if An operator is normal if and only if it is diagonalizable. An Hermitian operator is normal.

9 Unitary matrix A matrix/operator is unitary if Positive operator An operator is positive if and real for any vector. Any positive operator is Hermitian Any positive operator has real, positive eigenvalues and a spectral decomposition in terms of eigenvalues and orthonormal eigenvectors of

10 Tensor product A tensor product between vectors in is a vector in Example: A tensor product between operators/matrices Operation is denoted

11 Properties Matrix representation Example: For matrices we have the tensor product

12 Operator functions For a normal operator, written in the spectral decomposition we define the/operator matrix function Trace The trace of a matrix is Cyclic property Outer product formulation

13 Commutators The commutator between two operators/matrices is The anti-commutator between two operators/matrices is Matrix decompositions Polar decomposition: For a linear operator unitary operator and positive operators there exists a so that Singular value decomposition: For a square matrix exists unitary matrices and a diagonal matrix non-negative elements (singular values), such that there with

14 2.2 Postulates of quantum mechanics State space Postulate 1: Associated to any isolated physical system is a Hilbert space, known as the state space of the system. The system is completely described by its state vector, a unit vector in the state space Definitions/names A two-level, qubit state can generally be written as This is a superposition of the two basis states with amplitudes and The normalization condition gives and,

15 Evolution Postulate 2: The evolution of a quantum system is described by a unitary transformation. That is, the state of the system at time is related to the state of the system at time by a unitary operator as Postulate 2 : The evolution of state of a quantum system is described by the Schrödinger equation where is Plancks constant and Hermitian operator. the Hamiltonian, a

16 Closed system For a closed system the Hamiltonian and the system state is is independent on time where we define the unitary time evolution operator The Hamiltonian has the spectral decomposition where are the energy eigenstates and the energy. Effective Hamiltonian for open systems For many open systems we have an effective time dependent Hamiltonian acting on the system The solution to the Schrödinger equation is non-trivial

17 Measurement Projection measurement postulate A projective measurement is described by an observable,, an Hermitian operator on the state space of the system. The observable has the spectral decomposition The possible outcomes correspond to the eigenvalues Upon measuring the state is given by of, the probability of getting the result Given that has occured, the state immediatelly after the measurement is (wavefunction collapse) Measurement problem?.

18 General measurement Postulate 3: Quantum measurements are described by a collection of measurement operators, acting on the state space of the system. The index refers to the possible measurement outcomes. Upon measuring the state, the probability of getting the result is given by and the state after the measurement is The measurement operators satisfy the completeness relation Probabilities sum to one

19 Projective vs general measurement For a projective measurement The average measured value (over an ensemble of states The magnitude of the quantum fluctuations are Derivation: Heisenbergs uncertainty principle is )

20 Composite systems Postulate 4: The state space of a composite system is the tensor product of the state spaces of the component systems. If we have systems numbered through, and system prepared in state, the state of the total system is General measurement and projection II Derivation: Given projection measurements and an ancilla system, derive the general measurement principles. is

21 POVM measurement Common formulation of general measurement postulate. A measurement is described by measurement operators The probability to get the outcome is given by. We define the positive operator which has the properties We call the POVM-elements and the set a POVM. Distinghuishing quantum states Given a single copy of one of two non-orthogonal states not possible to determine which state by any measurement. Derivation: The example with, it is

22 Entanglement of two qubits A composite state of two qubits that can not be written as a tensor product of the states of the two qubits is entangled, that is Derivation: Show that the Bell state is entangled Entanglement is the energy for quantum information processing

23 2.3 Superdense coding Suppose that Alice wants to send two bits of classical information to Bob by only sending one qubit. Can she do it? 2 classical bits 1 qubit Alice Derivation: Superdense coding Bob

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