CHAPTER 2. The Quantum Recipe

P. C. W. Davies

Jochen Autschbach

Abstract Introduction of the postulates of quantum mechanics: Wavefunctions, operators, observables, commutating operators, expectation values, probabilities, Heisenberg uncertainty. The postulates are then used to set up a ‘quantum recipe’, i.e. a straightforward recipe by which to write down the (nonrelativistic) quantum Hamiltonian of a system of particles. This chapter also discusses the representation of quantum operators as matrices, in reference to a set of ‘basis’ functions, and the variation principle. The idea of a particle trajectory must be abandoned in quantum mechanics. Observable properties of a particle correspond to eigenvalues of the associated quantum operators. The chapter concludes with a brief discussion of the Schrodinger’s cat paradox, quantum entanglement, and other oddities.

Barry R. Clarke

M. S. Ramkarthik, Payal D. Solanki

This chapter contains recipes of all the major quantum gates used in performing various quantum information tasks. It also has some useful recipes on the construction of N qubit computational basis, N qubit Greenberger-Horne-Zeilinger states, N qubit W states, two-qubit Bell states, N qubit generalized Werner states. Apart from that, entropy and its quantification by various measures like Shannon entropy, linear and Relative entropy are given. It becomes very important in quantum information theory to know the closeness between two quantum states and to this end, recipes related to trace distance, fidelity, super fidelity, Bures distance. The chapter ends with two important recipes which are for computing the expectation values of observables and the quantum mechanical single qubit measurements.

M. S. Ramkarthik, Payal D. Solanki

This chapter contains the recipes for the operations which will be used in manipulating quantum states and operators. It contains recipes to perform normalization, inner and outer products, tensor products, commutators and anti-commutators, inter-conversion between binary and decimal, bit testing, swapping and shifts of qubits. These are the most important and fundamental operations involving quantum states and operators.