Linear Operators and Commutators

Paul Richard Halmos

Richard Cleve, Li Liu, William Slofstra

Linear system games are a generalization of Mermin’s magic square game introduced by Cleve and Mittal. They show that perfect strategies for linear system games in the tensor-product model of entanglement correspond to finite-dimensional operator solutions of a certain set of non-commutative equations. We investigate linear system games in the commuting-operator model of entanglement, where Alice and Bob’s measurement operators act on a joint Hilbert space, and Alice’s operators must commute with Bob’s operators. We show that perfect strategies in this model correspond to possibly infinite-dimensional operator solutions of the non-commutative equations. The proof is based around a finitely presented group associated with the linear system which arises from the non-commutative equations.

Ram Yatan Prasad, Pranita

In the present chapter, before discussing the mechanical operators, operator has been lucidly defined and examples are cited for its clear understanding. It includes linear and non-linear operator, commutator and commutation, and Hermitian operator and its properties. It also consists of Schmidt orthogonalisation, del and del 2 operators, linear momentum operator and it applications, angular momentum vector and commutation, and angular momentum in spherical polar co-ordinates. This chapter also provides a good discussion regarding ladder operator, Hamiltonian operator and its commutation, and relation of angular momentum operators with Hamiltonian operators. It also discusses projection operator clearly and solution of problem related to projection operator. The properties of projection operator have also been given. Suitable references have also been provided for the consultation. A large number of problems and their solutions relating to different operators have also been cited. Questions on concepts have also been provided for exercise purposes. The readers are advised to solve these problems for better understanding.

George C. Turrell

Franco Fagnola, Damiano Poletti

The generator of a Gaussian quantum Markov semigroup on the algebra of bounded operator on a [Formula: see text]-mode Fock space is represented in a generalized GKLS form with an operator [Formula: see text] quadratic in creation and annihilation operators and Kraus operators [Formula: see text] linear in creation and annihilation operators. Kraus operators, commutators [Formula: see text] and iterated commutators [Formula: see text] up to the order [Formula: see text], as linear combinations of creation and annihilation operators determine a vector in [Formula: see text]. We show that a Gaussian quantum Markov semigroup is irreducible if such vectors generate [Formula: see text], under the technical condition that the domains of [Formula: see text] and the number operator coincide. Conversely, we show that this condition is also necessary if the linear space generated by Kraus operators and their iterated commutator with [Formula: see text] is fully non-commutative.

Brian L. Silver