in this paper $s$ is a monoid with a left zero and $a_s$ (or $a$) is a unitary right $s$-act. it is shown that a monoid $s$ is right perfect (semiperfect) if and only if every (finitely generated) strongly flat right $s$-act is quasi-projective. also it is shown that if every right $s$-act has a unique zero element, then the existence of a quasi-projective cover for each right act implies that ...

Given a contractive linear system T which is represented as a causal linear operator on a Hilbert space H. We study the question: when does there exist an isometric or unitary causal extension of T to H⊕H?

1 The Axioms of Quantum Mechanics 1 1.1 The State Space Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Unitary Evolution Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 The Measurement Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 The Composite System Axiom . . . . . . . . . . . . . . . . . . ...

We construct a random unitary Gaussian circuit for continuous-variable (CV) systems subject to measurements. show that when the measurement rate is nonzero, steady state entanglement entropy saturates an area-law scaling. This different from many-body qubit system, where generic transition widely expected. Due unbounded local Hilbert space, time scale destroy always much shorter than one build ...