Quantum computational logic with mixed states

Abstract

Using an algebraic framework we solve a problem posed in [5] and [7] about the axiomatizability of a type quantum computational logic related to fuzzy logic. A Hilbert-style calculus is developed obtaining an algebraic strong completeness theorem.

An Holistic Extension for Classical Logic via Quantum Fredkin Gate

“…In this way, the state can be represented as a probability distribution over different pure states, i.e., it becomes a mixed state. These facts have motivated several authors to focus on a general model of quantum computation, where mixed states are in place of pure ones [6,[13][14][15][16]. In what follows, we provide a description of this model, which is better suited to our development.…”

“…Then, by considering relations between the input and the output of the quantum circuits, it is possible to define the notions of logical consequence. These logics have a common interpretation based on the probability values introduced in Equation ( 8 ) and have been studied by several authors [ 4 , 6 , 14 ].…”

Logical Structures Underlying Quantum Computing

“…To describe quantum computation from a logical and algebraic point of view, many formalisms were developed (see for example [ 3 , 4 , 5 , 6 , 7 , 8 , 9 ]). We refer to the logics studied in these works as quantum computational logics (QCL).…”

Holistic Type Extension for Classical Logic via Toffoli Quantum Gate

“…In recent years, an increasing interest in logical systems related to quantum mechanics has arisen. Most of these systems are not strictly related to the standard quantum logic, but they are motivated by concrete problems related to quantum information and quantum computation [ 1 , 2 , 3 , 4 , 5 , 6 , 7 ].…”