PMV-algebras with fix point negation: a probabilistic model for quantum computational logic based on density operators

H. FREYTES, G. DOMENECH

Darmstadt, Germany

Congreso; Algebra and Probability in Many-Valued Logics; 2009

Technische Universität

The structure of MV-algebras enriched with product operations are considered by several authors ([2], [3], [9]) as a basis for quantum structure models. In our case we use the structure of product MV-algebras ([6], [9]) (PMV-algebras for short) as the basis for a probabilistic model of quantum computation with density operators. In the usual representation of quantum computational processes, a quantum circuit is identified with an appropriate composition of quantum gates, i.e. unitary operators acting on pure states of a convenient (n-fold tensor product) Hilbert space. Consequently, quantum gates represent time reversible evolutions of pure states of the system. But for many reasons this restriction is unduly. On the one hand, it does not encompass realistic physical states described by mixtures. And on the other hand, there are interesting processes that cannot be encoded in unitary evolutions, such as measurements in middle of the process. Several authors ([1], [5], [11]) have paid attention to a more general model of quantum computational processes, where pure states and unitary operators are replaced by density operators and quantum operations, respectively. In this case, time evolution is no longer necessarily reversible. Quantum computational logic with mixed states may be presented as a logic whose language is an absolute free algebra Term and whose natural universe of interpretation is a set D of density operators. In this framework, two special operators P0 and P1 in D represent the falsity-property and the truth-property respectively. By applying the Born rule, the probability to obtain the truth-property P1 for a system being in the state represented by a density operator s is given by p(s) = trace(P1.s) On the other hand, connectives are naturally interpreted as certain quantum operations. More precisely, canonical interpretations are Term-homomorphisms from Term onto D. To define a relation of semantic consequence based on the probability assignment, it is necessary to introduce the notion of canonical valuations. In fact, canonical valuations are functions f from Term onto the unitary real interval [0, 1] such that f can be factorized by the commutative diagram f = p.e where p is the probability assignment. We will refer to these diagrams as probabilistic models. Then the semantical consequence related to this diagrams is given as follows: for each pair x, y in Term, "y is consequence of x" iff R[f(x), f(y)] where R is a type of relation that links f(x) and f(y). The connection between quantum computational logic with mixed states and fuzzy logic is given by the election of a system of quantum operations (or quantum gates) such that -when interpreted within probabilistic models- they turn into a type of operations in the real interval [0, 1] such as continuous t-norms, left-continuous t-norms, etc., associated to fuzzy logic. The systems presented in [4] and [5], precisely those that motivate our study, are of this kind. This quantum gate system is known as the Poincarè irreversible quantum computational algebra. An algebraic abstract frame of the Poincarè irreversible quantum computational algebra is a class of algebras admitting a reduct with the structure of PMV-algebra with fix point of the negation. This PMV-reduct represents in an abstract form the assignment of probability. We also remark that this structure results an extension of other structures associated to quantum computational structures known as square root quasi MV-algebras ([4], [5]). Under this framerwork, probabilistic models are refereed as PMV-models. The semantical consequence related to a PMV-model is given as follows: for each pair x, y in Term, "y is consequence of x" iff f(x) = 1 implies f(y) = 1. The fact that the logical consequence of these systems is related to functions f factorized through the PMV-models does not allow to use standard methods of algebrization to study the algebraic completeness of a Hilbert style calculus. In the present work, we will develop a Hilbert style calculus whose natural interpretation are the PMV-models. An algebraic completeness theorem that uses a variant of the mentioned methods of algebrization will be obtained. We recall that a "complicated" Hilbert style calculus for these kind of models has been previously presented. In our case, the use of PMV-algebras with fix point negation and the basic properties of injective objects in this class turns the logical system more adequate and simple. Bibliography [1] Aharanov D., Kitaev A., Nisan N Quantum circuits with mixed states, Proc. 13th Annual ACM Symp. on Theory of Computation (STOC, 1997). pp. 20-30. 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