Quantum computational logic with mixed states

HECTOR FREYTES Y GRACIELA DOMENECH

Canela

Congreso; VII Encuentro AFHIC; 2010

st1:*{behavior:url(#ieooui) } Standard quantum computing is based on quantum systems with finite dimensional Hilbert spaces, specially C2, the two-dimensional state space of a qbit. A qbit state (the quantum counterpart of the classical bit) is represented by a unit vector in C2 and, generalizing for a positive integer n, n-qbits are pure states represented by unit vectors in C2n. They conform the information units in quantum computation. These state spaces only concerned with the “static” part of quantum computing and possible logical systems can be founded in the Birkhoff and von Neumann quantum logic based on the Hilbert lattices L(C2n). Similarly to the classical computing case, we can introduce and study the behavior of a number of quantum logical gates (hereafter quantum gates for short) operating on qbits, giving rise to “new forms” of quantum logic. These gates are mathematically represented by unitary operators on the appropriate Hilbert spaces of qbits. In other words, standard quantum computation is mathematically founded on “qbits-unitary operators” and only takes into account reversible processes. This framework can be generalized to a powerful mathematical representation of quantum computation in which the qbit states are replaced by density operators over Hilbert spaces and unitary operators by linear operators acting over endomorphisms of Hilbert spaces called quantum operations. The new model “density operators-quantum operations” also called “quantum computation with mixed states” is equivalent in computational power to the standard one but gives a place to irreversible processes as measurements in the middle of the computation. In [1] a quantum gate system called Poincar´e irreversible quantum computational system (for short IP-system) was developed. Recently it was proved that the mentioned quantum gates system can be seen in the framework given by “density operators - quantum operations”. The IP system is an interesting quantum gates system since it is related to fuzzy logic of continuous t-norms [2] and subsequent generalizations allow to connect this system with sequential effect algebras introduced to study the sequential action of quantum effects which are unsharp versions of quantum events. Our study is motivated by the IP-system, and mainly by the following question proposed by the authors in [1]:  “The axiomatizability of quantum computational logic is an open problem”.  More precisely, in this work we give a Hilbert-style calculus obtaining a strong completeness theorem respect to probabilistic semantics associated with the IP-system.    [1] M. L. Dalla Chiara, R. Giuntini and R. Greechie, Reasoning in Quantum Theory. Kluwer, Dordrecht (2004).  [2] G. Domenech and H. Freytes, Fuzzy propositional logic associated with quantum computational gates. Int. J. Theor. Phys. 34, 228-261 (2006).