Quantum logics with classically determined states

Andrzej Janutka

In a concept of simulating the quantum logic with vector solitons by the author (Janutka 2006 J. Phys. A: Math. Gen. 39 12505 ), the soliton polarization is thought of as a state vector of a system of cebits (classical counterparts of qubits) switched via collisions with other solitons. The advantage of this method of information processing compared to schemes using linear optics is the possibility of the determination of the information-register state in a single measurement. Minimization of the information-processing error for different optical realizations of the logical systems is studied in the framework of a quantum analysis of soliton fluctuations. The problem is considered with relevance to general difficulties of the quantum error-correction schemes for the classical analogies of the quantum-information processing.

Heinrich Saller

Claudio Garola

Nicolas Renaud, Christian Joachim

By encoding the digital input of a classical logic gate on the Hamiltonian of a quantum system and driving the logic operation by preparing it in the same nonstationary state whatever the input, NOR and NAND classical logic gates are designed with a minimum of three quantum states. The outputs of one gate are obtained either by measuring the distance between the $\ensuremath{\rho}(t)$ periodic quantum trajectory and the output target state ${\ensuremath{\rho}}_{b}$, or by measuring the secular frequency of the $\ensuremath{\rho}(t)$ almost periodic trajectory in the ${\ensuremath{\rho}}_{b}$ direction. A comparison of the stability to noise between the two approaches demonstrates that the frequency approach is more immune to random fluctuations in the Hamiltonian than a distance control approach, opening the way to determine the logic output using a tunneling current passing through the gate.

Dietmar W. Dorninger, Helmut Länger

The probability [Formula: see text] of the occurrence of an event pertaining to a physical system which is observed in different states [Formula: see text] determines a function [Formula: see text] from the set [Formula: see text] of states of the system to [Formula: see text]. Function [Formula: see text] is called a numerical event or more precisely an [Formula: see text]-probability. When appropriately structured, sets [Formula: see text] of numerical events form so-called algebras of [Formula: see text]-probabilities, which are orthomodular posets that can serve as quantum logics. If one deals with a classical physical system, then [Formula: see text] will be a Boolean algebra. Starting with a supposed quantum logic or logics obtained by individual measurements, newly added numerical events may turn out to be crucial for the assumption of classicality or nonclassicality of the physical system. We will call those numerical events critical and will study their impacts among various classes of algebras of numerical events. Moreover, we will consider the situation of enlarging [Formula: see text] by new states and will describe how elements of an algebra of [Formula: see text]-probabilities contribute to the character of a logic when there exists a certain relation between them and the element that is newly added.

Dov M. Gabbay, Kurt Engesser, Daniel J. Lehmann