Obscure qubits and membership amplitudes

TL;DR

This paper introduces a novel quantum computing framework that integrates vagueness through obscure qubits, combining quantum probability with membership functions, and explores new types of entanglement and computation models.

Contribution

It proposes the concept of obscure qubits with membership amplitudes, and develops two models: product and Kronecker obscure qubits, expanding quantum computing to include vagueness.

Findings

Introduces obscure qubits with membership amplitudes.

Defines double obscure-quantum computation with mixed measurement.

Proposes new entanglement measures for obscure quantum states.

Abstract

We propose a concept of quantum computing which incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), in a natural way by introducing new entities, obscure qudits (e.g. obscure qubits), which are characterized simultaneously by a quantum probability and by a membership function. To achieve this, a membership amplitude for quantum states is introduced alongside the quantum amplitude. The Born rule is used for the quantum probability only, while the membership function can be computed from the membership amplitudes according to a chosen model. Two different versions of this approach are given here: the "product" obscure qubit, where the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the "Kronecker" obscure qubit, where quantum and vagueness computations are to be performed independently (i.e. quantum computation alongside truth evaluation). The latter is called a double obscure-quantum computation. In this case, the measurement becomes mixed in the quantum and obscure amplitudes, while the density matrix is not idempotent. The obscure-quantum gates act not in the tensor product of spaces, but in the direct product of quantum Hilbert space and so called membership space which are of different natures and properties. The concept of double (obscure-quantum) entanglement is introduced, and vector and scalar concurrences are proposed, with some examples being given.