**1. Introduction**

Nowadays, the development of quantum computing technique is governed by theoretical extensions of its ground concepts [1–3]. One of them is to allow two kinds of uncertainty, sometimes called randomness and vagueness/fuzziness (for a review, see, [4]), which leads to the formulation of combined probability and possibility theories [5] (see, also, [6–9]). Various interconnections between vagueness and quantum probability calculus were considered in [10–13], including the treatment of inaccuracy in measurements [14, 15], non-sharp amplitude densities [16] and the related concept of partial Hilbert spaces [17].

Relations between truth values and probabilities were also given in [18]. The hardware realization of computations with vagueness was considered in [19, 20]. On the fundamental physics side, it was shown that the discretization of space–time at small distances can lead to a discrete (or fuzzy) character for the quantum states themselves [21–24].

With a view to applications of the above ideas in quantum computing, we introduce a definition of quantum state which is described by both a quantum probability and a membership function, and thereby incorporate vagueness/fuzziness directly into the formalism. In addition to the probability amplitude we will define a membership amplitude, and such a state will be called an obscure/fuzzy qubit (or qudit).

In general, the Born rule will apply to the quantum probability alone, while the membership function can be taken to be an arbitrary function of all the amplitudes fixed by the chosen model of vagueness. Two different models of "obscurequantum computations with truth" are proposed below: (1) A "Product" obscure qubit, in which the resulting amplitude is the product (in ) of the quantum amplitude and the membership amplitude; (2) A "Kronecker" obscure qubit for which computations are performed "in parallel", so that quantum amplitudes and the membership amplitudes form "vectors", which we will call obscure-quantum amplitudes. In the latter case, which we call a double obscure-quantum computation, the protocol of measurement depends on both the quantum and obscure amplitudes, and in this case the density matrix need not be idempotent. We define a new kind of "gate", namely, the obscure-quantum gates, which are linear transformations in the direct product (not in the tensor product) of spaces: a quantum Hilbert space and a so-called membership space having special fuzzy properties. We introduce a new concept of double (obscure-quantum) entanglement, in which vector and scalar concurrences are defined and computed for some examples.
