https://arxiv.org/abs/2011.04370 A concept of quantum computing is proposed which naturally incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), by introducing obscure qudits (qubits), which are simultaneously characterized by a quantum probability and a membership function. Along with the quantum amplitude, a membership amplitude for states is introduced. The Born rule is used for the quantum probability only, while the membership function can be computed through the membership amplitudes according to a chosen model. Two different versions are given here: the "product" obscure qubit in which the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the "Kronecker" obscure qubit, where quantum and vagueness computations can be performed independently (i.e. quantum computation alongside truth). The measurement and entanglement of obscure qubits are briefly described.

1. arXiv:2011.04370v1[quant-ph]9Nov2020
Membership amplitudes and obscure qudits
Steven Duplij∗
and Raimund Vogl†
Center for Information Technology (WWU IT), University of M¨unster, D-48149 M¨unster, Germany
(Dated: start July 31, 2020, completion November 8, 2020)
A concept of quantum computing is proposed which naturally incorporates an additional kind of
uncertainty, i.e. vagueness (fuzziness), by introducing obscure qudits (qubits), which are simultane-
ously characterized by a quantum probability and a membership function. Along with the quantum
amplitude, a membership amplitude for states is introduced. The Born rule is used for the quantum
probability only, while the membership function can be computed through the membership ampli-
tudes according to a chosen model. Two different versions are given here: the “product” obscure
qubit in which the resulting amplitude is a product of the quantum amplitude and the membership
amplitude, and the “Kronecker” obscure qubit, where quantum and vagueness computations can
be performed independently (i.e. quantum computation alongside truth). The measurement and
entanglement of obscure qubits are briefly described.
I. INTRODUCTION
The modern development of the quantum computing technique implies various extensions of its ground concepts
[1–3]. One of these is the existence of two kinds of uncertainty, sometimes called randomness and vagueness/fuzziness
(for a review, see, [4]), which leads to the formulation of probability and possibility theories [5] (see, also, [6–9]).
Various interconnections between vagueness and quantum probability calculus were considered in [10–13], including
the inaccuracy of measurements [14, 15] and non-sharp amplitude densities [16] and partial Hilbert spaces [17]. The
hardware realization of computations with vagueness was considered in [18, 19]. On the other side, it was shown
that the discretization of space-time at small distances can lead to a discrete (or fuzzy) character for quantum states
themselves [20–23].
Reconsidering the ideas above with a view to applications in quantum computing, we introduce a definition of
quantum state which is described by quantum probability and a membership function simultaneously, in order to
incorporate vagueness/fuzziness directly into the formalism. In this way, in addition to the probability amplitude, we
define the membership amplitude, and we will call such a state an obscure/fuzzy qubit (qudit).
In general, we will imply the Born rule for the quantum probability only, while the membership function can be an
arbitrary function of all amplitudes fixed by the chosen vagueness model. We propose two different models of “obscure-
quantum computations with truth”: 1) “Product” obscure qubit, in which the resulting amplitude is a product (in
C) of the quantum amplitude and the membership amplitude; 2) “Kronecker” obscure qubit for which computations
are performed “in parallel”, such that quantum amplitudes and the membership amplitudes form “vectors”, the so
called obscure-quantum amplitudes. We will briefly consider measurement and entanglement for such obscure qubits.
II. PRELIMINARIES
Here we establish notations (see, e.g. [1, 2] and [24–26]). The standard qudit (in the computational basis and the
Dirac notation) in the d-dimensional Hilbert space H
(d)
q is (for a review, see, e.g. [27, 28])
ψ(d)
=
d−1
i=0
ai |i , ai ∈ C, |i ∈ H(d)
q , (2.1)
where ai are probability amplitudes. The probability pi to measure the ith state is pi = Fpi (a1, . . . , an), 0 ≤ pi ≤ 1,
0 ≤ i ≤ d − 1. The shape of the functions Fpi is governed by the Born rule Fpi (a1, . . . , ad) = |ai|2
, and d
i=0 pi = 1.
A one-qudit (L = 1) quantum gate is a unitary transformation U(d)
: H
(d)
q → H
(d)
q described by unitary d×d complex
matrices acting on the vector (2.1), and for a register containing L qudits quantum gates are unitary dL
×dL
matrices.
∗ douplii@uni-muenster.de, sduplij@gmail.com, https://ivv5hpp.uni-muenster.de/u/douplii
† rvogl@uni-muenster.de

2. 2
The concept of quantum computing is based on the quantum circuit model [29, 30], in which the quantum algorithms
are compiled as a sequence of elementary gates acting on a register containing L qubits (or qudits), followed by a
measurement [24, 31].
Further special details on qudits and their transformations can be found, e. g., in the reviews [27, 28] and refs
therein.
III. MEMBERSHIP AMPLITUDES
Let us introduce a new variable for the state |i : the membership amplitude αi. We define the obscure qudit with
d states by the following superposition (instead of (2.1))
ψ
(d)
ob =
d−1
i=1
αiai |i , (3.1)
where ai is a (complex) probability amplitude ai ∈ C, and αi is the (real) membership amplitude, αi ∈ [0, 1],
0 ≤ i ≤ d − 1. The probability pi to measure the ith state and the membership function µi (“of truth”) of the ith
state are functions of the corresponding amplitudes
pi = Fpi (a0, . . . , ad−1) , 0 ≤ pi ≤ 1, (3.2)
µi = Fµi (α0, . . . , αd−1) , 0 ≤ µi ≤ 1. (3.3)
The dependence of the probabilities of the ith states from the amplitudes, i.e. the form of the function Fpi is fixed
by the Born rule
Fpi (a1, . . . , an) = |ai|
2
, (3.4)
while the form of Fµi can be chosen arbitrarily for different obscurity assumptions. Here we confine ourselves to
the membership amplitudes and the membership functions being real (the complex obscure sets and numbers were
considered in [32–34]). So the real functions Fpi and Fµi , 0 ≤ i ≤ d − 1, contain the complete information about the
obscure qudit (3.1).
We demand the normalization conditions
d−1
i=0
pi = 1, (3.5)
d−1
i=0
µi = 1, (3.6)
where the first condition is standard in quantum mechanics, while the second condition is taken by analogy. It may
be that (3.6) is not satisfied, but we will not consider this case.
If d = 2, we obtain the obscure qubit in the general form (instead of (3.1))
ψ
(2)
ob = α0a0 |0 + α1a1 |1 , (3.7)
Fp0 (a0, a1) + Fp1 (a0, a1) = 1, (3.8)
Fµ0 (α0, α1) + Fµ1 (α0, α1) = 1. (3.9)
The Born probabilities to observe the states |0 and |1 are
p0 = FBorn
p0
(a0, a1) = |a0|2
, p1 = FBorn
p1
(a0, a1) = |a1|2
, (3.10)
and the membership functions are
µ0 = Fµ0 (α0, α1) , µ1 = Fµ1 (α0, α1) . (3.11)
If we assume the Born rule (3.10) for the membership functions as well
Fµ0 (α0, α1) = α2
0, Fµ1 (α0, α1) = α2
1, (3.12)

3. 3
(which is one of various possibilities depending on the chosen model), then
|a0|
2
+ |a1|
2
= 1, (3.13)
α2
0 + α2
1 = 1. (3.14)
Using (3.13)–(3.14) we can parametrize (3.7) as
ψ
(2)
ob = cos
θ
2
cos
θµ
2
|0 + eiϕ
sin
θ
2
sin
θµ
2
|1 , (3.15)
0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π, 0 ≤ θµ ≤ π. (3.16)
Therefore, geometrically, the obscure qubits (with Born-like rule for the membership functions) are described by
vectors inside two Bloch balls (not as vectors on the sphere, because “|sin| , |cos| ≤ 1”), one is for the probability
amplitude (ellipsoid inside the Bloch ball with θµ = const1), and another one for the membership amplitude (which
is reduced to the ellipse, being a slice inside the Bloch ball with θ = const2, ϕ = const3). The norm of the obscure
qubits is not constant, because
ψ
(2)
ob | ψ
(2)
ob =
1
2
+
1
4
cos (θ + θµ) +
1
4
cos (θ − θµ) . (3.17)
In the case θ = θµ the norm (3.17) becomes 1 − 1
2 sin2
θ, reaching its minimum 1
2 when θ = θµ = π
2 .
Note that for complicated functions Fµ0,1 (α0, α1) the condition (3.14) may be not satisfied, but the condition (3.6)
should nevertheless always be valid. The concrete form of the functions Fµ0,1 (α0, α1) depends upon the chosen model.
In the simplest case, we can identify two arcs on the Bloch ellipse for α0, α1 with the membership functions and obtain
Fµ0 (α0, α1) =
2
π
arctan
α1
α0
, (3.18)
Fµ1 (α0, α1) =
2
π
arctan
α0
α1
, (3.19)
such that µ0 + µ1 = 1 as it should be (3.6).
In [35, 36] a two stage special construction of quantum obscure/fuzzy sets was considered. The so-called classical-
quantum obscure/fuzzy registers were introduced (in n = 2 minimal case)
|s f = 1 − f |0 + f |1 , (3.20)
|s g = 1 − g |0 +
√
g |1 , (3.21)
where f, g ∈ [0, 1] are the classical-quantum membership functions. At the second stage their quantum superposition
is given by
|s = cf |s f + cg |s g , (3.22)
where cf and cg are the probability amplitudes of the fuzzy states |s f and |s g, respectively. It can be seen that the
state (3.22) is a particular case of (3.7) with
α0a0 = cf 1 − f + cg 1 − g, (3.23)
α1a1 = cf f + cg
√
g. (3.24)
It is important that the use of the membership amplitudes introduced here αi and (3.1) allows us to exploit the
standard quantum gates, but not to define new special ones, as in [35, 36].
Another possible form of Fµ0,1 (α0, α1) (3.11), such that the corresponding membership functions satisfy the stan-
dard fuzziness rules, can be found using the homeomorphism between the circle and the square. In [37, 38] this
transformation was applied to the probability amplitudes a0,1, but here we exploit it for the membership amplitudes
α0,1, which is natural in our approach.
Fµ0 (α0, α1) =
2
π
arcsin
α2
0 sign α0 − α2
1 sign α1 + 1
2
, (3.25)
Fµ1 (α0, α1) =
2
π
arcsin
α2
0 sign α0 + α2
1 sign α1 + 1
2
. (3.26)

4. 4
So for positive α0,1 we obtain (cf. [37])
Fµ0 (α0, α1) =
2
π
arcsin
α2
0 − α2
1 + 1
2
, (3.27)
Fµ1 (α0, α1) = 1. (3.28)
The equivalent membership functions for the outcome are
max (min (Fµ0 (α0, α1) , 1 − Fµ1 (α0, α1)) , min (1 − Fµ0 (α0, α1)) , Fµ1 (α0, α1)) , (3.29)
min (max (Fµ0 (α0, α1) , 1 − Fµ1 (α0, α1)) , max (1 − Fµ0 (α0, α1)) , Fµ1 (α0, α1)) . (3.30)
There are many different models for Fµ0,1 (α0, α1) which can be introduced in such a way that they satisfy the
obscure set axioms [7, 9].
IV. TRANSFORMATIONS OF OBSCURE QUBITS
Let us consider the obscure qubits in the vector representation, such that
|0 =
1
0
, |1 =
0
1
(4.1)
are basis vectors of H
(2)
q . Then a standard quantum computational process in the quantum register with L obscure
qubits (qudits (2.1)) is performed by sequences of unitary matrices U of size 2L
× 2L
(nL
× nL
), U†
U = I (here I is
the unit matrix of the corresponding size), which are called quantum gates. Thus, for one obscure qubit the quantum
gates are 2 × 2 unitary matrices.
In the vector representation, the standard qubit and the obscure qubit (3.7) differ by a 2 × 2 invertible diagonal
(but not unitary) matrix
ψ
(2)
ob = M (α0, α1) ψ(2)
, (4.2)
M (α0, α1) =
α0 0
0 α1
. (4.3)
We call M (α0, α1) a membership matrix which can optionally have the property
tr M2
= 1, (4.4)
if (3.14) holds.
Let us introduce the orthogonal commuting projection operators
P0 =
1 0
0 0
, P1 =
0 0
0 1
, (4.5)
P2
0 = P0, P2
1 = P1, P0P1 = P1P0 = 0, (4.6)
where 0 is the 2 × 2 zero matrix. Well-known properties of the projections are that
P0 ψ(2)
= a0 |0 , P1 ψ(2)
= a1 |0 , (4.7)
ψ(2)
P0 ψ(2)
= |a0|
2
, ψ(2)
P1 ψ(2)
= |a1|
2
. (4.8)
Therefore, the membership matrix (4.3) can be defined as a linear combination of the projection operators with the
membership amplitudes as coefficients
M (α0, α1) = α0P0 + α1P1. (4.9)
We compute
M (α0, α1) ψ
(2)
ob = α2
0a0 |0 + α2
1a1 |1 . (4.10)

5. 5
We can therefore treat applying the membership matrix (4.2) as being the origin of a reversible non-unitary “obscure
measurement” of the standard qubit to obtain the obscure qubit (cf. the “mirror measurement” [39, 40] and the origin
of ordinary qubit states on the fuzzy sphere [41]).
An obscure analog of the density operator (for a pure state) is (the density matrix in the vector representation)
ρ
(2)
ob = ψ
(2)
ob ψ
(2)
ob =
α2
0 |a0|2
α0a∗
0α1a1
α0a0α1a∗
1 α2
1 |a1|
2 (4.11)
with the obvious standard singularity property det ρ
(2)
ob = 0. But tr ρ
(2)
ob = α2
0 |a0|2
+ α2
1 |a1|2
= 1, and here there is no
idempotence ρ
(2)
ob
2
= ρ
(2)
ob .
V. KRONECKER OBSCURE QUBITS
Introducing the membership amplitude means that we should have an analog of quantum superposition for them,
which we call “obscure superposition” (cf. [42], and also [43]).
Here we consider quantum amplitudes and membership amplitudes separately in order to define an “obscure qubit”
in the form of the “double superposition” (cf. (3.7), and an analog of qudit (2.1) is straightforward)
|Ψob =
A0 |0 + A1 |1
√
2
, (5.1)
where the two-dimensional “vectors”
A0,1 =
a0,1
α0,1
(5.2)
are the (double) “obscure-quantum amplitudes” of the generalized states |0 , |1 . For the conjugated obscure qubit
we put (informally)
Ψob| =
A⋆
0 0| + A⋆
1 1|
√
2
, (5.3)
where we denote A⋆
0,1 = a∗
0,1 α0,1 , such that A⋆
0,1A0,1 = |a0,1|2
+ α2
0,1. The (double) obscure qubit is “normalized”
in such a way that, if (3.13)–(3.14) hold then
Ψob | Ψob =
|a0|
2
+ |a1|
2
2
+
α2
0 + α2
1
2
= 1. (5.4)
The measurement should be made separately and independently in the “probability space” and the “membership
space” by using an analog of the Kronecker product. Indeed, we could think that in the vector representation (4.1)
for the quantum states and the direct product amplitudes (5.2) we should have
|Ψob (0) =
1
√
2
A0 ⊗K
1
0
+ A1 ⊗K
0
1
, (5.5)
where the (left) Kronecker product is defined by (see (4.1))
a
α
⊗K
c
d
=




a
c
d
α
c
d



 =
a (ce0 + de1)
α (ce0 + de1)
, e0 =
1
0
, e1 =
0
1
, e0,1 ∈ H(2)
q . (5.6)
Informally, the wave function of the obscure qubit, in the vector representation, now “lives” in the four-dimensional
space (5.6) which has two dimensional spaces as blocks. The upper block, a quantum subspace, is the ordinary Hilbert
space H
(2)
q , but the lower block, if it is to be treated as an obscure (membership) subspace V
(2)
memb can have special
properties. Thus, the four-dimensional space, where “lives” Ψ
(2)
ob is not an ordinary tensor product of vector spaces,
because of (5.6), where the “vector” A on the l.h.s. has entries of different natures, that is the quantum amplitudes a0,1

6. 6
and the membership amplitudes α0,1. Despite the unit vectors in H
(2)
q and V
(2)
memb having the same form (4.1), they
belong to different spaces (as vector spaces over different fields). So instead of (5.6) we introduce a “Kronecker-like
product” ˜⊗K by
a
α
˜⊗K
c
d
=
a (ce0 + de1)
α (cε0 + dε1)
, (5.7)
e0 =
1
0
, e1 =
0
1
, e0,1 ∈ H(2)
q , (5.8)
ε0 =
1
0
(µ)
, ε1 =
0
1
(µ)
, ε0,1 ∈ V
(2)
memb. (5.9)
In this way, the obscure qubit (5.1) can be presented in the from
|Ψob =
1
√
2




a0
1
0
α0
1
0
(µ)



 +
1
√
2




a1
0
1
α1
0
1
(µ)



 =
1
√
2
a0e0
α0ε0
+
1
√
2
a1e1
α1ε1
. (5.10)
Therefore, we call the double obscure qubit (5.10) a “Kronecker obscure qubit” to distinguish it from the obscure
qubit (3.7). It can be also presented using the Hadamard product (element-wise or Schur product)
a
α
⊗H
c
d
=
ac
αd
(5.11)
in the following form
|Ψob =
1
√
2
A0 ⊗H E0 +
1
√
2
A1 ⊗H E1, (5.12)
where the unit vectors of the total four-dimensional space are
E0,1 =
e0,1
ε0,1
∈ H(2)
q × V
(2)
memb. (5.13)
The probabilities p0,1 and membership functions µ0,1 of the states |0 and |1 are computed through the corre-
sponding amplitudes by (3.10) and (3.11)
pi = |ai|
2
, µi = Fµi (α0, α1) , i = 0, 1, (5.14)
and in the particular case by (3.12) satisfying (3.14).
As an example we consider the Kronecker obscure qubit (with a real quantum part) in which the probability and
the membership function (measure of “trust”) of the state |0 are p and µ, and of the state |1 they are 1 − p and
1 − µ, respectively. In the model (3.18)–(3.19) for µi (not Born-like) we obtain
|Ψob =
1
√
2





√
p
0
cos
π
2
µ
0
(µ)





+
1
√
2





0√
1 − p
0
sin
π
2
µ
(µ)





=
1
√
2
e0
√
p
ε0 cos
π
2
µ
+
1
√
2
e1
√
1 − p
ε1 sin
π
2
µ
, (5.15)
where ei and εi are unit vectors defined in (5.8) and (5.9).
This can be compared e.g. with the “classical-quantum” approach (3.22) and [35, 36].
VI. OBSCURE-QUANTUM MEASUREMENT
Let us consider the case of one Kronecker obscure qubit register L = 1 (see (5.5)), or using (5.6) in the vector
representation (5.10). The standard (double) orthogonal commuting projection operators, “Kronecker projections”
are (cf. (4.5))
P0 =
P0 0
0 P
(µ)
0
, P1 =
P1 0
0 P
(µ)
1
, (6.1)

7. 7
where 0 is the 2 × 2 zero matrix, and P
(µ)
0,1 are the projections in the membership subspace V
(2)
memb (of the same form
as the ordinary quantum projections P0,1 (4.5))
P
(µ)
0 =
1 0
0 0
(µ)
, P
(µ)
1 =
0 0
0 1
(µ)
, P
(µ)
0 , P
(µ)
1 ∈ End V
(2)
memb, (6.2)
P
(µ)2
0 = P
(µ)
0 , P
(µ)2
1 = P
(µ)
1 , P
(µ)
0 P
(µ)
1 = P
(µ)
1 P
(µ)
0 = 0. (6.3)
For the double projections we have (cf. (4.6))
P 2
0 = P0, P 2
1 = P1, P0P1 = P1P0 = 0, (6.4)
where 0 is the 4 × 4 zero matrix, and P0,1 act on the Kronecker qubit (6.1) in the standard way (cf. (4.7))
P0 |Ψob =
1
√
2




a0
1
0
α0
1
0
(µ)



 =
1
√
2
a0e0
α0ε0
=
1
√
2
A0 ⊗H E0, (6.5)
P1 |Ψob =
1
√
2




a1
0
1
α1
0
1
(µ)



 =
1
√
2
a1e1
α1ε1
=
1
√
2
A1 ⊗H E1. (6.6)
Observe that for Kronecker qubits there exist in addition to (6.1) the following orthogonal commuting projection
operators
P01 =
P0 0
0 P
(µ)
1
, P10 =
P1 0
0 P
(µ)
0
, (6.7)
and we call these the “crossed” double projections. They satisfy the same relations as (6.4)
P 2
01 = P01, P 2
10 = P10, P01P10 = P10P01 = 0, (6.8)
but act on the obscure qubit in a different (“mixing”) way than (6.5) i.e.
P01 |Ψob =
1
√
2




a0
1
0
α1
0
1



 =
1
√
2
a0e0
α1ε1
, P10 |Ψob =
1
√
2




a1
0
1
α0
1
0



 =
1
√
2
a1e1
α0ε0
. (6.9)
The multiplication of the crossed double projections (6.7) and the double projections (6.1) is given by
P01P0 = P0P01 =
P0 0
0 0
≡ Q0, P01P1 = P1P01 =
0 0
0 P
(µ)
1
≡ Q
(µ)
1 , (6.10)
P10P0 = P0P10 =
0 0
0 P
(µ)
0
≡ Q
(µ)
0 , P10P1 = P1P10 =
P1 0
0 0
≡ Q1, (6.11)
where the operators Q0, Q1 and Q
(µ)
0 , Q
(µ)
1 satisfy
Q2
0 = Q0, Q2
1 = Q1, Q1Q0 = Q0Q1 = 0, (6.12)
Q
(µ)2
0 = Q
(µ)
0 , Q
(µ)2
1 = Q
(µ)
1 , Q
(µ)
1 Q
(µ)
0 = Q
(µ)
0 Q
(µ)
1 = 0, (6.13)
Q
(µ)
1 Q0 = Q
(µ)
0 Q1 = Q1Q
(µ)
0 = Q0Q
(µ)
1 = 0, (6.14)
and we call them “half Kronecker (double) projections”.
The above relations mean that the process of measurement using Kronecker obscure qubits (quantum computation
with truth or membership) is more complicated than in the standard (only quantum) case.

8. 8
Indeed, let us calculate the “obscure” analogs of the expected values for the above projections using the notation
¯A ≡ Ψob| A |Ψob . (6.15)
Then, using (5.1)–(5.3) for the projection operators Pi, Pij, Qi, Q
(µ)
i , i, j = 0, 1, i = j, we obtain (cf. (4.8))
¯Pi =
|ai|
2
+ α2
i
2
, ¯Pij =
|ai|
2
+ α2
j
2
, (6.16)
¯Qi =
|ai|
2
2
, ¯Q
(µ)
i =
α2
i
2
. (6.17)
So follows the relation between the “obscure” analogs of expected values of the projections
¯Pi = ¯Qi + ¯Q
(µ)
i , ¯Pij = ¯Qi + ¯Q
(µ)
j . (6.18)
Another approach to various relations between the truth values and probabilities was given in [44].
Using the “ket” of the “bra” Kronecker qubit (5.10) in the form
Ψob| =
1
√
2
a∗
0 1 0 , α0 1 0 +
1
√
2
a∗
1 0 1 , α1 0 1 , (6.19)
a Kronecker (4 × 4) obscure analog of the density matrix (for a pure state) is (cf. (4.11))
ρ
(2)
ob = |Ψob Ψob| =
1
2




|a0|
2
a0a∗
1 a0α0 a0α1
a1a∗
0 |a1|
2
a1α0 a1α1
α0a∗
0 α0a∗
1 α2
0 α0α1
α1a∗
0 α1a∗
1 α0α1 α2
1



 . (6.20)
If (3.13)–(3.14) are satisfied, the density matrix (6.20) is non-invertible, because det ρ
(2)
ob = 0 and has unit trace
tr ρ
(2)
ob = 1, but is not idempotent ρ
(2)
ob
2
= ρ
(2)
ob (as is the quantum density matrix).
VII. KRONECKER OBSCURE-QUANTUM GATES
In general, (double) “obscure-quantum computation” with L Kronecker obscure qubits (or qudits) can be performed
by a product of unitary (block) matrices U of the (double size to the standard one) size 2× 2L
× 2L
(or 2× nL
× nL
),
U†
U = I (here I is the unit matrix of the same size as U). We can also call such computation a “quantum computation
with truth” (or with membership).
Let us consider obscure-quantum computation with one Kronecker obscure qubit. Informally, we can present the
Kronecker obscure qubit (5.10) in the form
|Ψob =




1√
2
a0
a1
1√
2
α0
α1
(µ)



 . (7.1)
Thus, the state |Ψob can be interpreted as a “vector” in the direct product (not tensor product) space H
(2)
q ×V
(2)
memb,
where H
(2)
q is the standard two-dimenional Hilbert space of the qubit, and V
(2)
memb can be treated as the “membership
space” which has a different nature from the qubit space and can have a more complex structure. In general, one
can consider the obscure-quantum computation as a set of abstract rules, even without the introduction of the
corresponding spaces.
An elementary transformation of the obscure qubit (7.1) is called an obscure-quantum gate and is performed by
the unitary (block) matrices of the size 4 × 4 (over C) acting in the total space H
(2)
q × V
(2)
memb
U =
U 0
0 U(µ) , UU†
= U†
U = I, (7.2)
UU†
= U†
U = I, U(µ)
U(µ)†
= U(µ)†
U(µ)
= I, U ∈ End H(2)
q , U(µ)
∈ End V
(2)
memb, (7.3)

9. 9
where I is a unit 4 × 4 matrix, I is a unit 2 × 2 matrix, U and U(µ)
are unitary 2 × 2 matrices acting on the probability
and membership “subspaces” correspondingly. The matrix U (over C) is called a quantum gate, and we call the
matrix U(µ)
(over R) an “obscure gate”. We assume that the obscure gates U(µ)
are of the same shape as the standard
quantum gates, but have real elements (see, e.g. [1]). In this case, the obscure-quantum gate is characterized by the
pair U, U(µ)
, where the components are known gates (in various combinations), e.g., one qubit gates: Hadamard,
Pauli-X (NOT),Y,Z (or two qubit gates CNOT, SWAP, etc.). Then the transformed qubit becomes (informally)
U |Ψob =




1√
2
U
a0
a1
1√
2
U(µ) α0
α1
(µ)



 . (7.4)
So for the block diaginal form (7.2), the quantum and the membership parts are transformed independently. Some
examples of the latter can be found, e.g., in [35, 36, 45], and the differences between the parts were mentioned in
[46]. In this case, an obscure-quantum network is “physically” a device performing consequent elementary operations
on obscure qubits (product of matrices), such that the quantum and membership parts are synchronized in time (for
the obscure part of such physical devices, see [18, 19, 47, 48]). Then, the result of the obscure-quantum computation
consists, together with the quantum probabilities of the states, in the “level of truth” for each of them.
For example, the obscure-quantum gate UH,NOT = {Hadamard, NOT} acts on the state E0 (5.13) as follows
UH,NOTE0 = UH,NOT




1
0
1
0
(µ)



 =




1√
2
1
1
0
1
(µ)



 =
1√
2
(e0 + e1)
ε1
. (7.5)
It would be interesting to introduce the case, when U (7.2) is not block diagonal and consider possible “physical”
interpretations of the non-diagonal blocks.
VIII. DOUBLE ENTANGLEMENT
Let us consider a register consisting of two obscure qubits L = 2
Ψ
(n=2)
ob (L = 2) = |Ψob (2) =
B00′ |00′
+ B10′ |10′
+ B01′ |01′
+ B11′ |11′
√
2
, (8.1)
in the computational basis ij′
= |i ⊗ |j′
, with the two-dimensional “vectors” (obscure-quantum amplitudes)
Bij′ =
bij′
βij′
, i, j = 0, 1, j′
= 0′
, 1′
, (8.2)
where bij′ ∈ C are probability amplitudes and βij′ ∈ R are the membership amplitudes of the corresponding (pure)
states. The normalization factor in (8.1) is chosen by analogy with (5.1) and (5.4) such that
Ψob (2) | Ψob (2) = 1, (8.3)
if (cf. (3.13)–(3.14))
|b00′ |
2
+ |b10′ |
2
+ |b01′ |
2
+ |b11′ |
2
= 1, (8.4)
β2
00′ + β2
10′ + β2
01′ + β2
11′ = 1. (8.5)
A state of two qubits is “entangled”, if it cannot be decomposed into a product of two one-qubit states, and in the
other case it is “separable” (see, e.g. [1]). We define a product of two obscure qubits (5.1) as
|Ψob ⊗ |Ψ′
ob =
A0 ⊗H A′
0 |00′
+ A1 ⊗H A′
0 |10′
+ A0 ⊗H A′
1 |01′
+ A1 ⊗H A′
1 |11′
2
, (8.6)

10. 10
where ⊗H is the Hadamard product (5.11). Comparing (8.1) and (8.6) we obtain two sets of relations, for probability
amplitudes and for membership amplitudes
bij′ =
1
√
2
aiaj′ , (8.7)
βij′ =
1
√
2
αiαj′ , i, j = 0, 1, j′
= 0′
, 1′
. (8.8)
In this case, the relations (3.13)–(3.14) give (8.4)–(8.5).
Two obscure-quantum qubits are entangled, if their joint state (8.1) cannot be presented as a product of one qubit
states (8.6), and in the opposite case the states are called totally separable. It follows from (8.7)-(8.8), that there are
two general conditions for obscure qubits to be entangled
b00′ b11′ = b10′ b01′ , or det b = 0, b =
b00′ b01′
b10′ b11′
, (8.9)
β00′ β11′ = β10′ β01′ , or det β = 0, β =
β00′ β01′
β10′ β11′
· (8.10)
The first equation (8.9) is the entanglement relation for the standard qubit, while the second condition (8.10) is for the
membership amplitudes of the two obscure qubit joint state (8.1). Presence of two conditions (8.9)–(8.10) leads to new
additional possibilities (not existing for the ordinary qubits) for “partial” entanglement (or “partial” separability),
when only one from them is fulfilled. In this case, the states can be entangled in one subspace but not entangled in
another subspace (quantum or membership one).
The measure of entanglement is numerically characterized by the concurrence. Taking into account the two condi-
tions (8.9)–(8.10), we propose to generalize the notion of concurrence for two obscure qubits in two ways. First, we
introduce the “vector obscure concurrence”
Cvect =
Cq
C(µ) = 2
|det b|
|det β|
, (8.11)
where b and β are defined in (8.9)–(8.10), and 0 ≤ Cq ≤ 1, 0 ≤ C(µ)
≤ 1. The corresponding “scalar obscure
concurrence” can be defined as
Cscal =
|det b|
2
+ |det β|
2
2
, (8.12)
such that 0 ≤ Cscal ≤ 1. Thus, two obscure qubits are totally separable, if Cscal = 0.
For instance, for an obscure analog of the (maximally entangled) Bell state
|Ψob (2) =
1
√
2
1√
2
1√
2
|00′
+
1√
2
1√
2
|11′
(8.13)
we obtain
Cvect =
1
1
, Cscal = 1. (8.14)
A more interesting example is the “intermediately entangled” two obscure qubit state, e.g.
|Ψob (2) =
1
√
2
1
2
1√
2
|00′
+
1
4
√
5
4
|10′
+
√
3
4
1
2
√
2
|01′
+
1√
2
1
4
|11′
, (8.15)
where the amplitudes satisfy (8.4)–(8.5). If the Born-like rule (as in (3.12)) holds for the membership amplitudes,
then the probabilities and membership functions of the states in (8.15) are
p00′ =
1
4
, p10′ =
1
16
, p01′ =
3
16
, p11′ =
1
2
, (8.16)
µ00′ =
1
2
, µ10′ =
5
16
, µ01′ =
1
8
, µ11′ =
1
16
. (8.17)

11. 11
This means that, e.g., the state |10′
will be measured with the quantum probability 1/16 and the membership
function (“truth” value) 5/16. For the entangled obscure qubit (8.15) we obtain the concurrences
Cvect =
1
2
√
2 − 1
8
√
3
1
8
√
2
√
5 − 1
4
√
2
=
0.491
0.042
, Cscal =
53
128
−
1
16
√
5 −
1
16
√
2
√
3 = 0.348. (8.18)
In the vector representation (5.7)–(5.10) we have
ij′
= |i ⊗ |j′
=
ei ⊗K ej′
εi ⊗K εj′
, i, j = 0, 1, j′
= 0′
, 1′
, (8.19)
where ⊗K is the Kronecker product (5.6), and ei, εi are defined in (5.8)–(5.9). Using (8.2) and the Kronecker-like
product (5.7), we put (informally, with no summation)
Bij′ ij′
=
bij′ ei ⊗K ej′
βij′ εi ⊗K εj′
, i, j = 0, 1, j′
= 0′
, 1′
. (8.20)
To clarify our model, we show here a manifest form of the two obscure qubit state (8.15) in the vector representation
|Ψob (2) =
1
√
2






















1
2



1
0
1
0



1√
2



1
0
1
0



(µ)











+











1
4



0
1
1
0



√
5
4



0
1
1
0



(µ)











+











√
3
4



1
0
0
1



1
2
√
2



1
0
0
1



(µ)











+











1√
2



0
1
0
1



1
4



0
1
0
1



(µ)






















. (8.21)
We call the above states “symmetric two obscure qubit states”. As it is seen from the r.h.s. of (8.20) and (8.21),
there are more general possibilities, when the indices of the first row and second row do not coincide. This can lead to
more possible states, which we call“non-symmetric two obscure qubit states”, and it would be worthwhile to establish
their “physical” interpretation and further applications in obscure-quantum computing.
IX. CONCLUSIONS
We have proposed a new scheme of quantum computation, in which each state is characterized by a “measure of
truth” to include vagueness into consideration. This is done by introducing the membership amplitude in addition
to the probability amplitude and the concept of the obscure qubit. We consider two kinds of these: the “product”
obscure qubit in which the total amplitude is a product of quantum amplitude and membership amplitude, and the
“Kronecker” obscure qubit, when the amplitudes are manipulated separately. In latter case, the quantum part of
the computation is based on the Hilbert space, while the “truth” part needs a vague/fuzzy set formalism, which
can be performed even without introducing an analog of a vector space. Thus, in general, we can consider obscure-
quantum computation as a set of rules (obscure quantum gates) for managing quantum and membership amplitudes
independently. As a result we obtain not only the probabilities of final states, but also their membership functions,
i.e. how much “trust” we should assign to these probabilities.
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