**5. Kronecker obscure qubits**

We next introduce an analog of quantum superposition for membership amplitudes, called "obscure superposition" (cf. [43], and also [44]).

Quantum amplitudes and membership amplitudes will here be considered separately in order to define an "obscure qubit" taking the form of a "double superposition" (cf. (8), and a generalized analog for qudits (1) is straightforward)

$$|\Psi\_{ob}\rangle = \frac{\hat{A}\_0|\hat{\mathbf{0}}\rangle + \hat{A}\_1|\hat{\mathbf{1}}\rangle}{\sqrt{2}},\tag{43}$$

where the two-dimensional "vectors"

$$
\hat{A}\_{0,1} = \begin{bmatrix} a\_{0,1} \\ a\_{0,1} \end{bmatrix} \tag{44}
$$

are the (double) "obscure-quantum amplitudes" of the generalized states 0^ � � � , ^1 � � � . For the conjugate of an obscure qubit we put (informally)

$$
\langle \Psi\_{ab} \vert = \frac{\hat{A}\_0^\star \langle \hat{\mathbf{0}} \vert + \hat{A}\_1^\star \langle \hat{\mathbf{1}} \vert}{\sqrt{2}}, \tag{45}
$$

where we denote *<sup>A</sup>*^<sup>⋆</sup> 0,1 <sup>¼</sup> *<sup>a</sup>*<sup>∗</sup> 0,1 *<sup>α</sup>*0,1 � �, such that *<sup>A</sup>*^<sup>⋆</sup> 0,1*A*^ 0,1 <sup>¼</sup> j j *<sup>a</sup>*0,1 <sup>2</sup> <sup>þ</sup> *<sup>α</sup>*<sup>2</sup> 0,1. The (double) obscure qubit is "normalized" in such a way that, if the conditions (14)–(15) hold, then

$$
\langle \Psi\_{ab} | \Psi\_{ab} \rangle = \frac{\left| a\_0 \right|^2 + \left| a\_1 \right|^2}{2} + \frac{a\_0^2 + a\_1^2}{2} = \mathbf{1}. \tag{46}
$$

A measurement should be made separately and independently in the "probability space" and the "membership space" which can be represented by using an analog of the Kronecker product. Indeed, in the vector representation (32) for the quantum states and for the direct product amplitudes (44) we should have

$$|\Psi\_{ab}\rangle\_{(0)} = \frac{1}{\sqrt{2}}\hat{A}\_0 \otimes\_K \begin{pmatrix} 1\\0 \end{pmatrix} + \hat{A}\_1 \otimes\_K \begin{pmatrix} 0\\1 \end{pmatrix},\tag{47}$$

where the (left) Kronecker product is defined by (see (32))

$$
\begin{aligned}
\begin{bmatrix} a \\ a \end{bmatrix} \otimes\_K \begin{pmatrix} c \\ d \end{pmatrix} &= \begin{bmatrix} a \begin{pmatrix} c \\ d \end{pmatrix} \\ a \begin{pmatrix} c \\ d \end{pmatrix} \\ a \begin{pmatrix} c \\ d \end{pmatrix} \end{bmatrix} = \begin{bmatrix} a(c\hat{e}\_0 + d\hat{e}\_1) \\ a(c\hat{e}\_0 + d\hat{e}\_1) \end{bmatrix}, \\\ \begin{aligned} \dot{e}\_0 &= \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \ \dot{e}\_1 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \ \dot{e}\_{0,1} \in \mathcal{H}\_q^{(2)}. \end{aligned} \end{aligned} \tag{48}
$$

Informally, the wave function of the obscure qubit, in the vector representation, now "lives" in the four-dimensional space of (48) which has two two-dimensional spaces as blocks. The upper block, the quantum subspace, is the ordinary Hilbert space Hð Þ<sup>2</sup> *<sup>q</sup>* , but the lower block should have special (fuzzy) properties, if it is treated as an obscure (membership) subspace <sup>V</sup>ð Þ<sup>2</sup> *memb*. Thus, the four-dimensional space, where "lives" Ψð Þ<sup>2</sup> *ob* � � � E , is not an ordinary tensor product of vector spaces, because of (48), and the "vector" *A*^ on the l.h.s. has entries of different natures, that is the quantum amplitudes *a*0,1 and the membership amplitudes *α*0,1. Despite the unit vectors in Hð Þ<sup>2</sup> *<sup>q</sup>* and <sup>V</sup>ð Þ<sup>2</sup> *memb* having the same form (32), they belong to different spaces (as they are vector spaces over different fields). Therefore, instead of (48) we introduce a "Kronecker-like product" ⊗~ *<sup>K</sup>* by

$$
\begin{bmatrix} a \\ a \end{bmatrix} \tilde{\otimes}\_K \begin{pmatrix} c \\ d \end{pmatrix} = \begin{bmatrix} a(c\hat{e}\_0 + d\hat{e}\_1) \\ a(c e\_0 + d e\_1) \end{bmatrix}, \tag{49}
$$

$$
\hat{e}\_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad \hat{e}\_1 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \qquad \hat{e}\_{0,1} \in \mathcal{H}\_q^{(2)}, \tag{50}
$$

$$
\varepsilon\_0 = \begin{pmatrix} \mathbf{1} \\ \mathbf{0} \end{pmatrix}^{(\mu)}, \qquad \varepsilon\_1 = \begin{pmatrix} \mathbf{0} \\ \mathbf{1} \end{pmatrix}^{(\mu)}, \qquad \varepsilon\_{0,1} \in \mathcal{V}\_{mmb}^{(2)}.\tag{51}
$$

In this way, the obscure qubit (43) can be presented in the from

$$\begin{split} \left| \Psi\_{ab} \right\rangle &= \frac{1}{\sqrt{2}} \begin{bmatrix} a\_0 \begin{pmatrix} 1\\0 \end{pmatrix} \\ a\_0 \begin{pmatrix} 1\\0 \end{pmatrix}^{(\mu)} \end{bmatrix} + \frac{1}{\sqrt{2}} \begin{bmatrix} a\_1 \begin{pmatrix} 0\\1 \end{pmatrix} \\ a\_1 \begin{pmatrix} 0\\1 \end{pmatrix}^{(\mu)} \end{bmatrix} \\ &= \frac{1}{\sqrt{2}} \begin{bmatrix} a\_0 \dot{e}\_0\\a\_0 \varepsilon\_0 \end{bmatrix} + \frac{1}{\sqrt{2}} \begin{bmatrix} a\_1 \dot{e}\_1\\a\_1 \varepsilon\_1 \end{bmatrix}. \end{split} \tag{52}$$

Therefore, we call the double obscure qubit (52) a "Kronecker obscure qubit" to distinguish it from the obscure qubit (8). It can be also presented using the Hadamard product (the element-wise or Schur product)

$$
\begin{bmatrix} a \\ a \end{bmatrix} \otimes\_H \begin{pmatrix} c \\ d \end{pmatrix} = \begin{bmatrix} ac \\ ad \end{bmatrix} \tag{53}
$$

in the following form

$$|\Psi\_{ab}\rangle = \frac{1}{\sqrt{2}}\hat{A}\_0 \otimes\_H \hat{E}\_0 + \frac{1}{\sqrt{2}}\hat{A}\_1 \otimes\_H \hat{E}\_1,\tag{54}$$

where the unit vectors of the total four-dimensional space are

$$
\hat{E}\_{0,1} = \begin{bmatrix} \hat{e}\_{0,1} \\ \varepsilon\_{0,1} \end{bmatrix} \in \mathcal{H}\_q^{(2)} \times \mathcal{V}\_{mcmb}^{(2)}.\tag{55}
$$

The probabilities *<sup>p</sup>*0,1 and membership functions *<sup>μ</sup>*0,1 of the states <sup>0</sup>^ � � � and ^1 � � � are computed through the corresponding amplitudes by (11) and (12)

$$\left|p\_i = \left|a\_i\right|^2, \qquad \mu\_i = F\_{\mu\_i}(a\_0, a\_1), \qquad i = 0, 1,\tag{56}$$

and in the particular case by (13) satisfying (15).

*Obscure Qubits and Membership Amplitudes DOI: http://dx.doi.org/10.5772/intechopen.98685*

By way of example, consider a Kronecker obscure qubit (with a real quantum part) with probability *p* and membership function *μ* (measure of "trust") of the state 0^ � � � , and of the state ^1 � � � given by 1 � *p* and 1 � *μ* respectively. In the model (19)–(20) for *μ<sup>i</sup>* (which is not Born-like) we obtain

$$\begin{aligned} \left| \Psi\_{ob} \right\rangle &= \frac{1}{\sqrt{2}} \begin{bmatrix} \left( \sqrt{p} \\ 0 \right) \\ \left( \cos \frac{\pi}{2} \mu \right)^{(\mu)} \\ 0 \end{bmatrix} + \frac{1}{\sqrt{2}} \begin{bmatrix} 0 \\ \left( \sqrt{1-p} \right) \\ \left( \sqrt{1-p} \right) \\ \left( \sin \frac{\pi}{2} \mu \right)^{(\mu)} \end{bmatrix} \\ &= \frac{1}{\sqrt{2}} \begin{bmatrix} \left. \hat{e}\_{0} \sqrt{p} \\ \left. e\_{0} \cos \frac{\pi}{2} \mu \right. \end{bmatrix} + \frac{1}{\sqrt{2}} \begin{bmatrix} \left. \hat{e}\_{1} \sqrt{1-p} \\ \left. e\_{1} \sin \frac{\pi}{2} \mu \right. \end{bmatrix}, \end{aligned} \tag{57}$$

where ^*ei* and *ε<sup>i</sup>* are unit vectors defined in (50) and (51).

This can be compared e.g. with the "classical-quantum" approach (23) and [36, 37], in which the elements of columns are multiplied, while we consider them independently and separately.

### **6. Obscure-quantum measurement**

Let us consider the case of one Kronecker obscure qubit register *L* ¼ 1 (see (47)), or using (48) in the vector representation (52). The standard (double) orthogonal commuting projection operators, "Kronecker projections" are (cf. (36))

$$\mathbf{P}\_0 = \begin{bmatrix} \hat{P}\_0 & \hat{\mathbf{0}} \\ \hat{\mathbf{0}} & \hat{P}\_0^{(\mu)} \end{bmatrix}, \qquad \mathbf{P}\_1 = \begin{bmatrix} \hat{P}\_1 & \hat{\mathbf{0}} \\ \hat{\mathbf{0}} & \hat{P}\_1^{(\mu)} \end{bmatrix}, \tag{58}$$

where 0 is the 2 ^ � 2 zero matrix, and *<sup>P</sup>*^ð Þ *<sup>μ</sup>* 0,1 are the projections in the membership subspace <sup>V</sup>ð Þ<sup>2</sup> *memb* (of the same form as the ordinary quantum projections *<sup>P</sup>*^0,1 (36))

$$
\hat{P}\_0^{(\mu)} = \begin{pmatrix} \mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{pmatrix}^{(\mu)}, \qquad \hat{P}\_1^{(\mu)} = \begin{pmatrix} \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} \end{pmatrix}^{(\mu)}, \qquad \hat{P}\_0^{(\mu)}, \hat{P}\_1^{(\mu)} \in \mathbf{End}\mathcal{V}\_{memb}^{(2)}, \tag{59}
$$

$$
\hat{P}\_0^{(\mu)2} = \hat{P}\_0^{(\mu)}, \quad \hat{P}\_1^{(\mu)2} = \hat{P}\_1^{(\mu)}, \quad \hat{P}\_0^{(\mu)} \hat{P}\_1^{(\mu)} = \hat{P}\_1^{(\mu)} \hat{P}\_0^{(\mu)} = \hat{\mathbf{0}}.\tag{60}
$$

For the double projections we have (cf. (37))

$$P\_0^2 = P\_0, \; P\_1^2 = P\_1, \; P\_0 P\_1 = P\_1 P\_0 = \mathbf{0},\tag{61}$$

where **0** is the 4 � 4 zero matrix, and *P*0,1 act on the Kronecker qubit (58) in the standard way (cf. (38))

$$|\mathbf{P}\_0|\mathbf{V}\_{ab}\rangle = \frac{1}{\sqrt{2}} \begin{bmatrix} a\_0 \begin{pmatrix} 1\\0\\0 \end{pmatrix} \\ a\_0 \begin{pmatrix} 1\\0 \end{pmatrix}^{(\mu)} \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} a\_0 \hat{e}\_0\\a\_0 e\_0 \end{bmatrix} = \frac{1}{\sqrt{2}} \hat{A}\_0 \otimes\_H \hat{E}\_0,\tag{62}$$

$$\mathbf{P}\_{1}|\Psi\_{ab}\rangle = \frac{1}{\sqrt{2}} \begin{bmatrix} a\_{1} \begin{pmatrix} 0\\1 \end{pmatrix} \\\\ a\_{1} \begin{pmatrix} 0\\1 \end{pmatrix}^{(\mu)} \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} a\_{1}\hat{e}\_{1} \\\\ a\_{1}e\_{1} \end{bmatrix} = \frac{1}{\sqrt{2}}\hat{A}\_{1} \otimes\_{H} \hat{E}\_{1}. \tag{63}$$

Observe that for Kronecker qubits there exist in addition to (58) the following orthogonal commuting projection operators

$$\mathbf{P}\_{01} = \begin{bmatrix} \hat{P}\_0 & \hat{\mathbf{0}} \\ \hat{\mathbf{0}} & \hat{P}\_1^{(\mu)} \end{bmatrix}, \qquad \mathbf{P}\_{10} = \begin{bmatrix} \hat{P}\_1 & \hat{\mathbf{0}} \\ \hat{\mathbf{0}} & \hat{P}\_0^{(\mu)} \end{bmatrix}, \tag{64}$$

and we call these the "crossed" double projections. They satisfy the same relations as (61)

$$P\_{01}^2 = P\_{01}, \ P\_{10}^2 = P\_{10}, \ P\_{01} P\_{10} = P\_{10} P\_{01} = \mathbf{0},\tag{65}$$

but act on the obscure qubit in a different ("mixing") way than (62) i.e.

$$\mathbf{P}\_{01}|\Psi\_{ob}\rangle = \frac{1}{\sqrt{2}} \begin{bmatrix} a\_0 \begin{pmatrix} 1\\0 \end{pmatrix} \\ a\_1 \begin{pmatrix} 0\\1 \end{pmatrix} \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} a\_0 \dot{e}\_0\\a\_1 e\_1 \end{bmatrix},\tag{66}$$

$$\mathbf{P}\_{10}|\Psi\_{ab}\rangle = \frac{1}{\sqrt{2}} \begin{bmatrix} a\_1 \begin{pmatrix} 0\\1 \end{pmatrix} \\\\ a\_0 \begin{pmatrix} 1\\0 \end{pmatrix} \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} a\_1 \dot{e}\_1\\a\_0 e\_0 \end{bmatrix}.\tag{67}$$

The multiplication of the crossed double projections (64) and the double projections (58) is given by

$$\mathbf{P}\_{01}\mathbf{P}\_{0} = \mathbf{P}\_{0}\mathbf{P}\_{01} = \begin{bmatrix} \hat{\mathbf{P}}\_{0} & \hat{\mathbf{0}} \\ \hat{\mathbf{0}} & \hat{\mathbf{0}} \end{bmatrix} \equiv \mathbf{Q}\_{0}, \qquad \mathbf{P}\_{01}\mathbf{P}\_{1} = \mathbf{P}\_{1}\mathbf{P}\_{01} = \begin{bmatrix} \hat{\mathbf{0}} & \hat{\mathbf{0}} \\ \hat{\mathbf{0}} & \hat{P}\_{1}^{(\mu)} \end{bmatrix} \equiv \mathbf{Q}\_{1}^{(\mu)}, \tag{68}$$

$$\mathbf{P}\_{10}\mathbf{P}\_{0} = \mathbf{P}\_{0}\mathbf{P}\_{10} = \begin{bmatrix} \hat{\mathbf{0}} & \hat{\mathbf{0}} \\ \hat{\mathbf{0}} & \hat{P}\_{0}^{(\mu)} \end{bmatrix} \equiv \mathbf{Q}\_{0}^{(\mu)}, \qquad \mathbf{P}\_{10}\mathbf{P}\_{1} = \mathbf{P}\_{1}\mathbf{P}\_{10} = \begin{bmatrix} \hat{P}\_{1} & \hat{\mathbf{0}} \\ \hat{\mathbf{0}} & \hat{\mathbf{0}} \end{bmatrix} \equiv \mathbf{Q}\_{1}, \tag{69}$$

where the operators *<sup>Q</sup>*0, *<sup>Q</sup>*<sup>1</sup> and *<sup>Q</sup>*ð Þ *<sup>μ</sup>* <sup>0</sup> , *<sup>Q</sup>*ð Þ *<sup>μ</sup>* <sup>1</sup> satisfy

$$\mathbf{Q}\_0^2 = \mathbf{Q}\_0, \quad \mathbf{Q}\_1^2 = \mathbf{Q}\_1, \quad \mathbf{Q}\_1 \mathbf{Q}\_0 = \mathbf{Q}\_0 \mathbf{Q}\_1 = \mathbf{0},\tag{70}$$

$$\mathbf{Q}\_0^{(\mu)2} = \mathbf{Q}\_0^{(\mu)}, \quad \mathbf{Q}\_1^{(\mu)2} = \mathbf{Q}\_1^{(\mu)}, \quad \mathbf{Q}\_1^{(\mu)} \mathbf{Q}\_0^{(\mu)} = \mathbf{Q}\_0^{(\mu)} \mathbf{Q}\_1^{(\mu)} = \mathbf{0},\tag{71}$$

$$\mathbf{Q}\_1^{(\mu)} \mathbf{Q}\_0 = \mathbf{Q}\_0^{(\mu)} \mathbf{Q}\_1 = \mathbf{Q}\_1 \mathbf{Q}\_0^{(\mu)} = \mathbf{Q}\_0 \mathbf{Q}\_1^{(\mu)} = \mathbf{0},\tag{72}$$

and we call these "half Kronecker (double) projections".

*Obscure Qubits and Membership Amplitudes DOI: http://dx.doi.org/10.5772/intechopen.98685*

The relations above imply that the process of measurement when using Kronecker obscure qubits (i.e. for quantum computation with truth or membership) is more complicated than in the standard case.

To show this, let us calculate the "obscure" analogs of expected values for the projections above. Using the notation

$$\overline{\mathbf{A}} \equiv \langle \Psi\_{ob} | \mathbf{A} | \Psi\_{ob} \rangle. \tag{73}$$

Then, using (43)–(45) for the projection operators *Pi*, *Pij*, *Q<sup>i</sup>* , *Q*ð Þ *<sup>μ</sup> <sup>i</sup>* , *i*, *j* ¼ 0, 1, *i* ¼6 *j*, we obtain (cf. (39))

$$
\overline{P}\_i = \frac{\left|a\_i\right|^2 + a\_i^2}{2}, \qquad \overline{P}\_{ij} = \frac{\left|a\_i\right|^2 + a\_j^2}{2}, \tag{74}
$$

$$
\overline{\mathbf{Q}\_i} = \frac{\left| a\_i \right|^2}{2}, \qquad \qquad \qquad \overline{\mathbf{Q}\_i^{(\mu)}} = \frac{a\_i^2}{2}. \tag{75}
$$

So follows the relation between the "obscure" analogs of expected values of the projections

$$
\overline{\mathbf{P}}\_i = \overline{\mathbf{Q}}\_i + \overline{\mathbf{Q}}\_i^{(\mu)}, \qquad \overline{\mathbf{P}}\_{\overline{\mathbf{\cdot}}} = \overline{\mathbf{Q}}\_i + \overline{\mathbf{Q}}\_j^{(\mu)}.\tag{76}
$$

Taking the "ket" corresponding to the "bra" Kronecker qubit (52) in the form

$$\langle \Psi\_{ab} | = \frac{1}{\sqrt{2}} [a\_0^\*(\mathbf{1} \quad \mathbf{0}), \quad a\_0(\mathbf{1} \quad \mathbf{0})] + \frac{1}{\sqrt{2}} [a\_1^\*(\mathbf{0} \quad \mathbf{1}), \quad a\_1(\mathbf{0} \quad \mathbf{1})], \tag{77}$$

a Kronecker (4 � 4) obscure analog of the density matrix for a pure state is given by (cf. (42))

$$\begin{aligned} \left| \rho\_{ob}^{(2)} = |\Psi\_{ob}\rangle \langle \Psi\_{ob}| = \frac{1}{2} \begin{pmatrix} |a\_0|^2 & a\_0 a\_1^\* & a\_0 a\_0 & a\_0 a\_1 \\ a\_1 a\_0^\* & |a\_1|^2 & a\_1 a\_0 & a\_1 a\_1 \\ a\_0 a\_0^\* & a\_0 a\_1^\* & a\_0^2 & a\_0 a\_1 \\ a\_1 a\_0^\* & a\_1 a\_1^\* & a\_0 a\_1 & a\_1^2 \end{pmatrix} . \end{aligned} \tag{78}$$

If the Born rule for the membership functions (13) and the conditions (14)–(15) are satisfied, the density matrix (78) is non-invertible, because det *ρ* ð Þ2 *ob* ¼ 0 and has unit trace tr*ρ* ð Þ2 *ob* ¼ 1, but is not idempotent *ρ* ð Þ2 *ob* � �<sup>2</sup> 6¼ *ρ* ð Þ2 *ob* (as it holds for the ordinary quantum density matrix [1]).

### **7. 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 � <sup>2</sup>*<sup>L</sup>* � <sup>2</sup>*<sup>L</sup>* � � (or 2 � *<sup>n</sup><sup>L</sup>* � *nL* � �), *<sup>U</sup>*† *U* ¼ *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 (52) in the form

$$|\Psi\_{ab}\rangle = \begin{bmatrix} 1 \\ \frac{1}{\sqrt{2}} \begin{pmatrix} a\_0 \\ a\_1 \end{pmatrix} \\\\ \frac{1}{\sqrt{2}} \begin{pmatrix} a\_0 \\ a\_1 \end{pmatrix}^{(\mu)} \end{bmatrix}. \tag{79}$$

Thus, the state j i **Ψ***ob* can be interpreted as a "vector" in the direct product (not tensor product) space <sup>H</sup>ð Þ<sup>2</sup> *<sup>q</sup>* � Vð Þ<sup>2</sup> *memb*, where <sup>H</sup>ð Þ<sup>2</sup> *<sup>q</sup>* is the standard two-dimenional Hilbert space of the qubit, and <sup>V</sup>ð Þ<sup>2</sup> *memb* can be treated as the "membership space" which has a different nature from the qubit space and can have a more complex structure. For discussion of such spaces, see, e.g. [5, 6, 8, 9]. In general, one can consider obscure-quantum computation as a set of abstract computational rules, independently of the introduction of the corresponding spaces.

An obscure-quantum gate will be defined as an elementary transformation on an obscure qubit (79) and is performed by unitary (block) matrices of size 4 � 4 (over ) acting in the total space Hð Þ<sup>2</sup> *<sup>q</sup>* � Vð Þ<sup>2</sup> *memb*

$$\mathbf{U} = \begin{pmatrix} \hat{\mathbf{U}} & \hat{\mathbf{0}} \\ \hat{\mathbf{0}} & \hat{\mathbf{U}}^{(\mu)} \end{pmatrix}, \qquad \mathbf{U}\mathbf{U}^{\dagger} = \mathbf{U}^{\dagger}\mathbf{U} = \mathbf{I},\tag{80}$$

$$
\hat{\mathbf{U}}\mathbf{U}^{\dagger} = \hat{\mathbf{U}}^{\dagger}\hat{\mathbf{U}} = \hat{I}, \quad \hat{\mathbf{U}}^{(\mu)}\hat{\mathbf{U}}^{(\mu)\dagger} = \hat{\mathbf{U}}^{(\mu)\dagger}\hat{\mathbf{U}}^{(\mu)} = \hat{I}, \qquad \hat{\mathbf{U}} \in \text{End}\,\mathcal{H}\_{q}^{(2)}, \hat{\mathbf{U}}^{(\mu)} \in \text{End}\,\mathcal{V}\_{memb}^{(2)}, \tag{81}
$$

where *<sup>I</sup>* is the unit 4 � 4 matrix, ^*<sup>I</sup>* is the unit 2 � 2 matrix, *<sup>U</sup>*^ and *<sup>U</sup>*^ ð Þ *<sup>μ</sup>* are unitary 2 � 2 matrices acting on the probability and membership "subspaces" respectively. The matrix *<sup>U</sup>*^ (over ) will be called a quantum gate, and we call the matrix *<sup>U</sup>*^ ð Þ *<sup>μ</sup>* (over <sup>ℝ</sup>) an "obscure gate". We assume that the obscure gates *<sup>U</sup>*^ ð Þ *<sup>μ</sup>* are of the same shape as the standard quantum gates, but they act in the other (membership) space and have only real elements (see, e.g. [1]). In this case, an obscure-quantum gate is characterized by the pair *<sup>U</sup>*^ , *<sup>U</sup>*^ ð Þ *<sup>μ</sup>* n o, where the components are known gates (in various combinations), e.g., for one qubit gates: Hadamard, Pauli-X (NOT),Y,Z (or two qubit gates e.g. CNOT, SWAP, etc.). The transformed qubit then becomes (informally)

$$\mathbf{U}|\Psi\_{ab}\rangle = \begin{bmatrix} 1 \\ \frac{1}{\sqrt{2}} \hat{U} \begin{pmatrix} a\_0 \\ \\ a\_1 \end{pmatrix} \\\\ \frac{1}{\sqrt{2}} \hat{U}^{(\mu)} \begin{pmatrix} a\_0 \\\\ a\_1 \end{pmatrix}^{(\mu)} \end{bmatrix}.\tag{82}$$

Thus the quantum and the membership parts are transformed independently for the block diagonal form (80). Some examples of this can be found, e.g., in [36, 37, 45]. Differences between the parts were mentioned in [46]. In this case, an obscurequantum network is "physically" realised by a device performing elementary operations in sequence on obscure qubits (by a product of matrices), such that the quantum and membership parts are synchronized in time (for a discussion of the obscure part of such physical devices, see [19, 20, 47, 48]). Then, the result of

*Obscure Qubits and Membership Amplitudes DOI: http://dx.doi.org/10.5772/intechopen.98685*

the obscure-quantum computation consists of the quantum probabilities of the states together with the calculated "level of truth" for each of them (see, e.g. [18]).

For example, the obscure-quantum gate *UH*^ ,NOT ¼ f g Hadamard, NOT acts on the state *E*^<sup>0</sup> (55) as follows

$$\mathbf{U}\_{\hat{H}, \text{NOT}} \hat{\mathbf{E}}\_0 = \mathbf{U}\_{\hat{H}, \text{NOT}} \begin{bmatrix} \mathbf{1} \\ \mathbf{0} \\ \mathbf{0} \\ \mathbf{1} \\ \mathbf{0} \end{bmatrix} = \begin{bmatrix} \frac{1}{\sqrt{2}} \begin{pmatrix} \mathbf{1} \\ \mathbf{1} \end{pmatrix} \\\\ \begin{pmatrix} \mathbf{0} \\ \mathbf{0} \end{pmatrix}^{(\mu)} \end{bmatrix} = \begin{bmatrix} \frac{1}{\sqrt{2}} (\hat{e}\_0 + \hat{e}\_1) \\\\ \mathbf{e}\_1 \end{bmatrix}. \tag{83}$$

It would be interesting to consider the case when *U* (80) is not block diagonal and try to find possible "physical" interpretations of the non-diagonal blocks.
