**2. Preliminaries**

To establish notation standard in the literature (see, e.g. [1, 2, 25–27]) we present the following definitions. In an underlying *d*-dimensional Hilbert space, the standard qudit (using the computational basis and Dirac notation) Hð Þ *<sup>d</sup> <sup>q</sup>* is given by

$$\left|\boldsymbol{\nu}^{(d)}\right\rangle = \sum\_{i=0}^{d-1} a\_i |i\rangle, \qquad a\_i \in \mathbb{C}, |i\rangle \in \mathcal{H}\_q^{(d)},\tag{1}$$

where *ai* is a probability amplitude of the state j i*i* . (For a review, see, e.g. [28, 29]) The probability *pi* to measure the *i*th state is *pi* ¼ *Fpi* ð Þ *a*1, … , *an* , 0≤ *pi* ≤1, 0≤*i*≤ *d* � 1. The shape of the functions *Fpi* is governed by the Born rule *Fpi* ð Þ¼ *<sup>a</sup>*1, … , *ad ai* j j<sup>2</sup> , and P*<sup>d</sup> <sup>i</sup>*¼<sup>0</sup>*pi* <sup>¼</sup> 1. A one-qudit (*<sup>L</sup>* <sup>¼</sup> 1) quantum gate is a unitary transformation *<sup>U</sup>*ð Þ *<sup>d</sup>* : <sup>H</sup>ð Þ *<sup>d</sup> <sup>q</sup>* ! Hð Þ *<sup>d</sup> <sup>q</sup>* described by unitary *d* � *d* complex matrices acting on the vector (1), and for a register containing *L* qudits quantum gates are unitary *dL* � *dL* matrices. The quantum circuit model [30, 31] forms the basis for the standard concept of quantum computing. Here the quantum algorithms are compiled as a sequence of elementary gates acting on a register containing *L* qubits (or qudits), followed by a measurement to yield the result [25, 32].

For further details on qudits and their transformations, see for example the reviews [28, 29] and the references therein.

### **3. Membership amplitudes**

We define an obscure qudit with *d* states via the following superposition (in place of that given in (1))

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

$$\left|\boldsymbol{\mu}\_{ab}^{(d)}\right\rangle = \sum\_{i=1}^{d-1} a\_i a\_i |i\rangle,\tag{2}$$

where *ai* is a (complex) probability amplitude *ai* ∈ , and we have introduced a (real) membership amplitude *αi*, with *α<sup>i</sup>* ∈½ � 0, 1 , 0≤ *i*≤*d* � 1. The probability *pi* to find the *i*th state upon measurement, and the membership function *μ<sup>i</sup>* ("of truth") for the *i*th state are both functions of the corresponding amplitudes as follows

$$p\_i = F\_{p\_i}(a\_0, \dots, a\_{d-1}), \qquad 0 \le p\_i \le 1,\tag{3}$$

$$
\mu\_i = \mathcal{F}\_{\mu\_i}(a\_0, \dots, a\_{d-1}), \qquad 0 \le \mu\_i \le 1. \tag{4}
$$

The dependence of the probabilities of the *i*th states upon the amplitudes, i.e. the form of the function *Fpi* is fixed by the Born rule

$$F\_{p\_i}(a\_1, \dots, a\_n) = \left| a\_i \right|^2,\tag{5}$$

while the form of *Fμ<sup>i</sup>* will vary according to different obscurity assumptions. In this paper we consider only real membership amplitudes and membership functions (complex obscure sets and numbers were considered in [33–35]). In this context the real functions *Fpi* and *Fμ<sup>i</sup>* , 0≤*i*≤ *d* � 1 will contain complete information about the obscure qudit (2).

We impose the normalization conditions

$$\sum\_{i=0}^{d-1} p\_i = \mathbf{1},\tag{6}$$

$$\sum\_{i=0}^{d-1} \mu\_i = \mathbf{1},\tag{7}$$

where the first condition is standard in quantum mechanics, while the second condition is taken to hold by analogy. Although (7) may not be satisfied, we will not consider that case.

For *d* ¼ 2, we obtain for the obscure qubit the general form (instead of that in (2))

$$
\left|\boldsymbol{\mu}\_{ab}^{(2)}\right\rangle = a\_0 a\_0 |\mathbf{0}\rangle + a\_1 a\_1 |\mathbf{1}\rangle,\tag{8}
$$

$$F\_{p\_0}(a\_0, a\_1) + F\_{p\_1}(a\_0, a\_1) = \mathbf{1},\tag{9}$$

$$F\_{\mu\_0}(a\_0, a\_1) + F\_{\mu\_1}(a\_0, a\_1) = \mathbf{1}.\tag{10}$$

The Born probabilities to observe the states 0j i and 1j i are

$$p\_0 = F\_{p\_0}^{Born}(a\_0, a\_1) = \left| a\_0 \right|^2, \qquad p\_1 = F\_{p\_1}^{Born}(a\_0, a\_1) = \left| a\_1 \right|^2,\tag{11}$$

and the membership functions are

$$
\mu\_0 = F\_{\mu\_0}(a\_0, a\_1), \qquad \mu\_1 = F\_{\mu\_1}(a\_0, a\_1). \tag{12}
$$

If we assume the Born rule (11) for the membership functions as well

$$F\_{\mu\_0}(a\_0, a\_1) = a\_0^2, \qquad F\_{\mu\_1}(a\_0, a\_1) = a\_1^2,\tag{13}$$

(which is one of various possibilities depending on the chosen model), then

$$\left|\left|\mathfrak{a}\_{0}\right|^{2} + \left|\mathfrak{a}\_{1}\right|^{2} = \mathbf{1},\tag{14}$$

$$a\_0^2 + a\_1^2 = 1.\tag{15}$$

Using (14)–(15) we can parametrize (8) as

$$\left|\psi\_{ab}^{(2)}\right\rangle = \cos\frac{\theta}{2}\cos\frac{\theta\_{\mu}}{2}|0\rangle + e^{i\varphi}\sin\frac{\theta}{2}\sin\frac{\theta\_{\mu}}{2}|1\rangle,\tag{16}$$

$$0 \le \theta \le \pi, \qquad 0 \le \rho \le 2\pi, \qquad 0 \le \theta\_{\mu} \le \pi. \tag{17}$$

Therefore, obscure qubits (with Born-like rule for the membership functions) are geometrically described by a pair of vectors, each inside a Bloch ball (and not as vectors on the boundary spheres, because "j j sin , cos j j≤1"), where one is for the probability amplitude (an ellipsoid inside the Bloch ball with *θμ* ¼ *const*1), and the other for the membership amplitude (which is reduced to an ellipse, being a slice inside the Bloch ball with *θ* ¼ *const*2, *φ* ¼ *const*3). The norm of the obscure qubits is not constant however, because

$$
\left\langle \psi\_{ob}^{(2)} \middle| \psi\_{ob}^{(2)} \right\rangle = \frac{1}{2} + \frac{1}{4} \cos \left( \theta + \theta\_{\mu} \right) + \frac{1}{4} \cos \left( \theta - \theta\_{\mu} \right). \tag{18}
$$

In the case where *<sup>θ</sup>* <sup>¼</sup> *θμ*, the norm (18) becomes 1 � <sup>1</sup> <sup>2</sup> sin <sup>2</sup> *θ*, reaching its minimum <sup>1</sup> <sup>2</sup> when *<sup>θ</sup>* <sup>¼</sup> *θμ* <sup>¼</sup> *<sup>π</sup>* 2.

Note that for complicated functions *Fμ*0,1 ð Þ *α*0, *α*<sup>1</sup> the condition (15) may be not satisfied, but the condition (7) should nevertheless always be valid. The concrete form of the functions *Fμ*0,1 ð Þ *α*0, *α*<sup>1</sup> depends upon the chosen model. In the simplest case, we can identify two arcs on the Bloch ellipse for *α*0, *α*<sup>1</sup> with the membership functions and obtain

$$F\_{\mu\_0}(a\_0, a\_1) = \frac{2}{\pi} \arctan \frac{a\_1}{a\_0},\tag{19}$$

$$F\_{\mu\_1}(a\_0, a\_1) = \frac{2}{\pi} \arctan \frac{a\_0}{a\_1},\tag{20}$$

such that *μ*<sup>0</sup> þ *μ*<sup>1</sup> ¼ 1, as in (7).

In [36, 37] a two stage special construction of quantum obscure/fuzzy sets was considered. The so-called classical-quantum obscure/fuzzy registers were introduced in the first step (for *n* ¼ 2, the minimal case) as

$$|s\rangle\_f = \sqrt{\mathbf{1} - f}|\mathbf{0}\rangle + \sqrt{f}|\mathbf{1}\rangle,\tag{21}$$

$$|\mathfrak{s}\rangle\_{\mathfrak{g}} = \sqrt{\mathbf{1} - \mathbf{g}}|\mathbf{0}\rangle + \sqrt{\mathbf{g}}|\mathbf{1}\rangle,\tag{22}$$

where *f*, *g* ∈½ � 0, 1 are the relevant classical-quantum membership functions. In the second step their quantum superposition is defined by

$$|\mathfrak{s}\rangle = \mathfrak{c}\_f |\mathfrak{s}\rangle\_f + \mathfrak{c}\_\mathfrak{g} |\mathfrak{s}\rangle\_\mathfrak{g},\tag{23}$$

where *cf* and *cg* are the probability amplitudes of the fuzzy states j i*s <sup>f</sup>* and j i*s <sup>g</sup>* , respectively. It can be seen that the state (23) is a particular case of (8) with

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

$$a\_0 a\_0 = c\_f \sqrt{\mathbf{1} - f} + c\_\mathbf{g} \sqrt{\mathbf{1} - \mathbf{g}},\tag{24}$$

$$a\_1 a\_1 = c\_f \sqrt{f} + c\_\mathfrak{g} \sqrt{\mathfrak{g}}.\tag{25}$$

This gives explicit connection of our double amplitude description of obscure qubits with the approach [36, 37] which uses probability amplitudes and the membership functions. It is important to note that the use of the membership amplitudes introduced here *α<sup>i</sup>* and (2) allows us to exploit the standard quantum gates, but not to define new special ones, as in [36, 37].

Another possible form of *F<sup>μ</sup>*0,1 ð Þ *α*0, *α*<sup>1</sup> (12), with the corresponding membership functions satisfying the standard fuzziness rules, can be found using a standard homeomorphism between the circle and the square. In [38, 39] this transformation was applied to the probability amplitudes *a*0,1, but here we exploit it for the membership amplitudes *α*0,1

$$F\_{\mu\_0}(a\_0, a\_1) = \frac{2}{\pi} \arcsin \sqrt{\frac{a\_0^2 \* \text{sign}\, a\_0 - a\_1^2 \* \text{sign}\, a\_1 + 1}{2}},\tag{26}$$

$$F\_{\mu\_1}(a\_0, a\_1) = \frac{2}{\pi} \arcsin \sqrt{\frac{a\_0^2 \* \text{sign}\, a\_0 + a\_1^2 \* \text{sign}\, a\_1 + 1}{2}}.\tag{27}$$

So for positive *α*0,1 we obtain (cf. [38])

$$F\_{\mu\_0}(a\_0, a\_1) = \frac{2}{\pi} \arcsin \sqrt{\frac{a\_0^2 - a\_1^2 + 1}{2}},\tag{28}$$

$$F\_{\mu\_1}(a\_0, a\_1) = \mathbf{1}.\tag{29}$$

The equivalent membership functions for the outcome are

$$\max\left(\min\left(F\_{\mu\_0}(a\_0, a\_1), \mathbf{1} - F\_{\mu\_1}(a\_0, a\_1)\right), \min\left(\mathbf{1} - F\_{\mu\_0}(a\_0, a\_1)\right), F\_{\mu\_1}(a\_0, a\_1)\right),\tag{30}$$

$$\min\left(\max\left(F\_{\mu\_0}(a\_0, a\_1), \mathbf{1} - F\_{\mu\_1}(a\_0, a\_1)\right), \max\left(\mathbf{1} - F\_{\mu\_0}(a\_0, a\_1)\right), F\_{\mu\_1}(a\_0, a\_1)\right). \tag{31}$$

There are many different models for *F<sup>μ</sup>*0,1 ð Þ *α*0, *α*<sup>1</sup> which can be introduced in such a way that they satisfy the obscure set axioms [7, 9].
