Open access peer-reviewed chapter

# Obscure Qubits and Membership Amplitudes

By Steven Duplij and Raimund Vogl

Reviewed: June 2nd 2021Published: July 8th 2021

DOI: 10.5772/intechopen.98685

## 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.

### Keywords

• qubit
• fuzzy
• membership function
• amplitude
• Hilbert space

## 1. Introduction

Nowadays, the development of quantum computing technique is governed by theoretical extensions of its ground concepts [1, 2, 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, 7, 8, 9]). Various interconnections between vagueness and quantum probability calculus were considered in [10, 11, 12, 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, 22, 23, 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 “obscure-quantum computations with truth” are proposed below: (1) A “Product” obscure qubit, in which the resulting amplitude is the product (in C) 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.

## 2. Preliminaries

To establish notation standard in the literature (see, e.g. [1, 2, 25, 26, 27]) we present the following definitions. In an underlying d-dimensional Hilbert space, the standard qudit (using the computational basis and Dirac notation) Hqdis given by

ψd=i=0d1aii,aiC,iHqd,E1

where aiis a probability amplitude of the state i. (For a review, see, e.g. [28, 29]) The probability pito measure the ith state is pi=Fpia1an, 0pi1, 0id1. The shape of the functions Fpiis governed by the Born rule Fpia1ad=ai2, and i=0dpi=1. A one-qudit (L=1) quantum gate is a unitary transformation Ud:HqdHqddescribed by unitary d×dcomplex matrices acting on the vector (1), and for a register containing Lqudits quantum gates are unitary dL×dLmatrices. 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 Lqubits (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 dstates via the following superposition (in place of that given in (1))

ψobd=i=1d1αiaii,E2

where aiis a (complex) probability amplitude aiC, and we have introduced a (real) membership amplitude αi, with αi01, 0id1. The probability pito find the ith state upon measurement, and the membership function μi(“of truth”)for the ith state are both functions of the corresponding amplitudes as follows

μi=Fμiα0αd1,0μi1.E4

The dependence of the probabilities of the ith states upon the amplitudes, i.e. the form of the function Fpiis fixed by the Born rule

Fpia1an=ai2,E5

while the form of Fμiwill 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, 34, 35]). In this context the real functions Fpiand Fμi, 0id1will contain complete information about the obscure qudit (2).

We impose the normalization conditions

i=0d1pi=1,E6
i=0d1μi=1,E7

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=2, we obtain for the obscure qubit the general form (instead of that in (2))

ψob2=α0a00+α1a11,E8
Fp0a0a1+Fp1a0a1=1,E9
Fμ0α0α1+Fμ1α0α1=1.E10

The Born probabilities to observe the states 0and 1are

p0=Fp0Borna0a1=a02,p1=Fp1Borna0a1=a12,E11

and the membership functions are

μ0=Fμ0α0α1,μ1=Fμ1α0α1.E12

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

Fμ0α0α1=α02,Fμ1α0α1=α12,E13

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

a02+a12=1,E14
α02+α12=1.E15

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

ψob2=cosθ2cosθμ20+esinθ2sinθμ21,E16
0θπ,0φ2π,0θμπ.E17

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 “sin,cos1”), where one is for the probability amplitude (an ellipsoid inside the Bloch ball with θμ=const1), and the other for the membership amplitude (which is reduced to an ellipse, being a slice inside the Bloch ball with θ=const2, φ=const3). The norm of the obscure qubits is not constant however, because

ψob2ψob2=12+14cosθ+θμ+14cosθθμ.E18

In the case where θ=θμ, the norm (18) becomes 112sin2θ, reaching its minimum 12when θ=θμ=π2.

Note that for complicated functions Fμ0,1α0α1the condition (15) may be not satisfied, but the condition (7) should nevertheless always be valid. The concrete form of the functions Fμ0,1α0α1depends upon the chosen model. In the simplest case, we can identify two arcs on the Bloch ellipse for α0,α1with the membership functions and obtain

Fμ0α0α1=2πarctanα1α0,E19
Fμ1α0α1=2πarctanα0α1,E20

such that μ0+μ1=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=2, the minimal case) as

sf=1f0+f1,E21
sg=1g0+g1,E22

where f,g01are the relevant classical-quantum membership functions. In the second step their quantum superposition is defined by

s=cfsf+cgsg,E23

where cfand cgare the probability amplitudes of the fuzzy states sfand sg, respectively. It can be seen that the state (23) is a particular case of (8) with

α0a0=cf1f+cg1g,E24
α1a1=cff+cgg.E25

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 αiand (2) allows us to exploit the standard quantum gates, but not to define new special ones, as in [36, 37].

Another possible form of Fμ0,1α0α1(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 a0,1, but here we exploit it for the membership amplitudes α0,1

Fμ0α0α1=2πarcsinα02signα0α12signα1+12,E26
Fμ1α0α1=2πarcsinα02signα0+α12signα1+12.E27

So for positive α0,1we obtain (cf. [38])

Fμ0α0α1=2πarcsinα02α12+12,E28
Fμ1α0α1=1.E29

The equivalent membership functions for the outcome are

maxminFμ0α0α11Fμ1α0α1min1Fμ0α0α1Fμ1α0α1,E30
minmaxFμ0α0α11Fμ1α0α1max1Fμ0α0α1Fμ1α0α1.E31

There are many different models for Fμ0,1α0α1which can be introduced in such a way that they satisfy the obscure set axioms [7, 9].

## 4. Transformations of obscure qubits

Let us consider the obscure qubits in the vector representation, such that

0=10,1=01E32

are basis vectors of Hq2. Then a standard quantum computational process in the quantum register with Lobscure qubits (qudits (1)) is performed by sequences of unitary matrices Ûof size 2L×2L(nL×nL), ÛÛ=Î, which are called quantum gates (Îis the unit matrix). Thus, for one obscure qubit the quantum gates are 2×2unitary complex matrices.

In the vector representation, an obscure qubit differs from the standard qubit (8) by a 2×2invertible diagonal (not necessarily unitary) matrix

ψob2=M̂α0α1ψ2,E33
M̂α0α1=α000α1.E34

We call M̂α0α1a membership matrix which can optionally have the property

trM̂2=1,E35

if (15) holds.

Let us introduce the orthogonal commuting projection operators

P̂0=1000,P̂1=0001,E36
P̂02=P̂0,P̂12=P̂1,P̂0P̂1=P̂1P̂0=0̂,E37

where 0̂is the 2×2zero matrix. Well-known properties of the projections are that

P̂0ψ2=a00,P̂1ψ2=a10,E38
ψ2P̂0ψ2=a02,ψ2P̂1ψ2=a12.E39

Therefore, the membership matrix (34) can be defined as a linear combination of the projection operators with the membership amplitudes as coefficients

M̂α0α1=α0P̂0+α1P̂1.E40

We compute

M̂α0α1ψob2=α02a00+α12a11.E41

We can therefore treat the application of the membership matrix (33) as providing the origin of a reversible but non-unitary “obscure measurement” on the standard qubit to obtain an obscure qubit (cf. the “mirror measurement” [40, 41] and also the origin of ordinary qubit states on the fuzzy sphere [42]).

An obscure analog of the density operator (for a pure state) is the following form for the density matrix in the vector representation

ρob2=ψob2ψob2=α02a02α0a0α1a1α0a0α1a1α12a12E42

with the obvious standard singularity property detρob2=0. But trρob2=α02a02+α12a121, and here there is no idempotence ρob22ρob2, which distincts ρob2from the standard density operator.

## 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)

Ψob=Â00̂+Â11̂2,E43

where the two-dimensional “vectors”

Â0,1=a0,1α0,1E44

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

Ψob=Â00̂+Â11̂2,E45

where we denote Â0,1=a0,1α0,1, such that Â0,1Â0,1=a0,12+α0,12. The (double) obscure qubit is “normalized” in such a way that, if the conditions (14)(15) hold, then

ΨobΨob=a02+a122+α02+α122=1.E46

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

Ψob0=12Â0K10+Â1K01,E47

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

aαKcd=acdαcd=acê0+dê1αcê0+dê1,ê0=10,ê1=01,ê0,1Hq2.E48

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 Hq2, but the lower block should have special (fuzzy) properties, if it is treated as an obscure (membership) subspace Vmemb2. Thus, the four-dimensional space, where “lives” Ψob2, is not an ordinary tensor product of vector spaces, because of (48), and the “vector” Âon the l.h.s. has entries of different natures, that is the quantum amplitudes a0,1and the membership amplitudes α0,1. Despite the unit vectors in Hq2and Vmemb2having 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” ˜Kby

aα˜Kcd=acê0+dê1αcε0+dε1,E49
ê0=10,ê1=01,ê0,1Hq2,E50
ε0=10μ,ε1=01μ,ε0,1Vmemb2.E51

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

Ψob=12a010α010μ+12a101α101μ=12a0ê0α0ε0+12a1ê1α1ε1.E52

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)

aαHcd=acαdE53

in the following form

Ψob=12Â0HÊ0+12Â1HÊ1,E54

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

Ê0,1=ê0,1ε0,1Hq2×Vmemb2.E55

The probabilities p0,1and membership functions μ0,1of the states 0̂and 1̂are computed through the corresponding amplitudes by (11) and (12)

pi=ai2,μi=Fμiα0α1,i=0,1,E56

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

By way of example, consider a Kronecker obscure qubit (with a real quantum part) with probability pand membership function μ(measure of “trust”) of the state 0̂, and of the state 1̂given by 1pand 1μrespectively. In the model (19)(20) for μi(which is not Born-like) we obtain

Ψob=12p0cosπ2μ0μ+1201p0sinπ2μμ=12ê0pε0cosπ2μ+12ê11pε1sinπ2μ,E57

where êiand εiare 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=1(see (47)), or using (48) in the vector representation (52). The standard (double) orthogonal commuting projection operators, “Kronecker projections” are (cf. (36))

P0=P̂00̂0̂P̂0μ,P1=P̂10̂0̂P̂1μ,E58

where 0̂is the 2×2zero matrix, and P̂0,1μare the projections in the membership subspace Vmemb2(of the same form as the ordinary quantum projections P̂0,1(36))

P̂0μ=1000μ,P̂1μ=0001μ,P̂0μ,P̂1μEndVmemb2,E59
P̂0μ2=P̂0μ,P̂1μ2=P̂1μ,P̂0μP̂1μ=P̂1μP̂0μ=0̂.E60

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

P02=P0,P12=P1,P0P1=P1P0=0,E61

where 0is the 4×4zero matrix, and P0,1act on the Kronecker qubit (58) in the standard way (cf. (38))

P0Ψob=12a010α010μ=12a0ê0α0ε0=12Â0HÊ0,E62
P1Ψob=12a101α101μ=12a1ê1α1ε1=12Â1HÊ1.E63

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

P01=P̂00̂0̂P̂1μ,P10=P̂10̂0̂P̂0μ,E64

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

P012=P01,P102=P10,P01P10=P10P01=0,E65

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

P01Ψob=12a010α101=12a0ê0α1ε1,E66
P10Ψob=12a101α010=12a1ê1α0ε0.E67

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

P01P0=P0P01=P̂00̂0̂0̂Q0,P01P1=P1P01=0̂0̂0̂P̂1μQ1μ,E68
P10P0=P0P10=0̂0̂0̂P̂0μQ0μ,P10P1=P1P10=P̂10̂0̂0̂Q1,E69

where the operators Q0,Q1and Q0μ,Q1μsatisfy

Q02=Q0,Q12=Q1,Q1Q0=Q0Q1=0,E70
Q0μ2=Q0μ,Q1μ2=Q1μ,Q1μQ0μ=Q0μQ1μ=0,E71
Q1μQ0=Q0μQ1=Q1Q0μ=Q0Q1μ=0,E72

and we call these “half Kronecker (double) projections”.

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

A¯ΨobAΨob.E73

Then, using (43)(45) for the projection operators Pi, Pij, Qi, Qiμ, i,j=0,1, ij, we obtain (cf. (39))

P¯i=ai2+αi22,P¯ij=ai2+αj22,E74
Q¯i=ai22,Q¯iμ=αi22.E75

So follows the relation between the “obscure” analogs of expected values of the projections

P¯i=Q¯i+Q¯iμ,P¯ij=Q¯i+Q¯jμ.E76

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

Ψob=12a010,α010+12a101,α101,E77

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

ρob2=ΨobΨob=12a02a0a1a0α0a0α1a1a0a12a1α0a1α1α0a0α0a1α02α0α1α1a0α1a1α0α1α12.E78

If the Born rule for the membership functions (13) and the conditions (14)(15) are satisfied, the density matrix (78) is non-invertible, because detρob2=0and has unit trace trρob2=1, but is not idempotent ρob22ρob2(as it holds for the ordinary quantum density matrix [1]).

## 7. Kronecker obscure-quantum gates

In general, (double) “obscure-quantum computation” with LKronecker obscure qubits (or qudits) can be performed by a product of unitary (block) matrices Uof the (double size to the standard one) size 2×2L×2L(or 2×nL×nL), UU=I(here Iis 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

Ψob=12a0a112α0α1μ.E79

Thus, the state Ψobcan be interpreted as a “vector” in the direct product (not tensor product) space Hq2×Vmemb2, where Hq2is the standard two-dimenional Hilbert space of the qubit, and Vmemb2can 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 C) acting in the total space Hq2×Vmemb2

U=Û0̂0̂Ûμ,UU=UU=I,E80
ÛU=ÛÛ=Î,ÛμÛμ=ÛμÛμ=Î,ÛEndHq2,ÛμEndVmemb2,E81

where Iis the unit 4×4matrix, Îis the unit 2×2matrix, Ûand Ûμare unitary 2×2matrices acting on the probability and membership “subspaces” respectively. The matrix Û(over C) will be called a quantum gate, and we call the matrix Ûμ(over ) an “obscure gate”. We assume that the obscure gates Ûμare 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 ÛÛμ, 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)

UΨob=12Ûa0a112Ûμα0α1μ.E82

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 obscure-quantum 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 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 UĤ,NOT=HadamardNOTacts on the state Ê0(55) as follows

UĤ,NOTÊ0=UĤ,NOT1010μ=121101μ=12ê0+ê1ε1.E83

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.

## 8. Double entanglement

Let us introduce a register consisting of two obscure qubits (L=2) in the computational basis îj=îĵas follows

Ψobn=2L=2=Ψob2=B̂000̂0+B̂101̂0+B̂010̂1+B̂111̂12,E84

determined by two-dimensional “vectors” (encoding obscure-quantum amplitudes)

B̂ij=bijβij,i,j=0,1,j=0,1,E85

where bijCare probability amplitudes for a set of pure states and βijare the corresponding membership amplitudes. By analogy with (43) and (46) the normalization factor in (84) is chosen so that

Ψob2Ψob2=1,E86

if (cf. (14)(15))

b002+b102+b012+b112=1,E87
β002+β102+β012+β112=1.E88

A state of two qubits is “entangled”, if it cannot be decomposed as a product of two one-qubit states, and otherwise it is “separable” (see, e.g. [1]). We define a product of two obscure qubits (43) as

ΨobΨob=Â0HÂ00̂0+Â1HÂ01̂0+Â0HÂ10̂1+Â1HÂ11̂12,E89

where His the Hadamard product (53). Comparing (84) and (89) we obtain two sets of relations, for probability amplitudes and for membership amplitudes

bij=12aiaj',E90
βij=12αiαj,i,j=0,1,j=0,1.E91

In this case, the relations (14)(15) give (87)(88).

Two obscure-quantum qubits are entangled, if their joint state (84) cannot be presented as a product of one qubit states (89), and in the opposite case the states are called totally separable. It follows from (90)(91), that there are two general conditions for obscure qubits to be entangled

b00b11b10b01,ordetb0,b=b00b01b10b11,E92
β00β11β10β01,ordetβ0,β=β00β01β10β11E93

The first Eq. (92) is the entanglement relation for the standard qubit, while the second condition (93) is for the membership amplitudes of the two obscure qubit joint state (84). The presence of two different conditions (92)(93) leads to new additional possibilities (which do not exist for ordinary qubits) for “partial” entanglement (or “partial” separability), when only one of them is fulfilled. In this case, the states can be entangled in one subspace (quantum or membership) but not in the other.

The measure of entanglement is numerically characterized by the concurrence. Taking into account the two conditions (92)(93), we propose to generalize the notion of concurrence for two obscure qubits in two ways. First, we introduce the “vector obscure concurrence”

Ĉvect=CqCμ=2detbdetβ,E94

where band βare defined in (92)(93), and 0Cq1, 0Cμ1. The corresponding “scalar obscure concurrence” can be defined as

Cscal=detb2+detβ22,E95

such that 0Cscal1. Thus, two obscure qubits are totally separable, if Cscal=0.

For instance, for an obscure analog of the (maximally entangled) Bell state

Ψob2=1212120̂0'+12121̂1E96

we obtain

Ĉvect=11,Cscal=1.E97

A more interesting example is the “intermediately entangled” two obscure qubit state, e.g.

Ψob2=1212120̂0+14541̂0+341220̂1+12141̂1,E98

where the amplitudes satisfy (87)(88). If the Born-like rule (as in (13)) holds for the membership amplitudes, then the probabilities and membership functions of the states in (98) are

p00=14,p10=116,p01=316,p11=12,E99
μ00=12,μ10=516,μ01=18,μ11=116.E100

This means that, e.g., the state 1̂0will be measured with the quantum probability 1/16and the membership function (“truth” value) 5/16. For the entangled obscure qubit (98) we obtain the concurrences

Ĉvect=1221831825142=0.4910.042,Cscal=53128116511623=0.348.E101

In the vector representation (49)(52) we have

îj=îĵ=êiKêjεiKεj,i,j=0,1,j=0,1,E102

where Kis the Kronecker product (48), and êi,εiare defined in (50)(51). Using (85) and the Kronecker-like product (49), we put (informally, with no summation)

B̂ijîj=bijêiKêjβijεiKεj,i,j=0,1,j=0,1.E103

To clarify our model, we show here a manifest form of the two obscure qubit state (98) in the vector representation

Ψob2=12121010121010μ+140110540110μ+3410011221001μ+120101140101μ.E104

The states above may be called “symmetric two obscure qubit states”. However, there are more general possibilities, as may be seen from the r.h.s. of (103) and (104), when the indices of the first and second rows do not coincide. This would allow more possible states, which we call “non-symmetric two obscure qubit states”. It would be worthwhile to establish their possible physical interpretation.

The above constructions show that quantum computing using Kronecker obscure qubits can involve a rich structure of states, giving a more detailed description with additional variables reflecting vagueness.

## 9. Conclusions

We have proposed a new scheme for describing quantum computation bringing vagueness into consideration, in which each state is characterized by a “measure of truth” Ȧmembership amplitude is introduced in addition to the probability amplitude in order to achieve this, and we are led thereby to the concept of an obscure qubit. Two kinds of these are considered: the “product” obscure qubit, in which the total amplitude is the product of the quantum and membership amplitudes, and the “Kronecker” obscure qubit, where the amplitudes are manipulated separately. In latter case, the quantum part of the computation is based, as usual, in Hilbert space, while the “truth” part requires a vague/fuzzy set formalism, and this can be performed in the framework of a corresponding fuzzy space. Obscure-quantum computation may be considered as a set of rules (defining obscure-quantum gates) for managing quantum and membership amplitudes independently in different spaces. In this framework 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. Our approach considerably extends the theory of quantum computing by adding the logic part directly to the computation process. Future challenges could lie in the direction of development of the corresponding logic hardware in parallel with the quantum devices.

## Acknowledgments

The first author (S.D.) is deeply thankful to Geoffrey Hare and Mike Hewitt for thorough language checking.

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Steven Duplij and Raimund Vogl (July 8th 2021). Obscure Qubits and Membership Amplitudes, Topics on Quantum Information Science, Sergio Curilef and Angel Ricardo Plastino, IntechOpen, DOI: 10.5772/intechopen.98685. Available from:

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