## 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

## 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

where

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

where

The dependence of the probabilities of the

while the form of

We impose the normalization conditions

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

The Born probabilities to observe the states

and the membership functions are

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

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

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

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 “

In the case where

Note that for complicated functions

such that

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

where

where

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

Another possible form of

So for positive

The equivalent membership functions for the outcome are

There are many different models for

## 4. Transformations of obscure qubits

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

are basis vectors of

In the vector representation, an obscure qubit differs from the standard qubit (8) by a

We call

if (15) holds.

Let us introduce the orthogonal commuting projection operators

where

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

We compute

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

with the obvious standard singularity property

## 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)

where the two-dimensional “vectors”

are the (double) “obscure-quantum amplitudes” of the generalized states

where we denote

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

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

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

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

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)

in the following form

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

The probabilities

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

where

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

where

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

where

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

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

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

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

where the operators

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

Then, using (43)–(45) for the projection operators

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

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

a Kronecker (

If the Born rule for the membership functions (13) and the conditions (14)–(15) are satisfied, the density matrix (78) is non-invertible, because

## 7. Kronecker obscure-quantum gates

In general, (double) “obscure-quantum computation” with

Let us consider obscure-quantum computation with one Kronecker obscure qubit. Informally, we can present the Kronecker obscure qubit (52) in the form

Thus, the state

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

where

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

It would be interesting to consider the case when

## 8. Double entanglement

Let us introduce a register consisting of two obscure qubits (

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

where

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

where

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

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”

where

such that

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

we obtain

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

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

This means that, e.g., the state

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

where

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

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”

## Acknowledgments

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