A Rigorous Theory of Neutrosophic Quantum Channels

1. Introduction

Quantum information theory describes physical transformations of quantum states by linear maps on operator spaces. The standard mathematical model is a completely positive trace-preserving map, usually called a quantum channel [4,7]. If ${\mathcal{H}}_A$ and ${\mathcal{H}}_B$ are Hilbert spaces, a quantum channel has the form

$$\Phi :\mathcal{B}\left({\mathcal{H}}_A\right)⟶\mathcal{B}\left({\mathcal{H}}_B\right),$$

where $\Phi$ is linear, completely positive, and trace-preserving.

The Kraus representation theorem gives a fundamental description of such maps:

$$\Phi \left(\rho \right)=\sum _k{K}_k\rho K_k^{†},\sum _k{K}_k^{†}{K}_k=I.$$

This representation makes positivity and trace preservation particularly transparent [3,4,7].

On the other hand, neutrosophic theory was introduced as a framework for representing truth, indeterminacy, and falsity information. Single-valued neutrosophic sets provide a convenient finite-valued version in which

$$T,I,F\in [0,1]$$

are treated as separate components [5,6].

The purpose of this paper is to construct a mathematically rigorous interface between these two frameworks. We emphasize that the neutrosophic component is an additional information structure. It does not automatically replace the physical requirements of quantum mechanics.

Our main objectives are:

(i)

to define neutrosophic quantum states;

(ii)

to define neutrosophic quantum channels;

(iii)

to establish a Kraus representation for the quantum component;

(iv)

to define transformations of the neutrosophic component;

(v)

to study composition and tensor products;

(vi)

to formulate a Choi representation;

(vii)

to distinguish physical quantum operations from formal neutrosophic reweightings.

The framework developed below is therefore intended as a rigorous mathematical model rather than a claim that neutrosophic parameters constitute new physical observables.

2. Preliminaries on Quantum Channels

Let $\mathcal{H}$ be a finite-dimensional complex Hilbert space and let

$$\mathcal{B}\left(\mathcal{H}\right)$$

denote the algebra of linear operators on $\mathcal{H}$.

Definition 1.

A density operator on $\mathcal{H}$ is an operator $\rho \in \mathcal{B}\left(\mathcal{H}\right)$ satisfying

$$\rho \ge 0,Tr\left(\rho \right)=1.$$

The set of all density operators is denoted by

$$\mathcal{D}\left(\mathcal{H}\right).$$

Definition 2.

A linear map

$$\Phi :\mathcal{B}\left({\mathcal{H}}_A\right)⟶\mathcal{B}\left({\mathcal{H}}_B\right)$$

is positive if

$$X\ge 0⟹\Phi \left(X\right)\ge 0.$$

It is completely positive if, for every finite-dimensional auxiliary Hilbert space $\mathcal{K}$,

$${id}_{\mathcal{K}}\otimes \Phi$$

is positive.

Complete positivity is required because a quantum system may be entangled with an external system. Thus positivity must remain valid after adjoining an arbitrary identity channel [4,7].

Definition 3.

A linear map

$$\Phi :\mathcal{B}\left({\mathcal{H}}_A\right)⟶\mathcal{B}\left({\mathcal{H}}_B\right)$$

is trace-preserving if

$$Tr(\Phi (X\left)\right)=Tr\left(X\right)$$

for every $X\in \mathcal{B}\left({\mathcal{H}}_A\right)$. A completely positive trace-preserving map is called a quantum channel.

Theorem 1

(Kraus representation). A linear map Φ is completely positive and trace-preserving if and only if there exist operators

$$K_1,\dots ,K_r:{\mathcal{H}}_A⟶{\mathcal{H}}_B$$

such that

$$\Phi \left(X\right)={\displaystyle \sum _{k=1}^r}{K}_{k}X{K}_k^{†}$$

and

$${\displaystyle \sum _{k=1}^r}{K}_k^{†}{K}_k=I_{{\mathcal{H}}_A}.$$

The Kraus representation is fundamental in quantum information theory [3,4,7].

3. Single-Valued Neutrosophic Information

We use the single-valued version of neutrosophic information.

Definition 4.

A single-valued neutrosophic triple is a vector

$$\mathcal{N}=(T,I,F)\in {[0,1]}^3,$$

where T, I, and F denote, respectively, truth, indeterminacy, and falsity degrees.

Unlike probability vectors, the components need not satisfy

$$T+I+F=1.$$

In the standard single-valued neutrosophic framework, the three components are allowed to represent distinct information dimensions [6].

We shall therefore define the neutrosophic state space by

$$\mathfrak{N}={[0,1]}^3.$$

Definition 5.

A neutrosophic transformation is a map

$${\Phi }_{\mathcal{N}}:\mathfrak{N}⟶\mathfrak{N}.$$

It is called affine if

$${\Phi }_{\mathcal{N}}(\lambda {\mathcal{N}}_1+(1-\lambda ){\mathcal{N}}_2)=\lambda {\Phi }_{\mathcal{N}}\left({\mathcal{N}}_1\right)+(1-\lambda ){\Phi }_{\mathcal{N}}\left({\mathcal{N}}_2\right)$$

for $0\le \lambda \le 1$, whenever the convex combination belongs to $\mathfrak{N}$.

A simple example is the componentwise transformation

$${\Phi }_{\mathcal{N}}(T,I,F)=(f_T\left(T\right),f_I\left(I\right),f_F\left(F\right)),$$

where each $f_{\alpha }:[0,1]\to [0,1]$.

4. Neutrosophic Quantum States

We now combine a density operator with a neutrosophic information triple.

Definition 6.

A neutrosophic quantum state is an ordered pair

$$(\rho ,\mathcal{N}),$$

where

$$\rho \in \mathcal{D}\left(\mathcal{H}\right)$$

is a quantum density operator and

$$\mathcal{N}=(T,I,F)\in {[0,1]}^3$$

is its associated neutrosophic information state.

The set of neutrosophic quantum states is denoted by

$${\mathcal{D}}_{\mathcal{N}}\left(\mathcal{H}\right)=\mathcal{D}\left(\mathcal{H}\right)\times {[0,1]}^3.$$

Remark 1.

The neutrosophic triple is not itself a density matrix. In particular, the condition

$$T+I+F=1$$

is not imposed unless a particular application explicitly requires such a normalization. Consequently, $(T,I,F)$ should not be interpreted as a probability distribution without additional assumptions.

5. Definition of a Neutrosophic Quantum Channel

We can now give the central definition.

Definition 7.

A neutrosophic quantum channel from ${\mathcal{H}}_A$ to ${\mathcal{H}}_B$ is a pair

$$\Phi =(\Phi ,{\Phi }_{\mathcal{N}}),$$

where

$$\Phi :\mathcal{B}\left({\mathcal{H}}_A\right)⟶\mathcal{B}\left({\mathcal{H}}_B\right)$$

is a quantum channel and

$${\Phi }_{\mathcal{N}}:{[0,1]}^3⟶{[0,1]}^3$$

is a neutrosophic transformation.

The action of $\mathbf{\Phi }$ on a neutrosophic quantum state is defined by

$$\mathbf{\Phi }(\rho ,\mathcal{N})=\left(\Phi \left(\rho \right),{\Phi }_{\mathcal{N}}\left(\mathcal{N}\right)\right).$$

If

$$\mathcal{N}=(T,I,F),$$

we write

$${\Phi }_{\mathcal{N}}\left(\mathcal{N}\right)=(T^{\prime },I^{\prime },F^{\prime }).$$

Thus,

$$(\rho ,T,I,F)⟼\left(\Phi \left(\rho \right),T^{\prime },I^{\prime },F^{\prime }\right).$$

Remark 2.

The physical quantum evolution is carried by Φ, whereas ${\Phi }_{\mathcal{N}}$ describes the transformation of the associated neutrosophic information. This separation prevents an arbitrary neutrosophic scalar from being incorrectly identified with a physical quantum operation.

6. Kraus Representation

Suppose

$$\Phi \left(\rho \right)={\displaystyle \sum _{k=1}^r}{K}_k\rho K_k^{†},$$

where

$${\displaystyle \sum _{k=1}^r}{K}_k^{†}{K}_k=I.$$

Then a neutrosophic quantum channel has the form

$$\mathbf{\Phi }(\rho ,\mathcal{N})=\left({\displaystyle \sum _{k=1}^r}{K}_k\rho K_k^{†},{\Phi }_{\mathcal{N}}\left(\mathcal{N}\right)\right).$$

Proposition 1.

Let

$$\mathbf{\Phi }=(\Phi ,{\Phi }_{\mathcal{N}})$$

be a neutrosophic quantum channel. If

$$\Phi \left(\rho \right)={\displaystyle \sum _{k=1}^r}{K}_k\rho K_k^{†},$$

then the quantum component preserves positivity and trace, while the neutrosophic component remains in ${[0,1]}^3$ whenever

$${\Phi }_{\mathcal{N}}\left({[0,1]}^3\right)\subseteq {[0,1]}^3.$$

Proof. 

Since $\Phi$ is completely positive, each positive operator $\rho$ is mapped to a positive operator:

$$\rho \ge 0⟹\Phi \left(\rho \right)\ge 0.$$

Furthermore,

$$\begin{array}{cc}\hfill Tr(\Phi (\rho \left)\right)& =Tr\left(\sum _k{K}_k\rho K_k^{†}\right)\hfill \\ & =\sum _{k}Tr\left(K_k^{†}{K}_k\rho \right)\hfill \\ & =Tr\left(\left(\sum _k{K}_k^{†}{K}_k\right)\rho \right)\hfill \\ & =Tr\left(\rho \right).\hfill \end{array}$$

The neutrosophic component remains in ${[0,1]}^3$ by the assumed range condition on ${\Phi }_{\mathcal{N}}$. □

7. Composition of Neutrosophic Quantum Channels

Let

$$\mathbf{\Phi }=(\Phi ,{\Phi }_{\mathcal{N}})$$

and

$$\mathbf{\Psi }=(\Psi ,{\Psi }_{\mathcal{N}})$$

be neutrosophic quantum channels.

We define their composition by

$$\mathbf{\Psi }\circ \mathbf{\Phi }=\left(\Psi \circ \Phi ,{\Psi }_{\mathcal{N}}\circ {\Phi }_{\mathcal{N}}\right).$$

Theorem 2.

The composition of two neutrosophic quantum channels is again a neutrosophic quantum channel.

Proof. 

The composition of completely positive maps is completely positive, and the composition of trace-preserving maps is trace-preserving. Hence

$$\Psi \circ \Phi$$

is a quantum channel.

Moreover,

$${\Phi }_{\mathcal{N}}\left({[0,1]}^3\right)\subseteq {[0,1]}^3$$

and

$${\Psi }_{\mathcal{N}}\left({[0,1]}^3\right)\subseteq {[0,1]}^3.$$

Therefore

$$({\Psi }_{\mathcal{N}}\circ {\Phi }_{\mathcal{N}})\left({[0,1]}^3\right)\subseteq {[0,1]}^3.$$

Thus the pair

$$\left(\Psi \circ \Phi ,{\Psi }_{\mathcal{N}}\circ {\Phi }_{\mathcal{N}}\right)$$

is a neutrosophic quantum channel. □

8. Identity Channel

The identity neutrosophic quantum channel is

$$\mathbf{I}=(id,{id}_{\mathcal{N}}),$$

where

$$id\left(\rho \right)=\rho$$

and

$${id}_{\mathcal{N}}(T,I,F)=(T,I,F).$$

Consequently,

$$\mathbf{I}(\rho ,\mathcal{N})=(\rho ,\mathcal{N}).$$

Proposition 2.

For every neutrosophic quantum channel Φ,

$$\mathbf{\Phi }\circ \mathbf{I}=\mathbf{I}\circ \mathbf{\Phi }=\mathbf{\Phi }.$$

9. Tensor Products

Let

$${\mathbf{\Phi }}_A=({\Phi }_A,{\Phi }_{\mathcal{N},A})$$

and

$${\mathbf{\Phi }}_B=({\Phi }_B,{\Phi }_{\mathcal{N},B})$$

be neutrosophic quantum channels.

The quantum tensor product is

$${\Phi }_A\otimes {\Phi }_B.$$

For the neutrosophic component, we define

$${\Phi }_{\mathcal{N},A}{\otimes }_{\mathcal{N}}{\Phi }_{\mathcal{N},B}$$

according to a specified neutrosophic composition rule.

Thus,

$${\mathbf{\Phi }}_A{\otimes }_{\mathcal{N}Q}{\mathbf{\Phi }}_B=\left({\Phi }_A\otimes {\Phi }_B,{\Phi }_{\mathcal{N},A}{\otimes }_{\mathcal{N}}{\Phi }_{\mathcal{N},B}\right).$$

For example, if

$${\mathcal{N}}_A=(T_A,I_A,F_A)$$

and

$${\mathcal{N}}_B=(T_B,I_B,F_B),$$

one possible componentwise product is

$${\mathcal{N}}_A\odot {\mathcal{N}}_B=(T_A{T}_B,I_A{I}_B,F_A{F}_B).$$

Other neutrosophic product rules may be used, provided that they map ${[0,1]}^3\times {[0,1]}^3$ into ${[0,1]}^3$.

10. Choi Representation

The Choi representation provides an alternative characterization of complete positivity [1,2].

Let

$${\mathcal{H}}_A=span\{\left|1\right.〉,\dots ,\left|d_A\right.〉\}.$$

The Choi matrix of $\Phi$ is defined by

$$J(\Phi )={\displaystyle \sum _{i,j=1}^{d_A}}\left|i\right.〉\left.〈j\right|\otimes \Phi \left(\left|i\right.〉\left.〈j\right|\right).$$

Theorem 3

(Choi). A linear map Φ is completely positive if and only if

$$J(\Phi )\ge 0.$$

Furthermore, Φ is trace-preserving if and only if

$${Tr}_B\left(J(\Phi )\right)=I_A.$$

Therefore, the quantum component of a neutrosophic quantum channel can be characterized by

$$J(\Phi )\ge 0,{Tr}_B\left(J(\Phi )\right)=I_A.$$

The complete neutrosophic channel is then represented by

$$\mathbf{J}(\Phi )=\left(J(\Phi ),{\Phi }_{\mathcal{N}}\right).$$

This representation separates the operator-theoretic structure from the neutrosophic information transformation.

11. Neutrosophic Choi Data

Let

$${\mathcal{N}}_{\Phi }=(T_{\Phi },I_{\Phi },F_{\Phi })$$

be the neutrosophic information associated with a channel.

We define the neutrosophic Choi datum by

$${\mathfrak{C}}_{\mathcal{N}}(\mathbf{\Phi })=\left(J(\Phi ),T_{\Phi },I_{\Phi },F_{\Phi }\right).$$

Definition 8.

A neutrosophic Choi datum is a quadruple

$$\left(J,T,I,F\right)$$

such that

$$J\ge 0,{Tr}_B\left(J\right)=I_A,(T,I,F)\in {[0,1]}^3.$$

This gives a direct criterion for the quantum part and a separate admissibility condition for the neutrosophic part.

12. Measurement Through a Neutrosophic Channel

Let

$${\left\{M_m\right\}}_{m\in \mathcal{M}}$$

be a quantum measurement satisfying

$$\sum _{m\in \mathcal{M}}{M}_m^{†}{M}_m=I.$$

For a conventional quantum state $\rho$, the probability of outcome m is

$$p\left(m\right)=Tr\left(M_m^{†}{M}_m\rho \right).$$

If

$$\rho =\left|\psi \right.〉\left.〈\psi \right|,$$

then

$$p\left(m\right)=\left.〈\psi \right|M_m^{†}{M}_m\left|\psi \right.〉.$$

For a neutrosophic quantum state

$$(\rho ,\mathcal{N}),$$

we retain the conventional Born probability

$$p(m\mid \rho )=Tr\left(M_m^{†}{M}_m\rho \right)$$

and associate with the outcome the transformed neutrosophic information

$${\mathcal{N}}^{\prime }={\Phi }_{\mathcal{N}}\left(\mathcal{N}\right).$$

Thus, the measurement output may be represented as

$$\left(m,p(m\mid \rho ),{\mathcal{N}}^{\prime }\right).$$

Remark 3.

The neutrosophic triple is not automatically a probability distribution over measurement outcomes. The quantities T, I, and F describe an additional information layer. The normalization of the Born probabilities is governed by the measurement completeness relation.

13. Examples of Neutrosophic Quantum Channels

13.1. Unitary neutrosophic channel

Let U be a unitary operator:

$$U^{†}U=U{U}^{†}=I.$$

The corresponding quantum channel is

$${\Phi }_U\left(\rho \right)=U\rho U^{†}.$$

Let

$${\Phi }_{\mathcal{N},U}:{[0,1]}^3\to {[0,1]}^3$$

be a neutrosophic transformation. Then

$${\mathbf{\Phi }}_U=({\Phi }_U,{\Phi }_{\mathcal{N},U})$$

is a neutrosophic quantum channel.

For a pure state,

$$\left|\psi \right.〉⟼U\left|\psi \right.〉,$$

while

$$\mathcal{N}⟼{\Phi }_{\mathcal{N},U}\left(\mathcal{N}\right).$$

13.2. Depolarizing channel

For a d-dimensional system, consider

$${\Phi }_p\left(\rho \right)=(1-p)\rho +p{\displaystyle \frac{I}{d}},0\le p\le 1.$$

This is a quantum channel. A corresponding neutrosophic channel is

$${\mathbf{\Phi }}_p=({\Phi }_p,{\Phi }_{\mathcal{N},p}),$$

where

$${\Phi }_{\mathcal{N},p}:{[0,1]}^3\to {[0,1]}^3.$$

For example, one may choose

$${\Phi }_{\mathcal{N},p}(T,I,F)=\left((1-p)T,I+p(1-I),(1-p)F\right),$$

provided that the resulting triple remains in ${[0,1]}^3$.

The transformation increases the indeterminacy component according to

$$I^{\prime }=I+p(1-I),$$

and hence

$$I^{\prime }\ge I.$$

13.3. Bit-flip channel

The bit-flip channel is

$${\Phi }_p\left(\rho \right)=(1-p)\rho +pX\rho X,$$

where

$$X=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right).$$

A neutrosophic extension is

$${\mathbf{\Phi }}_p=({\Phi }_p,{\Phi }_{\mathcal{N},p}).$$

14. Complementary Channels

The Stinespring representation of a quantum channel provides an important connection with an environment. A channel can be written in the form

$$\Phi \left(\rho \right)={Tr}_E\left[V\rho V^{†}\right],$$

where

$$V:{\mathcal{H}}_A\to {\mathcal{H}}_B\otimes {\mathcal{H}}_E$$

is an isometry satisfying

$$V^{†}V=I_A.$$

The complementary channel is

$${\Phi }^c\left(\rho \right)={Tr}_B\left[V\rho V^{†}\right].$$

We may augment this structure by a neutrosophic transformation

$${\Phi }_{\mathcal{N}}^c:{[0,1]}^3\to {[0,1]}^3.$$

Hence the complementary neutrosophic channel is

$${\mathbf{\Phi }}^c=({\Phi }^c,{\Phi }_{\mathcal{N}}^c).$$

Remark 4.

The neutrosophic complementary component may be interpreted as information assigned to the environment, but such an interpretation requires an application-specific definition of the map ${\Phi }_{\mathcal{N}}^c$. It does not follow solely from the mathematical definition of a quantum complementary channel.

15. Information-Weighted Channels

One may wish to associate a scalar score with a neutrosophic triple. For example, define

$$S(T,I,F)={\displaystyle \frac{T}{T+I+F}},$$

whenever

$$T+I+F>0.$$

For

$$(T,I,F)=(0.95,0.03,0.02),$$

we obtain

$$S(T,I,F)={\displaystyle \frac{0.95}{0.95+0.03+0.02}}=0.95.$$

Such a score can be used to describe information quality, but it should not automatically be inserted as a multiplicative factor into a quantum density operator.

Indeed, suppose one formally defines

$$\tilde{\rho }=S(T,I,F)\rho .$$

Then

$$Tr\left(\tilde{\rho }\right)=S(T,I,F)Tr\left(\rho \right)=S(T,I,F),$$

which is generally not equal to 1.

Therefore, $\tilde{\rho }$ is not a density operator unless

$$S(T,I,F)=1.$$

This observation is important for a rigorous theory.

16. Normalization and Physical Admissibility

Proposition 3.

Let ρ be a density operator and let $s\ge 0$. Then

$$s\rho$$

is a density operator if and only if

$$s=1.$$

Proof. 

Since $\rho \ge 0$, we have

$$s\rho \ge 0.$$

However,

$$Tr\left(s\rho \right)=sTr\left(\rho \right)=s.$$

The trace equals 1 if and only if

$$s=1.$$

□

Thus an information score may be attached to a state without necessarily multiplying the physical density operator by that score.

A physically meaningful construction can instead be written as

$$(\rho ,\mathcal{N})⟼\left(\Phi \left(\rho \right),{\Phi }_{\mathcal{N}}\left(\mathcal{N}\right)\right).$$

17. A General Affine Neutrosophic Transformation

Consider an affine map

$${\Phi }_{\mathcal{N}}\left(\mathcal{N}\right)=A\mathcal{N}+b,$$

where

$$\mathcal{N}=\left(\begin{array}{c}T\\ I\\ F\end{array}\right),$$

$A\in {\mathbb{R}}^{3\times 3}$, and $b\in {\mathbb{R}}^3$.

Write

$$A=\left(\begin{array}{ccc}{a}_{TT}& a_{TI}& a_{TF}\\ a_{IT}& a_{II}& a_{IF}\\ a_{FT}& a_{FI}& a_{FF}\end{array}\right).$$

Then

$$\left(\begin{array}{c}{T}^{\prime }\\ I^{\prime }\\ F^{\prime }\end{array}\right)=\left(\begin{array}{ccc}{a}_{TT}& a_{TI}& a_{TF}\\ a_{IT}& a_{II}& a_{IF}\\ a_{FT}& a_{FI}& a_{FF}\end{array}\right)\left(\begin{array}{c}T\\ I\\ F\end{array}\right)+\left(\begin{array}{c}{b}_T\\ b_I\\ b_F\end{array}\right).$$

Proposition 4.

The affine transformation above defines an admissible neutrosophic transformation precisely when

$$A\mathcal{N}+b\in {[0,1]}^3$$

for every

$$\mathcal{N}\in {[0,1]}^3.$$

This condition is sufficient to guarantee that the neutrosophic component remains within its prescribed domain.

18. Monotonicity of Indeterminacy

Consider

$$I^{\prime }=I+\epsilon (1-I),0\le \epsilon \le 1.$$

Since

$$1-I\ge 0$$

and

$$\epsilon \ge 0,$$

we obtain

$$\epsilon (1-I)\ge 0.$$

Therefore,

$$I^{\prime }\ge I.$$

Moreover,

$$I^{\prime }=(1-\epsilon )I+\epsilon ,$$

so

$$0\le I^{\prime }\le 1.$$

Hence this transformation increases indeterminacy while preserving the admissible interval.

19. Structural Properties

Theorem 4.

The class of neutrosophic quantum channels forms a category whose objects are finite-dimensional Hilbert spaces equipped with neutrosophic information spaces and whose morphisms are pairs

$$(\Phi ,{\Phi }_{\mathcal{N}})$$

consisting of quantum channels and admissible neutrosophic transformations.

Proof. 

Identity morphisms are given by

$$(id,{id}_{\mathcal{N}}).$$

Composition is defined componentwise:

$$(\Psi ,{\Psi }_{\mathcal{N}})\circ (\Phi ,{\Phi }_{\mathcal{N}})=(\Psi \circ \Phi ,{\Psi }_{\mathcal{N}}\circ {\Phi }_{\mathcal{N}}).$$

Associativity follows from associativity of ordinary function composition:

$$(\Theta \circ \Psi )\circ \Phi =\Theta \circ (\Psi \circ \Phi )$$

and

$$({\Theta }_{\mathcal{N}}\circ {\Psi }_{\mathcal{N}})\circ {\Phi }_{\mathcal{N}}={\Theta }_{\mathcal{N}}\circ ({\Psi }_{\mathcal{N}}\circ {\Phi }_{\mathcal{N}}).$$

Thus the required categorical identities and associativity properties hold. □

20. Compatibility with Quantum Composition

The separation between the two components gives a natural compatibility principle.

Definition 9.

A neutrosophic quantum channel

$$\mathbf{\Phi }=(\Phi ,{\Phi }_{\mathcal{N}})$$

is calledcompatiblewith a family of quantum channels $\left\{{\Phi }_{\lambda }\right\}$ if there exists a corresponding family $\left\{{\Phi }_{\mathcal{N},\lambda }\right\}$ such that

$${\mathbf{\Phi }}_{\lambda }=({\Phi }_{\lambda },{\Phi }_{\mathcal{N},\lambda })$$

and

$${\mathbf{\Phi }}_{{\lambda }_2}\circ {\mathbf{\Phi }}_{{\lambda }_1}={\mathbf{\Phi }}_{{\lambda }_2\circ {\lambda }_1}.$$

This condition ensures that the evolution of the neutrosophic information is consistent with the composition law of the underlying quantum dynamics.

21. Relation to Ordinary Quantum Channels

The ordinary theory is recovered by taking the neutrosophic component to be trivial.

Let

$${\mathcal{N}}_0=(T_0,I_0,F_0)$$

be fixed and let

$${\Phi }_{\mathcal{N}}\left({\mathcal{N}}_0\right)={\mathcal{N}}_0.$$

Then

$$\mathbf{\Phi }(\rho ,{\mathcal{N}}_0)=(\Phi \left(\rho \right),{\mathcal{N}}_0).$$

Thus every ordinary quantum channel gives rise to a trivial neutrosophic quantum channel.

Proposition 5.

The ordinary category of finite-dimensional quantum channels embeds into the category of neutrosophic quantum channels.

Proof. 

Map each quantum channel $\Phi$ to

$$\Phi ⟼(\Phi ,{id}_{\mathcal{N}}).$$

Composition and identity are preserved componentwise. □

22. Discussion

The principal mathematical feature of the present framework is the separation

$$\mathrm{quantum}\mathrm{dynamics}+\mathrm{neutrosophic}\mathrm{information}.$$

The first component is governed by the established theory of completely positive trace-preserving maps. The second component is governed by an explicitly specified transformation of the neutrosophic state space.

This separation has several advantages.

First, complete positivity remains intact. Second, trace normalization of physical quantum states is not altered by arbitrary neutrosophic scores. Third, the neutrosophic component can be modified independently according to the intended information model.

At the same time, the framework makes clear that a neutrosophic extension is not unique. Different applications may require different choices of ${\Phi }_{\mathcal{N}}$, different aggregation operations, or different interpretations of T, I, and F.

Consequently, a physical implementation of a particular neutrosophic quantum channel would require an additional operational model explaining how the neutrosophic variables are encoded, transformed, and measured.

23. Conclusions

We have introduced a rigorous mathematical framework for neutrosophic quantum channels. A neutrosophic quantum channel is defined as a pair

$$\mathbf{\Phi }=(\Phi ,{\Phi }_{\mathcal{N}}),$$

where $\Phi$ is a completely positive trace-preserving quantum channel and ${\Phi }_{\mathcal{N}}$ is an admissible transformation of the single-valued neutrosophic information space.

Within this framework we established:

1.

a definition of neutrosophic quantum states;

2.

a Kraus representation of the quantum component;

3.

closure under composition;

4.

an identity channel;

5.

tensor-product constructions;

6.

a Choi representation;

7.

complementary-channel constructions;

8.

examples involving unitary, depolarizing, and bit-flip channels;

9.

normalization criteria for information-weighted states.

An important conclusion is that neutrosophic information should not be identified automatically with a physical quantum amplitude or with a probability distribution. The quantum component must retain the requirements of positivity, trace preservation, and complete positivity, while the neutrosophic component can provide an additional mathematical description of truth, indeterminacy, and falsity information.

The framework therefore provides a foundation for further studies of neutrosophic quantum measurements, quantum error correction, neutrosophic quantum communication, quantum algorithms under indeterminate information, and operational models connecting neutrosophic data with experimentally accessible quantum systems.