Quantum Fourier transform is the building block for creating entanglement

This study demonstrates entanglement can be exclusively constituted by quantum Fourier transform (QFT) blocks. A bridge between entanglement and QFT will allow incorporating a spectral analysis to the already traditional temporal approach of entanglement, which will result in the development of new more performant, and fault-tolerant protocols to be used in quantum computing as well as quantum communication, with particular emphasis in the future quantum Internet.

where 2 n = e i2 ∕2 n is the 2 n root of unity, while the inverse QFT is: The Hadamard matrix H is equivalent to the 1-qubit QFT and its inverse 24,25 , That is, for the 1-qubit QFT all its components are equivalent. Instead, for the 2-qubit QFT, the same does not happen, since On the other hand, the Feynman's gate 4 (also known as Controlled-X, CNOT, or CX gate), as well as its flipped version are respectively: and where the difference between them consists in that in Eq. (5a) the upper qubit is the control qubit, while the lower qubit is the target qubit. Instead, in the version of Eq. (5b) it is exactly the opposite, being: CNOT flipped = (H ⊗ H) × CNOT × (H ⊗ H) , "×" the matrix product, and "⊗" the Kronecker product 4 .
Multiplying both F 2 2 by itself and F −1 2 2 by itself, both multiplications result equal to the CNOT flipped gate of Eq. (5b): F 2 2 × F 2 2 = F −1 2 2 × F −1 2 2 = CNOT flipped . This can be easily verified by multiplying CNOT flipped by itself, and F 2 2 × F 2 2 by F −1 2 2 × F −1 2 2 and regrouping, However, and Therefore, √ CNOT ≠ F 2 2 and √ CNOT flipped ≠ F 2 2 . Finally, the CNOT gate is equal to the flipped version of the multiplication of QFT F 2 2 by itself, Equation (8) is fundamental in the creation of the entanglement for two or more qubits, as well as in all the applications that require it, as is the case of quantum teleportation 11 .

Bell states
Pauli's matrices 4 can be expressed in terms of the so-named Hadamard rotation gates 26  (4a)  Fig. 1a,c represents the implementation of the � � 00 ⟩ = �Φ + ⟩ Bell state in terms of its two original versions (direct and flipped), while Fig. 1c,d constitute their respective counterparts based exclusively on QFT blocks, confirming that these blocks are all that is needed to create entanglement while revealing its spectral nature.

N-qubits Greenberger-Horne-Zeilinger (GHZ N ) states
This family of configurations is the most commonly used in practice when it comes to entanglement between three or more particles 4,8-10 , being its general form as follows: Without loss of generality, in this study only � � GHZ 3 ⟩ and � � GHZ 4 ⟩ are implemented in terms of QFT blocks, where: Equations (13) and (14) are graphically represented in Fig. 2a   N , show that the equivalence of Fig. 1 is not a simple coincidence for a particular case like a Bell state, but actually, the entanglement in all its manifestations has a spectral nature, where the QFT is the essential instrument for a spectral tomography of it. It only remains to project this equivalence, in perhaps the most conspicuous application of entanglement, quantum teleportation 13 .

Teleportation
This protocol 13 is implemented in three different ways in Fig. 3, where a qubit � ⟩ to be teleported is prepared and introduced in the upper qubit on the left of the protocol. A Bell state like that of Fig. 1 is distributed between Alice and Bob. Subsequently, a module applied in the two upper qubits and constituted by a CNOT gate, an H gate, and two quantum measurement blocks (QuMe) constitute what in practice is known as a Bell State Measurement (BSM) module 4,[8][9][10] . The double lines at the output of each QuMe convey classical information from Alice to Bob in the form of two classical disambiguation or control bits. For this reason, this means of transport is known as a classic channel of disambiguation, control, or simply as an auxiliary channel.
A 2-qubits Controlled-ℤ gate or simply Cℤ gate can be constructed from QFT blocks according to the equivalence of Eq. (9d), or in terms of two H and one CNOT (Controlled-X or CX) gates from the following identity: As can be seen in Fig. 3c, except for the QuMe blocks, everything else in this protocol is representable using QFT blocks. This extends, with identical results, to all other protocols that are based on entanglement such as quantum secret sharing 14 , quantum key distribution 15 , quantum secure direct communication 16 , and quantum repeaters 17 , and that are used in quantum Internet 18 . Next, the complete timeline is developed according to Fig. 4, starting at t 1 , where | | | t 1 ⟩ is the wave-function in that instant,

Quantum spectral analysis.
is the upper qubit in Fig. 4, and � � � q l � t 1 �� = �0⟩ is the lower qubit in that figure. The qubits obtained � � q u ⟩ and � � q l ⟩ at time t 1 are completely independents 8 , and are used as inputs to the next step, which is made up of an H gate in � � q u ⟩ and an identity matrix in � � q l ⟩,   . Undoubtedly, this constitutes advance respect to the literature on the subject in force to date, which associated the aforementioned impossibility with the intervention of the CNOT gate, as a whole, in an exclusive way, or with an inappropriate coupling between the individual contributions of H and CNOT gates. In consequence, this analysis makes explicit an intermediate instance to the one already known for the non-separability and indistinguishability of the states during entanglement, which is exclusively the responsibility of a particular characteristic of the Discrete Fourier Transform 1 (DFT) and that is inherited by the QFT 3 . This characteristic refers to the fact that the DFT is a dense matrix, i.e. all its elements are different from zero, since, they are the N roots of the unit or twiddle factors 1 , which when they are multiplied by the input vector produce an output vector where each of its elements represents a mixture or weighted sum of the incoming vector. Finally, the intervention of the second flipped QFT 2 2 ×2 2 allows obtaining the wave-function at the time t 4 , where, as in the previous case, it is impossible to decompose | | | t 4 ⟩ into two independent states � � q u ⟩ and � � q l ⟩ , that is, is not factorable. This gives rise to a very particular state of null spin called entanglement 8 . The four density matrices associated with every wave-function of Fig. 4 are the following: where (•) * is the complex conjugate of (•), The four density matrices can be seen in Table 1, where only t 3 has an imaginary part. On the other hand, comparing the 3D bars of the density matrices at t 2 and t 4 , it can be seen that the consecutive action of both flipped QFT 2 2 ×2 2 has a stretching effect as far as the locations of the bars are concerned. This shows that both flipped QFT 2 2 ×2 2 are the architect of a bad copy of wave-function | | | t 2 ⟩ of Eq. (18), that is to say, This shows that entanglement is the result of an inadequate copy by a very inefficient copy machine embodied by both flipped QFT 2 2 ×2 2 . The aforementioned stretching effect added to the unification of the entanglement's own wave-function triggers its most conspicuous characteristic, that is to say, the impossibility of factoring the wave function of Eq. (20).
Moreover, given two subsystems (A, and B) that interact with each other, their density matrices treated individually are, and their von Neumann entropies are, where tr(•) is the trace of the square matrix (•), and log(•) is logarithm base 2 of (•). In the same way, for a composed system, the entropy is, S A∪B depends on the degree of correlation (completely independent, correlated, and entangled) between both subsystems. Besides, in the classical and the quantum worlds, the correlations between the subsystems are those established by the additional information. In the case of composite quantum systems, the mutual information S A∩B is introduced to quantify that additional information, allowing us to obtain the degree of correlation between both subsystems 8 , Therefore, the entropy of the composite system S A∩B indicates that the uncertainty of a state A∪B is less than the two subsystems S A and S B added together. Table 2 shows entropies in terms of the degree of correlations between both subsystems, in such a way that when S A∪B = 2 , the entropy of the composite system S A∩B = S A + S B − S A∪B = 1 + 1 − 2 = 0 , which means that both subsystems do not have mutual information, and this null degree of correlation corresponds to the case of Eq. (18) of Fig. 4 at time t 2 , where | | | t 2 ⟩ is factored into � � � q u � t 2 �� = �+⟩ , and � � � q l � t 2 �� = �0⟩ , that is, both subsystems are completely independents. Instead, when S A∪B = 1 , the entropy of the composite system S A∩B = S A + S B − S A∪B = 1 + 1 − 1 = 1 , this case corresponds to Eq.