On the spectral nature of entanglement

IET Quantum CommunicationVolume 2, Issue 1 p. 8-13 ORIGINAL RESEARCH PAPEROpen Access On the spectral nature of entanglement Mario Mastriani, Corresponding Author mmastria@fiu.edu orcid.org/0000-0002-5627-3935 School of Computing & Information Sciences, Florida International University, Miami, Florida, USA Correspondence Mario Mastriani, School of Computing & Information Sciences, Florida International University, 11200 S.W. 8th Street, Miami, FL 33199, USA. Email: mmastria@fiu.eduSearch for more papers by this author Mario Mastriani, Corresponding Author mmastria@fiu.edu orcid.org/0000-0002-5627-3935 School of Computing & Information Sciences, Florida International University, Miami, Florida, USA Correspondence Mario Mastriani, School of Computing & Information Sciences, Florida International University, 11200 S.W. 8th Street, Miami, FL 33199, USA. Email: mmastria@fiu.eduSearch for more papers by this author First published: 30 March 2021 https://doi.org/10.1049/qtc2.12005AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinked InRedditWechat Abstract This study establishes, for the first time in literature, that quantum entanglement, as well as the most important protocol derived from it, that is, quantum teleportation, completely rest on the Quantum Fourier Transform. This highlights the spectral nature of both protocols with important consequences on quantum computing, quantum communications, and quantum cryptography, with a particular emphasis on quantum Internet. 1 INTRODUCTION In addition to the myriad applications of the Fourier transform in the classical world, it is important to recognize its presence in the quantum world, where, the participation of its quantum version is essential in the implementation of the most important algorithms developed in the last three decades [1], being the Shor algorithm [2] and its most successful case, which uses the Quantum Fourier Transform (QFT) to perform phase estimation [1, 3]. In an easy to understand way, why quantum phase estimation is one of the most power tools inside the toolbox of quantum information processing, simply, because in this area there are decisions that can only be made in the spectral domain. In the last two decades, QFT has participated in the most diverse studies, such as the analysis of spectral effects in quantum teleportation [4], its experimental implementation using the local classical entanglement state between the polarization and orbital angular momentum [5], the relation between spectral theory and quantum mechanics [6], the teleportation of multiple qubits based on QFT [7, 8] where this tool is used to find orthogonal bases on which the sender projects the states to be teleported, the explanation about the surprising result that the Groverian entanglement of the periodic states built up during the preprocessing stage is only slightly affected by the QFT [9], its implementation in an entangled system of multilevel atoms [10], the realization of the two-qubit QFT and entangled protected using a Hamiltonian, which constitutes a matrix with circulant symmetry [11], its use in multiparty quantum telecommunication [12], its use as a tool to investigate phenomena such as quantum symmetry [13], and in an interesting study that showed herein that under the constraints imposed by quantum decoherence [14], only a parallel approach of QFT can guarantee a reliable solution or, alternatively, improve scalability [15] of this important tool. Based on what was said, the literature does not have a study about the spectral nature of the entanglement and its main application, the entanglement. Therefore, this is the ubiquity space of this work, that allows us to establish that all relative aspects of entanglement are covered by QFT, representing its most important tool for its finer analysis and future expansion. We hope this work provides an area of interest to theorists, experimentalists, and others who might want to study the underlying relationships behind all quantum algorithms, their respective implementations on a physical machine (based on superconductor technology [16, 17]) or an optical circuit [18], and an in-depth analysis of the respective outcomes. In Section II, the equivalences between 1 and 2-qubits quantum gates, and their respective representation via QFT are developed. Section III is especially dedicated to entanglement [19-21], and quantum teleportation [22-25]. Finally, Section IV has our conclusions. 2 EQUIVALENCES BETWEEN 1 AND 2-QUBIT GATES, AND QFT We know that the QFT represents an important family of quantum operations over the ring ℤ2n, where the n-qubit QFT coherently transforms an input state |x〉 in the computational basis as follows [26]: | x 〉 ↦ 1 2 n ∑ v = 0 2 n − 1 ω 2 n u . v | y 〉 , u = 0 , 1 , 2 , … , 2 n − 1 (1)where |x〉 represents a qubit-string |x1…xn〉 [26], and the kernel ω2n = ei2π/2n is the 2n root of unity [27], such that ω 2 n 2 n = 1, while the inverse QFT is: | y 〉 ↦ 1 2 n ∑ u = 0 2 n − 1 ω 2 n − v . u | x 〉 , v = 0 , 1 , 2 , … , 2 n − 1 (2) If from this point on, F2n represents the matrix of QFT [27] and F 2 n − 1 its reverse, the simple number 1 (unity), that is, QFT for 0-qubits, F 2 0 = F 2 0 − 1 = 1, will be present, per se, inside all the gates used in entanglement [19-21], and quantum teleportation [22-25]. In order to establish the preliminary equivalences that will later derive in the relationship between the entanglement and the QFT, we must define only three of the four Pauli's matrices [27], where these are as follows: I 2 × 2 = [ 1 0 0 1 ] , (3a) X = [ 0 1 1 0 ] , (3b) Z = [ 1 0 0 − 1 ] . (3c) On the other hand, it is known from literature [28, 29], that Hadamard matrix H is equivalent to the 1-qubit QFT and its inverse, that is, F 2 1 = H = 1 2 [ 1 1 1 − 1 ] = H − 1 = F 2 1 − 1 , (4)such that H H = H H−1 = I2×2. Now, we will establish some additional equivalences necessary for the rest of this study, Z = H X H , (5a) S = Z = [ 1 0 0 i ] (5b) V = X = H S H = [ ( 1 + i ) ( 1 − i ) ( 1 − i ) ( 1 + i ) ] / 2 . (5c) Equation (4) constitutes the last coincidence between both words, since for more qubits this coincidence disappears. We explore the relationship between QFT of 2-qubits F 2 2 , and the CNOT gate [27]. Then, being, F 2 2 = 1 2 [ 1 1 1 1 1 i − 1 − i 1 − 1 1 − 1 1 − i − 1 i ] (6a)and F 2 2 − 1 = 1 2 [ 1 1 1 1 1 − i − 1 i 1 − 1 1 − 1 1 i − 1 − i ] (6b)it is evident that QFT and its inverse do not coincide for the 2-qubit case ( F 2 2 ≠ F 2 2 − 1 ) as in the previous cases, in fact, this difference is deepened when considering cases for a greater number of qubits. It is necessary to establish with which of the two criteria present in Figure 1 we will work from here on, mainly in relation to the Feynman's gate [27, 30, 31], also known as Controlled-NOT, CNOT, Controlled-X, or simply CX. Figure 1 shows two associated matrices to CNOT gate, depending on which qubit the control is, as well as, which qubit the target is [32]. Different books, papers, and quantum platforms [16, 17] order their qubits differently. FIGURE 1Open in figure viewerPowerPoint Two options for the implementation of CNOT gate. (a) q [1] is the control qubit, and q[0] is the target qubit. (b) q[0] is the control qubit, and q [1] is the target qubit We choose the option of Figure 1b, precisely, because it is closer to that used in the main quantum platforms [16, 17]. Specifically, this option swaps amplitudes of |01⟩ and |11⟩ in the input vector [32], where the relation between both versions of Figure 1 is CXb) = (H ⨂ (h) CXa) (H ⨂ (h), being “⨂” is the Kronecker product [27, 30, 31]. Then, incorporating the option selected above, if we multiply F 2 2 by itself, and we do the same with F 2 2 2 − 1 , Equation (7) shows that both multiplications result identical and also equal to the CNOT gate of Figure 1b [27], F 2 2 × F 2 2 = F 2 2 − 1 × F 2 2 − 1 = C N O T = [ 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 ] (7)given that, multiplying F 2 2 × F 2 2 by F 2 2 − 1 × F 2 2 − 1 and regrouping, yields, ( F 2 2 × F 2 2 ) × ( F 2 2 − 1 × F 2 2 − 1 ) = F 2 2 × ( F 2 2 × F 2 2 − 1 ) × F 2 2 − 1 = F 2 2 × I × F 2 2 − 1 = F 2 2 × F 2 2 − 1 = I , (8)where I is the identity matrix. However, C N O T = [ 1 0 0 0 0 ( 1 + i ) / 2 0 ( 1 − i ) / 2 0 0 1 0 0 ( 1 − i ) / 2 0 ( 1 + i ) / 2 ] . (9) Therefore, considering Equations (6) and (9), it is evident that we will obtain CNOT≠ F 2 2 and CNOT≠ F 2 2 − 1 . Equation (9) has other associated equivalencies as a direct consequence of Equation (5), that is, those corresponding to the controlled gates of Equation (5), which concludes in another useful equality associated with Equation 9, C Z b = ( I ⊗ H ) C X b ( I ⊗ H ) = [ 1 1 1 1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 1 ] / 2 , (10a) C Z a = ( H ⊗ H ) C Z b ( H ⊗ H ) = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 − 1 ] , (10b) C S b = C Z b = [ 3 + i 1 − i 1 − i − 1 + i 1 − i 3 + i − 1 + i 1 − i 1 − i − 1 + i 3 + i 1 − i − 1 + i 1 − i 1 − i 3 + i ] / 4 , (10c) C S a = ( H ⊗ H ) C S b ( H ⊗ H ) = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 i ] , (10d) C V b = C X b = ( I ⊗ H ) C S b ( I ⊗ H ) = [ 1 0 0 0 0 ( 1 + i ) 0 ( 1 − i ) 0 0 1 0 0 ( 1 − i ) 0 ( 1 + i ) ] / 2 , (10e) C V a = ( H ⊗ H ) C V b ( H ⊗ H ) = [ 1 0 0 0 0 1 0 0 0 0 ( 1 + i ) ( 1 − i ) 0 0 ( 1 − i ) ( 1 + i ) ] / 2 . (10f)where the subscript a is associated to the CX version of Figure 1a, while subscript b is associated to the CX version of Figure 1b, demonstrating that they are perfectly interchangeable through a simple transformation based on Hadamard gates (h) Next, we will develop the last set of equivalences necessary for our study starting with the SWAP gate, S W = [ 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 ] , (11)so that SW SW = I4×4. On the other hand, expressing QFT4×4 in terms of its constitutive matrices [27], that is, Hadamard (H), Controlled-S and SWAP (SW), we will have both equivalent circuits of Figure 2. FIGURE 2Open in figure viewerPowerPoint Two equivalent representations of QFT, F 2 2 ∈ℂ22×22 expressed in terms of Hadamard (H), Controlled-S, and SWAP (SW) gates Finally, in Figure 3 the equivalences of Equation (10) and Figure 2 are used to verify the equality of Equation (7), thus demonstrating the equivalence between a CNOT gate and a QFT4×4×QFT4×4. FIGURE 3Open in figure viewerPowerPoint Complete transition, in terms of Equation (10) and Figure 2, between a CNOT gate and QFT4×4×QFT4×4 3 QUANTUM ENTAGLEMENT, AND TELEPORTATION We will focus our analysis on the intervention of the Fourier transform in Quantum Communications [33-35], by demonstrating its presence in Quantum Entanglement [19-21], and Quantum Teleportation [22-25]. Therefore, this section will make it clear that the two pillars of Quantum Information Processing (QIP), superposition and entanglement, are two attributes of unequivocally spectral characteristics. 3.1 Quantum Entanglement We will start with the so-called Bell states [27, 30, 31], which result from the combination of the Hadamard (H), and CNOT matrices of Equation (12), where, | β 00 〉 = 1 2 ( | 00 〉 + | 11 〉 ) = C X ( ( I ⊗ H ) | 00 〉 ) = [ 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 ] ( ( [ 1 0 0 1 ] ⊗ 1 2 [ 1 1 1 − 1 ] ) [ 1 0 0 0 ] ) = 1 2 [ 1 0 0 1 ] (12) The combination of matrices required by Equation (12), is carried out in optical circuits [18] by means of a beamsplitter, which is represented in Figure 4 thanks to {H, CNOT}, and { F 2 1 , F 2 2 × F 2 2 }. FIGURE 4Open in figure viewerPowerPoint Bell state via a beam-splitter implemented thanks to: (a) {H, CNOT}, and (b) { F 2 1 , F 2 2 × F 2 2 } It is evident that if we resort to all the equivalences between gates present in the literature [27, 30, 31], the Fourier expansion in other scopes outside the quantum entanglement and teleportation becomes obvious. This can be seen in detail in a recent work that links QFT with quantum information processing [36]. 3.2 Quantum Teleportation Using the equivalence of Figure 4, we can implement the famous quantum teleportation protocol [22-25] in terms of Fourier. Figure 5 presents three options to carry out this task. The first is the one that we can find in all the literature that deals with Quantum Teleportation [22-25], which is based on {H, CNOT, Quantum Measurement (QuMe), CZ}. In the second one, we replace CZ by {H, CNOT}, while in the last one, we replace all the gates of the latter by their Fourier representatives, that is, F 2 1 , and F 2 2 × F 2 2 . In the three cases, the double lines represent classical wires, which constitute the classical channel, while, the single lines represent the quantum wires. As we have already mentioned, the pair {H, CNOT} is known as the beamsplitter, and the group {H, CNOT, Quantum Measurement (QuMe)} is called Bell State Measurement (BSM), where the quantum measurement is carried out by Single-Photon Detectors (SPD) [23-25]. FIGURE 5Open in figure viewerPowerPoint Quantum teleportation protocol: (a) the traditional one thanks to {H, CNOT, QuMe}, (b) CZ = (I ⨂ (h) CX (I ⨂ (h), i.e., CZ is replaced by H and CNOT gates, and (c) in terms of { F 2 1 , F 2 2 × F 2 2 } Finally, both in the implementations of Figure 4, for the entanglement case, and in those of Figure 5, for the teleportation case, we have worked with |β00〉, although in reality we can do exactly the same with any of the other three Bell bases [19-21, 27, 30, 31]. 4 CONCLUSION We have demonstrated that both, quantum entanglement and quantum teleportation completely rests on QFT, making clear the spectral nature of both protocols, with an obvious projection on quantum communications [33-35], in general, and quantum Cryptography [37, 38] as well as quantum Internet [39-44], in particular, given that, having established the important link between the CNOT gate and QFT, it will now be easier to understand the genesis and incidence of the typical noises present in all quantum circuits due to quantum gates, particularly in those implementations carried out by means of optical elements [16]. These gates inherently generate a series of errors, specifically, three types: bit flip (X-errors), phase flip (Z-errors, or phase damping), and bit-phase flip [45]. The aforementioned link will also allow a better understanding of aspects related to decoherence [14, 15] in non-adiabatic environments, as well as a very common process present in all real quantum channels known as the amplitude damping [46]. Finally, the nexus between QFT, CNOT gate, entanglement, and teleportation, will bring us even closer to a more detailed understanding of the tools used in quantum communications known as quantum memories [47-49] and quantum repeaters [50, 51], which will play a key role in the development of the future quantum Internet [39-44]. ACKNOWLEDGMENTS Mario Mastriani thanks Prof. S.S. Iyengar, director of the School of Computing and Information Sciences of Florida International University for all his help and support. 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