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arXiv:1509.02806

Quantum entanglement and the Bell Matrix

Anna Chiara Lai, Marco Pedicini, Silvia Rognone

Published Sep 9, 2015• 0 citations
arXivHTMLPDF

Abstract

We present a class of maximally entangled states generated by a high-dimensional generalisation of the \textsc{cnot} gate. The advantage of our approach is the simple algebraic structure of both entangling operator and resulting entangled states. In order to show that the method can be applied to any dimension, we introduce new sufficient conditions for global and maximal entanglement with respect to Meyer and Wallach's measure.

DOI
10.48550/arxiv.1509.02806

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We present a class of maximally entangled states generated by a high-dimensional generalisation of the \textsc{cnot} gate. The advantage of our approach is the simple algebraic structure of both entangling operator and resulting entangled states. In order to show that the method can be applied to any dimension, we introduce new sufficient conditions for global and maximal entanglement with respect to Meyer and Wallach's measure.
The Bell basis is a distinctive set of maximally entangled two-particle quantum states that forms the foundation for many quantum protocols such as teleportation, dense coding and entanglement swapping. While the generation, manipulation, and measurement of two-level quantum states is well understood, the same is not true in higher dimensions. Here we present the experimental generation of a complete set of Bell states in a four-dimensional Hilbert space, comprising of 16 orthogonal entangled Bell-like states encoded in the orbital angular momentum of photons. The states are created by the application of generalized high-dimensional Pauli gates on an initial entangled state. Our results pave the way for the application of high-dimensional quantum states in complex quantum protocols such as quantum dense coding.
Abstract In the modern technological landscape, quantum information science is becoming an essential component of advanced computational systems, offering fundamentally new capabilities for solving complex problems. A central feature of quantum computing is the use of entangled states, where qubits become inseparably correlated in accordance with the principles of quantum entanglement. A key concept in this framework is the Bell state, a specific type of maximally entangled two-qubit quantum state used to characterize and measure entanglement. Bell states form the theoretical and practical foundation for numerous quantum information protocols, including secure communication and quantum teleportation. Bell States and Quantum Entanglement The four canonical Bell states represent maximally entangled two-qubit configurations. These states demonstrate non-classical correlations that cannot be explained by classical probability theory. In quantum cryptography and information security, Bell states enable: Detection of eavesdropping in quantum key distribution (QKD). Verification of entanglement integrity. Implementation of quantum communication protocols. Entanglement serves as a resource for security guarantees based on the laws of quantum mechanics rather than computational hardness assumptions. Role of the CNOT Gate A crucial building block for generating Bell states is the CNOT gate (controlled-NOT gate). The typical procedure for creating a Bell state involves: Applying a Hadamard gate to the first qubit (creating superposition). Applying the CNOT gate, which entangles the control and target qubits. This sequence transforms separable states into maximally entangled states, forming the core of many quantum algorithms and cryptographic protocols. Implications for Information Security The integration of entangled states and quantum logic gates provides a methodological basis for: Quantum key distribution (e.g., BB84, E91). Quantum authentication protocols. Secure multi-party computation. Detection of interception through disturbance measurement. Unlike classical cryptography, whose security relies on computational assumptions, quantum security mechanisms are grounded in physical principles such as the no-cloning theorem and measurement-induced disturbance. Strategic Perspective Understanding Bell states, entanglement, and controlled operations allows researchers and security specialists to: Design quantum-secure communication systems. Analyze vulnerabilities introduced by quantum computational capabilities. Develop hybrid classical-quantum security architectures. As quantum technologies evolve, the combination of entanglement theory, algorithmic design, and secure protocol engineering will play a decisive role in shaping next-generation information security frameworks.
Quantum information processing and communication techniques rely heavily upon entangled quantum states, and this dependence motivates the development of methods and systems to generate entanglement. Much research has been dedicated to state preparation for radix-2 qubits, and due to the pursuit of entangled states, the Bell state generator and its generalized forms where the number of entangled qubits is greater than two have been defined. In this work, we move beyond radix-2 and propose techniques for quantum state entanglement in high-dimensional systems through the generalization of the binary bipartite entanglement states. These higher-radix quantum informatic systems are composed of n quantum digits, or qudits, that are each mathematically characterized as elements of an r -dimensioned Hilbert vector space where r > 2. Consequently, the wave function is a time-dependent state vector of dimension r n . The generalization of the binary controlled-NOT to the controlled-modulo-addition gate, the concept of partial versus maximal entanglement, and architectures for generating higher-radix entangled states for the partial and maximal case are all presented.
Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly , from separable basis states, important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.