**1. Introduction**

Quantum entanglement is the foremost prediction of quantum mechanics and one of the resources needed in quantum computing [1, 2]. Nowadays, it is an approach to solving time and processing power-consuming problems using quantum computers. A qubit is a quantum bit, the counterpart in quantum computing to the binary digit or bit of classical computing [3–5]. An important distinguishing feature between a qubit and a classical bit is that qubits can exhibit quantum entanglement. Quantum entangle ment is a nonlocal property that allows a set of qubit to express a higher correlation that is not possible in classical systems. Qubits are employed in areas other than quantum computing, such as sensors. This helps in network and communication channels, improving the security of data stored in the quantum field and protecting it, as well as the speedier delivery of messages and encrypted networks for securityrelated information.

Einstein et al. began studying quantum entanglement in 1935 after presenting the EPR conundrum [6]. Bell [7] demonstrated the fault in EPR arguments by

demonstrating that the notion of locality invoked in the EPR conundrum was incompatible with the hidden variables interpretation of the quantum theory. On the other hand, Bell's theorem offers one of the potential ways to determine if two specific particles form an EPR pair or entangled state. The strong correlations of entangled particles are used as resources for quantum cryptography [8], quantum teleportation [9], and quantum computation [10] because entanglement allows multiple states of qubits to be acted on simultaneously unlike classical bits that can have one value at a time.

The bipartite entangled states of photons have been generated [11–13]. Being an excellent carrier of information, a photon is not suitable for long-term storage as it is immediately destroyed as soon as one tries to detect it. Some theories have been developed for studying bipartite states of electronic qubits produced by photoionization [14]. One of the simplest processes for producing EPR pairs of particles with non-zero rest mass is the simultaneous ejection of two electrons following the absorption of a single photon in an atom or molecule [15, 16]. This process is known as double photoionization (DPI). Here, entanglement is produced due to the process taking place inside an atom. We discuss DPI of an atom for generating bipartite states of two electronic qubits (say *e*<sup>1</sup> and *e*2). In the absence of spin-orbit interaction (SOI), these particles are entangled with regard to their spin-angular momenta. In addition, the electrons in DPI are released in all directions and with all kinetic energies (subject to the conservation of total energy). Additionally, any direction can be quantized for its spins. Utilizing the density matrix (DM) of DPI concurrence [17–19] allows us to measure the level of entanglement.

We first establish some pertinent conventions in Section 2 and then quickly discuss the density operator (DO) and states for an atom's DPI. This operator is appropriate when the target atom is in its ground state before DPI and the ionizing electromagnetic radiation is in a pure state of polarization. In the absence of SOI, this DO is used to develop an expression for the density matrix (DM) required to examine the quantum entanglement characteristics of the photoelectrons' and photons'spin states in Russell-Saunders coupling. In Section 3, we examine entanglement in a DPI system for qubits. Section 4 provides a quantitative application of the qubits for the DPI of the helium atom. The conclusion is given in Section 5, the last section.
