# Quantum Bell States in Wolfram

H Hannan

The Bell states, also known as EPR pairs, are specific quantum states in which two qubits that represent the simplest and, in many ways, most powerful form of quantum entanglement. This can be created in the Wolfram Language. You can represent these states using the Ket notation or by explicitly constructing the quantum state vectors. However, as far as I am aware, the Wolfram Language does not have specific built-in support for quantum computing primitives such as qubits and quantum gates simulation in the same way some specialized quantum computing frameworks do. Instead, quantum states can be represented using vectors and matrices.

Here are the four Bell states, expressed in their mathematical forms and how you might code them using lists in Wolfram Language to represent their state vectors:

Bell States

## Bell States

$$|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$$
$$|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle – |11\rangle)$$
$$|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$$
$$|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle – |10\rangle)$$

In the Wolfram Language, you can represent these states as follows:

(* Bell State |Φ+⟩ *)
phiPlus = 1/Sqrt[2] {{1, 0, 0, 1}};

(* Bell State |Φ-⟩ *)
phiMinus = 1/Sqrt[2] {{1, 0, 0, -1}};

(* Bell State |Ψ+⟩ *)
psiPlus = 1/Sqrt[2] {{0, 1, 1, 0}};

(* Bell State |Ψ-⟩ *)
psiMinus = 1/Sqrt[2] {{0, 1, -1, 0}};

Each state is represented as normalized vector on the basis of (|00\rangle), (|01\rangle), (|10\rangle), and (|11\rangle), corresponding to the tensor product space of the two qubits. Note that the basis vectors shown are ordered as (|00\rangle), (|01\rangle), (|10\rangle), and (|11\rangle), and each Bell state is a direct superposition of these basis states.

These vectors can be used in the Wolfram Language to perform further calculations, simulate quantum measurements easily, or analyze the properties of quantum entanglement. However, if you are looking for more advanced quantum computing simulations, you might need to use specialized quantum computing software or libraries that offer more comprehensive support for quantum operations and state manipulations. A reliable place to start looking is Qiskit.