What is the Bell state and how is it represented mathematically?
The Bell state, also known as the EPR (Einstein-Podolsky-Rosen) pair, is a fundamental concept in quantum information theory that exhibits the phenomenon of quantum entanglement. It was first introduced in a famous paper by John Bell in 1964, which challenged the classical understanding of physical reality.
Mathematically, the Bell state is represented as a superposition of two maximally entangled quantum states. In the standard notation, the Bell state is denoted as:
|Φ+⟩ = (|00⟩ + |11⟩)/√2
This state represents a system of two qubits, where |0⟩ and |1⟩ are the computational basis states for a single qubit. The subscripts indicate the state of each qubit, with the first subscript referring to the state of the first qubit and the second subscript referring to the state of the second qubit.
In the Bell state |Φ+⟩, both qubits are entangled such that the outcome of measuring one qubit is perfectly correlated with the outcome of measuring the other qubit, regardless of the physical distance between them. This property is known as non-locality and is a hallmark of quantum entanglement.
To understand the significance of the Bell state, let's consider an example. Suppose we have two particles, A and B, prepared in the Bell state |Φ+⟩. If we measure the state of particle A and find it to be |0⟩, then we know with certainty that the state of particle B is also |0⟩. Similarly, if we measure the state of particle A and find it to be |1⟩, then we know with certainty that the state of particle B is also |1⟩. This instantaneous correlation between the two particles, regardless of their separation, is what makes the Bell state so intriguing and useful for various applications in quantum information processing.
The Bell state is invariant under rotations in the computational basis. This means that if we apply a rotation operation to both qubits, the resulting state will still be a Bell state. For example, if we apply a Pauli-X gate to both qubits in the Bell state |Φ+⟩, we obtain:
X⨂X(|Φ+⟩) = X⨂X((|00⟩ + |11⟩)/√2) = (|11⟩ + |00⟩)/√2
This new state is also a Bell state, known as |Φ-⟩. Similarly, applying other rotation operations, such as the Pauli-Y or Pauli-Z gates, to the Bell state will also result in other Bell states.
The Bell state is a maximally entangled state that exhibits non-local correlations between the outcomes of measurements on its constituent qubits. It is represented mathematically as a superposition of two computational basis states, and it is invariant under rotations in the computational basis.
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