Why is the sy – state considered to have complete rotational invariance under all complex rotations?
The Bell state, also known as the maximally entangled state, is an important concept in the field of quantum information. It is a two-qubit state that exhibits a unique property known as rotational invariance under all complex rotations. This property makes it a valuable resource for various quantum information processing tasks, such as quantum teleportation and quantum cryptography.
To understand why the Bell state is considered to have complete rotational invariance, let us first define what we mean by complex rotations. In quantum mechanics, a complex rotation is a transformation that can be applied to a quantum state using a unitary operator. It involves rotating the state in the complex plane, rather than in physical space.
The Bell state, denoted as |Φ⁺⟩, is one of the four maximally entangled states that can be created from a pair of qubits. It can be expressed as:
|Φ⁺⟩ = (|00⟩ + |11⟩)/√2
where |00⟩ and |11⟩ represent the states of the individual qubits. This state is interesting because it exhibits a specific property under complex rotations.
When a complex rotation is applied to the Bell state, it affects both qubits simultaneously. The rotation can be represented by a unitary operator U, which acts on the state as:
U|Φ⁺⟩ = (U|00⟩ + U|11⟩)/√2
In order for the Bell state to be considered to have complete rotational invariance, it must satisfy two conditions. Firstly, the resulting state after the rotation should still be a valid Bell state. Secondly, the resulting state should be proportional to the original Bell state.
Let us examine these conditions in more detail. The first condition requires that the resulting state after the rotation is still entangled. In other words, it should not be possible to express the resulting state as a product of individual qubit states. If the resulting state can be written as |ψ⟩ = |a⟩⊗|b⟩, where |a⟩ and |b⟩ are the states of the individual qubits, then the entanglement is lost.
For the Bell state, applying a complex rotation does not break the entanglement. The resulting state remains entangled and cannot be expressed as a product of individual qubit states. Therefore, it satisfies the first condition for rotational invariance.
The second condition requires that the resulting state is proportional to the original Bell state. This means that the two states are related by a phase factor, which can be expressed as:
U|Φ⁺⟩ = e^(iθ)|Φ⁺⟩
where e^(iθ) represents the phase factor. If the resulting state is not proportional to the original Bell state, then the rotational invariance is not complete.
In the case of the Bell state, applying a complex rotation introduces a phase factor, but the resulting state remains proportional to the original Bell state. Therefore, it satisfies the second condition for rotational invariance.
The Bell state is considered to have complete rotational invariance under all complex rotations because it maintains its entanglement and remains proportional to the original state after the rotation. This property makes it a valuable resource for various quantum information processing tasks.
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