What is the concept of rotational invariance in the context of the Bell state?
In the field of quantum information, the concept of rotational invariance plays a important role in understanding the behavior of entangled states, such as the Bell state. To comprehend the concept fully, it is essential to have a solid grasp of quantum entanglement and the mathematical framework that describes it.
Quantum entanglement is a phenomenon in which two or more particles become correlated in such a way that the state of one particle cannot be described independently of the state of the other particles. The Bell state, also known as the maximally entangled state, is a specific type of entangled state that exhibits a high degree of correlation between two particles.
Rotational invariance refers to the property of a physical system that remains unchanged under rotations. In the context of the Bell state, rotational invariance implies that the entanglement between the particles is unaffected by rotations applied to the system. This means that the correlation between the particles remains the same, regardless of the orientation of the system.
To understand this concept further, let's consider a specific example. Suppose we have two entangled particles, labeled A and B, in a Bell state. The Bell state can be written as:
|Ψ⟩ = (1/√2)(|00⟩ + |11⟩),
where |00⟩ represents the state in which both particles are in the "0" state, and |11⟩ represents the state in which both particles are in the "1" state.
Now, let's apply a rotation to the system. We can represent a rotation in three-dimensional space using Euler angles, which describe the rotation around three axes: x, y, and z. For simplicity, let's consider a rotation around the z-axis by an angle θ.
The rotation operator for a single qubit can be written as:
R(θ) = exp(-iθσz/2),
where σz is the Pauli z matrix and i is the imaginary unit.
Applying this rotation to the Bell state, we obtain:
|Ψ'⟩ = (1/√2)(R(θ)|00⟩ + R(θ)|11⟩).
Now, the important point is that the rotational invariance of the Bell state implies that the correlation between the particles remains the same, regardless of the rotation angle θ. In other words, the probability of measuring both particles in the same state remains unchanged.
To see this, let's calculate the probability of measuring both particles in the "0" state for the rotated Bell state |Ψ'⟩. We can write this probability as:
P(00) = |⟨00|Ψ'⟩|^2,
where ⟨00| is the bra vector corresponding to the state |00⟩.
Expanding the expression and simplifying, we find:
P(00) = (1/2) * |⟨00|R(θ)|00⟩ + (1/2) * |⟨00|R(θ)|11⟩|^2.
Using the properties of the rotation operator and the fact that the Bell state |Ψ⟩ is an eigenstate of σz with eigenvalue 1, we can simplify further:
P(00) = (1/2) * |⟨00|00⟩ + (1/2) * |⟨00|11⟩|^2.
Since the Bell state |Ψ⟩ is defined as (1/√2)(|00⟩ + |11⟩), we have:
P(00) = (1/2) * 1 + (1/2) * 0 = 1/2.
This result shows that the probability of measuring both particles in the "0" state is independent of the rotation angle θ. Therefore, the rotational invariance of the Bell state is preserved.
Rotational invariance in the context of the Bell state refers to the property of the entangled state remaining unchanged under rotations applied to the system. This means that the correlation between the particles, as described by the Bell state, is unaffected by rotations. This concept is of fundamental importance in quantum information and provides insights into the behavior of entangled states.
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