How does measurement of one qubit affect the others?

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In quantum mechanics, the measurement of one qubit can have a profound and often counterintuitive effect on the state of other qubits in a quantum system, a phenomenon known as "quantum entanglement." This property arises from the fundamental principles of superposition and the non-commutativity of quantum observables.

When qubits are entangled, their quantum states become correlated in such a way that the measurement outcome of one qubit provides information about the measurement outcomes of the other qubits. This correlation persists even when the entangled qubits are physically separated from each other.

Here's a simplified explanation of how measurement of one qubit can affect the others:

Entanglement Creation: Let's consider a scenario where two qubits, A and B, become entangled. This can occur, for example, when two qubits are prepared in a quantum state that is an equal superposition of both qubits being in the states |0⟩ and |1⟩:

|Ψ⟩ = (|00⟩ + |11⟩) / √2

In this entangled state, the qubits are correlated in such a way that if you measure qubit A and find it in the state |0⟩, you know that qubit B will also collapse to the state |0⟩ when measured, and vice versa.

Measurement Outcome Correlation: When you measure qubit A, it "chooses" one of its possible states (|0⟩ or |1⟩) based on the probabilities determined by its quantum state |Ψ⟩. Once qubit A's state is determined through measurement, qubit B's state becomes instantaneously correlated with qubit A's measurement outcome. If qubit A is measured to be in the state |0⟩, then qubit B's state is now also |0⟩, even if qubit B is far away from qubit A.

Instantaneous Effect: The correlation between entangled qubits is instantaneous, regardless of the physical separation between them. This feature of quantum entanglement is sometimes referred to as "spooky action at a distance." It appears to violate classical notions of locality and suggests that quantum states cannot be fully described by classical physics.

It's important to note that entanglement and its effects are unique to the quantum realm and do not have direct analogs in classical physics. The measurement of one qubit collapses the joint state of the entangled system, leading to correlated measurement outcomes for the other qubits. This property is a fundamental aspect of quantum mechanics and has implications for quantum information processing, quantum communication, and quantum cryptography.