If we measure only the first qubit in the state (1/2) |01⟩ + (i/2) |11⟩, what is the new state after crossing out inconsistent possibilities?
In the field of Quantum Information, specifically in the context of Quantum Entanglement and Systems of two qubits, let us address the question of measuring the first qubit in a given state and determining the resulting state after eliminating inconsistent possibilities.
Consider the initial state (1/2) |01⟩ + (i/2) |11⟩, where |0⟩ and |1⟩ represent the computational basis states of a single qubit. This state is a superposition of two possible outcomes, where the first qubit is in the state |0⟩ and the second qubit is in the state |1⟩, and the first qubit is in the state |1⟩ and the second qubit is in the state |1⟩, respectively. The coefficients (1/2) and (i/2) represent the probability amplitudes associated with each possibility.
To determine the new state after measuring only the first qubit, we need to consider the rules of quantum measurement and entanglement. When a measurement is performed on a quantum system, the state "collapses" into one of the possible outcomes with a probability determined by the coefficients in the superposition.
In this case, if we measure the first qubit and obtain the outcome |0⟩, the resulting state will be |01⟩. Similarly, if we measure the first qubit and obtain the outcome |1⟩, the resulting state will be |11⟩. These outcomes are consistent with the initial state and correspond to the possibilities that were not crossed out.
However, it is important to note that once we measure the first qubit, the entanglement between the two qubits is broken, and the state of the second qubit becomes independent of the measurement outcome. Therefore, the measurement of the first qubit does not affect the state of the second qubit.
To summarize, if we measure only the first qubit in the state (1/2) |01⟩ + (i/2) |11⟩, the new state after crossing out inconsistent possibilities will be either |01⟩ or |11⟩, depending on the outcome of the measurement. The measurement outcome determines the state of the first qubit, while the second qubit remains unaffected.
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