How do we normalize the new state after measuring a specific outcome in a two-qubit system?
After measuring a specific outcome in a two-qubit system, it is necessary to normalize the new state in order to ensure that the probabilities of all possible outcomes add up to one. This process, known as state normalization, is important for maintaining the integrity of quantum information and preserving the principles of quantum mechanics.
To understand how to normalize the new state, let us first consider a general two-qubit system. In this system, each qubit can exist in a superposition of two basis states, usually denoted as |0⟩ and |1⟩. The state of the two-qubit system can be represented as a linear combination of the four possible basis states, such as |00⟩, |01⟩, |10⟩, and |11⟩.
When a measurement is performed on this system, it collapses the state into one of the possible outcomes. Let's say we measure the first qubit and obtain the outcome |0⟩. This means that the system has now collapsed into either the state |00⟩ or |01⟩, depending on the outcome of the measurement on the second qubit.
To normalize the new state, we need to calculate the probability amplitudes of the possible outcomes and divide them by the square root of the sum of their squared magnitudes. This ensures that the probabilities of all possible outcomes add up to one, as required by the principles of quantum mechanics.
Let's illustrate this with an example. Consider a two-qubit system in the state (1/√2)|00⟩ + (1/√2)|11⟩. If we measure the first qubit and obtain the outcome |0⟩, the new state will be |00⟩. To normalize this state, we calculate the probability amplitudes of the possible outcomes:
P(|00⟩) = |(1/√2)|² = 1/2
P(|01⟩) = |0|² = 0
To normalize the state |00⟩, we divide the probability amplitude of |00⟩ by the square root of the sum of the squared magnitudes:
|00⟩ normalized = (1/√2) / √(1/2) = 1/2
Therefore, the normalized state after measuring the specific outcome |0⟩ is |00⟩ normalized = 1/2|00⟩.
To normalize the new state after measuring a specific outcome in a two-qubit system, one needs to calculate the probability amplitudes of the possible outcomes and divide them by the square root of the sum of their squared magnitudes. This ensures that the probabilities of all possible outcomes add up to one, as required by the principles of quantum mechanics.
Other recent questions and answers regarding EITC/QI/QIF Quantum Information Fundamentals:
- Are amplitudes of quantum states always real numbers?
- How the quantum negation gate (quantum NOT or Pauli-X gate) operates?
- Why is the Hadamard gate self-reversible?
- If measure the 1st qubit of the Bell state in a certain basis and then measure the 2nd qubit in a basis rotated by a certain angle theta, the probability that you will obtain projection to the corresponding vector is equal to the square of sine of theta?
- How many bits of classical information would be required to describe the state of an arbitrary qubit superposition?
- How many dimensions has a space of 3 qubits?
- Will the measurement of a qubit destroy its quantum superposition?
- Can quantum gates have more inputs than outputs similarily as classical gates?
- Does the universal family of quantum gates include the CNOT gate and the Hadamard gate?
- What is a double-slit experiment?
View more questions and answers in EITC/QI/QIF Quantum Information Fundamentals