What are the properties of the unitary evolution?
In the realm of quantum information processing, the concept of unitary evolution plays a fundamental role in the dynamics of quantum systems. Specifically, when considering qubits – the basic units of quantum information encoded in two-level quantum systems, it is important to understand how their properties evolve under unitary transformations. One key aspect to consider is the preservation of the norm (scalar product) of qubits during unitary evolution.
In quantum mechanics, the norm of a quantum state represents the probability amplitude of the state and is important for ensuring the conservation of probabilities. For qubits, the norm is defined as the square root of the sum of the squares of the probability amplitudes of the two basis states (|0⟩ and |1⟩). Mathematically, the norm of a qubit state |ψ⟩ is given by ||ψ|| = √(|α|² + |β|²), where α and β are the probability amplitudes of the |0⟩ and |1⟩ states, respectively.
When a qubit undergoes a unitary transformation, its state evolves according to the Schrödinger equation, which ensures that the evolution is deterministic and reversible. Importantly, unitary transformations preserve the inner product (scalar product) of quantum states, which in turn guarantees the conservation of the norm of the qubit states. This property is important for maintaining the probabilistic interpretation of quantum mechanics and ensuring the consistency of quantum information processing protocols.
However, when dealing with a composite quantum system consisting of multiple qubits, the situation becomes more intricate. In the context of a general unitary evolution of a composite system, the evolution of an individual qubit may not preserve its norm. This is due to the entanglement that can arise between the qubits during the evolution, leading to correlations that can affect the individual qubit states.
For instance, consider a two-qubit system initially prepared in a separable state |ψ⟩ = |0⟩⊗|0⟩. If the system undergoes a unitary transformation that entangles the qubits, the resulting state may be an entangled state such as |ψ'⟩ = (|00⟩ + |11⟩)/√2. In this case, the norms of the individual qubit states are no longer preserved, as the entangled state cannot be factorized into separate qubit states. Consequently, the evolution of a qubit in a composite system can lead to a loss of norm preservation due to entanglement effects.
The norm of qubits is preserved under unitary evolution unless the qubit is part of a composite system undergoing a general unitary transformation that induces entanglement between the qubits. Understanding the preservation of norms under unitary transformations is essential for designing and analyzing quantum information processing tasks, ensuring the integrity of quantum protocols and computations.
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