Question

How is a qubit in superposition between $|0 angle$ and $|1 angle$ different from a mixed state of $|0 angle$ and $|1 angle$ (density matrix, quantumcomputing)?

Answer 1

In quantum computing, a qubit in a superposition state between $|0 angle$ and $|1 angle$ and a mixed state of $|0 angle$ and $|1 angle$ (represented by a density matrix) are distinct concepts with different properties.

1. Superposition State: A qubit in a superposition state exists in a coherent quantum state, meaning it can simultaneously be in a combination of the $|0 angle$ and $|1 angle$ states. Mathematically, it can be represented as $alpha|0 angle + eta|1 angle$, where $alpha$ and $eta$ are complex probability amplitudes satisfying the normalization condition $|alpha|^2 + |eta|^2 = 1$. The qubit exists in multiple states simultaneously, and upon measurement, it collapses into one of the basis states ($|0 angle$ or $|1 angle$) with probabilities proportional to the squared magnitudes of the amplitudes. Superposition is a key feature that allows quantum computers to perform parallel computations.

2. Mixed State (Density Matrix): A mixed state, on the other hand, represents a statistical mixture of different pure states. It arises when the observer has incomplete information about the true state of the qubit due to environmental noise or measurement uncertainty. A mixed state is described using a density matrix, denoted by the symbol $ ho$. For a qubit, the density matrix is a 2x2 matrix. In the case of a mixed state of $|0 angle$ and $|1 angle$, the density matrix takes the form:

$$\rho =\left(\begin{array}{cc}{p}_0& 0\\ 0& p_1\end{array}\right)$$