Does the basis with vectors called |+> and |-> represent a maximally non-orthogonal basis in relation to the computational basis with vectors called |0> and |1> (meaning that |+> and |-> are at 45 degrees in relation to 0> and | 1>)?
In quantum information science, the concept of bases plays a important role in understanding and manipulating quantum states. Bases are sets of vectors that can be used to represent any quantum state through a linear combination of these vectors. The computational basis, often denoted as |0⟩ and |1⟩, is one of the most fundamental bases in quantum computing, representing the basis states of a qubit. These basis vectors are orthogonal to each other, meaning they are at a 90-degree angle to each other in the complex plane.
When considering the basis with vectors |+⟩ and |−⟩, often referred to as the superposition basis, it is important to analyze their relationship with the computational basis. The vectors |+⟩ and |−⟩ represent superposition states that are obtained by applying the Hadamard gate to the |0⟩ and |1⟩ states, respectively. The |+⟩ state corresponds to a qubit in an equal superposition of |0⟩ and |1⟩, while the |−⟩ state represents a superposition with a phase difference of π between the |0⟩ and |1⟩ components.
To determine if the basis with |+⟩ and |−⟩ vectors is maximally non-orthogonal in relation to the computational basis with |0⟩ and |1⟩, we need to examine the inner product between these vectors. The orthogonality of two vectors can be determined by calculating their inner product, which is defined as the sum of the products of the corresponding components of the vectors.
For the computational basis vectors |0⟩ and |1⟩, the inner product is given by ⟨0|1⟩ = 0, indicating that they are orthogonal to each other. On the other hand, for the superposition basis vectors |+⟩ and |−⟩, the inner product is ⟨+|−⟩ = 0, showing that they are also orthogonal to each other.
In quantum mechanics, two vectors are said to be maximally non-orthogonal if their inner product is at its maximum value, which is 1 in the case of normalized vectors. In other words, maximally non-orthogonal vectors are as far away from being orthogonal as possible.
To determine if the basis with |+⟩ and |−⟩ vectors is maximally non-orthogonal in relation to the computational basis, we need to calculate the inner product between these vectors. The inner product between |+⟩ and |0⟩ is ⟨+|0⟩ = 1/√2, and the inner product between |+⟩ and |1⟩ is ⟨+|1⟩ = 1/√2. Similarly, the inner product between |−⟩ and |0⟩ is ⟨−|0⟩ = 1/√2, and the inner product between |−⟩ and |1⟩ is ⟨−|1⟩ = -1/√2.
From these calculations, we can see that the inner products between the superposition basis vectors and the computational basis vectors are not at their maximum value of 1. Therefore, the basis with |+⟩ and |−⟩ vectors is not maximally non-orthogonal in relation to the computational basis with |0⟩ and |1⟩.
The basis with vectors |+⟩ and |−⟩ does not represent a maximally non-orthogonal basis in relation to the computational basis with vectors |0⟩ and |1⟩. While the superposition basis vectors are orthogonal to each other, they are not maximally non-orthogonal with respect to the computational basis vectors.
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