Prove that a unitary transform preserves the inner product between two sets of vectors.
A unitary transform is a fundamental concept in quantum information processing that plays a important role in preserving the inner product between sets of vectors. In order to prove this, we need to understand the properties of unitary transforms and how they preserve the inner product.
A unitary transform is a linear operator that preserves the norm of a vector and the inner product between two vectors. Mathematically, a unitary transform U satisfies the condition U†U = I, where U† represents the conjugate transpose of U and I is the identity operator. This condition ensures that the inverse of U exists and is equal to its conjugate transpose.
Let's consider two sets of vectors, A = {a₁, a₂, …, aₙ} and B = {b₁, b₂, …, bₙ}, where aᵢ and bᵢ are complex vectors in an n-dimensional vector space. The inner product between two vectors a and b is defined as ⟨a, b⟩ = a†b, where † denotes the conjugate transpose.
To prove that a unitary transform preserves the inner product between A and B, we need to show that for any pair of vectors aᵢ and bᵢ in A and B respectively, the inner product remains unchanged under the unitary transform U. Mathematically, we need to prove that ⟨Uaᵢ, Ubᵢ⟩ = ⟨aᵢ, bᵢ⟩.
Let's expand the left-hand side of the equation:
⟨Uaᵢ, Ubᵢ⟩ = (Uaᵢ)†(Ubᵢ)
Using the properties of the conjugate transpose, we can rewrite this as:
⟨Uaᵢ, Ubᵢ⟩ = (aᵢ†U†)(Ubᵢ)
Since U is a unitary transform, U†U = I. Therefore, we can substitute U†U for I:
⟨Uaᵢ, Ubᵢ⟩ = (aᵢ†U†U)(Ubᵢ) = aᵢ†(U†U)bᵢ
Since U†U = I, we have:
⟨Uaᵢ, Ubᵢ⟩ = aᵢ†Ibᵢ = aᵢ†bᵢ
We can see that the left-hand side of the equation is equal to the right-hand side, which proves that the unitary transform U preserves the inner product between the sets of vectors A and B.
To illustrate this concept, let's consider a simple example. Suppose we have two vectors a = [1, 0] and b = [0, 1] in a two-dimensional vector space. We apply a unitary transform U given by the matrix:
U = [1/sqrt(2), 1/sqrt(2)] [1/sqrt(2), -1/sqrt(2)]
The inner product between a and b is ⟨a, b⟩ = a†b = [1, 0] [0, 1] = 0. Now, let's apply the unitary transform to the vectors:
Ua = [1/sqrt(2), 1/sqrt(2)] [1, 0] = [1/sqrt(2), 1/sqrt(2)] Ub = [1/sqrt(2), 1/sqrt(2)] [0, 1] = [1/sqrt(2), -1/sqrt(2)]
The inner product between Ua and Ub is ⟨Ua, Ub⟩ = [1/sqrt(2), 1/sqrt(2)] [1/sqrt(2), -1/sqrt(2)] = 0. We can see that the inner product is preserved under the unitary transform.
A unitary transform preserves the inner product between two sets of vectors. This property is a consequence of the unitarity condition U†U = I, which ensures that the inner product remains unchanged. This preservation of inner product is a fundamental property in quantum information processing and is essential for maintaining the integrity of quantum states during computations.
Other recent questions and answers regarding EITC/QI/QIF Quantum Information Fundamentals:
- What will be the continuous change to the interference pattern if we continue to move the detector away from the double slit in very small increments?
- Is the quantum Fourier transform exponentially faster than a classical transform, and is this why it can make difficult problems solvable by a quantum computer?
- What it means for mixed state qubits going below the Bloch sphere surface?
- What was the history of the double slit experment and how it relates to wave mechanics and quantum mechanics development?
- 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 you 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?
View more questions and answers in EITC/QI/QIF Quantum Information Fundamentals