What is a unitary transformation and how is it represented in a complex vector space?
A unitary transformation, in the context of quantum information processing, refers to a mathematical operation that preserves the inner product of vectors in a complex vector space. It is a fundamental concept in quantum mechanics and plays a important role in quantum information processing tasks such as quantum computation and quantum communication.
In a complex vector space, vectors are represented as column matrices, where each element of the matrix corresponds to a complex number. A unitary transformation is represented by a unitary matrix, which is a square matrix with complex entries that satisfies the condition of being Hermitian conjugate to its own transpose. In other words, the unitary matrix U satisfies the equation U†U = I, where U† denotes the Hermitian conjugate (also known as the adjoint) of U, and I represents the identity matrix.
The unitary matrix U acts on a vector |ψ⟩ in the complex vector space by left-multiplication, resulting in a transformed vector U|ψ⟩. The transformed vector is obtained by applying the unitary matrix to each element of the original vector. Mathematically, this can be expressed as:
U|ψ⟩ = |ψ'⟩,
where |ψ'⟩ represents the transformed vector. The unitary transformation preserves the inner product between vectors, meaning that ⟨ψ|ϕ⟩ = ⟨ψ'|ϕ'⟩, where ⟨ψ| and ⟨ϕ| are the bra vectors corresponding to |ψ⟩ and |ϕ⟩, respectively.
An important property of unitary transformations is that they are reversible. This means that for every unitary matrix U, there exists a unitary matrix U† (the Hermitian conjugate) such that U†U = I. Applying U† to the transformed vector U|ψ⟩ yields the original vector |ψ⟩:
U†(U|ψ⟩) = (U†U)|ψ⟩ = I|ψ⟩ = |ψ⟩.
This reversibility property is important in quantum computation, where quantum gates are implemented using unitary transformations. By applying a sequence of unitary transformations to a set of quantum bits (qubits), it is possible to perform complex computations efficiently.
To illustrate the concept of unitary transformations, consider the Hadamard gate, which is a commonly used quantum gate. The Hadamard gate is represented by the following unitary matrix:
H = 1/√2 * [1 1; 1 -1],
where √2 represents the square root of 2. When applied to a single qubit in the state |0⟩, the Hadamard gate transforms it into the superposition state:
H|0⟩ = 1/√2 * (|0⟩ + |1⟩).
This superposition state is a fundamental concept in quantum computation and allows for parallel processing of information.
A unitary transformation in a complex vector space is a mathematical operation that preserves the inner product of vectors. It is represented by a unitary matrix, which satisfies the condition U†U = I. Unitary transformations are reversible and play a important role in quantum information processing tasks such as quantum computation and quantum communication.
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