A unitary transformation is a type of linear transformation that preserves the inner product structure of a Hilbert space, meaning it maintains the lengths and angles of state vectors. This characteristic is crucial because it ensures that the probabilities associated with quantum states remain unchanged, making unitary transformations essential for describing quantum evolutions and symmetries. They are represented by unitary operators, which are complex matrices that satisfy the condition of being equal to their own adjoint (or Hermitian conjugate).
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