A unitary transformation is an isometry, as one can see by setting in this formula.

In other words, it is a unitary transformation.

Hence, the two total angular momentum bases are related by a unitary transformation.

Thus minimal Stinespring representations are unique up to a unitary transformation.

Recoupling coefficients are elements of a unitary transformation and their definition is given in the next section.

The attacker cannot perform arbitrary unitary transformations on the challenge quantum.

The relative minus sign appears because the beam splitter is a unitary transformation.

More generally the norm is invariant under a unitary transformation for complex matrices.

More precisely, a unitary transformation is an isomorphism between two Hilbert spaces.

A common first step beyond this is to remove the remaining time dependence in the Hamiltonian via another unitary transformation.

The unitary transformation preserves the commutator of the field operators, so:

Given one representation, another may be obtained by an arbitrary unitary transformation, since that leaves the commutator unchanged.

It follows that there must be a unitary transformation that satisfies:

Since the matrix is positive, it can be diagonalized with a unitary transformation u:

An additional unitary transformation can be applied on the system to accelerate the convergence in some computational schemes.

Most of the properties of the quantum Fourier transform follow from the fact that it is a unitary transformation.

By the fundamental theorem, we may replace the new set by the old set subject to a unitary transformation.

For instance, state transformations relating observers in different frames of reference are given by unitary transformations.

Such operations on the qubits are required to be unitary transformations on the initial state of the qubit.

At the minimum, the occupied orbitals are eigensolutions to the Fock operator via a unitary transformation between themselves.

The matrix elements of this unitary transformation are given by a scalar product and are known as recoupling coefficients.

This is because time-dependent unitary transformations relate operators in one picture to the analogous operators in the others.

The following theorem states that all systems of Kraus matrices which represent the same quantum operation are related by a unitary transformation:

If the unitary transformation is chosen correctly, several components of the combined system will evolve into approximate copies of the original system.

The hermiticity conditions are not invariant under the action of a Lorentz transformation because is not a unitary transformation.