Furthermore, equality is achieved if " U " * is the unitary operator in the polar decomposition of " A ".
42.
It follows that automorphisms of " Q " are in bijective correspondence to unitary or anti-unitary operators modulo multiplication by scalars of modulus 1.
43.
Mathematically, quantum amplification can be represented with a unitary operator, which entangles the state of the optical field with internal degrees of freedom of the amplifier.
44.
In that case, the operator is a partial isometry, that is, a unitary operator from the range of " T " to itself.
45.
Unitary operators emerge from the classical requirement of the conservation of energy of a classical wave propagating through lossless media that alter the polarization state of the wave.
46.
If time reversal were implemented as a unitary operator, it would reverse the sign of the energy just as space-reversal reverses the sign of the momentum.
47.
In operator theory the mapping takes self-adjoint operators " T " onto unitary operators " U " not containing 1 in their spectrum.
48.
Next, Hamiltonian simulation techniques are used to apply the unitary operator e ^ { iAt } to | b \ rangle for a superposition of different times t.
49.
The group of all unitary operators from a given Hilbert space to itself is sometimes referred to as the "'Hilbert group "'of, denoted.
50.
Then there is a measure space and a real-valued essentially bounded measurable function " f " on " X " and a unitary operator such that
