Quantum Information Processing, Vol. 5, No. 1, February 2006 (© 2006)
Quantum Computation Violates Mirror Symmetry
Gregor W. Bayer
Received July 5, 2005; accepted December 28, 2005
The CNOT gate is asymmetric with respect to parity. It requires interaction with
the environment, and cannot be realized as an isolated quantum collision.
KEY WORDS: Quantum computation; quantum gates; mirror symmetry;
PACS: 03.67.Lx; 03.65.Ta; 11.30.-j; 75.45.+j.
An elastic collision between spinning particles might be viewed as a kind
of logic gate, where the spin directions represent different bit values.
Since each particle is a representation of the Poincar
e group, character-
ized by mass, spin, and helicity, it is most natural to classify the vari-
ous transitions in terms of helicity amplitudes.
This formalism provides
a completely general description of quantum collisions involving spinning
For example, proton–proton scattering pp → pp is routinely analyzed
in terms of ﬁve independent amplitudes.
Using a polarized beam, a
polarized target, and polarized detectors, the cross section at a particu-
lar scattering angle is given by the square of the helicity amplitude for the
polarizations selected. Measurements taken at many different angles could
be converted into partial waves, but that is not necessary.
Consider the general case AB → CD, where A = C and B = D.For
particles with two possible spin states, there are a total of 16 helicity
amplitudes. In Table 1, the amplitudes F 1 through F 6 are independent,
while the other ten dependent amplitudes are related by mirror symmetry
(parity), which ﬂips all the helicities, and by time-reversal symmetry, which
reverses the order of the initial and ﬁnal states.
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