M1L5t.txt

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If I rephrase this theorem in modern language,
I would say we have a qubit, which
is a two-level system characterized
by two numbers, alpha, beta.
And we can do an arbitrary operation in the Hilbert space
of the qubit, which is called an arbitrary single qubit
operation.
So it can be performed by phase shifters and beam splitters.
Just to make sure that this doesn't lead to any confusion,
the phase shift, which is relevant for this Hilbert
space, is a phase shift which is relative between the two
states, and can be dictated by an interference experiment.
Those global phase shifts cannot be detected.
There is no procedure to observe those,
unless you would have a third mode.
But then you would expand the Hilbert space to three levels.
And then, of course, what is the global phase
shift in the two-dimensional Hilbert space
becomes a relative phase shift within the three-dimensional
Hilbert space.
And in modern language, those similar qubit operations
are called quantum gates.
So all gates, which act on a similar qubit,
can be realized with phase shifters and beam splitters.
Let me give you one example for that.
In quantum computation, quantum information science,
there is a very important gate, the Hadamard gate,
which is described by a transformation, which
is 1 minus 1 1 1.
And I said it can be realized with a beam splitter and phase
shifter, so let me just show it to you
in using our symbolic language.
So if you look at the matrix we had derived for the beam
splitter, you realize that we get the Hadamard gate by aiding
the phase shifter by pi.
OK, so in that sense, single qubit operations
are checked off.