View source for Interferometer Heisenberg limit

The shot noise limit is not fundamental.
Here is a simple argument that something better should be possible.
Recall that the desired signal at the output of our Mach-Zehnder
interferometer is <math>X=  a^\dagger  a -  b^\dagger  b </math>, and the noise is <math>Y= a^\dagger  b+ b^\dagger  a</math>.
If the inputs have <math>  a^\dagger  a =n</math> and <math>  b^\dagger  b =0</math>, and if <math>Y</math> were zero, then
the measured signal would be <math>n\cos\phi</math>.  And at the balanced
operating point <math>\phi=\pi/2</math>,
:<math> 
	\frac{|\Delta m|}{\Delta \phi} = n\sin\phi \leq n
\,.
</math>
Thus, if the smallest photon number change resolvable is <math>\Delta m=1</math>,
then <math>n\Delta\phi \geq 1</math>, from which it follows that
:<math> 
	\Delta \phi \geq \frac{1}{n}
\,.
</math>
This is known as the "Heisenberg limit" on interferometry.  There
are some general proofs in the literature that such a limit is the
best possible on interferometry.  It governs more than just
measurements of phase shifters; gyroscopes, mass measurements, and
displacement measurements all use interferometers, and obey a
Heisenberg limit.

The argument above only outlines a sketch for why <math>1/n</math> might be an
achievable limt, versus <math>1/\sqrt{n}</math>; it assumes that the noise <math>Y</math>
can be made zero, however, and does not provide a means for
accomplishing this in practice.
Many ways to reach the Heisenberg limit in interferometry are now
known.  Given the basic structure of a Mach-Zehnder interferometer,
::[[Image:chapter2-quantum-light-part-5-interferometry-l7-generic-mzi.png|thumb|400px|none|]]
one can consider changing the input state <math>|\psi_{in}{\rangle}</math>, changing the
beamsplitters, or changing the measurement.

Common to all of these approaches is the use of entangled states.  How
entanglement makes Heisenber-limited interferometry possible can be
demonstrated by the following setup.  Let us replace the beamsplitters
in the Mach-Zehnder interferometer with entangling and dis-entangling
devices:
::[[Image:chapter2-quantum-light-part-5-interferometry-l7-entangled-mzi.png|thumb|400px|none|]]

Conceptually, the unusual beamsplitters may be the nonlinear
Mach-Zehnder interferometers we discussed in Section~2.3.  They may
also be described by simple quantum circuits, using the Hadamard and
controlled-{\sc not} gate; for two qubits, the circuit is
::[[Image:chapter2-quantum-light-part-5-interferometry-l7-entangler1.png|thumb|200px|none|]]
Note how the output is one of the Bell states.  For three qubits, the
circuit is
::[[Image:chapter2-quantum-light-part-5-interferometry-l7-entangler2.png|thumb|400px|none|]]

This output state, <math>|000{\rangle}+|111{\rangle}</math> (suppressing normalization) is
known as a GHZ (Greenberger-Horne-Zeilinger) state.  Straightforward
generalization leads to larger "Schrodinger cat" states
<math>|00\cdots0{\rangle}+|11\cdots 1{\rangle}</math>, using one Hadamard gate and <math>n</math>
controlled-{\sc not} gates.  Note that the reversed circuit
unentangles the cat states to produce computational basis states.

The important feature of such <math>n</math>-qubit cat states, for our purpose,
is how they are transformed by phase shifters.  A single qubit
<math>|0{\rangle}+|1{\rangle}</math> becomes <math>|0{\rangle}+e^{i\phi}|1{\rangle}</math>.  Similarly, two entangled
qubits in the state <math>|00{\rangle}+|11{\rangle}</math>, when sent through two phase
shifters, becomes <math>|00{\rangle}+e^{2i\phi}|11{\rangle}</math>, since the phases add.  And
<math>n</math> qubits in the state <math>|00\cdots 0{\rangle}+|11\cdots1{\rangle}</math> sent through <math>n</math>
phase shifters becomes <math>|00\cdots 0{\rangle}+e^{ni\phi}|11\cdots 1{\rangle}</math>.

When such a phase shifted state is un-entangled, using the reverse of
the entangling circuit, the <math>n</math> controlled-{\sc not} gates leave the
state <math>|00\cdots 0{\rangle}[ |0 \rangle  + e^{ni\phi}|1 \rangle  ]</math>, where the last <math>n-1</math>
qubits are left in <math>|0{\rangle}</math>, and the first qubit (the qubit used as the
control for the {\sc cnot} gates) is
:<math> 
	\frac{|0 \rangle  + e^{ni\phi}|1{\rangle}}{\sqrt{2}}
\,.
</math>
Compare this state with that obtained from the single qubit
interferometer, Eq.(\ref{eq:l7-1qubitphase}); instead of a phase
<math>\phi</math>, the qubit now carries the phase <math>n\phi</math>.  This means that the
probability <math>P</math> of measuring a single photon at the output becomes
:<math> 
	P = \frac{1+\cos(n\phi)}{2}
\,.
</math>
The standard deviation, from repeating this experiment, on average,
would be
:<math> 
	\Delta P = \sqrt{P(1-P)} = \frac{\sin(n\phi)}{2}
\,.
</math>
Using <math>dP/d\phi = -n\sin(n\phi)/2</math>, we obtain for the uncertainty in <math>\phi</math>,
:<math> 
	\Delta\phi = \frac{\Delta\phi}
	{\left| \frac{dP}{d\phi} \right|} = \frac{1}{{n}}
\,,
</math>
which meets the Heisenberg limit.