Generalized measurements
A basic objective of quantum optics is to develop and improve the potential that light offers for the
realization of complete and accurate measurements. This potential has been enlarged by the
introduction of the so-called generalized measurements. The system to be observed is in a given
unknown state  s which is coupled to an auxiliary subsystem in a known state  aux fixed in
advance. After the coupling, a standard measurement is performed on the whole system.
S
aux
k
The statistics of the measurement P (k ) is given by projection on some family of orthonormal states
k eigenvectors of a Hermitian operator
2
P(k )  k U  aux  s
Since the state of the auxiliary system is known, the information provided by the measurement can
be regarded as information just about the state of the system. This interpretation can be expressed in
formulas by defining the system vectors
~
k   aux U  k
in such a way that the statistics of the measurement is given by the projection of the system state
~
 s on the states k
2
~
2
P(k )  k U  aux  s  k  s
In other words, the original measurement performed on the whole system is equivalent to an
effective measurement performed only on the system of interest.

S
k
~
In general, the states k that determine the effective measurement are not eigenvectors of any
Hermitian operator. As a matter of fact, in general it not clear at first sight the observable that they
might be representing. Therefore, it is necessary to examine each particular case in order to disclose
what is really measured in a generalized measurement and what kind of information about the
observed system provides.
We have examined this question in homodyne detectors with four, eight and twelve ports. On the
one hand, we have found that these measurements may be interpreted as noisy simultaneous
measurements of the Stokes operators. On the other hand we have demonstrated also that they can
provide exact noiseless information about many different observables, such as the quantum phase
operator, or even the complete and exact determination of the Wigner function. It is worth noting
that for most of such operators that there are no known alternative experimental implementations.
Generalized measurements in eight-port homodyne detection
A. Luis and J. Peřina, Quantum Semiclass. Opt. 8, 873 (1996)
Noisy simultaneous measurement of noncommuting observables in eight- and twelve-port
homodyne detection
A. Luis and J. Peřina, Quantum Semiclass. Opt. 8, 887 (1996)
Unbalanced homodyne detection with a weak local oscillator
A. Cives-Esclop, A. Luis and L. L. Sánchez-Soto, Opt. Commun. (aceptado)
An eight-port detector with a local oscillator of finite intensity
A. Cives-Esclop, A. Luis and L. L. Sánchez-Soto, J. Opt. B: Quantum Semiclass. Opt. 2, 526
(2000)
In quantum physics it is customary to link the idea of measurement with the statistics of Hermitian
operators. As we have shown above, this is not always the case and the generalized measurements
imply an extension of the usual understanding about quantum measurement. For example, it has
been shown that there are generalized measurements that provide complete information about the
state of the system, which amounts to know the statistics of any observable.
Furthermore, the idea of measurement can be extended to encompass the measurement of inputoutput processes or even the determination of practical measuring arrangements.
W(
W(k)
In this context we have demonstrated that any process (either transformation or measurement) can
be fully characterized by a function U (k ,  ) which is the analog of the impulse response function
~
W(k )  dU (k , ) W( )

If we are considering an input-output process this function relates the phase-space distributions
~
functions W( ), W (k ) (such as the Wigner function) associated to the input and output states
respectively. If the process is a measurement U (k ,  ) relates the phase-space distribution for the
~
input state W ( ) with the statistics of the measurement W (k ) .
Then we have demonstrated that this function U (k ,  ) can be easily determined experimentally.
This allows us to speak about the measurement of processes on an equal footing with the
measurement of states or observables. In particular, we have generalized the tomographic
reconstruction of quantum states to include also the determination of processes.
Complete characterization of arbitrary quantum measurement processes
A. Luis and L. L. Sánchez-Soto, Phys. Rev. Lett. 83, 3573 (1999)
Measuring quantum input-output processes: phase space representation of transformations
A. Luis and L. L. Sánchez-Soto, Phys. Lett. A 261, 12 (1999)
Quantum tomography of input-output processes
A. Luis, Phys. Rev. A 62, 054302 (2000)
Closely related to these measurement issues we find the practical generation special light states.
This is a very interesting problem since usually the optimization of practical measurements requires
the use of specific (and very often nonclassical) field states tightly adapted to the particular
measuring arrangement considered.
In the context of generalized measurements we have addressed the question of the practical
~
generation of the states k that determine the statistics In other words, the objective was to
transform the implicit formal definition
~
k   aux U 
k
into an explicit generation in practice.
We have demonstrated that this is always possible. To this end we can use the strong quantum
correlations generated in spontaneous parametric down-conversion in a nonlinear crystal. For every
generalized measurement it is possible to perform another measurement (usually they coincide) in
one of the down-converted modes in such a way that the reduced state in the other down-converted
~
mode is in the desired state k .
k’
vacuum
k
Conditional generation of field states in parametric down-conversion
A. Luis and L. L. Sánchez-Soto, Phys. Lett. A 244, 211 (1998)