Non classicality criteria
A key point of the quantum theory is the existence of nonclassical states displaying
properties incompatible with any classical theory. These states are clearly interesting
since they support the existence of the quantum theory. Thus, the discovery, detection
and characterization of nonclassical states is always an area of interest in quantum
physics, including quantum optics in particular.
At this stage it is convenient to recall that in quantum physics the system states play two
different but complementary roles.
A) They provide complete information about all properties of any system.
B) They determine the statistics of any measurements by projection on the state of the
measured system.
To distinguish both roles we may refer to them as measured states (case A) and
measuring states (case B).
In quantum physics the statistics p(B) of the result B of a measurement is usually
given by the formula
p ( B)   B  A
2
  AB
2
.
This is the scalar product between an state in role A and another in role B. There is a
complete symmetry between measuring and measured states despite their very different
roles.
It is worth noting that that the nonclassicality of measuring states (this is the
nonclassicallity of observables) has been rarely studied since all the effort in this subject
has been devoted to the nonclassical properties of measured states.
Since the outcomes of quantum measurements are almost by definition random the
detection of the nonclassicality depends on the evaluation of an statistical estimator.
Most often this is variance, as it is the case of squeezing of quadratures or
subPoissonian photon-number statistics. The variance X of the observable X is
X 2   x j  x 2 P( j ) ,
j
x   x j P( j ) ,
j
where x j are the possible values for X with probabilities P ( j ) that we have assumed
discrete for simplicity.
Although variance is an estimator with many good properties it has also some
noticeable drawbacks. The main difficulty is that it requires some more or less complex
calculus that in general will amplify unavoidable errors of practical origin. For example,
in the variance the outcomes with increasingly large | x j  x | are exceedingly
overvalued and multiply by very large values the probabilities P ( j ) of very unlikely
outcomes that then experimentally determined with large relative errors. We may refer
to this as tail effect. This leads to paradoxical results since distributions P ( j ) highly
concentrated may have diverging variances.
In this context the purpose of our work is twofold:
i) Discover the most simple non classicality criteria.
ii) Apply the above criteria to both measured and measuring states.
The clssical versión of the above formula for p(B) is
p( B)   dzPA ( z ) PB ( z )
where z represents coordinates in the phase space of the problem (set of variables that
competely specify the classical system state), and PA, B are the probability distribution
on the phase space associated to the system states A, B , whatever they are. Since they
are true probability distributions we have classically
PA, B ( z )  0
and
 dzPA, B ( z )  1 .
Since PA ( z)  0 we have PA ( z ) PB ( z )  PA ( z ) PB max where PB max is the maximum
of PB (z ) so that using that  dzPA ( z )  1 we get
p( B)   dzPA ( z ) PB ( z )  PB max .
Similarly, since PB ( z)  0 we have PA ( z ) PB ( z )  PB ( z ) PA max where PA max is the
maximum of PA (z ) so that using that  dzPB ( z )  1 we get
p( B)   dzPA ( z ) PB ( z )  PA max .
These equations provide upper bounds for the probability of the outcome B provided
that PA ( z)  0 and PB ( z)  0 , respectively. Therefore, the violation of these bounds
would mean that the key classical fact PA, B ( z )  0 is infringed so that the state mut be
nonclassical. The first bound above will be violated by nonclassical measured states
while the second bound will be infringed by nonclassical measuring states.
It is worth noting in passing that the definition of PA, B ( z ) in the quantum domain is far
from being simple problem. Nevertheless, when this is solved the conclusions obtained
above derived from simple reasoning are still valid.
The main properties the above nonclassicality criteria that distinguish this from other
approaches are:
▪ These nonclassical tests requires no data analysis whatsoever. This is just to perform
the measurement and compare the number of outcomes B with the corresponding
bound.
▪ This simplicity implies that the criteria are exceedingly robust under practical
imperfections such as losses, inneficient detection, or low number of repetitions of the
experiment. As a matter of fact, we have shown that, unlike other tests, in this case the
practical imperfections may help to observe nonclassical behaviours.
▪ These criteria are very general in the sense that they apply equally well to every
system and observable. They are equally valid for measuring and measured states.
▪ These criteria are independent of previous nonclassical critera.
Nonclassicality of states and measurements by breaking classical bounds on statistics
A. Rivas y A. Luis, Phys. Rev. A 79, 042105 (2009)
As we have discussed above, the definition of the phase space distributions PA, B ( z ) are
far from being trivial in the quantum case. A consequence of these difficulties is that
there is more than one definition depending on satisfaction of different properties. The
above analysis was based on the so called P and Q functions defined in terms of the
quadrature coherent states.
In a further work we have analyzed nonclassical effects PA, B ( z )  0 when the
definition used for PA, B ( z ) is the one introduced by Terletsky-Margenau-HillKikwood. This definition naturally arises when performing weak measurements so that
the measurement of a given observable is performed causing the minimum perturbation
on the observed system.
An important result is that in this approach the quadrature coherent states display
nonclassical behaviour. This is relevant since these states are always presented as a
basic universal example of classical states.
Nonclassicality in weak measurements
Lars M. Johansen, A. Luis, Phys. Rev. A 70, 052115 (2004)