Complementarity
The complementarity and the uncertainty relations are at the heart of the quantum
theory. Both imply that quantum systems have mutually excluding properties: the
precise knowledge of one of them is incompatible with the precise knowledge of the
other. A classic example is the wave-particle duality: the knowledge of the trajectory
followed by a particle within an interferometer is incompatible with the existence of
interference.
two slits
screen
D2
D1
Despite their common origin (i. e., the superposition principle) the relation between
complementarity and uncertainty relations has been controversial from the beginning of
the quantum theory until the present day. The classic examples of complementarity
were explained as consequences of position-momentum uncertainty relations associated
to the measuring apparatus: the destruction of the interference would be caused by the
random perturbation of the trajectory caused by the detection scheme.
However, recently some subtle examples of complementarity have been proposed and
carried our experimentally where seemingly the standard uncertainty relations would
play no role. Moreover, the detection scheme does not modify at all the trajectories
within the interferometer. These experiences have lead to many authors to suggest the
idea that complementarity is beyond uncertainty relations in the sense that it would not
be possible to explain its origin in terms of the alteration of the observed system caused
by the observing mechanism. The complementarity would be a consequence of quantum
correlations (entanglement) between the system and the apparatus, without involving
concepts such as fluctuations and uncertainty relations.
Nevertheless, we think it is still clear that the observation disturbs the observed system.
Actually, the system-apparatus correlations are the effective mechanism leading to such
disturbance. According to this reasoning, the demonstration of the enforcement of
complementarity by fluctuations and uncertainty relations reduces to the investigation
of the variables of the observed system that are disturbed (maybe not evident at a first
sight) and also to find the uncertainty relations at work in every example (maybe
different from the standard product of variances for position and momentum).
Following this approach we have been able to demonstrate that even in the most subtle
examples of complementarity it is possible to explain its enforcement by means of
suitable uncertainty relations applied to the relevant variables.
The above mentioned subtle examples of complementarity take place usually in twodimensional systems (the two internal trajectories for a double-beam interferometer).
For finite-dimensional spaces, the standard uncertainty relations in the form of product
of variances are always trivial, AB  0 , where (A)2  A2    A 2 . This is
trivial because A and B are greater than zero by definition. The origin of this is fact
is that for finite-dimensional systems A and B are always finite. Therefore, if
A  0 then AB  0 . The equality is reached by the eigenvectors of A or B. In the
quantum domain the situation is worse since the above relation does not exclude the
case A  0 and B  0 simultaneously, which is forbidden whenever A, B  0 .
In order to avoid this difficulty we have introduced a new measure of fluctuations better
behaved in this context than variance. This measure is based on the modulus of the
characteristic function: the Fourier transform of the probability distribution. We have
shown that this variable represents a degree of certainty one can have on the value of the
corresponding observable, so we refer to it as the certainty. We have demonstrated that
with this measure it is possible to derive meaningful certainty relations that show that it
is not possible to have maximum certainty for two complementary observables
simultaneously. With the help of these ideas, we have been able to explain the origin of
complementarity in finite-dimensional systems as enforced by certainty relations
verified by the apparatus variables, that decrease the certainty of the system variables.
We have investigated the maximum certainty states. We have generalized the duality
relations in order to encompass an arbitrary number of complementary observable
simultaneously.
Complementarity and certainty relations for two-dimensional systems
A. Luis, Phys. Rev. A. 64, 012103 (2001)
Complementarity and duality relations for finite-dimensional systems
A. Luis, Phys. Rev. A. 67, 032108 (2003)
The complementarity relations discussed above have been applies successfully to the
analysis of coherence and interferometric visibility of an arbitrary number of classical
waves, as detailed in other notes of this same web page.
Quantum-classical correspondence for visibility, coherence, and relative phase for
multidimensional systems
A. Luis, Phys. Rev. A 78, 025802 (2008)
We have addressed an study of complementarity as close as possible to the very basis of
this concept. The main ideas that distinguish our approach from previous analyses are:
i) We consider simultaneous measurements of the two complementary observables
in each realization of the observed system, always performed under the same
experimental conditions in all realizations.
ii) The information provided by the experiment is handled without any prior
assumption about the relation between the measurement outcomes and the
complementary observables studied. The statistics of the measurement is used directly
to infer the state of the system by using Bayesian techniques. Only after this state
inference is that conclusions are derived concerning the complementary observables.
The results are fully practical and operational complementarity relations that agree with
ideal theoretical analysis including the details of the observation procedure.
Operational approach to complementarity and duality relations
A. Luis, Phys. Rev. A 70, 062107 (2004)
Following the operation approach of the preceding work we have investigated the
emergence of complementarity where there is only one realization of the observed
system so that we can perform only one observation of the system. As before, this single
observation allows to infer the state of the system by Bayesian techniques and then to
derive conclusions concerning any observable.
A first result is that for single systems there are true complementarity relations. They
are formally the same that arise when the measurement is repeated many times. The
only difference is that they are more restrictive for single measurements.
We have applied the same analysis and procedure within the classical theory obtaining
results fully similar. This is true complementarity relations for single classical systems
even when the measurement is exact and noiseless for both observables. These relations
are less restrictive that the complementarity relations for single quantum systems but
more restrictive than the analogous relations for quantum systems under a large number
of measurements.
In the classical case these complementarity relations mean that it is not possible to
obtain full information with a single measurement. It can be expected that the quantum
and classical results will differ when considering increasing number of measurements.
Nevertheless, we think it is interesting that for single systems there are fully similar
complemetarity relations both in the classical and quantum domains.
Quantum and classical operational complementarity for single systems
A. Luis, Phys. Rev. A 72, 014106 (2005)
In most of the paradoxical examples of complementarity, the relevant variables which
are disturbed by the observation are not position or momentum. We have demonstrated
that in most cases the disturbed variable is the phase difference. As a matter of fact, the
alteration experienced by this variable is rather simply expressed by a convolution of
the original phase distribution P ( ) with a distribution of random phase shifts ( )
Pob ( )   d P(   )( ) .
This demonstrates that the observation of the trajectory increases the phase-difference,
fluctuations, which in turn wipe out the interference effect. This explains the mechanism
of the enforcement of complementarity in subtle examples of this quantum
phenomenon.
Complementarity enforced by random classical phase kicks
A. Luis and L. L. Sánchez-Soto, Phys. Rev. Lett. 81, 4031 (1998)
Randomization of quantum relative phase in welcher Weg measurements
A. Luis and L. L. Sánchez-Soto, J. Opt. B: Quantum Semiclass. Opt. 1, 668 (1999)
Reply to “Comment on ‘Complementarity enforced by random classical phase kicks’ ”
A. Luis and L. L. Sánchez-Soto, Phys. Rev. Lett. 84, 2041 (2000)
Usually, quantum observables are represented by Hermitian operators. However this is
not the most general approach. An observation implies a correspondence between
quantum states (density matrices  ) and probability distributions P(s ) . In the most
general case, this correspondence is established by a family of real and positive
operators ( s)  0 in the form P(s)  tr(s). In general, this correspondence does
not define any Hermitian operator. When we take into account these facts about the
definition of quantum observables we are able to demonstrate unexpected and
interesting results in the context of complementarity.
If we restrict ourselves to Hermitian operators complementarity would be a symmetric
property (if A is complementary to B then B is complementary to A) valid for all
observables irrespective of fluctuations measures. These seemingly natural properties
are lost when we consider the most general class of quantum observables (and there are
very common observables that belong to this category, such as the electromagnetic filed
phase). Specifically: i) The very same idea of complementarity depends on the measure
of fluctuations used: two observables can be complementary or not depending on how
fluctuations are measured. ii) Complementarity is not a symmetric relation: A can be
complementary to B while B may not be complementary to A. iii) There are observables
without complementary observables.
At the heart of this behavior we have found a surprising and appealing fact. In general,
the states that determine the statistics of a quantum measurement (via projection on the
state of the system) do not necessarily coincide with the minimum uncertainty states for
the same observable. In other words, the states that represent the observable when it is
measured are different from the states that carry precise information about the same
quantity.
Complementarity for Generalized Observables
A. Luis, Phys. Rev. Lett. 88, 230401 (2002)
A recent experiment has shown that in multiple-beam interferometers it is possible to
observe the trajectory of the particle within the interferometer without destroying the
interference. As a matter of fact, the visibility can even increase. We have demonstrated
that, contrary to what one might expect, the relative phase variable relevant in the
interferometer of the experiment, and the internal trajectories are not complementary
observables. Therefore, the common intuitions about complementarity cannot be
applied to this case.
Complementarity in multiple beam interference
A. Luis, J. Phys. A: Math. Gen. 34, 8597 (2001)