shannon entropy of the wigner function and position-momentum correlation in model systems
TRANSCRIPT
SHANNON ENTROPY OF THE WIGNER FUNCTION
AND POSITION-MOMENTUM CORRELATION
IN MODEL SYSTEMS
HUMBERTO G. LAGUNA* and ROBIN P. SAGARy
Departamento de Qu�{mica, Universidad Aut�onoma Metropolitana,
Apartado Postal 55-534, Iztapalapa, 09340 M�exico D.F., M�exico*[email protected]@xanum.uam.mx
Received 30 November 2009
Shannon entropies of the Wigner function are calculated for ground and excited stationarystates of the Particle-In-A-Box and Harmonic Oscillator model systems and examined as a
measure of the localization of the phase-space distribution. We show that their behavior is
consistent with that of the sum of the position and momentum space entropies as a function of
quantum number. Position-momentum correlation is then analyzed in these systems by de¯ningmutual information between position and momentum variables. This mutual information yields
non-zero values, in contrast to the quantum covariance, and increases with quantum number.
Keywords: Wigner function; position-momentum correlation; localization phase-space
distribution.
1. Introduction
One of the main interpretational issues in quantum mechanics is the Heisenberg
Uncertainty Principle (HUP), which implies a certain kind of correlation between the
position and momentum operators. This uncertainty relationship arises from the fact
that the operators do not commute with each other.
The usual formulation of the Schr€odinger equation in quantum mechanics is in
position space, but the problem can also be formulated in momentum space. The
wavefunctions in each space are related through a Dirac-Fourier transformation. One
of the quantum mechanical postulates is that the wavefunction in each space contains
all the available information of the system but that it does not have direct physical
signi¯cance. It is the squared norm of the wavefunction which can be interpreted as a
probability distribution.
It is also possible to formulate quantum mechanics in terms of a joint position-
momentum phase-space function which yields a classical-like picture of the problem.
International Journal of Quantum InformationVol. 8, No. 7 (2010) 1089�1100
#.c World Scienti¯c Publishing Company
DOI: 10.1142/S0219749910006484
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In this work, we restrict ourselves to one-dimensional, one-particle systems but
generalizations to multiple dimensions and particles can be done. Such a quantum
phase-space distribution must respect HUP and the non-commuting properties of the
position and momentum operators.
Among the many phase-space functions de¯ned in quantum mechanics, perhaps
the most widely known is the Wigner function.
It is de¯ned as,1
Wðx; pÞ ¼ 1
}�
Zdy ðxþ yÞ �ðx� yÞe2ipy: ð1Þ
The attractive properties of this distribution are that it is real-valued, its mar-
ginals are the Schr€odinger-representation densities in position [�ðxÞ] and momentum
[�ðpÞ] spaces, and hence the normalization condition is satis¯ed,1�4
Zdp W ðx; pÞ ¼ �ðxÞ;
Zdx W ðx; pÞ ¼ �ðpÞ;
Zdxdp Wðx; pÞ ¼ 1:
ð2Þ
The price to be paid in the attempt to formulate quantum mechanics with a joint
position-momentum distribution is that the Wigner function can assume negative
values. This does not permit its strict interpretation as a joint probability of position
and momentum (which is forbidden by the uncertainty principle). It is thus labeled as
a \quasi-density" or a \quasi-probability" density. Negative parts of the Wigner
distribution have been related with entanglement.5,6 There are however, quantum
mechanical systems which have positive de¯nite Wigner functions, including coher-
ent states and the ground state of the Harmonic Oscillator.
Position-momentum correlations are involved in the EPR phenomena7 and in
entanglement, and is inherent to quantum mechanics. The study of these correlations
in position-space or in momentum-space cannot be directly realized and thusmakes its
study di±cult. The Wigner function, being a joint position-momentum distribution,
provides one avenue of exploring correlations between position and momentum
although it cannot be strictly interpreted as a probabilistic one.
1.1. The uncertainty principle
The uncertainty principle between position and momentum establishes that the
knowledge of x precludes the knowledge of p, and vice versa, in order to maintain the
lower bound in the relation between their uncertainties. The HUP is usually written
in terms of the standard deviations (�x) and (�p),8,9 which are used to quantify
uncertainties.
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A stronger version of this form of the uncertainty principle was done by
Schr€odinger and includes a term that is the square of the quantum covariance between
position and momentum operators. Hence the uncertainty principle is related with the
statistical correlation between the operators.10
Uncertainties can also be measured by information entropies, which has led to the
formulation of an entropic uncertainty relation11 that has been shown to work better
than the usual standard deviation formulation in some aspects.12,13 This entropic
uncertainty relationship takes the form,11
st ¼ s� þ s� � 1þ ln�; ð3Þwith,
s� ¼ �Z
dx �ðxÞ ln �ðxÞ;
s� ¼ �Z
dp �ðpÞ ln �ðpÞ;ð4Þ
the Shannon information entropies in position and momentum space respectively.
These Shannon entropies measure the uncertainties or the extent of localization-
delocalization. Smaller values are indicative of a more localized distribution and
larger values associated with one that is more delocalized.
The entropic uncertainty relation can also be formulated in terms of the infor-
mation entropy of a separable phase-space distribution, ½�ðxÞ�ðpÞ�,14
st ¼ �Z
dxdp �ðxÞ�ðpÞ ln �ðxÞ�ðpÞ: ð5Þ
What is important here is that the separable form of a phase-space distribution
involves a certain kind of correlation between position and momentum which has as
its source an uncertainty relationship and is due to the noncommutativity of the
variables. This type of correlation is distinct from the statistical one since a separable
phase-space distribution would yield zero statistical correlation.15 The next logical
step would be to inquire as to the information entropy of a non-separable phase-space
distribution, such as the Wigner function, and its comparison with st to ascertain any
additional information which it might contain. This is done in the next section.
1.2. Shannon entropy of the Wigner function
The formulation of classical-like entropic expressions involving quantum mechanical
phase-space distributions have been reported in the literature.16�18 These have
usually been restricted to positive de¯nite distributions such as the Husimi func-
tion.19,20 Much less attention has been paid to entropic expressions based on the
Wigner function possibly due to the property that it can attain negative values.
The Shannon entropy of the Wigner function is de¯ned as,14
sw ¼ �Z
dxdp W ðx; pÞ lnW ðx; pÞ: ð6Þ
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In fact, the negative values of the Wigner function in some regions are not proble-
matic for the logarithm if we use the convention of working in the principal branch of
the logarithmic function.21,22 Thus the negative regions of the Wigner function imply
that its entropy is complex valued.
For practical calculations, Eq. (6) can be split into regions where the Wigner
function is positive (Wþ) and negative (W�),
sw ¼ �Z
dxdp W þðx; pÞ lnW þðx; pÞ
�Z
dxdp W �ðx; pÞ ln jW �ðx; pÞj � i�
Zdxdp W �ðx; pÞ; ð7Þ
where we have used the absolute value of the negative part (\j j") and that in the
principal branch of the logarithm we have lnð�1Þ ¼ i�. It is important to note that
the imaginary part of the entropy is proportional to the volume of the negative
regions of the Wigner function.
The Shannon entropy of the Wigner function is one measure of the structure or
localization-delocalization inherent in the phase-space distribution. The concept of
the localization of phase-space distributions and the link with uncertainty relation-
ships has been explored.23�27
Contributions to the real part of the entropy come from the ¯rst two terms in the
equation above. The ¯rst term (W þ) contributes as a normal entropy. If the function
possesses values in the interval ½0; 1�, its entropic contribution is positive, which leads
to larger values of the entropy. Values of the function larger than unity leads to
contributions which decrease the entropy. Note that the second term (W �) con-
tributes in an inverse manner to that of the ¯rst term due to the negativity of the
function. That is, values in the interval ½0; 1� lead to negative entropic contributions
and hence smaller values of the entropy.
1.3. Position-momentum correlation
Before proceeding, it would be appropriate to mention some issues of position-
momentum correlation. It has been shown that the correlation coe±cient for two
quantum variables that do not commute is lesser that unity, which is the classical
bound. In quantum systems, correlation between position and momentum is expected
to be lesser than in the classical ones.28
The study of correlation resembles statistical theory and functional relationships
between variables. Covariance or its normalized form, the correlation coe±cient, has
been used in the study of correlation in quantum systems, as in Ref. 29. Speci¯cally
for the case of position-momentum correlation, it can be studied by de¯ning a
quantum analogue as,30
Covðx; pÞ ¼ hx̂p̂ þ p̂x̂i2
� hx̂ihp̂i ð8Þ
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or the quantum correlation coe±cient
R ¼ Covðx; pÞ�x�p
: ð9Þ
However, if we measure position-momentum correlation through quantum
covariance, we have the problem that it vanishes for any stationary state.28,30,31 This
does not imply that there is no statistical correlation between variables, only that the
covariance is unable to detect it.
Position-momentum correlation can also be understood in terms of the localiz-
ation of a joint phase-space distribution, although localization in the distribution is
restricted by the uncertainty principle.15
1.4. Mutual information
Mutual information, from information theory, provides another measure of the
statistical correlation between variables. Mutual information between two variables
is de¯ned as the di®erence between Shannon entropies32,33 at one- and two- variable
levels, and is a measure of the statistical distance between a joint distribution from a
separable distribution formed by the product of their marginals.
For \true" probability distributions, i.e. positive de¯nite ones, it can be shown
that mutual information is non-negative, and only zero when the distribution is
separable.34 This result can be related, through the de¯nition of mutual information
involving conditional entropies, with the fact that the knowledge of one of the
variables can only diminish the uncertainty of the other.
The previous interpretation is a natural one in the classical theory of probability,
but what if the two variables do not commute as in the case of position and
momentum operators in quantum mechanics? For a distribution that can assume
negative values, the classical property that the mutual information is always greater
than or equal to zero would also not be strictly true.
A mutual information using the Wigner function can be de¯ned,
Iðx; pÞ ¼Z
dxdp W ðx; pÞ ln W ðx; pÞ�ðxÞ�ðpÞ ¼ s� þ s� � sw
¼ s� � sðxjpÞ ¼ s� � sðpjxÞ ð10Þwhere the conditional entropies are,
sðxjpÞ ¼ �Z
dxdp Wðx; pÞ ln W ðx; pÞ�ðpÞ
� �;
sðpjxÞ ¼ �Z
dxdp Wðx; pÞ ln W ðx; pÞ�ðxÞ
� �:
ð11Þ
Mutual information is attractive since it is the di®erence between st and sw. Thus
one of its terms (st), appears in the entropic uncertainty relationship.
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The purpose of this work is ¯rst to calculate the Shannon entropy of the Wigner
function in simple model systems (Particle-In-A-Box and Harmonic Oscillator) to
ascertain its ability to measure the localization-delocalization inherent in the phase-
space distribution. We will also compare it to st to examine if the interpretations and
results are consistent with the expected behavior in ground and excited stationary
states. We will then study position-momentum correlation by examination of the
mutual information in ground and excited stationary states of the model systems.
2. Results and Discussion
2.1. Shannon entropy of the Wigner function
Majernik et al.12,13 have investigated the formulation of the uncertainty relations in
terms of the position and momentum space entropies in ground and excited states of
the Particle-In-A-Box model (PIAB).
In the calculation of the Wigner function for this system35 one must make some
provisions for the integration limits, because ðx� yÞ are non-zero only in the
interval ½0; a�, and we have to satisfy that 0 � xþ y � a and 0 � x� y � a. This
leads to,
�x � y � þx; if 0 � x � a=2;
�ða� xÞ � y � þða� xÞ; if a=2 � x � a:ð12Þ
For the ½0; a=2� interval, it was shown35 that,
W ðx; pÞ ¼ 2
�}a
� �sin½2ðp=} � n�=aÞx�
4ðp=} � n�=aÞ þ sin½2ðp=} þ n�=aÞx�4ðp=} þ n�=aÞ
�
� cos2n�x
a
� �sin½2px=}�ð2p=}Þ
�: ð13Þ
To express the function in the interval ½a=2; a� we substitute x by ða� xÞ. The
Shannon entropy of the Wigner function was calculated using the above expression in
Eq. (6) and two-dimensional numerical integration with complex arithmetic. The
accuracy of the integrations was checked by monitoring the normalization of the
Wigner function and convergence of the integrals. Units of the entropic quantities are
in nats.
In order to obtain a real number from this entropy to measure localization or
delocalization in a given distribution, we can either use the real part or the norm of
the complex number.
In Fig. 1 we show the plots of Re½sw�, jswj and st for some states with box length
a ¼ 1. The imaginary part, proportional to the volume of the negative region(s) of
the function, increases with the quantum number. In general, all these quantities
increase with n. The position entropy of the PIAB model is constant with the state
[s� ¼ lnð2aÞ � 1] and its momentum entropy increases with n.12,13 Hence the beha-
vior of Re½sw� and jswj is consistent with the entropy sum.
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We must remark that in a given state all entropic quantities are constant with a,
the box length. This result is known for the entropic sum12,13 and admits an
interpretation in terms of the uncertainty principle, because localization in the
x-distribution implies delocalization in the p-distribution, and vice versa.
In the Harmonic Oscillator (HO) model, the Shannon entropies in each space and
the entropic uncertainty relation have been investigated.36�38 For this model, with
} ¼ m ¼ ! ¼ 1, the Wigner function is de¯ned as,39,40
W ðx; pÞ ¼ ð�1Þn�
e��2 Lnð�Þ ð14Þ
where n is the quantum number, � ¼ 2ðx2 þ p2Þ and Lnð�Þ is the nth-order Laguerrepolynomial.
In Fig. 2 we show the plot of Re½sw�, jswj and st for some states with potential
! ¼ 1. As in the PIAB case, all these quantities increase with the quantum number,
n. Moreover, the behavior of the two possible de¯nitions of the Shannon entropy of
the Wigner function is similar to that of the entropy sum.
Having established that the de¯nitions of the Shannon entropy of the Wigner
function behave similarly to st leads us to the next question: what is the di®erence
between sw and st? This di®erence, as quanti¯ed by mutual information, is the
subject of the next section.
Fig. 1. (Color on line) Plot of Shannon entropy of the Wigner function against n, the quantum number,
real part, red (�), its norm, green (/) and for the entropic sum, blue (n) for the Particle-In-A-Box model,
for a ¼ 1.
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2.2. Mutual information and position-momentum correlation
The possible (real valued) de¯nitions of the Shannon entropy in the previous section
lead to three di®erent options for the mutual information, i.e. Re½st � sw�, jstj � jswjand jst � swj.
We plot the three possible de¯nitions of mutual information for the PIAB model
in Fig. 3. The curve corresponding to jst � swj increases with n. If one extrapolates
this behavior to large n, its interpretation is consistent with that of a quantity that
measures position-momentum correlation, in the sense that quantum mechanics
reduces to classical mechanics (where position and momentum are more correlated)
for large quantum numbers, as in the case of the correlation coe±cient.28,31 This
curve is always greater than zero due to the de¯nition of the norm.
The curve Re½st � sw� does not have a lower bound of zero since the positivity of
mutual information is a consequence of the positivity of a \true" probability distri-
bution. It should be emphasized that this curve is, of the three de¯nitions, the one
which predicts lower correlation between position and momentum (is closer to zero).
There is no uniform tendency, although for large n the values are greater than for
small n. Thus in these cases it also yields more correlation in the excited states. The
possible shortcoming of this de¯nition is that it does not consider in a complete
manner the presence of the negative regions, taking into account only the absolute
value of the negative parts and neglecting the imaginary part of the entropy,
generated by this negativity.
Fig. 2. (Color on line) Plot of Shannon entropy of the Wigner function against n, the quantum number,
real part, red (�), its norm, green (/) and for the entropic sum, blue (n) for the Harmonic Oscillator model,
for w ¼ 1.
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For the analysis of the jstj � jswj plot, it is convenient to remember that the
positivity of (classical) mutual information implies that the \knowledge of p can only
diminish the uncertainty in x " [s� � sðxjpÞ] and vice versa, which is certainly con-
tradictory with quantum mechanical operator algebra. In this curve for n > 1, one
can say the opposite: \knowledge of p increases the uncertainty in x " [s� � sðxjpÞ],which is consistent with the quantum mechanical uncertainty principle. In this case
correlation between position and momentum also increases with n, consistent with
the quantum-classical comparison. We emphasize that the relationship between
[jstj � jswj] and [s� � sðxjpÞ] is illustrative in nature and not quantitative since the
¯rst is real valued and the second is complex valued.
The three di®erent de¯nitions of the mutual information of the HO model are
plotted in Fig. 4. All de¯nitions of mutual information are sensitive to position-
momentum correlation. In the ground state (n ¼ 0) mutual information is zero
because the Wigner function is separable. This does not imply that the operators
commute, but is indicative only of the Gaussian form of the eigenfunctions. All the
plots of the mutual information are consistent with the interpretation that for large
quantum numbers, the expected correlation between position and momentum is
greater than for the lower lying states. The jst � swj plot is also the one which
predicts the greatest position-momentum correlation and is entirely positive due to
the de¯nition of the norm.
Fig. 3. (Color on line) Plot of mutual information using the Shannon entropy of the Wigner function as a
function of n, the quantum number, for a ¼ 1, for the Particle-In-A-Box model. Re½st � sw�, red (�),jstj � jswj, green (/) and jst � swj, blue (n).
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To examine the di®erences between the HO and PIAB models, we mention that in
the HO case, the Re½st � sw� plot does not yield the minimum position-momentum
correlation. It is instead the jstj � jswj plot which gives the smallest position-
momentum correlation, except for the case n ¼ 1, where both curves are negative.
The interpretation concerning the knowledge precluded of one of the variables by the
knowledge of the other is maintained in the jstj � jswj plot, except for n ¼ 2 state.
At this point the following question should arise: what kind of correlation does the
mutual information between position and momentum measure? This question is
related to that of distinguishing between quantum and classical (or non-quantum)
correlations,41,42 which is an open problem.
We argue that the mutual information is measuring the total statistical corre-
lation. The mutual information is zero in cases where the Wigner function is separ-
able and the only correlation between position and momentum is manifested through
the uncertainty principle due to the noncommuting algebra.
In the cases where a particular de¯nition of the mutual information takes negative
values, the argument may be made that this behavior re°ects the noncommuting
operator algebra in terms of the informational quantities. That is, we can put forward
this fact as \the knowledge of one of the variables provokes the loss of information of
the other".
Fig. 4. (Color on line) Plot of mutual information using the Shannon entropy of the Wigner function as a
function of n, the quantum number, for the potential value w ¼ 1, for the Harmonic Oscillator model.
Re½st � sw�, red (�), jstj � jswj, green (/) and jst � swj, blue (n).
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It is also relevant that the quantum covariance is zero for stationary states and is
thus not able to detect the position-momentum correlations. On the other hand,
mutual information is non-zero and is capable of detecting these correlations.
3. Conclusion
The Shannon entropy of the Wigner function is calculated for ground and excited
stationary states of the Particle-In-A-Box and Harmonic Oscillator model systems
and probed as a suitable measure of the localization of a phase-space distribution.
This entropy is a complex valued quantity. Using the real part and also the norm, we
show that its behavior is similar to that of the entropy sum, as a function of quantum
number. This implies that the information encoded in the uncertainty principle,
through the entropy sum, is also present in the entropy of the Wigner function.
Position-momentum correlation is also probed in these model systems via calcu-
lation of the mutual information between x and p. Contrary to the quantum
covariance which is zero valued in stationary states, we show that mutual infor-
mation is capable of detecting the correlation between position and momentum.
Furthermore, the value of the mutual information increases with n. This is consistent
with the argument that in the limit of large n, one should recover classical behavior,
and that this correlation should be larger than that where quantum behavior
prevails. A discussion of the types of correlation measured by mutual information is
also presented.
Acknowledgments
The authors thank the Consejo Nacional de Ciencia y Tecnología (CONACyT) and
the PROMEP program of the Secretaría de Educación Pública in M�exico for support.
HGL thanks CONACyT for a graduate fellowship.
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