New Aspects of the Information Content of Classical and Quantized Electromagnetic Fields Edwin A. Marengo

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16 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

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New Aspects of the Information Content of Classical and Quantized
Electromagnetic Fields





Edwin A. Marengo


Department of Electrical Engineering

Technological University of Panama

Panama City, Panama

and

Department of Electrical and Computer Engineeri
ng

The University of Arizona

1230 E. Speedway

Tucson, Arizona 85721





Final Report to the Italian Consiglio Nazionalle della Ricerca (CNR)

as part of a CNR International Scholar Fellowship granted to Dr. Marengo
to visit Seconda Universita di Napoli fro
m September 8
-
22, 2000, where

he worked with Professor Rocco Pierri’s Inverse Scattering group.





February 12, 2001

I.

Introduction.


A problem of considerable practical interest in the wave inversion branches of
electromagnetics and acoustics is that
of characterizing quantitatively the information
content of electromagnetic fields. A number of recent papers by Professor Pierri and
coworkers at Seconda Universita di Napoli have addressed this topic in connection with
different scalar inverse scattering

models
[1, 2]
. The same topic is addressed by Bertero
[3]

and Basinger et al.
[4]
.

In the present report we discuss some results on this and related topics that we
derived during the course of our two
-
we
ek stay at Seconda Universita di Napoli. There
we had the very motivating opportunity of interacting with several of Professor Pierri’s
colleagues and students.

A problem that attracted (almost immediately) our attention was to generalize
some of Professo
r Pierri’s group’s results on the information content of fields into the
corresponding three
-
dimensional (3D) electromagnetic picture. This problem has not
been addressed in the literature, and it appears to be an interesting avenue for further
investigati
ons. In this report we present some results in connection with this aspect of the
information content of fields.

Another topic, closely connected to the former, and which we had the opportunity
to sketch, partly during this Italy visit, is the characteriz
ation of the information content
in the quantized electromagnetic field case. This topic is of major importance in the
current scientific and technology trends toward the realization of quantum computers,
quantum communication systems and possibly quantum
-
entangled imaging systems
which are part of our own current research ventures. In this report I shall therefore also
mention a number of new and exciting ideas in this other topical area, some of which also
originated in Italy. Most of the quantum portion
of the present report, however, matured
later on once we came back to America and had time to review the literature further. It
was then when we prepared the corresponding mathematical formalism, a summary of
which is contained in this report.


II.

Summary o
f the Two
-
Week Visit at Seconda Universita di Napoli, and
Future Projections of Our Research Interaction With the Italian Group.


During the two
-
week stay in Italy, I presented two seminars to the Inverse
Scattering academic community at Seconda Universita

di Napoli; one
on The Future of
Quantum Information and Its Applications
; and another seminar on
The Inverse Source
Problem and Nonradiating and Minimum Energy Sources
.

I also did research with Drs. Angelo Liseno and Francesco Soldovieri, Professor
Giova
nni Leone, Professor Rocco Pierri and Professor Adriana Brancaccio, all of them at
Seconda Universita di Napoli, on the subject of
The Information Content of
Electromagnetic Fields in Three
-
Dimensions
. The key ingredients of the associated
theory and resul
ts, some of which are still under current development, are outlined in this
report.

The interactions with Dr. Angelo Liseno also contributed to elaborate further
some key research concepts on quantum information with electromagnetic fields, some of
which
are contained in this report.

It is expected that at least one major joint publication to the international
electromagnetic community will result, in the long term, from this two
-
week research
interaction, with Professor Pierri and some of his coworkers (m
ainly Drs. Angelo Liseno
and Francesco Soldovieri and Professor Giovanni Leone, with whom there was a
significant and very fruitful amount of ideas
-
exchange during these two weeks).

We wish to note that, besides the above
-
summarized direct deliverables, t
he
present research opportunity has also indirectly opened the window for a new, fresh flow
of ideas from world
-
renowned workers in two different areas that are expected to merge
in the near future as a consequence of advances in quantum information techno
logy; in
particular, my own research team on quantum communications and imaging with
entangled states at the University of Arizona, which includes Professor Richard
Ziolkowski, and the group by Professor Pierri which has paved the way in recent years in
th
e related area of characterizing the information capabilities of classical fields. A merge
of the two topics into
an integrated quantum information theory of quantized
electromagnetic fields for communications, signal processing and imaging

will play an
es
sential role in future practical implementations of such quantum information
technology.

Another possibility is a future interaction for research to be carried out in Panama
by the USA Army in the area of ground penetrating radar, UXO and land mine detect
ion
and remediation. The latter is a major focus area of the Technological University of
Panama and Yuma Proving Ground, a major testing facility of the USA Army, located in
Yuma, Arizona. The work of Professor Pierri and coworkers in the general area of
i
nverse scattering is of immense practical impact in future applications of ground
penetrating radar for UXO and land mine remediation tests and experiments to be carried
out in tropical soils in Panama, and in which we are also heavily involved. The latter

are
part of a global USA effort to gather data and characterize performance measures and
detection/remediation strategies in tropical regions which have not been fully
characterized. The extension of land mine and UXO
-
contaminated regions all over the
wor
ld, including Europe, makes this research area a subject of much humanitarian
interest at both governmental and academic circles. We therefore envision the possibility
for further interactions with Professor Pierri and his coworkers not only on the two
top
ical research areas noted above but also, as the projects and resources permit, on
future deployments of ground penetrating radar equipment and tests in the UHF
frequency regime in the tropical, laterites
-
type soil of tropical
-
humid
-
heavily vegetated
Panam
a.


III.

Research Notes On The Information Content of Classical Electromagnetic
Fields.


General Scalar Inverse Scattering Formulation With Multipole Fields


We consider first the scalar inverse scattering problem to the scalar wave equation
in free space, i.
e.,

(

2

+ k
2
)


= V






(1)

where


is the total (incident plus scattered) field so that


(

2

+ k
2
)(

s

+

i
) = V(

s

+

i
).



(2)

In the Born approximation, the scattered field



s
=


d
r
G V

i





(3)

where G is the free space Green function and d
r

is
the (three
-
dimensional space)
differential element.

The scattered field can be expanded in the multipole expansion form



s
=

l,m

a
l,m

l,m






(4)

where the multipole field


l,m
= h
l
+
(kr)Y
l,m
(s)





(5)

where h
l
+
(kr) is the spherical Hankel function of

the first kind and order l, Y
l,m
(s) is the
spherical harmonic of degree l and order m, and s

r
/r is the unit vector in the direction of
the position vector
r
. The multipole moments a
l,m

of the scattered field are then given by


a
l,m

=


l,m
|V

i

X

=


d
r


l,m
*

V

i




(6)

where * denotes the complex conjugate and

l,m

is the source
-
free multipole field which
is defined by



l,m
= j
l
(kr) Y
l,m
(s)





(7)

where j
l
(kr) is the spherical Bessel function of order l.


In this formulation, the inverse scattering p
roblem can be stated as being that of
reconstructing a scattering potential V from knowledge of the multipole moments
a
l,m
(L,M) corresponding to scattering experiments involving

L,M

as incident fields (since
the most general incident field in a spherical
domain enclosing the source region can be
expanded as a sum of all such

L,M
spherical waves). Then we have a scattering matrix A
with entries


A(l,m,L,M) = a
l,m
(L,M) = “a
l,m

for

i

=

L,M


so that from (6) with

i

=

L,M


a
l,m
(L,M) =


d
r


l,m
*

V

L,M
.




(8)

Let us assume that the scattering potential is entirely confined within a spherical
volume D:r

R. Then we define the Hilbert spaces X and Y of the solution (scattering
potentials V) and of the data (scattering matrices A), respectively, and assign to

them the
usual L
2

inner products


V
1
| V
2

X

=


d
r

V
1
*

V
2


and


A
1
| A
2

Y

=

l,m,L,M
A
1
*
(l,m,L,M) A
2
(l,m,L,M).

(9)


By means of the usual formalism we define the forward mapping L:X

Y which
is found from (8)
-
(9) to be given by

A(l,m,L,M) =


l,m
|V

L,M

X

=


d
r


l,m
*

V

L,M
= (LV)(l,m,L,M).

(10)


The adjoint L
+
:Y

X is defined by



V|L
+
A

X

=

LV|A

Y






(11)

and is found from (9)
-
(11) to be defined by


(L
+
A)(
r

) = M(r )

l,m,L,M

A(l,m,L,M)

l,m


L,M
*


(12)

where M(r ) is a masking function that is 1 if
r

R and is 0 otherwise. The inverse
scattering solution, i.e., the scattering potential V corresponding to a suite of multipole
-
to
-
multipole mapping experiments from which we derive the scattering matrix A, is then
given by the usual pseudoinverse operator

for (L, L
+
). Here we are concerned with the
information content of the fields instead of on the inverse scattering solution, so we shall
come back later on to the results above from that other perspective. However, we wish to
furnish first the associated
formulation for the special case of a spherically symmetric
scattering potential, i.e., a radially
-
dependent scattering potential. This is done next.


Scalar Inverse Scattering Formulation With Multipole Fields for a Spherically Symmetric
Object



At this
point we can look at the special case of a radially
-
dependent scattering
potential V(
r

) = V(r ). To deal with this case, it is convenient to define the object Hilbert
space X of L
2

radially
-
dependent scattering potentials to which we assign the weighted
i
nner product



V
1
| V
2

X

=


dr r
2

V
1
*

V
2







(13)

where it is understood that the scattering potentials are functions of the radius r only.

The (V
-
to
-
A) forward mapping is defined by


A(l,m,L,M)

=


dr r
2

V(r ) j
l
(kr) j
L
(kr)


ds Y
l,m
*
(s) Y
L,M
(s)




=

l,L

m,M



dr r
2

V(r ) j
l
(kr) j
L
(kr)



(14)

where we have used the orthogonality property of the spherical harmonics. The scattering
matrix in (13) has only diagonal non
-
zero elements and off
-
diagonal zero elements. We
call


b
l

=


dr r
2

V(r ) j
l
2
(kr)






(15)

so that (13) reduces to


A(l,m,L,M) = b
l


l,L

m,M






(16)

and the corresponding multipole expansion of the scattered field for the l,m
-
incident
field is found from (4)
-
(5) and (16) to be



s
(l,m)

=

l,m

A(l,m,l,m)

l,m

=

l

j
l
(kr)

m

A(l,m,l,m)Y
l,m
(s)

=

l

b
l

j
l
(kr)

m
Y
l,m
(s).

In these expansions, the relevant terms are, as is well known, l=0,…,N

kR
whereas m goes from

N to N. The reason why the terms in l>N shall not be considered
is well known, and has to do with the fact that in the scattere
d field expansion above,
such terms will necessarily be negligible (and prone to high impact measurement errors
in the presence of noise), and this holds due to basic considerations of realizability of
fields for the associated inverse source problem that
can be shown readily to extend also
into the current inverse scattering problem. In view of (15)
-
(16), we see that the
independent pieces of information about the scatterer are provided only by the
coefficients b
l
, so there are only N+1 independent pieces
of information that we can
gather about the scatterer in this linearized forward model (the Born approximation), and
they are the projections of the object into the N+1 functions j
l
2
(kr), for l=0, N

kR. The
number of degrees of freedom (NDF) (as defined in

[1]
) in this simple case is then N

kR
where k is the wavenumber of the field and R the radius of the smallest source region
enclosing the unknown scatterer. The reconstructible scattering potential profiles V(r ) of
this model are simply
those for which all b
l

for l>N are negligible. Furthermore, we may
also want to know exactly which are the expansion functions that relate to those
coefficients, and for this purpose we look next into the associated adjoint problem and the
operator LL
+
:Y

Y
.

It is convenient to define next the Hilbert space of data vectors A(l) to which we
assign the inner product



A
1
| A
2

Y
=

l
A
1
*
(l)A
2
(l).

We have discarded the m
-
dependence; it is not relevant in this case due to the spherical
symmetry as was shown in (14)
-
(
16).

Now, from (13)
-
(16) and (11) with the inner products of X and Y as defined
above one obtains


(L
+
A)(r ) = M(r )

l

A(l) j
l
2
(kr)






(17)

where we recall from (16) that A(l) is equal to b
l
.

The reconstructed scattering potential is given by the formul
a


V(r ) = [L
+
(LL
+
)
-
1
A](r ).







(18)

We see from (13)
-
(18) that LL
+
:Y

Y is defined by


[(LL
+
)A](l) =

n
A(n)



dr M(r ) r
2

j
l
2
(kr) j
n
2
(kr).


The singular functions and singular values for this illustrative example are readily
determined. The singular

data vectors are, apart from a normalization, simply the vectors
A
l

having entries A
l
(n)=

l,n
b
l
. The singular scattering potential functions V
l
(the basis for
the Born
-
reconstructible profiles) are, apart from a normalization, simply

V
l
= M(r ) j
l
2
(kr).

The singular values are defined by the integrals

l
2

=


dr M(r ) r
2

j
l
4
(kr).


Summarizing, these manipulations in the multipole domain revealed the
information content of Born scattered fields and the class of reconstructible spherically
symmetric func
tions associated with that model. We consider next the special case of
scattering potentials having angular dependence only.


Information Content of Born Scattered Fields For Angularly
-
Dependent Objects



In this special case,



A(l,m,L,M) =


l,m
|V

L,
M

X

=


d
r


l,m
*

V

L,M

reduces to


A(l,m,L,M) = b
l,L



ds Y
l,m
*
(s) Y
L,M
(s) V(s)

where b
l,L

=


dr M(r ) r
2

j
l
(kr) j
L
(kr) where M(r ) is the masking function of the preceding
developments (and that is equal to 1 in the interior of the spherical volume of
radius R
and 0 outside).


We recall that
[5]


Y
l,m
(s) =

l
m

P
l
m
(cos

)exp(im

)

where

l
m

=(
-
1)
m

[(2l+1)/4


(l
-
m)

/(l+m)

] and P
l
m

(.) is the associated Legendre
polynomial of degree l and order m. In the most general case, any scattering p
otential that
depends only on the angular variables can be expanded as a series of the spherical
harmonics. In that case the independent pieces of information about the scattering
potential are contained in the coefficients of the associated spherical harm
onic expansion.
The relevant terms in the expansion from the point of view of stability are the first N+1 l
-
terms, which apply to both l and L, and the 2N+1 m and M
-
dependencies. This implies a
total of [(N+1)(2N+1)]
2
independent harmonic
-
expansion coeffic
ients. A special case of
interest for which the information content is more reduced than that general total, is the
special case of an object with separable azimuth
-

and polar
-
angle dependencies. We will
look first at the azimuth dependence separately. Lat
er we will look at the corresponding
polar dependence. We will find that the results below for the azimuth dependence agree
with those of the 2D analysis in
[2]
, although the present arguments are directly in the 3D
multipole domain which
is unique to this presentation. We will also outline the more
general theory which makes use of integrals of three spherical harmonics (as opposed to
the more familiar integrals involving only two spherical harmonics, and that make use of
the well known or
thogonality properties). The theory in question borrows from
developments in group theory and, in particular, the use of Clebsch
-
Gordan coefficients
[5]
.


Special Case: Azimuthal Dependence.


We write


V(

) =

q

C
q

exp(iq

)


so that


A(l,
m,L,M) = b
l,L



ds Y
l,m
*
(s) Y
L,M
(s) V(

)


=

l
m

b
l,L



d


sin


P
l
m
(cos

) P
L
M
(cos

)

q

C
q



d


exp[i(
-
m+M+q)

]


= 2


l
m

b
l,L



d


sin


P
l
m
(cos

) P
L
M
(cos

)

q

C
q


M+q
-
m,0
.

Since the M and the m indices go from

N to N in a realistic implementation of this
a
pproach, we deduce that the number of relevant coefficients Cq (the information
content) in the azimuthal dependence alone is of 4N +1 independent numbers. The
minimum occurs when M=N,m=
-
N, so that from the Kronecker delta above we have
N+q+N=0, i.e., q=
-
2
N. The maximum condition occurs when M=
-
N,m=N, and then
q=2N. The total number of relevant coefficients in the sum is then

2N, …,0,…, 2N
which is a total of 4N+1 numbers and this is the same result derived in
[2]

for the 2D
case, as expec
ted, since the azimuthal dependence has the same nature in the 2D and the
3D problems.


Special Case: Polar Dependence.


In this case we expand


V(

) =

q

C
q

P
q
0
(cos

).

Now

A(l,m,L,M) = b
l,L



ds Y
l,m
*
(s) Y
L,M
(s) V(

)

=

l
m

b
l,L

m,M

q

C
q



d


sin


P
l
m
(cos

) P
L
m
(cos

) P
q
0
(cos

).

The integral of the type above, involving three associated Legendre functions, can be
solved using group theory methods. We have carried out some manipulations in this
regard but will not include them here to avoid unessential

mathematical details in this
report. The general approach is described below, though, so as to deduce the number of
degrees of freedom in the polar angular variations alone. These results appear to be new.



If q=0 above, then the integral above reduces
to the usual orthogonality integral of
the associated Legendre functions. If q=1, we have, P
1
0
(cos

)=cos

. Now the integral
takes the form



d


sin


cos


P
l
m
(cos

) P
L
m
(cos

).

By applying the recurrence relations of the associated Legendre functions one ca
n
rewrite this integral in a form involving only two associated Legendre functions. This is
general, and it applies to the most general integral involving three Legendre functions.
Thus one can rewrite any such integral as a sum of other integrals that inv
olve only two
such Legendre functions. This is a key step since later on we can employ the basic
orthogonality relations and evaluate explicitly any such integral. The special case of q=1
yields by a simple manipulation the results in page 753, equations 1
2.189
-
12.192 of
Arfken
[5]
. Of particular interest to us is to know that the integral above for q=1 reduces
to two integrals involving only two associated Legendre functions. Due to orthogonality,
the only relevant contributions come from
the terms l,L+1 and the term l,L
-
1, i.e., the
terms in the polynomials P
l+1
m
(cos

) P
l
m
(cos

) and P
l
-
1
m
(cos

) P
l
m
(cos

). In general q, we
will have terms in the polynomials up to the point where l=L+q, l=L
-
q, so that q will go
from 0 to N (the maximum value

of l and L will be N, so that the maximum q will be
deduced via l=L+q, where l=N, L=0, q=N, and the minimum is obviously q=0). The
negative q
-
terms are not considered because they are not linearly independent to the
positive q
-
terms. In conclusion, the n
umber of independent pieces of information is in
this case N+1 (the N coefficients C
q

in the expansion above). The NDF is then N in this
case in the polar dependence.


Electromagnetic Case.


We also looked in detail into the associated EM case. It was con
cluded by means
of a multipole formalism analogous to the one above that in the case of scalar profiles in
the Born approximation, the information content is identical to that of the scalar
formulation since the object itself is scalar. In the general dyad
ic formulation where the
object is modelled as a dyadic scattering potential
[6]
, however, the information content
is doubled automatically, and in that case the information of the electric multipoles is
totally independent from the inform
ation in the magnetic multipoles. In the scalar object
interrogation with electromagnetic waves, one can use either type of multipole field but
both of them will not provide independent information. These questions where raised in
discussions with Dr. Ange
lo Liseno and Professor Giovanni Leone, and have now been
answered.








Research Notes On The Information Content of Quantized Electromagnetic Fields.


There is a growing interest in quantum information (QI) and its applications.
Areas of interest in
clude quantum computing
[7
-
48]
, quantum decoherence (due to
interactions of the quantum computer with the surrounding environment) and error
correction
[49
-
62]
, quantum information theory
[63
-
71]
, quantum
cryptography
[72
-
90]
,
quantum teleportation
[91
-
99]
, quantum communication networks
[100, 101]
, and
quantum system simulation
[102
-
107]
. There are several other interdisciplinary areas of
more recent interest such as quantum neural networks
[108]
, quantum control and
robotics
[109]
, and quantum game theory
[110
-
112]
.

These very valuable investigations have already established the backgroun
d to
create the required new quantum engineering principles and design skills needed for
future implementations of QI systems. However, to date, little work has been
accomplished on quantum information and its applications within the engineering
communitie
s. We are currently working on developing a new quantum communication
engineering and a new entangled
-
state imaging engineering as key application areas of a
more generalized quantum information engineering with quantized electromagnetic
fields. The subjec
t of characterizing and understanding the information nature of such
quantized fields becomes then pivotal. These new engineering paradigms will be
accomplished with the properties of truly quantized electromagnetic (QEM) fields as
opposed to the classical

electromagnetic fields that are employed in the usual microwaves
and antenna engineering. They will lead to new analog and digital quantum
-
wave
-
based
informatics. Many of these areas have not been investigated from the point of view of an
engineer whose

goal is to use the available base of scientific knowledge in order to
design practical, operational systems that meet a given set of system specifications (e.g.,
high data transfer rate, low probability of intercept, high performance, system
integrability
, low cost). We present next some ideas for these new research areas which
borrow from developments in the associated classical information theory for
electromagnetic fields. The latter, once again, is one of the main subject areas of
Professor Pierri’s gr
oup in Italy, and the need arises to generalize some of his
contributions to the quantum domain.

By writing the material contained in this section we expect to stimulate further a
productive research exchange between my group on this topic and Professor P
ierri’s
group in the related area of the information content of scattered fields, possibly with
communication and signal processing concepts in addition to the inverse scattering ones
(the latter being

usually
-

the most familiar to electromagneticians). A

large portion of
the material contained here is also a primer for a number of planned publications on this
area by my group on this topic. Interaction with Professor Pierri’s group is therefore very
welcome and encouraged, particularly for their expertise

on characterizing ‘information’
in a field
-
footing. It is worth emphasizing again that we have already interacted positively
with Dr. Angelo Liseno at Seconda Universita di Napoli in this related area of quantum
information, from our unique, combined inve
rse scattering and electromagnetic
perspective.




Quantum versus classical wave information
:

It is essential to understand what can and cannot be accomplished with classical
wave versus truly quantum information paradigms, from the points of view of
info
rmation storage, manipulation and communication. A basic understanding of
calculation/data processing capabilities of QI paradigms relative to other, non
-
quantal
information paradigms that are based on particles or waves, will also play a fundamental
role
in future QI
-
integrated wave
-
inversion algorithms with entangled states to be
explored in the final portion of the proposed work. Thus, the data/image processors to be
employed in the signal recovery and image recovery programs must generally be
inherently

quantal, in addition to the driving imaging fields that are employed in the
experiments (in order to be able to handle the entanglement
-
related information contents).
Yet another consideration is the need for some form of quantum error
-
correction
capabili
ty/strategy in both the communication and imaging applications (since the wave
states will generally decohere as they propagate through the medium). Moreover, since
throughout this work our emphasis will be on electromagnetic fields as
information/signal
-
c
arrying entities, a good understanding of the classical wave versus
the quantum information paradigms will be pivotal to derive truly novel (i.e., non
-
classical) techniques and results in our two main focus research areas: communications
and imaging. With
these motivations, we review next the classical wave versus the
quantum information paradigms, paying special attention to the nature of classical wave
versus quantum calculations, since they will have to be incorporated into fully
-
integrated
signal or ima
ge processing solutions.


Quantum entanglement
:

In the classical picture, a system is always in a product state of the individual
subsystem states. In quantum physics, however, we can have system states that do not
factor into such a simple form. In gener
al, the states of subsystems do not determine the
state of the entire system. From an information point of view, knowledge of the states of
subsystems provide little knowledge of the global state. This aspect of quantum
mechanics is known as entanglement
[113]

and has become a major theoretical and
experimental research area
[67, 96, 114
-
116]
. Part of the renewed interest in
entanglement comes from its key role in quantum information (see the reviews in
[117
]
).
It is, in fact, entanglement that is the main ingredient making classical wave informatics a
special case of a more general, quantum (entanglement
-
inclusive) wave informatics.


Quantum versus classical wave communications
:

From the point of view of s
ecure communications, quantum wave information
exchange is a collection of fundamentally quantum properties, namely, 1) entanglement,
2) Heisenberg uncertainty and 3) the no
-
cloning property of quantum states (the
impossibility of copying a quantum state w
ithout destroying the original); they make it
possible to teleport quantum states
[91
-
99]

and to safely distribute quantum
cryptographic keys
[72
-
90]
. Teleportation provides a new way to transmit information
whereas quantu
m cryptography enables one to safely transmit cryptographic keys even in
the presence of eavesdroppers in the communication channel. The kinds of physical
phenomena that can be taken advantage of in quantum communication have no classical
wave counterpart
whatsoever. Consequently, they cannot be manufactured with classical
wave transmit
-
receive systems. From a communication point of view, only the true QI
paradigm opens new avenues for flexible, data
-
compressed, and extremely secure
communication systems.



Quantum versus classical wave calculations
:

Let us qualitatively compare the quantum versus classical wave computation
paradigms from the points of view of complexity, robustness and energetic efficiency. It
is important to note that the most basic form
of quantum parallelism, i.e., that giving rise
to superpositions of individual quantum bits (qubits), is also present in classical
electromagnetic wavefields (this is Thomas Young’s wave interference). However, as the
number of qubits in an n
-
qubit system
(e.g., a quantum register) grows, the state vectors
that can be built via such Young superpositions will account only for a small subset of the
entire Hilbert space. The really interesting superpositions in the quantum picture are
those in the basis of the

entire Hilbert space, not those in one of the basis corresponding
to a subspace of it. This opens the possibility of entanglement among different
subsystems of a quantum system, such as different particles (atoms) representing
different qubits of a quantu
m register or a serial sequence of communicated qubits. In
fact, it is entanglement that truly distinguishes the quantum wave information paradigm
from its classical wave version. Quantum computers that take advantage of their
entanglement capabilities are

inherently built (by nature) to be able to solve complex
computational tasks more efficiently than any classical computer, be it based on particles
or waves.

The possible decoherence of quantum states due to interactions with the
surrounding environment
in the course of useful computations has been the subject of
many investigations
[59]
. Quantum decoherence is perhaps the major obstacle toward the
practical realization of a quantum computer. Significant progress has been made via
quantum

error correction codes
[56
-
58]
. The latter are quantum generalizations of error
correction codes employed in classical digital computers that make use of redundancy.
Classical systems are physically more immune to loss of information due
to environment
interactions (including human
-
observer interactions). However, this is accomplished at
the expense of reduced inherent computational complexity. Quantum systems are
inherently more powerful, computationally, at the expense of increased fragi
lity to
external inputs.

Finally, from the point of view of energetic computational efficiency, we note that
reversible unitary evolution can also be present in classical wave optics (in the form of
wave propagation in a lossless medium). Thus, reversible

computations are not applicable
only to quantum systems (actually, one does not even need a wave paradigm since
ballistic digital computers made of Toffoli (control
-
control
-
not (CCN)) gates are also
reversible).

In summary, the classical wave computation

paradigm is a special, more limited
case of the more general quantum computation paradigm. The former exhibits first
-
order
superpositions at the single
-
qubit level, but each qubit acts as a separate, independent
physical entity. Classical wave computers c
an thus be useful for computational tasks that
require massive parallelism. The qubits are individually in superpositions of different
states, but they are independent of each other, both physically and computationally. In
problems that can be efficiently
solved via an algorithm that does not utilize
entanglement, the algorithm can be run successfully in a classical wave computer without
extra overhead relative to a quantum computer. In that case, what we have is simply a
vast number of identical physical q
ubits for each computational qubit, so any program
run is essentially a simulation of the quantum system. In problems that can be solved
efficiently with a classical wave computer, a classical wave computer will have the
advantage (over its full quantum an
alogue) of being physically more robust, i.e., more
immune to small imperfections and interactions with the environment. In a quantum
computer, spurious interactions of the quantum computer with the surrounding
environment will irreversibly destroy coheren
ce and entanglement, and the remaining
running of the algorithm will be non
-
unitary. However, classical wave computers cannot
be utilized to efficiently simulate quantum algorithms requiring entanglement among
different qubits. In order to do so, the class
ical wave computer would have to grow
exponentially in size to accommodate for its lack of physical entanglement among the
qubits.

Finally, we note that in the classical wave optics case, the different linear optical
components (e.g., beam splitters, pola
rizers) with which one can build the required logic
gates also maintain the optical coherence of the information (thereby preserving the phase
information). In particular, the transfer functions or matrices of these devices are complex
probability amplitud
es rather than just classical probabilities. On the other hand, if one
uses incoherent light and/or optical components with random transfer functions or
matrices, then the classical wave computer would operate rather as a classical
probabilistic ballistic
computer (a probabilistic Turing machine). In this case the
computational trajectories are described by classical probabilities in the physical form of
(intensity
-
only) power transfer functions, instead of true field transfer functions which
would preserve

the phase information. We thus see how the classical wave information
paradigm incorporates both the classical deterministic and the classical probabilistic
ballistic information paradigms as special cases. It also incorporates the probability
amplitude
-
b
ased information paradigm up to the entanglement
-
inclusive level that is the
exclusive domain of a truly quantum mechanical computer.


Information
-

and Signal Carrying
-
Capabilities of Quantized Electromagnetic Fields
:

The most general one
-
photon state can

be written in the Fock basis

k,s

|n

k,s
>
[118]

in the general form

|

> =

k,s

C(k,s) |1

k,s
>

(1)

where the C(k,s) are complex probability amplitudes that must be normalized, whereas k
(with k=

/c) defines the wave vector of the corresponding (propagating/evanescent)
plane wave mode, and s=1,2 is a short
-
hand notation for the polarization state in a
suitable orthogonal polarization basis (e.g., linear, circular). The summation in (1) is over
all
the relevant modes (which can be infinite). The three degrees of freedom of the
electromagnetic field are contained in (1): the oscillation frequency

; the spatial mode
(which can be characterized by the wave vector k for plane wave modes, or by the
multi
pole order (L,M) for spherical wave modes); and the polarization state s=1 or 2 in a
suitable polarization basis (in the spherical wave case, s=1 or 2 can denote the electric or
magnetic multipole field, respectively). Sometimes we will use a different not
ation that is
more self
-
suggestive of these three degrees of freedom; in particular, we will also denote
a photon state with definite

, k and s in the form |k>|s>, where it is understood that
k=

/c. This notation will more clearly reveal the ties between
the present quantized field
analysis and our proposed novel approaches and applications to quantum
communications, signal processing and imaging. In this special notation, Eq.(1) becomes

|

> =

k,s

C(k,s) |k>|s >

(2)

This notation is useful in explaining
a form of ‘self
-
entanglement’,
within a single
photo
n, that occurs among its different available degrees of freedom. This form of self
-
entanglement is precisely what is taken advantage of in the proposed CN gates that utilize
‘which
-
way’ variables entangle
d to polarization states, and has been discussed in a few
papers in
[93, 119, 120]

and in a recent book
[121]
. In particular, the state defined in
Eq.(2) can be entangled in the sense that its propagation direction (the w
hich
-
way
variable) and its polarization state can be glued together in non
-
separable ways. For
example, a particular form the state in Eq.(2) could be the maximally (self
-
) entangled
state

|

> = (|k
1
>|2>


|k
2
>|1>)/


.

(3)

Thus, whenever this photon travel
s in the propagation direction k
1

(or k
2
), it will
necessarily exhibit the polarization state which we have labeled 2 (or 1), and there are
equal probabilities for these two alternative occurrences.

The true many
-
body picture is developed next. The most g
eneral (i.e., generally
entangled) quantized electromagnetic field can be expressed in the Fock basis in the
general form

|

> =


C(k
1
,s
1
,k
2
,s
2
,…, k
n
,s
n
) |k
1
>|s
1
>|k
2
>|s
2
>

|k
n
>|s
n
>

(4)


k
1
,s
1
,k
2
,s
2
,…,k
n
,s
n

where the summation occurs over all n relevant mo
des (k
i
,s
i
) (which can be infinite, or
finite
-
approximated in certain practical settings). In the radiation analysis of certain
quantum communication, scattering and imaging problems with entangled states we may
find more convenient to work in the more com
pact, multipole representation, i.e., the
generally
-
entangled many
-
photon state in Eq.(4) can then be expressed as

|

>=

C(L
1
,M
1
,

1
,s
1
,L
2
,M
2
,

2
,s
2
,…,L
n
,M
n
,

n
,s
n
)|L
1
,M
1
>|

1
>|s
1
>|L
2
,M
2
>|

2
>|s
2
>…|L
n
,M
n
>|

n
>|s
n
>





L
1
,M
1
,

1
,s
1
,L
2
,M
2
,

2
,s
2
,…,L
n
,M
n
,

n
,s
n

(5)



where the s=1 or 2 refers to the electric or magnetic multipole field, respectively.

The state of the quantized electromagnetic field described in Eq.(4) will generally
exhibit entanglement among the different degrees of freedom of ea
ch photon state, but,
more importantly, it will also exhibit entanglement among the different degrees of
freedom of different photon states. Also, in the state in Eq.(4), there may be several
photons in a given, identical mode (k
i
,s
i
), and some or all of s
uch photons may be
entangled to each other, as well as to other photons corresponding to completely different
modes. This is the key to our proposed, novel strategies in quantum communication,
signal processing and imaging with such quantized, entangled fi
elds. For the quantum
communication and signal processing, the main form of entanglement that we propose to
elucidate and engineer involves polarization and frequency, whereas for imaging the
main form of entanglement that we propose to elucidate and engin
eer is that leading to
confocal excitations and it occurs in the wave vectors degree of freedom (i.e., in the wave
propagation directions).

From the perspective of communications, we find from Eqs.(4)
-
(5) that the field
state as seen at a communication re
ceiver, corresponding to a particular propagation
mode (i.e., the one that links the transmitter to the receiver e.g., as defined by a line
-
of
-
sight in a wireless satellite communication link, or by the appropriate propagating mode
in a waveguide or transm
ission line/optical fiber link), will be expressible in the general
form

|

rec.
> =


C(

1
,s
1
,

2
,s
2
,…,

n
,s
n
) |

1
>|s
1
>|

2
>|s
2
>

|

n
>|s
n
>

(6)


1
,s
1
,

2
,s
2
,…,

n
,s
n


which is the projection of the overall field state in Eqs.(4,5) into the subspace
corresponding

to all the photon modes that can be received/detected at the receiver’s
location.

In the important special case of digital communications, we simply discard the
signal point of view (as described by the spectral information that is present in Eq.(6))
and

focus on the polarization state information which is of a discrete (i.e., binary) nature.
In the corresponding analog or combined analog/digital communications picture, we
could still use the information
-
carrying
-
capabilities of the polarization degree of

freedom.
However, the main surprise comes from the rather subtle signal
-
carrying
-
capabilities
associated with a combined polarization/frequency entanglement, which, to our
knowledge, are waiting to be investigated. We shall briefly outline in the followin
g a few
key ideas for the work which we propose to carry out in these, most fundamental
engineering areas of quantum information, which, to our knowledge, have not been
looked at with the present, fresh engineering signal processing perspective.

Consider,
for the sake of illustration, the following special (and very practical)
version of Eq.(6)

|

rec.
> =


C(

1
,s
1
,

2
,s
2
,…,

n
,s
n
) |

1
>|s
1
>|

2
>|s
2
>

|

n
>|s
n
>

(7)

s
1
,s
2
,…,s
n

where we can have polarization
-
entanglement among different photons, corresponding to
the same or to different frequencies. A simple example is the maximally entangled state

|

rec.
> = (|

1
>|2>|

2
>|1>


|

1
>|1>|

2
>|2>)/


.






(8)

In this elementary example, we can already see that the polarization and the frequency
components can become m
ixed together in a very non
-
classical way, i.e., the polarization
for each frequency does not have a definite state, and its measured state will depend on
the polarization for the other frequency. Only one particular realization out of the four
possible re
alizations (

1
,1), (

1
,2), (

2
,1), (

2
,2) will actually occur (i.e., be measured) at
the receiver (in this case the probabilities for the occurrences (

1
,1) and (

2
,2) will be
zero, whereas the probabilities for the occurrences (

1
,2) and (

2
,1) will each
be 50%).
This is illustrated in Fig.1, which schematically depicts the signal representation in the
relevant 2




dimensional (s,

) space (see also Fig.2 for the entire quantum signals in
this signal
-
space). Furthermore, although this simple example does
not illustrate this
point, the order in which the frequencies are interrogated will also play a very non
-
classical and central role in more general versions of Eq.(7), corresponding to many
-
photon states. This is analogous to what we have already pointed
out earlier in connection
with quantum digital communications. In particular, in our unpublished analyses of QEM
signals from the point of view of photodetection
[122]
, where the measured signal can be
a photon number (an ‘intensity’) vers
us polarization and frequency (i.e., a signal in a 2



dimensional space), we have been able to show that one can visualize quantum signals in
a stochastic processes framework as long as care is exercised in taking into account the
additional quantum featu
re of entanglement among all the available signal degrees of
freedom (in this case, the spectral/temporal and polarization variables of the process).
The basic idea is summarized next, and will be elaborated further in this project in
connection with the t
heoretical aspects of quantum signal processing.

We wish to emphasize that it is not possible to know the field state |

rec.
>
completely via measurements at the receiver (indeed, little knowledge of the state itself
will, generally, be available). We can
only obtain a particular realization (a signal)
corresponding to a given field observable. The signal in question will be just one among
an infinite myriad of allowed, permissible signals, all of which have their respectively
assigned probabilistic measure
. The way we interrogate the received state plays a key role
in the probabilistic behavior of the signals. In general, the available quantum
electromagnetic field at the receiver (seen here as a global system) will contain
simultaneously an infinitude of p
ossible signals for each physical continuous observable
(say, the photon absorption operator). It is the particular way we interrogate the system
through its available degrees of freedom (i.e., the particular protocolar ordering which we
alluded to earlier

in connection with digital communications, but generalized now to both
polarization and frequency) that will eventually decide
-
probabilistically
-

which one will
be our final, measured outcome (identified here with a photon count per polarization and
fre
quency for the sake of illustration, but other observable variables can be used in
alternative, more general descriptions). So, unlike the case when we use classical signals,
in the quantum case, our observations contribute to build the yet
-
to
-
be
-
measured
signal
itself. Again, this is due to entanglement among the many
-
photon states. Any
measurement, any interrogation of a particular aspect of the field will irreversibly force
the entire state to collapse into a subspace of its entire permitted space, and t
his will, in
turn, affect all future (conditional) probabilities for the future measurements. This is how
protocoling

plays a key role in extracting information or, in the present context, the signal
content from quantized fields. A non
-
intended recipient,

lacking the required signal
processing capabilities and communication protocols, will sense the very same quantum
signals pretty much like noise.












Fig.1. Schematic representation of the polarization
-
frequency signal cont
ent of the state
in Eq.(7), which illustrates the mixture of polarization
-
entanglement between different
frequencies.






s=1


1

n(

)


1


2

n(

)


2

competing
alternative
s


s=1


s=2













Fig.2. Schematization of the polarization
-
frequency signal content of a general state of
the form Eq.(6). The
figure depicts a particular realization of the signal which can be,
e.g., a photon count n(

,s) in its 2



dimensional space (s,

). In general, we can also
deal with the higher
-
order correlation functions of the photon count, and this will be
investigated
in detail during the course of this effort.


In the classical case, a stochastic process X(

) can be characterized by a
collection of all possible realizations, which is referred to as the ensemble of X(

). In a
given experiment one will obtain a given sa
mple of the process, say the signal X(

;

)
where


is a label assigned to the particular, obtained sample. Each possible signal has an
assigned probability of being chosen, i.e., of being the particular sensed realization of the
process. In the quantum cas
e, the way the final signal realization emerges is much more
complex, as illustrated in Fig.3, which depicts the expectation value of the photon count
n(s,

) conditioned to a priori knowledge of a given polarization
-
frequency component of
n(s,

), e.g., th
e value of n(s,

) in a given, narrow frequency band. Here we depict four
possible expectation values of the random quantum signal n(s,

). Each expectation value
is conditional to a particular measurement of the signal for a given polarization and
frequenc
y, which are indicated by a bar in the different plots. We see that entirely
different conditional expectation values are obtained after each measurement suite; the
expectation values will, in general, depend strongly on the a priori knowledge, i.e., the
e
xpectation values are conditional due to the entanglement (statistical dependence)
among different realizations of the process. Thus we see that in the quantum case, all the
polarization and frequency components of the signal can be entangled to each other
, and
this manifests in the statistical dependence of the different components. In the classical
random case, each polarization and frequency component is orthogonal to all different
polarization and frequency components, and the random signal would be onl
y a function
of s,

,


where


denotes the particular realization of the ensemble. On the other hand, in
the quantum random case, each component is, in general, dependent on the others, so that
the random signal would be a function of 1) s,

, and 2) the way

and order in which the
measurements are made, which we shall denote as an additional variable

. In addition,
the realizations of the ensemble are now associated with the three values (s,

,

), so that
the measured random signal could be thought of as a fu
nction of the four variables s,

,

,

. The way we gather pieces of the puzzle will influence what the puzzle will turn out to
be, and in this case the puzzle is a signal that we may want to protect from unintended

s=1





n(

)

n(

)


s=2

recipients (in the communication case), or

simply a signal that contains much physical
information about an scattering object that we want to decipher (in the imaging case).

Similar considerations apply to the digital case. Then the digital signal can be
contained physically in a sequence of qub
its corresponding to polarization states of
different photons. The latter travel in the configuration space in the direction of the
intended receiver. The qubit sequence is stored first, physically, and later processed
according to a particular measurement

protocol so as to collapse the intended serial
sequence of qubits. A particular packet can be of 24 or 32 photons by analogy with
classical bit packets used in conventional digital communications
[123]
. Part of the
packet can be used in
protocol and security issues, whereas the remaining bits are
reserved for the actual digital signal. An analogous approach can be employed in
quantum digital communications which we shall explore in detail for the proposed effort.



E[
n
(s,

)]
S=1
S=1
S=2
S=2


Fig.3. Illustration of the dependence of the measured quantum signal on the wavepacket
-
collapsing order. Here we depict four possible expectation values of the random quantum
signal n(s,

). Each expectation value is conditional to a particular measu
rement of the
signal for a given polarization and frequency, which are indicated by a bar in the different
plots. We see that entirely different conditional expectation values are obtained after each
measurement suite.


The digital communication state vec
tor is defined by analogy with Eq.(7) as

|

rec.
> =


C(s
1
,s
2
,…,s
n
) |s
1
>|s
2
>

|s
n
>

(9)


s
1
,s
2
,…,s
n

where the probability amplitudes C(s
1
,s
2
,…,s
n
), where P(|0>|0>

|0>) = |C(0,0,…,0)|
2
,

P(|1>|0>

|0>) = |C(1,0,…,0)|
2
, and so on, have to

be normalized according to



|C(s
1
,s
2
,…,s
n
) |
2
=1.

(10)

s
1
,s
2
,…,s
n

We thus see that, unlike a classical bit sequence, each possible qubit sequence
(seen as a whole) has its own probability P. We thus have to deal with a joint discrete
probability distribu
tion instead of the marginal probability distributions for each bit, in
contrast to digital bit sequences. This also explains why the probabilities for future
measurements of the bit sequence are affected by previous measurements since there is a
built
-
in
conditional probability distribution involving all or several of the qubits of the
sequence. Several basic results readily follow. For example, the expectation values of a
given pair of qubits are not necessarily equal to the products of the individual exp
ectation
values of each qubit (they are correlated, and the future expectation values are affected by
the already collapsed (measured) qubits).

Finally, we wish to conclude this QEM field information
-
theoretic section by
emphasizing the fundamental concept

that in the full entanglement
-
inclusive picture of
the QEM field, the latter cannot be described by a collection of independent harmonic
oscillators (photons). Instead, several photons may be entangled. Entanglement can be
produced by physical interaction

among the entangled subsystems. For example, we may
massively entangle an array of quantum harmonic oscillators by enabling them to interact
in prescribed entanglement
-
producing ways (by creating coupling among them). As is
well known, the quantum harmoni
c oscillators will, in general, remain entangled, even
after the interactions in question (couplings) cease. However, a fundamentally quantum
mechanical aspect of entanglement is that it can be produced without direct physical
interaction among the to
-
be
-
e
ntangled subsystems. In particular, there are alternative,
practical entanglement
-
producing mechanisms that do not rely on interactions among the
entangled subsystems. One such mechanism is the quantum eraser effect
[124]
. The latter
also
illustrates the possibility of entangling cavity quantum electrodynamic (QED)
systems with atoms
[125]
, and, in general, the possibility of both entangling and
teleporting different states associated with different kinds of physical system
s. One can
store, for example, a given photon state in an atom or another physical entity, or vice
versa, in order to ease the associated physical manipulations. Photons could be used for
the communication stage, whereas atoms or other systems could be emp
loyed in the
storage and data processing stages of a unified transmit/receive/data
manipulation/processing system. An idea, that we plan to pursue further in the future, is
to design, evaluate and investigate various integrated communication/computation
ar
chitectures based on this concept. For example, in order to synthesize two
-
qubit
control
-
not (CN) gates, or to carry out joint Bell measurements for many
-
photon systems,
perhaps it may be possible to teleport the polarization state of a given photon (or ph
otons)
of interest into a which
-
way variable; furthermore, one can later carry out the required
CN operations or Bell measurements by means of the now trivial approach discussed in
[93, 119
-
121]

of encoding qubits in which
-
way variables of

a given physical photon (a
qubit is stored in the polarization state whereas a second qubit is stored in a which
-
way
variable, both of which are encoded in the same physical photon, so as to ease
entanglement
-
simulation and manipulation of entangled state
s via joint Bell
measurements).


IV.

Acknowledgments.


We wish to thank the Italian Consiglio Nazionalle della Ricerca (CNR) for
providing the financial support for this two
-
week research interaction in Italy. We also
wish to specially thank Professor Rocco P
ierri and Dr. Angelo Liseno at Seconda
Universita di Napoli for their arrangements with the CNR along with their most
appreciated research interaction and hospitality during this two
-
week stay at their
academic institution. We also wish to thank Professors

Giovanni Leone and Adriana
Brancaccio and Dr. Francesco Soldovieri for their very enlightening discussions and
hospitality during this visit. Finally, we extend our gratitude to the entire Pierri’s pack of
inverse scattering researchers, for their candid
attention and time during our visit.

REFERENCES

1.

Pierri, R., Soldovieri, F.,
On the information content of the radiated fields in the
near zone over bounded domains.

Inv. Probl., 1998.
14
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