A POVM view of the ensemble approach to polarization optics
arXiv:0704.0147v2  [physics.optics]  20 Jun 2007
A POVM view of the ensemble approach to polarization optics
Sudha,1 A.V. Gopala Rao,2 A. R. Usha Devi,3 and A.K. Rajagopal4
1Department of P.G. Studies in Physics,
Kuvempu University, Shankaraghatta-577 451, India∗
2Department of Studies in Physics, Manasagangothri,
University of Mysore, Mysore 570 006, India
3 Department of Physics, Jnanabharathi Campus,
Bangalore University, Bangalore-560 056, India
4 Department of Computer Science and Center for Quantum Studies,
George Mason University, Fairfax, VA 22030, USA
and Inspire Institute Inc., McLean, VA 22101, USA
Statistical ensemble formalism of Kim, Mandel and Wolf (J. Opt. Soc. Am. A
4, 433 (1987)) offers a realistic model for characterizing the effect of stochastic
non-image forming optical media on the state of polarization of transmitted
light. With suitable choice of the Jones ensemble, various Mueller transforma-
tions - some of which have been unknown so far - are deduced. It is observed
that the ensemble approach is formally identical to the positive operator val-
ued measures (POVM) on the quantum density matrix. This observation, in
combination with the recent suggestion by Ahnert and Payne (Phys. Rev. A
71, 012330, (2005)) - in the context of generalized quantum measurement on
single photon polarization states - that linear optics elements can be employed
in setting up all possible POVMs, enables us to propose a way of realizing
different types of Mueller devices.
c© 2018 Optical Society of America
OCIS codes: 030.0030, 230.0230.
∗Corresponding author: arss@rediffmail.com
http://arxiv.org/abs/0704.0147v2
1. Introduction
The intensity and polarization of a beam of light passing through an isolated optical de-
vice undergoes a linear transformation. But this is an ideal situation because, in general,
the optical system is embedded in some media such as atmosphere or other ambient mate-
rial, which further modifies the polarization properties of the light beam passing through
it. A statistical ensemble model describing random linear optical media was formulated
two decades ago by Kim, Mandel and Wolf [1], but is not examined in any detail in the
literature, to the best of our knowledge. The purpose of the present paper is to pursue
this avenue in a new way arising from the realization of a relationship, presented here, with
the positive operator valued measures (POVM) of quantum measurement theory. This is
because the transformation of the polarization states of a light beam propagating through
an ensemble of deterministic optical devices exhibits a structural similarity with the POVM
transformation of quantum density matrices. This connection motivates, in view of the re-
cent interest in the implementations of POVMs on single photon density matrix employing
linear optics elements [2], identification of experimental schemes to realize various kinds of
Muller transformations. The properties of the transformation of the polarization states of
light form a much studied topic in literature [3 – 17]. Thus the power of the ensemble
approach becomes evident in elucidating the known optical devices as well as some hitherto
unknown types [17], which had remained only a mathematical possibility.
The contents of this paper are organized as follows. In Sec. 2, a concise formulation of
the Jones and Mueller matrix theory, along with a summary of main results of Gopala Rao
et al. [17] is given. Based on the approach of Kim, Mandel and Wolf [1] suitable Jones en-
sembles, corresponding to various types of Mueller transformations are identified in Sec. 3.
In Sec. 4, a structural equivalence between Jones ensemble and POVMs of quantum mea-
surement theory is established. Following the linear optics scheme of Ahnert and Payne [2]
for the implementation of POVMs on single photon density matrix, experimental setup for
realizing Mueller matrices of types I and II are suggested in Sec. 5. The final section has
some concluding remarks.
2. Brief summary of known results on the Jones and the Mueller formalism.
Following the standard procedure, let E1 and E2, defined here as a column matrix E =
, denote two components of the transverse electric field vector associated with a light
beam. The coherency matrix (or the polarization matrix) of the light beam is a positive
semidefinite 2x2 hermitian matrix defined by,
C = 〈E⊗E†〉. (1)
Expressing this in terms of the standard Pauli matrices σ1 =
 , σ3 =
 and the unit matrix σ0 =
, we have
siσi =
s0 + s3 s1 − is2
s1 + is2 s0 − s3
 (2)
The physical significance of the quantities arising here are
s0 = Tr (Cσ0) = Intensity of the beam
si = Tr (Cσi) = Components of Polarization vector ~s of the beam
Thus the coherency matrix completely specifies the physical properties of the light beam.
The four-vector S =
 defined by Eq. (3) is the well known Stokes vector, which
represents the state of polarization of the light beam. Because C is hermitian, the Stokes
vector is real. The positive semidefiniteness of C implies that the Stokes vector must satisfy
the properties
s0 > 0, s
0 − |~s|2 ≥ 0 (4)
A 2x2 complex matrix J, called the Jones matrix, represents the so-called deterministic
optical device [18] or medium. When a light beam represented by E passes through such a
medium, the transformed light beam is given by E′ = JE. Correspondingly, the coherency
matrix C transforms as
′ = JCJ† (5)
(Here J† is the hermitian conjugate of J.)
Alternatively, instead of the 2 × 2 matrix transformation of the coherency matrix, as
given by Eq. (5), a transformation
′ = MS (6)
of the four componental Stokes column S through a real 4x4 matrix M, called the Mueller
matrix, is found be more useful [18].
Using Eq. (3) and Eq. (5 we have,
s′i = Tr(C
′σi) = Tr(JCJ
† σi) =
Tr(J†σiJσj)sj
which leads to the well-known relationship [1]
Mij =
Tr(J†σiJσj)
between the elements of a Jones matrix and that of corresponding Mueller matrix.
But in the case where medium cannot be represented by a Jones matrix, it is not possible
to characterize the change in the state of polarization of the light beam through Eq. (5).
In such a situation, Mueller formalism provides a general approach for the polarization
transformation of the light beam. The Mueller matrix M is said to be non-deterministic
when it has no corresponding Jones characterization.
Mathematically, a Mueller device can be represented by any 4 × 4 matrix such that the
Stokes parameters of the outgoing light beam satisfy the physical constraint Eq. (4). In other
words, a Mueller matrix is any 4×4 real matrix that transforms a Stokes vector into another
Stokes vector. There are many aspects of the relationships between these two formulations
of the polarization optics and a complete characterization of Mueller matrices has been the
subject matter of Ref. [1, 3-17]. It was Gopala Rao et al. [17] who presented a complete set
of necessary and sufficient conditions for any 4x4 real matrix to be a Mueller matrix. In so
doing, they found that there are two algebraic types of Mueller matrices called type I and
type II; and it has been shown [17] that only a subset of the type-I Mueller matrices - called
deterministic or pure Mueller matrices - have corresponding Jones characterization. All the
known polarizing optical devices such as retarders, polarizers, analyzers, optical rotators are
pure Mueller type and are well understood. Mueller matrices of the Type II variety are yet
to be physically realized and have remained as mere mathematical possibility. For the sake
of completeness, we present here the characterization as well as categorization of these two
types of Mueller matrices as is given in Ref. [17]. This will enable us to show that both
Type I and II Mueller devices are realizable in an unified manner in terms of the proposed
ensemble approach [1].
I. A 4× 4 real matrix M is called a type-I Mueller matrix iff
(i) M00 ≥ 0
(ii) The G-eigenvalues ρ0, ρ1, ρ2, ρ3 of the matrix N=M̃GM are all real. (Here, M̃
stands for the transpose of M; G-eigenvalues are the eigenvalues of the matrix GN,
with G = diag(1, −1, −1, −1)).
iii) The largest G-eigenvalue ρ0 possesses a time-like G-eigenvector and the G-eigenspace
of N contains one time-like and three space-like G-eigenvectors.
II. A 4× 4 real matrix M is called a type-II Mueller matrix iff
(i) M00 > 0.
(ii) The G-eigenvalues ρ0, ρ1, ρ2, ρ3 of N=M̃GM are all real.
(iii) The largest G-eigenvalue ρ0 possesses a null G-eigenvector and the G-eigenspace of N
contains one null and two space-like G-eigenvectors.
(iv) If X0 = e0 + e1 is the null G-eigenvector of N such that e0 is a time-like vector
with positive zeroth component, e1 is a space-like vector G-orthogonal to e0 then
ẽ0Ne0 > 0.
Despite the knowledge of these new category of Mueller matrices [15, 17], not much
attention is paid for realizing the corresponding devices. An experimental arrangement
involving a parallel combination of deterministic (pure Mueller) optical devices is proposed
in Ref. [17] for realizing type-II Mueller devices. The physical situations, where the beam of
light is subjected to the influence of a medium such as atmosphere was addressed in Ref. [1].
In the next section, we discuss this ensemble approach for random optical media, proposed
by Kim, Mandel and Wolf [1] .
3. Mueller matrices as ensemble of Jones devices
Kim et. al. [1] associate a set of probabilities {pe,
pe = 1} to describe the stochastic
medium. Then a Jones device Je associated with each element e of the ensemble gives a
corresponding coherency matrix C′e = JeCJ
e. The ensemble averaged coherency matrix
Cav =
pe(JeCJ
e) (7)
then describes the effects of the medium on the beam of light. In a similar fashion, the
corresponding ensemble of Mueller matrices {Me} associated with the ensemble of Jones
matrices {Je} is constructed and its ensemble averaged Mueller matrix is similarly formed
as Mav =
e peMe. Since a linear combination of Mueller matrices with non-negative
coefficients is also a Mueller matrix, the ensemble averaged Mueller matrix Mav is a Mueller
matrix 1.
We now turn to the question of constructing an appropriate ensemble designed to describe
a given physical situation. The simplest example of an ensemble is one where the elements
are chosen entirely randomly, i.e., the system is described by a chaotic ensemble where the
probabilities are all equal, pe =
, where n denotes the number of elements in the ensemble.
The coherency matrix Cav of the light beam passing through such a chaotic assembly is just
an arithmetic average of the coherency matrices C′e = JeCJ
e and hence
Cav =
e (8)
More general models can be constructed depending on the medium for the propagation of
the beam of light. For example, one may employ various types of filters or solid state systems
through which the light passes; the assignment of the Jones matrices and the corresponding
probabilities will then differ depending on the weights placed on these elements.
Restricting ourselves to an ensemble consisting of only two Jones devices which occur
with equal probability p1 = 1/2, p2 = 1/2, we have found out that the resultant Mueller
matrices can either be deterministic or non-deterministic. We give in the foregoing (see Table
I) some examples of Mueller matrices corresponding to different choices of Jones matrices
in an ensemble Je, e = 1, 2, for some representative cases. This will also serve to show the
1 This is because, each Mueller matrix M
transforms an initial Stokes vector into a final Stokes vector and
a linear combination of Stokes vectors with non-negative coefficients p
is again a Stokes vector.
generality of the ensemble procedure in capturing the physical realizations for the Mueller
devices discussed in Ref. [17].
Table 1. Mueller matrices resulting from 2-element Jones ensemble.
J1 J2 M = p1M1 + p2M2, Type of M
p1 = p2 =
1. 1√
1 1− i
1 + i −1
1 1− i
1 + i −1
3 0 0 0
0 −1 2 2
0 2 −1 2
0 2 2 −1
Pure Mueller
1 0 0 0
1 0 0 0
0 0 0 0
0 0 0 0
Type-I
1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 −1
Type-I
4. 1√
3 0 0 0
0 −1 0 2
0 0 −1 0
0 2 0 −1
Type-I
1 1− i
1 + i −1
5 0 0 0
0 −1 2 4
0 2 −3 2
0 4 2 −1
Type-I
1 −1 0 0
0 0 0 0
0 0 0 0
0 0 0 0
Type-II
2 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
Type-II
In Table I, the Jones matrices chosen are so as to give pure Mueller (deterministic)
and non-deterministic type-I, type-II matrices respectively. We observe that an assembly
of Jones matrices can result in a pure Mueller matrix if and only if all elements of the
assembly correspond to the same optical device. This is because, with all Je’s are same,
a transformation of the form Cav =
e pe(JeCJ
e) is equivalent to a transformation of the
Stokes vector S through a Mueller matrix Mav =
peMe = Mpure. When the medium is
represented by a pure Mueller matrix, the outgoing light beam will have the same degree of
polarization as the incoming light beam. In fact, pure Mueller matrix is the simplest among
type-I Mueller matrices. Not all type-I Mueller matrices preserve the degree of polarization
of the incident light beam. To see this, note that the type-I matrix of example 2 (see Table
I) converts any incident light beam into a linearly polarized light beam; the other three
type-I matrices (examples 3 to 5) transform completely polarized light beams into partially
polarized light beams. Similarly, type-II Mueller matrices do not, in general, preserve the
degree of polarization of the incident light beam. It may be seen that the type-II Mueller
matrix of example 7 is a depolarizer matrix, since it converts any incident light beam into
an unpolarized light beam.
Though one cannot a priori state which choices of Jones matrices result in type-I or type-
II, it is interesting to observe that all types of Mueller matrices result - even in 2-element
ensembles. It is not difficult to conclude that an ensemble, with more Jones devices and
with different weight factors, can give rise to a variety of Mueller matrices of all possible
algebraic types. It would certainly be interesting to physically realize such systems.
In the following section, a connection between the ensemble approach for optical devices
and the POVMs of quantum measurement theory is established.
4. A connection to Positive Operator Valued Measures
We will now show that the phenomenology of the ensemble construction of Kim, Mandel
and Wolf [1] described above has a fundamental theoretical underpinning, if we make a
formal identification of the coherency matrix with the density matrix description of the
subsystem of a composite quantum system. The coherency matrix defined by Eqs. (1)
and (2) resembles a quantum density matrix in that both describe a physical system by
a hermitian, trace-class, and positive semi-definite matrix. While the quantum density
matrix has unit trace, the coherency matrix has intensity of the beam as the value of the
trace. The Jones matrix transformation is a general transformation of the coherency matrix,
which preserves its hermiticity and positive semi-definiteness - but changes the values of the
elements of the coherency matrix. The most general transformation of the density matrix
ρ, which preserves its hermiticity, positive semi-definiteness and also the unit trace is the
positive operator valued measures (POVM) [19]:
iVi = I (9)
where Vi’s are general matrices and I is the unit element in the Hilbert space. More generally,
one could relax the condition of preservation of the unit trace of the density matrix by
examining the possibility of a contracting transformation, where the unit matrix condition
on the POVM operators is replaced by an inequality.
This mathematical theorem has a physical basis in the Kraus operator formalism [19]
when we consider the Hamiltonian description of a composite interacting system A, B
described by a density matrix ρ(A, B) and deduce the subsystem density matrix of A given
by, ρ(A) = TrB ρ(A, B). In this case, the Kraus operators are the explicit expressions of
the POVM operators and contain the effects of interaction between the systems A and B
in the description of the subsystem A. It is thus clear that the phenomenology of Ref. [1]
has a correspondence with the Kraus formulation and the POVM theory. In order to make
this association complete, we compare Eq. (9) with the expression given by Eq. (7). Apart
from a phase factor, the Kraus operators {Vi}, associated with POVMs, may be related to
the Jones assembly {Ji}, chosen in the form
iVi =
iJi (10)
In the construction of the Table I presented earlier, a simple model was proposed where all
probabilities were chosen to be equal and the condition on the sum over the Jones matrix
combinations was set equal to unit matrix. In such cases, the intensity of the beam gets
reduced by 1/n and the polarization properties of the beam gets changed as was described
earlier. With this identification, we have provided here an important interpretation and
meaning to the phenomenology of the ensemble approach of Kim et al.[1].
Recently Ahnert and Payne [2] proposed an experimental scheme to implement all possible
POVMs on single photon polarization states using linear optical elements. In view of the
connection between the ensemble formalism for Jones and Mueller matrices with the POVMs,
a possible experimental realization of the two types of Mueller matrices is suggested in the
next section.
5. Possible experimental realization of types I and II Mueller matrices.
We first observe that the density matrix of a single photon polarization state,
ρ = ρHH |H〉 〈H|+ ρHV |H〉 〈V |+ ρ∗HV |V 〉 〈H|+ ρV V |V 〉 〈V | (11)
is nothing but the coherency matrix of the photon [20]
〈â†H âH〉 〈â
H âV 〉
〈â†V âH〉 〈â
V âV 〉
 , (12)
where âH and âV are the creation operators of the polarization states of the single photon;
{|H〉, |V 〉} denote the transverse orthogonal polarization states of photon. This is seen
explicitly by noting that the average values of the Stokes operators are obtained as,
s0 = 〈Ŝ0〉 = 〈(â†H âH + â
V âV )〉 = ρHH + ρV V = Tr(ρ),
s1 = 〈Ŝ1〉 = 〈(â†H âV + â
V âH)〉 = ρHV + ρ∗HV = Tr(ρ σ1),
s2 = 〈Ŝ2〉 = i 〈(â†V âH − â
H âV )〉 = i (ρHV − ρ∗HV ) = Tr(ρ σ2),
s3 = 〈Ŝ3〉 = 〈(â†H âH − â
V âV )〉 = ρHH − ρV V = Tr(ρ σ3). (13)
Hence the proposed setup [2], involving only linear optics elements such as polarizing beam
splitters, rotators and phase shifters, that promises to implement all possible POVMs on a
single photon polarization state leads to all possible ensemble realizations for the Mueller
matrices. More specifically, this provides a general experimental scheme to realize varieties
of Mueller matrices - including the hitherto unreported type-II Mueller matrices. We briefly
describe the scheme proposed in Ref. [2] and illustrate, by way of examples, how it leads to
both type-I and type-II Mueller matrices.
In Ref. [2], a module corresponds to an arrangement having polarization beam splitters,
polarization rotators, phase shifters and unitary operators. For an n element POVM, a
setup involving n − 1 modules are needed. That means, a single module is enough for a 2
element POVM; a setup involving two modules is required for a 3 element POVM and so on.
We describe two, three element POVMs by specifying the optical elements in the respective
modules and by specifying the corresponding Kraus operators in terms of these elements.
For any two operator POVM, the Kraus operators V1, V2 are given by V1 = U
and V2 = U
′′D2U. Here U, U
′, U′′ are the three unitary operators in a single module.
Denoting θ, φ as the angles of rotation of the two variable polarization rotators and γ, ξ,
the angles of the two variable phase shifters in the module, the diagonal matrices D1, D2
are given by,
eiγ cos θ 0
0 cos φ
 , D2 =
eiξ sin θ 0
0 sin φ
 (14)
The POVM elements
F1 = V
1V1 = U
1D1U, F2 = V
2V2 = U
2D2U (15)
satisfy the condition
i=1,2 Fi = F1 + F2 = I.
For any three operator POVM, the Kraus operators are given by
V1 = U
IDIUI,
V2 = U
IIDIIUIIU
V3 = U
IIUIIU
Here, the diagonal D matrices are
eiγI cos θI 0
0 cosφI
 , D′I =
eiξI sin θI 0
0 sinφI
 (17)
DII =
eiγII cos θII 0
0 cos φII
 , D′II =
eiξII sin θII 0
0 sinφII
 (18)
(θI, φI), (γI, ξI) are respectively the pair of angles corresponding to variable polarization
rotators and variable phase shifters in the first module. Similarly, (θII, φII), (γII, ξII) are the
pairs of angles corresponding to variable polarization rotators and variable phase shifters
respectively in the second module. UI, U
I are the unitary operators used in the first
module and UII, U
II, U
II are the unitary operators used in the second module. (Notice
that all the unitary operators in the above schemes are arbitrary and a particular choice of
the associated unitary operators gives rise to a different experimental arrangement). The
extension of this scheme to n operator POVM involving n-1 modules is quite similar and is
given in [2].
We had identified, in Sec. 3, that an ensemble average of Jones devices will lead to all
possible types of Mueller matrices, some examples of which are given in Table 1. We now
show that the experimental set up proposed in Ref. [2] can also be used to realize varieties
of Mueller devices. To substantiate our claim, we identify here the linear optical elements
needed in the single module set up of Ahnert and Payne [2], which lead to the physical
realization of two typical Mueller matrices given in Table 1.
To obtain the type-I Mueller matrix M= 1

1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 −1

of example 3 (see Table I), we
use U = I, U′ =
 and U′′ =
 as the required unitary Jones devices and
both the variable polarization rotators are set with their rotation angles θ=φ = π/4. There
is no need of phase shifter devices in this case i.e, γ=ξ=0.
Similarly for the type-II Mueller matrix M= 1

2 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0

of example 7, we find that
U = I, U′ = 1√
 andU′′ = 1√
 are the required unitary Jones devices. The
rotation angles of the variable polarization rotators are, as in the earlier case, θ=φ = π/4
and there is no need of phase shifter devices i.e., γ=ξ=0. Notice that in both the above
examples the unitary operators U′, U′′ correspond to linear and circular retarders [18].
These two examples illustrate that the experimental set up given in Ref. [2] may be
utilized to realize the required non-determinisitc Mueller devices. In fact Mueller matri-
ces corresponding to an ensemble with more than two Jones devices may also be realized
by employing larger number of modules as given in the experimental scheme proposed by
Ref. [2].
6. Conclusion
We have established here a connection between the phenomenological ensemble approach [1]
for the coherency matrix and the POVM transformation of quantum density matrix. This
opens up a fresh avenue to physically realize types I and II of the Mueller matrix classification
of Ref. [17]. We have also given experimental setup to implement Mueller transformations
corresponding to ensemble average of Jones devices by employing the POVM scheme on the
single photon density matrix suggested in Ref. [2], in the context of quantum measurement
theory. It is gratifying to note that two decades after the introduction of the ensemble
approach, which had remained obscure and only received passing reference in textbooks such
as [20], its value is revealed in this paper through its connection with the new developments
in quantum measurement theory. We plan on exploring further the POVM transformation
in the description of quantum polarization optics.
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