Quantum polarization characterization and tomography

We present a complete polarization characterization of any quantum state of two orthogonal polarization modes, and give a systematic measurement procedure to collect the necessary data. Full characterization requires measurements of the photon number in both modes and linear optics. In the situation where only the photon-number difference can be determined, a limited but useful characterization is obtained. The characteristic Stokes moment profiles are given for several common quantum states.


Introduction
Far from its source, any freely propagating electromagnetic field can be considered to a good approximation as a plane wave, with its electric field lying in a plane perpendicular to the direction of propagation. This simple observation is the root of the notion of polarization. At first glance, it may seem rather obvious how to translate such a concept into the realm of quantum optics. However, hurdles such as hidden polarization [1], the fact that the Poincaré sphere is too small to accommodate states with excitation larger than one photon [2], and the difficulties in defining polarization properties of two-photon entangled fields [3], to cite only a few examples, show that the classical theory, mainly based on first-order polarization moments, is insufficient for quantized fields.
Here, we outline a systematic method for polarization characterization of quantum fields. The method is based on a simple premise; namely, that if we can predict the mth-order moment of the Stokes operator in any direction on the Poincaré sphere, we know all there is to be known about the state polarization of this order [4], including any correlations between the Stokes operators. A tensor representation of the polarization information is based on such correlations. However, expressing the Stokes moments as functions of the measurement directions gives a more compact representation and provides a natural visualization. The Stokes profile representation also gives a relevant characterization for passive interferometry. Our analysis below makes use of both representations.
As a state's polarization properties do not require the full density matrix to be determined, it allows polarization tomography to be more easily performed than full quantum tomography [5][6][7]. Considering polarization tomography with ideal detection, we show that the number of measurement directions can be made equal to the number of independent parameters.
The remaining material of the article is organized as follows. After recalling the fundamentals on the quantum description of polarization in section 2, we present our scheme for characterization of quantum polarization properties in section 3. In section 4, we consider how the necessary data can be obtained experimentally. We thus arrive at an efficient method, which is feasible for polarization tomography of few-photon states. In section 5, we apply our characterization to several classes of states. Finally, our conclusions are presented in section 6.

Setting the scene
In the following, we consider monochromatic plane waves. Such fields can be decomposed into two orthogonal transverse modes, such as the horizontally and vertically polarized modes. For highly focused beams or waves in a waveguide, the plane-wave description is often inadequate, as the field is not longer transverse. It is our belief that the concepts discussed in this paper can also be extended to such non-plane waves, and several proposals have already appeared in the literature [8][9][10]. However, we shall not discuss such generalizations here.
The classical theory for the polarization of plane waves was established by Stokes already more than 150 years ago [11]. We shall build on his theory as the basis of our treatment will be the Stokes operators, whose expectation values are the Stokes parameters [12]. Following the conventions used in the quantum theory of angular momentum [13] and in quantum optics [14,15], we define the Stokes operators aŝ whereâ H andâ V are the annihilation operators of the modes associated with horizontally and vertically oscillating fields, respectively. With this choice, the usual ordering of the Stokes parameters I = Ŝ 0 , Q = Ŝ 3 , U = Ŝ 1 , and V = Ŝ 2 differs from that of the indices of the operators. However, as far as the theory below is concerned, we could just as well have associated any other pair of orthogonal polarization modes to these operators. That would only influence the interpretation of the theory and not the theory itself. As the annihilation and creation operators obey the bosonic commutation relations [â α ,â † β ] = δ αβ , for α, β ∈ {H,V }, the Stokes operators satisfy the commutation relations of an su(2) algebra where the latin indices run from 1 to 3 and ε jkℓ is the fully antisymmetric Levi-Civita tensor. The noncommutability of these operators precludes the simultaneous exact measurement of the corresponding physical quantities. The variances (∆S j ) 2 = Ŝ 2 j − Ŝ j 2 are found to obey the uncertainty relation Moreover, while the Stokes operators are all Hermitian, the noncommutability makes "mixed," non-symmetric products (such asŜ 1Ŝ2 ) non-Hermitian, also precluding their direct measurement.
The standard definition of the degree of polarization for a quantum stateρ is and Ŝ is the Stokes vector. Note that only first-order moments of the Stokes operators are used in this definition. In a more elaborated characterization, the degree of polarization can be subdivided into excitation manifolds according to the total photon number N. This makes physical sense because since the corresponding observablê S 0 commutes with all the other Stokes operators a complete set of simultaneous eigenstates ofŜ 0 and any ofŜ 1 ,Ŝ 2 , andŜ 3 does exist. In fact, the statistics of the latter three operators is usually determined by a set of wave plates, a polarizing beam splitter, and two photodetectors, giving (in the ideal case) information not only aboutŜ 1 ,Ŝ 2 , orŜ 3 , but simultaneously ofŜ 0 . Let us here take a quick look at excitation manifolds N = 1 and N = 2. One can readily convince oneself that any pure single-photon state |Ψ 1 satisfies P S (|Ψ 1 ) = 1, i.e., any such state is fully polarized according to the definition (2.4). In fact, for an arbitrary single-photon stateρ 1 , the degree of polarization is related to the purity Tr(ρ 2 ) according to However, this relation does not hold for other excitation manifolds. For example, any pure state in excitation manifold N = 2 of the form [16] |Ψ(a, θ ) = ae −iθ |2, 0 where a and θ are real numbers and 0 ≤ a ≤ 1/ √ 2, satisfies P S (|Ψ(a, θ ) ) = 0, which indicates that it is unpolarized. However, as we shall see below, these states have polarization structure (they are not isotropic in the polarization sense) and cannot be regarded as unpolarized.

Higher-order polarization properties
In order to characterize the polarization properties of a state, we shall employ measurements of higher-order moments of the Stokes operators. This is very close to the Glauber correlation functions in quantum coherence theory [17], and has common grounds with Klyshko generalized coherence matrices [1]. In a recent paper [4], we have used the central moments for higher-order polarization characterization. Whereas the central moments may be preferred by some readers, the raw moments used in the present work seem to allow for an easier and more systematic approach.
As we have already discussed, one can perform a measurement of the total photon number without disturbing the measurement of any other Stokes operator. In classical optics, this is tantamount to the fact that the state of polarization is independent of the intensity. This suggests that the polarization properties are given byŜ. However, an ideal measurement of polarization provides some information about the total energy and vice versa. For example, an even (odd) measured eigenvalue of any of the observablesŜ 1 ,Ŝ 2 , andŜ 3 implies an even (odd) total number of photons. Also, determining the probability distribution for the total number of photons p N simultaneously sets bounds on the polarization properties in accordance with the inequalities (2.3).
Taking these observations into account, we distinguish polarization properties for different numbers of photons, and let full polarization characterization refer to complete knowledge of the expectation values of all possible combinations of the Stokes operators.
The rth-order polarization information of a stateρ is then given by p N and the expectation values of the form T (r,N) where j k ∈ {1, 2, 3} andρ N denotes the normalized two-mode, N-photon state obtained by projectingρ onto the Nth excitation manifold Using the Fock basis, the projector can thus be expressed as1 1 N = ∑ N n=0 |n, N − n n, N − n| and p N = Tr(1 1 Nρ ). For any given order r and excitation manifold N, the elements (3.1) form a Cartesian tensor T (r,N) (ρ) of rank r. Due to the Hermiticity of the Stokes operators, theses tensors satisfy T (r,N) We leaveρ N and T (r,N) undefined for any N such that p N = 0, and employ the convention that they then do not contribute to sums. When (µ, ν, j k ) is a cyclic permutation of (1, 2, 3), the commutation relation (2.2) implies that polarization tensor elements of neighboring ranks are related according to Hence, T (r−1,N) can be determined from T (r,N) and, consequently, T (R,N) determines all T (r,N) such that r < R. Complete polarization information of order R is thus equivalent to complete polarization information of all orders r ≤ R.
Using the relations (2.1) and (2.2), it is also straightforward to show that the polarization information carried by T (R,N) is equivalent to that contained in the set of generalized coherence matrices of orders 2r (r ≤ R), whose elements are of the form [1] Having complete polarization information of all orders about a stateρ is therefore equivalent to knowing its block-diagonal projection [5,7,16,18] 5) whereρ N is given by (3.2). This is the so called the polarization sector (or polarization density matrix). The number of parameters characterizing a block-diagonal state limited to the excitation manifolds In particular, when a state is limited to the manifolds 0, 1, . . . , N, the number of parameters simplifies to For such a state, complete polarization information of order N is sufficient to determine its block-diagonal projection (3.5). The general density matrix for a state with no more than N photons is determined by N( N + 3)( N 2 + 3 N + 4)/4 independent real numbers. Hence, the polarization share of this information quickly decreases with N.

Polarization tomography
We now turn to the question of how to characterize polarization properties experimentally. Since our interest is limited to the information contained in the block-diagonal projection (3.5), it is clear that we are not required to do full quantum state tomography [5][6][7].
As complete polarization information corresponds to doing quantum tomography of all Nphoton Hilbert spaces excited by the considered state, one can make use of the methods developed for finite-dimensional systems [19][20][21]. However, some recently proposed higherorder intensity measurements [22] seem to be closest related to the ones we present below. We will assume ideal measurements and that the total photon number and its probability distribution p N can be determined. This is obviously a severe restriction apart from the lowest excitation manifolds. However, the situation where no information about the total photon number can be obtained is described by simply summing over the different manifolds as discussed in section 4.4.
Below, we also treat the experimental determination of different moments of an observable as different measurements. In principle, each moment requires an infinite number of measurement runs in order to be determined exactly. This would also give us the full probability distribution of the eigenvalues and thus all the moments. However, for the vast majority of realistic probability distributions, a lower moment requires fewer runs to be accurately determined.

Moment measurements
The fact that the classical Stokes parameters are easily determined experimentally makes them highly practical. Also in quantum optics, the measurement setups corresponding to the fundamental Stokes operators are simple. These setups are composed only by phase shifters, beam splitters and photon-number measurements. The effects of linear optical devices are described by SU(2) transformations [14], which can be expressed aŝ where Φ, Θ, and Ξ are the Euler angles. Any such transformation can be easily realized using linear optics [23] and they are lossless, so they leaveŜ 0 unaffected. Let us now introduce the Stokes operator in an arbitrary direction characterized by the unit vector n ∈ R 3 aŝ The effect of an arbitrary SU(2) transformation onŜ n , can then be expressed aŝ where R k (φ ) denotes the matrix describing a rotation of φ around the e k -axis, e.g.
Hence, any SU(2) transformation corresponds to a proper rotation in R 3 [14]. We note that S 3 , which gives the photon-number difference, is transformed according tô That is,Û(Φ, Θ, 0) is the standard displacement on the sphere and the transformation parameters Θ and Φ equal the spherical coordinates of the vector n characterizing the transformed Stokes operator. We see that anyŜ n is related toŜ 3 by an SU(2) transformation corresponding to a polarization rotation of Θ/2 followed by a differential phase shift of Φ/2. Expectation values of the form Ŝ r n can thus be straightforwardly determined experimentally. Ideally, we can simultaneously measure the total photon number N, so that also expectation values of the form Ŝ r n N can be determined. The tensor T (r,N) gives any expectation value of the form When all vectors are the same, (4.6) simplifies considerably. For a given state, the relation between Ŝ r n N and the direction n will be referred to as the N-photon Stokes moment profile of order r. These profiles can be expressed as where the moment component M is the sum of all tensor elements of the form T (r,N) j 1 ... j r that have k ones and ℓ twos as subscripts. Due to (3.1), every moment component is thus the expectation value of the Hermitian operator formed by the sum of the Stokes-operator products corresponding to its tensor elements. The number of such elements is given by the trinomial coefficient 321 . We note that the sum of all tensor elements of order r in excitation manifold N can be written as where n diag = (1, 1, 1)/ √ 3. Since the polarization tensor satisfies the Hermiticity condition (3.3), the moment components are real, and consequently the real part of T (r,N) is sufficient to determine Ŝ r n N in any direction n. Naturally, knowing the Stokes moment profile (4.7) is equivalent to knowing the k,ℓ . Moreover, using the commutation relation (2.2), it is possible to determine the differences between the tensor elements belonging to the same moment component M Making repeated use of the commutation relation (2.2), the moment components can then be expressed as k,ℓ is a sum over Stokes-operator products of orders smaller than r. The Casimir operatorŜ implies that, for r ≥ 2, we have where the last term can again be written as a sum over Stokes-operator products of orders smaller than r. Equations (4.12) and (4.14) show that there is a relation between moment components of orders r and r − 2, and that the number of independent moment components of order r is m r − m r−2 = 2r + 1. For a general N-photon state, the number of independent moment components to determine is thus ∑ N r=1 (2r + 1) = N(N + 2). For a general blockdiagonal state, we also have to determine the probability distribution for the total number of photons. Assuming that the excited manifolds are known to be limited to N 1 , N 2 , . . . , N ν , we find the number of independent parameters to be ν − 1 + ∑ ν k=1 N k (N k + 2), which is in agreement with (3.6).

General single-photon state
In the basis (|1, 0 , |0, 1 ), the density matrix of a general single-photon state can be written aŝ where R 2 + I 2 ≤ π 0 (1 − π 0 ). Using a superscript to identify state-specific average values, the first-order Stokes moment profile is given by Hence, the three moment components are seen to be independent.

General two-photon state
Using the basis (|2, 0 , |1, 1 , |0, 2 ), the density matrix of a general two-photon state can be written aŝ (4.17) The first-and second-order Stokes moment profiles can then be expressed as Ŝ n which makes it easy to identify the moment components. As implied by (4.13), the moment components of the three first terms of (4.19) are determined by two parameters. Hence, there are only five independent second-order moment components.

Choosing measurement directions
We have seen that the information content of the moment components allows us to do polarization tomography by only measuring moments. Performing the moment measurements in increasing order, the 2r + 1 independent moment components for each order r and manifold N can be determined by choosing equally many directions n such that (4.7) gives linearly independent equations for the unknown moment components.

First order
Quite naturally, both the first-order moment components and the firstorder polarization tensors are given by the manifold-specific Stokes parameters The three sets of information are hence identical and are obtained by determining the expectation value Ŝ n N for the directions n = (1, 0, 0), (0, 1, 0), and (0, 0, 1). As the operatorŝ S n andŜ −n only differ by the signs of their eigenvalues, the corresponding measurements will give the same information. Hence, equivalent measurements correspond to a line through the origin. Choosing three orthogonal directions as above thus results in a uniform distribution of the measurements on the Poincaré sphere.

Second order
We have seen that there are five independent moment components of second order. Thinking of the measurements as lines, we choose the directions as which maximizes the minimum angle between the lines [24,25] and thus in some sense spreads out the measurements over the Poincaré sphere as much as possible. The six secondorder moment components are then given by Note that their determination does not require any first-order measurement. The second-order polarization tensors can be expressed in the first-and second-order moment components as When writing tensors, we let larger entities and rows correspond to tensor indices placed to the left of those corresponding to smaller entities and columns.
As an aside, we decompose the Stokes-operator covariance matrix into different excitation manifolds Γ = ∑ ∞ N=0 p N Γ N and note that the matrix elements are given by For any state that satisfies P S = 0, we thus have Γ N = Re(T (2,N) ).

Third order
We know that there are seven independent third-order moment components. Maximizing the minimum angle between seven lines [25], we find that the measurements should correspond toŜ 1 ,Ŝ 2 ,Ŝ 3 , and the directions However, this choice gives only four independent measurements, since we havê and similar relations forŜ 3 2 andŜ 3 3 . By choosing directions close toŜ 1 ,Ŝ 2 , andŜ 3 , it is possible to determine all third-order moment components. However, this choice would make it hard to obtain the necessary data, since the corresponding expectation values differ only slightly from the known Ŝ 3 1 N , Ŝ 3 2 N , and Ŝ 3 3 N . Consequently, although highly symmetric polyhedrons have been successfully applied in protocols for tomography of multi-qubit states [26], the related method considered here fails. How to optimally choose the measurement directions for higher-order polarization tomography thus appears to be a complicated problem. This notwithstanding, the third-order polarization tensors can be expressed as

Recurrence relation for Stokes moment profiles
Above, we have seen that the N lowest-order Stokes moment profiles { Ŝ r n N } N r=1 contain all polarization information of an N-photon state. In particular, we show in the appendix that the higher-order profiles are determined by the recurrence relation where µ is a non-negative integer and f (n, k) are the central factorial numbers of the first kind given by [27] f (n, k) = (4.34) For the lowest excitation manifolds, we thus get Ŝ r n 0 = 0, Ŝ r n 1 = 1, r even, Ŝ n 1 , r odd, , r even, , r odd.

Non-resolved photon numbers
As pointed out above, apart from the few lowest excitation manifolds, it is difficult to distinguish different excitation manifolds experimentally. In case there is no information about the total photon number available, the measured expectation values are weighted averages over the manifolds of the form Â = ∑ ∞ N=0 p N Â N . Due to linearity, it is clear that the corresponding polarization tensors, Stokes moment profiles, and moment components, which are given by respectively, enjoy most of the properties of their manifold-specific counterparts. However, due to the factor N(N + 2) appearing in (4.14), the relations between moment components of different orders depend on the excitation manifold. In general, this makes all photon-numberaveraged moment components M (r) k,ℓ independent and Rth-order polarization tomography then requires ∑ R r=1 m r = R(R 2 + 6R + 11)/6 parameters to be determined. However, the knowledge of the average photon number Ŝ 0 and its variance Ŝ 2 0 − Ŝ 0 2 is sufficient to remove the redundancy for the second order, since we then know the right-hand side of the relation M (2) 2,0 + M (2) 0,2 + M (2) 0,0 = Ŝ 0 (Ŝ 0 + 2) obtained from (4.13). With this partial knowledge about the photon distribution, we can thus determine the second-order moment components using the five measurement settings given in section 4.2.2.
Now, assume that we know that a state is limited to the first three manifolds, i.e., that the number of photons cannot exceed two. In this case, the determination of Ŝ 0 , Ŝ 2 0 , and the three lowest-order Stokes moment profiles is sufficient for complete photon-resolved polarization characterization. Explicitly, we have p 1 = 2 Ŝ 0 − Ŝ 2 0 and p 2 = ( Ŝ 2 0 − Ŝ 0 )/2, which together with (4.35) give (4.38)

A menagerie of states and their polarization properties
We next apply the characterization developed above to some classes of states. In most cases, we give only the Stokes moment profiles for the states, as these provide the most compact presentation of the polarization properties. It should be straightforward to obtain the moment components and polarization tensors if needed. Using (4.5), we can write the Stokes moment profiles of an arbitrary stateρ as where the SU(2) transformationÛ(Φ, Θ, Ξ) is given by (4.1). Now, consider the stateρ ′ obtained by applying an SU(2) transformation toρ according tô As the trace of a product is invariant under cyclic permutations, (4.3) ensures that the Stokes moment profiles of the stateρ ′ are related to those ofρ by rotations. Indeed, we find that Ŝ r n ρ ′ N is obtained from Ŝ r n ρ N by rotating the latter ξ around the e 3 -axis, followed by a rotation of ϑ around the e 2 -axis and another of ϕ around the e 3 -axis. That is, we have Naturally, the sequence of rotations appearing in (5.3) is the inverse of the one described above. Because of these simple rotations, the determination of the polarization properties of a stateρ, implicitly gives the polarization properties of all states related toρ by an SU(2) transformation, although these states may appear very different. We note that common, passive, two-mode interferometers are described by SU(2) transformations too. The considerations below are therefore relevant to interferometry.

SU(2) coherent states
The SU(2) coherent states are the eigenstates of the operatorsŜ n . They are also the only states that minimize the variance sum, i.e., that saturate the left inequality in the uncertainty relation (2.3). Using the spherical coordinates (4.5), we have the eigenequationŜ n |N; Θ, Φ = N|N; Θ, Φ . The N-photon, SU(2) coherent states are of the form Since an overall phase factor does not have any physical significance, they are thus all related to the state |N, 0 by an SU(2) transformation (4.1). As discussed above, such transformations correspond to simple rotations of the Poincaré sphere, so we limit our explicit treatment to the states |N, 0 . Making use of well-known results for the beam splitter, we easily find the Stokes moment profiles to be The lowest-order polarization tensors are For N > 0, the SU(2) coherent states thus satisfy P S = 1.

Two-mode coherent states
Since any pair of two-mode coherent states with the same average total energy are related by an SU(2) transformation, it suffices to study states of the form The block-diagonal projection is clearly independent of the phase of α, and is given by a Poissonian mixture of SU(2) coherent states that all belong to different excitation manifolds. Hence, the manifold-specific expectation values coincide with those of the SU (2) In accordance with classical optics, P S = 1 for any two-mode coherent state with a finite average photon numberN. We note that whenN ≫ 1, the lowest Stokes moments satisfy Ŝ r n |α,0 n 3 =1 ≈ (Nn 3 ) r . For n 3 = 1, the Stokes moments (5.10) are directly given by the Poissonian photon distribution, and the approximation corresponds to the classical deterministic limit. The n 3 dependence of the approximation describes the transmission through a classical beam splitter or linear polarizer. In particular, Malus' law is obtained for r = 1.

|m, m states
When considering the two-mode Fock states |m, m , which allow for Heisenberglimited interferometry [28], we implicitly treat all states obtained from these by SU (2) transformations. The latter states can be expressed aŝ where ⌊x⌋ denotes the largest integer that is smaller than or equal to x. We have also assumed that the binomial coefficients are defined through the gamma function, so that negative integers are allowed as arguments. The Stokes moment profiles of the states |m, m are given by where F(n, k) denotes the central factorial numbers of the second kind (A.2). As both arguments are even in our case, we have [27] F(r, 2 j) = 2 In particular, we get The photon-number symmetry makes all odd-order Stokes profiles vanish. This is in stark contrast to the states |N, 0 and |α, 0 , and makes P S = 0. However, the so-called hidden polarization of the |m, m states appears in the even-order Stokes profiles. As sin 2 Θ = 1 − n 2 3 , these profiles have common features with rotated ones for |2m, 0 . This is in agreement with the fact that more elaborate measurements than those considered here are required to achieve Heisenberg resolution when employing the |m, m states [29].

Two-mode squeezed vacuum
Using the process of spontaneous parametric down-conversion, one can straightforwardly generate two-mode squeezed vacuum states. These have thermal photon-pair distributions and take the form

NOON states
Finally, let us consider the NOON states (|N, 0 + |0, N )/ √ 2, where N > 0. These are in some sense optimal for interferometry [30]. Their Stokes moment profiles are found to be Ŝ r We note that when r is even and N is odd, the effect of the second term within square brackets vanishes. For each of the first six NOON states, we have plotted the Stokes moment profile Ŝ where the polynomial [31] Q j (n) = 2 j−2n satisfies Q 0 (n) = 1 and the recurrence relation Q j+1 (n) = 2n 2 Q j (n) − n(2n − 1)Q j (n − 1). Hence, the de Broglie wavelength of the NOON states, which scales as N −1 and is the reason for their superiority [32], can be seen in these measurements.
Let us now return to the pure two-photon states that satisfy P S = 0. These are given in (2.7) and are found to be related to the two-photon NOON state by appropriate SU(2) transformations according to Hence, any Stokes moment profile of the state |Ψ(a, θ ) is related to the corresponding NOON profile by simple rotations. Since R 3 (π/2 + θ )·R 2 (χ)·R 3 (−π/2) = R 3 (θ )·R 1 (−χ), Ŝ 2 n |Ψ(a,θ ) 2 is obtained from Ŝ 2 n NOON 2 in figure 1 by applying a rotation of − arccos( √ 2a) around e 1 followed by a rotation of θ around e 3 . Since Ŝ n |Ψ(a,θ ) 2 = 0 is independent of n, the states |Ψ(a, θ ) lack first-order polarization structure. However, they do all have a second-order polarization structure.

Conclusions
Using expectation values of Stokes-operator products, we have developed a systematic scheme for characterizing higher-order polarization properties of two-mode quantized fields. Polarization tensors and Stokes moment profiles were introduced as two representations of the polarization information. The latter show how passive interferometry affects the moments of photon difference. This viewpoint was taken as polarization properties of different states were compared.
Other possible representations of the polarization information include central moments [4], quasi-probability distributions [33] and excitation-specific generalized coherence matrices. Complete polarization characterization requires the excitation manifolds to be addressed separately. For situations where this cannot be achieved, our characterization coincide with the one provided by Klyshko's generalized coherence matrices [1].
Assuming ideal photon-number resolving detectors, we have shown that it is possible to efficiently collect the data through Stokes moment measurements in different directions. In an experiment, it may be more practical to use more measurement directions than the minimum required, but our method should serve as a guide. In particular, we expect the introduced moment components to be useful. Another advantage of the described method is that it treats the Stokes moments order by order. Hence, if only the first few polarization orders are of interest, it makes the measurements easier.
Since the different excitation manifolds are treated separately, losses have drastic consequences in that higher excitation manifolds then contribute to the lower ones. Furthermore, whereas linear losses often model imperfections of single-photon detectors well, photon-number resolving detectors, which are required for full polarization characterization, are more complex and may call for nonlinear modeling.
On the other hand, the separation of data into excitation manifolds and moment orders may be useful when developing methods for efficient determination of polarization characteristics. For example, one can take into account that all state projectionsρ N in the different excitation manifolds must be physical states. In this way, it should be possible to develop efficient maximum likelihood methods similar to those regularly employed in common quantum tomography.