Abstract

We show that metrological resolution in the detection of small phase shifts provides a suitable generalization of the degrees of coherence and polarization. Resolution is estimated via Fisher information. Besides the standard two-beam Gaussian case, this approach provides also good results for multiple field components and nonGaussian statistics. This works equally well in quantum and classical optics.

© 2012 Optical Society of America

1. Introduction

Interference and polarization are leading manifestations of coherence [1, 2]. In spite of being a very basic topic the complete relationship between coherence, interference and polarization for arbitrary fields is still an open question and is currently the subject of very active investigation [324].

For two electromagnetic fields E1,2 we may say that interference reflects coherence between fields in the same vibration mode, while polarization displays coherence between fields in orthogonal vibration modes. Interference and polarization are easily interchangeable by means of simple energy-preserving transformations, such as phase plates, so that they are not independent field properties.

More specifically, coherence manifests in interference through the modulation of the intensity of the superposition 〈|E1e1 + E2e2|2〉, where ϕj are the phases acquired by the waves within the interferometer. This modulation can be conveniently assessed by the fringe visibility

V(E)=2|E1E2*||E1|2+|E2|2|μ(E)|=|E1E2*|(|E1|2|E2|2)1/2,
where μ is the complex second-order degree of coherence, and the angle brackets represent ensemble averages. In the polarization context, coherence manifests in the purity of the polarization state assessed through the degree of polarization
𝒫=|IMIm|IM+Im=[14detΓ(trΓ)2]1/2,Γ(E)=(|E1|2E1E2*E1*E2|E2|2),
where IM,m are the eigenvalues of the correlation matrix Γ. A key point is that 𝒫 is invariant under deterministic unitary transformations EE′ =UE, Γ(E) → Γ(E′) =UΓ(E)U, being U 2 × 2 unitary matrices, so we have 𝒫(E′) = 𝒫(E) while V(E′) ≠ V(E) and |μ(E′)| ≠ |μ(E)|. More specifically, it holds that V and 𝒫 satisfy the equality [21]
V2+R2=𝒫2,R(E)=||E1|2|E2|2|E1|2+|E2|2|.
This, along with Eq. (1), implies that 𝒫 is the maximum degree of coherence that can be obtained for a given field state after any deterministic unitary transformation
|μ(UE)|𝒫.

The standard definitions of the degrees of coherence and polarization in Eqs. (1) and (2) hold provided that three conditions are satisfied: i) we are restricted to two scalar electromagnetic fields, ii) we perform statistical evaluations of second order in the amplitudes, and iii) the fields obey Gaussian statistics. Beyond these three restrictions the standard approach may be undetermined or lack usefulness when applied to more general situations involving either more than two electric fields or nonGaussian statistics, as it is the case of the problems addressed in Refs. [333]. A particularly clear example of these limitations is provided by quantum field states reaching maximum resolution in phase-shift detection, since they present vanishing degrees of polarization and coherence μ = 𝒫 = 0 in spite of their interferometric usefulness [22, 32].

In this work we look for a more general approach to coherence that can be applied to any situation with multi-component fields and nonGaussian statistics, including naturally Eqs. (1) and (2) in the particular case of just two electric fields in a thermal-chaotic light state. To this end we focus on the practical application of interference and polarization to the detection of small phase shifts, as the cornerstone of the metrological applications of optics.

The key point is the idea of coherence as equivalent to good interference. This idea emerges after Eq. (1), where coherence is the maximum fringe visibility, as well as after Eq. (4), where visibility reaches its maximum provided that the unitary transformation U is properly chosen. This suggests that the degrees of coherence and polarization indicate the best interference that can be achieved with a suitable preparation of the interference for a given input field state.

Therefore, we will identify the generalized amount of coherence χ of a given field state with the maximum phase-shift resolution after a suitable optimization of the detection arrangement, so that the phase-shift measurement exploits all the interfering capabilities of the field state. Note that this approach can be equally well applied to quantum and classical fields.

The resolution can be properly assessed by means of the Fisher information entering in the Cramér-Rao lower bound for the uncertainty of phase-shift estimators [34,35]. It is worth pointing out that the Fisher information has been extensively used to provide original analyses and interpretations of many different problems in science, from the formation of basic physical laws, to biology and financial economy, for example [36, 37].

2. Fisher information and detection of small phase shifts

2.1. Phase shifts and phase-shift detection

By phase shifts we will understand linear, energy-preserving, unitary transformations of the complex amplitudes E of an arbitrary number k of complex electric fields, EUϕE, where Uϕ is a k × k unitary matrix. In general terms we may express Uϕ as Uϕ = exp(iϕG), where ϕ is the phase shift, and G an Hermitian k × k matrix G = G.

In quantum optics the amplitudes E become complex-amplitude operators a, with [ai, aj] = δi,j and [ai, aj] = 0. The phase-shift transformation reads aUϕa = TaT, with T = exp(iϕĜ) and Ĝ = aGa [38].

Both in classical and quantum optics the objective is to infer the value of the unknown phase shift ϕ from measurements performed on the transformed electric fields UϕE. Let us call P(θ|ϕ) the statistics of the measurement outcomes θ when the true value of the phase shift is ϕ. From a Bayesian perspective P(θ|ϕ) defines a posteriori probability distribution P(ϕ̃|θ) for the inferred value of the signal ϕ̃ depending on the N outcomes θ obtained after N repetitions of the measurement. This is of the form

P(ϕ˜|θ)Πj=1NP(θj|ϕ˜)Φ(ϕ˜),
where Φ(ϕ̃) is the prior distribution for the signal.

2.2. Fisher information

A convenient assessment of the uncertainty Δϕ on the value of ϕ after the measurement is the Fisher information F entering in the Cramér-Rao lower bound

(Δϕ)21NF,F=dθ1P(θ|ϕ)[dP(θ|ϕ)dϕ]2.
It is worth noting that the Cramér-Rao lower bound is not necessarily reachable in every situation, i. e., there is no guarantee that an efficient estimator can be found that reaches the minimum variance 1/(NF), specially for small N. However, for increasing N, the maximum likelihood estimator [the ϕ̃ value that maximizes P(ϕ̃|θ)] is unbiased and asymptotically efficient, so that Δϕ˜1/NF when N → ∞ [34]. After a large enough number of repetitions, 1/(NF) becomes the width of P(ϕ̃|ϕ) as a function of ϕ̃ since P(ϕ̃|ϕ) ∝ exp[−NF(ϕ̃ϕ)2/2], which is consistent with the above mentioned asymptotic efficiency.

Besides, the Fisher information is also the approximate form of several distances, or divergences [39, 40], between the statistics associated with two close enough phase shifts ϕ and ϕ + δϕ, such as the Hellinger distance,

H=dθ[P(θ|ϕ+δϕ)P(θ|ϕ)]2(δϕ)24F,
the Kullback-Liebler divergence
K=dθP(θ|ϕ+δϕ)lnP(θ|ϕ+δϕ)P(θ|ϕ)(δϕ)22F,
or the Rényi-Chernoff distance
Cs=lndθPs(θ|ϕ+δϕ)P1s(θ|ϕ)s(1s)2(δϕ)2F,
that becomes the Bhattacharyya distance for s = 1/2. All them have been already applied for the assessment of coherence and polarization in different contexts [2931, 41, 42].

2.3. Generalized measure of coherence

In general, F depends on the light state, on the generator G, on the measurement performed, and also on ϕ. Following the idea of coherence as optimum visibility, we will consider that the measurement performed extracts from the transformed fields UϕE the maximum information possible about ϕ. Throughout we assume ϕ → 0 since precision metrology is always interested in the detection of small phase shifts, which can be, without loss of generality, brought back around a zero phase-shift value. Nevertheless, there would be no fundamental difficulty to apply this approach beyond precision metrology to large phase shifts, maybe with the drawback of lacking simple expressions and obtaining different conclusions depending on the particular values of ϕ considered.

The larger F the lesser the uncertainty and the larger the resolution so we may say that larger F means larger coherence. Since F is not bounded from above, ∞ ≥ F ≥ 0, it might be convenient to normalize it, so we may introduced a generalized assessment of coherence χ comprised between 0 and 1 in the form

χ2=FF+4tr(G2).
The factor 4tr(G2) is introduced so that χ becomes exactly the degree of coherence for two electric fields in a classical thermal-chaotic light state [see Eq. (21) below]. The dependence on G means essentially that the usefulness of interference fringes depends on whether the phase-shift transformation exploits all the interfering capabilities of the light state. A coherence assessment depending just on the field state can be obtained as the maximum of χ when G is varied. This might be regarded as the generalization of the degree of polarization as the ultimate measure of coherence in accordance with the standard result for two electric fields and Gaussian statistics recalled in the introduction.

Quantum and classical statistics differ in very fundamental issues, such as the uncertainty relation for example. Thus, at this point the analysis must be split into classical and quantum channels, although we will see that they largely run parallel.

3. Classical sector

In classical optics every measurement statistics P(θ|ϕ) can be derived as a marginal distribution of the probability distribution for the transformed amplitudes W(E|ϕ)=W(UϕE), where W(E) is the probability distribution for the complex amplitudes in the field state before the signal-dependent transformation Uϕ. More specifically, taking into account that any measured observable is a function of the complex amplitudes θ(E) we have

P(θ|ϕ)=d2kEW(UϕE)δ[θθ(E)].
Formally we can consider a suitable change of variables including θ within the new set of arguments of the transformed distribution 𝒲, so that
P(θ|ϕ)=dϑ𝒲(θ,ϑ),
where ϑ represents the rest of variables required for a complete specification of the system.

After Eqs. (11) and (12) we have that any measurement statistics P(θ|ϕ) is extracted as a marginal distribution from the complete distribution W(UϕE). Therefore, W(UϕE) contains no less information than any P(θ|ϕ), so that (see Appendix A)

FFC=d2kE1W(UϕE)[dW(UϕE)dϕ]2.
This is to say that FC is the maximum Fisher information reachable by any measurement. Therefore,
χ2=FCFC+4tr(G2),
provides a suitable estimation of the degree of coherence, understood as metrological usefulness under phase shifts generated by G. Let us consider some meaningful examples.

3.1. Thermal-chaotic light

A fundamental example is provided by the coherence between two electric fields in a thermal-chaotic light state. This allows us to check the compatibility of this approach with the standard definition. Thermal-chaotic light is fully characterized by second-order moments of the field amplitudes as

W(E)=detMπkexp(EME),
where M = Γ−1 and Γij=EiEj*, with Γj,j = 〈|Ej|2〉 ≠ 0 since otherwise we shall accordingly reduce the number k of electric fields. A simple calculus leads to (see Appendix B)
FC=2[tr(ΓGΓ1G)tr(G2)].

The dependence on G reflects the fact that for a given field state the usefulness of the interference depends on the particular interferometric arrangement considered and whether it exploits the capabilities of the incoming light state. To exploit all the coherence conveyed by the field state we have to look for the maximum of FC by varying G. To this end we particularize the above expressions to the field basis where M is diagonal, leading to

FC=4i,j=1k|Gi,j|2𝒫i,j21𝒫i,j24tr(G2)𝒫max21𝒫max2,
where only the terms ij contribute, 𝒫i,j is the two-field degree of polarization of modes i, j, and 𝒫max is the maximum of 𝒫i,j for all pairs i, j:
𝒫i,j=|IiIj|Ii+Ij𝒫max=IMImIm+Im,
where IM,m are the maximum and minimum intensities. Therefore, after Eq. (14)
χ𝒫max
The equality in Eqs. (17) and (19) is reached when G just mixes the two electric fields with the maximum degree of polarization.

The upperbound in Eq. (19) agrees with previous results showing that the maximum visibility in the interference of two partially polarized transversal waves is reached by combining just the two components with the largest degree of polarization [14]. The bound in Eq. (17) reproduces also basic results of classical metrology: i) the resolution increases without limit as 𝒫max → 1, and ii) resolution is independent of the total field intensity. We will see below that these two properties no longer hold in the quantum sector.

3.1.1. Interference between two scalar fields

Let us particularize this result to the standard case with two electric fields k = 2

Γ=(|E1|2E1E2*E1*E2|E2|2),G(1001)
where G generates phase-difference shifts. In this case, after Eqs. (10) and (16) we obtain
FC=4tr(G2)|μ|21|μ|2,χ=|μ|,
where μ is the standard two-mode degree of coherence in Eq. (1).

3.1.2. Interference between two partially polarized fields

Let us consider the interference between two partially polarized fields in schemes of the Young type. This is k = 4 and

G(I00I),
where I is the 2 × 2 identity matrix. We are considering the field basis in which E1 = E1,x, E2 = E1,y denote the two components, say x and y, of one of the fields, while E3 = E2,x, E4 = E2,y represent the components of the other one. Note that as reflected by the form of G in this scheme the two components x, y of each wave experience the same phase shift. The corresponding 4×4 correlation matrix can be expressed as
Γ=(Γ1ϒϒΓ2)=Γ01/2U(IΩΩI)UΓ01/2,
with
Γ0=(Γ100Γ2),Γj=(|Ej,x|2Ej,xEj,y*Ej,x*Ej,y|Ej,y|2)
and
U=(U100U2),Ω=(μS00μI),
where U1,2 are the unitary matrices performing the singular-value decomposition of Γ11/2ϒΓ21/2, this is Ω=U1Γ11/2ϒΓ21/2U2, and μS,I are the corresponding real and positive singular values, with μSμI for example. These singular values were introduced in Ref. [10] to represent the coherence between two partially polarized waves independently of their polarization states.

Applying Eq. (16) to this case and taking into account that Γ01/2GΓ01/2=G and UGU = G we get

FC=2tr(G2)(11μS2+11μI22),χ2=μS2+μI22μS2μI22μS2μI2.
For example, if μS = μI we get χ = μS which is exactly the case of two scalar fields in Eq. (21). On the other hand FC diverges when μS → 1. All this is fully consistent with the properties of μS,I as measures of coherence between two partially polarized waves [43, 44].

3.2. Fields with vanishing second-order degrees of coherence and polarization

The above equivalences in Eqs. (21) and (26) between resolution and the standard degree of coherence apply just to two fields in a thermal-chaotic light state, but fails to be true in general for other more sophisticated field statistics, as already shown in Ref. [22].

This is the case of field states whose probability distribution for the field amplitudes is an even function of any of the amplitudes, say W(−E1, E2) = W(E1, E2), since Eq. (1) leads automatically to μ = 0 irrespectively of any other statistical properties of the field state. Some examples are provided by most of the quantum field states reaching maximum phase-shift resolution that have EiEj*=Cδi,j, for all i, j, where C is a constant [32] leading to μ = 𝒫 = 0. This includes the so-called N00N states [45] that are examined in more detail in the quantum sector.

4. Quantum sector

In the quantum sector the upper bound FC in Eq. (13) cannot be defined because there is no proper probability distribution W(E) providing observable statistics via marginals. Actually, this is the distinctive feature of nonclassical light [1, 46].

Nevertheless, it is possible to find an upper bound to F in terms of the so-called quantum Fisher information FQ [35],

FFQ=2i,j(pipj)2pi+pj|ψi|G^|ψj|2,
where pi and |ψi〉 are the eigenvalues and eigenstates of the density matrix ρ representing the field state
ρ=ipi|ψiψi|.
In Eq. (27) the sum is extended to all pairs i, j with pi + pj ≠ 0. When the field state is pure, i. e., ρ = |ψ〉 〈ψ| we have that Eq. (27) greatly simplifies becoming the variance of the generator Ĝ in the field state |ψ
FQ=4(ΔG^)2.
Thus, in any case FQ is the quantum analog of FC in Eq. (13). Accordingly, a suitable estimation of the degree of coherence conveyed by the field state under phase shifts generated by Ĝ in the quantum sector is
χ2=FQFQ+4tr(G2).

4.1. Thermal-chaotic light

In order to compare the classical and quantum sectors with a meaningful example let us consider quantum thermal-chaotic light. In the mode basis where the correlation matrix is diagonal aiaj=n¯iδi,j the field state factorizes ρ=i=1kρi, with [1]

ρi=11+n¯in=0(n¯i1+n¯i)n|niin|,
where |ni are the corresponding photon-number states.

In order to check compatibility with the classical sector as simply and clearly as possible let us consider the two-field case k = 2. By symmetry, any generator of the form G^exp(iϕ)a1a2+exp(iφ)a1a2 is optimum for any φ, leading to a maximum Fisher information (see Appendix C)

FQ=4tr(G2)𝒫21𝒫2+2/n¯,χ=𝒫(n¯n¯+2)1/2,
where 𝒫 is the standard degree of polarization in Eq. (2), and is the total mean number of photons:
𝒫=|n¯1n¯2|n¯1+n¯2,n¯=n¯1+n¯2.
It can be appreciated that for 𝒫 ≠ 1 the classical result in Eq. (17) is obtained in the limit → ∞, as it should be expected. On the other hand, for 𝒫 → 1 we have that: i) the Fisher information does not diverge, contrary to the equivalent situation in the classical sector, but remains always finite, and ii) the Fisher information depends on the field intensity, so that for 𝒫 = 1 it holds that FQ = 2tr(G2). Actually, the scaling FQ is known as the standard quantum limit, which is the largest Fisher information that can be reached with classical light in quantum optics [47,48]. This may be compared with the scaling FQ2 obtained for suitable nonclassical light, such as N00N states, as shown next in more detail.

4.2. N00N state

In the photon-number basis, the N00N states are

|ψ=12(|n1|02+|01|n2).
This is an example of field states with vanishing degrees of polarization and coherence μ = 𝒫 = 0, since for all n > 1 it holds that E1E2*=0, and 〈|E1|2〉 = 〈|E2|2〉. Despite this, these states are extremely useful in interference since they provide much larger resolution than other states with larger values of μ and 𝒫.

To show this we note that the maximum quantum Fisher information holds for G^a1a1a2a2 leading to FQ = 2tr(G2)2, where = n is the total mean number of photons [32]. The scaling FQ2 is known as Heisenberg limit, that clearly surpasses the scaling FQ that can be obtained with classical light [49].

The optimum Fisher information can be approached via ideal phase measurement whose statistics is given by projection on the relative-phase states [50]

P(θ)=|θ|ψ|2,|θ=12πm=0Neimθ|nm1|m2,
leading to the phase-difference distribution
P(θ|ϕ)=1πcos2[n(θϕ)/2],
with Fisher information F = n2.

The relative phase states emerge from a polar decomposition of the product of complex amplitudes a1a2 into a product of amplitude modulus and exponential of the phase difference. This problem has unitary solution for the exponential of the relative phase at difference with the equivalent single-mode polar decompositions [50].

It is worth noting that the total number n and the relative phase ϕ are commuting observables as the phase-number counterpart of commutation between total momentum and relative position for two particles [p1 + p2, x1x2] = 0 since, roughly speaking, we may say that n = n1 + n2 and ϕ = ϕ1ϕ2, where n1,2 and ϕ1,2 are the corresponding variables in the respective field mode. This does not contradict that single-mode number and phase do not commute, [nj, ϕj] ≠ = 0.

5. Conclusions

The classic degrees of coherence and polarization may admit different legitimate generalizations focusing on different field properties. For example, the metrological-based approach presented here is naturally different from global assessments of coherence and polarization in terms of distances of the light state to fully incoherent or unpolarized light [2528, 51]. This is also different from the degree of polarization in Ref. [52] defined in terms of the minimum overlap between the original and transformed field state under any SU(2) transformation. In this work we are considering exclusively infinitesimal transformations in the spirit of precision metrology, while typically minimum overlap is obtained for large phase shifts.

We highlight the applicability of one and the same formalism both to the classical and quantum optical realms, obtaining parallel and consistent results in both of them [18, 5356].

We have checked the consistency of this approach in the case of thermal light and N00N states. We think that there will be no consistency problems in any situation with multi-component fields and nonGasussian statistics because of the practical nature of this formalism and its focus on the observables consequences of coherence.

A. Upper bound to Fisher information in classical optics

In this Appendix we demonstrate the upper bound for Fisher information in classical optics in Eq. (13). To simplify notation let us consider discrete indices instead of continuous variables so that the measured statistics Pj of the j variable can be expressed in terms of the joint statistics of the j and k variables Wj,k as

Pj=kWj,k,
where for simplicity we omit expressing explicitly the ϕ dependence. Let us demonstrate that FFC where
F=jPj2Pj=jPj(lnPj)=j,kWj,k(lnPj),FC=j,kWj,k2Wj,k=j,kWj,k(lnWj,k),
where primes indicate derivation with respect to ϕ. Then
FCF=j,kWj,k(lnWj,kPj)=jPjkΛj,k(lnΛj,k),
where Λj,k is the conditional probability of k given j
Λj,k=Wj,kPj,kΛj,k=1.
Finally
FCF=jPjkΛj,k2Λj,k0.

As a further alternative demonstration let us show that the Hellinger distance in Eq. (7) HP between the probability distributions Pj = Pj(ϕ) and j = j(ϕ + δϕ) is smaller than the Hellinger distance HW between Wj,k = Wj,k(ϕ) and j,k = j,k(ϕ + δϕ)

HP=2(1jPjP˜j),HW=2(1j,kWj,kW˜j,k).
This is because PjP˜jkWj,kW˜j,k. To show this let us consider the square of the left-hand side
PjP˜j=k,Wj,kW˜j,=12k,(Wj,kW˜j,+Wj,W˜j,k),
and the square of the right-hand side
(kWj,kW˜j,k)2=k,Wj,kW˜j,Wj,W˜j,k,
so that PjP˜jkWj,kW˜j,k holds after comparing the right-hand side of Eqs. (43) and (44) because a+b2ab for all a, b ≥ 0. Finally this implies HPHW and thus FFC after Eq. (7).

B. Fisher information for classical thermal-chaotic light

Let us particularize FC in Eq. (13) to the case of thermal-chaotic light. After Eq. (15) we have

W(UϕE)=detMπkexp(EUϕMUϕE),
so that
W(UϕE)dϕ|ϕ=0=EMEW(E),
where
M=d(UϕMUϕ)dϕ|ϕ=0=i[G,M].
Thus Eq. (13) leads to
FC=d2kE(EME)2W(E)=i,j,,m=1kMi,jM,md2kEEi*EjE*EmW(E).
Using the Gaussian momentum theorem [1]
EjEmEi*E*=EjEi*EmE*+EmEi*EjE*,
we readily get
FC=[tr(MM1)]2+tr[(MM1)2]=tr[(MM1)2],
leading to
FC=2[tr(M1GMG)tr(G2)].
In the field basis where M is diagonal we get
FC=i,j(IiIj)2IiIj|Gi,j|2,
where Ii = 〈|Ei|2〉. Note the formal similarity with the quantum Fisher information in Eq. (27). We finally get
FC=4i,j=1k|Gi,j|2𝒫i,j21𝒫i,j24tr(G2)𝒫max21𝒫max2,
where 𝒫i,j is the two-mode degree of polarization of modes i, j, and 𝒫max is the maximum of 𝒫i,j for all pairs i, j:
𝒫i,j=|IiIj|Ii+Ij𝒫max=IMImIM+Im,
where IM,m are the maximum and minimum intensities. The equality in Eq. (53) is reached when G mixes just the electric fields with the maximum and minimum intensities.

C. Quantum Fisher information for thermal-chaotic light

Since the thermal-chaotic states in Eq. (31) are mixed we have to resort to Eq. (27) to compute its quantum Fisher information. Since the product of number states |n1〉 |n2〉 are the eigenstates of ρ, for G^=exp(iφ)a1a2+exp(iφ)a1a2 we have to compute

|ψi|G^|ψj|2=|n1|n2|[exp(iφ)a1a2+exp(iφ)a1a2]|n1|n2|2.
In the first place we have
n1|n2|[exp(iφ)a1a2+exp(iφ)a1a2]|n1|n2=exp(iφ)(n1+1)n2δn1=n1+1δn2=n21+exp(iφ)n1(n2+1)δn1=n11δn2=n2+1.
The two Kronecker deltas are incompatible so that in the modulus square there are no crossed contributions and
FQ=2n1,n2=0(pn1+1pn21pn1pn2)2pn1+1pn21+pn1pn2(n1+1)n2+exchange12,
where
pni=11+n¯i(n¯i1+n¯i)ni.
After some simple algebra
FQ=4(n¯1n¯2)22n¯1n¯2+n¯1+n¯2.
This expression can be fruitfully simplified using the total mean number of photons = 1 + 2 and the degree of polarization in Eq. (2) 𝒫 = |12|/(1 + 2) where we have taken into account that for the thermal state in Eq. (31) E1E2*=0, so that
FQ=4𝒫2n¯2n¯2(1𝒫2)/2+n¯=4tr(G2)𝒫21𝒫2+2/n¯,
where we have used that tr(G2) = 2.

Acknowledgments

This work has been supported by Project No. FIS2008-01267 of the Spanish Dirección General de Investigación del Ministerio de Ciencia e Innovación, and from Project QUITEMAD S2009-ESP-1594 of the Consejería de Educación de la Comunidad de Madrid. I thank the anonymous reviewers for one of the demonstrations in Appendix A and other useful suggestions.

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15. J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007). [CrossRef]  

16. I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for nxn covariance matrices,” arXiv:0807.2171v1 [physics.optics].

17. P. Réfrégier, “Mean-square coherent light,” Opt. Lett. 33, 1551–1553 (2008). [CrossRef]   [PubMed]  

18. A. Luis, “Quantum-classical correspondence for visibility, coherence, and relative phase for multidimensional systems,” Phys. Rev. A 78, 025802 (2008). [CrossRef]  

19. P. Réfrégier and A. Luis, “Irreversible effects of random unitary transformations on coherence properties of partially polarized electromagnetic fields,” J. Opt. Soc. Am. A 25, 2749–2757 (2008). [CrossRef]  

20. R. Martínez-Herrero and P.M. Mejías, “Maximizing Youngs fringe visibility under unitary transformations for mean-square coherent light,” Opt. Express 17, 603–610 (2009). [CrossRef]   [PubMed]  

21. A. Luis, “Coherence and visibility for vectorial light,” J. Opt. Soc. A 27, 1764–1769 (2010). [CrossRef]  

22. A. Luis, “Coherence versus interferometric resolution,” Phys. Rev. A 81, 065802 (2010). [CrossRef]  

23. A. Luis, “An overview of coherence and polarization properties for multicomponent electromagnetic waves,” in Advances in Information Optics and Photonics, International Commission for Optics, vol. VI, A. T. Friberg and R. Dändliker, eds. (SPIE, 2009) pp. 171–188.

24. J. Tervo, T. Setälä, and A. T. Friberg, “Phase correlations and optical coherence,” Opt. Lett. 37, 151–153 (2012). [CrossRef]   [PubMed]  

25. A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002). [CrossRef]  

26. A. Luis, “Degree of polarization of type-II unpolarized light,” Phys. Rev. A 75, 053806 (2007) [CrossRef]  

27. A. Luis, “Polarization distributions and degree of polarization for quantum Gaussian light fields,” Opt. Commun. 273, 173–181 (2007). [CrossRef]  

28. A. Luis, “Ensemble approach to coherence between two scalar harmonic light vibrations and the phase difference,” Phys. Rev. A 79, 053855 (2009). [CrossRef]  

29. A. Picozzi, “Entropy and degree of polarization for nonlinear optical waves,” Opt. Lett. 29, 1653–1655 (2004). [CrossRef]   [PubMed]  

30. P. Réfrégier, “Polarization degree of optical waves with non-Gaussian probability density functions: Kullback relative entropy-based approach,” Opt. Lett. 30, 1090–1092 (2005). [CrossRef]   [PubMed]  

31. P. Réfrégier and F. Goudail, “Kullback relative entropy and characterization of partially polarized optical waves,” J. Opt. Soc. A 23, 671–678 (2006). [CrossRef]  

32. A. Rivas and A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A 77, 022105 (2008). [CrossRef]  

33. A. Luis, “Quantum-limited metrology with nonlinear detection schemes,” SPIE Reviews 1, 018006 (2010).

34. H. Cramér, Mathematical Methods of Statistics (Asia Publishing House, 1962).

35. S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994). [CrossRef]   [PubMed]  

36. Exploratory Data Analysis Using Fisher Information, B. R. Frieden, ed. (Springer-Verlag, 2007). [CrossRef]  

37. B. R. Frieden, Physics from Fisher information: A Unification, (Cambridge U. Press, 1999).

38. A. Luis and L. L. Sánchez-Soto, “A quantum description of the beam splitter,” Quantum Semiclass. Opt. 7, 153–160 (1995). [CrossRef]  

39. T. M. Cover and J. A. Thomas, Elements of Information Theory, (Wiley Interscience, 1991). [CrossRef]  

40. A. D. C. Nascimento, R. J. Cintra, and A. C. Frery, “Hypothesis testing in speckled data with stochastic distances,” IEEE Trans. Geos. Remot. Sens. 48, 373–385 (2010). [CrossRef]  

41. F. Goudail, P. Réfrégier, and G. Delyon, “Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images,” J. Opt. Soc. Am. A 21, 1231–1240 (2004). [CrossRef]  

42. G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010). [CrossRef]  

43. P. Réfrégier, “Mutual information-based degrees of coherence of partially polarized light with Gaussian fluctuations,” Opt. Lett. 30, 3117–3119 (2005). [CrossRef]   [PubMed]  

44. P. Réfrégier and A. Roueff, “Visibility interference fringes optimization on a single beam in the case of partially polarized and partially coherent light,” Opt. Lett. 32, 1366–1368 (2007). [CrossRef]   [PubMed]  

45. I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010). [CrossRef]   [PubMed]  

46. H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995). [CrossRef]  

47. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981). [CrossRef]  

48. A. Rivas and A. Luis, “Precision quantum metrology and nonclassicality in linear and nonlinear detection schemes,” Phys. Rev. Lett. 105, 010403 (2010). [CrossRef]   [PubMed]  

49. Z. Y. Ou, “Fundamental quantum limit in precision phase measurement,” Phys. Rev. A 55, 2598–2609 (1997). [CrossRef]  

50. A. Luis and L. L. Sánchez-Soto, “Quantum phase difference, phase measurements and Stokes operators,” in Progress in Optics, vol. 41, E. Wolf, ed. (Elsevier, Amsterdam, 2000), pp. 421–482. [CrossRef]  

51. A. Luis, “Visibility for anharmonic fringes,” J. Phys. A: Math. Gen. 35, 8805–8815 (2002). [CrossRef]  

52. A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005). [CrossRef]  

53. R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987). [CrossRef]   [PubMed]  

54. A. Luis, “Quantum mechanics as a geometric phase: phase-space interferometers,” J. Phys. A 34, 7677–7684 (2001). [CrossRef]  

55. A. Luis, “Classical mechanics and the propagation of the discontinuities of the quantum wave function,” Phys. Rev. A 67, 024102 (2003). [CrossRef]  

56. B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010). [CrossRef]   [PubMed]  

References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  2. J. W. Goodman, Statistical Optics (John Wiley and Sons Inc., 1985).
  3. B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
    [Crossref]
  4. H. M. Ozaktas, S. Yüksel, and M. A. Kutay, “Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence,” J. Opt. Soc. Am. A 19, 1563–1571 (2002).
    [Crossref]
  5. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
    [Crossref]
  6. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields” Opt. Express 11, 1137–1143 (2003).
    [Crossref] [PubMed]
  7. T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328–330 (2004).
    [Crossref] [PubMed]
  8. E. Wolf, “Comment on Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1712–1712 (2004).
    [Crossref] [PubMed]
  9. T. Setälä, J. Tervo, and A. T. Friberg, “Reply to comment on Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1713–1714 (2004).
    [Crossref]
  10. P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 6051–6060 (2005).
    [Crossref] [PubMed]
  11. F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007).
    [Crossref] [PubMed]
  12. R. Martínez-Herrero and P.M. Mejías, “Maximum visibility under unitary transformations in two-pinhole interference for electromagnetic fields,” Opt. Lett. 32, 1471–1473 (2007).
    [Crossref] [PubMed]
  13. A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A 24, 1063–1068 (2007).
    [Crossref]
  14. A. Luis, “Maximum visibility in interferometers illuminated by vectorial waves,” Opt. Lett. 32, 2191–2193 (2007).
    [Crossref] [PubMed]
  15. J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
    [Crossref]
  16. I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for nxn covariance matrices,” arXiv:0807.2171v1 [physics.optics].
  17. P. Réfrégier, “Mean-square coherent light,” Opt. Lett. 33, 1551–1553 (2008).
    [Crossref] [PubMed]
  18. A. Luis, “Quantum-classical correspondence for visibility, coherence, and relative phase for multidimensional systems,” Phys. Rev. A 78, 025802 (2008).
    [Crossref]
  19. P. Réfrégier and A. Luis, “Irreversible effects of random unitary transformations on coherence properties of partially polarized electromagnetic fields,” J. Opt. Soc. Am. A 25, 2749–2757 (2008).
    [Crossref]
  20. R. Martínez-Herrero and P.M. Mejías, “Maximizing Youngs fringe visibility under unitary transformations for mean-square coherent light,” Opt. Express 17, 603–610 (2009).
    [Crossref] [PubMed]
  21. A. Luis, “Coherence and visibility for vectorial light,” J. Opt. Soc. A 27, 1764–1769 (2010).
    [Crossref]
  22. A. Luis, “Coherence versus interferometric resolution,” Phys. Rev. A 81, 065802 (2010).
    [Crossref]
  23. A. Luis, “An overview of coherence and polarization properties for multicomponent electromagnetic waves,” in Advances in Information Optics and Photonics, International Commission for Optics, vol. VI, A. T. Friberg and R. Dändliker, eds. (SPIE, 2009) pp. 171–188.
  24. J. Tervo, T. Setälä, and A. T. Friberg, “Phase correlations and optical coherence,” Opt. Lett. 37, 151–153 (2012).
    [Crossref] [PubMed]
  25. A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
    [Crossref]
  26. A. Luis, “Degree of polarization of type-II unpolarized light,” Phys. Rev. A 75, 053806 (2007)
    [Crossref]
  27. A. Luis, “Polarization distributions and degree of polarization for quantum Gaussian light fields,” Opt. Commun. 273, 173–181 (2007).
    [Crossref]
  28. A. Luis, “Ensemble approach to coherence between two scalar harmonic light vibrations and the phase difference,” Phys. Rev. A 79, 053855 (2009).
    [Crossref]
  29. A. Picozzi, “Entropy and degree of polarization for nonlinear optical waves,” Opt. Lett. 29, 1653–1655 (2004).
    [Crossref] [PubMed]
  30. P. Réfrégier, “Polarization degree of optical waves with non-Gaussian probability density functions: Kullback relative entropy-based approach,” Opt. Lett. 30, 1090–1092 (2005).
    [Crossref] [PubMed]
  31. P. Réfrégier and F. Goudail, “Kullback relative entropy and characterization of partially polarized optical waves,” J. Opt. Soc. A 23, 671–678 (2006).
    [Crossref]
  32. A. Rivas and A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A 77, 022105 (2008).
    [Crossref]
  33. A. Luis, “Quantum-limited metrology with nonlinear detection schemes,” SPIE Reviews 1, 018006 (2010).
  34. H. Cramér, Mathematical Methods of Statistics (Asia Publishing House, 1962).
  35. S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
    [Crossref] [PubMed]
  36. Exploratory Data Analysis Using Fisher Information, B. R. Frieden, ed. (Springer-Verlag, 2007).
    [Crossref]
  37. B. R. Frieden, Physics from Fisher information: A Unification, (Cambridge U. Press, 1999).
  38. A. Luis and L. L. Sánchez-Soto, “A quantum description of the beam splitter,” Quantum Semiclass. Opt. 7, 153–160 (1995).
    [Crossref]
  39. T. M. Cover and J. A. Thomas, Elements of Information Theory, (Wiley Interscience, 1991).
    [Crossref]
  40. A. D. C. Nascimento, R. J. Cintra, and A. C. Frery, “Hypothesis testing in speckled data with stochastic distances,” IEEE Trans. Geos. Remot. Sens. 48, 373–385 (2010).
    [Crossref]
  41. F. Goudail, P. Réfrégier, and G. Delyon, “Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images,” J. Opt. Soc. Am. A 21, 1231–1240 (2004).
    [Crossref]
  42. G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
    [Crossref]
  43. P. Réfrégier, “Mutual information-based degrees of coherence of partially polarized light with Gaussian fluctuations,” Opt. Lett. 30, 3117–3119 (2005).
    [Crossref] [PubMed]
  44. P. Réfrégier and A. Roueff, “Visibility interference fringes optimization on a single beam in the case of partially polarized and partially coherent light,” Opt. Lett. 32, 1366–1368 (2007).
    [Crossref] [PubMed]
  45. I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
    [Crossref] [PubMed]
  46. H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
    [Crossref]
  47. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [Crossref]
  48. A. Rivas and A. Luis, “Precision quantum metrology and nonclassicality in linear and nonlinear detection schemes,” Phys. Rev. Lett. 105, 010403 (2010).
    [Crossref] [PubMed]
  49. Z. Y. Ou, “Fundamental quantum limit in precision phase measurement,” Phys. Rev. A 55, 2598–2609 (1997).
    [Crossref]
  50. A. Luis and L. L. Sánchez-Soto, “Quantum phase difference, phase measurements and Stokes operators,” in Progress in Optics, vol. 41, E. Wolf, ed. (Elsevier, Amsterdam, 2000), pp. 421–482.
    [Crossref]
  51. A. Luis, “Visibility for anharmonic fringes,” J. Phys. A: Math. Gen. 35, 8805–8815 (2002).
    [Crossref]
  52. A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
    [Crossref]
  53. R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
    [Crossref] [PubMed]
  54. A. Luis, “Quantum mechanics as a geometric phase: phase-space interferometers,” J. Phys. A 34, 7677–7684 (2001).
    [Crossref]
  55. A. Luis, “Classical mechanics and the propagation of the discontinuities of the quantum wave function,” Phys. Rev. A 67, 024102 (2003).
    [Crossref]
  56. B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
    [Crossref] [PubMed]

2012 (1)

2010 (8)

A. Luis, “Quantum-limited metrology with nonlinear detection schemes,” SPIE Reviews 1, 018006 (2010).

A. D. C. Nascimento, R. J. Cintra, and A. C. Frery, “Hypothesis testing in speckled data with stochastic distances,” IEEE Trans. Geos. Remot. Sens. 48, 373–385 (2010).
[Crossref]

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

A. Luis, “Coherence and visibility for vectorial light,” J. Opt. Soc. A 27, 1764–1769 (2010).
[Crossref]

A. Luis, “Coherence versus interferometric resolution,” Phys. Rev. A 81, 065802 (2010).
[Crossref]

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref] [PubMed]

A. Rivas and A. Luis, “Precision quantum metrology and nonclassicality in linear and nonlinear detection schemes,” Phys. Rev. Lett. 105, 010403 (2010).
[Crossref] [PubMed]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

2009 (2)

R. Martínez-Herrero and P.M. Mejías, “Maximizing Youngs fringe visibility under unitary transformations for mean-square coherent light,” Opt. Express 17, 603–610 (2009).
[Crossref] [PubMed]

A. Luis, “Ensemble approach to coherence between two scalar harmonic light vibrations and the phase difference,” Phys. Rev. A 79, 053855 (2009).
[Crossref]

2008 (4)

A. Rivas and A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A 77, 022105 (2008).
[Crossref]

P. Réfrégier, “Mean-square coherent light,” Opt. Lett. 33, 1551–1553 (2008).
[Crossref] [PubMed]

A. Luis, “Quantum-classical correspondence for visibility, coherence, and relative phase for multidimensional systems,” Phys. Rev. A 78, 025802 (2008).
[Crossref]

P. Réfrégier and A. Luis, “Irreversible effects of random unitary transformations on coherence properties of partially polarized electromagnetic fields,” J. Opt. Soc. Am. A 25, 2749–2757 (2008).
[Crossref]

2007 (8)

2006 (1)

P. Réfrégier and F. Goudail, “Kullback relative entropy and characterization of partially polarized optical waves,” J. Opt. Soc. A 23, 671–678 (2006).
[Crossref]

2005 (4)

2004 (5)

2003 (3)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields” Opt. Express 11, 1137–1143 (2003).
[Crossref] [PubMed]

A. Luis, “Classical mechanics and the propagation of the discontinuities of the quantum wave function,” Phys. Rev. A 67, 024102 (2003).
[Crossref]

2002 (3)

A. Luis, “Visibility for anharmonic fringes,” J. Phys. A: Math. Gen. 35, 8805–8815 (2002).
[Crossref]

H. M. Ozaktas, S. Yüksel, and M. A. Kutay, “Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence,” J. Opt. Soc. Am. A 19, 1563–1571 (2002).
[Crossref]

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[Crossref]

2001 (1)

A. Luis, “Quantum mechanics as a geometric phase: phase-space interferometers,” J. Phys. A 34, 7677–7684 (2001).
[Crossref]

1997 (1)

Z. Y. Ou, “Fundamental quantum limit in precision phase measurement,” Phys. Rev. A 55, 2598–2609 (1997).
[Crossref]

1995 (2)

H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[Crossref]

A. Luis and L. L. Sánchez-Soto, “A quantum description of the beam splitter,” Quantum Semiclass. Opt. 7, 153–160 (1995).
[Crossref]

1994 (1)

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
[Crossref] [PubMed]

1987 (1)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[Crossref] [PubMed]

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[Crossref]

1963 (1)

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
[Crossref]

Afek, I.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref] [PubMed]

Ambar, O.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref] [PubMed]

Björk, G.

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
[Crossref]

Borghi, R.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007).
[Crossref] [PubMed]

Braunstein, S. L.

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
[Crossref] [PubMed]

Caves, C. M.

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
[Crossref] [PubMed]

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[Crossref]

Cintra, R. J.

A. D. C. Nascimento, R. J. Cintra, and A. C. Frery, “Hypothesis testing in speckled data with stochastic distances,” IEEE Trans. Geos. Remot. Sens. 48, 373–385 (2010).
[Crossref]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory, (Wiley Interscience, 1991).
[Crossref]

Cramér, H.

H. Cramér, Mathematical Methods of Statistics (Asia Publishing House, 1962).

Delyon, G.

Espinoza, P.

A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
[Crossref]

Frery, A. C.

A. D. C. Nascimento, R. J. Cintra, and A. C. Frery, “Hypothesis testing in speckled data with stochastic distances,” IEEE Trans. Geos. Remot. Sens. 48, 373–385 (2010).
[Crossref]

Friberg, A. T.

Frieden, B. R.

B. R. Frieden, Physics from Fisher information: A Unification, (Cambridge U. Press, 1999).

Ghiu, I.

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

Gil, J. J.

J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[Crossref]

I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for nxn covariance matrices,” arXiv:0807.2171v1 [physics.optics].

Goodman, J. W.

J. W. Goodman, Statistical Optics (John Wiley and Sons Inc., 1985).

Gori, F.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007).
[Crossref] [PubMed]

Goudail, F.

Karczewski, B.

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
[Crossref]

Klimov, A. B.

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
[Crossref]

Kutay, M. A.

Lee, H.-W.

H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[Crossref]

Luis, A.

A. Rivas and A. Luis, “Precision quantum metrology and nonclassicality in linear and nonlinear detection schemes,” Phys. Rev. Lett. 105, 010403 (2010).
[Crossref] [PubMed]

A. Luis, “Quantum-limited metrology with nonlinear detection schemes,” SPIE Reviews 1, 018006 (2010).

A. Luis, “Coherence and visibility for vectorial light,” J. Opt. Soc. A 27, 1764–1769 (2010).
[Crossref]

A. Luis, “Coherence versus interferometric resolution,” Phys. Rev. A 81, 065802 (2010).
[Crossref]

A. Luis, “Ensemble approach to coherence between two scalar harmonic light vibrations and the phase difference,” Phys. Rev. A 79, 053855 (2009).
[Crossref]

A. Rivas and A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A 77, 022105 (2008).
[Crossref]

A. Luis, “Quantum-classical correspondence for visibility, coherence, and relative phase for multidimensional systems,” Phys. Rev. A 78, 025802 (2008).
[Crossref]

P. Réfrégier and A. Luis, “Irreversible effects of random unitary transformations on coherence properties of partially polarized electromagnetic fields,” J. Opt. Soc. Am. A 25, 2749–2757 (2008).
[Crossref]

A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A 24, 1063–1068 (2007).
[Crossref]

A. Luis, “Maximum visibility in interferometers illuminated by vectorial waves,” Opt. Lett. 32, 2191–2193 (2007).
[Crossref] [PubMed]

A. Luis, “Degree of polarization of type-II unpolarized light,” Phys. Rev. A 75, 053806 (2007)
[Crossref]

A. Luis, “Polarization distributions and degree of polarization for quantum Gaussian light fields,” Opt. Commun. 273, 173–181 (2007).
[Crossref]

A. Luis, “Classical mechanics and the propagation of the discontinuities of the quantum wave function,” Phys. Rev. A 67, 024102 (2003).
[Crossref]

A. Luis, “Visibility for anharmonic fringes,” J. Phys. A: Math. Gen. 35, 8805–8815 (2002).
[Crossref]

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[Crossref]

A. Luis, “Quantum mechanics as a geometric phase: phase-space interferometers,” J. Phys. A 34, 7677–7684 (2001).
[Crossref]

A. Luis and L. L. Sánchez-Soto, “A quantum description of the beam splitter,” Quantum Semiclass. Opt. 7, 153–160 (1995).
[Crossref]

A. Luis, “An overview of coherence and polarization properties for multicomponent electromagnetic waves,” in Advances in Information Optics and Photonics, International Commission for Optics, vol. VI, A. T. Friberg and R. Dändliker, eds. (SPIE, 2009) pp. 171–188.

A. Luis and L. L. Sánchez-Soto, “Quantum phase difference, phase measurements and Stokes operators,” in Progress in Optics, vol. 41, E. Wolf, ed. (Elsevier, Amsterdam, 2000), pp. 421–482.
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Marian, P.

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

Marian, T. A.

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

Martínez-Herrero, R.

Mejías, P.M.

Mukunda, N.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[Crossref] [PubMed]

Nascimento, A. D. C.

A. D. C. Nascimento, R. J. Cintra, and A. C. Frery, “Hypothesis testing in speckled data with stochastic distances,” IEEE Trans. Geos. Remot. Sens. 48, 373–385 (2010).
[Crossref]

Ou, Z. Y.

Z. Y. Ou, “Fundamental quantum limit in precision phase measurement,” Phys. Rev. A 55, 2598–2609 (1997).
[Crossref]

Ozaktas, H. M.

Picozzi, A.

Réfrégier, P.

Rivas, A.

A. Rivas and A. Luis, “Precision quantum metrology and nonclassicality in linear and nonlinear detection schemes,” Phys. Rev. Lett. 105, 010403 (2010).
[Crossref] [PubMed]

A. Rivas and A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A 77, 022105 (2008).
[Crossref]

Roueff, A.

San José, I.

I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for nxn covariance matrices,” arXiv:0807.2171v1 [physics.optics].

Sánchez-Soto, L. L.

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
[Crossref]

A. Luis and L. L. Sánchez-Soto, “A quantum description of the beam splitter,” Quantum Semiclass. Opt. 7, 153–160 (1995).
[Crossref]

A. Luis and L. L. Sánchez-Soto, “Quantum phase difference, phase measurements and Stokes operators,” in Progress in Optics, vol. 41, E. Wolf, ed. (Elsevier, Amsterdam, 2000), pp. 421–482.
[Crossref]

Santarsiero, M.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007).
[Crossref] [PubMed]

Sehat, A.

A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
[Crossref]

Setälä, T.

Silberberg, Y.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref] [PubMed]

Simon, B. N.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

Simon, R.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[Crossref] [PubMed]

Simon, S.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

Soderholm, J.

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

Söderholm, J.

A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
[Crossref]

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[Crossref] [PubMed]

Tervo, J.

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory, (Wiley Interscience, 1991).
[Crossref]

Wolf, E.

E. Wolf, “Comment on Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1712–1712 (2004).
[Crossref] [PubMed]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Yüksel, S.

Eur. Phys. J. Appl. Phys. (1)

J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[Crossref]

IEEE Trans. Geos. Remot. Sens. (1)

A. D. C. Nascimento, R. J. Cintra, and A. C. Frery, “Hypothesis testing in speckled data with stochastic distances,” IEEE Trans. Geos. Remot. Sens. 48, 373–385 (2010).
[Crossref]

J. Opt. Soc. A (2)

A. Luis, “Coherence and visibility for vectorial light,” J. Opt. Soc. A 27, 1764–1769 (2010).
[Crossref]

P. Réfrégier and F. Goudail, “Kullback relative entropy and characterization of partially polarized optical waves,” J. Opt. Soc. A 23, 671–678 (2006).
[Crossref]

J. Opt. Soc. Am. A (4)

J. Phys. A (1)

A. Luis, “Quantum mechanics as a geometric phase: phase-space interferometers,” J. Phys. A 34, 7677–7684 (2001).
[Crossref]

J. Phys. A: Math. Gen. (1)

A. Luis, “Visibility for anharmonic fringes,” J. Phys. A: Math. Gen. 35, 8805–8815 (2002).
[Crossref]

Opt. Commun. (2)

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

A. Luis, “Polarization distributions and degree of polarization for quantum Gaussian light fields,” Opt. Commun. 273, 173–181 (2007).
[Crossref]

Opt. Express (3)

Opt. Lett. (12)

F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007).
[Crossref] [PubMed]

R. Martínez-Herrero and P.M. Mejías, “Maximum visibility under unitary transformations in two-pinhole interference for electromagnetic fields,” Opt. Lett. 32, 1471–1473 (2007).
[Crossref] [PubMed]

A. Luis, “Maximum visibility in interferometers illuminated by vectorial waves,” Opt. Lett. 32, 2191–2193 (2007).
[Crossref] [PubMed]

P. Réfrégier, “Mean-square coherent light,” Opt. Lett. 33, 1551–1553 (2008).
[Crossref] [PubMed]

T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328–330 (2004).
[Crossref] [PubMed]

E. Wolf, “Comment on Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1712–1712 (2004).
[Crossref] [PubMed]

T. Setälä, J. Tervo, and A. T. Friberg, “Reply to comment on Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 1713–1714 (2004).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Phase correlations and optical coherence,” Opt. Lett. 37, 151–153 (2012).
[Crossref] [PubMed]

A. Picozzi, “Entropy and degree of polarization for nonlinear optical waves,” Opt. Lett. 29, 1653–1655 (2004).
[Crossref] [PubMed]

P. Réfrégier, “Polarization degree of optical waves with non-Gaussian probability density functions: Kullback relative entropy-based approach,” Opt. Lett. 30, 1090–1092 (2005).
[Crossref] [PubMed]

P. Réfrégier, “Mutual information-based degrees of coherence of partially polarized light with Gaussian fluctuations,” Opt. Lett. 30, 3117–3119 (2005).
[Crossref] [PubMed]

P. Réfrégier and A. Roueff, “Visibility interference fringes optimization on a single beam in the case of partially polarized and partially coherent light,” Opt. Lett. 32, 1366–1368 (2007).
[Crossref] [PubMed]

Phys. Lett. (1)

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
[Crossref]

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[Crossref]

Phys. Rep. (1)

H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[Crossref]

Phys. Rev. A (10)

A. Luis, “Classical mechanics and the propagation of the discontinuities of the quantum wave function,” Phys. Rev. A 67, 024102 (2003).
[Crossref]

A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
[Crossref]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[Crossref] [PubMed]

Z. Y. Ou, “Fundamental quantum limit in precision phase measurement,” Phys. Rev. A 55, 2598–2609 (1997).
[Crossref]

A. Luis, “Quantum-classical correspondence for visibility, coherence, and relative phase for multidimensional systems,” Phys. Rev. A 78, 025802 (2008).
[Crossref]

A. Rivas and A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A 77, 022105 (2008).
[Crossref]

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[Crossref]

A. Luis, “Degree of polarization of type-II unpolarized light,” Phys. Rev. A 75, 053806 (2007)
[Crossref]

A. Luis, “Ensemble approach to coherence between two scalar harmonic light vibrations and the phase difference,” Phys. Rev. A 79, 053855 (2009).
[Crossref]

A. Luis, “Coherence versus interferometric resolution,” Phys. Rev. A 81, 065802 (2010).
[Crossref]

Phys. Rev. D (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[Crossref]

Phys. Rev. Lett. (3)

A. Rivas and A. Luis, “Precision quantum metrology and nonclassicality in linear and nonlinear detection schemes,” Phys. Rev. Lett. 105, 010403 (2010).
[Crossref] [PubMed]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref] [PubMed]

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
[Crossref] [PubMed]

Quantum Semiclass. Opt. (1)

A. Luis and L. L. Sánchez-Soto, “A quantum description of the beam splitter,” Quantum Semiclass. Opt. 7, 153–160 (1995).
[Crossref]

Science (1)

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[Crossref] [PubMed]

SPIE Reviews (1)

A. Luis, “Quantum-limited metrology with nonlinear detection schemes,” SPIE Reviews 1, 018006 (2010).

Other (9)

H. Cramér, Mathematical Methods of Statistics (Asia Publishing House, 1962).

Exploratory Data Analysis Using Fisher Information, B. R. Frieden, ed. (Springer-Verlag, 2007).
[Crossref]

B. R. Frieden, Physics from Fisher information: A Unification, (Cambridge U. Press, 1999).

A. Luis, “An overview of coherence and polarization properties for multicomponent electromagnetic waves,” in Advances in Information Optics and Photonics, International Commission for Optics, vol. VI, A. T. Friberg and R. Dändliker, eds. (SPIE, 2009) pp. 171–188.

I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for nxn covariance matrices,” arXiv:0807.2171v1 [physics.optics].

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

J. W. Goodman, Statistical Optics (John Wiley and Sons Inc., 1985).

T. M. Cover and J. A. Thomas, Elements of Information Theory, (Wiley Interscience, 1991).
[Crossref]

A. Luis and L. L. Sánchez-Soto, “Quantum phase difference, phase measurements and Stokes operators,” in Progress in Optics, vol. 41, E. Wolf, ed. (Elsevier, Amsterdam, 2000), pp. 421–482.
[Crossref]

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Equations (60)

Equations on this page are rendered with MathJax. Learn more.

V ( E ) = 2 | E 1 E 2 * | | E 1 | 2 + | E 2 | 2 | μ ( E ) | = | E 1 E 2 * | ( | E 1 | 2 | E 2 | 2 ) 1 / 2 ,
𝒫 = | I M I m | I M + I m = [ 1 4 det Γ ( tr Γ ) 2 ] 1 / 2 , Γ ( E ) = ( | E 1 | 2 E 1 E 2 * E 1 * E 2 | E 2 | 2 ) ,
V 2 + R 2 = 𝒫 2 , R ( E ) = | | E 1 | 2 | E 2 | 2 | E 1 | 2 + | E 2 | 2 | .
| μ ( U E ) | 𝒫 .
P ( ϕ ˜ | θ ) Π j = 1 N P ( θ j | ϕ ˜ ) Φ ( ϕ ˜ ) ,
( Δ ϕ ) 2 1 N F , F = d θ 1 P ( θ | ϕ ) [ d P ( θ | ϕ ) d ϕ ] 2 .
H = d θ [ P ( θ | ϕ + δ ϕ ) P ( θ | ϕ ) ] 2 ( δ ϕ ) 2 4 F ,
K = d θ P ( θ | ϕ + δ ϕ ) ln P ( θ | ϕ + δ ϕ ) P ( θ | ϕ ) ( δ ϕ ) 2 2 F ,
C s = ln d θ P s ( θ | ϕ + δ ϕ ) P 1 s ( θ | ϕ ) s ( 1 s ) 2 ( δ ϕ ) 2 F ,
χ 2 = F F + 4 tr ( G 2 ) .
P ( θ | ϕ ) = d 2 k E W ( U ϕ E ) δ [ θ θ ( E ) ] .
P ( θ | ϕ ) = d ϑ 𝒲 ( θ , ϑ ) ,
F F C = d 2 k E 1 W ( U ϕ E ) [ d W ( U ϕ E ) d ϕ ] 2 .
χ 2 = F C F C + 4 tr ( G 2 ) ,
W ( E ) = det M π k exp ( E M E ) ,
F C = 2 [ tr ( Γ G Γ 1 G ) tr ( G 2 ) ] .
F C = 4 i , j = 1 k | G i , j | 2 𝒫 i , j 2 1 𝒫 i , j 2 4 tr ( G 2 ) 𝒫 max 2 1 𝒫 max 2 ,
𝒫 i , j = | I i I j | I i + I j 𝒫 max = I M I m I m + I m ,
χ 𝒫 max
Γ = ( | E 1 | 2 E 1 E 2 * E 1 * E 2 | E 2 | 2 ) , G ( 1 0 0 1 )
F C = 4 tr ( G 2 ) | μ | 2 1 | μ | 2 , χ = | μ | ,
G ( I 0 0 I ) ,
Γ = ( Γ 1 ϒ ϒ Γ 2 ) = Γ 0 1 / 2 U ( I Ω Ω I ) U Γ 0 1 / 2 ,
Γ 0 = ( Γ 1 0 0 Γ 2 ) , Γ j = ( | E j , x | 2 E j , x E j , y * E j , x * E j , y | E j , y | 2 )
U = ( U 1 0 0 U 2 ) , Ω = ( μ S 0 0 μ I ) ,
F C = 2 tr ( G 2 ) ( 1 1 μ S 2 + 1 1 μ I 2 2 ) , χ 2 = μ S 2 + μ I 2 2 μ S 2 μ I 2 2 μ S 2 μ I 2 .
F F Q = 2 i , j ( p i p j ) 2 p i + p j | ψ i | G ^ | ψ j | 2 ,
ρ = i p i | ψ i ψ i | .
F Q = 4 ( Δ G ^ ) 2 .
χ 2 = F Q F Q + 4 tr ( G 2 ) .
ρ i = 1 1 + n ¯ i n = 0 ( n ¯ i 1 + n ¯ i ) n | n i i n | ,
F Q = 4 tr ( G 2 ) 𝒫 2 1 𝒫 2 + 2 / n ¯ , χ = 𝒫 ( n ¯ n ¯ + 2 ) 1 / 2 ,
𝒫 = | n ¯ 1 n ¯ 2 | n ¯ 1 + n ¯ 2 , n ¯ = n ¯ 1 + n ¯ 2 .
| ψ = 1 2 ( | n 1 | 0 2 + | 0 1 | n 2 ) .
P ( θ ) = | θ | ψ | 2 , | θ = 1 2 π m = 0 N e im θ | n m 1 | m 2 ,
P ( θ | ϕ ) = 1 π cos 2 [ n ( θ ϕ ) / 2 ] ,
P j = k W j , k ,
F = j P j 2 P j = j P j ( ln P j ) = j , k W j , k ( ln P j ) , F C = j , k W j , k 2 W j , k = j , k W j , k ( ln W j , k ) ,
F C F = j , k W j , k ( ln W j , k P j ) = j P j k Λ j , k ( ln Λ j , k ) ,
Λ j , k = W j , k P j , k Λ j , k = 1 .
F C F = j P j k Λ j , k 2 Λ j , k 0 .
H P = 2 ( 1 j P j P ˜ j ) , H W = 2 ( 1 j , k W j , k W ˜ j , k ) .
P j P ˜ j = k , W j , k W ˜ j , = 1 2 k , ( W j , k W ˜ j , + W j , W ˜ j , k ) ,
( k W j , k W ˜ j , k ) 2 = k , W j , k W ˜ j , W j , W ˜ j , k ,
W ( U ϕ E ) = det M π k exp ( E U ϕ M U ϕ E ) ,
W ( U ϕ E ) d ϕ | ϕ = 0 = E M E W ( E ) ,
M = d ( U ϕ M U ϕ ) d ϕ | ϕ = 0 = i [ G , M ] .
F C = d 2 k E ( E M E ) 2 W ( E ) = i , j , , m = 1 k M i , j M , m d 2 k E E i * E j E * E m W ( E ) .
E j E m E i * E * = E j E i * E m E * + E m E i * E j E * ,
F C = [ tr ( M M 1 ) ] 2 + tr [ ( M M 1 ) 2 ] = tr [ ( M M 1 ) 2 ] ,
F C = 2 [ tr ( M 1 G M G ) tr ( G 2 ) ] .
F C = i , j ( I i I j ) 2 I i I j | G i , j | 2 ,
F C = 4 i , j = 1 k | G i , j | 2 𝒫 i , j 2 1 𝒫 i , j 2 4 tr ( G 2 ) 𝒫 max 2 1 𝒫 max 2 ,
𝒫 i , j = | I i I j | I i + I j 𝒫 max = I M I m I M + I m ,
| ψ i | G ^ | ψ j | 2 = | n 1 | n 2 | [ exp ( i φ ) a 1 a 2 + exp ( i φ ) a 1 a 2 ] | n 1 | n 2 | 2 .
n 1 | n 2 | [ exp ( i φ ) a 1 a 2 + exp ( i φ ) a 1 a 2 ] | n 1 | n 2 = exp ( i φ ) ( n 1 + 1 ) n 2 δ n 1 = n 1 + 1 δ n 2 = n 2 1 + exp ( i φ ) n 1 ( n 2 + 1 ) δ n 1 = n 1 1 δ n 2 = n 2 + 1 .
F Q = 2 n 1 , n 2 = 0 ( p n 1 + 1 p n 2 1 p n 1 p n 2 ) 2 p n 1 + 1 p n 2 1 + p n 1 p n 2 ( n 1 + 1 ) n 2 + exchange 1 2 ,
p n i = 1 1 + n ¯ i ( n ¯ i 1 + n ¯ i ) n i .
F Q = 4 ( n ¯ 1 n ¯ 2 ) 2 2 n ¯ 1 n ¯ 2 + n ¯ 1 + n ¯ 2 .
F Q = 4 𝒫 2 n ¯ 2 n ¯ 2 ( 1 𝒫 2 ) / 2 + n ¯ = 4 tr ( G 2 ) 𝒫 2 1 𝒫 2 + 2 / n ¯ ,

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