Unified treatment of the total angular momentum of single photons via generalized quantum observables

In this paper we provide a consistent framework to address the notorious decomposition of the single-photon total angular momentum (TAM) into a spin (SAM) and an orbital (OAM) component. In particular, we find that the canonical SAM and OAM, generators of internal and spatial rotations respectively in the space of physical states, are mutually compatible but unsharp quantum observables, therefore POVM (Positive Operator-Valued Measures) describe their joint measurements. On the other hand, a non-canonical SAM and OAM can be constructed that is mutually incompatible but represent sharp quantum observables, thus PVM (Projector-Valued Measurements), reflecting their consistency with the transversality condition characterizing singlephoton wavefunctions. We discuss the implementations on joint measurements for both decompositions and provide an explicit calculation of all these quantities for circularly polarized Gaussian single-photon states.


I. INTRODUCTION
Understanding and manipulating the angular momentum of single photons is a central goal of modern physics, due to theoretical, experimental and even technical implications. On the experimental side, the angular momentum of light has been recently recognized as a novel powerful resource for implementing quantum-information protocols [1][2][3][4][5][6]. Moreover, experiments with light at the single-photon level have historically been at the forefront of fundamental tests of quantum mechanics [7][8][9][10][11][12][13].
The problem of introducing a physically unambiguous separation of the total angular momentum (TAM) of photons into a spin part (SAM) and an orbital part (OAM) is a controversial and debated subject [15][16][17][18][19][20][21] since it was first proposed by J. Humblet in 1943 [14].
The root of this long-standing problem lies in the transversality condition on the electromagnetic field and the single-photon wavefunction, that introduces an interdependence between the spatial and internal degrees of freedom of those quantities, hampering the possibility to define spin and orbital rotations separately.
It is well known from both the quantum mechanical first-quantization description of the photon [16,22] and the classical electromagnetic theory [17,23,24] that only the total angular momentumĴ of light is a well-defined and physically relevant quantity, subject to a conservation law stemming from rotational invariance.Ĵ can be separated in two partsL andŜ that satisfy the commutation relations characterizing the Lie algebra so(3), thus representing the correct generators of orbital and spin rotations, respectively. However,L andŜ are both inconsistent with the transversality condition, i.e. they do not leave the subspaces of longitudinal and transversal wavefunctions invariant, this fact leads to difficulties in their physical interpretation.
The problem of providing an alternative to the canonical decompositionĴ =L +Ŝ of the TAM was addressed in the secondquantization framework and in the paraxial limit by Van Enk and Nienhuis [17]. Bliokh et al. in Ref. [20,21] continued this discussion within the first-quantization framework and beyond the paraxial limit. In both cases, the authors proposed an alternative or non-canonical decompositionĴ =L ′ +Ŝ ′ of the TAM, in which the new orbital and spin components are consistent with the transversality condition [16,18] and therefore directly measurable. However,L ′ andŜ ′ do not satisfy the commutation relations of the so(3) algebra, therefore no longer representing the generators of rotations in spatial and internal degrees of freedom, respectively. In a recent work [25] we presented a general formalism based on Kraus operators [26] which allows in treating every singlephoton observable, including position and spin or momentum and helicity, in a unified picture. Consequently, making it possible to construct the corresponding probability distributions in terms of Positive Operator-Valued Measures (POVMs), a tool of paramount importance in the fields of quantum information science and open quantum systems. In particular, we show how the transversality condition categorizes single-photon observables into two classes: observables that are consistent with the transversality condition (e.g. momentum, energy and helicity) are sharp quantum observables, described by Projector-Valued Measures (PVMs), while observables that are not consistent with the transversality condition find a natural description in terms of POVMs, i.e. they are unsharp quantum observables [27,28]. Considering the Heisenberg product ∆X ∆P this explains the increase via the unsharpness of the position observable.
The purpose of the present work is instead to face the problem of separating the TAM into a spin and an orbital component, and of consistently describing these quantities in terms of PVMs and POVMs, in a unified picture with position, spin, momentum and helicity. We show that our generalization of Kraus' treatment allows to treat both the canonical and the non-canonical decomposition of the TAM in a consistent way, endowing them with a clear quantum information-theoretical characterization.
In particular, we find that the canonical OAM and SAM,Ĵ =L+Ŝ, are mutually compatible but unsharp quantum observables, and we provide the explicit expression for the POVM describing their joint measurements. On the other hand, the non-canonical OAM and SAM,Ĵ =L ′ +Ŝ ′ , represent mutually incompatible but sharp quantum observables, reflecting their consistency with the transversality condition for single-photon wavefunctions. Finally, we give explicit examples for both decompositions.
The paper is organized as follows. In Section II the properties of the single-photon Hilbert space and the definition of single-photon observables as POVMs are briefly recalled. In Section III the TAM observable is presented, and its canonical and non-canonical decomposition in an OAM and in a SAM part are discussed in detail. The differences between the two decompositions are assessed with a study of Gaussian states in Section V C, and Conclusions are drawn in the last Section VI.

II. SINGLE-PHOTON STATES AND OBSERVABLES
In the present Section we briefly recall the description of single-photon observables in terms of POVMs, which generalizes the treatment of the position observable proposed by K. Kraus [26]. The interested reader is referred to [25] for all the details in merit.
The proper quantum-mechanical description of a single photon in free space takes place in the Hilbert space where the set {e i (p)} i=1,2,3 denotes the so-called intrinsic frame, i.e. a reference frame such that one of the axis is directed along the direction of the momentum, i.e. e 3 (p) = p |p| , and e 1 (p) × e 2 (p) = e 3 (p). H S as defined in Eq. (1) is equipped with the positive definite inner product Note that H S defined in Eq. (1) is isomorphic to L 2 R 3 , d 3 p |p| ⊗ C 2 despite the photon spin number is s = 1. This traduces the well-known transversality condition, i.e. the condition ψ(p) · p = 0 according to which the longitudinal component of the photon state, namely the one along the direction of e 3 (p), is suppressed [25,26,[29][30][31]. Consequently, the degrees of momentum and spin get entangled in a fuddling way, which leads to the rich physics of photons. We stress that this result can be derived by only requiring the single-photon Hilbert space to carry an irreducible representation of the Poincaré group uniquely characterized by spin s = 1 and mass m = 0 Casimir invariants. In particular, the mass-shell condition, which also implies the transversality condition, naturally selects H S as the proper subspace of a spin s = 1 irreducible representation of Poincaré group carrying a positive definite inner product (the latter being a necessary ingredient in order to endow the theory of a probabilistic character) [25].
The main idea behind a neat treatment of all single photon observables in a unified way is to formalize the suppression of the longitundinal component of the wavefunction in terms of the action of a projection operatorπ(p) : where is isomorphic to ≃ L 2 R 3 , d 3 p |p| ⊗ C 3 and consists of wave-functions that differ from those of H S simply by the presence of a longitudinal component. Physically, the projector (3) can be interpreted as a quantum analogue of the Helmholtz projection which is used to decompose the electric and magnetic field into a longitudinal and a transversal component. The introduction of the Hilbert space H A is the key for a unified treatment of all single-photon observables. In particular, any observable O can be associated to a self-adjoint operatorÔ defined upon it which remarkably retains the same structure as in the case of a relativistic massive spin s = 1 particles [26,[29][30][31], opportunely adapted to the massless case of photons by the constraint p 0 = |p|. This means, for example, thatP represent the momentum (generator of spatial translations), Newton-Wigner position (generator of boosts) and TAM (generator of rotations) operators, respectively. In general, given that the system is described by a state |φ , the probability for any observable O to have an outcome in a generic measurable set M belonging to a suitable σ−algebra is given by the familiar expression whereÊ O (M) is the associated PVM. When we consider photons, we need to move from the extended Hilbert space H A , where all these observables are well-defined, to the physical Hilbert space H S in order to cope with the transversality condition (1). Thus projecting the associated PVM M →Ê O (M) defined on H A onto H S through the operatorπ, we obtain where |ψ ∈ H S describes the single-photon wave-function and The Observables described by POVMs are referred to as unsharp [27,[33][34][35], since their emergence reflects either practical limits in the precision of measurements (in which case POVMs are coarse-grained versions of PVMs) [36,37] or the inherent impossibility of realizing a preparation in which the value of an observable is perfectly defined [27,28,33]. This statement can be made more quantitative by simply showing that where |φ =Ôπ|ψ . Since the operator 1 −π is positive, the variance Var povm (O) is always larger than the variance Var pvm (O) that would arise if the POVM operators were idempotent, i.e. in the case of a PVM. In this sense, POVMs increase the statistical character of quantum observables [44]. A final remark is worth to be made at this point concerning Eq. (9). It is evident in fact that if  [25].
The incompatibility with the transversality condition has led to the introduction of opportunely modified position [40] and spin [17,20] operators. It is quite straightforward to show that such modified operators correspond to the projected version, throughπ, of the familiar operators defined in Eq. (5). Such modified operators thus become compatible by construction with Eq. (10) but no longer represent the generators of boosts and rotations, respectively.
for all M and thus Eq.(10) holds. Indeed, one can writê It is finally worth remarking that there may exist specific states |ψ * ∈ H S such that the mean value ψ * | Ô ,π |ψ * = 0, even though Eq. (10) is not satisfied. This is a general property of the uncertainty relations of the Robertson form [41] and can be circumvent by transforming uncertainty relations to entropic ones [42,43]. If a single photon is prepared in such state, the extra variance in (9) disappears despite O being unsharp.
In the next Sections III, IV we will show how this final subtle point can be related to the behavior of the canonical and non-canonical decompositions of the TAM in the paraxial limit. We will now see how this formalism, when applied to single photon angular momentum, provides the two decompositions with a novel and unified interpretation in a quantum-information perspective and allows to evaluate the corresponding probability distributions according to Eq. (8).

III. CANONICAL DECOMPOSITION OF THE TOTAL ANGULAR MOMENTUM
Since the main focus of the present work is on the angular momentum, it is essential to remind that H S hosts the irreducible representation of the roto-translation groupÛ with R ∈ SO(3) being a rotation matrix and a ∈ R 3 a translation vector. This representation is consistently maintained also on the extended Hilbert space H A , where all the spin and angular momentum operators can be properly defined. The spin s = 1 of the photon has a deep consequence on the connection between spin and rotations (and thus between internal and configurational degrees of freedom), which is compressed in the following key relation whereŜ are the generators of the SO(3) vector rotations [46] andV is an opportune unitary matrix. The matrixV has also the remarkable role to show the equivalence between the condition of transversality (1) and that of non-zero helicity, which is also known to characterize single-photon wavefunctions [25]. In fact a straightforward calculation shows that the eigenfunctions of the helicity operatorǫ are given byV e 3 (p) with eigenvalue 0 andVẽ ± (p) =V e1(p)∓ie2(p) √ 2 with eigenvalue ±1. The suppression of the longitudinal component is therefore unitarily equivalent to the suppression of zero-helicity eigenstates.
It is important now to notice that any choice of a particular representation of these generators such that the su(2) algebra is satisfied (i.e. Ŝ i ,Ŝ j = ǫ ijkŜk ) uniquely determines the matrixV according to (14). Equivalently said, (14) uniquely fixes the couple Ŝ ,V . As an example, if we choose the spin matrices to be of the form thenV is equal to Alternatively, we can chooseV = 1, this way fixing the three relevant spin matrices to have the following representation which has the merit to immediately show that they represent the anti-symmetric subset of the su (3) Gell-Mann matrices. This obviously has to be the case, because only in this case e i φ n·Ŝ becomes real and, consequently, can be identified with the rotation matrix R in the real space.
As discussed in the previous Section, the generator of rotations, i.e. the single-photon TAM, can be canonically decomposed on The OAM,L, and SAM,Ŝ, involved in the canonical decomposition obey the familiar angular momentum commutation relations characterizing the Lie algebra so(3) of the rotation group SO (3), thus allowing to regard them as the generators of internal and spatial rotations, respectively. In the light of (5), we can also consistently express the canonical OAM in terms of the Newton-Wigner position operator aŝ This form makes immediately evident that the OAM and SAM involved in the canonical decomposition of the TAM satisfy equation (10)  The canonical OAM and SAM are therefore inconsistent with the transversality condition, except in the paraxial limit, where the wavefunction ψ(p) is concentrated around a certain value p 0 . To make this statement quantitative, let us consider a family of wavefunctions ψ a (p) = u(p) f a (p) ∈ H S such that p · u(p) = 0 (to comply with the transversality condition), normalized u(p) 2 = 1 and which satisfies lim a→0 |f 2 a (p)| 2 = |p 0 | δ(p − p 0 ). For the observableŜ · n, the extra variance corresponding to the second term of (10) reads so that, ifV u(p 0 ) is an eigenfunction ofŜ · n with eigenvalue ±1, An identical observation characterizes the canonical OAM in the paraxial limit. Finally, let us remark that the TAM defined in (19), despite being given by the sum of two unsharp observables, is a sharp observable consistent with the transversality condition (10) (the proof of this fact can be found in Appendix A). Therefore, its statistics is described in terms of a PVM. An example is provided in section V A.

IV. NON-CANONICAL DECOMPOSITION OF THE TAM
As stressed before, the operatorsL k andŜ k which stem from the canonical decomposition of the TAM are not compatible with the transversality (or equivalently with the non-zero helicity) condition, which means that their action on a transverse wavefunction (i.e. a physical singe photon state) results in a non-vanishing longitudinal component [16,17,20]. This fact has led to introduce an alternative decomposition of the TAM in such a way that the two resulting components would be consistent with the transversality condition, therefore becoming direct observables on the physical single-photon Hilbert space H S , but no longer representing the generators of rotations and translations [20,38,39].
We recall here this non-canonical decomposition and endow it with a clear interpretation in a quantum-information theoretical perspective. Making use of the identity In equation (25)ǫ denotes the helicity operator defined in equation (15). We stress that both observables are well-defined in terms of self-adjoint operators only on the extended Hilbert space H A . Since the operatorŜ ′ satisfies the transversality condition (10), alsoL ′ has to have a vanishing commutation relation with the projection onto the physical Hilbert space, i.e.
BothŜ ′ k andL ′ k therefore classify as sharp observables [27,35] and their probability distributions are simply obtained as the mean values of the associated family of PVMs.
The non-canonical decompositon in equations (25)- (26) has several remarkable differences with respect to the canonical one. First of all, the components of the SAM operatorŜ ′ k commute with each other, while the components of the OAM operator L ′ k do not commute with each other. Thus the rotation and translation degrees of freedoms are no longer treated on equal footing. Moreover, the SAM and OAM operatorsŜ ′ k andL ′ k do not commute with each other, thus being mutually incompatible quantum observables [20], in strong contrast to the canonical decomposition into SAM and OAM. The explicit expressions of the commutators are given in Appendix A. It is finally important to point out that, despite the decomposition of the TAM into a SAM and an OAM part is highly not unique and further different decompositions could be taken into account, we have analyzed the two most relevant from the physical point of view. The canonical decomposition of the TAM is in fact dictated by the additional constraint that the resulting OAM and SAM represent the correct generators of spatial and internal rotations. The non-canonical decomposition is instead fixed by the constraint that both components are compatible with the transversality condition, i.e. for Eq. (10) to be satisfied, which for the SAM part means that the eigenspace relative to the eigenvalue 0 coincides with that of helicity.

V. IMPLICATIONS ONTO OBSERVABLES
We illustrate the two different decompositions now by two explicit physical examples, i.e. for photons with a fixed spatial direction and for Gaussian states with definite circular polarization.
A. Joint POVM of the canonical SAM and OAM along a fixed spatial direction We have shown that both the SAM and OAM are unsharp observables for which a PVM consistent with the transversality condition cannot be given (see also equation (A11) of Appendix A for details). In the present Section we explicitly derive the joint probability distribution and marginals of the single-photon OAM and SAM along a generic spatial axis. This is achieved through the concrete construction of the join POVMs of such observables, according to the procedure outlined in [25] and in the previous Section. In the following calculation we consider, without any loss of generality, the spin and orbital angular momenta along the z-axis of a suitable Cartesian frame in real space. We make use of spherical coordinates (p, θ, φ) relative to the z-axis in momentum space, denoting through {e s } 3 s=1 the canonical basis of R 3 . On the extended Hilbert space H A , the joint eigenfunctions of the compatible observablesL z andŜ z have the familiar form u m,ms (p) = f (p, θ) e imφ √ 2π e s ,L z u m,ms = mu m,ms ,Ŝ z u m,ms = m s u m,ms , where m s = 2 − s, and f (p, θ) is a square-integrable function of the variables p, θ ensuring the proper normalization of (28) ∞ 0 dp p π 0 dθ cos(θ) |f (p, θ)| 2 = 1 .
The joint PVM ofL z andŜ z is then given by The correspondent POVM on the physical single-photon Hilbert space H S is therefore obtained by the application of Eq. (8) and reads The joint probability distribution of such observables is readily obtained using Eq. (7) p Lz,Sz (m, m s ) = ∞ 0 dp p The marginals of (32) read and B. PVMs of the non-canonical SAM and OAM along a fixed spatial direction As another example of the versatility and generality of the formalism outlined above, and for conformity with Section V A, we explicitly derive the joint probability distribution of the non-canonical SAM and OAM operators along a fixed spatial axis, which again we consider to be the z-axis. Let us begin by reminding that the vectorsVẽ ± (p), withẽ ± (p) =ẽ 1 (p)∓iẽ2(p) √ 2 , are the eigenvectors of the helicity operatorǫ relative to the eigenvalues ±1, and depend on p only through the angles θ, φ. Therefore, the joint generalized eigenfunctions of helicity andŜ ′ z are the following elements of H A : where s 0 ∈ [−1, 1] and f (p, φ) is a properly normalized function. It is clear from Eq. (35) that Notice that cos(θ) = s 0 andẽ + (p) lead toŜ ′ z = s 0 , as well as cos(θ) = −s 0 andẽ − (p). As a consequence, the eventŜ ′ z ∈ M, where M is a Borel subset of [−1, 1], happens if and only if cos(θ) ∈ M + and ǫ = 1 or cos(θ) ∈ M − and ǫ = −1, the sets M ± being: The probability thatŜ ′ z ∈ M then clearly reads: where f ∈ H A . The probability can be written as the average over f of the projectorÊ S ′ z ,ǫ (M, 1) +Ê S ′ z ,ǫ (M, −1). In the light of (39), the joint PVM associated toŜ ′ z and helicity on H A is (40) and the PVM associated toŜ ′ z is the marginal of (40) over the helicity degrees of freedom. The projection (8) of the PVM (46) onto the physical Hilbert space H S preserves the idempotence property which characterizes a PVM, i.e.,F 2 , and therefore the probability distribution ofŜ ′ z reads: We remark that since H S is left invariant by the projector π, ( On the extended Hilbert space H A , the generalized eigenfunctions ofL ′ z are readily worked out starting from (43), details are reported in the Appendix C. They read where s 0 ∈ [−1, 1], f (|p|) is a properly normalized function, v n,j (θ, φ) is detailed in Appendix C andL ′ z u n,j (p) = (j s 0 + n) u n,j (p).
The choice j = 0 leads to unphysical eigenfunctions such thatπV † u n,0 (p) = 0 (i.e.V † u n,0 (p) is a longitudinal function). On the other hand, the choices j = ±1 produce eigenfunctions such thatπV † u n,0 (p) = u n,0 (p), i.e.V † u n,0 (p) ∈ H S . In the light of these observations, the PVM associated to the OAM observable on H A is: where M is a Borel subset of R and M n,j is the set of angles θ such that j cos(θ) + n ∈ M. The corresponding POVM on H S reads and the probability distribution ofL ′ z is As in the case of the SAM observable, since H S is left unchanged by the projectors (45), (46) is a PVM.

C. Application to Gaussian states
For illustration, we show a direct application of the above formalism by explicitly calculating the first two cumulants (mean value and variance) of the probability distributions for both the canonical and the non-canonical SAM and OAM over circularly polarized Gaussian single-photons, i.e. wavefunctions ψ(p) ∈ H S of the form where p 0 = p 0 e z and a = (∆p) 2 2p 2 0 (49) denoting a positive, dimensionless parameter which takes into account the spread of the wavefunction in momentum space. We stress that, while any of the three components x, y, z of these operators can be in principle evaluated, we will show the result for the z-component of the observables L, S, L ′ and S ′ , i.e. the one parallel to the chosen p 0 . Remarkably, the mean values of the two decompositions of the TAM are equal to each other on this particular state, plotted also in Fig. 1 (a), where This shows that, for this particular class of single-photon states, there is no quantitative difference between the two decompositions of the total angular momentum at the level of the mean value. Moreover, the values of spin and momentum equal for a = 1 2 corresponding to ∆p = p 0 , in this case both contributions to the TAM are maximal uncertain. For the paraxial limit a → 0 + the mean value of the TAM is dominated by the spin part, whereas for increasing a the mean value of the TAM is basically the angular momentum, showing the dominant physical behaviour for high and low energetic photons (for fixed ∆p), respectively.
The departure of the two decompositions is instead witnessed at the level of the respective variances, shown in Fig. 1 (b), which contains all the crucial information about their statistical character. In accordance with all the formal construction outlined above, the unsharpness of the canonical OAM and SAM brought by the introduction of the POVMs is in fact reflected in a larger value of the variance with respect to the (sharp) non-canonical decomposition for every value of the parameter a. One can finally notice that, in the paraxial limit a → 0 + , the two groups of variances correctly vanish, as the wavefunction is by construction perfectly defined in the momentum space.

VI. CONCLUSIONS
We have studied in detail two different separations of the single-photon angular momentum into a spin and an orbital part, relying on the generalization [25] of Karl Kraus' construction of the position observable. The canonical decomposition of the total angular momentum (TAM) into the spin angular momentum (SAM) and the orbital angular momentum (OAM) are both compatible observables, however, have to be described by POVMs (positive-operator valued measures) due to their incompatibility with the transversality condition. The non-canonical decomposition of the TAM proposed firstly by S. J. van Enk and G Nienhuis [17] guarantees the compatibility with the transversality condition by paying the prize that the spatial and internal spin degrees of freedom get coupled. Thanks to our unified and general framework for dealing with generic single-photon observables, we could deduce the form of the non-canonical OAM and SAM as projector-valued measurements (PVMs). This proves how the transversality condition categorizes into principally unsharp or sharp observables.
Last but not least we have quantitatively shown the difference between the two above-mentioned decompositions of the TAM by calculating the first two cumulants (mean values and variances) on circularly polarized Gaussian states. The results allowed to clearly emphasize the unsharp character of the canonical OAM and SAM, in contrast with the sharpness of the non-canonical OAM and SAM, as is reflected by their larger variances. Moreover, it shows that independently of the decomposition polarisation becomes a well defined property for low energetic photons since the mean z-component becomes small.
This work could pave a new unified framework to handle also more photons in a quantum information theoretic way.