# Non-quantum Entanglement Resolves a Fundamental Issue in Polarization Optics

###### Abstract

The issue raised in this Letter is classical in the sense of being quite ancient: which subset of real matrices should be accepted as physical Mueller matrices in polarization optics? Non-quantum entanglement between the polarization and spatial degrees of freedom of an electromagnetic beam is shown to provide the physical basis to resolve this issue in a definitive manner.

###### pacs:

42.25.Ja, 42.25.Kb, 03.65.Ud, 03.67.-aEntanglement is traditionally studied almost exclusively in the context of quantum systems. However, this notion is basically kinematic, and so is bound to present itself whenever and wherever the state space of interest is the tensor product of two (or more) vector spaces. Polarization optics of paraxial electromagnetic beams happens to have precisely this kind of a setting, and so one should expect entanglement to play a significant role in this situation. It turns out that entanglement in this non-quantum setup is not just a matter of academic curiosity: we shall show in this paper that consideration of this non-quantum entanglement resolves a fundamental issue in classical polarization optics. It will appear that this issue could not have been resolved without explicit consideration of entanglement. We begin by outlining the structure of classical polarization optics MandelWolf ; Brosseau ; Simon82 ; Kim-Mandel-Wolf

The Mueller-Stokes Formalism : Traditional Mueller-Stokes formalism applies to plane electromagnetic waves or, more generally, to elementary beams (see below). If the wave propagates along the positive -axis, the components of the transverse electric field along the and directions can be arranged into a column vector

(1) |

called the Jones vector, analogous to the state vector of a qubit. [A scalar factor of the form has been suppressed.] While is (a measure of) the intensity, the ratio of the (complex) components specifies the state of polarization.

When is not deterministic, the state of polarization is described by the coherency or polarization matrix

(2) |

where denotes ensemble average. The two defining properties of the coherency matrix are hermiticity, , and positivity, : every matrix obeying these two conditions is a valid coherency matrix.

It is clear that the intensity corresponds to , and fully polarized (pure) states describable by Jones vectors correspond to . Partially polarized or mixed states correspond to Thus coherency matrices are analogous to the density operators of a qubit.

Since is hermitian, it can be conveniently described as real linear combination of the four hermitian matrices which are mutually orthogonal,

(3) |

The coefficients define the components of the Stokes vector . The intensity equals .

While hermiticity of is equivalent to reality of the Stokes vector , the positivity conditions read, respectively, . Thus, permissible polarization states correspond to the positive light cone and its interior (solid cone). Pure states live on the surface of this cone.

Typical systems of interest in polarization optics are spatially homogeneous (in the transverse plane), in the sense that their action is independent of the coordinates . If such a system is deterministic and acts linearly on the field amplitude, it is described by a complex numerical matrix , the Jones matrix of the system:

(4) | |||||

Such Jones systems are analogous to hamiltonian evolutions of a qubit; since the intensity need not be preserved, need not be unitary. It is clear that Jones systems map pure states () into pure states.

We can go from a pair of indices, each running over and , to a single index running over to and vice versa. Thus, the elements of can be written as an associated column vector with , , , and . The one-to-one relationship (3) between and may thus be written as the vector equation

(5) |

Thus, while the associated column vector . The numerical matrix exhibited above is essentially unitary: .

Optical systems of interest can be more general than the ones described by Jones matrices. Such a general system is said to be non-deterministic, and acts directly on the Stokes vector rather than through the Jones vector. It is specified by a real matrix called the Mueller matrix, transforming the Stokes vectors linearly:

(6) |

Mueller matrix of a Jones system will be called Mueller-Jones matrix .

Since produces a linear transformation on , the linear invertible relationship (3) or (5) between and implies that will induce a linear transformation on , which we may write in the form :

(7) |

The fact that needs to be hermitian for all hermitian demands that the map or super-operator , viewed as a matrix with (going over to ) labeling the rows and labeling the rows, be hermitian. Let us define a new matrix by permuting the indices of :

(8) |

i.e., is obtained from by simply interchanging with , with , with , and with . In terms of , Eq. (7) transcribes into the vector equation , and in view of (5) we have the invertible relationship

(9) |

Thus, the correspondence between real matrices and hermitian matrices , established through (8) and (9), is indeed one-to-one. Elements of in terms of those of can be found in Eq. (8) of Ref. Simon82 .

If the system described by is a Jones system with Jones matrix , it is clear from the transformation law given in (4) that and, consequently, , where is the column vector associated with the matrix . Thus we arrive at the following result of fundamental importance Simon82 .

Proposition 1 : A Mueller matrix represents a Jones system if and only if the associated hermitian matrix is a one-dimensional projection. If is such a projection , then , being the matrix associated with the column vector .

As a consequence, which is mathematically trivial but quite important for the issue on hand, we have Kim-Mandel-Wolf

Proposition 2 : A real matrix can be realized as a positive sum (ensemble) of Mueller-Jones matrices if and only if the associated hermitian matrix is positive semidefinite. If , then where is the Mueller-Jones matrix associated with .

With this brief outline, we are now ready to describe the fundamental issue being addressed in this Letter.

The Issue : The Mueller-Stokes formalism takes as state space the collection of all Stokes vectors:

(10) |

where

Let us denote by the collection of matrices which can be realized as positive sum of Mueller-Jones systems . It is clear that is contained in . The structure of is fairly simple: it is clear from Proposition 1 Simon82 that elements of are in one-to-one correspondence with nonnegative matrices Kim-Mandel-Wolf . But the structure of is considerably more involved. Owing to a sequence of developments Sanjay92 ; Givens93 ; Mee93 ; Sridhar94 ; Rao98 , which are surprisingly recent in relative terms, we now have a complete characterization of . The basic tool has been orbits of under double-coseting by Sridhar94 .

That elements of are Mueller matrices is clear, for they are realized as convex sums of Jones systems. That matrices which fall outside are not Mueller matrices is also clear, for they fail to map the state space into itself. Thus the issue is really one about the grey domain ‘in between’, the complement of in : are these matrices physical Mueller matrices?

By definition, members of this domain cannot be realized as positive sums of Jones systems; but they map into itself. No one has come up with a scheme to realize them physically. On the other hand there are Mueller matrices, extracted from actual experiments, which fall deep into this grey domain (see Ref. Sridhar94 for examples from Ref. Zyl87 ).

There are two difficulties in simply dismissing these matrices as unphysical: first, the experimenters did not realize them as convex sums of Jones systems, and so the fact that they fall outside cannot be enough reason to dismiss them; and secondly, within the Mueller-Stokes formalism there seems to exist no additional qualification we can demand of a Mueller matrix, over and above the requirement that it should map into itself.

As a simple illustration of this grey region between and , let us consider matrices of the diagonal form will map into , and hence be in , if and only if . It is clear that

The hermitian matrix associated with is, from (10),

(11) |

Clearly, if and only if . i.e., iff is in the solid tetrahedron with vertices at , and

Proposition 3 : For matrices of the restricted form corresponds to the cubical region with vertices at corresponds to the inscribed solid tetrahedron with vertices at , whereas ,

In this Letter we present a compelling physical ground which judges every matrix which is not an element of as unphysical. And this physical ground comes from consideration of entanglement between the polarization and spatial degrees of freedom of an electromagnetic beam.

Non-quantum Entanglement : Let us now go beyond plane waves and consider paraxial electromagnetic beams. The simplest beam field has, in a transverse plane constant described by coordinates the form , where are complex constants, and the scalar-valued function may be assumed to be square-integrable over the transverse plane: . are unit vectors along the axes. It is clear that the polarization part and the spatial dependence or modulation part of such a beam are well separated, allowing one to focus attention on one aspect at a time. When one is interested in only the modulation aspect, the part may be suppressed, thus leading to ‘scalar optics’. On the other hand, if the spatial part is suppressed we are led to the traditional polarization optics or Mueller-Stokes formalism for plane waves.

Beams whose polarization and spatial modulation separate in the above manner will be called elementary beams. Suppose we superpose or add two such elementary beam fields and . The result is not of the elementary form , for any unless either is proportional to so that one gets committed to a common polarization, or and are proportional so that one gets committed to a fixed spatial mode. Thus, the set of elementary fields is not closed under superposition.

Since superposition principle is essential for optics, we are led to consider beam fields of the more general form , and consequently to pay attention to the implications of inseparability or entanglement of polarization and spatial variation. This more general form is obviously closed under superposition. We may write as a (generalised) Jones vector

(12) |

The intensity at location corresponds to . This field is of the elementary or separable form iff and are linearly dependent (proportional to one another). Otherwise, polarization and spatial modulation are inseparably entangled.

The point is that the set of possible beam fields in a transverse plane constitutes the tensor product space . But the set of all elementary fields constitutes just the set product of and , and hence forms a measure zero subset of the tensor product . In other words, in beam fields represented by generic elements of polarization and spatial modulation should be expected to be entangled : Entanglement is not an exception; it is the rule in , the space of pure states appropriate for electromagnetic beams.

To handle fluctuating beams, we need the beam-coherence-polarization (BCP) matrix Gori98 :

(13) |

As the name suggests, the BCP matrix describes both the coherence and polarization properties. It is a generalization of the numerical coherency matrix of Eq. (2), now to the case of beam fields.

It is clear from the very definition (13) of BCP matrix that this matrix kernel, viewed as an operator from , is hermitian nonnegative:

(14) |

. These are the defining properties of the BCP matrix: every matrix of two-point functions meeting just these two conditions is a valid BCP matrix of some beam of light.

Resolution of the Issue : In the BCP matrix (16), each of the four blocks is an (infinite-dimensional) operator . For the issue on hand, however, it turns out to be sufficient to restrict the spatial dependence to just two orthonormal spatial modes . This amounts to considering in place of the two-dimensional space , the linear span of and , so that the BCP matrix is a positive operator mapping , the first being for the polarization degree of freedom and the second for the spatial degree of freedom.

Choice of a product basis in transcribes the BCP matrix into a hermitian nonnegative numerical matrix of four blocks, with the polarization (Roman) indices labeling the blocks and the spatial (Greek) indices labeling entries within each block Santarsiero07 . The following orthonormal product basis suggests itself naturally:

(15) |

for . Thus a Jones vector in necessarily has the form . It is of the separable or elementary form if and only if . It can be identified with a four-dimensional numerical column vector whose entries are the expansion coefficients :

(16) |

Note that the Greek indices are left unaffected, consistent with the (transverse) spatial homogeneity of .

Now consider the special Jones vector corresponding to

Given an matrix, let be the result of the action of on . We have from (16)

(17) |

This is our final result. Suppose is not positive then , the result of acting on the entangled Jones vector , fails to be positive and hence is unphysical, showing in turn that could not have been physical. Thus is a necessary condition for to be a physical Mueller matrix. On the other hand, we have seen that if then can be realized as a convex sum of Jones systems, showing that is a sufficient condition for to be a Mueller matrix. We thus have

Theorem : The necessary and sufficient condition for to be a physical Mueller matrix is that the associated hermitian matrix . Every physical Mueller matrix is a convex sum of Mueller-Jones matrices.

Thus matrices which map into itself should be called pre-Mueller matrices rather than Mueller matrices. For, to be promoted to the status of Mueller matrices they needs to meet the stronger condition arising from consideration of entanglement. In view of the rapidly growing current interest in an unified approach to coherence and polarization in optics, it is hoped that our result will stimulate further research into the kinematic role of entanglement in classical optics.

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