Abstract

The growing recognition that entanglement is not exclusively a quantum property, and does not even originate with Schrödinger’s famous remark about it [Proc. Cambridge Philos. Soc. 31, 555 (1935) [CrossRef]  ], prompts the examination of its role in marking the quantum-classical boundary. We have done this by subjecting correlations of classical optical fields to new Bell-analysis experiments and report here values of the Bell parameter greater than B=2.54. This is many standard deviations outside the limit B=2 established by the Clauser–Horne–Shimony–Holt Bell inequality [Phys. Rev. Lett. 23, 880 (1969) [CrossRef]  ], in agreement with our theoretical classical prediction, and not far from the Tsirelson limit B=2.828. These results cast a new light on the standard quantum-classical boundary description, and suggest a reinterpretation of it.

© 2015 Optical Society of America

1. INTRODUCTION

For many decades the term “entanglement” has been attached to the world of quantum mechanics [1]. However, it is true that nonquantum optical entanglement can exist (realized very early by Spreeuw [2]) and its applications have concrete consequences. These are based on entanglements between two, or more than two, degrees of freedom (DOFs), which are easily available classically [26]. Multientanglements of the same kind are also being explored quantum mechanically [7]. Applications in the classical domain have included, for example, the resolution of a long-standing issue concerning Mueller matrices [8], an alternative interpretation of the degree of polarization (DOP) [9], introduction of the Bell measure as a new index of coherence in optics [10], and innovations in polarization metrology [11]. Here we present theoretical and experimental results extending these results by showing that probabilistic classical optical fields can exhibit violations of the Clauser–Horne–Shimony–Holt (CHSH) Bell inequality [12] of quantum strength. This is evidence of a new kind that asks for reconsideration of the common understanding that Bell violation signals quantum physics. We emphasize that our discussion focuses on the nonquantum entanglement of nondeterministic classical optical fields and does not engage issues such as nonlocality that are important for some applications in quantum information.

The observations and applications of nonquantum wave entanglement noted above [26,811] exploited nonseparable correlations among two or more modes or DOFs of optical wave fields. Nonseparable correlations among modes are an example of entanglement, but are not enough for our present purpose. In addition, we want to conform to three criteria that Shimony has identified for Bell tests [13], facts of quantum nature that must be satisfied when examining possible tests of the quantum-classical border. Fortuitously, the ergodic stochastic optical fields of the classical theory of partial coherence and partial polarization (see Wolf [14]) satisfy these criteria fully (see Supplement 1), and we have used such fields as our test bed.

2. BACKGROUND THEORY

We will deal here only with the simplest suitable example, the theory of completely unpolarized classical light, and have explained elsewhere (see [15] and Supplement 1) the generalizations needed to treat partially polarized fields, which lead to the same conclusions. In all cases there are only two DOFs to deal with, namely, the direction of polarization and the temporal amplitude of the optical field. In both classical and quantum theories, these are fundamentally independent attributes. An electric field, for a beam traveling in z direction, is written as

E⃗(t)=x^Ex(t)+y^Ey(t).
In the classical theory of unpolarized light [16], an optical field’s two amplitudes Ex and Ey are statistically completely uncorrelated and are treated as vectors in a stochastic function space. A scalar product of the vectors in this space corresponds physically to observable correlation functions such as ExEy. For unpolarized light we have ExEx=EyEy and ExEy=0.

Now it is possible to talk of entanglement of the classical field. This is because entangled states are superpositions of products of vectors from different vector spaces, whenever the superpositions cannot be rearranged into a single product that separates the two spaces [1]. Looking again at Eq. (1) we see that this is the case because we have taken E⃗ to be unpolarized. That is, by the definition of unpolarized light, there is no direction u^ of polarization that captures the total intensity, so E⃗(t) cannot, for any direction u^, be written in the form E⃗(t)=u^F(t), which would factorably separate the polarization and amplitude DOFs [17].

Beyond its probabilistic indeterminacy, the E⃗ in Eq. (1) has other quantum-like attributes—it has the same form as a quantum state superposition and can be called a pure state in the same sense. More precisely, it is a two-party state living in two vector spaces at once, a polarization space for x^ and y^, and an infinitely continuous stochastic function space for Ex and Ey.

The Bell inequality most commonly used for correlation tests is due to Clauser et al. [12]. It deals with correlations between two different DOFs when each is two dimensional. The Schmidt theorem of analytic function theory [18] ensures two-dimensionality by guaranteeing that among the infinitely many dimensions available to the amplitudes in Eq. (1), only two dimensions are active. This is a consequence arising just from the fact that the partner polarization vectors x^ and y^ live in a two-dimensional space.

For convenience, we introduce e⃗, the field normalized to the intensity I=ExEx+EyEy,

e⃗(t)E⃗(t)/I={x^ex(t)+y^ey(t)},
where now e⃗·e⃗=exex+eyey=1.

For some simplification in writing, we will use Dirac notation for the vectors without, of course, imparting any quantum character to the fields. The unit polarization vectors x^ and y^ will be renamed as x^|u1 and y^|u2 and the unit amplitudes will be rewritten as ex|f1 and ey|f2. If desired, the Dirac notation can be discarded at any point and the vector signs and hats re-installed. For the case of unpolarized light, we have u1|u2=0 and f1|f2=0. Unit projectors in the two spaces take the form 1=|u1u1|+|u2u2| and 1=|f1f1|+|f2f2|. In this notation, and in the original notation for comparison, the field takes the form

E⃗/I=x^ex+y^ey=|e=(|u1|f1+|u2|f2)/2.
In this notation, the field actually looks like what it is, a two-party superposition of products in independent vector spaces, i.e., an entangled two-party state (actually a Bell state). Here the two parties are the independent polarization and amplitude DOFs.

The notation for a CHSH correlation coefficient C(a,b) implies that arbitrary rotations of the unit vectors |uj and |fk(j,k=1,2) through angles a and b can be managed independently in the two spaces. An arbitrary rotation through angle a of the polarization vectors |u1 and |u2 takes the forms

|u1a=cosa|u1sina|u2and|u2a=sina|u1+cosa|u2.
For function space rotations, we have |f1b and |f2b defined similarly:
|f1b=cosb|f1sinb|f2and|f2b=sinb|f1+cosb|f2,
where the rotation angles a and b are unrelated.

Next, the correlation between the polarization (u) and function (f) DOFs is given by the standard average

C(a,b)=e|Zu(a)Zf(b)|e,
where Z is shorthand for the difference projection: Zu(a)|u1au1a||u2au2a|, analogous to a σz spin operation. C(a,b) is thus a combination of four joint projections such as
P11(a,b)=e|(|u1a|f1bf1b|u1a|)|e=|f1b|u1a|e|2.
This is all classical and all of the correlation projections Pjk(a,b), with j,k=1,2, have familiar roles in classical optical polarization theory [16].

Gisin [19] observed that any quantum state entangled in the same way as the classical pure state Eq. (2) will lead to a violation of the CHSH inequality, which takes the form B2, where

B=|C(a,b)C(a,b)+C(a,b)+C(a,b)|.
The same result will be found here, as one uses only DOF independence and properties of positive functions and normed vectors to arrive at it (see details in Supplement 1). We note again that the issue of entanglement itself is pertinent to the discussion, but the usefulness of entanglement as a resource for particular applications is not. Thus we have reached the main goal of our theoretical background sketch. This was to demonstrate the existence of a purely classical field theory that can exhibit a violation of the CHSH Bell inequality.

3. EXPERIMENTAL TESTING

The remaining task is to show that experimental observation confirms this theoretical prediction, in effect shifting one’s interpretation of tests of the quantum-classical border by showing that, along with quantum fields, classical fields conforming to the Shimony Bell-test criteria are capable of Bell violation. In order to make such a demonstration, a classical field source must be used. This means a source producing a field that is quantum mechanical (since we believe all light fields are intrinsically quantum), but a field whose quantum statistics are not distinguishable from classical statistics. This is only necessary up to second order in the field because the CHSH procedure engages no higher order statistics. Such sources are easily available. Since the earliest testing of laser light it has been known that a laser operated below threshold has a statistical character not distinguishable from classical thermal statistics. So in our experiments we have used a broadband laser diode operated below threshold.

Our experiment repeatedly records the correlation function C(a,b) defined in Eq. (6) for four different angles in order to construct the value of the Bell parameter B. This is done through measurements of the joint projections Pjk(a,b). We will describe explicitly only the recording of P11(a,b), identified in Eq. (7), but the others are done similarly in an obvious way. In the classical context that we are examining, the optical field is macroscopic and correlation detection is essentially calorimetric (i.e., using a power meter, not requiring or employing individual photon recognition).

4. POLARIZATION TOMOGRAPHY

The first step is to tomographically determine the polarization state of the test field. A polarization tomography setup is shown in Fig. 1. Using a polarizing beam splitter (PBS) and half-wave plates (HWP) and a quarter-wave plate (QWP) to project onto circular and diagonal bases, the Stokes parameters (S1,S2,S3), relative to S0=1, are found to be (0.0827,0.0920,0.0158), providing a small nonzero DOP equal to 0.125. This departure from 0 requires a slight modification of the theory presented above (see Supplement 1) and reduces the maximum possible value of B able to be achieved for our specific experimental field to B=2.817, below but close to B=22=2.828, the theoretical maximum for completely unpolarized light.

 figure: Fig. 1.

Fig. 1. Experimental setup consists of a source of unpolarized light and a measurement using a modified MZ interferometer. HWP and a QWP control the polarization of the source. All beam splitters are 50:50 unless marked as a PBS. Intensities needed for obtaining the required joint projections are measured at detector D1. Shutters S independently block the arms of the interferometer in order to measure light through the arms separately. A removable mirror (RM) directs the light to a polarization tomography setup where the orthogonal components of the polarization in the basis determined by the wave plate are measured at detectors D2 and D3.

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5. EXPERIMENTAL BELL TEST

The experimental test has two major components, as shown in Fig. 1: a source of light to be measured and a Mach–Zehnder (MZ) interferometer. The source utilizes a 780 nm laser diode, operated in the multimode region below threshold, giving it a short coherence length of the order of 1 mm. The beam is assumed to be statistically ergodic, stable and stationary, as commonly delivered from such a multimode below-threshold diode. It is incident on a 50:50 beam splitter and recombined on a PBS after adequate delay so that the light to be studied is an incoherent mix of horizontal and vertical polarizations before being sent to the measurement area via a single-mode fiber. A HWP in one arm controls the relative power, and thus, the DOP. QWP and HWP help correct polarization changes introduced by the fiber.

In Fig. 1 the partially polarized beam entering the MZ is separated by a 50:50 beam splitter into a primary test beam |E and an auxiliary beam |E¯. The two beams inherit the same statistical properties from their mother beam, and thus, both can be expressed as in Eq. (3), with intensities I and I¯. The phase of the auxiliary beam |E¯ is shifted by an unimportant factor i at the beam splitter.

To determine the joint projection P11(a,b) of the test beam |E, the first step is to project the field to obtain |E1a|u1au1a|E. This can be realized by the polarizer labeled a on the bottom arm of the MZ. The transmitted beam retains both |f components in function space:

|E1a=I1a|u1a(c11|f1b+c12|f2b),
where I1a is the intensity and c11 and c12 are normalized amplitude coefficients with |c11|2+|c12|2=1. Here c11 relates to P11 in an obvious way: P11(a,b)=I1a|c11|2/I. One sees that the intensities I and I1a can be measured directly, but not the coefficient c11.

For P11(a,b) our aim is to produce a field that combines a projection onto |f1b in function space with the |u1a projection in polarization space. The challenge of overcoming the lack of polarizers for the projection of a nondeterministic field in an arbitrary direction in its independent infinite-dimensional function space is managed by a “stripping” technique [Supplement 1] applied to the auxiliary E¯ field in the left arm. We pass E¯ through a polarizer rotated from the initial |u1|u2 basis by a specially chosen angle s so that the statistical component |f2b is stripped off. The transmitted beam |E¯1s then has only the |f1b component, as desired: |E¯1s=iI¯1s|u1s|f1b. Here I¯1s is the corresponding intensity and the special stripping angle s is given by tans=(κ1/κ2)tanb (see [15] and Supplement 1).

The function-space-oriented beam |E¯1s is then sent through another polarizer a to become |E¯1a=|u1au1a|E¯1a=iI¯1a|u1a|f1b, where I¯1a is the corresponding intensity. Finally, the beams |E1a and |E¯1a are combined by a 50:50 beam splitter, which yields the outcome beam |E1T=(|E¯1a+i|E1a)/2. The total intensity I1T of this outcome beam can be easily expressed in terms of the needed coefficient c11.

Some simple arithmetic will immediately provide the joint projection P11(a,b) in terms of various measurable intensities:

P11(a,b)=(2I1TI¯1aI1a)2/4II¯1a.
Other Pjk(a,b) values can be obtained similarly by rotations of polarizers a and s. To make our measurements, polarizers a were simultaneously rotated using motorized mounts, whereas the third polarizer s was fixed at different values in a sequence of runs.

6. RESULTS

For each angle, measurements were made at detector D1 for the total intensity IT and for the separate intensities from each arm Ia and I¯a by closing the shutters S alternately. In this way, the measurements of the polarization space and statistical amplitude space are carried out separately. From these measurements, the needed correlations C(a,b) were determined and Eq. (8) was used to evaluate the CHSH parameter B.

Figure 2 shows C(a,b) obtained by measuring the joint projections Pjk(a,b) for a complete rotation of polarizer a, with different curves corresponding to b (and thus s) fixed at different values. It is apparent from the near-identity of the curves that, to good approximation, the correlations are a function of the difference in angles, i.e., C(a,b)=C(ab). The maximum value for B can then be found straightforwardly from any one of the curves in Fig. 2. Among them, the smallest and largest values of B (obtained for curves 1 and 4) are 2.548±0.004 and 2.679±0.007, respectively.

 figure: Fig. 2.

Fig. 2. Plots of the correlation functions C(a,b) obtained by rotating polarizer a in the polarization space and keeping angle b in the function space constant. Curves 1–4 correspond to different fixed values of b separated by π/4. The invariant cosine function required to violate the Bell inequality is clearly present. Error bars are included but are scarcely visible.

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To be careful, we note that in our experiments the field was almost but not quite completely unpolarized; thus, not quite the same field was sketched in the Background Theory section. Thus, we could not expect to get the maximum quantum result B=22=2.828. for the Bell parameter, although the values achieved also present a strong violation. The background theory is mildly more complicated for partially polarized rather than unpolarized light, but when worked out for the DOP of our light beams (see [15] and Supplement 1) it supports the values we observed.

7. SUMMARY

In summary, we first sketched the purely classical theory of optical beam fields (1) that satisfy the Bell-test criteria of Shimony [13,15]. Their bipartite pure state form shows the entanglement of their two independent DOFs [20]. The classical theory defines them as dynamically probabilistic fields, meaning that individual field measurements yield values that cannot be predicted except in an average sense, which is another feature shared with quantum systems but also associated for more than 50 years with the well-understood and well-tested optical theory of partial coherence [16]. Our theoretical sketch for the simplest case, unpolarized light, indicated that such fields or states are predicted to possess a range of correlation strengths equal to that of two-party quantum systems, that is, outside the bound B2 of the CHSH Bell inequality and potentially as great as B=22. In our experimental test, we used light whose statistical behavior (field second-order statistics) is indistinguishable from classical, viz., the light from a broadband laser diode operating below threshold. Our detections of whole-beam intensity are free of the heralding requirements familiar in paired-photon CHSH experiments. Repeated tests confirmed that such a field can strongly violate the CHSH Bell inequality and can attain Bell-violating levels of correlation similar to those found in tests of maximally entangled quantum systems.

One naturally asks, how are these results possible? We know that a field with classically random statistics is a local real field, and we also know that Bell inequalities prevent local physics from containing correlations as strong as what quantum states provide. But the experimental results directly contradict this. The resolution of the apparent contradiction is not complicated, but does mandate a shift in the conventional understanding of the role of Bell inequalities, particularly as markers of a classical-quantum border. Bell himself came close to addressing this point. He pointed out [21] that even adding classical indeterminism would still not be enough for any type of hidden variable system to overcome the restriction imposed by his inequalities. This is correct as far as it goes, but fails to engage the point that local fields can be statistically classical and exhibit entanglement at the same time. For the fields under study, the entanglement is a strong correlation that is intrinsically present between the amplitude and polarization DOFs, and it is embedded in the field from the start (as it also is embedded ab initio in any quantum states that violate a Bell inequality). The possibility of such pre-existing structural correlation is bypassed in a CHSH derivation. Thus one sees that Bell violation has less to do with quantum theory than previously thought, but everything to do with entanglement.

FUNDING INFORMATION

Defense Advanced Research Projects Agency (DARPA) (W31P4Q-12-1-0015); National Science Foundation (NSF) (PHY-0855701, PHY-1203931).

ACKNOWLEDGMENT

The authors acknowledge helpful discussions with many colleagues, particularly including M. Lewenstein, as well as A. F. Abouraddy, A. Aspect, C. J. Broadbent, L. Davidovich, E. Giacobino, G. Howland, A. N. Jordan, P. W. Milonni, R. J. C. Spreeuw, and A. N. Vamivakas.

 

See Supplement 1 for supporting content.

REFERENCES AND NOTES

1. In the paper where he introduced the English word entanglement [E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935)], [CrossRef]   Schrödinger referred to the math-physics textbook by R. Courant and D. Hilbert, Methoden der mathematischen Physik, 2nd ed. (specifically p. 134) where the mathematical character of entanglement appears in connection with the Schmidt theorem of analytic function theory.

2. R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998). [CrossRef]  

3. P. Ghose and M. K. Samal, “EPR type nonlocality in classical electrodynamics,” arXiv:quant-ph/0111119 (2001).

4. K. F. Lee and J. E. Thomas, “Entanglement with classical fields,” Phys. Rev. Lett. 88, 097902 (2002). [CrossRef]  

5. C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010). [CrossRef]  

6. M. A. Goldin, D. Francisco, and S. Ledesma, “Simulating Bell inequality violations with classical optics encoded qubits,” J. Opt. Soc. Am. A 27, 779–786 (2010). [CrossRef]  

7. A recent example is A. Tiranov, J. Lavoie, A. Ferrier, P. Goldner, V. B. Verma, S. W. Nam, R. P. Mirin, A. E. Lita, F. Marsili, H. Herrmann, C. Silberhorn, N. Gisin, M. Afzelius, and F. Bussieres, “Storage of hyperentanglement in a solid-state quantum memory,” Optica 2, 279–287 (2015). [CrossRef]  

8. 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]  See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992). [CrossRef]  

9. X. F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011). [CrossRef]   See also T. Setälä, A. Shevchenko, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002), and [CrossRef]  J. Ellis and A. Dogariu, “Optical polarimetry of random fields,” Phys. Rev. Lett. 95, 203905 (2005). [CrossRef]  

10. K. H. Kagalwala, G. DiGiuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013). [CrossRef]  

11. F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014). [CrossRef]  

12. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969). [CrossRef]  

13. A. Shimony, “Contextual hidden variables theories and Bell’s inequalities,” Br. J. Philos. Sci. 35, 25–45 (1984).See also A. Shimony, “Bell’s Theorem,” in Stanford Encyclopedia of Philosophy, http:/plato.stanford.edu/entries/bell-theorem (2009).

14. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).

15. See X.-F. Qian and J. H. Eberly, “Entanglement is sometimes enough,” arXiv:1307.3772 (2013).

16. See C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998) and E. Wolf, ed. Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

17. Elliptical polarization needs different notation, but the results are the same.

18. The Schmidt theorem is a continuous-space version of the singular-value decomposition theorem for matrices. For background, see M. V. Fedorov and N. I. Miklin, “Schmidt modes and entanglement,” Contemp. Phys. 55, 94–109 (2014). [CrossRef]   The original paper is E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. 1. Entwicklung willküriger Funktionen nach Systeme vorgeschriebener,” Math. Ann. 63, 433–476 (1907). [CrossRef]   See also A. Ekert and P. L. Knight, “Entangled quantum-systems and the Schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995). [CrossRef]  

19. N. Gisin, “Bell’s inequality holds for all non-product states,” Phys. Lett. A 154, 201–202 (1991). [CrossRef]  

20. Classical fields are often thought of as the many-photon macroscopic limit of quantum coherent states. It is interesting to propose connections between our work, and the CHSH Bell violations discussed theoretically for maximally entangled macroscopic coherent states in C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009) [CrossRef]  .Following the number-state treatments of C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, “Strong violations of Bell-type inequalities for path-entangled number states,” Phys. Rev. A 76, 052101 (2007). [CrossRef]  

21. J. S. Bell, “Quantum mechanical ideas,” Science 177, 880–881 (1972). [CrossRef]  

References

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  1. In the paper where he introduced the English word entanglement [E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935)], Schrödinger referred to the math-physics textbook by R. Courant and D. Hilbert, Methoden der mathematischen Physik, 2nd ed. (specifically p. 134) where the mathematical character of entanglement appears in connection with the Schmidt theorem of analytic function theory.
    [Crossref]
  2. R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
    [Crossref]
  3. P. Ghose and M. K. Samal, “EPR type nonlocality in classical electrodynamics,” arXiv:quant-ph/0111119 (2001).
  4. K. F. Lee and J. E. Thomas, “Entanglement with classical fields,” Phys. Rev. Lett. 88, 097902 (2002).
    [Crossref]
  5. C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
    [Crossref]
  6. M. A. Goldin, D. Francisco, and S. Ledesma, “Simulating Bell inequality violations with classical optics encoded qubits,” J. Opt. Soc. Am. A 27, 779–786 (2010).
    [Crossref]
  7. A recent example is A. Tiranov, J. Lavoie, A. Ferrier, P. Goldner, V. B. Verma, S. W. Nam, R. P. Mirin, A. E. Lita, F. Marsili, H. Herrmann, C. Silberhorn, N. Gisin, M. Afzelius, and F. Bussieres, “Storage of hyperentanglement in a solid-state quantum memory,” Optica 2, 279–287 (2015).
    [Crossref]
  8. 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).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
    [Crossref]
  9. X. F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011). See also T. Setälä, A. Shevchenko, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002), and J. Ellis and A. Dogariu, “Optical polarimetry of random fields,” Phys. Rev. Lett. 95, 203905 (2005).
    [Crossref]
  10. K. H. Kagalwala, G. DiGiuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
    [Crossref]
  11. F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
    [Crossref]
  12. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
    [Crossref]
  13. A. Shimony, “Contextual hidden variables theories and Bell’s inequalities,” Br. J. Philos. Sci. 35, 25–45 (1984).See also A. Shimony, “Bell’s Theorem,” in Stanford Encyclopedia of Philosophy, http:/plato.stanford.edu/entries/bell-theorem (2009).
  14. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
  15. See X.-F. Qian and J. H. Eberly, “Entanglement is sometimes enough,” arXiv:1307.3772 (2013).
  16. See C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998) and E. Wolf, ed. Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  17. Elliptical polarization needs different notation, but the results are the same.
  18. The Schmidt theorem is a continuous-space version of the singular-value decomposition theorem for matrices. For background, see M. V. Fedorov and N. I. Miklin, “Schmidt modes and entanglement,” Contemp. Phys. 55, 94–109 (2014). The original paper is E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. 1. Entwicklung willküriger Funktionen nach Systeme vorgeschriebener,” Math. Ann. 63, 433–476 (1907). See also A. Ekert and P. L. Knight, “Entangled quantum-systems and the Schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995).
    [Crossref]
  19. N. Gisin, “Bell’s inequality holds for all non-product states,” Phys. Lett. A 154, 201–202 (1991).
    [Crossref]
  20. Classical fields are often thought of as the many-photon macroscopic limit of quantum coherent states. It is interesting to propose connections between our work, and the CHSH Bell violations discussed theoretically for maximally entangled macroscopic coherent states in C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009).Following the number-state treatments of C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, “Strong violations of Bell-type inequalities for path-entangled number states,” Phys. Rev. A 76, 052101 (2007).
    [Crossref]
  21. J. S. Bell, “Quantum mechanical ideas,” Science 177, 880–881 (1972).
    [Crossref]

2015 (1)

2014 (2)

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

The Schmidt theorem is a continuous-space version of the singular-value decomposition theorem for matrices. For background, see M. V. Fedorov and N. I. Miklin, “Schmidt modes and entanglement,” Contemp. Phys. 55, 94–109 (2014). The original paper is E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. 1. Entwicklung willküriger Funktionen nach Systeme vorgeschriebener,” Math. Ann. 63, 433–476 (1907). See also A. Ekert and P. L. Knight, “Entangled quantum-systems and the Schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995).
[Crossref]

2013 (1)

K. H. Kagalwala, G. DiGiuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

2011 (1)

2010 (3)

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).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

M. A. Goldin, D. Francisco, and S. Ledesma, “Simulating Bell inequality violations with classical optics encoded qubits,” J. Opt. Soc. Am. A 27, 779–786 (2010).
[Crossref]

2009 (1)

Classical fields are often thought of as the many-photon macroscopic limit of quantum coherent states. It is interesting to propose connections between our work, and the CHSH Bell violations discussed theoretically for maximally entangled macroscopic coherent states in C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009).Following the number-state treatments of C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, “Strong violations of Bell-type inequalities for path-entangled number states,” Phys. Rev. A 76, 052101 (2007).
[Crossref]

2002 (1)

K. F. Lee and J. E. Thomas, “Entanglement with classical fields,” Phys. Rev. Lett. 88, 097902 (2002).
[Crossref]

1998 (1)

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

1991 (1)

N. Gisin, “Bell’s inequality holds for all non-product states,” Phys. Lett. A 154, 201–202 (1991).
[Crossref]

1984 (1)

A. Shimony, “Contextual hidden variables theories and Bell’s inequalities,” Br. J. Philos. Sci. 35, 25–45 (1984).See also A. Shimony, “Bell’s Theorem,” in Stanford Encyclopedia of Philosophy, http:/plato.stanford.edu/entries/bell-theorem (2009).

1972 (1)

J. S. Bell, “Quantum mechanical ideas,” Science 177, 880–881 (1972).
[Crossref]

1969 (1)

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[Crossref]

1959 (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).

1935 (1)

In the paper where he introduced the English word entanglement [E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935)], Schrödinger referred to the math-physics textbook by R. Courant and D. Hilbert, Methoden der mathematischen Physik, 2nd ed. (specifically p. 134) where the mathematical character of entanglement appears in connection with the Schmidt theorem of analytic function theory.
[Crossref]

Abouraddy, A. F.

K. H. Kagalwala, G. DiGiuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

Afzelius, M.

Aiello, A.

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Bell, J. S.

J. S. Bell, “Quantum mechanical ideas,” Science 177, 880–881 (1972).
[Crossref]

Benmoussa, A.

Classical fields are often thought of as the many-photon macroscopic limit of quantum coherent states. It is interesting to propose connections between our work, and the CHSH Bell violations discussed theoretically for maximally entangled macroscopic coherent states in C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009).Following the number-state treatments of C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, “Strong violations of Bell-type inequalities for path-entangled number states,” Phys. Rev. A 76, 052101 (2007).
[Crossref]

Borges, C. V. S.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[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).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

Brosseau, C.

See C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998) and E. Wolf, ed. Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Bussieres, F.

Clauser, J. F.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[Crossref]

DiGiuseppe, G.

K. H. Kagalwala, G. DiGiuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

Eberly, J. H.

Fedorov, M. V.

The Schmidt theorem is a continuous-space version of the singular-value decomposition theorem for matrices. For background, see M. V. Fedorov and N. I. Miklin, “Schmidt modes and entanglement,” Contemp. Phys. 55, 94–109 (2014). The original paper is E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. 1. Entwicklung willküriger Funktionen nach Systeme vorgeschriebener,” Math. Ann. 63, 433–476 (1907). See also A. Ekert and P. L. Knight, “Entangled quantum-systems and the Schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995).
[Crossref]

Ferrier, A.

Francisco, D.

M. A. Goldin, D. Francisco, and S. Ledesma, “Simulating Bell inequality violations with classical optics encoded qubits,” J. Opt. Soc. Am. A 27, 779–786 (2010).
[Crossref]

Gerry, C. C.

Classical fields are often thought of as the many-photon macroscopic limit of quantum coherent states. It is interesting to propose connections between our work, and the CHSH Bell violations discussed theoretically for maximally entangled macroscopic coherent states in C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009).Following the number-state treatments of C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, “Strong violations of Bell-type inequalities for path-entangled number states,” Phys. Rev. A 76, 052101 (2007).
[Crossref]

Ghose, P.

P. Ghose and M. K. Samal, “EPR type nonlocality in classical electrodynamics,” arXiv:quant-ph/0111119 (2001).

Giacobino, E.

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Gisin, N.

Goldin, M. A.

M. A. Goldin, D. Francisco, and S. Ledesma, “Simulating Bell inequality violations with classical optics encoded qubits,” J. Opt. Soc. Am. A 27, 779–786 (2010).
[Crossref]

Goldner, P.

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).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

Herrmann, H.

Holt, R. A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[Crossref]

Hor-Meyll, M.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

Horne, M. A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[Crossref]

Huguenin, J. A. O.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

Kagalwala, K. H.

K. H. Kagalwala, G. DiGiuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

Khoury, A. Z.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

Lavoie, J.

Ledesma, S.

M. A. Goldin, D. Francisco, and S. Ledesma, “Simulating Bell inequality violations with classical optics encoded qubits,” J. Opt. Soc. Am. A 27, 779–786 (2010).
[Crossref]

Lee, K. F.

K. F. Lee and J. E. Thomas, “Entanglement with classical fields,” Phys. Rev. Lett. 88, 097902 (2002).
[Crossref]

Leuchs, G.

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Lita, A. E.

Marquardt, C.

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Marsili, F.

Miklin, N. I.

The Schmidt theorem is a continuous-space version of the singular-value decomposition theorem for matrices. For background, see M. V. Fedorov and N. I. Miklin, “Schmidt modes and entanglement,” Contemp. Phys. 55, 94–109 (2014). The original paper is E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. 1. Entwicklung willküriger Funktionen nach Systeme vorgeschriebener,” Math. Ann. 63, 433–476 (1907). See also A. Ekert and P. L. Knight, “Entangled quantum-systems and the Schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995).
[Crossref]

Mimih, J.

Classical fields are often thought of as the many-photon macroscopic limit of quantum coherent states. It is interesting to propose connections between our work, and the CHSH Bell violations discussed theoretically for maximally entangled macroscopic coherent states in C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009).Following the number-state treatments of C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, “Strong violations of Bell-type inequalities for path-entangled number states,” Phys. Rev. A 76, 052101 (2007).
[Crossref]

Mirin, R. P.

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).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

Nam, S. W.

Qian, X. F.

Qian, X.-F.

See X.-F. Qian and J. H. Eberly, “Entanglement is sometimes enough,” arXiv:1307.3772 (2013).

Saleh, B. E. A.

K. H. Kagalwala, G. DiGiuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

Samal, M. K.

P. Ghose and M. K. Samal, “EPR type nonlocality in classical electrodynamics,” arXiv:quant-ph/0111119 (2001).

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).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

Schrödinger, E.

In the paper where he introduced the English word entanglement [E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935)], Schrödinger referred to the math-physics textbook by R. Courant and D. Hilbert, Methoden der mathematischen Physik, 2nd ed. (specifically p. 134) where the mathematical character of entanglement appears in connection with the Schmidt theorem of analytic function theory.
[Crossref]

Shimony, A.

A. Shimony, “Contextual hidden variables theories and Bell’s inequalities,” Br. J. Philos. Sci. 35, 25–45 (1984).See also A. Shimony, “Bell’s Theorem,” in Stanford Encyclopedia of Philosophy, http:/plato.stanford.edu/entries/bell-theorem (2009).

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[Crossref]

Silberhorn, C.

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).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

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).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

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).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

Spreeuw, R. J. C.

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

Thomas, J. E.

K. F. Lee and J. E. Thomas, “Entanglement with classical fields,” Phys. Rev. Lett. 88, 097902 (2002).
[Crossref]

Tiranov, A.

Töppel, F.

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Verma, V. B.

Wolf, E.

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).

Br. J. Philos. Sci. (1)

A. Shimony, “Contextual hidden variables theories and Bell’s inequalities,” Br. J. Philos. Sci. 35, 25–45 (1984).See also A. Shimony, “Bell’s Theorem,” in Stanford Encyclopedia of Philosophy, http:/plato.stanford.edu/entries/bell-theorem (2009).

Contemp. Phys. (1)

The Schmidt theorem is a continuous-space version of the singular-value decomposition theorem for matrices. For background, see M. V. Fedorov and N. I. Miklin, “Schmidt modes and entanglement,” Contemp. Phys. 55, 94–109 (2014). The original paper is E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. 1. Entwicklung willküriger Funktionen nach Systeme vorgeschriebener,” Math. Ann. 63, 433–476 (1907). See also A. Ekert and P. L. Knight, “Entangled quantum-systems and the Schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995).
[Crossref]

Found. Phys. (1)

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

J. Opt. Soc. Am. A (1)

M. A. Goldin, D. Francisco, and S. Ledesma, “Simulating Bell inequality violations with classical optics encoded qubits,” J. Opt. Soc. Am. A 27, 779–786 (2010).
[Crossref]

Nat. Photonics (1)

K. H. Kagalwala, G. DiGiuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

New J. Phys. (1)

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Nuovo Cimento (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).

Opt. Lett. (1)

Optica (1)

Phys. Lett. A (1)

N. Gisin, “Bell’s inequality holds for all non-product states,” Phys. Lett. A 154, 201–202 (1991).
[Crossref]

Phys. Rev. A (2)

Classical fields are often thought of as the many-photon macroscopic limit of quantum coherent states. It is interesting to propose connections between our work, and the CHSH Bell violations discussed theoretically for maximally entangled macroscopic coherent states in C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009).Following the number-state treatments of C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, “Strong violations of Bell-type inequalities for path-entangled number states,” Phys. Rev. A 76, 052101 (2007).
[Crossref]

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

Phys. Rev. Lett. (3)

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).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

K. F. Lee and J. E. Thomas, “Entanglement with classical fields,” Phys. Rev. Lett. 88, 097902 (2002).
[Crossref]

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[Crossref]

Proc. Cambridge Philos. Soc. (1)

In the paper where he introduced the English word entanglement [E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935)], Schrödinger referred to the math-physics textbook by R. Courant and D. Hilbert, Methoden der mathematischen Physik, 2nd ed. (specifically p. 134) where the mathematical character of entanglement appears in connection with the Schmidt theorem of analytic function theory.
[Crossref]

Science (1)

J. S. Bell, “Quantum mechanical ideas,” Science 177, 880–881 (1972).
[Crossref]

Other (4)

See X.-F. Qian and J. H. Eberly, “Entanglement is sometimes enough,” arXiv:1307.3772 (2013).

See C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998) and E. Wolf, ed. Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Elliptical polarization needs different notation, but the results are the same.

P. Ghose and M. K. Samal, “EPR type nonlocality in classical electrodynamics,” arXiv:quant-ph/0111119 (2001).

Supplementary Material (1)

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Figures (2)

Fig. 1.
Fig. 1. Experimental setup consists of a source of unpolarized light and a measurement using a modified MZ interferometer. HWP and a QWP control the polarization of the source. All beam splitters are 50:50 unless marked as a PBS. Intensities needed for obtaining the required joint projections are measured at detector D1. Shutters S independently block the arms of the interferometer in order to measure light through the arms separately. A removable mirror (RM) directs the light to a polarization tomography setup where the orthogonal components of the polarization in the basis determined by the wave plate are measured at detectors D2 and D3.
Fig. 2.
Fig. 2. Plots of the correlation functions C(a,b) obtained by rotating polarizer a in the polarization space and keeping angle b in the function space constant. Curves 1–4 correspond to different fixed values of b separated by π/4. The invariant cosine function required to violate the Bell inequality is clearly present. Error bars are included but are scarcely visible.

Equations (10)

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

E⃗(t)=x^Ex(t)+y^Ey(t).
e⃗(t)E⃗(t)/I={x^ex(t)+y^ey(t)},
E⃗/I=x^ex+y^ey=|e=(|u1|f1+|u2|f2)/2.
|u1a=cosa|u1sina|u2and|u2a=sina|u1+cosa|u2.
|f1b=cosb|f1sinb|f2and|f2b=sinb|f1+cosb|f2,
C(a,b)=e|Zu(a)Zf(b)|e,
P11(a,b)=e|(|u1a|f1bf1b|u1a|)|e=|f1b|u1a|e|2.
B=|C(a,b)C(a,b)+C(a,b)+C(a,b)|.
|E1a=I1a|u1a(c11|f1b+c12|f2b),
P11(a,b)=(2I1TI¯1aI1a)2/4II¯1a.

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