Shifting the quantum-classical boundarytheory and experiment for statistically classicaloptical fieldsXIAO-FENG QIAN BETHANY LITTLE JOHN C HOWELL AND J H EBERLY

Center for Coherence and Quantum Optics and the Department of Physics amp Astronomy University of Rochester Rochester New York 14627 USACorresponding author xfqianpasrochesteredu

Received 15 December 2014 revised 21 April 2015 accepted 2 June 2015 (Doc ID 230588) published 29 June 2015

The growing recognition that entanglement is not exclusively a quantum property and does not even originate withSchroumldingerrsquos famous remark about it [Proc Cambridge Philos Soc 31 555 (1935)] prompts the examination of itsrole in marking the quantum-classical boundary We have done this by subjecting correlations of classical optical fieldsto new Bell-analysis experiments and report here values of the Bell parameter greater than B 254 This is manystandard deviations outside the limit B 2 established by the ClauserndashHornendashShimonyndashHolt Bell inequality [PhysRev Lett 23 880 (1969)] in agreement with our theoretical classical prediction and not far from the Tsirelson limitB 2828hellip These results cast a new light on the standard quantum-classical boundary description and suggest areinterpretation of it copy 2015 Optical Society of America

OCIS codes (0300030) Coherence and statistical optics (2605430) Polarization

httpdxdoiorg101364OPTICA2000611

1 INTRODUCTION

For many decades the term ldquoentanglementrdquo has been attached tothe world of quantum mechanics [1] However it is true thatnonquantum optical entanglement can exist (realized very earlyby Spreeuw [2]) and its applications have concrete consequencesThese are based on entanglements between two or more thantwo degrees of freedom (DOFs) which are easily available clas-sically [2ndash6] Multientanglements of the same kind are also beingexplored quantum mechanically [7] Applications in the classicaldomain have included for example the resolution of a long-standing issue concerning Mueller matrices [8] an alternative in-terpretation of the degree of polarization (DOP) [9] introductionof the Bell measure as a new index of coherence in optics [10] andinnovations in polarization metrology [11] Here we present theo-retical and experimental results extending these results by showingthat probabilistic classical optical fields can exhibit violations ofthe ClauserndashHornendashShimonyndashHolt (CHSH) Bell inequality [12]of quantum strength This is evidence of a new kind that asksfor reconsideration of the common understanding that Bellviolation signals quantum physics We emphasize that our discus-sion focuses on the nonquantum entanglement of nondetermin-istic classical optical fields and does not engage issues such asnonlocality that are important for some applications in quantuminformation

The observations and applications of nonquantum waveentanglement noted above [2ndash68ndash11] exploited nonseparablecorrelations among two or more modes or DOFs of optical wavefields Nonseparable correlations among modes are an example of

entanglement but are not enough for our present purpose Inaddition we want to conform to three criteria that Shimonyhas identified for Bell tests [13] facts of quantum nature thatmust be satisfied when examining possible tests of the quan-tum-classical border Fortuitously the ergodic stochastic opticalfields of the classical theory of partial coherence and partialpolarization (see Wolf [14]) satisfy these criteria fully (seeSupplement 1) and we have used such fields as our test bed

2 BACKGROUND THEORY

We will deal here only with the simplest suitable example thetheory of completely unpolarized classical light and have ex-plained elsewhere (see [15] and Supplement 1) the generalizationsneeded to treat partially polarized fields which lead to the sameconclusions In all cases there are only two DOFs to deal withnamely the direction of polarization and the temporal amplitudeof the optical field In both classical and quantum theories theseare fundamentally independent attributes An electric field for abeam traveling in z direction is written as

Et xExt yEyt (1)

In the classical theory of unpolarized light [16] an optical fieldrsquostwo amplitudes Ex and Ey are statistically completely uncorre-lated and are treated as vectors in a stochastic function spaceA scalar product of the vectors in this space corresponds physicallyto observable correlation functions such as hExEyi For unpolar-ized light we have hExExi hEyEyi and hExEyi 0

2334-253615070611-05$150$1500 copy 2015 Optical Society of America

Research Article Vol 2 No 7 July 2015 Optica 611

Now it is possible to talk of entanglement of the classical fieldThis is because entangled states are superpositions of products ofvectors from different vector spaces whenever the superpositionscannot be rearranged into a single product that separates the twospaces [1] Looking again at Eq (1) we see that this is the casebecause we have taken E to be unpolarized That is by the def-inition of unpolarized light there is no direction u of polarizationthat captures the total intensity so Et cannot for any directionu be written in the form Et uF t which would factorablyseparate the polarization and amplitude DOFs [17]

Beyond its probabilistic indeterminacy the E in Eq (1) hasother quantum-like attributesmdashit has the same form as a quan-tum state superposition and can be called a pure state in the samesense More precisely it is a two-party state living in two vectorspaces at once a polarization space for x and y and an infinitelycontinuous stochastic function space for Ex and Ey

The Bell inequality most commonly used for correlation tests isdue to Clauser et al [12] It deals with correlations between twodifferent DOFs when each is two dimensional The Schmidt theo-remof analytic function theory [18] ensures two-dimensionality byguaranteeing that among the infinitely many dimensions availableto the amplitudes in Eq (1) only two dimensions are active This isa consequence arising just from the fact that the partner polariza-tion vectors x and y live in a two-dimensional space

For convenience we introduce e the field normalized to theintensity I hExEx EyEyi

et equiv Et∕ffiffiffiI

p fxext yeytg (2)

where now he middot ei hexex eyeyi 1For some simplification in writing we will use Dirac notation

for the vectors without of course imparting any quantum char-acter to the fields The unit polarization vectors x and y will berenamed as x rarr ju1i and y rarr ju2i and the unit amplitudes willbe rewritten as ex rarr jf 1i and ey rarr jf 2i If desired the Diracnotation can be discarded at any point and the vector signsand hats re-installed For the case of unpolarized light we havehu1ju2i 0 and hf 1jf 2i 0 Unit projectors in the two spacestake the form 1 ju1ihu1j ju2ihu2j and 1 jf 1ihf 1jjf 2ihf 2j In this notation and in the original notation forcomparison the field takes the form

E∕ffiffiffiI

p xex yey jei ju1ijf 1i ju2ijf 2i∕

ffiffiffi2

p (3)

In this notation the field actually looks like what it is a two-partysuperposition of products in independent vector spaces ie anentangled two-party state (actually a Bell state) Here the twoparties are the independent polarization and amplitude DOFs

The notation for a CHSH correlation coefficient Ca bimplies that arbitrary rotations of the unit vectors juji andjf kij k 1 2 through angles a and b can be managed inde-pendently in the two spaces An arbitrary rotation through angle aof the polarization vectors ju1i and ju2i takes the forms

jua1i cos aju1i minus sin aju2i and

jua2i sin aju1i cos aju2i (4)

For function space rotations we have jf b1i and jf b

2i definedsimilarly

jf b1i cos bjf 1i minus sin bjf 2i and

jf b2i sin bjf 1i cos bjf 2i (5)

where the rotation angles a and b are unrelatedNext the correlation between the polarization (u) and func-

tion (f ) DOFs is given by the standard average

Ca b hejZua otimes Zf bjei (6)

where Z is shorthand for the difference projectionZua equiv jua1ihua1j minus jua2ihua2j analogous to a σz spin operationCa b is thus a combination of four joint projections such as

P11a b hejjua1ijf b1ihf b

1jhua1jjei jhf b1jhua1jeij2 (7)

This is all classical and all of the correlation projections Pjka bwith j k 1 2 have familiar roles in classical optical polarizationtheory [16]

Gisin [19] observed that any quantum state entangled in thesame way as the classical pure state Eq (2) will lead to a violationof the CHSH inequality which takes the form B le 2 where

B jCa b minus Ca 0 b Ca b 0 Ca 0 b 0j (8)

The same result will be found here as one uses only DOF inde-pendence and properties of positive functions and normed vectorsto arrive at it (see details in Supplement 1) We note again that theissue of entanglement itself is pertinent to the discussion but theusefulness of entanglement as a resource for particular applica-tions is not Thus we have reached the main goal of our theoreticalbackground sketch This was to demonstrate the existence of apurely classical field theory that can exhibit a violation of theCHSH Bell inequality

3 EXPERIMENTAL TESTING

The remaining task is to show that experimental observation con-firms this theoretical prediction in effect shifting onersquos interpre-tation of tests of the quantum-classical border by showing thatalong with quantum fields classical fields conforming to theShimony Bell-test criteria are capable of Bell violation In orderto make such a demonstration a classical field source must beused This means a source producing a field that is quantummechanical (since we believe all light fields are intrinsically quan-tum) but a field whose quantum statistics are not distinguishablefrom classical statistics This is only necessary up to second orderin the field because the CHSH procedure engages no higher orderstatistics Such sources are easily available Since the earliest test-ing of laser light it has been known that a laser operated belowthreshold has a statistical character not distinguishable fromclassical thermal statistics So in our experiments we have useda broadband laser diode operated below threshold

Our experiment repeatedly records the correlation functionCa b defined in Eq (6) for four different angles in order toconstruct the value of the Bell parameter B This is done throughmeasurements of the joint projections Pjka b We will describeexplicitly only the recording of P11a b identified in Eq (7) butthe others are done similarly in an obvious way In the classicalcontext that we are examining the optical field is macroscopicand correlation detection is essentially calorimetric (ie usinga power meter not requiring or employing individual photonrecognition)

Research Article Vol 2 No 7 July 2015 Optica 612

4 POLARIZATION TOMOGRAPHY

The first step is to tomographically determine the polarizationstate of the test field A polarization tomography setup is shownin Fig 1 Using a polarizing beam splitter (PBS) and half-waveplates (HWP) and a quarter-wave plate (QWP) to project ontocircular and diagonal bases the Stokes parameters (S1 S2 S3) rel-ative to S0 1 are found to be (minus00827 minus00920 minus00158)providing a small nonzero DOP equal to 0125 This departurefrom 0 requires a slight modification of the theory presentedabove (see Supplement 1) and reduces the maximum possiblevalue of B able to be achieved for our specific experimental fieldto B 2817 below but close to B 2

ffiffiffi2

p 2828hellip thetheoretical maximum for completely unpolarized light

5 EXPERIMENTAL BELL TEST

The experimental test has two major components as shown inFig 1 a source of light to be measured and a MachndashZehnder(MZ) interferometer The source utilizes a 780 nm laser diodeoperated in the multimode region below threshold giving it ashort coherence length of the order of 1 mm The beam is as-sumed to be statistically ergodic stable and stationary as com-monly delivered from such a multimode below-thresholddiode It is incident on a 5050 beam splitter and recombinedon a PBS after adequate delay so that the light to be studied

is an incoherent mix of horizontal and vertical polarizations beforebeing sent to the measurement area via a single-mode fiber AHWP in one arm controls the relative power and thus the DOPQWP and HWP help correct polarization changes introduced bythe fiber

In Fig 1 the partially polarized beam entering the MZ is sep-arated by a 5050 beam splitter into a primary test beam jEi andan auxiliary beam jEi The two beams inherit the same statisticalproperties from their mother beam and thus both can be ex-pressed as in Eq (3) with intensities I and I The phase of theauxiliary beam jEi is shifted by an unimportant factor i at thebeam splitter

To determine the joint projection P11a b of the test beamjEi the first step is to project the field to obtainjEa

1i equiv jua1ihua1jEi This can be realized by the polarizer labeleda on the bottom arm of the MZ The transmitted beam retainsboth jf i components in function space

jEa1i

ffiffiffiffiffiI a1

pjua1ic11jf b

1i c12jf b2i (9)

where I a1 is the intensity and c11 and c12 are normalized amplitudecoefficients with jc11j2 jc12j2 1 Here c11 relates to P11 in anobvious way P11a b I a1jc11j2∕I One sees that the intensitiesI and I a1 can be measured directly but not the coefficient c11

For P11a b our aim is to produce a field that combines aprojection onto jf b

1i in function space with the jua1i projectionin polarization space The challenge of overcoming the lack ofpolarizers for the projection of a nondeterministic field in an ar-bitrary direction in its independent infinite-dimensional functionspace is managed by a ldquostrippingrdquo technique [Supplement 1] ap-plied to the auxiliary E field in the left arm We pass E through apolarizer rotated from the initial ju1i minus ju2i basis by a speciallychosen angle s so that the statistical component jf b

2i is strippedoff The transmitted beam jE s

1i then has only the jf b1i compo-

nent as desired jEs1i i

ffiffiffiffiffiI s1

pjus1ijf b

1i Here I s1 is the corre-sponding intensity and the special stripping angle s is given bytan s κ1∕κ2 tan b (see [15] and Supplement 1)

The function-space-oriented beam jEs1i is then sent through

another polarizer a to become jEa1i jua1ihua1jEa

1i i

ffiffiffiffiffiI a1

pjua1ijf b

1i where I a1 is the corresponding intensityFinally the beams jEa

1i and jEa1i are combined by a 5050 beam

splitter which yields the outcome beam jET1 i jEa

1iijEa

1i∕ffiffiffi2

p The total intensity IT1 of this outcome beam can

be easily expressed in terms of the needed coefficient c11Some simple arithmetic will immediately provide the joint

projection P11a b in terms of various measurable intensities

P11a b 2IT1 minus I a1 minus Ia12∕4I I a1 (10)

Other Pjka b values can be obtained similarly by rotations ofpolarizers a and s To make our measurements polarizers a weresimultaneously rotated using motorized mounts whereas thethird polarizer s was fixed at different values in a sequence of runs

6 RESULTS

For each angle measurements were made at detector D1 for thetotal intensity IT and for the separate intensities from each arm I a

and I a by closing the shutters S alternately In this way the mea-surements of the polarization space and statistical amplitude spaceare carried out separately From these measurements the neededcorrelations Ca b were determined and Eq (8) was used toevaluate the CHSH parameter B

Fig 1 Experimental setup consists of a source of unpolarized lightand a measurement using a modified MZ interferometer HWP and aQWP control the polarization of the source All beam splitters are 5050unless marked as a PBS Intensities needed for obtaining the requiredjoint projections are measured at detector D1 Shutters S independentlyblock the arms of the interferometer in order to measure light through thearms separately A removable mirror (RM) directs the light to a polari-zation tomography setup where the orthogonal components of the polari-zation in the basis determined by the wave plate are measured at detectorsD2 and D3

Research Article Vol 2 No 7 July 2015 Optica 613

Figure 2 shows Ca b obtained by measuring the joint pro-jections Pjka b for a complete rotation of polarizer a with dif-ferent curves corresponding to b (and thus s) fixed at differentvalues It is apparent from the near-identity of the curves thatto good approximation the correlations are a function of the dif-ference in angles ie Ca b Ca minus b The maximum valuefor B can then be found straightforwardly from any one of thecurves in Fig 2 Among them the smallest and largest valuesof B (obtained for curves 1 and 4) are 2548 0004 and2679 0007 respectively

To be careful we note that in our experiments the field wasalmost but not quite completely unpolarized thus not quite thesame field was sketched in the Background Theory section Thuswe could not expect to get the maximum quantum result B 2

ffiffiffi2

p 2828hellip for the Bell parameter although the valuesachieved also present a strong violation The background theoryis mildly more complicated for partially polarized rather thanunpolarized light but when worked out for the DOP of ourlight beams (see [15] and Supplement 1) it supports the valueswe observed

7 SUMMARY

In summary we first sketched the purely classical theory of opticalbeam fields (1) that satisfy the Bell-test criteria of Shimony[1315] Their bipartite pure state form shows the entanglementof their two independent DOFs [20] The classical theory definesthem as dynamically probabilistic fields meaning that individualfield measurements yield values that cannot be predicted except inan average sense which is another feature shared with quantumsystems 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 unpolarizedlight indicated that such fields or states are predicted to possessa range of correlation strengths equal to that of two-party quan-tum systems that is outside the bound B le 2 of the CHSH Bellinequality and potentially as great as B 2

ffiffiffi2

p In our experi-

mental test we used light whose statistical behavior (fieldsecond-order statistics) is indistinguishable from classical vizthe light from a broadband laser diode operating below threshold

Our detections of whole-beam intensity are free of the heraldingrequirements familiar in paired-photon CHSH experimentsRepeated tests confirmed that such a field can strongly violatethe CHSH Bell inequality and can attain Bell-violating levelsof correlation similar to those found in tests of maximallyentangled quantum systems

One naturally asks how are these results possible We knowthat a field with classically random statistics is a local real fieldand we also know that Bell inequalities prevent local physics fromcontaining correlations as strong as what quantum states provideBut the experimental results directly contradict this The resolu-tion of the apparent contradiction is not complicated but doesmandate a shift in the conventional understanding of the role ofBell inequalities particularly as markers of a classical-quantumborder Bell himself came close to addressing this point Hepointed out [21] that even adding classical indeterminism wouldstill not be enough for any type of hidden variable system to over-come the restriction imposed by his inequalities This is correct asfar as it goes but fails to engage the point that local fields can bestatistically classical and exhibit entanglement at the same timeFor the fields under study the entanglement is a strong correla-tion that is intrinsically present between the amplitude and polari-zation DOFs and it is embedded in the field from the start (as italso is embedded ab initio in any quantum states that violate a Bellinequality) The possibility of such pre-existing structural corre-lation is bypassed in a CHSH derivation Thus one sees that Bellviolation has less to do with quantum theory than previouslythought but everything to do with entanglement

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

The authors acknowledge helpful discussions with many col-leagues particularly including M Lewenstein as well as A FAbouraddy A Aspect C J Broadbent L Davidovich EGiacobino G Howland A N Jordan P W Milonni R J CSpreeuw and A N Vamivakas

See Supplement 1 for supporting content

REFERENCES AND NOTES

1 In the paper where he introduced the English word entanglement [ESchroumldinger ldquoDiscussion of probability relations between separated sys-temsrdquo Proc Cambridge Philos Soc 31 555ndash563 (1935)] Schroumldingerreferred to the math-physics textbook by R Courant and D HilbertMethoden der mathematischen Physik 2nd ed (specifically p 134)where the mathematical character of entanglement appears in connec-tion with the Schmidt theorem of analytic function theory

2 R J C Spreeuw ldquoA classical analogy of entanglementrdquo Found Phys28 361ndash374 (1998)

3 P Ghose and M K Samal ldquoEPR type nonlocality in classical electrody-namicsrdquo arXivquant-ph0111119 (2001)

4 K F Lee and J E Thomas ldquoEntanglement with classical fieldsrdquo PhysRev Lett 88 097902 (2002)

5 C V S Borges M Hor-Meyll J A O Huguenin and A Z Khoury ldquoBell-like inequality for the spin-orbit separability of a laser beamrdquo Phys RevA 82 033833 (2010)

6 M A Goldin D Francisco and S Ledesma ldquoSimulating Bell inequalityviolations with classical optics encoded qubitsrdquo J Opt Soc Am A 27779ndash786 (2010)

7 A recent example is A Tiranov J Lavoie A Ferrier P Goldner V BVerma S W Nam R P Mirin A E Lita F Marsili H HerrmannC Silberhorn N Gisin M Afzelius and F Bussieres ldquoStorage of

Fig 2 Plots of the correlation functions Ca b obtained by rotatingpolarizer a in the polarization space and keeping angle b in the functionspace constant Curves 1ndash4 correspond to different fixed values of bseparated by π∕4 The invariant cosine function required to violatethe Bell inequality is clearly present Error bars are included but arescarcely visible

Research Article Vol 2 No 7 July 2015 Optica 614

hyperentanglement in a solid-state quantum memoryrdquo Optica 2279ndash287 (2015)

8 B N Simon S Simon F Gori M Santarsiero R Borghi N Mukundaand R Simon ldquoNonquantum entanglement resolves a basic issue inpolarization opticsrdquo Phys Rev Lett 104 023901 (2010)See also MSanjay Kumar and R Simon ldquoCharacterization of Mueller matrices inpolarization opticsrdquo Opt Commun 88 464ndash470 (1992)

9 X F Qian and J H Eberly ldquoEntanglement and classical polarizationstatesrdquo Opt Lett 36 4110ndash4112 (2011) See also T Setaumllauml AShevchenko and A Friberg ldquoDegree of polarization for optical nearfieldsrdquo Phys Rev E 66 016615 (2002) and J Ellis and A DogariuldquoOptical polarimetry of random fieldsrdquoPhys Rev Lett 95 203905 (2005)

10 K H Kagalwala G DiGiuseppe A F Abouraddy and B E A SalehldquoBellrsquos measure in classical optical coherencerdquo Nat Photonics 7 72ndash78(2013)

11 F Toumlppel A Aiello C Marquardt E Giacobino and G LeuchsldquoClassical entanglement in polarization metrologyrdquo New J Phys 16073019 (2014)

12 J F Clauser M A Horne A Shimony and R A Holt ldquoProposed experi-ment to test local hidden-variable theoriesrdquo Phys Rev Lett 23 880ndash884(1969)

13 A Shimony ldquoContextual hidden variables theories and Bellrsquos inequal-itiesrdquo Br J Philos Sci 35 25ndash45 (1984)See also A ShimonyldquoBellrsquos Theoremrdquo in Stanford Encyclopedia of Philosophy httpplatostanfordeduentriesbell‑theorem (2009)

14 E Wolf ldquoCoherence properties of partially polarized electromagneticradiationrdquo Nuovo Cimento 13 1165ndash1181 (1959)

15 See X-F Qian and J H Eberly ldquoEntanglement is sometimes enoughrdquoarXiv13073772 (2013)

16 See C Brosseau Fundamentals of Polarized Light A StatisticalOptics Approach (Wiley 1998) and E Wolf ed Introduction to theTheory of Coherence and Polarization of Light (Cambridge University2007)

17 Elliptical polarization needs different notation but the results are thesame

18 The Schmidt theorem is a continuous-space version of the singular-value decomposition theorem for matrices For background see M VFedorov and N I Miklin ldquoSchmidt modes and entanglementrdquoContemp Phys 55 94ndash109 (2014) The original paper is E SchmidtldquoZur Theorie der linearen und nichtlinearen Integralgleichungen 1Entwicklung willkuumlriger Funktionen nach Systeme vorgeschriebenerrdquoMath Ann 63 433ndash476 (1907) See also A Ekert and P L KnightldquoEntangled quantum-systems and the Schmidt decompositionrdquo Am JPhys 63 415ndash423 (1995)

19 N Gisin ldquoBellrsquos inequality holds for all non-product statesrdquo Phys Lett A154 201ndash202 (1991)

20 Classical fields are often thought of as the many-photon macroscopiclimit of quantum coherent states It is interesting to propose connectionsbetween our work and the CHSH Bell violations discussed theoreticallyfor maximally entangled macroscopic coherent states in C C Gerry JMimih and A Benmoussa ldquoMaximally entangled coherent states andstrong violations of Bell-type inequalitiesrdquo Phys Rev A 80 022111(2009) Following the number-state treatments of C F WildfeuerA P Lund and J P Dowling ldquoStrong violations of Bell-type inequalitiesfor path-entangled number statesrdquo Phys Rev A 76 052101(2007)

21 J S Bell ldquoQuantum mechanical ideasrdquo Science 177 880ndash881(1972)

Research Article Vol 2 No 7 July 2015 Optica 615