Entanglement-enabled quantum holography

Hugo Defienne,1, ˚ Bienvenu Ndagano,1 Ashley Lyons,1 and Daniele Faccio1,˚

1School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK(Dated: November 5, 2019)

Quantum features of light enable newimaging technologies1 including interaction-free schemes2,3, induced-coherence imaging4,sensitivity-enhanced5 and super-resolution tech-niques6,7. However, quantum imaging systemscan often be replicated using classical lightwith similar performance8,9 as a result of thefact that, nonlocal entanglement, a unique fea-ture of quantum mechanics is not employedin an essential manner10,11. Here we describeand experimentally demonstrate a holographicimaging approach that is genuinely based onquantum entanglement. Alice encodes an imagein the phase component of space-polarisationhyperentangled photons using a spatial lightmodulator. The image can be reconstructedby Bob using a purposely-developed full-fieldquantum holographic technique. The experimen-tal results show that the presence of nonlocalpolarisation entanglement between photons is es-sential to the holographic encoding and decoding.Moreover, we show that entanglement-encodedphase images propagate without disturbancethrough dynamic phase disorder and can even beretrieved in the presence of strong classical noisee.g. stray light, with practical advantages overclassical holographic protocols.

Holography is an essential tool of modern optics, at theorigin of many applications ranging from microscopy12 tosecure data storage13. In this respect, holographic inter-ferometry is a widely-used technique that exploits opticalinterference to retrieve the phase component of a classicaloptical field through intensity measurements. For exam-ple, phase-shifting holography14 uses four intensity im-ages Iθ (θ P t0, π{2, π, 3π{2u) of a reference optical fieldaeiθ interfering with an unknown field beiφ to reconstructthe phase profile

φ “ arg“

I0 ´ Iπ ` ipIπ{2 ´ I3π{2q‰

. (1)

Maintaining optical coherence between interfering fieldsis therefore essential in all holographic protocol. Mechan-ical instabilities, random phase disorder and the pres-ence of stray light are examples of phenomena that de-grade light coherence and hinder the phase reconstruc-tion process. In the following, we present a quantumholographic technique that circumvents these detrimen-tal effects. Phase images are encoded in the polarisation-entanglement of hyperentangled photons and retrievedthrough spatial intensity correlation measurements (i.e.photon coincidence counting). The nonlocal nature ofour measurements removes the need for path overlap and

shields us from mechanical instabilities, while polarisa-tion encoding provides robustness against random phasedisorder. Furthermore, the measurement of two-photoncorrelations allows us to be insensitive to the presence ofstray light. While holograms of non-classical light havebeen predicted15,16 and observed with single-photons andphoton pairs17,18, or used for quantum state character-isation and manipulation19–21, we note no experimentaldemonstration of entanglement-based holographic inter-ferometry heretofore.

The conceptual arrangement of our quantum holo-graphic scheme is illustrated in Fig. 1a. Photon pairsentangled in space and polarisation22 interact with twospatial light modulators (‘Alice SLM’ and ‘Bob SLM’)and are then detected by two single-photon imaging de-vices, for example two electron multiplied charge coupleddevice cameras (‘Alice EMCCD’ and ‘Bob EMCCD’).The transverse momentum k of the photons is mappedonto separated pixels of the SLMs and re-imaged onto thecameras. Alice and Bob shape and detect photons withmomentum of negative x-component pkx ă 0q and pos-itive x-component pkx ą 0q, respectively. The quantumstate of the photon pair after the SLMs is thus

ÿ

k

”

|V yk|V y´k ` eiΨpkq|Hyk|Hy´k

ı

(2)

in which Ψ is a relative phase, |Hy and |V y represent hori-zontal and vertical polarisation states of the photons. Fora given momentum k pkx ą 0q, Ψpkq is the sum of threephase terms Ψ0pkq, θAp´kq and θBpkq. Ψ0pkq is a staticphase distortion produced during the photon generationprocess23 that is characterised beforehand (see Methods).The phases θAp´kq and θBpkq are actively controlled byAlice and Bob by programming pixels at coordinates ´kand k of their SLMs. This is made possible by the use ofparallel aligned nematic liquid-crystal SLMs, that enablethe manipulation of the horizontal polarisation of incom-ing photons but leave the vertical component unchanged.

First, Alice encodes an image θAp´kq in the phasecomponent of entangled photons by programming herSLM with the corresponding phase pattern. Figure 1bshows the pattern used in our experiments, correspond-ing to the letters UofG. On the other hand, Bob displayson his SLM a phase mask θBpkq “ ´Ψ0pkq to compen-sate for the phase distortion Ψ0 (Fig. 1e). This correct-ing phase remains superimposed to all phase masks thatBob programs throughout the experiment. As a result,the phase of the quantum state after the SLMs equalsexactly the encoded image Ψpkq “ θAp´kq. In the ex-ample shown in Fig. 1b, pixels associated with the lettersU and o are encoded as the Bell states |V V y ` |HHy

arX

iv:1

911.

0120

9v1

[qu

ant-

ph]

4 N

ov 2

019

2

00

a

100

-1000

0

100

-100

k

-k

BobEMCCD

AliceEMCCD

Alice SLM

Bob SLM

kx (pixel)

ky (pixel)

100

-100

100

-100

HyperentangledPhoton pair

-100 0

0

-100

100 0

2π

Phase (rad

)kx (pixel)

ky (

pix

el)

Alice image encoding

k

-k

0 100

0

-100

100 0

Phase (rad

)

kx (pixel)

ky (

pix

el)

0 1000

440

Inten

sity (arb.u

nits)

Bob image decoding

b

-100 00

440

Inten

sity (arb. u

nits)

kx (pixel)

c

+π2π

0 100 0 100 0 100

d

kx (pixel) kx (pixel)0 100

+0f j

+π/2 +3π/2

VH

300

Inten

sity correlations

(arb.u

nits)

00 100

-8

1

Phase (rad

)

kx (pixel)

e g h i

45°-polariser

θA(-k)

θB(k)

Figure 1. Schematic of the quantum holographic reconstruction. a, Space-polarisation hyperentangled photon pairspropagate through two spatial light modulators (Alice SLM and Bob SLM) and are detected by two electron multiplied chargecouple device cameras (Alice EMCCD and Bob EMCCD). Transverse momentums k of photons with negative x-componentpkx ă 0q are mapped to pixels on Alice’s SLM and camera, while those with positive x-component pkx ą 0q are mapped topixels on Bob’s SLM and camera. Parallel aligned nematic liquid-crystal SLMs allow Alice and Bob to modulate at any pixelthe horizontal polarisation of incoming photons with spatial phases θA and θB . Two polarisers oriented at 45 degrees areinserted between SLMs and cameras. b, Phase image θAp´kq displayed on Alice SLM. c and d, Intensity images measuredby Alice and Bob on their cameras, respectively. e, SLM pattern displayed on Bob SLM to compensate for the static phasedistorsion Ψ0. f-i, Intensity correlation images measured by Bob for different constant phase shift programmed on his SLM: `0(f), `π{2 (g), `π (h) and `3π{2 (i). Each image is obtained by measuring intensity correlations between Bob camera pixelsk and their symmetric on Alice camera ´k. j, Phase image reconstructed by Bob, with a signal-to-noise ratio (SNR) over 19and a normalised mean square error (NMSE) of 5%. A total of 2.5ˆ 106 frames was acquired to retrieve the phase. Intensityand intensity correlation values are in arbitrary units and the same scales are used in all the figures of the manuscript.

(Ψ “ 0), while f and G are encoded as |V V y ´ |HHy(Ψ “ π). After programming Alice’s phase, we observethat the intensity images measured by both Alice andBob, shown in Figs. 1c and e, are homogeneous and donot reveal the phase-encoded image. This observationremains valid when including polarisers in front of thecameras, at any orientation.

In the holographic reconstruction step of the process,Bob decodes the image by performing intensity corre-lation measurements between pixels at k of his cameraand symmetric pixels at ´k on Alice’s camera24, with thetwo polarisers oriented at 45 degrees (see Methods). Thismeasurement is repeated four times for four different con-stant phase shifts θ applied on Bob’s SLM, resulting inintensity correlation image Rθpkq9 1`cospθApkq`θq (seeMethods). Figures 1f-i show four intensity correlationimages measured for θ P t0, π{2, π, 3π{2u that partiallyreveal the hidden phase. Following a similar approachto classical holography, Bob then reconstructs the en-

coded image by using equation 1 after replacing Iθ byRθ. As shown in Fig. 1j, the retrieved image is 180 de-grees rotated and is of high quality, with a signal-to-noiseratio (SNR) over 19 and a normalised mean square error(NMSE) of 5%. While the SNR measures the intrinsicquality of the image retrieved by Bob in term of noiselevel, the NMSE quantifies its resemblance to the origi-nal image encoded by Alice (see Methods).

In our protocol, the photon pair spatial correlationsprovide the high-dimensional image space25 while polar-isation entanglement carries the grey-scale informationat each pixel. The presence of polarisation entanglementis therefore essential to this scheme. For example, Fig. 2shows results of quantum holography performed with thesame encoded image as in Fig. 2a but using a source ofphoton pairs that are entangled in space but not in polar-isation (see Methods). As in the previous case, intensityimages measured by Alice and Bob in Fig. 2b and c do notreveal information about the encoded phase. However,

3

Inten

sity correlation (A

rb. U

nits)

kx (pixel)

-100

100

0

1000

ky (

pix

el)

0

193

0

-100

100

ky (

pix

el)

0

2π

Phase (rad

)

da b c

0

380

Inte

nsi

ty (

arb.u

nits)

-100 100 1000 0-100kx (pixel)

f

1000 1000 1000

+π+0 +π/2 +3π/2

0

g h i

Figure 2. Quantum holography without polarisationentanglement. a, Phase image encoded by Alice. b andc, Intensity images measured by Alice and Bob. d, Phasereconstructed by Bob that does not reveal the encoded im-age (NMSE=95%). f-i, Intensity correlation images used inthe phase reconstruction process measured for different phaseshifts: `0 (f), `π{2 (g), `π (h) and `3π{2 (i). 2.5 ˆ 106

frames were acquired in total.

Figs. 2f-i show that the intensity correlation images ac-quired during the phase-shifting process do not reveal anyimage information either, and the phase image cannot beretrieved (NMSE=95%), as shown in Fig. 2d. Non-zerovalues in intensity correlation images also confirm that(classical) correlations between photon polarisations arepresent without entanglement; the existence of a phaseΨ is conditioned on the coherence, and thus the entan-glement, between the two-qubit terms |V V y and |HHyof the state26.

Beyond the intrinsic interest in imaging schemes thatrely exclusively on nonlocal entanglement, our quantumholographic protocol is also robust against dynamic phasedisorder. Figure 3 describes an experimental apparatusin which space-polarisation entangled photons propagatethrough a thin diffuser (figure inset) positioned on a mo-torised translation stage that is placed in the image planeof both SLMs and cameras. In this configuration, the dif-fuser introduces a time varying random phase Φtpkq inall spatial modes k (see Methods). At time t, the statedetected by Alice and Bob thus becomes

ÿ

k

eirΦtpkq`Φtp´kqs”

|Hyk|Hy´k ` eiΨpkq|V yk|V y´k

ı

.

(3)The phase disorder terms factorise in equation 3 leav-ing the term Ψ undisturbed. Figures 4a-b show the ex-perimental reconstruction of a phase image through adynamic diffuser. The diffuser induces a small blur of

the edges in the intensity images measured by Alice andBob (Figs. 4b and c), yet the image encoded by Alice asshown in Fig. 4a is very accurately reconstructed by Bobin Fig. 4d (SNR“ 21 and NMSE“ 2%). It is importantto note that each photon of a pair experiences a phasedisorder independent of that experienced by its twin (i.e.they propagate through spatially separated parts of thediffuser). Therefore, our holographic protocol can op-erate with Alice and Bob placed at any distance fromeach other, without being sensitive either to mechanicalinstability and vibrations.

Finally, we demonstrate that our approach also worksin the presence of stray, classical light falling on the de-tectors. As illustrated in Fig. 3, images of two differentcat-shaped objects illuminated by a laser are superim-posed onto both Alice’s and Bob’s sensors. These clas-sical images are clearly visible in the intensity images asshown in Figs. 4f and g. However, because photons emit-ted by the classical source are spatially uncorrelated, thecat-shaped images do not appear in the intensity correla-tion images27 used for phase reconstruction, as shown inFigs. 4i and j. Therefore, a high quality (SNR“ 20 andNMSE“ 4%) phase image, encoded by Alice (Fig. 4e), isretrieved by Bob (Fig. 4h). This experiment highlightsanother practical advantage of the protocol, namely thatit remains operational in natural environments contain-ing sources of stray light.

In summary, we have presented a quantum imagingtechnique enabled by nonlocal entanglement. Grey scaleinformation about the image is encoded in the relativephases between the entangled polarisation qubits and dis-tributed over many spatial positions through the high-dimensional structure of spatial entanglement. Bob can-not retrieve the phase information encoded by Alice byclassical holography. Furthermore, because the hologra-phy happens in a degree of freedom that is not affectedby phase disorder in the image basis (i.e. polarisation)and is based on correlation measurements, it is robustagainst the presence of dynamic diffusers and stray light,two factors that would respectively disrupt and inhibit aphase reconstruction process using other techniques.

Beyond imaging, the use of quantum entanglementas an information carrier holds potential for developingquantum-secured imaging applications28. In particular,the possibility to arbitrarily enlarge the distance be-tween Alice and Bob makes our scheme promising forquantum communication. The measurements performedby Bob in the holographic process correspond exactlyto projections in the diagonal (θB P t0, πu) and circular(θB P tπ{2, 3π{2u) polarisation basis. Similarly, Alicecan use her SLM to perform measurements in thecorresponding rotated basis θA P tπ{4, 3π{4, 5π{4, 7π{4u,instead of encoding an image. As shown in Fig. 5, onemay use these measurement settings to show spatially-resolved violation of the Clauser-Horne-Shimony-Holt(CHSH) inequality between Alice and Bob (see Meth-ods). This implies that in principle Ekert’s quantumkey distribution protocol29 could be implemented to

4

share secret keys across many spatial channels (i.e. pixelpairs). The ability to exploit many channels in parallelcould considerably increase the secret key rates of QKDprotocols30 compared to the conventional single channelapproach.

Acknowledgements. The Authors acknowledgediscussions with M. Barbieri. D.F. acknowledges finan-cial support from the UK Engineering and PhysicalSciences Research Council (grants EP/M01326X/1 andEP/R030081/1) and from the European Union’s Horizon2020 research and innovation programme under grantagreement No 801060. H.D. acknowledges financialsupport from the EU Marie-Curie Sklodowska Actions(project 840958).

METHODS

Experimental layout. A paired set of BBO crystalshave dimensions of 0.5 ˆ 5 ˆ 5 mm each and arecut for type I SPDC at 405 nm. They are opticallycontacted with one crystal rotated by 90 degrees aboutthe axis normal to the incidence face. Both crystals areslightly rotated around horizontal and vertical axis toensure near-collinear phase matching of photons at theoutput (i.e. rings collapsed into disks). The pump is acontinuous-wave laser at 405 nm (Coherent OBIS-LX)with an output power of approximately 200 mW anda beam diameter of 0.8 ˘ 0.1 mm. A 650 nm-cut-offlong-pass filter is used to block pump photons afterthe crystals, together with a band-pass filter centredat 810 ˘ 5 nm. The SLM is a phase only modulator(Holoeye Pluto-2-NIR-015) with 1920ˆ1080 pixels and a8µm pixel pitch. The camera is an EMCCD (Andor IxonUltra 897) that operates at ´60˝C, with a horizontalpixel shift readout rate of 17 MHz, a vertical pixel shiftevery 0.3µs, a vertical clock amplitude voltage of `4Vabove the factory setting and an amplification gain setto 1000. It has a 16µm pixel pitch. Exposure time isset to 3 ms. The classical source is a superluminescentdiode laser (Qphotonics) with a spectrum of 810˘15 nmthat is filtered using a band-pass filter at 810˘ 5 nm tomatch the photon pair’s spectrum. The lens f1 has aneffective focal lenght of f1 “ 54 mm that is created by aseries of three lenses of focal lengths 45 mm - 125mm -150 mm positioned into a Fourier imaging configuration.Focal lengths of the other lenses are f2 “ 150 mm,f3 “ f4 “ 75 mm, f4 “, f5 “ 100 mm and f6 “ 175mm. The diffuser is a plastic sleeve layer of thicknessă 100µm, roughness of 46µm and has a decorrelationtime of 183 ms. See the Supplementary Information forfurther details on the diffuser properties.

Intensity correlation images. The camera sensor issplit in two identical regions of interest composed of 201ˆ

101 pixels associated with Alice and Bob. To measureintensity correlations, the camera first acquires a set ofN images. Then, values of intensity correlation Rpkqbetween pixel at k on Bob’s side and the symmetric at´kon Alice’s side are calculated by subtracting the productof intensity values measured in the same frame by theproduct of intensity values measured in successive frames,and averaging over all the frames:

Rpkq “1

N

Nÿ

l“1

rIlpkqIlp´kq ´ IlpkqIl`1p´kqs . (4)

in which Il denotes the lth frame24. See the Supplemen-tary Information for further details on the intensity cor-relation measurement.Quantum holography. Intensity correlation measure-ment performed between pixels k and ´k with two po-larisers oriented at 45 degrees positioned in front of thecameras can be associated with the following measure-ment operator:

1

2

“

|HyxH| ` |V yxV | ` |HyxV | ` |V yxH|‰

k

b“

|HyxH| ` |V yxV | ` |HyxV | ` |V yxH|‰

´k(5)

For a given pair of pixels (´k,k), the expectation valueof this operator in the state described by equation 2 is

Rpkq “1

2r1` cospΨpkqqs (6)

During the holographic process, Alice encodes a phaseθAp´kq and Bob applies a phase shift θ superim-posed over the phase compensation pattern ´Ψ0pkq.As a result, intensity correlation measurements per-formed by Bob for a given θ are given by Rθpkq “12 r1 ` cospθApkq ` θqs. As in classical holography(equation 1), Bob then reconstructs the phase im-age θApkq image using four successive measurements:θApkq “ arg

“

R0pkq ´Rπpkq ` i`

Rπ{2pkq ´R3π{2pkq˘‰

.Note that, to take into account a more general case, thestate in equation 2 can be re-written as:

ÿ

k

”

eiΨpkq|Hyk|Hy´k ` α|V yk|V y´k

ı

(7)

with α Ps0, 1s. In this case, the expectation value of theoperator in equation 5 changes into 1

2

“

1` α2 cospΨpkqq‰

,but θApkq is still retrieved using equation 1 (albeit withvisibility equal to α2).Spatial and polarisation entanglement. Spatialentanglement in the photon source is character-ized by performing intensity correlation measure-ments between positions and momentum of pho-tons31,32. Correlation width measurements returnvalues of σr “ 10.85 ˘ 0.06µm for position andσk “ r2.033 ˘ 0.001s ˆ 103 rad.m´1 for momen-tum. These values show violation of EPR criteriaσrσk “ r2.21 ˘ 0.01s ˆ 10´2 ă 1

2 and allows estimation

5

Motorizedtranslation

stage

BobEMCCD

Alice EMCCD

f4

f5

f6

Laser810nm

Cat-shapedobjects

Lens HWP22.5°

Polariser45°

Filter

Laser 405nm

AliceSLM

Paired BBOs

f1BobSLM

Diffusers

f2f3

f7

BS 90T/10R

V45°

-k

k

Figure 3. Detailled experimental setup with dynamic diffusers and stray light. Light emitted by a laser diode at405nm and polarised at 45 degrees illuminates a pair of β-Barium Borate (BBO) crystals (0.5mm thickness each) whose opticalaxes are perpendicular to each other to produce pairs of photons entangled in space and polarisation by type-I spontaneousparametric down-conversion (SPDC). After the crystals, pump photons are filtered out by a combination of long-pass andband-pass filters. The momentum of red photons is mapped onto an SLM divided in two parts (Alice SLM and Bob SLM)by Fourier imaging with lens f1. Lenses f2 ´ f3 image the SLM plane onto a thin diffuser (inset) positioned on a motorisedtranslation stage and lenses f4 ´ f5 image it on an EMCCD camera split in two parts (Alice EMCCD and Bob EMCCD). Apolariser oriented at 45 degrees is positioned between lenses f4 and f5. Two cat-shaped objects illuminated by a laser (810nm)are imaged on the camera and superimposed on top of quantum light using two lenses f6´ f5 and an unbalanced beam splitter(BS 90T/10R). For clarity, only two propagation paths of entangled photons at k and ´k are represented, while they havea higher dimensional spatial structure (ą 500 modes) ; SLMs, EMCCD cameras and diffusers are represented by pairs, whilethey are single devices spatially divided into two independent parts; the SLM is represented in transmission, while it operatesin reflection. See Methods for further information.

da b c

100-1000

450

Inten

sity (arb.u

nits)

0

100-100

-100 0

-100100

100

π

-π

0

π

-π

0 0

-100

100

ky (

pix

el)

kx ky kx ky

kx (pixel)

Phase (rad

)π

-π

Phase (rad

)

π

-π00

0

-100

100

ky (

pix

el)

0

2π

Phase (rad

)

0

500

Inte

nsi

ty (

arb.u

nits)

-100 100 1000 0-100 0

he f g

Inten

sity correlation (A

rb. U

nits)

kx (pixel)100

0

211

1000

+π+0 ji

Figure 4. Quantum holography through dynamic phase disorder and in the presence of stray light. a, Phaseimage encoded by Alice. b and c, Intensity images measured respectively by Alice and Bob through the dynamic diffuserwith no stray light. d, Phase image reconstructed by Bob with SNR=21 and NMSE=2%. e, Phase image by Alice. f and g,Intensity images measured by Alice and Bob through the dynamic diffuser and in the presence of stay light taking the form oftwo cat-shaped images. h, Phase image reconstructed by Bob with SNR=20 and NMSE=4%. i-j, Intensity correlation imagesmeasured by Bob for phase shifts: `0 (i) and `π (j). 5ˆ 106 frames were acquired in total for each case.

6

kx (pixel)

ky (

pix

el)

200-200 0

200

-200

0

b

S

2

2.8

aR

L

VH

A

D

θA

R

L

VH

A

D

θB

Figure 5. Spatially-resolved Clauser-Horne-Shimony-Holt (CHSH) inequality violation. a, Poincar spheresrepresenting the projections performed by Alice and Bob withtheir SLMs. |Dy and |Ay are two eigenstates of the diagonalpolarisation basis and |Ry and |Ly are the two eigenstates ofthe circular polarisation basis. Bob performs measurementsdirectly in the diagonal and circular basis, while Alice oper-ates in a similar basis rotated by π{4. b, Values of S measuredat different pairs of pixel of Alice’s and Bob’s sensors. Theimage is calculated from intensity correlation measured for 16combinations of angles θA and θB . Spatial averaged value isS “ 2.23˘ 0.02 ą 2. See Methods for further details.

of the number of entangled modes K “ 514 ˘ 5 25.Polarisation entanglement is certified by demonstratingviolation of Bell inequalities between all spatially corre-lated pairs of pixels at the output, with a mean valueof S “ 2.23 ˘ 0.02 ą 2 (Fig. 5). See the SupplementaryInformation for further details on the characterisation ofspatial and polarisation entanglement.

Phase distortion characterisation. The phasedistortion Ψ0pkq originates from the birefringence ofthe paired BBO crystals used to generate photonpairs23. Ψ0pkq is measured beforehand by perform-ing a holographic measurement between a flat phasepattern programmed on Alice SLM and successivephase shifts displayed on Bob SLM. This character-isation process results in a phase distortion of theform Ψ0pkx, kyq “ 4.69k2

x ` 5.04k2y ` 0.02. See the

Supplementary Information for further details on thephase distortion characterisation.

Signal-to-noise, normalised mean square error

and spatial resolution. Signal-to-noise ratio (SNR)is obtained by calculating an averaged value of the phasein a region of the retrieved image where it is constant,and then dividing it by the standard deviation of thenoise in the same region. To have a common reference,SNR values are calculated using areas where the phaseis constant and equals π. For a fixed exposure timeand pump power, the SNR varies as

?N , where N is

the number of images used to reconstruct the intensitycorrelation images33. The normalised mean square error(NMSE)34 quantifies the resemblance between an imagereconstructed by Bob and the ground truth image en-coded by Alice. The NMSE is calculated using the for-mula:

NMSE “|M0 ´M8|

M0(8)

where M0 is the mean square error (MSE) measuredbetween the ground truth and the retrieved image andM8 is an average value of MSE measured between theground truth and a set of images composed of phasevalues randomly distributed between 0 and 2π. TheMSE between two images composed of P pixels withvalues denoted respectively txiuiPrr1,P ss and tyjujPrr1,P ss

is defined as M “ 1{PřPi“1 |xi ´ yi|

2. Values of NMSErange between 1 (retrieved image is a random phaseimage) and 0 (retrieved image is exactly the groundtruth). Spatial resolution in the retrieved image isdetermined by the spatial correlation width of entangledphotons. In our experiment, its value is estimated tod “ 45 ˘ 3µm, which corresponds to approximately 3camera pixels. See the Supplementary Information forfurther details on the signal-to-noise ratio and spatialresolution characterisation.

Photons without polarisation entanglement. Re-sults shown in Fig. 3 are obtained using a quantum statedefined by the following density operator:

1

2

ÿ

k

“

|Hyk|Hy´kxH|kxH|´k ` |V yk|V y´kxV |kxV |´k

‰

(9)Experimentally, it is produced by switching the polari-sation of the pump laser between vertical and horizontalpolarisations, which is equivalent of using a unpolarisedpump. Because entanglement originates fundamentallyfrom a transfer of coherence properties between the pumpand the down-converted fields in SPDC35–37, the lack ofcoherence in the pump polarisation induces the absenceof polarisation entanglement in the produced two-photonstate, while spatial and temporal entanglement are main-tained26. See the Supplementary Information for furtherdetails on state entangled in space but not in polarisa-tion.Spatially-resolved Clauser-Horne-Shimony-Holt(CHSH) measurement. A set of 16 intensity corre-lations images RθA,θB is first measured using all combi-nations of uniform phases θA P tπ{4, 3π{4, 5π{4, 7π{4u

7

and θB P t0, π{2, π, 3π{2u programmed on Alice and BobSLMs. Then, a correlation image EθA,θB is calculatedusing the following formula38:

EθA,θB “RθA,θB ´RθA,θB`π ´RθA`π,θB `RθA`π,θB`πRθA,θB `RθA,θB`π `RθA`π,θB `RθA`π,θB`π

(10)Finally, the image of S values shown in Fig. 5.b is ob-tained using the following equation:

S “ |Eπ{2,π{4 ´ Eπ{2,5π{4| ` |E0,π{4 ` E0,5π{4| (11)

As shown in Fig. 5, almost all values of S measured be-tween in Alice and Bob correlated pixels show violationof CHSH inequality S ą 2. A spatial averaged value ofS “ 2.23˘0.015 ą 2 is estimated by calculating the meanand variance of S values over a region of a smaller regionof the sensor where S is homogeneous (kx P r10, 50s andky P r´40, 40s). See the Supplementary Information forfurther details on the CHSH measurement.

8

Supplementaryinformation

I. DETAILS ON INTENSITY CORRELATIONMEASUREMENT

This section provides more details about the intensitycorrelation measurement performed by the EMCCD cam-era. Further theoretical details can be found in24.

An EMCCD camera can be used to reconstruct thespatial (a) intensity distribution Ipkq and (b) intensitycorrelation distribution Γpk1,k2q of photon pairs, wherek, k1 and k2 correspond to positions of camera pixels.To do that, the camera first acquires a set of N framestIlulPrr1,Nss using a fixed exposure time. Then:

(a) The intensity distribution is reconstructed by aver-aging over all the frames:

Ipkq “1

N

Nÿ

l“1

Ilpkq (12)

(b) The intensity correlation distribution is recon-structed by performing the following substation:

Γpk1,k2q “1

N

Nÿ

l“1

Ilpk1qIlpk2q´1

N ´ 1

N´1ÿ

l“1

Ilpk1qIl`1pk2q

(13)Under illumination by the photon pairs, intensitycorrelations in the left term of the subtraction origi-nate from detections of both real coincidences (twophotons from the same entangled pair) and acci-dental coincidences (two photons from two differententangled pairs), while intensity correlations in thesecond term originate only from photons from dif-ferent entangled pairs (accidental coincidence) be-cause there is zero probability for two photons fromthe same entangled pair to be detected in two suc-cesive images. A subtraction between these twoterms leaves only genuine coincidences, that is pro-portional to the spatial joint probability distribu-tion of the pairs.

In our work, we use this technique for measuring cor-relation between pairs of symmetric pixels k and ´k toreconstruct intensity correlation images Rpkq. These im-ages correspond exactly to the anti-diagonal componentof the complete intensity correlation distribution Rpkq “Γpk,´kq. Fig. 6 illustrates these different types of mea-surements in the case of the experiment described in Fig.1of the manuscript when Bob displays a phase shift `π onhis SLM. Fig. 6.a and b show respectively intensity im-ages measured by Alice (pixels k “ pkx ă 0, kyq) and Bob(pixels k “ pkx ą 0, kyq). These images do not provideany information about the correlation between photonpairs. Fig. 6.c is the conditional image Γpk,k1q that rep-resents the probability of measuring a photon from a pair

at pixel k in Alice side conditioned by the detection of itstwin photon at pixel k1 in Bob side. We observe a strongpeak of correlation centred around the symmetric pixel´k1 due to the strong anti-correlation between momen-tum of the pairs (zoom in inset). Similarly, Fig. 6.d isthe conditional image Γpk,k2q relative to position k2. Inthis last case, the peak of correlation is very weak (zoomin inset). Finally, Fig. 6.e shows the intensity correlationimage Rpkq “ Γpk,´kq. In this last image, the value atpixel k1 corresponds to the value of the peak of corre-lation at ´k1 shown in Fig. 6.c and the value at pixelk2 corresponds to the value of the peak of correlation at´k2 in Fig. 6.d.

II. DETAILS ON SPATIAL ANDPOLARISATION ENTANGLEMENT OF THE

SOURCE

As described in Fig.3 of the manuscript, entangled pho-ton pairs are generated by type I SPDC using a pair ofBBO crystals. These pairs are entangled in both theirpolarisation and spatial degree of freedom22.

A. Spatial entanglement

Spatial entanglement between photons is characterisedby performing intensity correlation measurement be-tween (a) positions and (b) momentum of photons25,31,32.Fig. 7 describes the corresponding experimental appara-tus:

(a) Positions r of photons are mapped onto pixels ofthe camera using a two-lens imaging configurationf1 ´ f2. After measuring the intensity correla-tion distribution Γpr1, r2q, its projection along theminus-coordinate axis r1 ´ r2 is shown in Fig. 7.b.The peak of correlation at its center is a signatureof strong correlations between positions of photons.Its width σr “ 10.85˘ 0.06µm is measured by fit-ting with a Gaussian model39.

(b) Momentum k of photons are mapped onto pix-els of the camera by replacing the lens f2 by alens with twice its focal length. After measuringthe intensity correlation distribution Γpk1,k2q, itsprojection along the sum-coordinate axis k1 ` k2

is shown in Fig. 7.c. The peak of correlation atits center is a signature of strong anti-correlationsbetween momentum of photons. Its width σk “r2.033 ˘ 0.001s ˆ 103 rad.m´1 is measured by fit-ting with a Gaussian model39.

Under certain assumptions that are discussed in25,the presence of spatial entanglement between photonsis certified by violating a EPR-type inequality: σrσk “r2.21 ˘ 0.01s ˆ 10´2 ă 1

2 . Moreover, the dimension of

9

r|A)

-100 0kx (pixel)

a

0

-100

100

ky (

pix

el)

0 1000

440

Inten

sity (arb.u

nits)

k2

k1

c

-100 0kx (pixel)

-100 0

d

-100 0kx (pixel)

0

350In

tensity correlation

s (arb

.units)

k2

e

-k2

-k1 k1

b

Figure 6. Intensity and intensity correlation measurements between photon pairs with an EMCCD camera. aand b, Intensity images measured respectively by Alice and Bob. Two pixels k1 and k2 are arbitrarily selected in Bob side. c,Conditional image relative to position k1. d, Conditional image relative to position k2. Zooms of areas centred around pixels´k1 and ´k2 are shown in inset. e, Intensity correlation image reconstructed by Bob. Values at k1 and k2 correspond tointensity correlation values measured at ´k1 and ´k2 in images c and d.

entanglement K “ 514 ˘ 5 is calculated using the for-mula40:

K “1

4

„

σrσk `1

σrσk

2

(14)

B. Polarisation entanglement

The presence of polarisation entanglement betweenphotons can be demonstrated by violating Clauser-Horne-Shimony-Holt (CHSH) inequality38. A simplifiedversion of the experimental apparatus used to performthis measurement is shown in Fig. 7.d. In this case, thecombination of Alice and Bob SLMs with a 45 degreespolariser positioned in front of the cameras play the roleof the rotating polarisers used in a conventional CSHSviolation experiment41. When a constant phase θB (θA)is programmed onto Bob SLM (Alice SLM), each pixelof Bob camera (Alice) performs a measurement that cor-responds to an operator of the form:

BθB “1

2

“

|HyxH| ` |V yxV | ` eiθB |HyxV | ` e´iθB |V yxH|‰

(15)In the first step of this experiment, Alice and Bob mea-sure intensity correlation images RθA,θB for 16 com-binations of phase values θA and θB programmed ontheir SLMs. On the one hand, Bob uses the same setthan the one used in the holographic process θB “

t0, π{2, π, 3π{2u, where θB “ 0 and θB “ π correspondto measurements performed in the diagonal polarisationbasis B˘0{π “ p|V y ˘ |HyqpxV | ˘ xH|q and values θB “

π{2 and θB “ 3π{2 correspond to measurements per-

formed in the circular polarisation basis B˘π2 {

3π2

“ p|V y˘

i|HyqpxV | ¯ ixH|q. On the other hand, Alice programsphase values θB “ tπ{4, 3π{4, 5π{4, 7π{4u of her SLMto perform measurements in diagonal and circular basisrotated by π{4: A˘π

4 {5π4

“ p|V y ˘ 1?2|HyqpxV | ˘ 1?

2xH|q

and A˘3π4 {

7π4

“ p|V y ˘ i?2|HyqpxV | ¯ i?

2xH|q. In the sec-

ond step, Alice and Bob combine these 16 intensity cor-relation images to compute EθA,θB using the followingformula38:

EθA,θB “RθA,θB ´RθA,θB`π ´RθA`π,θB `RθA`π,θB`πRθA,θB `RθA,θB`π `RθA`π,θB `RθA`π,θB`π

(16)Finally, an image of S values is obtained using the fol-lowing equation:

S “ |Eπ{2,π{4 ´ Eπ{2,5π{4| ` |E0,π{4 ` E0,5π{4| (17)

This image is shown in Fig.5.b of the manuscript and inFig. 7.e. We observe that almost all values of S mea-sured between in Alice and Bob correlated pixels showviolation of CHSH inequality S ą 2. A spatial averagedvalue of S “ 2.23 ˘ 0.02 ą 2 is estimated by calculat-ing the mean and variance of S values over a region ofa smaller region of the sensor where S is homogeneous(kx P r10, 50s and ky P r´40, 40s). Violation of CHSHinequality demonstrates the presence of polarisation en-tanglement between photon pairs.

III. DETAILS ON SIGNAL-TO-NOISE ANDSPATIAL RESOLUTION

This section provides more details about the signal-to-noise ratio (SNR) and spatial resolution of the phaseimage retrieved by quantum holography.

10

kx1+kx2

ky1+

ky2

100-100 0

100

-100

0

Laser 405nmPaired BBOs

f1V

45°

a

EMCCD

b c

x1-x2

y 1-y

2

100-100 0

100

-100

0

d

kx (pixel)

ky (

pix

el)

200-200 0

200

-200

0

Laser 405nm

AliceSLM

Paired BBOs

f1BobSLM

f2f3

V45°

-k

k

e

Lens HWP22.5°

Filter Polariser45°

Inten

sity correlation (A

rb. U

nits)

0

193

S

2

2.8

BobEMCCD

AliceEMCCD

f2

Figure 7. Spatial and polarisation entanglement characterisation. a, Experimental setup used for spatial entanglementcharacterisation. f1 “ 75mm and f2 “ 100mm form a two-lenses imaging system. b, Minus-coordinate projection of theintensity correlation distribution measured between positions of photons. Zoom of the central area is in inset. c, Sum-coordinate projection of the intensity correlation distribution measured between momentum of photons. In this case, the lensf2 is replaced by a lens of twice smaller focal length f 12 “ 50mm to map momentum of photons onto pixels of the camera.Zoom of the central area is in inset. d, Experimental setup used for polarisation entanglement characterisation. f1 “ 54mm ispositioned in a Fourier imaging configuration and f2 “ 150 and f3 “ 100mm for a two-lenses imaging configuration. e, Imageof measured Clausen-Horne-Shimony-Holt (CHSH) inequality values, denoted S.

A. Signal-to-noise ratio.

In our work, we define the SNR as the ratio betweenπ and the standard deviation of the noise measured ina region of the image in which phase values equal π. Itis for example the case for the area within the lettersU and o in the phase image shown in Fig.1.b. For aconstant source intensity and a fixed exposure time, thefactor that most influences the SNR is the total numberof images N acquired to measure the four intensity corre-lation images used to reconstruct the phase image. Eachphase image shown in the manuscript has been retrievedusing N “ 2.5ˆ106 images and show SNR values rangingbetween 19 and 21. Fig.8.a shows SNR values measuredfor different values of N (black crosses). As predictedby the theory24 and demonstrated by fitting the data(blue dashed curve), the SNR evolves as

?N . Fig.8.b-c

show three images of the retrieved phase for respectivelyN “ 2.5ˆ 104 images, N “ 2.5ˆ 105 and N “ 2.5ˆ 107

images.

B. Spatial resolution.

To measure the spatial resolution in the retrieved phaseimage, a radial resolution target (Siemens star with 16branches) is programmed by Alice. Fig. 8.e shows the

retrieved image. The area of the resolution target thatis not spatially resolved is a disk of diameter 29 pixels,which corresponds to a spatial resolution of d “ 45˘3µmor approximately 3 pixels.

IV. DETAILS ON PHASE DISTORTIONCHARACTERISATION

This section provides more details about the charac-terisation of the phase distortion Ψ0.

Phase distortion Ψpkq originates from the SPDC pro-cess used to produce pairs of photons23,42. To charac-terise it, Alice and Bob perform a quantum holographicexperiment using the scheme described in Fig.1.a. Inthis case, Alice programs a flat phase pattern θA “ 0(Fig. 9.a) and Bob programs successive phase shifts pat-terns with values `0 (Fig. 9.b), `π{2 (Fig. 9.b), `π(Fig. 9.b) and `3π{2 (Fig. 9.b), without any correc-tion phase mask superimposed on them. Fig. 9.f-i showintensity correlations images measured for each of thefour SLM patterns displayed by Bob. Therefore, the re-sulting reconstructed phase image corresponds exactlyto the phase distortion Ψ0pkq (Fig. 9.j). The phase im-age is then fitted by a quadratic function of the formΨ0pkx, kyq “ 4.69k2

x ` 5.04k2y ` 0.02 (Fig. 9.k). Result of

the fit is then used to construct and program the phase

11

r|A)

kx (pixel)0 1000 100 0 100

0

-100

100

ky (

pix

el)

N (Number of images)

SN

R (

log.

scal

e)

29 p

ixels

a

b e

0 100

Experiment

Theory

N=2.5e4 N=2.5e5 N=2.5e7

x107

c d

Figure 8. Signal-to-noise and spatial resolution. a,Values of single-to-noise ratio (SNR) in the retrieved phaseimage measured for different total number of images acquiredN (black crosses) together with a theoretical fit of the form:

SNR“ 0.074?N (blue dashed line). b, Phase image recon-

structed using N “ 2.5 ˆ 104 images. b, Phase image re-constructed using N “ 2.5 ˆ 105images. b, Phase image re-constructed using N “ 2.5 ˆ 107images. e, Reconstructedphase image of a radial resolution target (Siemens star with16 branches). Resulting spatial resolution is d “ 45˘ 3µm, ,which corresponds to approximately to 3 pixels

compensation pattern on Bob SLM (Fig. 9.l).

V. DETAILS ON QUANTUM HOLOGRAPHYWITHOUT POLARISATION ENTANGLEMENT

This section provides more details about quantumholography performed with photons entangled in spacebut not in polarisation.

Fig.2 of the manuscript shows results of quantum holo-graphic experiment performed with photons that are en-tangled in space but not in polarisation. In this experi-ment, the state generated at the output of the crystals isdefined by the density operators:

1

2

ÿ

k

“

|Hyk|Hy´kxH|kxH|´k ` |V yk|V y´kxV |kxV |´k

‰

(18)This state is composed of a balanced statistical mixtureof two pure states:

ř

k |Hyk|Hy´k andř

k |V yk|V y´k.As shown in Fig. 10.a, this state is produced by alter-nating the polarisation of the pump laser between ver-

tical and horizontal polarisations. Frames acquired bythe camera in each configuration are then summed to-gether (Fig. 10.b). Then, the intensity and intensitycorrelation images shown in Fig.2 of the manuscript arereconstructed using the set of summed frames. This ex-periment is equivalent of generating the photons with apump that is coherent in space and time but not in po-larisation (i.e. unpolarised). Due to the fundamentaltransfer of coherence that occurs between the pump andthe down-converted two-photon field in SPDC35–37, theproduced state is entangled in space and time, but notin polarisation.

VI. DETAILS ON THE CHARACTERISATIONOF THE DYNAMIC PHASE DISORDER

This section provides more details about the dynamicphase disorder and its characterisation.

Fig. 11.a shows the experimental setup used to char-acterise properties of the phase disorder introduce bythe presence of the diffuser, that is a plastic sleeve layerof thickness ă 100µm. Without diffuser, a sine-shapedphase image programmed on the SLM (Fig. 11.b) gen-erates a very specific diffraction pattern on the camera(Fig. 11.c). After introducing the static diffuser, thediffraction pattern becomes a speckle pattern (Fig. 11.c).The presence of the diffuser erases all information aboutthe phase image, that cannot be retrieved by classi-cal holographic techniques such as phase retrieval43 andphase-stepping holography14. When the diffuser is mov-ing, the speckle pattern takes the form of a diffuse haloif the exposure time of the camera is larger (0.5s) thanthe typical decorrelation time of the speckle. The halowidth is estimated to 1.3mm by Gaussian fitting whichcorresponds approximately to a diffusing angle of 1 de-gree and a surface roughness of 46µm. Moreover, the dy-namic properties of the disorder are estimated by mea-suring the speckle decorrelation time. Speckle correla-tion coefficients are calculated by acquiring a series ofspeckle patterns using short exposure times (3ms) andcorrelating them with a reference speckle image. Fig. 11shows the decrease of speckle correlation with time (blackcrosses). A typical decorrelation time of 183ms is mea-sured by fitting data with an exponential model (dashedblue curve).

12

kx (pixel)0 100 100 100 1000 0 0

1000

-100 0

0

-100

100

kx (pixel)

ky (

pix

el)

+π+0 +π/2 +3π/2

0 100kx (pixel)

0 100 0 100 0 100

0

2π

Phase (rad

)

-8

1

Phase (rad

)

kx (pixel)

0

-100

100

ky (

pix

el)

-8

Phase (rad

)

1

kx (pixels)0 100

Phas

e (r

ad)

kx (pixel)ky (pixel)

kFit akx

2+bky2+c

Data

0

-100

100

ky (

pix

el)

l

250

Inten

sity correlations

(arb.u

nits)

0

f jg h i

ba c d e

Figure 9. Phase distortion characterisation. a, Flat phase pattern programmed on Alice SLM. b-e, Phase-shifted patternsprogrammed on Bob SLM: `0 (b), `π{2 (c), `π (d) and `3π{2 (e). f-i, Intensity correlation images measured for each phasemask displayed on Bob SLM. j, Phase image retrieved by Bob. k, Fit of the phase image by a quadratic function of the form:Ψ0pkx, kyq “ 4.69k2x ` 5.04k2y ` 0.02. l, Correction phase pattern resulting from the fitting process.

13

Laser 405nm

Paired BBOs

V

+

Laser 405nm

Paired BBOs

H

a

-k

k

-k

k

b

0

-100

100

ky (

pix

el)

1000 0-100kx (pixel)

0

-100

100

ky (

pix

el)

1000 0-100kx (pixel)

+

29

2860Inten

sity (arb. u

nits)

Figure 10. Generation of a mixed state without polarisation entanglement. a, Experimental configurations usedto generate the mixed state:

ř

k

“

|Hyk|Hy´kxH|kxH|´k ` |V yk|V y´kxV |kxV |´k

‰

. Polarisation of pump laser is alternatedbetween vertical and horizontal. b, Frames acquired in each configuration are summed. Intensity images shown in Fig.2.b.c ofthe manuscript and intensity correlation images shown in Fig.2.e-f of the manuscript are reconstructed from the set of summedframes.

14

Motorized translation stageEMCCD

f3

Laser 810nm

SLM

Diffuser

f1f2

a

0

-100

100

y (p

ixel

)

-100 0 100x (pixel)

0

-0.2

0.2

ky (

pix

el-1

)

-0.2 0 0.2kx (pixel-1)

time (s)

Spec

kle

corr

elat

ion

0

-0.2

0.2

ky (

pix

el-1

)

-0.2 0 0.2kx (pixel-1)

0

-0.2

0.2

ky (

pix

el-1

)

-0.2 0 0.2kx (pixel-1)

b c

d e fExperimentTheory

Figure 11. Characterisation of the dynamic phase disorder. a, A classical laser (810nm ; beam diameter 0.8mm)illuminates an SLM that is imaged onto a the diffuser by two lenses f1 “ 150mm and f2 “ 75mm. Lens f3 “ 75mm Fourier-images the diffuser onto the camera. b, Sine-shaped phase pattern programmed on the SLM. c, Intensity image measured onthe camera without diffuser. d, Intensity image measured on the camera with the diffuser maintained static. e, Intensity imagemeasured on the camera with the moving diffuser using an exposure time of 0.5s. Width of the diffuse halo is estimated to1.3mm by fitting with a Gaussian function. f, Measurement of speckle correlation values with time (black crosses) together

with a theoretical fit of the form 1´ 0.77p1´ e´t{0.183q (blue dashed line). Typical decorrelation time equals to 183ms.

15

˚ Corresponding authors: hugo.defienne@glasgow.ac.uk ;daniele.faccio@glasgow.ac.uk.

1 P.-A. Moreau, E. Toninelli, T. Gregory, and M. J. Padgett,Nature Reviews Physics 1, 367 (2019), arXiv: 1908.03034.

2 A. G. White, J. R. Mitchell, O. Nairz, and P. G. Kwiat,Physical Review A 58, 605 (1998).

3 T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V.Sergienko, Physical Review A 52, R3429 (1995).

4 G. B. Lemos, V. Borish, G. D. Cole, S. Ramelow, R. Lap-kiewicz, and A. Zeilinger, Nature 512, 409 (2014).

5 G. Brida, M. Genovese, and I. R. Berchera, Nature Pho-tonics 4, 227 (2010).

6 T. Ono, R. Okamoto, and S. Takeuchi, Nature communi-cations 4, 2426 (2013).

7 R. Tenne, U. Rossman, B. Rephael, Y. Israel,A. Krupinski-Ptaszek, R. Lapkiewicz, Y. Silberberg, andD. Oron, Nature Photonics , 1 (2018).

8 A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato,Physical Review Letters 93, 093602 (2004).

9 J. H. Shapiro, D. Venkatraman, and F. N. C. Wong, Sci-entific Reports 5, 10329 (2015).

10 A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, andM. C. Teich, Physical Review Letters 87, 123602 (2001).

11 M. J. Padgett and R. W. Boyd, Philosophical Transactionsof the Royal Society A: Mathematical, Physical and Engi-neering Sciences 375, 20160233 (2017).

12 P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche,Y. Emery, T. Colomb, and C. Depeursinge, Optics Letters30, 468 (2005).

13 P. Refregier and B. Javidi, Optics Letters 20, 767 (1995).14 I. Yamaguchi and T. Zhang, Optics Letters 22, 1268

(1997).15 A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and

M. C. Teich, Optics Express 9, 498 (2001).16 S. Asban, K. E. Dorfman, and S. Mukamel, Proceedings

of the National Academy of Sciences 116, 11673 (2019).17 R. Chrapkiewicz, M. Jachura, K. Banaszek, and

W. Wasilewski, Nature Photonics 10, 576 (2016).18 F. Devaux, A. Mosset, F. Bassignot, and E. Lantz, Phys-

ical Review A 99, 033854 (2019).19 K. Wang, J. G. Titchener, S. S. Kruk, L. Xu, H.-P. Chung,

M. Parry, I. I. Kravchenko, Y.-H. Chen, A. S. Solntsev,Y. S. Kivshar, D. N. Neshev, and A. A. Sukhorukov, Sci-ence 361, 1104 (2018).

20 T. Stav, A. Faerman, E. Maguid, D. Oren, V. Kleiner,E. Hasman, and M. Segev, Science 361, 1101 (2018).

21 C. Altuzarra, A. Lyons, G. Yuan, C. Simpson, T. Roger,

J. S. Ben-Benjamin, and D. Faccio, Physical Review A99, 020101 (2019).

22 J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G.Kwiat, Physical Review Letters 95, 260501 (2005).

23 S. F. Hegazy and S. S. A. Obayya, JOSA B 32, 445 (2015).24 H. Defienne, M. Reichert, and J. W. Fleischer, Physical

review letters 120, 203604 (2018).25 J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W.

Boyd, Physical Review Letters 92, 210403 (2004).26 P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin,

Nature 409, 1014 (2001).27 H. Defienne, M. Reichert, J. W. Fleischer, and D. Faccio,

Science Advances 5, eaax0307 (2019).28 M. Malik, O. S. Magaa-Loaiza, and R. W. Boyd, Applied

Physics Letters 101, 241103 (2012).29 A. Ekert and P. L. Knight, American Journal of Physics

63, 415 (1995).30 J. Fang, P. Huang, and G. Zeng, Physical Review A 89,

022315 (2014).31 P.-A. Moreau, J. Mougin-Sisini, F. Devaux, and E. Lantz,

Physical Review A 86, 010101 (2012).32 D. S. Tasca, F. Izdebski, G. S. Buller, J. Leach, M. Agnew,

M. J. Padgett, M. P. Edgar, R. E. Warburton, and R. W.Boyd, Nature Communications 3, 984 (2012).

33 M. Reichert, H. Defienne, and J. W. Fleischer, PhysicalReview A 98, 013841 (2018).

34 R. C. Gonzalez and P. Wintz, Reading, Mass., Addison-Wesley Publishing Co., Inc.(Applied Mathematics andComputation , 451 (1977).

35 A. K. Jha and R. W. Boyd, Physical Review A 81, 013828(2010).

36 G. Kulkarni, V. Subrahmanyam, and A. K. Jha, PhysicalReview A 93, 063842 (2016).

37 G. Kulkarni, P. Kumar, and A. K. Jha, JOSA B 34, 1637(2017).

38 J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt,Physical Review Letters 23, 880 (1969).

39 M. V. Fedorov, Y. M. Mikhailova, and P. A. Volkov, Jour-nal of Physics B: Atomic, Molecular and Optical Physics42, 175503 (2009).

40 M. V. Fedorov, Physica Scripta 90, 074048 (2015).41 A. Aspect, P. Grangier, and G. Roger, Physical review

letters 49, 91 (1982).42 S. F. Hegazy, Y. A. Badr, and S. S. A. Obayya, Optical

Engineering 56, 026114 (2017).43 J. R. Fienup, Applied Optics 21, 2758 (1982).