Quantum image distillation

Images composed of both quantum and classical light can be distilled so as to separate the quantum from the classical information.


THEORY
This section provides a brief overview of the theory that underlies our image processing technique. A complete description can be found in [27].
We consider the case of a camera illuminated by a source of spatially-entangled photon-pairs, similar to the one shown in Figure 1. Photon-pairs are described by a two-photon wavefunction φpr 1 , r 2 q, where r 1 and r 2 are camera pixels positions. At each camera pixel, photons are converted into intensity values in two steps: 1. Photons are transformed into photo-electrons by a photo-sensitive screen of quantum efficiency η 2. Photo-electrons are transformed into intensity values I k by an amplification register. For k photoelectrons at the input of the register, the camera returns an average grey value that is proportional to k: I k Ak x 0 , where x 0 is an electronic noise mean value and A is an amplification parameter.
The camera acquires a set of N images tI l u lrr1,N ss using a fixed exposure time. xIprqy is defined as the mean intensity value measured at pixel r: xIpr 1 qIpr 2 qy is defined as the mean intensity product value measured between pixels r 1 and r 2 : The theoretical analysis is performed under the following assumptions: i. Pump laser is operating above threshold to ensure a Poisson distribution of pump photons ii. Pump laser power is low enough to ensure that ¡ 2 photons generation process in the crystal are negligible iii. Coherence time of photon-pairs is much smaller than the time between two successive images iv. Cross-talk between pixels is negligible Following the reasoning detailed in the Appendix E of the supplementary document of [27], xIprqy and xIpr 1 qIpr 2 qy are written in function of the camera parameters (x 0 and A) and the joint probability distribution of photon-pairs |φpr 1 , r 2 q| 2 . On the one hand, xIprqy is written as wherem is the mean photon-pair rate and P m prq ³ |φpr, r I q| 2 dr I is the probability of detecting a photon at pixel r (i.e. marginal probability). On the other hand, for r 1 $ r 2 , xIpr 1 qIpr 2 qy is written as Finally, the joint probability distribution |φpr 1 , r 2 q| 2 can be written as |φpr 1 , r 2 q| 2 xIpr 1 qIpr 2 qy ¡ xIpr 1 qyxIpr 2 qy While it is commonly thought that photon counting is necessary to compute the joint probability distribution, this result shows that simple operation of a camera without threshold also enables its measurement.
However, this result is only valid for r 1 $ r 2 . As described in Appendix H of the supplementary document of [27], xIprq 2 y can be written as: where σ 0 is the standard deviation of the camera electronic noise. As a result, xIprq 2 y $ xIprqy 2 and equation (4) is not valid for r 1 r 2 . In our experiment, Γpr, rq is estimated using the approximation Γpr, rq Γpr, r δrq, where δr ¡δ e x with δ 16µm (pixel width) and e x is an unit vector. .

MEASUREMENT OF Γpr rq
In our experiment, the camera is an EMCCD Andor . is set to 6ms. All assumptions enumerated in section I are verified: pump laser operates above threshold with a power of 50mW [(i) and (ii)], coherent time of the pairs ( 1ps) is much smaller than the time between two successive frames ( 4ms) (iii) and cross-talk between pixels is negligible (iv). In the following, we describe step-by-step the technique to measure Γpr, rq: 1. Acquisition of a set of N images tI l u lrr1,N ss at fixed exposure time τ 6ms, with N on the order of 10 6¡7 .
2. Estimation of the first term of equation (2)  3. Estimation of the second term of equation (2) by multiplying pixel values in the l th image by those of the following image l 1 th and average over the set: xIpr 1 qyxIpr 2 qy 1 N 2 Ņ l1 I l pr 1 qI l 1 pr 2 q By definition, xIpr 1 qyxIpr 2 qy equals the limit N Ñ V for the following summation: I l pr 1 qI l pr 2 q 1 N 2 Ņ l$l 1 I l pr 1 qI l 1 pr 2 q (8) The first term in equation 8 can be written as: because by definition of xIpr 1 qIpr 2 qy, the serie 1 N°N l1 I l pr 1 qI l pr 2 q. The second term in equation 8 is an estimation of the mean value of intensity product between different frames xI l pr 1 qI l$l 1 pr 2 qy. Because the probability for two photons of the same pair to be detected in two different frames is null (coherent time much smaller than camera readout time), intensity values in different frames are independent with each other.
In consequence, xI l pr 1 qI l$l 1 pr 2 qy can be estimated using only successive frames by calculating the sum 1 N 2°N l1 I l pr 1 qI l 1 pr 2 q. Experimentally, the use of successive frames to estimate xIpr 1 qyxIpr 2 q rather than the complete set has the advantage of reducing artifacts as spatial distortions in the measured Γ, mainly due to fluctuations of the amplification gain of the camera [27].
4. Subtraction between these two terms: Γpr 1 , r 2 q 1 N Ņ l1 I l pr 1 qI l pr 2 q ¡ 1 N 2 Ņ l$l 1 I l pr 1 qI l 1 pr 2 q (10) 5. As shown in Section I, equation 5 is only valid for r 1 $ r 2 . Estimation of the intensity correlation values Γpr, rq from those measured between pixel r px, yq is then performed using neighbouring pixels r I px ¡ δ, yq [δ 16µm pixel size]: Γpr, rq Γppx, yq, px ¡ δ, yqq In our experiment, this approximation is valid because the fill factor of the Andor Ixon Ultra is near 100% and the position correlation width on the camera is estimated from the thickness of the crystal to be σ r 10µm [28].
. PROJECTIONS OF Γ Γpr 1 , r 2 q Γppx 1 , y 1 q px 2 , y 2 qq is a 4-dimensional matrix. Its information content can be visualized using two types of projections: 1. Conditional projection relative to an arbitrarily chosen position r I , defined as Γpr|r I q Γpr, r I q°r 1 Γpr, r I q It represents the probability of detecting a photon from a pair at position r under the condition that another photon is detected at r I .
2. The minus-coordinate projection, defined as P Γ ¡ pr ¡ q ŗ Γpr ¡ r, rq It represents the probability for two photons of a pair to be detected in coincidence between pairs of pixels separated by an oriented distance r ¡ .

CTION S SE
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