Entropic Reciprocity in Time-Reversed Young Interferometry

The gist: time-reversed Young interferometry reorganizes, rather than reverses, optical entropy.

How it works

1. Standard Young interference measures a detector marginal distribution pstdθ(x) [Page 1]. This involves a point source illuminating two paths and the detector plane revealing the interference through a marginal distribution over the detector coordinate [Page 1].

2. Time-reversed Young interferometry exchanges operational roles: a laterally addressable source plane replaces the usual observation screen, while the detector is held fixed [Page 1]. The resulting interference is read as a source-label response conditioned on a detection event rather than a detector-plane fringe pattern [Page 1].

3. The two geometries are statistical reductions of the same reciprocal propagation kernel pθ(x, y) [Page 1]. Standard Young interference marginalizes over the source and reads out detector-space entropy, while time-reversed Young interference fixes the detector and reads out conditioned source-label entropy [Page 1].

Key Findings on Information Invariants

** The central result of this Letter is that the reciprocal invariant is not a marginal entropy, but the mutual information between source and detector coordinates [Page 1]. This mutual information Iθ(X; Y) measures the amount of source information contained in the detector coordinate, or equivalently the amount of detector information contained in the source label [Page 1]. The standard experiment naturally measures a detector entropy HX(θ), whereas the time-reversed experiment measures a conditioned source entropy HYx0(θ) [Page 2]. There is no general reciprocity principle requiring HX = HYx0, as they refer to different probability spaces: a detector marginal versus a source conditional [Page 2]. The reciprocal quantity is the mutual information Iθ(X; Y), which is symmetric under exchange of X and Y, measuring the amount of source information contained in the detector coordinate, or equivalently the amount of detector information contained in the source label [Page 2].**

Source-Space Information Processing

1. The time-reversed Young readout is interpreted as a source-space information processor with no analogue in ordinary detector-plane fringe readout [Page 1]. It reveals the complementary conditioning of the same joint optical information [Page 1].

2. For a uniform source prior over an allowed source support of measure Ay, Eq. (9) gives a particularly transparent entropy identity: ln Ay − HYx0 = DKL[pTRYθ(yx0)∥π(y)] [Page 4]. This shows that TRY converts a single detector event into source-space information gain [Page 4].

3. The TRY score function s TRYθ(yx0) is defined as ∂θ ln Rθ(y; x0) − ⟨∂θ ln Rθ⟩Yx0, which is forced by normalization and guarantees that ⟨s TRYθ⟩Yx0 = 0 [Page 11].

Fisher Information Enhancement

** The second central result is the TRY Fisher information F TRYθ(x0) = VarYx0[∂θ ln Rθ(y; x0)] [Page 3]. This proves that TRY sensitivity is controlled by the source-space variation of the logarithmic detector response [Page 11]. Sensitivity is enhanced when the source basis samples regions where the logarithmic response has large contrast, and source programming can therefore shape the score function directly [Page 3].**

Regularized Null-Response Analysis

1. Near a destructive response, the measured response is modeled as Rθ(y; x0) ≃ ϵ(y) + θq(y)2 + b, where b > 0 is a background floor [Page 11].

2. The useful regime is therefore a structured null: the nominal response is suppressed, while the perturbation response changes nonuniformly across the source labels [Page 11]. When this condition is satisfied, the logarithmic derivative ∂θ ln Rθ can become large and highly structured, leading to enhanced Fisher information per detected event [Page 11].

3. The enhancement implied by Eq. (12) requires more than a small denominator; if q(y) is locally proportional to ϵ(y) over the accepted source region, then ∂θ ln Rθ is approximately common-mode and the variance in Eq. (S27) remains small [Page 11].

Coherence and Thermal Fields

** The coherence entropy Scoh = −Tr(ρcoh ln ρcoh) is unchanged under ideal lossless propagation because the eigenvalues λn are preserved by unitary similarity transformation [Page 12]. Entropy changes in a realistic TRY experiment should be attributed to filtering, coarse graining, conditioning, loss, or postselection rather than a reversal of thermodynamic time [Page 12].**

The results establish that marginal entropies are not reciprocal invariants; mutual information is the invariant connecting standard and time-reversed geometries [Page 13]. The operational advantage of TRY lies in its ability to sculpt conditioned source-label entropy and Fisher information using a fixed detector and a programmable source basis [Page 13]. This framework clarifies why a dark response can be useful without implying unbounded information, as conditioning can enhance the Fisher information per detected event while the full photon budget remains regularized by background, finite aperture, and detection probability [Page 12]. The results provide a compact theoretical basis for source-space null sensing, reciprocal autofocus, defocus estimation, wavelength discrimination, and programmable fixed-detector interferometric metrology [Page 12].

reference:Binghamton University through Startup funds and Watson College through the internal award (No. 1201479)

1. Source-Space Information Processing Capability: The improved AI system can function as a source-space information processor by utilizing time-reversed Young interferometry to reorganize, rather than reverse, optical entropy. This allows the system to condition the source label distribution based on a fixed detector, enabling source-space conditioning which is not an analogue in ordinary detector-plane fringe readout.

2. Reciprocal Invariant Measurement: The system can calculate the reciprocal invariant, which is defined as the mutual information between source and detector coordinates, instead of relying on marginal entropies. This provides a more fundamental measure of optical reciprocity, as marginal entropies in the standard and time-reversed geometries are generally unequal.

3. Programmable Statistical Sensitivity: The AI system can engineer measurement sensitivity by controlling the score function, defined as s TRY θ (yx0) = ∂θ ln Rθ(y; x0) − ⟨∂θ ln Rθ⟩Y x0, which is forced by the normalization of the conditional distribution. This enables source programming to shape the measurement, as sensitivity is enhanced when the source basis samples regions where the logarithmic response has large contrast.

4. Dark Response Signal Enhancement: The system can exploit structured nulls near a destructive response to achieve enhanced Fisher information per detected event. Specifically, when the nominal response is suppressed, while the perturbation response changes nonuniformly across the source labels, it can achieve enhanced Fisher information per detected event through the mechanism described by Equation (S38).

5. Robust Null Sensing: The AI system can perform source-space null sensing where a null operation redistributes information into the conditional source-label record, with physical regularization ensuring the result is finite. This allows for reciprocal autofocus, defocus estimation, wavelength discrimination, and programmable fixed-detector interferometric metrology.

We show that time-reversed Young interferometry reorganizes, rather than reverses, optical entropy. A fixed detector conditions the reciprocal source--detector Green function and produces a source-label probability distribution. Marginal entropies in the standard and time-reversed geometries are generally unequal; the reciprocal invariant is instead the mutual information between source and detector coordinates. Near a destructive response, the conditioned source-label entropy can decrease while Fisher information for small phase, tilt, or defocus perturbations increases. The result identifies time-reversed Young interferometry as a source-space information processor with no analogue in ordinary detector-plane fringe readout.