Visibility Correlation in Interferometry

• Visibility correlation in interferometry is a quantitative measure linking fringe contrast with coherence, quantum entanglement, and systematic error influences.
• It integrates methodologies from quantum information and statistical optics to extract key parameters such as mutual information, quantum discord, and covariance structures.
• Empirical covariance analysis and analytic fitting techniques enable precise diagnosis of decoherence mechanisms and noise impacts in varied interferometric setups.

Visibility correlation in interferometry quantifies the statistical and physical relationships between the interference contrast observed in an interferometric signal and underlying parameters such as coherence, quantum and classical correlations, decoherence mechanisms, and noise sources. These correlations, while often appearing as direct relationships between measured visibilities across baselines, spectral channels, time, or modalities, encode key physical quantities including entanglement, quantum discord, mutual information, spatial or temporal coherence, and the effect of systematic or stochastic errors. The modern framework integrates concepts from quantum information, statistical optics, and signal processing, and is foundational in optical, radio, and quantum interferometry.

1. Visibility and Correlation in Quantum and Classical Interferometry

Fringe visibility $V$ is the primary operational measure of interference in two-path (Mach–Zehnder, double-slit), multi-path, and multiphoton interferometers. Formally, for output probability $p(\phi)$ as a function of phase $\phi$, the standard visibility is

$$V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}},$$

where $I_{\max}$, $I_{\min}$ are the extremal output intensities or detection probabilities. In the canonical Mach–Zehnder configuration, with an arbitrary input qubit (Bloch vector $S$) and an asymmetric exit beamsplitter (implemented via the unitary $U_B(\beta)$), the visibility takes the explicit form

$$V = \frac{A\,\sin\beta\,\sqrt{S_y^2 + S_z^2}}{1 + S_x \cos\beta},$$

with $A = |\operatorname{Tr}_D(U \rho^D_{\mathrm{in}})|$ denoting detector-induced coherence reduction, and the denominator encoding suppression from input imbalance and BS2 asymmetry. Full visibility ($p(\phi)$0) is achieved only for specific choices: pure detector state ($p(\phi)$1), $p(\phi)$2 and $p(\phi)$3 tuned for ideal interference, and maximal path superposition (Liu et al., 2022). The measured visibility is thus directly sensitive to both physical setup—beamsplitter ratios, phase shifters—and quantum properties of the input and detector states.

In intensity interferometry, the squared modulus of the complex coherence $p(\phi)$4 between spatial points, given by the van Cittert–Zernike theorem, is accessed via the second-order correlation function $p(\phi)$5:

$p(\phi)$6

where $p(\phi)$7 is the fringe visibility at baseline $p(\phi)$8 (Guerin et al., 2018). This relation establishes a central correspondence between experimentally measured intensity correlations and the spatial coherence structure of the source.

2. Information-Theoretic and Quantum Correlation Measures

Visibility correlation connects closely to classical and quantum information-theoretic quantities. For the joint output state $p(\phi)$9 of a particle and which-path detector in an interferometer, key measures are:

• Quantum mutual information (total correlation):

$\phi$0

where $\phi$1 is the von Neumann entropy.

• Classical correlation (CC):

$\phi$2

optimized over all detector measurements.

• Quantum discord (QD):

$\phi$3

These measures, with closed-form expressions in the pure and environment-mixed output states, encode the degree of classical and quantum correlations between the particle and detector. In pure-state output, $\phi$4, and the complementarity

$\phi$5

reflects a rigorous tradeoff between accessible visibility and correlation—maximal visible interference precludes detector-path correlation, and vice versa (Liu et al., 2022).

Quantum optical networks extend this to multipath and multiparticle settings, where the $\phi$6th-order Glauber correlation function $\phi$7 and associated multiphoton visibilities encapsulate high-order quantum coherence, distinguishability, and entanglement (Tamma et al., 2014, Biswas et al., 2017).

3. Visibility Correlation Functions, Covariances, and Systematic Error Modeling

Beyond scalar visibility, real-world and multipath setups feature ensembles of visibilities indexed by baseline, time, frequency, or control parameter. Their statistical correlations—empirical covariance matrices and higher-order moments—are crucial for model fitting, detection thresholds, and error propagation.

For example, in O/IR long-baseline interferometry, measured (squared) visibilities across spectral channels and baselines are affected by both statistical noise and systematic, often multiplicative, correlated errors (e.g., fluctuating transfer function, P2VM coefficients). The full empirical covariance includes terms such as

$\phi$8

where $\phi$9 is uncorrelated noise and $$V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}},$$0 describes common-mode systematics (Kammerer et al., 2020, Lachaume, 2021). Proper incorporation of this structure in least-squares or Bayesian fits improves sensitivity (up to a factor of 2 in real data) and suppresses false positives by orders of magnitude, while naive, diagonal error models result in underestimated uncertainties and biased parameter recovery.

Visibilities and their covariances may also encode physically meaningful correlations beyond noise statistics—pairwise or higher-order visibility-correlation (covariance) functions can serve as operational coherence monotones, directly related to the probe state's off-diagonal density matrix elements (Biswas et al., 2017).

4. Decoherence, Noise, and Environmental Correlations

Visibility correlation analysis is integral to diagnosing and differentiating decoherence mechanisms. In multiphoton or double-Fock superposition interferometry, visibility as a function of decoherence parameters (distinguishability $$V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}},$$1, mixing $$V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}},$$2, dephasing $$V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}},$$3) enables discrimination of the underlying physical process via analytic relationships among experimental visibilities:

• Single-photon: $$V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}},$$4
• N00N: $$V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}},$$5
• Hong-Ou-Mandel (HOM) (two photons): $$V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}},$$6

By extracting multiple visibilities $$V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}},$$7 one can invert to obtain the quantities $$V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}},$$8, and so attribute loss of coherence to a specific source (Tichy et al., 2014). Multiparameter fits to visibility landscapes thus realize a full "differential diagnostic" of interferometric decoherence.

In the presence of noise, e.g., thermal backgrounds in induced-coherence interferometry, visibility is suppressed by incoherent pedestals. For instance, in a ZWM setup with mean background photons $$V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}},$$9, the singles visibility is

$I_{\max}$0

with thermal terms appearing additively in the denominator. Passive (attenuation, additional crystals) or active (heralded detection) schemes restore visibilities up to theoretical limits by eliminating or projecting out noise contributions (Theerthagiri et al., 5 Nov 2025).

5. Spatial, Spectral, and Temporal Correlations of Visibility Functions

Visibility correlation extends to spatiotemporal and spectral domains in both classical and quantum interferometry. In radio VLBI, correlations of visibility functions in the delay domain are diagnostic of interstellar medium scattering—central unresolved spikes, correlated across baselines within a single diffraction spot, encode spatial coherence lengths, while envelope fitting (single or double Lorentzian) distinguishes isotropic versus anisotropic scattering regimes (Popov et al., 2019). The spatial correlation coefficient $I_{\max}$1 of these spikes drops precipitously for baselines exceeding the diffraction scale.

In quantum interferometry, the joint spectral and temporal structure of photon pairs or higher-order multiplets yields visibility functions whose correlations depend on the underlying joint spectral amplitude (JSA), joint temporal intensity (JTI), and applied spectral phases. Conjugate-Franson interferometry enables direct measurement of phase-sensitive temporal correlations via the visibility

$I_{\max}$2

which is uniquely sensitive to the biphoton spectral phase, whereas Franson or HOM visibilities are insensitive (Chen et al., 2021).

In multi-path networks, the permanents of overlap matrices determine entire "landscapes" of $I_{\max}$3-photon visibility, with cross-correlations between fringes in different parameter directions or configurations encoding high-order coherence and indistinguishability (Tamma et al., 2014).

6. Operational Role: Coherence Quantification and Model-Invariant Physical Measures

Optimized visibilities and their correlation functions, when maximized over measurement basis, serve as coherence monotones—functionals that satisfy the axioms of a resource theory of coherence (strong monotonicity under strictly incoherent operations, convexity). For instance, in $I_{\max}$4-path interferometry, the supremum of the maximal fringe visibility is the $I_{\max}$5 norm of coherence, while covariances of pairwise visibilities correspond to products of off-diagonal elements. These monotones are empirically estimable and provide lower bounds on standard measures such as robustness and relative entropy of coherence (Biswas et al., 2017).

This framework applies across architectures, from spatial to spectral and multiphoton entanglement, ensuring that visibility correlation is not merely a noise model but a fundamental operational quantifier of coherence, correlation, and interference phenomena in diverse quantum and classical systems.

---

References

• "Fringe visibility and correlation in Mach-Zehnder interferometer with an asymmetric beam splitter" (Liu et al., 2022)
• "Heralded Induced-Coherence Interferometry in a Noisy Environment" (Theerthagiri et al., 5 Nov 2025)
• "Experimental demonstration of conjugate-Franson interferometry" (Chen et al., 2021)
• "Increasing the achievable contrast of infrared interferometry with an error correlation model" (Kammerer et al., 2020)
• "Spatial intensity interferometry on three bright stars" (Guerin et al., 2018)
• "Validating the Angular Sizes of Red Clump Stars with Intensity Interferometry" (Kim et al., 2 Feb 2026)
• "Probing Quantum Correlation Functions Through Energy Absorption Interferometry" (Withington et al., 2016)
• "Coherence and visibility in Fano interferometers" (Lauble et al., 2018)
• "Multiboson Correlation Interferometry with arbitrary single-photon pure states" (Tamma et al., 2014)
• "Bias-free model fitting of correlated data in interferometry" (Lachaume, 2021)
• "Substructure of visibility functions from scattered radio emission of pulsars through space VLBI" (Popov et al., 2019)
• "Closure statistics in interferometric data" (Blackburn et al., 2019)
• "Radius measurement in binary stars: simulations of intensity interferometry" (Rai et al., 2021)
• "Double-Fock Superposition Interferometry for Differential Diagnosis of Decoherence" (Tichy et al., 2014)
• "Interferometric visibility and coherence" (Biswas et al., 2017)