Multi-Detector Quantum Interference

• Multi-detector quantum interference is the study of interference among multiple quantum paths via correlated detection events, revealing key features of quantum coherence and path indistinguishability.
• Experimental setups such as multi-slit interferometers, linear optical networks, and electronic Mach–Zehnder interferometers enable precise measurements of interference and coherence parameters.
• Quantitative metrics like the l₁-norm coherence and Sorkin parameter allow rigorous testing of quantum theory and support applications in quantum certification and secure information protocols.

Multi-detector quantum interference encompasses the study and utilization of interference phenomena between multiple quantum paths, as resolved or revealed by multiple, often correlated, detection events in interferometric, linear optical, or other multi-output quantum systems. This subject connects fundamental questions around complementarity, quantum coherence, the nature and limitations of interference order, experimental certification of quantum superpositions, and protocols in quantum information and technology relying on multi-detector readouts. Multi-detector setups—ranging from generalized multi-slit interferometry to high-order coincidence detection in optical networks—not only probe the foundations of quantum mechanics, such as the exclusivity of pairwise interference implied by Born’s rule, but also serve as essential tools for assessing indistinguishability and coherence in practical systems including quantum networks and photonic quantum computers.

1. Theoretical Foundations: Quantum Coherence and High-Order Interference

The mathematical description of multi-detector quantum interference begins with the density matrix formalism in a path basis $\{|\psi_i\rangle\}$. The quantum coherence of a system traversing $n$ distinguishable paths is quantified by

$C = \frac{1}{n-1} \sum_{i\ne j} |\rho_{ij}|,$

where $\rho_{ij} = \langle\psi_i|\rho|\psi_j\rangle$ are the off-diagonal elements of the quanton’s density matrix. $C$ serves as a normalized measure of wave-like character, attaining $C=1$ for maximally coherent superpositions and $C=0$ for fully incoherent mixtures (Paul et al., 2017).

When path-detecting modules ("which-path" detectors) are included, the overall system evolves as

$$|\Psi\rangle_{QD} = \sum_{i=1}^n c_i |\psi_i\rangle \otimes |d_i\rangle,$$

and tracing out the detector degrees of freedom yields a reduced quanton state where the coherence is suppressed by detector state overlaps $|\langle d_j|d_i\rangle|$. Thus, the ability to distinguish paths directly degrades multi-path quantum interference as observed in the off-diagonal $\rho_{ij}$ and in $C$.

Standard quantum mechanics, as enforced by Born's rule, only permits pairwise (second-order) interference; the two-path interference term is $I_{ij}=P_{ij} - P_i - P_j$, with $P_{ij}$ the joint probability for two paths open. Higher-order or multi-path-only terms, such as the Sorkin parameter for third-order interference,

$$I_3 = P_{123} - (P_{12}+P_{13}+P_{23}) + (P_1+P_2+P_3) - P_0,$$

are strictly zero. Experiments with multi-detector readout confirm that the third- and higher-order Sorkin parameters are bounded below $10^{-2}$ of the expected two-path interference, constraining or ruling out non-quantum generalizations (Sinha et al., 2010).

2. Experimental Arrangements: Multi-Slit and Linear Optical Architectures

Multi-detector quantum interference is realized in a variety of experimental modalities:

• Multi-slit interferometry: Each slit can be equipped with which-path modules, such as polarization tags or sub-interferometers, preparing detector states $|d_i\rangle$ and permitting simultaneous recording of both interference fringe intensities ($I(x)$) and detector click statistics. This dual readout allows measurement of the coherence parameter $C$ and direct tests of duality relations such as $D_Q + C \leq 1$, where $D_Q$ captures path distinguishability through detector outcomes (Paul et al., 2017).
• Linear optical networks: High-order interference is observed by sending $N$-photon Fock states or coherent state approximations through cascades of $50/50$ beam splitters and monitoring multi-fold coincidences across detector banks at multiple output ports (Gard et al., 2011, Navarrete et al., 2017). For instance, the generalized Hong–Ou–Mandel setup extends two-photon coalescence to multi-photon, multi-output scenarios with "tritters" and beyond.
• Electronic Mach–Zehnder interferometers: Multi-terminal configurations with additional quantum point contact (QPC) detectors allow for joint readout of the interference current and weakly coupled which-path observables. The cross-correlation of three detector signals forms a robust, parameter-free quantum interference witness that identically vanishes for all classical models, revealing underlying quantum coherence (Esin et al., 2019).

A summary of exemplary systems is presented below:

Platform	N (paths/photons)	Readout	Key Observable
Multi-slit interferometer	2 – 6	Screen + path detectors	$C$, $V$, $D$, Sorkin $I_n$
Beam splitter network	2 – 16+	Bank of output detectors	$G^{(2)}, G^{(3)}, \ldots$
Electronic MZI+QPC	2	3 current readouts	3-point correlation, WWS signal

3. Quantifying and Measuring Coherence in Multi-Detector Scenarios

Classical fringe visibility $V = (I_{max}-I_{min})/(I_{max}+I_{min})$ becomes inadequate in multi-path scenarios because it may be non-monotonic under selective decoherence or path-dependent phase shifts. In contrast, the $l_1$-norm coherence $\mathcal{C}$, as defined above, maintains a rigorous correspondence with quantum coherence. For arbitrary $n$,

$\mathcal{C} = \frac{2\beta}{n} + \frac{n-2}{n},$

where $\beta$ encodes the distinguishability of one path with respect to the rest. $\mathcal{C}$ decreases monotonically as decoherence or path distinguishability increases, in contrast to $V$, which can paradoxically increase when one path acquires a $\pi$ phase and decoherence selectively (Mishra et al., 2019).

Operationally, $C$ or $\mathcal{C}$ can be extracted experimentally either directly via "coherent vs. incoherent" intensity measurements or via pairwise visibility protocols—blocking all but two slits at a time, measuring associated visibilities, and reconstructing $\mathcal{C}$ via averaging:

$\mathcal{C} = \frac{2}{n(n-1)} \sum_{i<j} v_{ij}$

(Paul et al., 2017, Mishra et al., 2019).

4. Multi-Photon and High-Order Coincidence Detection

Multi-detector setups facilitate the exploration of true $n$-photon quantum interference. In quantum random walks on cascaded beam splitter arrays, one-, two-, and three-fold coincidence detection schemes access increasing orders of joint quantum correlations, denoted by

$$P^{(n)}(m_1,\dots,m_n) = \left\langle a_{m_1}^\dagger \cdots a_{m_n}^\dagger a_{m_n} \cdots a_{m_1} \right\rangle,$$

where $a_m$ are output-mode annihilation operators (Gard et al., 2011).

In practical characterization, threshold single-photon detectors and decoy-state analysis enable tight estimation of multi-photon event probabilities in the presence of imperfect sources and detectors:

• By varying mean photon number (input decoys) and detection efficiency (detector decoys), one builds a linear system to bound the true $P(\mathbf{x}|\mathbf{n})$—the conditional multi-photon event rates—using only “click/no-click” data and linear programming. This renders feasible the measurement of generalized coincidence probability (e.g. the three-photon coincidence in a tritter) and visibility even with imperfect hardware (Navarrete et al., 2017).

5. Quantum Certification and Security via Multi-Detector Protocols

Multi-detector interference provides a robust witness of quantum coherence fundamentally inaccessible to classical models. For example, three-point cross-correlations in triple-detector architectures cannot be mimicked by any classical wave or particle process: the "weak–weak–strong" (WWS) protocol based on joint cross-correlators yields a signal strictly zero classically and generically nonzero for quantum interference, providing a quantum "certificate" even with the interferometer held at a fixed working point (Esin et al., 2019). This property underpins practical security in cryptographic protocols, establishing measurement-device-independent QKD immune to detector attacks, as multi-detector signatures correlated with quantum interference cannot be replicated by any side-channel manipulation (Rubenok et al., 2013).

6. Implications for Quantum Information, Metrology, and Foundational Tests

Multi-detector quantum interference is crucial for:

• Quantum information processing: Many schemes in photonic quantum computation, BosonSampling, and entanglement distribution rely on the precise characterization and exploitation of high-order interference between indistinguishable particles, accessible only through complex multi-detector measurements (Navarrete et al., 2017, Gard et al., 2011).
• Quantum cryptography: Bell-state measurement protocols for entanglement swapping and networked key distribution fundamentally depend on the correct operation and monitoring of two or more detector coincidences to ensure both functionality and security (Rubenok et al., 2013).
• Foundational experiments: Ruling out non-pairwise interference to high precision provides experimental constraints on potential extensions to quantum theory and on the possible existence of "super-quantum" resources (Sinha et al., 2010).

Experimental constraints on higher-order Sorkin parameters, tests of multi-path coherence decay under decoherence, and the precision measurement of $C$ and $\mathcal{C}$ offer tools for probing the boundaries and consistency of quantum mechanics.

7. Practical Considerations and Extensions

Accurate multi-detector quantum interference measurements demand meticulous control:

• Phase stability: Path-length and phase fluctuations directly affect observed coherence and must be actively compensated.
• Detector calibration: Non-uniform efficiencies, dark counts, and finite response times (including convolution of detector response functions) can bias measured multi-photon probabilities and visibilities. Calibration and correction protocols are essential (Paul et al., 2017, Flagg et al., 2010).
• Data analysis: Multi-detector readouts require statistical techniques to address Poissonian counting fluctuations, systematic drift, and truncation errors in decoy-based inversion (Navarrete et al., 2017).
• Scalability: As photon number or detector complexity increases, the combinatorial growth of possible multi-photon paths necessitates computational methods such as matrix permanents and Feynman-diagrammatic expansions (Gard et al., 2011).

Multi-detector approaches generalize to higher-order quantum walks, number-resolving detection, and are applicable across platforms including photons, electrons (edge-state interferometry), and cold atoms. These advances enable refined studies of decoherence, path indistinguishability, and the practical exploitation of quantum coherence as a technological resource.