Sorkin Test: Quantum Interference Hierarchy

• Sorkin test is a null test that isolates third-order interference by subtracting single-path and pairwise contributions to confirm the quadratic probability rule of standard quantum mechanics.
• It employs various methodologies in experimental settings, including triple-slit, energy-space, and quantum-computing protocols, ensuring precise measurement of interference terms.
• The test emphasizes careful control of noise and systematic errors, with implementations in helium photoionization and qubit-based simulations serving as deep-quantum benchmarks.

The Sorkin test is, in the Born-rule literature, a hierarchy test for quantum interference: it asks whether a three-path experiment contains an irreducible third-order interference term after all single-path and pairwise contributions have been removed. Standard quantum mechanics predicts that this term vanishes identically, because probabilities are quadratic in amplitudes and therefore generate only pairwise cross terms. The same label, however, is not uniform across all Sorkin-associated literatures. In causal set theory it can be confused with the Sorkin–Johnston construction, where the relevant object is not an interference diagnostic but a vacuum-selection criterion; in Lorentzian topology change it can refer to Borde–Sorkin criteria for causal continuity; and in measurement theory for quantum field theory it can denote a causality-adequacy test based on Sorkin’s “impossible measurements” argument (Förderer et al., 16 Jan 2025, Jones, 2024, Papageorgiou et al., 2023).

1. Core meaning in the interference hierarchy

In its standard meaning, the Sorkin test probes Born’s rule by isolating interference of order higher than two. For two alternatives $a,b$, standard quantum mechanics gives

$P_{ab}=|A_a+A_b|^2=P_a+P_b+I_{ab},$

with $P_j=|A_j|^2$ and $I_{ab}=A_a^*A_b+A_aA_b^*$. For three alternatives $a,b,c$,

$$P_{abc}=|A_a+A_b+A_c|^2=P_a+P_b+P_c+I_{ab}+I_{ac}+I_{bc},$$

so there is no genuinely new three-way term. Sorkin’s third-order quantity is therefore defined by

$I_{abc}=P_{abc}-P_{ab}-P_{ac}-P_{bc}+P_a+P_b+P_c,$

and Born’s rule requires $I_{abc}=0$ (Förderer et al., 16 Jan 2025).

A closely related projector-based formulation appears in quantum-computing implementations. There one encodes three orthogonal “paths” as three basis states and defines probabilities $p_{123},p_{12},p_{13},p_{23},p_1,p_2,p_3$ by projection onto equal superpositions and single-path states. The corresponding Sorkin combination is

$$\kappa_3=3p_{123}-2(p_{12}+p_{23}+p_{13})+p_1+p_2+p_3,$$

which is identically zero in standard quantum mechanics. The identity expresses the same structural claim: the three-path pattern is fully reconstructible from one-path data and pairwise interference alone (Sadana et al., 2021).

The physical content is therefore a null test. A result consistent with $P_{ab}=|A_a+A_b|^2=P_a+P_b+I_{ab},$0 or $P_{ab}=|A_a+A_b|^2=P_a+P_b+I_{ab},$1 confirms the ordinary quadratic probability rule. A statistically significant nonzero value, once ordinary systematics are excluded, would indicate genuine third-order interference and hence a deviation from the standard Born-rule structure (Förderer et al., 16 Jan 2025).

2. Operational forms and normalization conventions

The operational content of a Sorkin test depends on how “paths” are realized. In slit-like or multipath settings one measures the full family of configurations with paths open or closed. In the helium photoionization proposal the required data are the eight probabilities

$P_{ab}=|A_a+A_b|^2=P_a+P_b+I_{ab},$2

where $P_{ab}=|A_a+A_b|^2=P_a+P_b+I_{ab},$3 is the all-paths-closed background signal. Background subtraction is implemented by

$P_{ab}=|A_a+A_b|^2=P_a+P_b+I_{ab},$4

This yields corrected second- and third-order interference terms

$P_{ab}=|A_a+A_b|^2=P_a+P_b+I_{ab},$5

The manuscript’s printed normalized parameter $P_{ab}=|A_a+A_b|^2=P_a+P_b+I_{ab},$6 has a garbled denominator, but the intended meaning is a dimensionless ratio in which the residual third-order term is normalized by the ordinary two-path interference scale (Förderer et al., 16 Jan 2025).

In the quantum-computing formulation, by contrast, one does not literally open and block slits. “Opening” a subset of paths is represented by choosing the corresponding measurement projector. The three-path term comes from projection onto an equal superposition such as $P_{ab}=|A_a+A_b|^2=P_a+P_b+I_{ab},$7; pairwise terms come from equal two-path superpositions; single-path probabilities come from computational-basis counts. The paths must be encoded as orthogonal states, because otherwise the simple amplitude decomposition is no longer exact (Sadana et al., 2021).

These conventions are mathematically equivalent only after attention to normalization, background, and encoding. The essential invariant content is the same inclusion–exclusion structure: remove all one-path and pairwise contributions and test whether a residual third-order term survives.

3. Experimental and algorithmic realizations

A recent implementation proposal uses laser-assisted attosecond photoionization of helium as a controllable three-path interferometer in energy space. The atom starts in the helium ground state $P_{ab}=|A_a+A_b|^2=P_a+P_b+I_{ab},$8, a broadband XUV pulse creates a continuum wave packet, and three narrow infrared components $P_{ab}=|A_a+A_b|^2=P_a+P_b+I_{ab},$9 complete three alternative two-photon routes to the same final photoelectron energy $P_j=|A_j|^2$0: $P_j=|A_j|^2$1 The “paths” are therefore not spatial slits but spectrally distinct ionization routes. By turning individual IR components on or off, one measures all eight required configurations. The proposal uses an angle-integrated photoelectron spectrum sampled over $P_j=|A_j|^2$2 energies in a window $P_j=|A_j|^2$3, and a Monte Carlo simulation with amplitude noise, timing jitter, dark counts, finite efficiency, and Poisson counting predicts an averaged Sorkin precision at the $P_j=|A_j|^2$4 level, summarized as

$P_j=|A_j|^2$5

which is fully consistent with zero and “at the same level as most Sorkin tests to date” (Förderer et al., 16 Jan 2025).

Quantum computers provide a different realization. In the binary-encoding scheme of the Rigetti experiment, three paths are encoded in a two-qubit register, random qutrit-like states are prepared by a unitary $P_j=|A_j|^2$6, and the required probabilities are extracted by applying inverse basis-change circuits before computational-basis measurement. The implementation uses $P_j=|A_j|^2$7 qubits for the three-path case, $P_j=|A_j|^2$8 shots per circuit, and bootstrap-based confidence intervals. The reported $P_j=|A_j|^2$9 values are statistically compatible with zero, as expected from standard quantum mechanics (Sadana et al., 2021).

These platforms illustrate that a Sorkin test is not tied to literal triple-slit hardware. It can be realized in energy space, in encoded Hilbert-space subspaces, or, more generally, wherever three controllable alternatives can be made to interfere and all lower-order contributions can be measured separately.

4. Generalizations beyond the single-particle three-path case

The conventional Sorkin test is the $I_{ab}=A_a^*A_b+A_aA_b^*$0 member of a broader interference hierarchy. For many-particle interference, the natural observables are not single-particle intensities but $I_{ab}=A_a^*A_b+A_aA_b^*$1-particle correlation functions

$I_{ab}=A_a^*A_b+A_aA_b^*$2

In this setting the relevant alternatives are $I_{ab}=A_a^*A_b+A_aA_b^*$3-particle paths, and pairwise interference between such paths can collectively involve more than two slits. The central result is that for $I_{ab}=A_a^*A_b+A_aA_b^*$4 particles,

$I_{ab}=A_a^*A_b+A_aA_b^*$5

Equivalently, the first necessarily vanishing term is

$I_{ab}=A_a^*A_b+A_aA_b^*$6

The associated generalized Sorkin parameter is

$I_{ab}=A_a^*A_b+A_aA_b^*$7

For $I_{ab}=A_a^*A_b+A_aA_b^*$8, this reduces to the usual normalized third-order Sorkin parameter (Pleinert et al., 2018).

This generalization removes a common misconception. Nonzero higher-order terms for many particles do not automatically imply a Born-rule violation. For example, in the two-particle case the theory allows $I_{ab}=A_a^*A_b+A_aA_b^*$9 and $a,b,c$0, while still requiring

$a,b,c$1

The reason is that Born’s rule still permits only pairwise interference of amplitudes, but the amplitudes are now amplitudes of many-particle paths rather than single-particle slit alternatives (Pleinert et al., 2018).

The same work argues that the many-particle parameters are exponentially more sensitive to deviations from Born’s rule than the single-particle test, with reported gains of about a factor of $a,b,c$2 for $a,b,c$3, about one order of magnitude for $a,b,c$4, and about two orders of magnitude for $a,b,c$5 (Pleinert et al., 2018).

5. Statistical interpretation, noise, and false positives

A nonzero Sorkin parameter is not self-interpreting. In the helium proposal, the two-photon model is perturbative and explicitly neglects higher-order processes; the first neglected non-suppressed corrections appear effectively at fourth order, and these can generate an apparent nonzero $a,b,c$6 without any true violation of Born’s rule. The protocol therefore includes background subtraction through $a,b,c$7, Monte Carlo propagation of amplitude fluctuations and timing jitter, multinomial sampling of detected counts, and a bias-avoidance procedure in which the weighted arithmetic means of the numerator and denominator are computed separately before forming the averaged ratio (Förderer et al., 16 Jan 2025).

On quantum computers, the same point appears as a hardware problem rather than a dynamical one. Simulations of Sorkin circuits with explicit noise models show that readout noise, depolarizing noise, thermal relaxation, and finite sampling can all produce spurious $a,b,c$8. Readout error gives a quadratic dependence on the flip probability $a,b,c$9, with $$P_{abc}=|A_a+A_b+A_c|^2=P_a+P_b+P_c+I_{ab}+I_{ac}+I_{bc},$$0 at both $$P_{abc}=|A_a+A_b+A_c|^2=P_a+P_b+P_c+I_{ab}+I_{ac}+I_{bc},$$1 and $$P_{abc}=|A_a+A_b+A_c|^2=P_a+P_b+P_c+I_{ab}+I_{ac}+I_{bc},$$2. Depolarizing noise first drives $$P_{abc}=|A_a+A_b+A_c|^2=P_a+P_b+P_c+I_{ab}+I_{ac}+I_{bc},$$3 away from zero and then back toward zero as the state approaches the maximally mixed state. Thermal relaxation generates nonzero $$P_{abc}=|A_a+A_b+A_c|^2=P_a+P_b+P_c+I_{ab}+I_{ac}+I_{bc},$$4 when $$P_{abc}=|A_a+A_b+A_c|^2=P_a+P_b+P_c+I_{ab}+I_{ac}+I_{bc},$$5 is short compared with gate times and disappears as $$P_{abc}=|A_a+A_b+A_c|^2=P_a+P_b+P_c+I_{ab}+I_{ac}+I_{bc},$$6 becomes large. These effects make the Sorkin test a sensitive deep-quantum benchmark as well as a foundational null test (2207.13585).

The interpretive rule is therefore restrictive. A null result is consistent with standard quantum mechanics. A nonzero result must survive detailed scrutiny against background counts, higher-order dynamics, readout bias, dephasing, relaxation, state-preparation error, and ratio-estimation bias before it can be read as evidence for genuine higher-order interference (Förderer et al., 16 Jan 2025, 2207.13585).

6. Distinct Sorkin-associated usages and terminological cautions

The label “Sorkin test” is not semantically stable across all literatures using Sorkin’s name. The following usages are distinct.

Context	Meaning of the “test”	Representative criterion
Born-rule interference	Null test for third-order interference	$$P_{abc}=|A_a+A_b+A_c|^2=P_a+P_b+P_c+I_{ab}+I_{ac}+I_{bc},$$7 or $$P_{abc}=|A_a+A_b+A_c|^2=P_a+P_b+P_c+I_{ab}+I_{ac}+I_{bc},$$8
Sorkin–Johnston vacuum	Vacuum-selection rule, not an experiment	purity, positivity, hermiticity, symmetry
Morse topology change	Causal admissibility criterion	causal continuity of Morse spacetime
QFT measurement theory	No-signalling adequacy test	$$P_{abc}=|A_a+A_b+A_c|^2=P_a+P_b+P_c+I_{ab}+I_{ac}+I_{bc},$$9

In the Sorkin–Johnston literature, a 2024 causal-set paper explicitly states that “Sorkin Test” there does not name a practical diagnostic or interference experiment. The nearest analogue is a selection criterion for a global quasifree vacuum: given the causal propagator and the chosen field space, one asks whether a candidate state satisfies positivity, hermiticity, purity, and symmetry. Under the paper’s assumptions—globally hyperbolic setting, normally hyperbolic equation, quasifree real scalar field, the Peierls relation, and the natural $I_{abc}=P_{abc}-P_{ab}-P_{ac}-P_{bc}+P_a+P_b+P_c,$0 field space—the unique solution is the Sorkin–Johnston state,

$I_{abc}=P_{abc}-P_{ab}-P_{ac}-P_{bc}+P_a+P_b+P_c,$1

obtained as the positive part of $I_{abc}=P_{abc}-P_{ab}-P_{ac}-P_{bc}+P_a+P_b+P_c,$2 (Jones, 2024).

In the topology-change literature, the relevant notion is the Borde–Sorkin conjecture: a Morse spacetime was conjectured to be causally continuous iff the index and coindex of every critical point are different from $I_{abc}=P_{abc}-P_{ab}-P_{ac}-P_{bc}+P_a+P_b+P_c,$3. Recent work proves a special case under small anisotropy and Euclidean background metric, but another paper gives a four-dimensional counterexample with index $I_{abc}=P_{abc}-P_{ab}-P_{ac}-P_{bc}+P_a+P_b+P_c,$4, coindex $I_{abc}=P_{abc}-P_{ab}-P_{ac}-P_{bc}+P_a+P_b+P_c,$5, and sufficiently large anisotropy, showing that the original conjecture is false in full generality. The operative “test” in that domain is therefore a causal-continuity criterion for Morse spacetimes, not an interference measurement (García-Heveling, 2022, Dahinden et al., 2024).

In relativistic quantum measurement theory, Sorkin’s “impossible measurements” argument yields a different kind of test altogether. One considers three regions $I_{abc}=P_{abc}-P_{ab}-P_{ac}-P_{bc}+P_a+P_b+P_c,$6, with $I_{abc}=P_{abc}-P_{ab}-P_{ac}-P_{bc}+P_a+P_b+P_c,$7 spacelike to $I_{abc}=P_{abc}-P_{ab}-P_{ac}-P_{bc}+P_a+P_b+P_c,$8 and $I_{abc}=P_{abc}-P_{ab}-P_{ac}-P_{bc}+P_a+P_b+P_c,$9 intermediate. Microcausality alone does not prevent superluminal signalling if one naively imports Lüders-rule state update into QFT. The sharp no-signalling criterion, in the formulation quoted from Borsten, Jubb, and Kells, is

$I_{abc}=0$0

for all $I_{abc}=0$1 and $I_{abc}=0$2. Here the “Sorkin test” is a causality-adequacy test for any proposed local measurement scheme in QFT (Papageorgiou et al., 2023).

Unqualified use of the term is therefore potentially misleading. In interference and Born-rule studies it denotes a null test for higher-order interference; in the other Sorkin-associated programs it denotes, at most, a criterion of admissibility, selection, or causal consistency rather than an interference experiment.