Routed Bell Tests

• Routed Bell tests are certification protocols that direct one subsystem of an entangled pair to either local or remote measurement devices.
• They redistribute Bell-test tasks by using high-efficiency short-range data to certify distant quantum correlations under significant loss.
• Architectures include one-sided near/far-path setups and entanglement-swapping designs, which support robust DIQKD and semi-device-independent protocols.

Searching arXiv for recent work on routed Bell tests and related DIQKD/security papers. Routed Bell tests are Bell-type certification protocols in which a routing variable directs one subsystem of an entangled resource either to a nearby test device or to a distant measurement stage. Their central purpose is to redistribute the certification task: rather than demanding a loophole-free long-distance Bell violation on the full communication link, they use short-range or local Bell data to constrain the source or measurement structure and then leverage those constraints to certify remote correlations under substantial loss. In the recent literature, routed Bell tests appear in one-sided near/far-path architectures, in entanglement-swapping or distant Bell-state-measurement (BSM) networks, and in extensions to semi-device-independent witnesses and multi-party local self-testing (Lobo et al., 2023, Roy-Deloison et al., 2024, Sekatski et al., 17 Feb 2025, Chaturvedi et al., 24 Apr 2025, Koßmann et al., 9 Oct 2025, Vijayaraj et al., 24 Jun 2026).

1. Architectural forms and operational definition

In the one-sided routed architecture, Alice receives one subsystem from the source, while Bob’s subsystem is actively routed by a switch to one of two measurement stations: a nearby device close to the source and a distant device suffering transmission loss. A representative formulation writes the observed correlations as

$$p(a,b|x,y,i)=\mathrm{tr}\!\left(\rho_{AB}\,A_a^x\otimes B_b^{y,i}\right),$$

where the routing input $i\in\{0,1\}$ determines whether Bob’s particle is sent to the near station $B_0$ or the distant station $B_1$ (Chaturvedi et al., 24 Apr 2025). In closely related notation, the routed Bell experiment studied in the near/far-path setting uses

$p(a,b\mid x,y,z),\qquad a,b\in\{\pm1\},$

with $z\in\{s,\ell\}$ selecting the short or long path and corresponding CHSH-type expressions

$$C_z=\langle A_0 B_{0z}\rangle+\langle A_0 B_{1z}\rangle+\langle A_1 B_{0z}\rangle-\langle A_1 B_{1z}\rangle$$

for the short-path and long-path branches (Lobo et al., 2023).

A second major architecture is the entanglement-swapping or distant-BSM variant. There, each party has a source and a routing switch that can send one half of a generated pair either to a distant BSM unit or to a local test device near the source. The BSM publicly announces a bit $z$, with $z=1$ for a successful projection and $z=0$ for failure or inconclusive detection. The key is generated from the $i\in\{0,1\}$0 rounds, while Bell certification is performed locally through CHSH tests near the sources (Vijayaraj et al., 24 Jun 2026).

The standard Bell inequality used in this setting is CHSH. In the entanglement-swapping version one convenient expression is

$i\in\{0,1\}$1

and analogously on Bob’s side (Vijayaraj et al., 24 Jun 2026). In the near/far-path literature, the same certification role is played by CHSH on the short path, combined with either CHSH-like or more specialized Bell functionals on the distant path (Lobo et al., 2023, Sekatski et al., 17 Feb 2025).

Historically, routed and destination-randomized Bell scenarios have antecedents in work on random destination sources. That line established that local postselection is not automatically illegitimate: if the selection rule depends only on local information and is independent of the local setting, Bell inequalities remain valid on the selected ensemble. This earlier point is conceptually adjacent to later routed Bell designs, although the modern routed literature focuses on detection-loophole mitigation and device-independent cryptography rather than on postselection alone (Sciarrino et al., 2010).

2. Detection loopholes, correlation classes, and what is actually certified

The primary motivation for routed Bell tests is the detection-efficiency loophole. Over long distances, losses reduce the fraction of detected events, and without fair sampling the detected sample can be biased. In DIQKD this is not merely an implementation nuisance: if the observed Bell statistics can be explained by postselected or loss-conditioned classical behavior, the security proof fails (Roy-Deloison et al., 2024, Koßmann et al., 9 Oct 2025).

A decisive conceptual refinement was the distinction between short-range quantum (SRQ) and long-range quantum (LRQ) correlations. SRQ correlations are those for which the long-path channel is entanglement-breaking; equivalently, the distant observables become jointly measurable after the long-path channel. LRQ correlations are the complement, $i\in\{0,1\}$2. This definition is stricter than simply excluding source-predetermined outputs for the remote device. A central point of the reanalysis in (Lobo et al., 2023) is that the earlier $i\in\{0,1\}$3-marginal notion is too restrictive: one can send only classical information to the remote station after a local quantum measurement and still violate the older bound, so “not predetermined at the source” does not yet certify that a quantum system reached the distant device (Lobo et al., 2023).

This distinction is directly tied to security. For active-attack-resistant routed DIQKD, it is not enough to exclude a local-hidden-variable model for the remote device; one must certify that its measurements are non-jointly measurable (NJM). If the remote measurements were jointly measurable, an eavesdropper could perform the parent measurement in advance and classically reproduce the apparent measurement outcomes. In the routed setting analyzed through parallel repetition, LRQ correlations are exactly those incompatible with a jointly measurable model at the distant device, which is the security-relevant condition (Chaturvedi et al., 24 Apr 2025).

The certification consequences are subtle. Under the SRQ notion, a maximal short-path CHSH value does not improve the long-path CHSH threshold itself; the improvement appears only for appropriately chosen long-path Bell expressions. One family is

$i\in\{0,1\}$4

with local and quantum bounds

$i\in\{0,1\}$5

and, under maximal short-path CHSH, the stronger SRQ bound

$i\in\{0,1\}$6

For $i\in\{0,1\}$7, this is strictly below the local bound $i\in\{0,1\}$8, so the short-path test genuinely strengthens long-path certification (Lobo et al., 2023).

A common misconception is therefore that routed Bell tests automatically permit arbitrarily low distant detection efficiency in the standard two-setting CHSH configuration. The literature does not support that statement. In the SRQ framework there is a universal lower bound

$i\in\{0,1\}$9

and hence $B_0$0 for a two-setting distant device (Lobo et al., 2023). Later arbitrary-loss results circumvent this finite-setting obstruction by moving to many-setting or continuous-setting constructions rather than by invalidating the bound (Sekatski et al., 17 Feb 2025).

3. Security proofs and routed DIQKD formulations

The first detailed DIQKD analysis built directly on routed Bell tests considers an entangled source, Alice’s device $B_0$1, Bob’s distant device $B_0$2, an additional nearby testing device $B_0$3, and an actively controlled switch. In each round, Bob’s particle is routed either to $B_0$4 for a short-range Bell test or to $B_0$5 for long-range key-generation and testing data. The asymptotic key rate takes the Devetak–Winter form

$B_0$6

and the single-round conditional entropy is lower-bounded with the Brown–Fawzi–Fawzi method combined with noncommutative polynomial optimization and semidefinite relaxations in the NPA hierarchy. At the many-round level, the reduction is handled through the Generalized Entropy Accumulation Theorem (Roy-Deloison et al., 2024).

That framework supports both CHSH-based and BB84-based routed protocols. A notable feature is that BB84-like correlations, which are local in a standard Bell setting, become certifiable in the routed setting because the short-range CHSH test self-tests the source and Alice’s measurement structure. This produces the routed BB84 protocol, or rBB84, within a fully device-independent analysis (Roy-Deloison et al., 2024).

A more symmetric routed-DIQKD formulation introduces local partners for both communicating parties—Alice with Fred $B_0$7 and Bob with George $B_0$8—along with an entanglement-swapping apparatus $B_0$9 that produces the long-range state $B_1$0. The key structural assumption is the pair of marginal constraints

$B_1$1

which identify the reduced states seen in the long-range branch with those certified locally in each laboratory. The security proof then uses robust self-testing: if Alice’s CHSH winning probability is $B_1$2, the certified deviation from the ideal qubit model is $B_1$3, and the BB84 key rate degrades only by an $B_1$4 term (Koßmann et al., 9 Oct 2025).

The resulting asymptotic lower bound is

$B_1$5

with $B_1$6 and $B_1$7 the observed bit error rates in the $B_1$8- and $B_1$9-bases. In the ideal limit $p(a,b\mid x,y,z),\qquad a,b\in\{\pm1\},$0, the rate approaches the standard BB84 asymptotic expression

$p(a,b\mid x,y,z),\qquad a,b\in\{\pm1\},$1

This is the precise sense in which perfect local Bell tests can, in principle, overcome the detection-efficiency barrier: the asymptotic rate becomes limited only by standard bit-flip errors, as in the device-dependent case (Koßmann et al., 9 Oct 2025).

In the entanglement-swapping/BSM architecture, the protocol remains device-independent in the sense that the devices are treated as uncharacterized, memoryless, and non-signalling, and the routing choice must be hidden from both the source and the devices. The BSM outcome $p(a,b\mid x,y,z),\qquad a,b\in\{\pm1\},$2 is publicly announced only after all rounds are completed, so it cannot influence the measurement choices $p(a,b\mid x,y,z),\qquad a,b\in\{\pm1\},$3. This preserves the logic of DI security in a routed network, although the paper also emphasizes that finite local efficiencies remain a critical limitation (Vijayaraj et al., 24 Jun 2026).

4. Thresholds, loss tolerance, and asymptotic scaling laws

For simple two-qubit routed DIQKD protocols, routing substantially lowers the detection-efficiency threshold of the distant device when the short-range test is of high quality. In the protocol family analyzed in (Roy-Deloison et al., 2024), the routed CHSH protocol tolerates $p(a,b\mid x,y,z),\qquad a,b\in\{\pm1\},$4 at $p(a,b\mid x,y,z),\qquad a,b\in\{\pm1\},$5, about an $p(a,b\mid x,y,z),\qquad a,b\in\{\pm1\},$6 improvement over the non-routed CHSH counterpart. The routed CHSH-BB84 combination reaches $p(a,b\mid x,y,z),\qquad a,b\in\{\pm1\},$7 at $p(a,b\mid x,y,z),\qquad a,b\in\{\pm1\},$8, about a $p(a,b\mid x,y,z),\qquad a,b\in\{\pm1\},$9 improvement over the non-routed version. For routed BB84, the paper finds positive key rate down to about $z\in\{s,\ell\}$0 with binning of test outcomes on $z\in\{s,\ell\}$1, and down to about $z\in\{s,\ell\}$2 without binning on $z\in\{s,\ell\}$3; that $z\in\{s,\ell\}$4 point is emphasized as the minimal threshold for any QKD protocol with two untrusted measurements (Roy-Deloison et al., 2024).

The more conservative reanalysis of routed Bell experiments identifies bounded but genuine improvements for long-range certification. In analytically accessible families, the threshold can drop from the standard $z\in\{s,\ell\}$5 to $z\in\{s,\ell\}$6, and at best to $z\in\{s,\ell\}$7 when no-click outcomes are kept as an extra outcome and the distant side has two settings. The same work also reports that routing helps only when the short-path efficiency is itself very high, specifically for CHSH only for $z\in\{s,\ell\}$8 in the binned case or $z\in\{s,\ell\}$9 in the no-binning case (Lobo et al., 2023).

A different route to lower thresholds is parallel repetition. In the repeated routed BB84 and routed CHSH strategies, $$C_z=\langle A_0 B_{0z}\rangle+\langle A_0 B_{1z}\rangle+\langle A_1 B_{0z}\rangle-\langle A_1 B_{1z}\rangle$$0 entangled pairs are tested in parallel, the nearby devices are assumed to achieve the maximal violation of the $$C_z=\langle A_0 B_{0z}\rangle+\langle A_0 B_{1z}\rangle+\langle A_1 B_{0z}\rangle-\langle A_1 B_{1z}\rangle$$1-product CHSH inequality,

$$C_z=\langle A_0 B_{0z}\rangle+\langle A_0 B_{1z}\rangle+\langle A_1 B_{0z}\rangle-\langle A_1 B_{1z}\rangle$$2

and self-testing of the $$C_z=\langle A_0 B_{0z}\rangle+\langle A_0 B_{1z}\rangle+\langle A_1 B_{0z}\rangle-\langle A_1 B_{1z}\rangle$$3-pair maximally entangled target is then used to analyze the distant device. The central threshold law is

$$C_z=\langle A_0 B_{0z}\rangle+\langle A_0 B_{1z}\rangle+\langle A_1 B_{0z}\rangle-\langle A_1 B_{1z}\rangle$$4

This scaling is described as exponential, optimal, and robust: for $$C_z=\langle A_0 B_{0z}\rangle+\langle A_0 B_{1z}\rangle+\langle A_1 B_{0z}\rangle-\langle A_1 B_{1z}\rangle$$5, $$C_z=\langle A_0 B_{0z}\rangle+\langle A_0 B_{1z}\rangle+\langle A_1 B_{0z}\rangle-\langle A_1 B_{1z}\rangle$$6; for $$C_z=\langle A_0 B_{0z}\rangle+\langle A_0 B_{1z}\rangle+\langle A_1 B_{0z}\rangle-\langle A_1 B_{1z}\rangle$$7, $$C_z=\langle A_0 B_{0z}\rangle+\langle A_0 B_{1z}\rangle+\langle A_1 B_{0z}\rangle-\langle A_1 B_{1z}\rangle$$8; and each additional repetition halves the required efficiency (Chaturvedi et al., 24 Apr 2025).

The strongest loss-tolerance statement currently in the routed literature uses a continuous family of distant measurements and a steering reduction. The long-path Bell expression is

$$C_z=\langle A_0 B_{0z}\rangle+\langle A_0 B_{1z}\rangle+\langle A_1 B_{0z}\rangle-\langle A_1 B_{1z}\rangle$$9

with average click rate

$z$0

and the strong routed Bell inequality

$z$1

For the maximally entangled two-qubit state with optimal short-path CHSH, one obtains $z$2, and violation occurs for every $z$3. This establishes certification of long-range quantum correlations at arbitrary nonzero transmission, but it does so by using a continuous family of distant measurements rather than the two-setting scenario constrained by the universal $z$4 bound (Sekatski et al., 17 Feb 2025).

5. Entanglement swapping, multiple local tests, and conditional consistency

In the BSM-based routed architecture, a basic vulnerability arises because the key is extracted only from successful BSM rounds. The observed Bell value can then be written as an adversarial convex mixture,

$z$5

where $z$6 is the Bell value in key-generating rounds, $z$7 is the probability that the state is routed to the BSM, and $z$8 is the BSM efficiency. This decomposition expresses the possibility that ideal Bell violation is hidden in discarded rounds while the successful BSM rounds contain degraded correlations (Vijayaraj et al., 24 Jun 2026).

The remedy proposed in (Vijayaraj et al., 24 Jun 2026) is the introduction of decoy Bell tests (DBTs). Instead of using a single local test device per side, each side has many local devices, and the switches route qubits so that $z$9 local tests are performed in every round, including rounds in which the distant BSM succeeds. The paper states that “DBTs occur with randomly selected devices” and that one can arrange that “exactly $z=1$0 photons reach $z=1$1 local test devices in each round.” The protocol then requires consistency between ordinary local Bell violations and the Bell violations conditioned on successful BSM events, ideally

$z=1$2

If the conditional violations in the $z=1$3 subset are systematically weaker, the protocol interprets that as evidence of selective manipulation of the key-generating rounds (Vijayaraj et al., 24 Jun 2026).

This conditional-consistency requirement yields concrete threshold improvements. For Bell values around $z=1$4, the paper reports that no key is possible below about $z=1$5 BSM efficiency without the extra decoy Bell tests, whereas with DBTs and a $z=1$6-port BSM the threshold drops to about $z=1$7. More generally, for high-visibility states and Bell values in the range $z=1$8, the improvement in critical efficiency is about $z=1$9. The same work notes that DBTs remain useful even when efficiencies are around $z=0$0, a regime relevant to long-distance entanglement-swapping experiments, while also stressing that finite local efficiencies impose a demanding local requirement, around $z=0$1, for security (Vijayaraj et al., 24 Jun 2026).

This line of work changes the role of entanglement swapping in routed Bell tests. Event-ready approaches delay measurements until a successful BSM projection, typically requiring quantum memories. The routed protocol instead keeps local Bell certification near the sources and uses the distant BSM as a heralding mechanism for key rounds. A plausible implication is that routed Bell tests and event-ready architectures address the same loss problem from opposite directions: one by localizing certification, the other by delaying nonlocal measurements until heralding succeeds.

6. Generalizations, semi-device-independent extensions, and related terminology

Routed Bell tests now support several generalizations beyond the original one-sided near/far-path design. One is the extension to arbitrarily many local parties on each side: the switch can route systems to multiple local partners, allowing many local Bell tests and many correlation checks per laboratory. In that setting, the marginal-constraint idea becomes a family of equalities requiring that all Alice-side marginals agree with one another, and likewise on Bob’s side (Koßmann et al., 9 Oct 2025).

Another extension moves from Bell nonlocality to dimension certification. In the semi-device-independent variant proposed in (Vijayaraj et al., 24 Jun 2026), the sources are assumed only to be qubit-bounded and to emit one qubit per round, while the states are mostly sent to the BSM and only sometimes tested locally. The witness takes a CHSH-like form,

$z=0$2

with maximal quantum value $z=0$3 for perfectly encoded BB84 states and suitable local measurements. Here the routing logic is unchanged, but the certification target is different: maximal witness violation is interpreted as certifying that the source produces correctly encoded qubit states, so the protocol becomes semi-device-independent rather than fully device-independent (Vijayaraj et al., 24 Jun 2026).

The arbitrary-distance DIQKD proposal is another important generalization. Using $z=0$4 measurement settings on each side,

$z=0$5

and similarly on the short and long paths, the short-path statistics maximally violate the $z=0$6-input chained Bell inequality and self-test the source and Alice’s measurements. The long path then becomes equivalent to a prepare-and-measure problem, and the protocol tolerates loss whenever $z=0$7. By choosing $z=0$8 large enough, one can in principle obtain a nonzero key rate for any fixed $z=0$9 (Sekatski et al., 17 Feb 2025). This suggests that routed Bell tests interpolate between finite-setting Bell certification and high-setting steering-like certification as loss becomes extreme.

Finally, the phrase “routed Bell test” has also appeared in a distinct, stabilizer-structured sense in work on Bell inequalities tailored to the $i\in\{0,1\}$00 toric code. There the measurement settings are “routed” through the code geometry rather than through a near/far-path switch. That usage is conceptually separate from the DIQKD and long-distance-loss literature, but it indicates that the term has begun to denote a broader design principle: Bell certification whose measurement architecture is explicitly structured by an auxiliary routing or coding constraint (Vallée et al., 28 Nov 2025).