Tripartite Nonlocal Magic

• Tripartite non-local magic is defined as irreducible three-party quantum correlations that cannot be reproduced by any model combining pairwise nonlocal resources and shared randomness.
• It unifies approaches from Hardy-type paradoxes, LOSR network experiments, and stabilizer magic theory to certify genuinely multipartite nonclassicality.
• The framework bridges theoretical resource optimization and experimental protocols by isolating basis-invariant non-stabilizerness linked to multipartite entanglement.

Searching arXiv for the cited papers and closely related work on tripartite nonlocality and non-local magic. Tripartite non-local magic denotes a family of closely related concepts at the intersection of multipartite nonlocality and quantum-resource theory. In one line of work, it refers to irreducible three-party correlations that cannot be reproduced by any hybrid model that is local across some bipartition, by any non-signaling version of such a model, or, more stringently, by any network model built only from bipartite nonlocal resources together with globally shared randomness (Yu et al., 2013, Paul et al., 2015, Huang et al., 2022). In another line of work, it refers to non-stabilizerness whose persistence or spreading is locked to multipartite entanglement structure, so that GHZ-type and Bell-pair correlations act as conduits for globally delocalized magic (Hou et al., 26 Mar 2025); the experimentally established bipartite notion of non-local magic, defined as the part of magic that survives optimization over local unitaries, provides a direct resource-theoretic precursor for multipartite extensions (Ahmad et al., 19 Nov 2025). Taken together, these works suggest that tripartite non-local magic is best understood as irreducible three-party nonclassicality that is neither reducible to pairwise nonlocality nor removable by local basis changes.

1. Formal scope and competing notions of irreducibility

The tripartite literature distinguishes several nonequivalent notions of nonclassicality. For three parties Alice, Bob, and Charlie, a fully local hidden-variable model has the form

$$P(a,b,c|x,y,z)=\sum_{\lambda}q_{\lambda}P_{\lambda}(a|x)P_{\lambda}(b|y)P_{\lambda}(c|z),$$

where $a,b,c\in\{0,1\}$ are outputs and $x,y,z\in\{0,1\}$ are inputs (Paul et al., 2015). Genuine tripartite nonlocality asks for more than the failure of full locality: it asks whether the correlations can still be written as a mixture of terms that are local with respect to some bipartition. In Svetlichny’s hybrid form,

$$P(abc|xyz) =\sum_{\lambda}q_{\lambda}P_{\lambda}(ab|xy ) P_{\lambda}(c|z)+ \sum_{\mu} q_{\mu}P_{\mu}(ac|xz) P_{\mu}(b|y ) +\sum_{\nu} q_{\nu}P_{\nu}(bc|yz) P_{\nu}(a|x),$$

and correlations that admit such a decomposition are called $S_2$ local or Svetlichny local (Paul et al., 2015).

A stricter operational variant is $NS_2$ nonlocality, in which the bipartite terms in the hybrid decomposition must themselves satisfy non-signaling constraints. A correlation that cannot be decomposed in this way is genuinely tripartite nonlocal in the $NS_2$ sense (Paul et al., 2015). The Hardy-type theorem for pure states adopts the same physical principle: the hidden-variable model may contain bipartite nonlocal subcorrelations, but those subcorrelations must remain non-signaling (Yu et al., 2013).

The network-resource formulation goes further. In the LOSR setting used in the strict-locality photonic experiment, the question is whether observed correlations can be simulated using only bipartite nonlocal resources $\delta_{ij}$ and a common random variable $\lambda_{ABC}$. Violation of the corresponding network Bell-type inequality excludes all such decompositions and certifies genuine LOSR tripartite nonlocality (Huang et al., 2022). The conceptual significance is that Svetlichny-type, $NS_2$, and LOSR/network notions are related but not identical: the last is formulated explicitly as a resource-theoretic exclusion of every bipartite-resource explanation under shared randomness.

The magic literature introduces a different irreducibility criterion. For a bipartite pure state, non-local magic is defined as the part of stabilizer Rényi entropy that cannot be erased by local unitaries,

$a,b,c\in\{0,1\}$0

with local magic defined by subtraction,

$a,b,c\in\{0,1\}$1

The paper emphasizes that entanglement and magic are distinct resources: a state can be highly entangled but carry only local magic, and non-local magic appears only when magic and entanglement cooperate (Ahmad et al., 19 Nov 2025). This suggests that a tripartite analogue would not be exhausted either by entanglement alone or by ordinary nonlocality alone.

2. Pure-state tripartite nonlocality and Hardy-type paradoxes

A central theorem states that every tripartite entangled pure state is genuinely tripartite nonlocal with respect to non-signaling hybrid local/nonlocal models (Yu et al., 2013). The setting is three distant observers $a,b,c\in\{0,1\}$2, each choosing one of two dichotomic observables $a,b,c\in\{0,1\}$3 or $a,b,c\in\{0,1\}$4 with outcomes $a,b,c\in\{0,1\}$5. The corresponding hybrid model allows one party to be local while the other two may share arbitrary bipartite nonlocal correlations: $a,b,c\in\{0,1\}$6 subject to non-signaling constraints on every $a,b,c\in\{0,1\}$7 term (Yu et al., 2013).

The theorem is established through a Hardy-type test without inequalities. The six conditions are

$a,b,c\in\{0,1\}$8

$a,b,c\in\{0,1\}$9

These conditions are incompatible with any non-signaling bi-local model, but are satisfied by all tripartite entangled asymmetric pure states and, together with an alternative symmetric-state Hardy test, by every fully entangled $x,y,z\in\{0,1\}$0-qubit pure state (Yu et al., 2013).

The proof uses a magic basis representation

$x,y,z\in\{0,1\}$1

with $x,y,z\in\{0,1\}$2, and constructs measurements such that the zero-probability constraints hold while the success probability remains strictly positive. The key success amplitude is

$x,y,z\in\{0,1\}$3

For a concrete gedanken realization with two electrons and a positron, the success probability is

$x,y,z\in\{0,1\}$4

For the three-party case, the maximal success probability is stated as

$x,y,z\in\{0,1\}$5

where $x,y,z\in\{0,1\}$6 is the unique positive root of

$x,y,z\in\{0,1\}$7

The theorem extends beyond qubits. By local projections, any tripartite entangled pure state in arbitrary local dimensions can be reduced to a genuine $x,y,z\in\{0,1\}$8-qubit entangled subspace, so the conclusion persists: tripartite entanglement implies genuine tripartite nonlocality under non-signaling hybrid models (Yu et al., 2013). The significance is that, for pure states, irreducible three-party nonlocality is not exceptional but generic.

3. Hidden genuine tripartite nonlocality and sequential activation

The pure-state theorem does not eliminate contextual subtleties. A separate line of work shows that genuine tripartite nonlocality can remain hidden in standard Bell scenarios and then be revealed by a sequential protocol based on stochastic local operations and classical communication (Paul et al., 2015). The Bell scenario is the standard binary-input, binary-output case with $x,y,z\in\{0,1\}$9 parties, $$P(abc|xyz) =\sum_{\lambda}q_{\lambda}P_{\lambda}(ab|xy ) P_{\lambda}(c|z)+ \sum_{\mu} q_{\mu}P_{\mu}(ac|xz) P_{\mu}(b|y ) +\sum_{\nu} q_{\nu}P_{\nu}(bc|yz) P_{\nu}(a|x),$$0 measurements per party, and $$P(abc|xyz) =\sum_{\lambda}q_{\lambda}P_{\lambda}(ab|xy ) P_{\lambda}(c|z)+ \sum_{\mu} q_{\mu}P_{\mu}(ac|xz) P_{\mu}(b|y ) +\sum_{\nu} q_{\nu}P_{\nu}(bc|yz) P_{\nu}(a|x),$$1 outcomes per measurement. In that scenario, three explicitly constructed genuine tripartite entangled $$P(abc|xyz) =\sum_{\lambda}q_{\lambda}P_{\lambda}(ab|xy ) P_{\lambda}(c|z)+ \sum_{\mu} q_{\mu}P_{\mu}(ac|xz) P_{\mu}(b|y ) +\sum_{\nu} q_{\nu}P_{\nu}(bc|yz) P_{\nu}(a|x),$$2-states,

$$P(abc|xyz) =\sum_{\lambda}q_{\lambda}P_{\lambda}(ab|xy ) P_{\lambda}(c|z)+ \sum_{\mu} q_{\mu}P_{\mu}(ac|xz) P_{\mu}(b|y ) +\sum_{\nu} q_{\nu}P_{\nu}(bc|yz) P_{\nu}(a|x),$$3

$$P(abc|xyz) =\sum_{\lambda}q_{\lambda}P_{\lambda}(ab|xy ) P_{\lambda}(c|z)+ \sum_{\mu} q_{\mu}P_{\mu}(ac|xz) P_{\mu}(b|y ) +\sum_{\nu} q_{\nu}P_{\nu}(bc|yz) P_{\nu}(a|x),$$4

$$P(abc|xyz) =\sum_{\lambda}q_{\lambda}P_{\lambda}(ab|xy ) P_{\lambda}(c|z)+ \sum_{\mu} q_{\mu}P_{\mu}(ac|xz) P_{\mu}(b|y ) +\sum_{\nu} q_{\nu}P_{\nu}(bc|yz) P_{\nu}(a|x),$$5

can each be $$P(abc|xyz) =\sum_{\lambda}q_{\lambda}P_{\lambda}(ab|xy ) P_{\lambda}(c|z)+ \sum_{\mu} q_{\mu}P_{\mu}(ac|xz) P_{\mu}(b|y ) +\sum_{\nu} q_{\nu}P_{\nu}(bc|yz) P_{\nu}(a|x),$$6 local and, for suitable parameter ranges, $$P(abc|xyz) =\sum_{\lambda}q_{\lambda}P_{\lambda}(ab|xy ) P_{\lambda}(c|z)+ \sum_{\mu} q_{\mu}P_{\mu}(ac|xz) P_{\mu}(b|y ) +\sum_{\nu} q_{\nu}P_{\nu}(bc|yz) P_{\nu}(a|x),$$7 local as well (Paul et al., 2015).

The relevant Svetlichny inequality is

$$P(abc|xyz) =\sum_{\lambda}q_{\lambda}P_{\lambda}(ab|xy ) P_{\lambda}(c|z)+ \sum_{\mu} q_{\mu}P_{\mu}(ac|xz) P_{\mu}(b|y ) +\sum_{\nu} q_{\nu}P_{\nu}(bc|yz) P_{\nu}(a|x),$$8

with

$$P(abc|xyz) =\sum_{\lambda}q_{\lambda}P_{\lambda}(ab|xy ) P_{\lambda}(c|z)+ \sum_{\mu} q_{\mu}P_{\mu}(ac|xz) P_{\mu}(b|y ) +\sum_{\nu} q_{\nu}P_{\nu}(bc|yz) P_{\nu}(a|x),$$9

The maximal Svetlichny values for the initial states are

$S_2$0

$S_2$1

$S_2$2

For $S_2$3 locality, the analysis uses all $S_2$4 facet inequalities of the $S_2$5-local polytope, of which the Svetlichny inequality is one facet (Paul et al., 2015).

The revealing mechanism is a Sequential Measurement Protocol interpreted as a WCCPI protocol and explicitly as an SLOCC process in the preparation stage. Each party initially holds one qubit from each of the three states, performs Bell-basis measurements on two qubits, communicates the outcomes, and postselects on Bell outcomes such as

$S_2$6

The resulting tripartite state is

$S_2$7

with

$S_2$8

A notable structural feature is that $S_2$9 is independent of $NS_2$0 and $NS_2$1, reflecting the entanglement-swapping and postselected SLOCC character of the preparation (Paul et al., 2015).

The final state can violate Svetlichny’s inequality or, more strongly, one of the $NS_2$2 $NS_2$3 facets even when each initial state satisfies all of them. The corresponding genuine multipartite concurrence is

$NS_2$4

and for the postselected state

$NS_2$5

The significance is that genuine tripartite nonlocality is a resource that can be hidden in ordinary Bell tests and operationally revealed by sequential local measurements and classical communication (Paul et al., 2015).

4. Network certification under strict locality conditions

The photonic triangular-network experiment provides an explicit experimental realization of irreducible tripartite nonlocality under strict locality constraints (Huang et al., 2022). The setup contains three spatially separated observers, Alice, Bob, and Charlie, and two independent EPR sources $NS_2$6 and $NS_2$7 generating Bell pairs

$NS_2$8

By interfering one photon from each pair at Charlie’s station on a PBS and postselecting suitable four-fold events, the network prepares the three-photon GHZ state

$NS_2$9

or equivalently

$NS_2$0

The experimental claim is not merely that the postselected state is entangled, but that its measured correlations violate a network Bell-type inequality satisfied by every model composed only of bipartite nonlocal resources together with shared randomness. In that sense the certified resource is genuinely LOSR tripartite nonlocality (Huang et al., 2022). The inequality combines a Bell game with a same-output consistency game, and for the ideal GHZ strategy the corresponding score can reach

$NS_2$1

The experiment observes

$NS_2$2

which violates the bound $NS_2$3 by $NS_2$4 standard deviations (Huang et al., 2022).

The locality architecture is central. Relevant events are spacelike separated, measurement bases are chosen in real time using quantum random number generators, and a fast single-photon polarization modulation system switches measurement bases at $NS_2$5 MHz with about $NS_2$6 fidelity. Settings and detections are recorded locally and combined only later. The spacetime analysis reports, for example, that Alice’s measurement event $NS_2$7 is $NS_2$8 ns outside Charlie’s QRNG event, and Charlie’s measurement $NS_2$9 is $\delta_{ij}$0 ns outside Alice’s QRNG event (Huang et al., 2022). Alice and Charlie use $\delta_{ij}$1 and $\delta_{ij}$2 measurements, while Bob uses $\delta_{ij}$3, $\delta_{ij}$4, and $\delta_{ij}$5.

The prepared postselected GHZ state has fidelity

$\delta_{ij}$6

with a tomography cross-check of

$\delta_{ij}$7

The paper notes that, for a GHZ state mixed with white noise, the inequality requires a fidelity threshold of about $\delta_{ij}$8, so the observed state is just above threshold (Huang et al., 2022).

This experiment is often taken as the clearest operational meaning of tripartite non-local magic in a resource-theoretic sense: the observed correlations cannot be simulated by pairwise nonclassical links plus global shared randomness. The result is also explicitly stronger than ordinary Svetlichny-type claims, because it excludes all bipartite-resource explanations allowed by the LOSR network framework of Coiteux-Roy, Wolfe, and Renou (Huang et al., 2022).

5. Non-local magic as basis-invariant irreducible non-stabilizerness

The resource-theoretic notion of non-local magic isolates a different kind of irreducibility. For an $\delta_{ij}$9-qubit pure state $\lambda_{ABC}$0 of dimension $\lambda_{ABC}$1, the $\lambda_{ABC}$2-stabilizer Rényi entropy is

$\lambda_{ABC}$3

where $\lambda_{ABC}$4 is the $\lambda_{ABC}$5-qubit Pauli group (Ahmad et al., 19 Nov 2025). For pure states, $\lambda_{ABC}$6 with $\lambda_{ABC}$7 is a valid magic monotone, and the experimentally accessible case is $\lambda_{ABC}$8. Non-local magic is the minimal residual magic after optimizing over local basis changes, while local magic is the removable part. The basic physical point is that non-local magic is not synonymous with entanglement: maximally entangled states can possess only local magic, and states of the form

$\lambda_{ABC}$9

with $NS_2$0 an entangled stabilizer state can be entangled yet not host non-local magic (Ahmad et al., 19 Nov 2025).

For pure two-qubit states written in Schmidt form,

$NS_2$1

the non-local magic depends only on the Schmidt spectrum: $NS_2$2 Equivalently, in terms of reduced density-matrix purity

$NS_2$3

one has

$NS_2$4

In the Schmidt-parameter form,

$NS_2$5

with $NS_2$6 (Ahmad et al., 19 Nov 2025). The importance of these formulas is operational: non-local magic can be inferred directly from subsystem purity rather than from an explicit local-unitary optimization.

The corresponding experiment uses Randomized Clifford Measurements. The protocol is: prepare the target state, apply random single-qubit Clifford unitaries, measure in the computational basis, and average over Clifford instances to estimate the purity $NS_2$7 and stabilizer purity $NS_2$8. The estimator is

$NS_2$9

For two qubits, the experiment samples $a,b,c\in\{0,1\}$00 random Clifford settings (Ahmad et al., 19 Nov 2025).

Although the demonstration is bipartite, its relevance to tripartite non-local magic is explicit. The paper proposes that multipartite generalization should proceed by minimizing total stabilizer Rényi entropy over local unitaries on each subsystem and separating magic into locally removable and irreducible non-local components (Ahmad et al., 19 Nov 2025). This does not yet yield a general tripartite closed form, but it fixes the resource-theoretic template.

6. Tripartite stabilizer entanglement as a magic highway

A complementary development treats tripartite non-local magic dynamically, as the spreading of injected non-stabilizerness through pre-existing multipartite entanglement (Hou et al., 26 Mar 2025). The initial state is a stabilizer pure state with no initial magic in the injection region. Magic is then injected by a local Haar random unitary. The central quantity is the linear stabilizer entropy

$a,b,c\in\{0,1\}$01

with associated stabilizer Rényi entropy

$a,b,c\in\{0,1\}$02

More generally,

$a,b,c\in\{0,1\}$03

A structural fact used throughout is that $a,b,c\in\{0,1\}$04 is invariant under Clifford unitaries and multiplicative on tensor products (Hou et al., 26 Mar 2025).

For bipartite stabilizer states, local Clifford equivalence yields the canonical form

$a,b,c\in\{0,1\}$05

where $a,b,c\in\{0,1\}$06 is an integer stabilizer entanglement entropy. Under a Haar random unitary on region $a,b,c\in\{0,1\}$07,

$a,b,c\in\{0,1\}$08

and in the $a,b,c\in\{0,1\}$09 limit

$a,b,c\in\{0,1\}$10

Under factorized local randomization $a,b,c\in\{0,1\}$11,

$a,b,c\in\{0,1\}$12

These formulas express the “magic highway” effect: entanglement throttles how effectively locally injected magic becomes globally delocalized (Hou et al., 26 Mar 2025).

The tripartite extension uses the standard local-unitary normal form consisting of $a,b,c\in\{0,1\}$13 GHZ states, Bell pairs $a,b,c\in\{0,1\}$14, and local product qubits $a,b,c\in\{0,1\}$15. For the factorized unitary $a,b,c\in\{0,1\}$16, the paper derives

$a,b,c\in\{0,1\}$17

up to subleading corrections $a,b,c\in\{0,1\}$18 and $a,b,c\in\{0,1\}$19 (Hou et al., 26 Mar 2025). The leading suppression depends on $a,b,c\in\{0,1\}$20 and on the entanglement linking $a,b,c\in\{0,1\}$21 to $a,b,c\in\{0,1\}$22, namely $a,b,c\in\{0,1\}$23, while the correction term measures proximity to full Haar-random saturation. In the stronger tripartite result for $a,b,c\in\{0,1\}$24,

$a,b,c\in\{0,1\}$25

and global magic again approaches the Haar-random value if the parties are sufficiently entangled (Hou et al., 26 Mar 2025).

An important qualitative statement is that pairwise entanglement is not strictly necessary in all channels: large $a,b,c\in\{0,1\}$26 and $a,b,c\in\{0,1\}$27 can drive the result toward Haar-random behavior even if $a,b,c\in\{0,1\}$28 and $a,b,c\in\{0,1\}$29 are not directly entangled (Hou et al., 26 Mar 2025). The same qualitative picture persists for shallow-depth brickwork circuits and for initial states with non-stabilizer entanglement such as

$a,b,c\in\{0,1\}$30

This line of work therefore interprets tripartite non-local magic as a dynamical, multi-terminal transport property of entanglement structure: GHZ and Bell correlations determine whether local magic injection becomes globally indistinguishable from Haar-random magic.

7. Caveats, misconceptions, and current research directions

Several common identifications are explicitly rejected by the literature. First, tripartite entanglement is not by itself the same thing as tripartite non-local magic. The pure-state Hardy theorem proves genuine tripartite nonlocality for every tripartite entangled pure state under non-signaling hybrid models (Yu et al., 2013), but the non-local magic framework emphasizes that entanglement alone is insufficient for irreducible magic: a state may be entangled and still host only local magic (Ahmad et al., 19 Nov 2025). These statements concern different resource orderings and are not contradictory.

Second, genuine tripartite nonlocality is not a unique notion. Svetlichny nonlocality, $a,b,c\in\{0,1\}$31 nonlocality, and genuine LOSR tripartite nonlocality differ in what kinds of bipartite resources they permit inside a hidden-variable or network model (Paul et al., 2015, Huang et al., 2022). A correlation can therefore be nonclassical in one sense and unclassified in another, which is why the network-based photonic violation is presented as stronger than ordinary Svetlichny-type certification (Huang et al., 2022).

Third, hidden-nonlocality and strict-locality experiments involve distinct operational assumptions. The SLOCC-based activation protocol relies on a preparation stage with Bell measurements, classical communication, and postselection before the final Bell test (Paul et al., 2015). The photonic triangular-network experiment enforces spacelike separation during each trial, but it also explicitly uses postselection and the fair sampling assumption; it does not close detection or postselection loopholes. The paper is careful to note that conditioning on a subset of events can introduce selection bias, so the claim of genuine tripartite nonlocality is made under fair-sampling and postselection assumptions (Huang et al., 2022).

Fourth, experimental non-local magic has only been demonstrated directly in a two-qubit superconducting processor. The paper itself does not give a full tripartite formula, but states that its logic generalizes naturally to partitioned many-body systems and that, for tripartite systems, one would expect optimization over $a,b,c\in\{0,1\}$32 together with a richer structure because different bipartitions can carry different amounts of non-local magic (Ahmad et al., 19 Nov 2025). This is a programmatic extension, not an already realized tripartite experiment.

The forward directions stated across the cited works are correspondingly varied. They include larger $a,b,c\in\{0,1\}$33-party GHZ states, genuinely LOSR multipartite nonlocality in more complex networks, and closing the postselection loophole using heralded event-ready sources or high-efficiency single-photon technologies (Huang et al., 2022); multipartite optimization-based decompositions of local and non-local magic accessible through randomized measurements (Ahmad et al., 19 Nov 2025); and recursive Hardy-type tests for $a,b,c\in\{0,1\}$34-party genuine nonlocality of the form

$a,b,c\in\{0,1\}$35

with

$a,b,c\in\{0,1\}$36

or, equivalently,

$a,b,c\in\{0,1\}$37

where $a,b,c\in\{0,1\}$38 (Yu et al., 2013).

In this combined perspective, tripartite non-local magic is not a single settled formalism but an emerging convergence of ideas: irreducible three-party nonlocality, hidden or activated genuine tripartite nonlocality, and basis-invariant non-stabilizerness whose global form is controlled by multipartite entanglement structure. The literature converges on one robust conclusion: pairwise resources plus classical coordination are insufficient to capture the full three-party structure exhibited by GHZ-class correlations and by entanglement-assisted magic spreading (Huang et al., 2022, Hou et al., 26 Mar 2025).