Quantum Causal Index: Concepts & Measures

• Quantum Causal Index encompasses operator-valued interventional channels, comb orderings, and QCMI-based measures that formalize quantum causal structure.
• It leverages methods like recursive comb algorithms and cross-entropy metrics to quantify causal influence in many-body and monitored dynamics.
• The framework bridges theory and experiment by enabling causal order discovery, directional flow analysis, and state-dependent assessment of nonclassical effects.

Searching arXiv for papers on "Quantum Causal Index" and closely related quantum causal influence measures. The expression Quantum Causal Index has been used in the recent quantum-causality literature to denote several distinct but related objects: an ordered list of input–output slots in a quantum comb, an asymmetric conditional-mutual-information-based directional measure in many-body systems, an operator-valued interventional channel reconstructed from observational data, and, in monitored dynamics, an experimentally accessible cross-entropy quantum causal influence that functions as a quantitative causal index (Bai et al., 2020, Ghosh, 16 Aug 2025, Friend et al., 2023, Wang et al., 18 Jan 2026). Across these usages, the common objective is to formalize how local interventions, measurements, or perturbations modify downstream statistics or process structure in settings where classical light-cone intuition, state-independence, or purely classical causal assumptions are inadequate.

1. Multiple formal meanings of the term

In the quantum-comb literature, the Quantum Causal Index is an ordering object. For a process consistent with a comb ordering

$$(A_{\sigma(1)},B_{\pi(1)}) \to (A_{\sigma(2)},B_{\pi(2)}) \to \cdots \to (A_{\sigma(n)},B_{\pi(n)}),$$

the index is the ordered list

$$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$

or simply the pair of permutations $(\sigma,\pi)$ (Bai et al., 2020). In this usage, the index encodes causal order discovery for a black-box process represented by a comb Choi operator.

In a quantum-many-body usage, the Quantum Causal Index is identified with an asymmetric quantum conditional mutual information:

$$QCI(A\to B|C)\equiv I(A;B|C)=\sum_x p_x\,I(B:C)_{\rho^x},$$

where a quantum instrument acts on $A$, one conditions on the outcome $x$, and one measures residual correlations between $B$ and $C$ in the post-measurement branches (Ghosh, 16 Aug 2025). Here the index is explicitly directional, because the intervention is performed only on $A$.

A third usage is operator-valued rather than scalar. The relevant quantity is the interventional channel

$${\mathcal C}_{A\to B}:=\Phi_{B\mid \mathsf{do}(A)}\in\mathrm{CPTP}(A\to B),$$

or equivalently its Choi operator

$$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$0

which is reconstructed under structural assumptions such as a quantum front-door criterion (Friend et al., 2023). In that formulation, the “index” is the entire causal map rather than a single number.

A fourth line of work does not use the exact term in the title but introduces a quantity that the source material explicitly describes as a quantitative “quantum causal index”: the cross-entropy quantum causal influence (XEQCI), defined for monitored dynamics with measurements and post-selection (Wang et al., 18 Jan 2026). This measure was proposed to quantify causal influence operationally and to remain experimentally accessible on near-term hardware.

This suggests that the phrase functions less as a uniquely standardized invariant than as a family label for objects that quantify or encode quantum causal structure under different operational assumptions.

2. Comb-based causal order discovery

In the theory of quantum combs, a multipartite process with definite causal order is represented by a positive operator

$$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$1

subject to nested link-product normalization constraints such as

$$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$2

continuing recursively down to

$$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$3

(Bai et al., 2020). These constraints guarantee that the global map can be built by wiring together $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$4 CPTP maps in series, each with memory.

Within this formalism, the Quantum Causal Index is the recovered slot order of the black-box process. A correct index $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$5 is one for which there exists a positive operator $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$6 satisfying the comb-normalization conditions in the corresponding slot order and such that $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$7 for small $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$8, in trace norm on Choi operators or diamond norm on channels (Bai et al., 2020).

The central algorithmic subroutine is FindLast$$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$9, which determines which input/output pair occupies the last tooth in the unknown order. It tests candidate pairs $(\sigma,\pi)$0 by checking whether $(\sigma,\pi)$1 is independent of all wires except $(\sigma,\pi)$2 using informationally complete inputs, maximally entangled states on other inputs, and comparisons of reduced outputs after tracing over $(\sigma,\pi)$3 (Bai et al., 2020). Once the last pair is found, it is removed by feeding the maximally mixed state into the input and tracing out the associated output, and the procedure is repeated recursively.

The stated query complexity is

$(\sigma,\pi)$4

with the polynomial-time guarantee tied to low Kraus rank or small memory dimension (Bai et al., 2020). For the noiseless, unitary-memory setting, the source gives

$(\sigma,\pi)$5

which for fixed $(\sigma,\pi)$6 is $(\sigma,\pi)$7 (Bai et al., 2020).

A concrete three-tooth example in the source takes unknown order $(\sigma,\pi)$8 and $(\sigma,\pi)$9, corresponding to

$$QCI(A\to B|C)\equiv I(A;B|C)=\sum_x p_x\,I(B:C)_{\rho^x},$$0

and the recursive procedure recovers the index

$$QCI(A\to B|C)\equiv I(A;B|C)=\sum_x p_x\,I(B:C)_{\rho^x},$$1

tooth by tooth (Bai et al., 2020).

3. Directional QCMI as a scalar causal index

A distinct proposal identifies the Quantum Causal Index with an asymmetric QCMI derived from a measurement-based intervention on subsystem $$QCI(A\to B|C)\equiv I(A;B|C)=\sum_x p_x\,I(B:C)_{\rho^x},$$2 (Ghosh, 16 Aug 2025). Starting from a tripartite state $$QCI(A\to B|C)\equiv I(A;B|C)=\sum_x p_x\,I(B:C)_{\rho^x},$$3, one considers Kraus operators $$QCI(A\to B|C)\equiv I(A;B|C)=\sum_x p_x\,I(B:C)_{\rho^x},$$4 on $$QCI(A\to B|C)\equiv I(A;B|C)=\sum_x p_x\,I(B:C)_{\rho^x},$$5 with $$QCI(A\to B|C)\equiv I(A;B|C)=\sum_x p_x\,I(B:C)_{\rho^x},$$6, producing branch states

$$QCI(A\to B|C)\equiv I(A;B|C)=\sum_x p_x\,I(B:C)_{\rho^x},$$7

where

$$QCI(A\to B|C)\equiv I(A;B|C)=\sum_x p_x\,I(B:C)_{\rho^x},$$8

The directional quantity is then

$$QCI(A\to B|C)\equiv I(A;B|C)=\sum_x p_x\,I(B:C)_{\rho^x},$$9

and the paper identifies

$A$0

(Ghosh, 16 Aug 2025).

The asymmetry is operational rather than formal: one intervenes only on $A$1, then inspects how much correlation remains between $A$2 and $A$3. The quantity vanishes whenever a measurement on $A$4 destroys any residual $A$5–$A$6 correlations (Ghosh, 16 Aug 2025). This differs from the standard symmetric quantum conditional mutual information

$A$7

which obeys $A$8 (Ghosh, 16 Aug 2025).

For lattice systems, the construction yields an effective causal propagation protocol. One sets $A$9 site $x$0, $x$1 site $x$2, and $x$3 the spins in between, performs a projective measurement on $x$4 at $x$5, evolves under $x$6, and computes $x$7. The arrival time is defined by

$x$8

with effective velocity

$x$9

or equivalently by fitting $B$0 and taking $B$1 (Ghosh, 16 Aug 2025).

The source reports several spin-chain examples. For a transverse-field Ising chain with $B$2, $B$3, and $B$4, $B$5 is nearly zero for small $B$6, then rises and exhibits damped oscillations (Ghosh, 16 Aug 2025). For an XX chain with $B$7, $B$8, $B$9, initial state $C$0, and $C$1, the time dependence again starts near zero, then after a finite delay begins to grow and oscillate (Ghosh, 16 Aug 2025). For an XX chain with $C$2 and $C$3, the source gives $C$4 bits, $C$5, and therefore $C$6, with comparison to $C$7 and $C$8 (Ghosh, 16 Aug 2025).

4. Interventional-channel and operator-valued formulations

In the identification framework of quantum causal inference, the central object is not necessarily a scalar. The process-theoretic proposal treats the interventional channel itself as the Quantum Causal Index:

$C$9

with equivalent Choi-matrix representation

$A$0

(Friend et al., 2023). Scalar indices can then be extracted from this channel, including the diamond-norm variation

$A$1

or an average causal effect under a prior $A$2 (Friend et al., 2023).

This framework emphasizes identifiability from observational data obtained by restricting “observations” to projective measurements. The source explicitly notes that quantum measurements do not trivialize the identification problem: there exist scenarios for which quantum causal identification is not possible, and sufficient conditions are then provided for cases where it is (Friend et al., 2023).

Two structural results are highlighted. The first is a quantum analogue of the front-door criterion, under assumptions that $A$3 $A$4 is the only path from $A$5 to $A$6 and $A$7 a latent $A$8 is a common cause of $A$9 and $${\mathcal C}_{A\to B}:=\Phi_{B\mid \mathsf{do}(A)}\in\mathrm{CPTP}(A\to B),$$0 but not of $${\mathcal C}_{A\to B}:=\Phi_{B\mid \mathsf{do}(A)}\in\mathrm{CPTP}(A\to B),$$1 (Friend et al., 2023). Under these assumptions, the interventional state $${\mathcal C}_{A\to B}:=\Phi_{B\mid \mathsf{do}(A)}\in\mathrm{CPTP}(A\to B),$$2 is identified by an explicit adjustment formula involving observational projectors $${\mathcal C}_{A\to B}:=\Phi_{B\mid \mathsf{do}(A)}\in\mathrm{CPTP}(A\to B),$$3 and $${\mathcal C}_{A\to B}:=\Phi_{B\mid \mathsf{do}(A)}\in\mathrm{CPTP}(A\to B),$$4 and a CPTP map $${\mathcal C}_{A\to B}:=\Phi_{B\mid \mathsf{do}(A)}\in\mathrm{CPTP}(A\to B),$$5 recovered from observational statistics (Friend et al., 2023).

The second is a more general single-intervention criterion, referred to in the source as Theorem 8.1 of [JKZ19], which expresses the interventional channel $${\mathcal C}_{A\to B}:=\Phi_{B\mid \mathsf{do}(A)}\in\mathrm{CPTP}(A\to B),$$6 as a composition $${\mathcal C}_{A\to B}:=\Phi_{B\mid \mathsf{do}(A)}\in\mathrm{CPTP}(A\to B),$$7 of operator expressions reconstructed from observational data across two cuts of the comb (Friend et al., 2023).

The process-theoretic significance lies in a unification strategy: classical causal identification lives in a Markov category, whereas quantum theory does not because of no broadcasting; the proposed extension instead works in an arbitrary symmetric monoidal category with discarding, such as CPM, and represents causal assumptions by formal string diagrams realized by symmetric-monoidal functors (Friend et al., 2023). In this setting, the Quantum Causal Index is operationally the CPTP map one would obtain under a surgical intervention on $${\mathcal C}_{A\to B}:=\Phi_{B\mid \mathsf{do}(A)}\in\mathrm{CPTP}(A\to B),$$8 and subsequent observation at $${\mathcal C}_{A\to B}:=\Phi_{B\mid \mathsf{do}(A)}\in\mathrm{CPTP}(A\to B),$$9.

5. Cross-entropy quantum causal influence in monitored dynamics

In monitored many-body dynamics with measurements and post-selection, Wang, Tang, and Qi introduce the cross-entropy quantum causal influence (XEQCI) as a practical measure of causal influence (Wang et al., 18 Jan 2026). Let $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$00 and $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$01 be spacetime regions of a quantum system undergoing possibly non-unitary monitored evolution. One perturbs region $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$02 by a unitary $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$03 and probes region $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$04 by a POVM $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$05. Writing $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$06 for the Born probability of outcome $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$07 in $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$08 under perturbation $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$09, the cross-entropy causal index for one choice of $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$10 is

$$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$11

Averaging over $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$12 from a unitary 1-design on $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$13 and over monitored circuits gives

$$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$14

(Wang et al., 18 Jan 2026).

The measure is motivated by the difficulty of directly estimating nonlinear functionals of the final super-density operator in monitored circuits. According to the source, trace distances and related quantities require multi-copy post-selection and become exponentially costly, whereas the cross-entropy construction extracts a linear functional of the state trajectory on $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$15 and thereby avoids the sampling explosion (Wang et al., 18 Jan 2026). Operationally, one runs the circuit, records the measurement trajectory $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$16 and outcome $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$17, computes offline

$$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$18

and averages the observed $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$19 under the sampling distribution $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$20 to reproduce $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$21 (Wang et al., 18 Jan 2026).

A central feature of the XEQCI program is state dependence on both initial and final boundary states. For a pure product initial state and no post-selection, the XEQCI landscape shows a conventional forward-directed light cone: only $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$22 in the future of $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$23 has nontrivial $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$24 (Wang et al., 18 Jan 2026). If one instead begins with maximally mixed $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$25 and post-selects onto pure $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$26, then by time reversal the light cone is inverted: only $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$27 to the past of $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$28 has nonzero influence (Wang et al., 18 Jan 2026). More generally, partially mixed $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$29 and $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$30 interpolate between forward, backward, or two-sided causal futures (Wang et al., 18 Jan 2026).

The source further reports that numerical Clifford-circuit studies reveal that the direction in which $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$31 decays most slowly points from low entropy toward high entropy (Wang et al., 18 Jan 2026). In hybrid circuits with a measurement-induced entanglement transition, the spatial range over which $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$32 remains non-zero peaks at the critical measurement rate $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$33 (Wang et al., 18 Jan 2026).

6. Entropy, analytical models, and comparison with classical causal indices

The monitored-dynamics literature connects the directionality of quantum causal influence to entropy gradients. In a Brownian Gaussian Unitary Ensemble model, the long-time difference between forward and backward influence obeys

$$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$34

where $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$35 is the second Rényi entropy and $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$36 is a dimension-dependent prefactor (Wang et al., 18 Jan 2026). For disjoint regions a similar result holds with a different prefactor, and at arbitrary time separation one recovers an overall $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$37 decay modulated by factors depending on $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$38 (Wang et al., 18 Jan 2026). The sign of

$$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$39

therefore determines whether forward or backward influence dominates, which the source summarizes as a precise statement of “time flows toward increasing entropy” (Wang et al., 18 Jan 2026).

In dual-unitary circuits, inserting a finite patch $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$40 of non-dual gates and post-selection inside otherwise dual-unitary dynamics produces state-dependent reconfiguration of causal cones (Wang et al., 18 Jan 2026). The source distinguishes three cases: if only one quadrant is blocked by $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$41, then $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$42 everywhere; if two quadrants are blocked, the causal future reduces to two $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$43 light rays; if three quadrants are blocked, the remaining quadrant gives a full $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$44 cone of nonzero $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$45, with “time” radiating outward from $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$46 (Wang et al., 18 Jan 2026).

These results contrast sharply with classical light-cone causal indices. The source states that classical causal influence is state-independent and strictly confined within a future light cone determined by a Lieb–Robinson bound, whereas quantum causal influence is generically nonlocal, state-dependent through $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$47, and can violate classical notions of future versus past by exhibiting inverted cones, bisected cones, or null rays (Wang et al., 18 Jan 2026). In the QCMI-based spin-chain setting, one still finds a finite-speed causal spread with $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$48, together with coherent oscillations (Ghosh, 16 Aug 2025). Taken together, these results suggest a spectrum of quantum-causal behavior ranging from modified but still cone-like propagation to causality landscapes whose geometry depends on measurement, post-selection, and boundary conditions.

7. Relation to average causal effect and nonclassical causal influence

Several related works quantify quantum causal influence using generalizations of average causal effect rather than the term Quantum Causal Index. These are conceptually relevant because they provide scalar interventional measures against which QCI-type quantities can be compared.

One proposal defines the quantum average causal effect by replacing total-variation distance with trace distance:

$$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$49

or, for qubit interventions, by averaging over pairs of pure inputs on the Bloch sphere (Hutter et al., 2022). The source states the bounds

$$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$50

with $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$51 corresponding to no causal influence and $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$52 to perfectly distinguishable output states (Hutter et al., 2022). It also gives explicit values for paradigmatic two-qubit gates: local unitaries have QACE $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$53, CNOT and CZ have $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$54, the $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$55-gate has approximately $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$56, $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$57 has approximately $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$58, and SWAP has $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$59 (Hutter et al., 2022). In measurement-based quantum computation, the ideal two-qubit graph state yields QACE $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$60, separable resources satisfy

$$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$61

and pure entangled resources of the family $$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$62 exceed the separable bound (Hutter et al., 2022).

A related but distinct framework considers qACE in the presence of a quantum common cause in an instrumental scenario:

$$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$63

(Gachechiladze et al., 2020). In that setting, the source shows that every pure bipartite entangled state violates classical ACE bounds even in the simplest instrumental scenario where Bell inequalities cannot be violated (Gachechiladze et al., 2020). It also gives the hierarchy

$$[(\sigma(1),\pi(1)),(\sigma(2),\pi(2)),\ldots,(\sigma(n),\pi(n))],$$64

for classical, quantum, and nonsignaling common-cause models (Gachechiladze et al., 2020).

A common misconception is that quantum causal quantification is exhausted either by Bell nonlocality or by fixed causal order. The collected literature does not support that view. The instrumental-scenario results show nonclassical causal signatures without Bell violation (Gachechiladze et al., 2020); the comb results show efficient recovery of hidden causal order under low Kraus rank (Bai et al., 2020); the identification framework shows that interventional effects can sometimes be reconstructed from observational projective data under structural assumptions (Friend et al., 2023); the QCMI formulation yields directional finite-speed propagation in spin chains (Ghosh, 16 Aug 2025); and the XEQCI program shows state-dependent, post-selection-sensitive causal geometries, including inverted light cones, in monitored dynamics (Wang et al., 18 Jan 2026).

In this literature, the Quantum Causal Index is therefore best understood not as a single universally adopted invariant but as a cluster of operational formalisms for encoding or quantifying quantum causal structure under different experimental, information-theoretic, and process-theoretic regimes.