NPA-like Hierarchy for Quantum Instruments

• The paper introduces an NPA-like hierarchy that extends static Bell scenario analyses to sequential protocols using a Heisenberg-picture framework.
• It employs *-homomorphisms to encode sequential quantum instrument actions, preserving causal order and enforcing non-signaling constraints.
• The hierarchy’s structure, with moment matrices and flatness results, underpins finite-dimensional certification and rigorous soundness bounds in compiled nonlocal games.

Searching arXiv for the cited works and related context. An NPA-like hierarchy for quantum instruments is a semidefinite hierarchy that extends the Navascués–Pironio–Acín paradigm from static Bell scenarios to sequential protocols in which parties act one after another and earlier parties apply quantum instruments rather than a single POVM. In the formulation introduced for multipartite compiled nonlocal games, the hierarchy is expressed in the Heisenberg picture, uses $*$-homomorphisms to encode the action of sequential instruments, and characterizes correlations from operationally-non-signaling sequential strategies. It is complete, composable across parties, and equipped with flatness and strict-feasibility results (Baroni et al., 29 Sep 2025). Earlier device-independent work on quantum instruments established a self-testing and certification framework for the pre-measurement state and for each conditional post-measurement state, but did not formulate a moment-matrix relaxation, an SDP hierarchy, or a convergence theory; that earlier framework nonetheless supplies several ingredients that are structurally relevant to a later hierarchy-based treatment (Wagner et al., 2018).

1. Motivation and departure from standard NPA

The standard NPA hierarchy is built for ordinary Bell scenarios in which each party performs a single POVM on a shared state and operators associated with different parties commute. In that setting one introduces operator symbols $f^{(i)}_{a_i|x_i}$ satisfying the usual PVM relations,

$$\bigl(f^{(i)}_{a_i|x_i}\bigr)^* = f^{(i)}_{a_i|x_i},\qquad f^{(i)}_{a_i|x_i}f^{(i)}_{a_i'|x_i}=\delta_{a_i,a_i'}f^{(i)}_{a_i|x_i},\qquad \sum_{a_i} f^{(i)}_{a_i|x_i}=1,$$

together with inter-party commutation,

$$[f^{(i)}_{a_i|x_i},f^{(j)}_{a_j|x_j}]=0\qquad (i\neq j).$$

Words in these symbols index a moment matrix $\Gamma^{(n)}$, and positivity plus the algebraic relations define the SDP relaxation (Baroni et al., 29 Sep 2025).

That static framework cannot model sequential protocols. In a compiled game, earlier players do not merely measure; they apply quantum instruments, namely completely positive maps, that transform the system before the next player acts. The hierarchy for quantum instruments is therefore built around a different operator-algebraic object: a Heisenberg-picture sequential hierarchy that keeps track of the transformations themselves via $*$-homomorphisms, preserving the full algebraic structure of sequential instruments. This is the central reason it generalizes the bipartite sequential NPA of Klep et al. $(2025)$, which was formulated in a Schrödinger-picture, subnormalized-state language and does not extend cleanly beyond two parties (Baroni et al., 29 Sep 2025).

A common misconception is that any instrument-aware Bell analysis already constitutes an NPA hierarchy. The sequential hierarchy is more specific: it is an SDP relaxation with a moment matrix, algebraic word system, affine constraints, and a completeness theorem. By contrast, a certification framework based on Bell inequalities and fidelity bounds can treat quantum instruments without yet becoming an NPA-like optimization hierarchy.

2. Sequential operator algebra and Heisenberg-picture instruments

For a tripartite scenario $A\to B\to C$, the hierarchy models Charlie as the last player with ordinary POVM symbols $f_{c|z}$, while Bob and Alice are represented by instrument maps that pull back later operators. Charlie’s universal measurement algebra is

$A_C = C^*(\{f_{c|z}\}_{c,z}\mid R_C),$

where $f^{(i)}_{a_i|x_i}$0 encodes the projection relations (Baroni et al., 29 Sep 2025).

Bob’s sequential algebra is

$f^{(i)}_{a_i|x_i}$1

generated by symbols satisfying

$f^{(i)}_{a_i|x_i}$2

together with the non-signaling-after-Bob constraint

$f^{(i)}_{a_i|x_i}$3

and normalization

$f^{(i)}_{a_i|x_i}$4

Alice’s algebra $f^{(i)}_{a_i|x_i}$5 obeys analogous projection relations and the causality constraints

$f^{(i)}_{a_i|x_i}$6

The sequential action is encoded by $f^{(i)}_{a_i|x_i}$7-homomorphisms

$f^{(i)}_{a_i|x_i}$8

$f^{(i)}_{a_i|x_i}$9

These maps encode quantum instruments in the Heisenberg picture (Baroni et al., 29 Sep 2025).

For general $$\bigl(f^{(i)}_{a_i|x_i}\bigr)^* = f^{(i)}_{a_i|x_i},\qquad f^{(i)}_{a_i|x_i}f^{(i)}_{a_i'|x_i}=\delta_{a_i,a_i'}f^{(i)}_{a_i|x_i},\qquad \sum_{a_i} f^{(i)}_{a_i|x_i}=1,$$0, the construction becomes a chain of universal $$\bigl(f^{(i)}_{a_i|x_i}\bigr)^* = f^{(i)}_{a_i|x_i},\qquad f^{(i)}_{a_i|x_i}f^{(i)}_{a_i'|x_i}=\delta_{a_i,a_i'}f^{(i)}_{a_i|x_i},\qquad \sum_{a_i} f^{(i)}_{a_i|x_i}=1,$$1-algebras $$\bigl(f^{(i)}_{a_i|x_i}\bigr)^* = f^{(i)}_{a_i|x_i},\qquad f^{(i)}_{a_i|x_i}f^{(i)}_{a_i'|x_i}=\delta_{a_i,a_i'}f^{(i)}_{a_i|x_i},\qquad \sum_{a_i} f^{(i)}_{a_i|x_i}=1,$$2 linked by maps $$\bigl(f^{(i)}_{a_i|x_i}\bigr)^* = f^{(i)}_{a_i|x_i},\qquad f^{(i)}_{a_i|x_i}f^{(i)}_{a_i'|x_i}=\delta_{a_i,a_i'}f^{(i)}_{a_i|x_i},\qquad \sum_{a_i} f^{(i)}_{a_i|x_i}=1,$$3. The resulting architecture is composable: adding a new party appends a new algebra layer and one more non-signaling constraint. This makes the hierarchy suitable for multipartite sequential settings rather than only bipartite ones (Baroni et al., 29 Sep 2025).

3. Moment matrices, word sets, and sequential SDP constraints

As in standard NPA, the hierarchy is organized around a moment matrix indexed by operator words, but the words are generated sequentially so that the algebra respects instrument order. In the tripartite case, the level-$$\bigl(f^{(i)}_{a_i|x_i}\bigr)^* = f^{(i)}_{a_i|x_i},\qquad f^{(i)}_{a_i|x_i}f^{(i)}_{a_i'|x_i}=\delta_{a_i,a_i'}f^{(i)}_{a_i|x_i},\qquad \sum_{a_i} f^{(i)}_{a_i|x_i}=1,$$4 word sets are

$$\bigl(f^{(i)}_{a_i|x_i}\bigr)^* = f^{(i)}_{a_i|x_i},\qquad f^{(i)}_{a_i|x_i}f^{(i)}_{a_i'|x_i}=\delta_{a_i,a_i'}f^{(i)}_{a_i|x_i},\qquad \sum_{a_i} f^{(i)}_{a_i|x_i}=1,$$5

$$\bigl(f^{(i)}_{a_i|x_i}\bigr)^* = f^{(i)}_{a_i|x_i},\qquad f^{(i)}_{a_i|x_i}f^{(i)}_{a_i'|x_i}=\delta_{a_i,a_i'}f^{(i)}_{a_i|x_i},\qquad \sum_{a_i} f^{(i)}_{a_i|x_i}=1,$$6

$$\bigl(f^{(i)}_{a_i|x_i}\bigr)^* = f^{(i)}_{a_i|x_i},\qquad f^{(i)}_{a_i|x_i}f^{(i)}_{a_i'|x_i}=\delta_{a_i,a_i'}f^{(i)}_{a_i|x_i},\qquad \sum_{a_i} f^{(i)}_{a_i|x_i}=1,$$7

For general $$\bigl(f^{(i)}_{a_i|x_i}\bigr)^* = f^{(i)}_{a_i|x_i},\qquad f^{(i)}_{a_i|x_i}f^{(i)}_{a_i'|x_i}=\delta_{a_i,a_i'}f^{(i)}_{a_i|x_i},\qquad \sum_{a_i} f^{(i)}_{a_i|x_i}=1,$$8, the hierarchy uses word sets

$$\bigl(f^{(i)}_{a_i|x_i}\bigr)^* = f^{(i)}_{a_i|x_i},\qquad f^{(i)}_{a_i|x_i}f^{(i)}_{a_i'|x_i}=\delta_{a_i,a_i'}f^{(i)}_{a_i|x_i},\qquad \sum_{a_i} f^{(i)}_{a_i|x_i}=1,$$9

with sequential inclusion

$$[f^{(i)}_{a_i|x_i},f^{(j)}_{a_j|x_j}]=0\qquad (i\neq j).$$0

The level-$$[f^{(i)}_{a_i|x_i},f^{(j)}_{a_j|x_j}]=0\qquad (i\neq j).$$1 moment matrix is

$$[f^{(i)}_{a_i|x_i},f^{(j)}_{a_j|x_j}]=0\qquad (i\neq j).$$2

for a normalized positive linear functional $$[f^{(i)}_{a_i|x_i},f^{(j)}_{a_j|x_j}]=0\qquad (i\neq j).$$3 associated with a state $$[f^{(i)}_{a_i|x_i},f^{(j)}_{a_j|x_j}]=0\qquad (i\neq j).$$4. Positivity of $$[f^{(i)}_{a_i|x_i},f^{(j)}_{a_j|x_j}]=0\qquad (i\neq j).$$5 is equivalent to

$$[f^{(i)}_{a_i|x_i},f^{(j)}_{a_j|x_j}]=0\qquad (i\neq j).$$6

The observed correlation is read from one-letter entries,

$$[f^{(i)}_{a_i|x_i},f^{(j)}_{a_j|x_j}]=0\qquad (i\neq j).$$7

and the Bell objective is

$$[f^{(i)}_{a_i|x_i},f^{(j)}_{a_j|x_j}]=0\qquad (i\neq j).$$8

These are direct sequential analogues of the usual NPA bookkeeping, but now with words adapted to instrument composition (Baroni et al., 29 Sep 2025).

The SDP constraints consist of the familiar NPA identities together with genuinely sequential constraints:

1. PSD

$$[f^{(i)}_{a_i|x_i},f^{(j)}_{a_j|x_j}]=0\qquad (i\neq j).$$9

2. Normalization

$\Gamma^{(n)}$0

3. Hankel / consistency

$\Gamma^{(n)}$1

The new ingredient is operational non-signaling at the level of instruments. For Alice, the hierarchy enforces

$\Gamma^{(n)}$2

for all $\Gamma^{(n)}$3 and appropriate words $\Gamma^{(n)}$4, $\Gamma^{(n)}$5, with total degree at most $\Gamma^{(n)}$6. For Bob, it enforces

$\Gamma^{(n)}$7

for all $\Gamma^{(n)}$8, with $\Gamma^{(n)}$9 and $*$0, again subject to the degree cutoff. The $*$1-party version inserts the corresponding party-$*$2 non-signaling condition inductively. This is the formal sequential analogue of no signaling from later to earlier parties (Baroni et al., 29 Sep 2025).

4. Completeness, flatness, and finite-dimensional strategies

The hierarchy is complete. For the general $*$3-party hierarchy,

$*$4

More explicitly, for a correlation $*$5, the following are equivalent:

1. It arises from a $*$6-partite sequential operationally-non-signaling strategy.
2. It arises from a $*$7-partite commuting-operator strategy.
3. There exists a family of feasible SDP solutions $*$8 to the level-$*$9 hierarchy such that

$(2025)$0

The proof follows an NPA-style compactness argument adapted to the sequential algebra. From a feasible family $(2025)$1, one takes a weak-$(2025)$2 convergent subsequence using Banach–Alaoglu; the limit defines a state $(2025)$3 on the universal sequential $(2025)$4-algebra $(2025)$5; the Hankel and positivity constraints guarantee that $(2025)$6 is a legitimate positive functional; the sequential non-signaling constraints imply that the averaged instrument maps become independent of the earlier input; and the GNS construction then yields a commuting-operator realization, equivalently a $(2025)$7-algebraic sequential strategy. Conversely, any commuting-operator or sequential operationally-non-signaling strategy gives a feasible moment matrix through

$(2025)$8

These statements establish that the hierarchy is a characterization theorem for sequential instrument correlations rather than merely a one-sided relaxation (Baroni et al., 29 Sep 2025).

Two auxiliary structural results sharpen this characterization. First, if at some level $(2025)$9,

$A\to B\to C$0

then the solution is flat. The existence of a flat optimal solution at some finite level is equivalent to the existence of a finite-dimensional optimal quantum strategy. Second, for every level $A\to B\to C$1, the SDP is strictly feasible: there exists a feasible matrix $A\to B\to C$2. The proof embeds the sequential hierarchy as a principal submatrix of a standard NPA moment matrix, which is known to be strictly feasible (Baroni et al., 29 Sep 2025).

5. Relation to device-independent characterization of quantum instruments

Before the introduction of a sequential NPA-like SDP hierarchy, quantum instruments had already been treated in a fully device-independent Bell-test setting. In that earlier framework, an ideal instrument $A\to B\to C$3 with $A\to B\to C$4 outcomes acting on a $A\to B\to C$5-dimensional system is defined by Kraus operators $A\to B\to C$6 satisfying

$A\to B\to C$7

with outcome map

$A\to B\to C$8

outcome probability

$A\to B\to C$9

and normalized conditional post-measurement state

$f_{c|z}$0

The instrument is also packaged as a trace-preserving map with a classical register $f_{c|z}$1,

$f_{c|z}$2

For an unknown actual device,

$f_{c|z}$3

with

$f_{c|z}$4

A key point is that the post-measurement state is not fixed solely by the Kraus operators in the projective-measurement sense; it can depend on the input state, so certification must address both statistics and conditional output states (Wagner et al., 2018).

The device-independent scenario is bipartite. An unknown source produces $f_{c|z}$5 shared by Alice and Bob; Alice has two binary measurements $f_{c|z}$6; Bob has four binary measurements $f_{c|z}$7; and Bob applies the instrument before the later measurement choices relevant for post-measurement certification. The certification proceeds in two steps. First, Alice and Bob estimate the CHSH value

$f_{c|z}$8

which certifies that the shared state is close to $f_{c|z}$9 up to local extraction maps. Second, conditioned on the instrument outcome, Alice and Bob use $A_C = C^*(\{f_{c|z}\}_{c,z}\mid R_C),$0 together with $A_C = C^*(\{f_{c|z}\}_{c,z}\mid R_C),$1 to self-test the conditional post-measurement state. For outcome $A_C = C^*(\{f_{c|z}\}_{c,z}\mid R_C),$2,

$A_C = C^*(\{f_{c|z}\}_{c,z}\mid R_C),$3

and a Bell inequality $A_C = C^*(\{f_{c|z}\}_{c,z}\mid R_C),$4 with maximal quantum value $A_C = C^*(\{f_{c|z}\}_{c,z}\mid R_C),$5 certifies closeness to

$A_C = C^*(\{f_{c|z}\}_{c,z}\mid R_C),$6

through

$A_C = C^*(\{f_{c|z}\}_{c,z}\mid R_C),$7

A related Bell test certifies

$A_C = C^*(\{f_{c|z}\}_{c,z}\mid R_C),$8

for the other outcome. The global post-measurement state with register is

$A_C = C^*(\{f_{c|z}\}_{c,z}\mid R_C),$9

The central certification target is

$f^{(i)}_{a_i|x_i}$00

and the combined fidelity bound is

$f^{(i)}_{a_i|x_i}$01

That framework provides explicit Bell inequalities, local extraction channels, and a Choi–Jamiołkowski viewpoint,

$f^{(i)}_{a_i|x_i}$02

but it does not present an NPA hierarchy, a moment matrix, a semidefinite relaxation, a noncommutative polynomial optimization program, or a hierarchy of outer approximations to the quantum set (Wagner et al., 2018).

This difference is conceptually important. The earlier work certifies the pre-measurement state and the family of conditional post-measurement states, then combines these into certification of the instrument. The later hierarchy instead optimizes over sequential behaviors by means of moment matrices and affine constraints. A plausible implication is that the earlier certification framework supplies natural target data and operator relations, while the later paper supplies the actual hierarchy machinery.

6. Soundness applications and conceptual significance

The hierarchy was introduced as the main technical tool in the proof of quantitative quantum soundness for all multipartite compiled nonlocal games. In the compiled setting, an efficient quantum prover yields a matrix

$f^{(i)}_{a_i|x_i}$03

that is PSD and normalized but only approximately satisfies the sequential non-signaling constraints. The paper describes this as a weakly non-signaling or pseudo-feasible solution. To convert such an object into an actual feasible point, it develops a geometric correction argument (Baroni et al., 29 Sep 2025).

The correction starts from a linear map $f^{(i)}_{a_i|x_i}$04 that collects all affine constraints, namely Hankel consistency and all party-$f^{(i)}_{a_i|x_i}$05 sequential non-signaling conditions. Feasibility is equivalent to $f^{(i)}_{a_i|x_i}$06. The compiled matrix satisfies

$f^{(i)}_{a_i|x_i}$07

so it is close to the affine constraint set. One then projects onto the affine subspace,

$f^{(i)}_{a_i|x_i}$08

which restores the linear constraints but may destroy positivity. Positivity is then restored by convex mixing with a strictly feasible point,

$f^{(i)}_{a_i|x_i}$09

where $f^{(i)}_{a_i|x_i}$10 is the smallest eigenvalue of $f^{(i)}_{a_i|x_i}$11 and $f^{(i)}_{a_i|x_i}$12 is the projection error. After renormalization,

$f^{(i)}_{a_i|x_i}$13

one obtains a feasible point satisfying

$f^{(i)}_{a_i|x_i}$14

This closeness is then used to upper-bound the compiled score by the SDP optimum (Baroni et al., 29 Sep 2025).

The resulting soundness bounds are quantitative. Every compiled strategy satisfies

$f^{(i)}_{a_i|x_i}$15

If the original game has a finite-dimensional optimal quantum strategy, then for some finite level $f^{(i)}_{a_i|x_i}$16,

$f^{(i)}_{a_i|x_i}$17

and hence

$f^{(i)}_{a_i|x_i}$18

Within this application, the hierarchy serves both as a characterization theorem for correlations from operationally-non-signaling sequential strategies and as the technical engine behind the finite-$f^{(i)}_{a_i|x_i}$19 soundness bounds (Baroni et al., 29 Sep 2025).

The main conceptual significance is therefore twofold. First, the hierarchy replaces static commuting POVMs by a layered algebra of sequential instrument maps, so it models transformation rather than only observation. Second, it shows that the appropriate extension of NPA to quantum instruments is not merely a matter of adding more outcome symbols; it requires encoding causal order, operational non-signaling, and the Heisenberg-picture action of instruments directly into the moment-matrix formalism.