IC Preservability: Completeness Under Transformation

• IC Preservability is defined as the property ensuring that a complete object retains full information after undergoing restrictions, transformations, or execution processes.
• In quantum settings, it quantifies whether measurements or channels maintain full informational completeness via metrics like Q_IC and polynomial counting techniques.
• Across domains like local tomography, process verification, and logic programming, methods such as bi-convex optimization and coverage criteria ensure that completeness is preserved under operational constraints.

Informational Completeness-Preservability (IC-preservability) denotes a family of preservation properties concerning whether a notion of completeness survives a restriction, transformation, or execution. In quantum measurement theory, it refers to whether a continuous-variable measurement remains informationally complete after truncation to a finite-dimensional subspace, or whether a quantum channel preserves the informational completeness of measurements under the Heisenberg action (Sych et al., 2012, Ghai et al., 2 Jun 2026). Closely related work studies when local reduced density matrices determine all non-local observables, exactly or up to an error controlled by local conditional entropies (Kim, 2014). In data-centric processes and in logic programming, analogous preservation questions concern whether query answers remain complete along process executions, and whether pruning of SLD-trees preserves completeness of required answers (Razniewski et al., 2013, Drabent, 2014).

1. Conceptual scope

Taken together, these formulations suggest a common schema: a complete object is subjected to a restriction, channel, pruning operation, or process evolution; the central question is whether the induced object remains complete for the target specification.

Setting	Complete object	Preserving transformation
Continuous-variable tomography	POVM on $\mathcal H$	restriction to $\mathcal H_d$
Qubit channels	IC POVM $A$	$\mathcal E^\dagger(A)$
Local observables	local reduced density matrices	reconstruction of the global state
Quality-aware processes	complete query answer	all paths in a QATS
Logic programs	completeness w.r.t. $S$	pruning of SLD-trees

The quantum-information usages are the most literal. A POVM $A=\{A_x\}$ is informationally complete if the outcome probabilities $p_x(\rho)=\mathrm{Tr}[\rho A_x]$ separate all density operators, equivalently if the measurement operators span the full operator space (Ghai et al., 2 Jun 2026). In continuous variables, the same injectivity criterion is applied after projecting onto a truncated Fock subspace (Sych et al., 2012).

The process-theoretic and logic-programming formulations use the vocabulary of completeness rather than informational completeness in the POVM sense. Even so, they address the same structural problem: whether a completeness property survives a controlled loss of operational alternatives, such as delayed data copying in a process or branch pruning in SLD-resolution (Razniewski et al., 2013, Drabent, 2014).

2. Continuous-variable truncation and finite-setting tomography

For continuous-variable systems, a measurement scheme on a Hilbert space $\mathcal H$ is described by a POVM $M=\{E_\lambda\}_{\lambda\in\Lambda}$ with

$$\rho \mapsto p(\lambda)=\mathrm{Tr}[\rho E_\lambda].$$

It is informationally complete on $\mathcal H_d$0 if this map is injective on density operators. Fixing the $\mathcal H_d$1-dimensional subspace $\mathcal H_d$2 spanned by the first $\mathcal H_d$3 Fock states and writing $\mathcal H_d$4 for the projector onto $\mathcal H_d$5, the restriction of $\mathcal H_d$6 is $\mathcal H_d$7. The measurement remains informationally complete on $\mathcal H_d$8 iff these operators span the full $\mathcal H_d$9-dimensional operator space on $A$0. A continuous-variable measurement is IC-preservable up to dimension $A$1 if, when restricted to any $A$2-dimensional subspace $A$3, it remains informationally complete on $A$4 (Sych et al., 2012).

The main theorem in this setting concerns homodyne tomography. For

$A$5

let $A$6 be the POVM formed by homodyne projections at $A$7 distinct phases $A$8. Then $A$9 is informationally complete on $\mathcal E^\dagger(A)$0 iff

$\mathcal E^\dagger(A)$1

The number of linearly independent operators generated on $\mathcal E^\dagger(A)$2 by $\mathcal E^\dagger(A)$3 quadrature cuts is

$\mathcal E^\dagger(A)$4

so injectivity is reached precisely when $\mathcal E^\dagger(A)$5 (Sych et al., 2012).

The proof uses polynomial counting in the Fock basis. For a truncated state, the homodyne probability density at phase $\mathcal E^\dagger(A)$6 is, up to a Gaussian factor, a finite sum of terms

$\mathcal E^\dagger(A)$7

where $\mathcal E^\dagger(A)$8 is the Hermite polynomial of degree $\mathcal E^\dagger(A)$9 and $S$0. A single cut yields $S$1 independent real parameters; subsequent cuts contribute $S$2, then $S$3, and so on, until saturation at $S$4 (Sych et al., 2012).

The same polynomial-counting method generalizes to other continuous-variable schemes. If the Born probabilities in $S$5 can be written as linear combinations of monomials $S$6, then sampling at $S$7 distinct values can generate

$S$8

for $S$9, saturating at $A=\{A_x\}$0 when $A=\{A_x\}$1 exceeds the ceiling of $A=\{A_x\}$2. IC on $A=\{A_x\}$3 occurs iff $A=\{A_x\}$4 and the samples are chosen so that the Vandermonde-type matrices are nondegenerate (Sych et al., 2012).

Several examples delineate the notion sharply. Heterodyne detection is IC-preservable, but on $A=\{A_x\}$5 a finite grid of points $A=\{A_x\}$6 must be large enough, at least $A=\{A_x\}$7 distinct points in general position, to recover all $A=\{A_x\}$8 matrix elements. Photon-number measurement yields only diagonal elements $A=\{A_x\}$9, never any off-diagonal information; hence it is not IC on any $p_x(\rho)=\mathrm{Tr}[\rho A_x]$0 subspace, and it is not IC-preservable as defined. Displacement-plus-photon-counting becomes informationally complete on $p_x(\rho)=\mathrm{Tr}[\rho A_x]$1 after sufficiently many displacements, roughly requiring $p_x(\rho)=\mathrm{Tr}[\rho A_x]$2 different $p_x(\rho)=\mathrm{Tr}[\rho A_x]$3 values. Prior information can reduce the number of settings: if only levels $p_x(\rho)=\mathrm{Tr}[\rho A_x]$4 are occupied, then roughly $p_x(\rho)=\mathrm{Tr}[\rho A_x]$5 quadrature cuts suffice instead of $p_x(\rho)=\mathrm{Tr}[\rho A_x]$6 (Sych et al., 2012).

3. Qubit channels and quantitative IC-preservability

In finite dimensions, and specifically for qubits, IC-preservability is defined as a property of quantum channels. For a POVM $p_x(\rho)=\mathrm{Tr}[\rho A_x]$7 on a $p_x(\rho)=\mathrm{Tr}[\rho A_x]$8-dimensional Hilbert space $p_x(\rho)=\mathrm{Tr}[\rho A_x]$9, the IC-power of $\mathcal H$0 is

$\mathcal H$1

One has

$\mathcal H$2

so $\mathcal H$3 is a faithful IC-measure. For a channel $\mathcal H$4 acting in the Heisenberg picture, its IC-preservability is

$\mathcal H$5

Thus $\mathcal H$6 exactly when $\mathcal H$7 is incomplete for every POVM $\mathcal H$8 (Ghai et al., 2 Jun 2026).

The paper also gives the equivalent form

$\mathcal H$9

hence

$M=\{E_\lambda\}_{\lambda\in\Lambda}$0

This can be cast as a bi-convex optimization or, in principle, as an SDP (Ghai et al., 2 Jun 2026).

Several structural properties are established. The quantity is nonnegative and faithful. It is unitarily invariant:

$M=\{E_\lambda\}_{\lambda\in\Lambda}$1

for unitary conjugations $M=\{E_\lambda\}_{\lambda\in\Lambda}$2. If $M=\{E_\lambda\}_{\lambda\in\Lambda}$3, then

$M=\{E_\lambda\}_{\lambda\in\Lambda}$4

If $M=\{E_\lambda\}_{\lambda\in\Lambda}$5 for a statistical morphism $M=\{E_\lambda\}_{\lambda\in\Lambda}$6, then

$M=\{E_\lambda\}_{\lambda\in\Lambda}$7

If $M=\{E_\lambda\}_{\lambda\in\Lambda}$8, then

$M=\{E_\lambda\}_{\lambda\in\Lambda}$9

The same work explicitly evaluates the informational completeness of qubit symmetric informationally complete measurements and shows that it is an upper bound for all qubit minimal informationally complete measurements (Ghai et al., 2 Jun 2026).

For qubit CPTP maps, the characterization is expressed in Bloch form. Every qubit channel $$\rho \mapsto p(\lambda)=\mathrm{Tr}[\rho E_\lambda].$$0 can be written as

$$\rho \mapsto p(\lambda)=\mathrm{Tr}[\rho E_\lambda].$$1

with signed singular values $$\rho \mapsto p(\lambda)=\mathrm{Tr}[\rho E_\lambda].$$2 and translation $$\rho \mapsto p(\lambda)=\mathrm{Tr}[\rho E_\lambda].$$3. By unitary invariance one may reduce to the diagonal channel. The resulting bounds are

$$\rho \mapsto p(\lambda)=\mathrm{Tr}[\rho E_\lambda].$$4

with $$\rho \mapsto p(\lambda)=\mathrm{Tr}[\rho E_\lambda].$$5 (Ghai et al., 2 Jun 2026).

Important special cases are explicit. For unital channels $$\rho \mapsto p(\lambda)=\mathrm{Tr}[\rho E_\lambda].$$6,

$$\rho \mapsto p(\lambda)=\mathrm{Tr}[\rho E_\lambda].$$7

For isotropic depolarizing channels, where all $$\rho \mapsto p(\lambda)=\mathrm{Tr}[\rho E_\lambda].$$8,

$$\rho \mapsto p(\lambda)=\mathrm{Tr}[\rho E_\lambda].$$9

For the phase-flip or dephasing channel, with $\mathcal H_d$00 and $\mathcal H_d$01,

$\mathcal H_d$02

For unitary channels,

$\mathcal H_d$03

so all informational completeness is perfectly preserved (Ghai et al., 2 Jun 2026).

A further relation links IC-preservability to coherence. With absolute output coherence defined as

$\mathcal H_d$04

Theorem 13 proves

$\mathcal H_d$05

This means that any channel preserving informational completeness up to some finite amount must necessarily generate at least that much absolute coherence in its outputs (Ghai et al., 2 Jun 2026).

4. Local marginals as informationally complete data

A closely related notion arises in multipartite tomography from local observables. Let $\mathcal H_d$06 be an $\mathcal H_d$07-partite state. For each site $\mathcal H_d$08, choose two small shields $\mathcal H_d$09 and $\mathcal H_d$10, and define

$\mathcal H_d$11

Set

$\mathcal H_d$12

Because each term depends only on reduced density matrices $\mathcal H_d$13 and $\mathcal H_d$14, this is a purely local condition (Kim, 2014).

The main theorem states that if two states $\mathcal H_d$15 agree exactly on each local cluster,

$\mathcal H_d$16

then

$\mathcal H_d$17

In particular, if $\mathcal H_d$18, then the global state is uniquely fixed by its local marginals. The expectation value of any non-local observable $\mathcal H_d$19 can then be approximated from the maximum-entropy completion $\mathcal H_d$20, with error bounded by

$\mathcal H_d$21

In the exact Markov case, one also has the explicit Petz-recovery formula

$\mathcal H_d$22

These results show that suitable local observables can be informationally complete for global properties, although the paper does not use the channel-based term IC-preservability (Kim, 2014).

The reconstruction procedure is formulated as a convex program:

$\mathcal H_d$23

For tomography, the number of samples needed per marginal is $\mathcal H_d$24, and because there are $\mathcal H_d$25 blocks, the total sample complexity is polynomial in $\mathcal H_d$26 and $\mathcal H_d$27 (Kim, 2014).

The principal application is to unique ground states of 2D local Hamiltonians with a uniform gap, assuming the area law

$\mathcal H_d$28

for simply connected regions $\mathcal H_d$29. Choosing shields of thickness $\mathcal H_d$30 after coarse-graining into blocks of size $\mathcal H_d$31 yields

$\mathcal H_d$32

and therefore

$\mathcal H_d$33

By taking $\mathcal H_d$34, the error can shrink polynomially in $\mathcal H_d$35 (Kim, 2014).

5. Query completeness over quality-aware processes

In data quality and business-process theory, completeness is defined over pairs of databases. The real world is represented by a database $\mathcal H_d$36, and the information system by a database $\mathcal H_d$37. For a conjunctive query

$\mathcal H_d$38

query completeness means

$\mathcal H_d$39

A quality-aware transition system (QATS) augments an ordinary labelled transition system with real-world effects and copy effects, thereby modeling how data are created in reality and later copied into the information system (Razniewski et al., 2013).

The process-level notion of IC-preservability is stated explicitly. For a QATS $\mathcal H_d$40 and conjunctive query $\mathcal H_d$41,

$\mathcal H_d$42

Equivalently, no matter how the process executes in the real world and in the information system, the answer to $\mathcal H_d$43 over $\mathcal H_d$44 always coincides with the answer over $\mathcal H_d$45 (Razniewski et al., 2013).

The central characterization is based on risky real-world effects and their repair by subsequent copy effects. A real-world effect

$\mathcal H_d$46

is risky for $\mathcal H_d$47 if there is some way to insert a fresh $\mathcal H_d$48-fact under $\mathcal H_d$49 that would change $\mathcal H_d$50's answer. Using the $\mathcal H_d$51-projection $\mathcal H_d$52 of the query, the paper states that $\mathcal H_d$53 is risky iff

$\mathcal H_d$54

is satisfiable. A risky $\mathcal H_d$55 is repaired by later copy effects $\mathcal H_d$56 exactly when

$\mathcal H_d$57

This yields the action-sequence criterion

$\mathcal H_d$58

and the process-level characterization

$\mathcal H_d$59

Here $\mathcal H_d$60 is the set of all copy effects occurring on any path after the occurrence of $\mathcal H_d$61 (Razniewski et al., 2013).

The verification problem reduces to finitely many containment tests. By duplicate removal, it is enough to consider normal action sequences of length at most $\mathcal H_d$62. For each realized normal sequence and for each real-world effect in it, one checks whether the suffix copy effects repair the risky contribution. Reachability of a normal sequence is a graph problem in PTime; the main source of complexity is union-of-conjunctive-queries containment. The resulting bounds are:

• for linear relational queries, IC-preservability can be decided in coNP;
• for linear conjunctive queries, it is coNP-complete;
• for relational conjunctive queries, it is in $\mathcal H_d$63;
• for full conjunctive queries with comparisons, it is $\mathcal H_d$64-complete (Razniewski et al., 2013).

The school-enrollment example illustrates the semantics. For the query

$\mathcal H_d$65

the only real-world effect that can introduce a new Hofer pupil is risky, but the subsequent copy effect copies all newly enrolled Hofer pupils, so

$\mathcal H_d$66

Hence every path preserves completeness for $\mathcal H_d$67. By contrast, the cross-school query

$\mathcal H_d$68

can become incomplete along partial interleavings where one school has copied but the other has not, so the process is not IC-preservable for $\mathcal H_d$69 (Razniewski et al., 2013).

6. Completeness preservation in logic programs under pruning

In logic programming, correctness and completeness are defined relative to a specification $\mathcal H_d$70, where $\mathcal H_d$71 is the Herbrand base and $\mathcal H_d$72 is the least Herbrand model of a definite program $\mathcal H_d$73:

$\mathcal H_d$74

Operationally, for any atomic query $\mathcal H_d$75 and any ground answer substitution $\mathcal H_d$76,

$\mathcal H_d$77

expresses correctness, while

$\mathcal H_d$78

expresses completeness. The paper studies sufficient conditions for completeness and the preservation of completeness under pruning of SLD-trees, including pruning due to Prolog's cut (Drabent, 2014).

The basic sufficient condition is coverage. A ground atom $\mathcal H_d$79 is covered by a clause $\mathcal H_d$80 w.r.t. $\mathcal H_d$81 if there is a ground instance

$\mathcal H_d$82

of $\mathcal H_d$83 with all $\mathcal H_d$84. If every atom of $\mathcal H_d$85 is covered by $\mathcal H_d$86 w.r.t. $\mathcal H_d$87, then $\mathcal H_d$88 is semi-complete w.r.t. $\mathcal H_d$89: for any query whose SLD-tree is finite, $\mathcal H_d$90 produces all answers in $\mathcal H_d$91 demanded by the query. Semi-completeness becomes full completeness if, in addition, either every $\mathcal H_d$92 has some finite SLD-derivation, or $\mathcal H_d$93 is recurrent, or $\mathcal H_d$94 is acceptable (Drabent, 2014).

A direct completeness criterion uses level mappings. With a partial level mapping $\mathcal H_d$95, a ground atom $\mathcal H_d$96 is recurrently covered by $\mathcal H_d$97 w.r.t. $\mathcal H_d$98 and $\mathcal H_d$99 if there is a ground instance

$A$00

with all $A$01, all levels defined, and $A$02 for all $A$03. If all atoms of $A$04 are recurrently covered by $A$05, then $A$06 is complete w.r.t. $A$07 (Drabent, 2014).

For pruning, the paper introduces clause-selection SLD-trees. If $A$08 is partitioned into subprograms $A$09, a c-selection rule picks at each node a selected atom and one of the subprograms, producing a pruned SLD-tree of the full program. If the specification is correspondingly split as $A$10, then completeness of the pruned tree is preserved under three conditions:

1. each atom in $A$11 is covered by $A$12 w.r.t. $A$13;
2. the csSLD-tree is compatible with the split;
3. either $A$14 is recurrent or acceptable, or the tree itself is finite.

Under these conditions, every answer demanded by $A$15 for the root query appears in the pruned tree (Drabent, 2014).

A separate theorem handles cuts in the last clause of each predicate. Using a call-success specification $A$16 and the refined notion of adjustable coverage, one obtains: if the pruned LD-tree is finite and its root query is in $A$17, and each atom of $A$18 is adjustably covered by $A$19 w.r.t. $A$20 and $A$21, then the pruned tree is complete w.r.t. $A$22. The IN/2 example with $A$35 shows that any finite pruned tree where the cut fires still yields all ground solutions in the target specification (Drabent, 2014).

The methodology emphasizes approximate specifications. Instead of fixing the exact least Herbrand model, one may use two specifications:

• $A$23: the atoms that must be computed;
• $A$24: the atoms that are allowed to be computed;

and require

$A$25

The recommended workflow is to first write a pure logical program and prove correctness and completeness declaratively with respect to approximate specifications, then add control such as cuts or delays, and finally use pruning-preservation theorems to verify that no required answers are lost (Drabent, 2014).

7. Scope, limitations, and recurring interpretive issues

A recurring issue across the literature is that completeness before transformation does not by itself imply preserved completeness after transformation. In continuous variables, heterodyne detection is complete on the full infinite-dimensional space, yet after truncation it still requires at least $A$26 sample points in general position to recover all $A$27 matrix elements on $A$28; photon-number measurement remains non-IC on every $A$29 subspace because it never accesses off-diagonal terms (Sych et al., 2012).

In local-marginal reconstruction, local data are not automatically globally complete. The guarantee depends on the locally checkable quantity

$A$30

If $A$31, the global state is uniquely fixed by its local marginals; if $A$32 is merely small, the theorem yields only an approximate trace-distance guarantee. The result is therefore a certificate of approximate global determinacy rather than an unconditional statement about arbitrary local observables (Kim, 2014).

In the channel setting, IC-preservability is quantitative and channel-dependent. Unitary channels preserve informational completeness perfectly, while for the dephasing family $A$33 and vanishes at the fully dephasing point $A$34. The lower and upper bounds for general qubit channels depend on the smallest Bloch singular value and on the translation vector, so the relevant obstruction is geometric rather than purely combinatorial (Ghai et al., 2 Jun 2026).

In process verification, the formalism is restricted to monotonic conjunctive queries and to data-addition effects. The framework does not cover negation or full SQL features such as nested subqueries, and it does not model deletions or updates. Integrity constraints can be integrated, but at the cost of higher complexity. Runtime, instance-aware completeness may succeed even when design-time IC-preservability fails, and a full characterization of that case remains open (Razniewski et al., 2013).

In logic programming, pruning is not completeness-preserving by default. The preservation theorems require coverage conditions, compatibility of the csSLD-tree with the split specification, and either finiteness of the pruned tree or recurrence or acceptability of the program. For cuts in last clauses, c-s-correctness and adjustable coverage are additionally required. The results therefore justify pruning only under explicit structural hypotheses, not as a purely operational heuristic (Drabent, 2014).

These domain-specific formulations suggest a stable underlying theme: IC-preservability is not a single universal invariant, but a class of preservation statements in which completeness is evaluated relative to a specified representation, transformation, and target object. In each setting, the operative question is whether reduced observational access still suffices to recover everything that the specification demands.