--- title: 'IC Preservability: Completeness Under Transformation' url: https://www.emergentmind.com/topics/informational-completeness-preservability-ic-preservability type: topic --- # IC Preservability: Completeness Under Transformation 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 [1211.4967] [2606.03964]. 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 [1405.0137]. 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 [1306.1689] [1411.3015]. ## 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 [2606.03964]. In continuous variables, the same injectivity criterion is applied after projecting onto a truncated Fock subspace [1211.4967]. 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 [1306.1689] [1411.3015]. ## 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 $\mathcal H$ if this map is injective on density operators. Fixing the $d$-dimensional subspace $\mathcal H_d \cong \mathbb C^d$ spanned by the first $d$ Fock states and writing $\Pi_d$ for the projector onto $\mathcal H_d$, the restriction of $M$ is $\{\Pi_d E_\lambda \Pi_d\}$. The measurement remains informationally complete on $\mathcal H_d$ iff these operators span the full $d^2$-dimensional operator space on $\mathcal H_d$. A continuous-variable measurement is IC-preservable up to dimension $d$ if, when restricted to any $d$-dimensional subspace $\mathcal H_d$, it remains informationally complete on $\mathcal H_d$ [1211.4967]. The main theorem in this setting concerns homodyne tomography. For $$ X_\theta=\frac{a e^{-i\theta}+a^\dagger e^{i\theta}}{\sqrt 2}, $$ let $M_m=\{|x;\theta_k\rangle\langle x;\theta_k|:x\in\mathbb R,\ k=1,\dots,m\}$ be the POVM formed by homodyne projections at $m$ distinct phases $\theta_1,\dots,\theta_m$. Then $M_m$ is informationally complete on $\mathcal H_d$ iff $$ m\ge d. $$ The number of linearly independent operators generated on $\mathcal H_d$ by $m$ quadrature cuts is $$ N(m,d)= \begin{cases} m(2d-m),& m1$ subspace, and it is not IC-preservable as defined. Displacement-plus-photon-counting becomes informationally complete on $\mathcal H_d$ after sufficiently many displacements, roughly requiring $O(d^2)$ different $\alpha$ values. Prior information can reduce the number of settings: if only levels $\{0,4,8,\dots,4(d-1)\}$ are occupied, then roughly $m^*\approx\lfloor d/2\rfloor+1$ quadrature cuts suffice instead of $d$ [1211.4967]. ## 3. Qubit channels and quantitative IC-preservability In finite dimensions, and specifically for qubits, IC-preservability is defined as a property of quantum channels. For a POVM $A=\{A_x\}$ on a $d$-dimensional Hilbert space $H$, the IC-power of $A$ is $$ \mathrm{IC}(A)\equiv \mathcal D(A):=\inf_{\rho\ne\sigma} \frac{\sum_x\bigl|\mathrm{Tr}[\rho A_x]-\mathrm{Tr}[\sigma A_x]\bigr|} {\|\rho-\sigma\|_1}\ge 0. $$ One has $$ \mathcal D(A)=0 \Longleftrightarrow A \text{ is informationally incomplete}, $$ so $\mathcal D(A)$ is a faithful IC-measure. For a channel $\mathcal E$ acting in the Heisenberg picture, its IC-preservability is $$ Q_{IC}(\mathcal E):=\sup_A \mathcal D\bigl(\mathcal E^\dagger(A)\bigr) =\sup_A \mathrm{IC}\bigl(\mathcal E^\dagger(A)\bigr). $$ Thus $Q_{IC}(\mathcal E)=0$ exactly when $\mathcal E^\dagger(A)$ is incomplete for every POVM $A$ [2606.03964]. The paper also gives the equivalent form $$ \mathcal D(A)= \min_{\substack{X=X^\dagger,\ \mathrm{Tr}\,X=0\\ \|X\|_1=1}} \sum_x\bigl|\mathrm{Tr}[X A_x]\bigr|, $$ hence $$ Q_{IC}(\mathcal E)= \sup_A \min_{\substack{X=X^\dagger,\ \mathrm{Tr}\,X=0\\ \|X\|_1=1}} \sum_x\Bigl|\mathrm{Tr}\bigl[X\,\mathcal E^\dagger(A_x)\bigr]\Bigr|. $$ This can be cast as a bi-convex optimization or, in principle, as an SDP [2606.03964]. Several structural properties are established. The quantity is nonnegative and faithful. It is unitarily invariant: $$ Q_{IC}(\mathcal E)=Q_{IC}\bigl(\mathcal U\circ \mathcal E\circ \mathcal V\bigr) $$ for unitary conjugations $\mathcal U,\mathcal V$. If $\mathcal E_2=\Phi\circ\mathcal E_1$, then $$ Q_{IC}(\mathcal E_2)\ge Q_{IC}(\mathcal E_1). $$ If $\mathcal E_1=\mathcal N\circ\mathcal E_2$ for a statistical morphism $\mathcal N$, then $$ Q_{IC}(\mathcal E_2)\ge Q_{IC}(\mathcal E_1). $$ If $\mathcal E_2\succeq_{\mathrm{asymp}}\mathcal E_1$, then $$ Q_{IC}(\mathcal E_2)=0 \Longrightarrow Q_{IC}(\mathcal E_1)=0. $$ The same work explicitly evaluates the informational completeness of qubit symmetric informationally complete measurements and shows that it is an upper bound for all qubit minimal informationally complete measurements [2606.03964]. For qubit CPTP maps, the characterization is expressed in Bloch form. Every qubit channel $\Lambda$ can be written as $$ r\mapsto r'=Mr+t,\qquad M=O_1\,\mathrm{diag}(\lambda_1,\lambda_2,\lambda_3)\,O_2, $$ with signed singular values $\{\lambda_i\}$ and translation $t\in\mathbb R^3$. By unitary invariance one may reduce to the diagonal channel. The resulting bounds are $$ \frac{\max(0,\ |\lambda_3|-|t|)}{\sqrt 6} \le Q_{IC}(\Lambda)\le \min_{i=1,2,3}|\lambda_i| \equiv \lambda_{\min}, $$ with $|\lambda_1|\ge |\lambda_2|\ge |\lambda_3|$ [2606.03964]. Important special cases are explicit. For unital channels $(t=0)$, $$ \frac{|\lambda_3|}{\sqrt 6}\le Q_{IC}(\Lambda)\le |\lambda_3|. $$ For isotropic depolarizing channels, where all $\lambda_i=1-p$, $$ Q_{IC}(\Lambda_{\mathrm{dep}(p)})=1-p. $$ For the phase-flip or dephasing channel, with $\lambda_1=\lambda_2=1-2p$ and $\lambda_3=1$, $$ Q_{IC}(\Lambda_{\mathrm{deph}(p)})=|1-2p|. $$ For unitary channels, $$ Q_{IC}(\mathcal U\,\cdot\,\mathcal U^\dagger)=1, $$ so all informational completeness is perfectly preserved [2606.03964]. A further relation links IC-preservability to coherence. With absolute output coherence defined as $$ \mathcal C(\Lambda):=\min_{\text{basis }E}\ \max_\rho C_E\bigl(\Lambda(\rho)\bigr), $$ Theorem 13 proves $$ Q_{IC}(\Lambda)\le \mathcal C(\Lambda). $$ This means that any channel preserving informational completeness up to some finite amount must necessarily generate at least that much absolute coherence in its outputs [2606.03964]. ## 4. Local marginals as informationally complete data A closely related notion arises in multipartite tomography from local observables. Let $\rho$ be an $N$-partite state. For each site $k$, choose two small shields $\mathcal M_k\subset\{1,\dots,k-1\}$ and $\mathcal M'_k\subset\{k+1,\dots,N\}$, and define $$ S(k\mid \mathcal M_k)_\rho=S(\rho_{k\cup \mathcal M_k})-S(\rho_{\mathcal M_k}), \qquad S(k\mid \mathcal M'_k)_\rho=S(\rho_{k\cup \mathcal M'_k})-S(\rho_{\mathcal M'_k}). $$ Set $$ \Delta=\sum_{k=1}^N \Bigl[ S(k\mid \mathcal M_k)_\rho+ S(k\mid \mathcal M'_k)_\rho \Bigr]\ll 1. $$ Because each term depends only on reduced density matrices $\rho_{k\cup\mathcal M_k}$ and $\rho_{k\cup\mathcal M'_k}$, this is a purely local condition [1405.0137]. The main theorem states that if two states $\rho,\sigma$ agree exactly on each local cluster, $$ \rho_{k\cup \mathcal M_k}=\sigma_{k\cup \mathcal M_k}, \qquad \rho_{k\cup \mathcal M'_k}=\sigma_{k\cup \mathcal M'_k} \quad\forall k, $$ then $$ \|\rho-\sigma\|_1\le 2^{3/2}\sqrt{\Delta}. $$ In particular, if $\Delta=0$, then the global state is uniquely fixed by its local marginals. The expectation value of any non-local observable $O$ can then be approximated from the maximum-entropy completion $\sigma$, with error bounded by $$ \bigl|\mathrm{Tr}[\rho O]-\mathrm{Tr}[\sigma O]\bigr| \le \|O\|\,2^{1/2}\,\Delta^{1/4}. $$ In the exact Markov case, one also has the explicit Petz-recovery formula $$ \rho_{ABC}= \rho_{BC}^{1/2}\, \rho_B^{-1/2}\, \rho_{AB}\, \rho_B^{-1/2}\, \rho_{BC}^{1/2}. $$ These results show that suitable local observables can be informationally complete for global properties, although the paper does not use the channel-based term IC-preservability [1405.0137]. The reconstruction procedure is formulated as a convex program: $$ \sigma= \arg\max_{\tau\ge 0,\ \mathrm{Tr}\tau=1} \Bigl\{ S(\tau)\ \Big|\ \tau_{k\cup \mathcal M_k}=\rho_{k\cup \mathcal M_k}, \ \tau_{k\cup \mathcal M'_k}=\rho_{k\cup \mathcal M'_k} \ \forall k \Bigr\}. $$ For tomography, the number of samples needed per marginal is $O(d^{2|k\cup\mathcal M_k|}/\varepsilon^2)$, and because there are $O(N)$ blocks, the total sample complexity is polynomial in $N$ and $1/\varepsilon$ [1405.0137]. The principal application is to unique ground states of 2D local Hamiltonians with a uniform gap, assuming the area law $$ S(A)_\rho=\alpha |\partial A|-\gamma+O(e^{-L/\xi}) $$ for simply connected regions $A$. Choosing shields of thickness $O(\xi)$ after coarse-graining into blocks of size $L\gg \xi$ yields $$ S(k\mid \mathcal M_k)_\rho=O(e^{-L/\xi}), \qquad \Delta=N\,O(e^{-L/\xi}), $$ and therefore $$ \|\rho-\sigma\|_1\le O\!\bigl(\sqrt N\,e^{-L/(2\xi)}\bigr). $$ By taking $L\sim \xi\log N$, the error can shrink polynomially in $N$ [1405.0137]. ## 5. Query completeness over quality-aware processes In data quality and business-process theory, completeness is defined over pairs of databases. The real world is represented by a database $D^{\mathrm{rw}}$, and the information system by a database $D^{\mathrm{is}}\subseteq D^{\mathrm{rw}}$. For a conjunctive query $$ Q(\bar z):\ R_1(t_1)\wedge\cdots\wedge R_n(t_n)\wedge M, $$ query completeness means $$ \bigl(D^{\mathrm{rw}},D^{\mathrm{is}}\bigr)\models \mathrm{Compl}\,Q \quad\Longleftrightarrow\quad Q(D^{\mathrm{rw}})=Q(D^{\mathrm{is}}). $$ A quality-aware transition system (QATS) augments an ordinary labelled transition system with real-world effects and copy effects, thereby modeling how data are created in reality and later copied into the information system [1306.1689]. The process-level notion of IC-preservability is stated explicitly. For a QATS $T$ and conjunctive query $Q$, $$ \mathrm{ICPres}(T,Q)\Longleftrightarrow \forall\text{ paths }\pi:\ \pi\models \mathrm{Compl}\,Q. $$ Equivalently, no matter how the process executes in the real world and in the information system, the answer to $Q$ over $D^{\mathrm{is}}$ always coincides with the answer over $D^{\mathrm{rw}}$ [1306.1689]. The central characterization is based on risky real-world effects and their repair by subsequent copy effects. A real-world effect $$ r:\ R(\bar x,\bar y)\leftsquigarrow G_1(\bar x,\bar z_1) $$ is risky for $Q$ if there is some way to insert a fresh $R$-fact under $r$ that would change $Q$'s answer. Using the $R$-projection $Q^R$ of the query, the paper states that $r$ is risky iff $$ G_1(\bar x,\bar z_1)\wedge \Bigl(\bigwedge_{i=1}^n R_i(t_i)\Bigr)\wedge M\wedge \bigvee_{R_i=R}\bigl((\bar x,\bar y)=t_i\bigr) $$ is satisfiable. A risky $r$ is repaired by later copy effects $c_1,\dots,c_m$ exactly when $$ P_r\cap Q^R\subseteq \bigcup_{j=1}^m P_{c_j}. $$ This yields the action-sequence criterion $$ \alpha\models \mathrm{Compl}\,Q \iff \text{every risky real-world effect in }\alpha \text{ is repaired by later copy effects,} $$ and the process-level characterization $$ \mathrm{ICPres}(T,Q) \Longleftrightarrow \forall\text{ real-world effect }r\text{ of }T: \bigl(P_r\cap Q^R\subseteq \bigcup_{c\in \mathrm{Post}(r)}P_c\bigr). $$ Here $\mathrm{Post}(r)$ is the set of all copy effects occurring on any path after the occurrence of $r$ [1306.1689]. The verification problem reduces to finitely many containment tests. By duplicate removal, it is enough to consider normal action sequences of length at most $|A|$. For each realized normal sequence and for each real-world effect in it, one checks whether the suffix copy effects repair the risky contribution. Reachability of a normal sequence is a graph problem in PTime; the main source of complexity is union-of-conjunctive-queries containment. The resulting bounds are: - for linear relational queries, IC-preservability can be decided in coNP; - for linear conjunctive queries, it is coNP-complete; - for relational conjunctive queries, it is in $\Pi_2^P$; - for full conjunctive queries with comparisons, it is $\Pi_2^P$-complete [1306.1689]. The school-enrollment example illustrates the semantics. For the query $$ Q_{\mathit{Hofer}}(n):\ pupil(n,1,\mathit{HoferSchool}), $$ the only real-world effect that can introduce a new Hofer pupil is risky, but the subsequent copy effect copies all newly enrolled Hofer pupils, so $$ P_{rW}\cap Q_{\mathit{Hofer}}^R \subseteq P_{rH}. $$ Hence every path preserves completeness for $Q_{\mathit{Hofer}}$. By contrast, the cross-school query $$ Q_{\mathit{all}}(n):\ pupil(n,1,s) $$ can become incomplete along partial interleavings where one school has copied but the other has not, so the process is not IC-preservable for $Q_{\mathit{all}}$ [1306.1689]. ## 6. Completeness preservation in logic programs under pruning In logic programming, correctness and completeness are defined relative to a specification $S\subseteq HB$, where $HB$ is the Herbrand base and $M_P$ is the least Herbrand model of a definite program $P$: $$ P\text{ is correct w.r.t. }S \iff M_P\subseteq S, \qquad P\text{ is complete w.r.t. }S \iff M_P\supseteq S. $$ Operationally, for any atomic query $A$ and any ground answer substitution $\theta$, $$ P\models A\theta \Rightarrow S\models A\theta $$ expresses correctness, while $$ S\models A\theta \Rightarrow P\models A\theta $$ expresses completeness. The paper studies sufficient conditions for completeness and the preservation of completeness under pruning of SLD-trees, including pruning due to Prolog's cut [1411.3015]. The basic sufficient condition is coverage. A ground atom $H$ is covered by a clause $C$ w.r.t. $S$ if there is a ground instance $$ H\leftarrow B_1,\dots,B_m $$ of $C$ with all $B_i\in S$. If every atom of $S$ is covered by $P$ w.r.t. $S$, then $P$ is semi-complete w.r.t. $S$: for any query whose SLD-tree is finite, $P$ produces all answers in $S$ demanded by the query. Semi-completeness becomes full completeness if, in addition, either every $A\in S$ has some finite SLD-derivation, or $P$ is recurrent, or $P$ is acceptable [1411.3015]. A direct completeness criterion uses level mappings. With a partial level mapping $|\cdot|:HB\to \mathbb N$, a ground atom $H$ is recurrently covered by $P$ w.r.t. $S$ and $|\cdot|$ if there is a ground instance $$ H\leftarrow B_1,\dots,B_m $$ with all $B_i\in S$, all levels defined, and $|H|>|B_i|$ for all $i$. If all atoms of $S$ are recurrently covered by $P$, then $P$ is complete w.r.t. $S$ [1411.3015]. For pruning, the paper introduces clause-selection SLD-trees. If $P$ is partitioned into subprograms $\Pi_1,\dots,\Pi_n$, a c-selection rule picks at each node a selected atom and one of the subprograms, producing a pruned SLD-tree of the full program. If the specification is correspondingly split as $S=S_1\cup\cdots\cup S_n$, then completeness of the pruned tree is preserved under three conditions: 1. each atom in $S_i$ is covered by $\Pi_i$ w.r.t. $S$; 2. the csSLD-tree is compatible with the split; 3. either $P$ is recurrent or acceptable, or the tree itself is finite. Under these conditions, every answer demanded by $S$ for the root query appears in the pruned tree [1411.3015]. A separate theorem handles cuts in the last clause of each predicate. Using a call-success specification $(pre,post)$ and the refined notion of adjustable coverage, one obtains: if the pruned LD-tree is finite and its root query is in $pre$, and each atom of $S$ is adjustably covered by $P$ w.r.t. $S$ and $(pre,post)$, then the pruned tree is complete w.r.t. $S$. The IN/2 example with ```prolog in([],L). in([H|T],L) :- m(H,L), !, in(T,L). ``` shows that any finite pruned tree where the cut fires still yields all ground solutions in the target specification [1411.3015]. The methodology emphasizes approximate specifications. Instead of fixing the exact least Herbrand model, one may use two specifications: - $S_c$: the atoms that must be computed; - $S_{\mathrm{corr}}$: the atoms that are allowed to be computed; and require $$ S_c\subseteq M_P\subseteq S_{\mathrm{corr}}. $$ The recommended workflow is to first write a pure logical program and prove correctness and completeness declaratively with respect to approximate specifications, then add control such as cuts or delays, and finally use pruning-preservation theorems to verify that no required answers are lost [1411.3015]. ## 7. Scope, limitations, and recurring interpretive issues A recurring issue across the literature is that completeness before transformation does not by itself imply preserved completeness after transformation. In continuous variables, heterodyne detection is complete on the full infinite-dimensional space, yet after truncation it still requires at least $d^2$ sample points in general position to recover all $d^2$ matrix elements on $\mathcal H_d$; photon-number measurement remains non-IC on every $d>1$ subspace because it never accesses off-diagonal terms [1211.4967]. In local-marginal reconstruction, local data are not automatically globally complete. The guarantee depends on the locally checkable quantity $$ \Delta=\sum_k\bigl[S(k\mid \mathcal M_k)_\rho+S(k\mid \mathcal M'_k)_\rho\bigr]. $$ If $\Delta=0$, the global state is uniquely fixed by its local marginals; if $\Delta$ 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 [1405.0137]. In the channel setting, IC-preservability is quantitative and channel-dependent. Unitary channels preserve informational completeness perfectly, while for the dephasing family $Q_{IC}=|1-2p|$ and vanishes at the fully dephasing point $p=\tfrac12$. 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 [2606.03964]. 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 [1306.1689]. 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 [1411.3015]. 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. Source: https://www.emergentmind.com/topics/informational-completeness-preservability-ic-preservability