Quantum Mechanics as Information Preservation: Channels, Recoverability, and Higher Descent

1. Scope and the informational thesis

The thesis studied here is: quantum mechanics constrains the preparation, preservation, transformation, and retrieval of information carried by physical systems. Storage is the preservation of a specified logical system or observable algebra through an intervening process. A storage claim must state what information, which process, which recovery operations, what duration, and what error tolerance are intended.

Spatial size and Hilbert-space dimension are different parameters. No spatial-size variable occurs in the definitions below. Deriving a storage-density bound requires a physical model connecting dimension, energy, locality, and matter. Mathematical abstraction makes these distinctions explicit; small physical objects are not an axiom of the categorical semantics.

Categorical process semantics [1-3] provides the operational setting. The channel category is not identified with a topos. Trace-preserving maps need not inherit the dagger compact structure of a larger category of completely positive maps.

Storage alone does not select quantum mechanics: classical theories also store information. Reconstructions require additional principles [13]. We assume finite-dimensional channels and the Born rule; cited information-theoretic results are established.

2. Registers, channels, and exact storage

Write $M_d=\operatorname{End}_{\mathbb C}(\mathbb C^d)$. A state is $\rho\in M_d$ with $\rho\geq0$ and $\operatorname{Tr}\rho=1$. A channel $\Phi:M_d\to M_e$ is a completely positive, trace-preserving complex-linear map. Complete positivity requires $\Phi\otimes\operatorname{id}_{M_r}$ to preserve positivity for every $r$. Channels compose and tensor to form a symmetric monoidal category $\mathsf{QChan}$.

A measurement has positive operators $F_y$ with $\sum_yF_y=I$, giving $p(y\mid\rho)=\operatorname{Tr}(F_y\rho)$. Classical outputs can be included through finite-dimensional commutative algebras [3,4].

Definition 2.1 (memory). A memory consists of channels $$ M_\ell\xrightarrow{\mathcal E}M_p\xrightarrow{\mathcal N}M_q\xrightarrow{\mathcal R}M_\ell. $$ Its effective logical channel is $\mathcal T=\mathcal R\mathcal N\mathcal E$. The memory is exact when $$ \boxed{\mathcal R\mathcal N\mathcal E=\operatorname{id}_{M_\ell}.} \tag{1} $$ The physical process may be $\mathcal N_t$ for a specified duration $t$. A claim uniform in time must specify whether recovery may depend on $t$.

Categorically, (1) makes $M_\ell$ a retract of $M_q$, with section $\mathcal N\mathcal E$. The encoding is also split monic, with left inverse $\mathcal R\mathcal N$. More explicitly, let $\mathbb M$ be the category presented by three objects $L,P,Q$, arrows $e:L\to P$, $n:P\to Q$, $r:Q\to L$, and relation $rne=1_L$. Exact memories are functors $\mathbb M\to\mathsf{QChan}$.

Proposition 2.2 (finite exactness test). For $E_{ab}=|a\rangle\langle b|$, the following are equivalent: (i) $\mathcal T=\operatorname{id}$; (ii) $\mathcal T(E_{ab})=E_{ab}$ for all $a,b$; (iii) $\mathcal T\otimes\operatorname{id}$ preserves every joint state with every finite reference register.

*Proof.* Matrix units are a complex-linear basis, giving (i) iff (ii). Tensoring the identity gives (iii). Conversely, (iii) includes preservation of all logical states; density matrices span $M_\ell$, giving (i). $\square$

Preserving basis populations alone misses coherence and reference correlations.

3. Approximate storage and composition

For a linear map $\Lambda$, set $$ \|\Lambda\|_\diamond=\sup_{r\geq1}\sup_{X\ne0} \frac{\|(\Lambda\otimes\operatorname{id}_{M_r})(X)\|_1}{\|X\|_1}. $$ Let $d_\diamond(\Phi,\Psi)=\tfrac12\|\Phi-\Psi\|_\diamond$. An $\varepsilon$-memory satisfies $\delta(\mathcal T):=d_\diamond(\mathcal T,\operatorname{id})\leq\varepsilon$, with $0\leq\varepsilon\leq1$. The reference system in this norm makes the criterion stable in larger experiments [4].

Proposition 3.1 (composition and perturbation). Compatible channels satisfy $$ \delta(\mathcal T_2\mathcal T_1)\leq\delta(\mathcal T_1)+\delta(\mathcal T_2),\qquad \delta(\mathcal T_1\otimes\mathcal T_2)\leq\delta(\mathcal T_1)+\delta(\mathcal T_2). $$ For two memory triples with the same dimensions, $$ d_\diamond(\mathcal R\mathcal N\mathcal E,\mathcal R'\mathcal N'\mathcal E') \leq d_\diamond(\mathcal R,\mathcal R')+d_\diamond(\mathcal N,\mathcal N')+d_\diamond(\mathcal E,\mathcal E'). $$

*Proof.* Expand the differences by telescoping. For example, $$ \mathcal T_2\mathcal T_1-\operatorname{id} =\mathcal T_2(\mathcal T_1-\operatorname{id})+(\mathcal T_2-\operatorname{id}), $$ and $$ \mathcal T_1\otimes\mathcal T_2-\operatorname{id}\otimes\operatorname{id} =(\mathcal T_1-\operatorname{id})\otimes\mathcal T_2 +\operatorname{id}\otimes(\mathcal T_2-\operatorname{id}). $$ The triple difference expands into three terms by replacing one factor at a time. Apply the triangle inequality and submultiplicativity; every channel has diamond norm one. $\square$

4. Specifying the information to preserve

Define the Heisenberg adjoint by $\operatorname{Tr}(\mathcal T(\rho)a)=\operatorname{Tr}(\rho\,\mathcal T^\dagger(a))$.

Definition 4.1 (retained algebra). For a unital $C^*$-subalgebra $\mathcal A\subseteq M_\ell$, a memory preserves $\mathcal A$ if $\mathcal T^\dagger(a)=a$ for every $a\in\mathcal A$. Equivalently, it preserves all expectation values specified by $\mathcal A$ for every input.

The decomposition $\mathcal A\cong\bigoplus_jM_{d_j}$ describes classical sector information and quantum information inside each sector. A commutative algebra specifies classical data; a full matrix algebra specifies a complete quantum register.

For a channel $\mathcal F(\rho)=\sum_iK_i\rho K_i^\dagger$, the established full-support correctability criterion is $$ \mathcal A\text{ is correctable for }\mathcal F \quad\Longleftrightarrow\quad [a,K_i^\dagger K_j]=0\quad\text{for all }a\in\mathcal A,i,j. \tag{2} $$ This is the full-support specialization of operator algebra quantum error correction [5, Theorem 9]; it is an imported result. To analyze an encoded memory, take $\mathcal F=\mathcal N\mathcal E$ and its recovery to have output $M_\ell$.

Example 4.2 (classical information survives dephasing). Let $\Delta_Z(\rho)=(\rho+Z\rho Z)/2$. Its adjoint fixes $\operatorname{span}\{I,Z\}$, so computational-basis probabilities survive. Yet $$ \Delta_Z(|+\rangle\langle+|)=\Delta_Z(|-\rangle\langle-|)=I/2. $$ No recovery can send this one output to both required input states. Hence the full qubit is not recoverable.

Example 4.3 (a recoverable quantum register). Set $c_0=000$, $c_1=111$, and encode $V|a\rangle=|c_a\rangle$. For $S=\{000,001,010,100\}$, define $X_e|x\rangle=|x\oplus e\rangle$. Consider $$ \mathcal N(\rho)=\sum_{e\in S}p_eX_e\rho X_e^\dagger, \qquad p_e\geq0,\quad\sum_ep_e=1. $$ Define $D_e=\sum_{a=0}^1|a\rangle\langle c_a\oplus e|$ and $\mathcal R(\sigma)=\sum_eD_e\sigma D_e^\dagger$. The eight strings $c_a\oplus e$ are distinct, so $\sum_eD_e^\dagger D_e=I_8$. Also $D_fX_eV=\delta_{fe}I_2$. Therefore $$ \mathcal R\mathcal N(V\rho V^\dagger)=\sum_ep_e\rho=\rho. \tag{3} $$ This preserves all matrix units, including coherences. It illustrates the established theory of quantum error-correcting codes [6]. The code corrects the specified single-bit-flip error set; it does not correct arbitrary single-qubit errors.

5. Constraints on copying and classical readability

Proposition 5.1 (no universal pure-state cloning). No channel copies every pure state when $d\geq2$.

No-broadcasting extends the obstruction to noncommuting mixed states [7].

Proposition 5.2 (distinguishability bound). A $d$-dimensional register encodes at most $d$ perfectly distinguishable classical messages without additional correlated resources.

*Proof.* Suppose a POVM distinguishes states $\rho_x$ perfectly. Then $\operatorname{Tr}(F_x\rho_x)=1$ and $\operatorname{Tr}(F_x\rho_y)=0$ for $x\ne y$. Positivity implies $\operatorname{supp}\rho_x\subseteq\ker(I-F_x)$ and $\operatorname{supp}\rho_y\subseteq\ker F_x$. These supports are pairwise orthogonal. At most $d$ nonzero mutually orthogonal supports fit in dimension $d$. $\square$

Corollary 5.3 (storage bottleneck). Exact storage implies $\ell\leq\min\{p,q\}$.

*Proof.* The $\ell$ logical basis inputs remain perfectly distinguishable at either intermediate stage: apply the remaining channels and measure the recovered basis. Proposition 5.2 applies to each intermediate register. $\square$

The established Holevo bound further gives, for an ensemble followed by a measurement, $$ I(X:Y)\leq S\!\left(\sum_xp_x\rho_x\right)-\sum_xp_xS(\rho_x)\leq\log_2d, \tag{4} $$ without preshared entanglement or correlated side information [8]. Here $S(\rho)=-\operatorname{Tr}(\rho\log_2\rho)$. Continuous amplitudes are not an arbitrarily long classical record readable from one specimen. Tomography on repeated preparations consumes additional specimens and is a different resource model.

6. Operational equality and deformation

Proposition 6.1 (path collapse). The topological space of channels of fixed input and output dimensions is contractible.

*Proof.* It is convex: $(1-t)\Phi+t\Psi$ remains completely positive and trace preserving. Interpolating to a fixed channel gives a contraction. $\square$

Thus treating arbitrary continuous deformation as operational equality identifies all channels of fixed dimensions. In particular it identifies the identity channel with the depolarizing channel $\rho\mapsto\operatorname{Tr}(\rho)I/d$, despite their different storage behavior. Deformation can be meaningful family data, but cannot automatically certify equality of physical operations.

We retain channels as coefficient data and use specified changes of frame as representation equivalences. A continuous family of coefficient values is not an internal identity between its endpoint operations.

7. Higher semantics of memory families

Fix $\ell,p,q,\varepsilon$. In the finite-dimensional topology of Choi matrices let $$ S_\varepsilon=\{(\mathcal E,\mathcal N,\mathcal R): d_\diamond(\mathcal R\mathcal N\mathcal E,\operatorname{id})\leq\varepsilon\}. $$ Let $G=PU(\ell)\times PU(p)\times PU(q)$. Projective unitaries suffice because scalar phases do not change conjugation channels. Write $\operatorname{Ad}_u(a)=uau^\dagger$. Define $$ \begin{aligned} \mathcal E'&=\operatorname{Ad}_v\mathcal E\operatorname{Ad}_{u^{-1}},\\ \mathcal N'&=\operatorname{Ad}_w\mathcal N\operatorname{Ad}_{v^{-1}},\\ \mathcal R'&=\operatorname{Ad}_u\mathcal R\operatorname{Ad}_{w^{-1}}. \end{aligned} \tag{5} $$ Then $\mathcal T'=\operatorname{Ad}_u\mathcal T\operatorname{Ad}_{u^{-1}}$. Unitary invariance of diamond norm makes $S_\varepsilon$ invariant.

Choose a small skeleton $\mathcal C$ of second-countable manifolds with continuous maps, equipped with the open-cover topology. Work in the infinity-topos $\mathcal X=\operatorname{Shv}(\mathcal C;\mathcal S)$ of space-valued sheaves [9]. Define the internally $0$-truncated coefficient sheaves $$ \mathbf S_\varepsilon(T)=C(T,S_\varepsilon),\qquad\mathbf G(T)=C(T,G). $$

Definition 7.1 (memory stack). The stack of $\varepsilon$-memories is the quotient $$ \boxed{\mathbf{Mem}_\varepsilon=[\mathbf S_\varepsilon/\mathbf G].} \tag{6} $$ An object over $T$ is a locally trivial principal $G$-bundle $P\to T$ with an equivariant map $f:P\to S_\varepsilon$, using $f(pg)=g^{-1}f(p)$. Arrows are equivariant bundle isomorphisms preserving $f$. This is a $1$-stack inside an infinity-topos. It does not posit additional higher physical state variables. Its families have higher topology through their topological classifying spaces.

The operational category supplies processes; the infinity-topos supplies a universe for varying specifications, equivalences, and descent. These roles must remain distinct. In the exact case, unitary natural isomorphisms of the functors in Section 2 give the change-of-frame arrows.

Theorem 7.2 (effective descent). For an open cover $\{T_i\to T\}$, let $m_i:T_i\to S_\varepsilon$ and $g_{ij}:T_i\cap T_j\to G$ satisfy $m_i=g_{ij}\cdot m_j$, $g_{ii}=1$, and $g_{ij}g_{jk}=g_{ik}$. These data determine an object of $\mathbf{Mem}_\varepsilon(T)$, unique up to equivalence compatible with the specified descent data.

*Proof.* The cocycle glues trivial principal bundles to a principal $G$-bundle. The transformation equations glue the local coefficient maps to an equivariant map into $S_\varepsilon$. Conversely, local trivializations recover these data, and changes of trivialization give their equivalences. Gauge invariance of the storage inequality makes correctness well defined throughout the glued family. $\square$

Pairwise equivalences alone are insufficient. On a nonempty triple overlap, $g_{12}=g_{23}=1$ and $g_{13}=h\ne1$ can preserve the same local object when $h$ is its automorphism, but fail the cocycle equation. This invalidates those identifications, not necessarily the existence of a different gluing.

Tensor products give coherent operations between the stacks: two tolerances $\varepsilon,\eta$ give tolerance at most $\min\{1,\varepsilon+\eta\}$, by Proposition 3.1. Tensoring unitary frames defines the required group homomorphisms.

8. Perfect storage without a global frame

Theorem 8.1 (unitary exact-memory sector). For $\ell=p=q=d$, the substack where all three channels are unitary and their composite is the identity is equivalent to $B\mathbf{PU(d)}$.

*Proof.* The $PU(d)^3$ action is transitive: choose frames successively along the unitary encoding and evolution; exactness makes recovery the identity too. The stabilizer of $(\operatorname{id},\operatorname{id},\operatorname{id})$ is the diagonal $PU(d)$. The quotient stack of a transitive action is the classifying stack of its stabilizer. $\square$

Corollary 8.2 (nontrivial qubit families). Over $S^4$ there are perfectly preserving memory families, locally identity triples, with no global simultaneous frame.

*Proof.* Theorem 8.1 identifies them with principal $PU(2)$-bundles. The double cover $SU(2)\cong S^3\to PU(2)\cong SO(3)$ gives $\pi_3(PU(2))\cong\mathbb Z$. Clutching along the equator of $S^4$ yields nontrivial bundles. A global frame would trivialize the bundle. $\square$

The associated topological space has $\pi_4(BPU(2))\cong\mathbb Z$. This obstructs global framing, not global memory. The logical register bundle varies here; a fixed global logical frame and unitary encoding trivialize the other bundles. This topology adds no readable classical capacity to an individual qubit.

9. Measurement contextuality is another gluing problem

For a set $U$ of binary observables, let $\mathcal O(U)=\{+1,-1\}^{U}$, with restriction by forgetting values. An empirical model supplies distributions and support constraints on jointly measurable contexts. For two qubits consider $$ \begin{array}{ccc} X\otimes I&I\otimes X&X\otimes X\\ I\otimes Z&Z\otimes I&Z\otimes Z\\ X\otimes Z&Z\otimes X&Y\otimes Y. \end{array} \tag{7} $$ Here $X,Y,Z$ are Pauli matrices. Every row and column is commuting. All row products and the first two column products equal $I$; the final column product equals $-I$.

Proposition 9.1 (parity obstruction). There is no context-independent assignment of values $\pm1$ to these nine observables satisfying all six product constraints.

*Proof.* Multiplying the constraints counts each assigned value twice, giving $+1$ on the left and $-1$ on the right. $\square$

This is the Mermin parity argument [10], within the sheaf account of contextuality [11]. The unrestricted assignment presheaf has global assignments; the quantum support constraints exclude every one. This does not imply absence of a quantum state: density matrices supply compatible Born distributions on the commuting contexts.

Topos approaches to algebraic quantum theory organize commutative observational contexts, with internal spectra and probability valuations [12]. Their context-based constructions differ from the parameter-based memory stack above. Contextuality obstructs a global noncontextual outcome assignment; a nontrivial frame bundle obstructs a global coordinate choice. They are not interchangeable obstructions.

10. Interpretation, evidence, and limitations

Storage is relational: a specification, encoding, dynamics, and recovery determine what survives. Higher semantics records representation equivalences and their coherence.

The assumptions here do not derive the Born rule, single out quantum theory among all information theories, or establish an ontology of information. Infinite-dimensional systems, quantum fields, energy budgets, locality, thermal stability, and fault-tolerant engineering require further hypotheses. No storage density or lifetime is inferred from Hilbert-space dimension alone.

The attached Python script uses integer-pair complex arithmetic: 16 code compression identities, 16 recovery identities, 18 commuting pairs, six product signs, and all 512 outcome assignments were checked. Zero assignments satisfy the six constraints. These checks verify finite examples, not the general proofs.

Authorship and automation. Anonymous author is the public byline of the responsible human contributor requesting publication. An AI assistant assisted with drafting, proof development, source checking, and finite calculations. The contribution is a synthesis and explicit memory-family semantics; no priority claim is made for the established results cited below.

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