En05-C: Tensor-Free Snapshots and Causal Locality for Gauge-Invariant Observable Algebras

We reformulate the snapshot–channel–adjoint comparison for unital local observable subalgebras without a spatial tensor decomposition. For a unital completely positive Heisenberg channel, equality of input state restrictions determines the output restriction exactly when the pulled-back output algebra is contained in the declared input algebra. The induced local channel is unique, completely positive, and stable under independent matrix references. The proof uses separation and Jordan decomposition of functionals, not a complementary-state preparation channel.

We retain central-sector probabilities, characterize the weaker conclusion available from restricted preparations, and prove the normal von Neumann algebra version. Conditional expectations are optional; their state-preserving existence is not assumed. An explicit constrained ${\mathbb{Z}}_2$ graph supplies centerful regional algebras, localized complete instruments, successful and failed containment tests, and an obstruction to independently changing shared boundary data.

For nets with local commutativity, specified internal Kraus devices give one-layer containment, algebraic ancestral factorization, metric support bounds, matched replacement invariance, and unconditioned no-signalling. A regional channel restriction alone need not determine its global device extension; this additional interface is essential. Finally, we construct a bounded-sequence observable limit and an exact subdivision limit of the finite gauge model, including compatible states, references, records, and matched devices. These are observable-level constructions, not a relativistic field-theory limit. No general gauge-theory transport compilation, Lorentz covariance, gravitational dynamics, or physical speed calibration is inferred.

The comparison lemmas in Cheung En05-A, En05-B, and En06-A identify three descriptions of finite hybrid-channel locality: factorization through a snapshot, determinacy by that snapshot, and containment of the Heisenberg pullback CheungEn05A,CheungEn05B,CheungEn06A. Their complementary-state proof uses a spatial tensor interface. The present paper removes that assumption from the comparison theorem itself.

Gauge constraints make this distinction useful. A physical regional observable algebra can have a center and need not define a tensor factor of the global physical algebra. Moreover, assigning an observable algebra to a region involves a boundary convention CasiniHuertaRosabal. Introducing boundary variables or edge modes is a different construction; it is not required by the theorem proved here DonnellyFreidel.

The elementary operator-algebra ingredients are standard: Hahn–Banach separation, Jordan decomposition of bounded Hermitian functionals, inherited matrix positivity of a $C^{\ast }$-subalgebra, and completely positive maps Pedersen,Arveson,Stinespring. We do not claim a new general separation theorem. The purpose is to give a complete operational formulation, its gauge-compatible implementation interface, explicit counterexamples, and compatible observable-limit constructions.

The supplied plan contains both theorem targets and construction targets. We address each, with its domain stated before its proof.

{}1.12

The last boundary is substantive. Completing an observable-level refinement construction for a specified model is not the construction of a continuum Yang–Mills theory, nor the discharge of every observable-limit or covariance hypothesis in En05-A or En06-A.

First, proving that an operation fixes exterior observables does not, without an additional argument, place a pulled-back observable in a chosen regional algebra. It may only place it in a commutant. Our implementation theorem proves membership by multiplication inside a declared larger regional algebra.

Second, specifying the restriction of a global device to its own regional algebra need not specify its action on observables extending across the boundary. We give an explicit gauge example in Section sec:deviceextension. Matched replacements therefore retain complete device actions or specified realizations, not merely their regional restrictions.

These corrections are needed to turn the plan into a valid tensor-free causal-network argument.

All $C^{\ast }$-algebras are complex and unital. A unital subalgebra has the same unit as its ambient algebra and is norm closed.

An interface consists of global algebras ${\mathfrak{M}}_{\mathrm{in}},{\mathfrak{M}}_{\mathrm{out}}$, local algebras ${\mathfrak{A}}_{\mathrm{in}}(I),{\mathfrak{A}}_{\mathrm{out}}(Y)$, unital injective $\ast$-homomorphisms

and a Heisenberg channel

that is unital and completely positive.

We ordinarily identify the local algebras with their images. No complementary algebra or tensor decomposition is part of this definition.

For a state $\omega \in \mathsf{S}({\mathfrak{M}}_{\mathrm{in}})$, the output state is

The input snapshot in $I$ is

The output snapshot in $Y$ is

Restriction is affine. It does not require a density operator, a trace, a partial trace, or a conditional expectation. For bounded functionals we use the same notation for the dual restriction map

On a declared collection of regions, a local net assigns unital subalgebras

It is isotone if $I\subseteq J$ implies $\mathfrak{A}(I)\subseteq \mathfrak{A}(J)$. For a discrete support convention, local commutativity means

A spacetime version may instead impose this condition on spacelike separated regions.

Our scheduled implementation theorem uses the discrete condition (eq:localcomm). The abstract comparison theorem does not require local commutativity.

A gauge-model specification must identify:

1. the global physical algebra and its representation;

2. which constraint-preserving observables are assigned to $I$;

3. the treatment of boundary electric flux or other central data;

4. the allowed complete devices and preparations.

These choices precede every locality conclusion.

Local commutativity gives

for disjoint $I,J$. It does not give equality. We never replace membership in $\mathfrak{A}(I)$ by commutation with a declared exterior algebra unless such an equality has been proved.

We do not assume

A gauge-invariant loop observable can belong to the algebra of a union without being generated by the algebras of its separate pieces. Thus equal one-piece snapshots need not determine the snapshot on the union.

We use the standard Jordan decomposition of a bounded Hermitian functional:

In a von Neumann algebra, the decomposition of a normal Hermitian functional is normal. These are background functional-analysis facts Pedersen,Sakai.

Let $\mathfrak{A}\subseteq \mathfrak{M}$ be a unital $C^{\ast }$-subalgebra. For $h=h^{\ast }\in \mathfrak{M}$, the following are equivalent:

1. $h\in \mathfrak{A}$;

2. $\omega (h)=\nu (h)$ whenever $\omega ,\nu \in \mathsf{S}(\mathfrak{M})$ have equal restrictions to $\mathfrak{A}$.

Proof. The forward implication follows by restriction.

Suppose $h\notin \mathfrak{A}$. The real closed linear subspace ${\mathfrak{A}}_{\mathrm{sa}}$ does not contain $h$. Real Hahn–Banach separation gives a bounded real functional $\ell$ on ${\mathfrak{M}}_{\mathrm{sa}}$ such that

Its complexification is a bounded Hermitian functional $f$ on $\mathfrak{M}$, vanishing on $\mathfrak{A}$.

Write $f=f_{+}-f_{-}$. Because $1\in \mathfrak{A}$, we have

The nonzero value $f(h)$ implies $m>0$. Moreover,

Thus

are states with equal snapshots on $\mathfrak{A}$, but

This contradicts (ii).

For the interface of Section sec:interfaces, the following are equivalent:

1. There exists a UCP map

   $$K:{\mathfrak{A}}_{\mathrm{out}}(Y)⟶{\mathfrak{A}}_{\mathrm{in}}(I)$$

   such that

   $$T\circ {\iota }_Y={\iota }_I\circ K.$$

2. For every pair of global input states,

   $$\omega \circ {\iota }_I=\nu \circ {\iota }_I⟹\omega \circ T\circ {\iota }_Y=\nu \circ T\circ {\iota }_Y.$$

3. The Heisenberg pullback satisfies

   $$T({\iota }_Y({\mathfrak{A}}_{\mathrm{out}}(Y)))\subseteq {\iota }_I({\mathfrak{A}}_{\mathrm{in}}(I)).$$

When these conditions hold, $K$ is unique and

on bounded functionals, and hence on states. No spatial complement or state-extension channel is used.

Proof. Equation (eq:algebrafactor) gives (eq:snapshotdet) by evaluation against states, and also gives (eq:containment).

Assume (ii). For a self-adjoint $b\in {\mathfrak{A}}_{\mathrm{out}}(Y)$, complete positivity makes $T({\iota }_Y(b))$ self-adjoint. Its expectation is equal on every pair of states with equal input snapshots. Lemma lem:separation places it in ${\iota }_I({\mathfrak{A}}_{\mathrm{in}}(I))$. Every output element is a complex linear combination of two self-adjoint elements, proving (iii).

Assume (iii). Define

It is linear and unital. For every matrix size $r$, the inclusion ${{\iota }_I}^{(r)}$ is a complete order embedding. If $[b_{ij}]\ge 0$ in $M_r({\mathfrak{A}}_{\mathrm{out}}(Y))$, then

in $M_r({\mathfrak{M}}_{\mathrm{in}})$. Its entries lie in the input local algebra, where the inherited positive cone is the same. Therefore $K$ is completely positive.

Injectivity of ${\iota }_I$ proves uniqueness. Taking dual maps in (eq:algebrafactor) gives (eq:dual-factor).

Nothing in Theorem thm:comparison makes $K$ multiplicative. For example, dephasing on $M_2$ sends the Pauli operator $X$ to zero and $X^2=1$ to $1$.

For finite tensor-product hybrid interfaces, use the local inclusions by identity extension and let $T={\Phi }^{\ast }$. Then (eq:dual-factor) is precisely

Thus Theorem thm:comparison recovers lem:comparison in En05-A, En05-B, and En06-A without their complementary-state proof.

Proof. Under finite-dimensional trace duality, the dual of identity extension is partial trace together with classical marginalization. The dual of a UCP map is a complete trace-preserving channel. Apply Theorem thm:comparison.

A state on a unital $C^{\ast }$-subalgebra has a state extension to the ambient algebra by the usual positive extension theorem. This does not provide a canonical extension, an affine extension channel, or independently compatible extensions for several regions. None is required in the proof above.

A finite-dimensional regional algebra has an intrinsic decomposition

Let $e_{\alpha }$ be its minimal central projections.

Every state on $\mathfrak{A}(I)$ is uniquely represented by positive matrices

Its sector probabilities and conditional states are

The complete array is affine in the state. The collection of normalized conditional states alone is not a complete snapshot.

Proof. Trace duality on each matrix summand gives the unique positive matrix ${\omega }_{\alpha }$. Evaluation on the unit gives the normalization, and evaluation on $e_{\alpha }$ gives $p_{\alpha }$. Conditional normalization introduces division by $p_{\alpha }$. Without the probabilities, distinct arrays can have the same normalized matrices.

This is the direct-sum snapshot convention already retained in En05-A and En06-A CheungEn05A,CheungEn06A. Gauge-theory discussions of local operational access similarly require care with superselection sectors VanAcoleyenEtAl.

Let $\mathfrak{A}=M_2\oplus M_2$, and fix a density matrix $\rho$. The states

have the same normalized state $\rho$ in both sectors. Nevertheless the central projection $e_{+}=(1,0)$ has

Thus a complete central readout distinguishes $p=1/4$ from $p=3/4$. The finite gauge example below realizes these states as restrictions of physical global states.

For an admitted set ${\mathcal{S}}_{\mathrm{adm}}\subseteq \mathsf{S}(\mathfrak{M})$, define

Its annihilator is

Snapshot determinacy tested only on ${\mathcal{S}}_{\mathrm{adm}}$ is equivalent to

It implies literal regional containment if

For the full global state space, (eq:fiberseparation) holds.

Proof. Equality of the two output restrictions is exactly

for the generating pairs in the definition. Taking their linear span gives the first assertion. The second is immediate. For the full state space, local elements belong to the annihilator, and Lemma lem:separation excludes every self-adjoint element outside the local algebra. Complexification gives equality.

For a restricted state class, ${{\mathcal{D}}_{\mathcal{I}}}^{\perp }$ is a closed linear space containing the local algebra. It need not be closed under multiplication. Nor need the globally undetectable space

be an ideal. An operational quotient is not automatically a $C^{\ast }$-algebra quotient.

Take $\mathfrak{M}=M_2$, $\mathfrak{A}=\mathbb{C}1$, and ${\mathcal{S}}_{\mathrm{adm}}=\{{\omega }_0\}$. The identity channel passes every equal-snapshot test on this preparation class, but

All-state quantification, or an explicit fiber-separation condition, is therefore essential.

If preparations are confined to a superselection sector, the appropriate comparison theorem is formulated in that sector's represented ambient algebra and its represented regional subalgebras. The sector unit replaces the original unit. The output channel must preserve the stated sector interface. A full-algebra conclusion cannot be recovered from untested sectors.

Under Theorem thm:comparison, for every $r\ge 1$,

Consequently, equal joint restrictions of two states on ${\mathfrak{M}}_{\mathrm{in}}\otimes M_r$ to ${\mathfrak{A}}_{\mathrm{in}}(I)\otimes M_r$ give equal joint output restrictions on ${\mathfrak{A}}_{\mathrm{out}}(Y)\otimes M_r$.

Proof. Amplify the linear identity (eq:algebrafactor). Complete positivity ensures positivity at every matrix size. Evaluation against the joint states gives the conclusion.

This tensor product represents an independent external reference, not a spatial complement of the gauge system. It is the matrix-level version of the reference clause in the supplied comparison and soundness interfaces CheungEn05A,CheungEn05B,CheungEn06A,CheungEn02C. The operator-system framework distinguishes matrix-level positivity from mere scalar positivity Arveson.

Equal restrictions on $\mathfrak{A}(I)$ alone do not imply equal restrictions on $\mathfrak{A}(I)\otimes M_r$. Two different Bell states have the same system marginal but different joint system–reference states. Theorem thm:comparison does not erase those correlations.

If the reference shares a Gauss constraint with the system, an unrestricted tensor product is not its declared physical interface. Specify the joint constrained global and local algebras and apply Theorem thm:comparison to those inclusions. No claim of independent reference preparation then follows.

A finite instrument with outcomes $y$ consists of CP maps

Its recorded Heisenberg channel is

The map $\hat{T}$ is UCP. The outcome probability in state $\omega$ is $\omega (T_y(1))$. Discarding the record gives the nonselective channel

These constructions preserve all independent matrix references.

Proof. At every matrix level, positivity in a direct sum is blockwise. Each summand is mapped positively, so their sum is completely positive. The stated completeness identity is unitality. The central unit of outcome block $y$ pulls back to $T_y(1)$, giving its probability. Embedding an unrecorded observable diagonally gives $T_{\mathrm{ns}}$. Amplification preserves the identities.

Selective normalized states are not substituted for this complete channel. This is the same complete-history distinction used in the finite program and readout accounting of Landmark I, En04-A, En02-C, and Hawk04-B CheungI,CheungEn04A,CheungEn02C,CheungHawk04B.

A conditional expectation onto $\mathfrak{A}\subseteq \mathfrak{M}$ is a UCP map $E:\mathfrak{M}\to \mathfrak{A}$ that fixes $\mathfrak{A}$ pointwise and satisfies

When convenient, its inclusion into $\mathfrak{M}$ is understood.

If such an $E$ exists, then the conditions of Theorem thm:comparison are equivalent to

In that case $K=ET{|}_{{\mathfrak{A}}_{\mathrm{out}}(Y)}$. Also, every local state $\eta$ has the common affine extension $\eta \circ E$.

Proof. The fixed-point range of $E$ is exactly $\mathfrak{A}$. Thus the displayed equality is equivalent to containment. Composition with $E$ then gives the unique local channel. Since $E{|}_{\mathfrak{A}}=\mathrm{id}$, $\eta \circ E$ restricts to $\eta$.

This is additional infrastructure, not a premise of Theorem thm:comparison.

Suppose $\mathfrak{M}=\mathcal{B}(\mathcal{H})$ and

Then

is a trace-preserving conditional expectation.

Proof. Central-block pinching, partial normalized trace, and identity extension are CP. The formula is unital and fixes $\mathfrak{A}$. Moving the left and right matrix factors through the partial trace gives bimodularity. The ordinary trace of the resulting block equals that of the original diagonal block, and off-diagonal blocks have zero trace.

For a von Neumann subalgebra and a faithful normal state $\phi$, the existence of a normal $\phi$-preserving conditional expectation is characterized by invariance of the subalgebra under the modular automorphism group of $\phi$. We use this only as a stated external criterion Takesaki. Neither a regional declaration nor gauge averaging automatically verifies it.

The finite example in Section sec:finiteexpectexample explicitly distinguishes gauge averaging, regional expectation, and state-preserving expectation.

Let $\Gamma =(V,E)$ be a finite connected graph without self-loops; parallel links are allowed. Associate a qubit to each link:

Let $X_e,Z_e$ be its Pauli operators. The Gauss operators are

They commute, and ${\prod }_v{G}_v=1$.

Define

The tensor product above is kinematic. We do not assume that the constrained physical Hilbert space factors across a link-region partition.

This finite pure-gauge choice is a specified discrete model. The broader Hamiltonian lattice-gauge setting is reviewed in the original formulation of Kogut and Susskind KogutSusskind; no Hamiltonian from that work is a premise here.

For $S\subseteq E$, let

Its physical image is

The map $q_{\mathrm{phys}}$ is a unital $\ast$-homomorphism on the global constraint-preserving algebra. Each $\mathfrak{A}(S)$ is a unital $C^{\ast }$-subalgebra of $\mathfrak{M}$. The net is isotone and locally commuting on disjoint link sets. Its global regional algebra satisfies $\mathfrak{A}(E)=\mathfrak{M}$. Every state on $\mathfrak{M}$ is an admitted physical state in this model.

Proof. A constraint-preserving observable commutes with $P$. Hence its restriction preserves products and adjoints:

The image is a finite-dimensional unital $\ast$-algebra, therefore norm closed. Isotony follows from kinematic support inclusion. Disjoint kinematic supports commute, and their physical images continue to commute because restriction is multiplicative.

For $c\in \mathcal{B}({\mathcal{H}}_{\mathrm{phys}})$, extend it by zero on ${{\mathcal{H}}_{\mathrm{phys}}}^{\perp }$. Every $G_v$ is the identity on ${\mathcal{H}}_{\mathrm{phys}}$ and preserves its orthogonal complement, so this extension commutes with every $G_v$. Its restriction is $c$, proving $\mathfrak{A}(E)=\mathfrak{M}$.

For arbitrary kinematic operators that do not preserve the physical subspace, $a\mapsto PaP$ need not preserve products. The proof above uses $[a,P]=0$ explicitly.

Write, up to a scalar phase,

Let $\partial :{{\mathbb{Z}}_2}^E\to {{\mathbb{Z}}_2}^V$ be the graph incidence map.

The algebra $\mathfrak{F}(S)$ is spanned by Pauli monomials supported in $S$ whose $Z$-exponent satisfies

For connected $\Gamma$,

Proof. Conjugation by $G_v$ multiplies a Pauli monomial by $(-1{)}^{(\partial \beta {)}_v}$. Since Pauli monomials form a basis, the invariant subspace is exactly the stated span.

In the joint $X$-eigenbasis, a physical basis vector has eigenvalue $(-1{)}^{f_e}$ on link $e$, with $\partial f=0$. The incidence map of a connected graph has rank $|V|-1$ over ${\mathbb{Z}}_2$, so its kernel has dimension $|E|-|V|+1$.

Thus all supported electric operators $X_e$ are retained. Supported closed $Z$-strings are retained when their full cycles lie in the region. Open charged strings are not inserted. This is the boundary convention for the present paper.

Take two vertices joined by three distinct links $1,2,3$. Both vertex constraints are

The physical Hilbert space has dimension four.

Use the $X$-eigenbasis with eigenvalues $a,b,c\in \{+1,-1\}$. The constraint is $abc=1$. Define an orthonormal physical basis

This identifies the vector space with a sector label and a logical qubit. It is not a spatial factorization into the region $S=\{1,2\}$ and its complement $\{3\}$.

On ${\mathcal{H}}_{\mathrm{phys}}$ set

Then

Let $e_f=(1+fF)/2$.

For the theta graph,

and

Consequently

No spatial tensor-factor decomposition of $\mathfrak{M}$ realizes these two regional inclusions as complementary tensor factors.

Proof. An invariant Pauli monomial supported on links $1,2$ can have $\beta =(0,0,0)$ or $\beta =(1,1,0)$. Hence it is generated by $X_1,X_2,Z_1{Z}_2$. On the physical space $X_2=FQ$. Within each $f$ sector, $Q,R$ are a Pauli pair and generate $M_2$. The projection $e_f$ selects the sector, proving (eq:A12).

On link $3$ alone, invariance permits only $1,X_3$, proving (eq:A3). The center and intersection follow. Complementary tensor factors with common unit intersect only in scalars; here the intersection contains the non-scalar $F$.

In this example the commutant identity $\mathfrak{A}(\{3\}{)}^{\prime }\cap M_4=\mathfrak{A}(\{1,2\})$ happens to hold. The general theory does not assume it.

Set $P_{\pm }=(1\pm Q)/2$. The supported Kraus operators $P_{\pm }$ commute with the Gauss constraint. The complete recorded instrument is

On the local recorded output $\mathfrak{A}(\{1,2\})\oplus \mathfrak{A}(\{1,2\})$, its range is in $\mathfrak{A}(\{1,2\})$. The scalar report channel has effects $P_{+},P_{-}$ and distinguishes the physical inputs $|f,+⟩$ and $|f,-⟩$ with gap one. Thus containment does not imply a constant output.

The nonselective channel acts on the local basis by

These formulas test containment on a complete spanning set.

The channel from a two-outcome report algebra is

It has range in every unital regional algebra and always reports $+$. A neutral local pointer initialized in $+$ implements this report without resetting the gauge system. Declared ancestry need not be influential.

In the coordinates (eq:theta-basis), take

Its full local snapshot is

The normalized states within both sectors are ${\rho }_q$, but the local central readout $e_{+},e_{-}$ has law $(p,1-p)$. A smaller algebra $1_f\otimes M_2$ also fails to distinguish these inputs. It is not a replacement for the declared full regional algebra (eq:A12).

Let

It is gauge invariant but anticommutes with $F$. Thus $W\notin \mathfrak{A}(\{1,2\})$. Define a binary readout by

The global density matrices

are positive and normalized. Every element of $\mathfrak{A}(\{1,2\})$ is block diagonal in $f$, whereas $W$ is off diagonal, so ${\rho }_{+}$ and ${\rho }_{-}$ have equal regional snapshots. Their output laws under $T_W$ are respectively ${\delta }_{+}$ and ${\delta }_{-}$. This is an exact witness for the failure of containment.

Restricting Example ex:failure to the singleton preparation class $\{1/4\}$ would conceal the failure, providing the fourth, preparation-class test.

The regional conditional expectation onto $\mathfrak{A}(\{1,2\})$ is

It preserves the ordinary trace. Gauge averaging by $G$ acts as the identity on the physical algebra and is therefore not $E_{12}$.

There are faithful physical states for which no state-preserving conditional expectation onto $\mathfrak{A}(\{1,2\})$ exists.

Proof. Equation (eq:thetaexpect) is block pinching onto the two full matrix blocks, hence a conditional expectation by Proposition prop:finiteexpect. The constraint $G$ is the identity on ${\mathcal{H}}_{\mathrm{phys}}$.

Any conditional expectation onto these full blocks is forced by bimodularity to kill every off-diagonal block $e_{+}a{e}_{-}$ and $e_{-}a{e}_{+}$, and to fix every diagonal block. It must therefore equal $E_{12}$.

Choose

This state is faithful, and

The unique expectation does not preserve it.

There are two gauge-invariant devices supported on $\{1,2\}$ that agree on $\mathfrak{A}(\{1,2\})$ but disagree on $\mathfrak{M}$.

Proof. The kinematic unitary $U=X_1{X}_2$ is supported on $\{1,2\}$ and commutes with every Gauss operator. Its physical image is $F$. The channels

agree on $\mathfrak{A}(\{1,2\})$, because $F$ is central there. But for the cross-boundary observable $W=Z_1{Z}_3$,

The factor $F$ is not central in the global algebra. This example prevents a proof of matched replacement invariance from agreement of regional restrictions alone.

A specified device includes its complete global UCP action. A localized Kraus realization is one sufficient way to specify it:

A chosen Kraus family is realization data, not a canonical observable associated to the channel. Different Kraus families may specify the same global channel.

This distinction preserves En06-A's warning that arbitrary Kraus representatives are not intrinsic observables CheungEn06A.

Suppose $I,J$ are disjoint regions and $z\in \mathfrak{A}(I)\cap \mathfrak{A}(J)$. Every internal Kraus device localized in $I$ fixes $z$. In the theta graph, no device that fixes $\mathfrak{A}(\{3\})$ can induce the sector-swap automorphism

on $\mathfrak{A}(\{1,2\})$.

Proof. Local commutativity implies that each Kraus operator in $\mathfrak{A}(I)$ commutes with $z\in \mathfrak{A}(J)$. Hence

For the theta graph, the proposed swap sends $F$ to $-F$, whereas exterior fixation requires $F$ to remain $F$.

The sector-swap map is a valid abstract UCP automorphism of $M_2\oplus M_2$. It is not thereby an admissible exterior-fixing physical device. This is an explicit failure of arbitrary independent regional operation closure.

Likewise, two separately assigned snapshots on $\mathfrak{A}(\{1,2\})$ and $\mathfrak{A}(\{3\})$ cannot be restrictions of one global state if they assign different probabilities to their common projection $e_{+}$.

Every controller, classical record, seed, resource, and quantum observable used by a device belongs to its declared localization interface. The device action is fixed before the unknown input state is supplied.

If an auxiliary neutral apparatus is used, its local algebra, preparation, record outputs, and downstream control paths are included in the experiment. No unknown-state description or undeclared remote scheduler input is admitted.

This is the operational support discipline compared with VII-G and the supplied En05 series CheungVIIG,CheungEn05A,CheungEn05B,CheungEn06A.

Let $\mathfrak{A}$ be an isotone locally commuting net. For the device (eq:internaldevice),

If $b\cap Y=\varnothing$, then

Devices with disjoint localization regions commute as global superoperators.

Proof. For $a\in \mathfrak{A}(Y)$, isotony places $a$ and all $V_{b,\mu }$ in $\mathfrak{A}(Y\cup b)$. This algebra is closed under products and sums, proving (eq:enlargement).

For disjoint $b,Y$, local commutativity gives

For disjoint devices $b,c$, their Kraus operators and adjoints commute. Expanding both compositions and reordering those operators proves $T_b{T}_c=T_c{T}_b$.

Membership in (eq:enlargement) follows from multiplication inside the declared algebra. It is not inferred solely from (eq:exteriorfix).

In the finite gauge net (eq:gaugenet), choose kinematic Kraus operators $v_{b,\mu }\in \mathfrak{F}(b)$ satisfying

Their physical restrictions satisfy (eq:internaldevice) and all conclusions of Lemma lem:devicesupport. Complete selected outcomes may be grouped and recorded as in Proposition prop:records.

Proof. Constraint preservation makes $q_{\mathrm{phys}}$ a $\ast$-homomorphism, so the kinematic completeness identity restricts to the physical completeness identity. The physical Kraus operators lie in $\mathfrak{A}(b)$ by definition. Apply Lemma lem:devicesupport and Proposition prop:records.

Let a finite site set $X$ have a partition ${\mathcal{P}}_s$ at tick $s$. The sites may be links of the gauge graph. Each block $b$ carries a specified internal Kraus device. Define the global Heisenberg layer by composition:

The order is irrelevant by Lemma lem:devicesupport. Equation (eq:layercomposition) is not a spatial tensor product.

For $Y\subseteq X$, put

For the layer (eq:layercomposition),

Its restriction to $\mathfrak{A}(Y)$ depends only on the specified global devices in the blocks meeting $Y$.

Proof. Let ${\mathcal{P}}_s(Y)$ be the meeting blocks. By commutation, compose them first. Repeated use of (eq:enlargement) places the result in $\mathfrak{A}(U_s(Y))$. Every remaining block is disjoint from $U_s(Y)$, so its device fixes that entire algebra by (eq:exteriorfix). It makes no contribution.

If finite-region algebras generate a quasi-local global algebra and every finite region meets only finitely many blocks, the same formula defines a layer on the local union. Enlarging a region adds only devices that fix the smaller calculation, so the definition is consistent. The finite compositions are UCP contractions at every matrix level; norm completion gives a UCP global map. Finite horizons still have finite backward regions under the usual finite-ball and finite-block hypotheses. No infinite-past network semantics is inferred.

For a finite experiment, gauge-neutral pointer and controller registers can be reserved locally. Tensoring the constrained system with these declared neutral registers does not impose a spatial tensor decomposition on the gauge degrees of freedom.

At finite matrix interfaces, a local instrument can be recorded using a local pointer and the isometry

Completeness gives the isometry identity. Recording or dephasing the pointer yields the direct-sum instrument of Proposition prop:records. For finite-dimensional local algebras, this local isometry can be completed blockwise to a unitary on a sufficiently large reserved neutral apparatus. All such apparatus factors must be retained in the locality ledger Stinespring,Watrous.

Consequently Theorem thm:onelayer applies to the complete record-retaining implementation, including later locally controlled continuations. It is not a theorem about an isolated selectively normalized branch.

For terminal region $Y$ and terminal tick $T$, define

Let

For the specified layers,

There is a unique UCP channel

whose inclusion equals the displayed restriction of $T_{s:T}$. The terminal snapshot therefore depends only on the initial state restriction to $\mathfrak{A}(N_s(Y))$, including joint restrictions with any independent matrix reference.

Proof. Apply Theorem thm:onelayer successively in reverse tick order. The set recursion is exactly (eq:backwardsets). Theorem thm:comparison gives the local channel and snapshot factorization, and Corollary cor:reference gives amplification.

Consider two runs on the same net, partitions, and record interfaces. Suppose:

1. their initial states agree on $\mathfrak{A}(N_{t_0}(Y))$;

2. at every tick $s$, their specified global block devices agree on every block meeting $N_{s+1}(Y)$.

Then their terminal snapshots on $\mathfrak{A}(Y)$ agree. Devices outside the backward recursion may be replaced by arbitrary devices from the stated localized class.

For the reference version, the initial restrictions must agree jointly on $\mathfrak{A}(N_{t_0}(Y))\otimes M_r$.

Proof. Pull an arbitrary terminal observable backward. At each layer only the meeting block devices contribute by Theorem thm:onelayer, and these are common. Thus the two terminal pullbacks are equal and belong to $\mathfrak{A}(N_{t_0}(Y))$. Their expectations agree by (i). The amplified calculation is identical.

Agreement of chosen complete kinematic block realizations is a sufficient certificate for condition (ii). Agreement only on $\mathfrak{A}(b)$ is not sufficient, by Proposition prop:extensiongap. A weaker sufficient condition is agreement of the layer restrictions on every algebra $\mathfrak{A}(N_{s+1}(Y))$ used in the backward proof.

The theorem is the gauge-net counterpart of the prescribed-support comparisons in En03-A, VII-E, VII-J, Psy04-B, and En06-A CheungEn03A,CheungVIIE,CheungVIIJ,CheungPsy04B,CheungEn06A. Their model-specific tensor wiring is not imported.

For a finite sequence of localized internal devices with envelopes $b_1,\dots ,b_m$, start from $S_m=Y$ and put

The complete pullback of $\mathfrak{A}(Y)$ lies in $\mathfrak{A}(S_0)$. Replacing excluded devices does not change the terminal snapshot when the initial restriction and the included complete device actions agree.

Proof. Apply Lemma lem:devicesupport at each backward step. Then apply Theorem thm:comparison.

This formulation uses declared algebras at the successive cuts. It does not assert that every arbitrary graph of regional UCP maps has a uniquely defined global wiring operation.

Let $d$ be an integer-valued metric on $X$, and suppose every partition block has diameter at most $r$.

For finite $Y$,

Equal initial state restrictions on the algebra of that metric neighborhood, and common included complete device actions, give equal terminal snapshots.

Any admitted matched local signalling pair whose changes are excluded from this backward region has zero unconditioned law gap. For ticks of physical duration $\Delta t$, the resulting scalar metric propagation upper bound is $r/\Delta t$. No lower attainment statement follows.

Proof. If $x$ is added through a block meeting $N_{s+1}(Y)$ at $y$, then

Backward induction proves (eq:metricenvelope). Isotony permits restriction from the larger metric-neighborhood algebra to the ancestor algebra. Theorem thm:matched gives the snapshot statement. The speed assertion is the same upper-bound comparison in physical units.

Gauss constraints can prevent an independently adjustable single-site perturbation from existing. A declared metric envelope is therefore not an attainment theorem. Nor is algebraic localization necessarily unique: two distinct regions can share the same central observable. The admissible intervention, not a label alone, determines whether a local variation is possible.

Let a finite dependency partial order compare every pair of devices that are not declared independent. Suppose incomparable devices commute as complete global maps; in particular, disjoint internal Kraus devices satisfy this condition. Every linear extension gives the same complete composite channel.

Proof. Any two linear extensions are connected by adjacent swaps of incomparable events. Each swap leaves the composition unchanged.

With complete local records and their dependencies included, this also preserves the joint labelled record law. It does not assert boost covariance or unrestricted inter-frame device composition.

Relative to a protected input algebra $\mathfrak{A}(I)\subseteq \mathfrak{M}$, an exterior-fixing intervention is a UCP map $L:\mathfrak{M}\to \mathfrak{M}$ with

If $T({\mathfrak{A}}_{\mathrm{out}}(Y))\subseteq {\mathfrak{A}}_{\mathrm{in}}(I)$ and $L$ is exterior fixing, then

Thus $\omega$ and $\omega \circ L$ give the same unconditioned output snapshot for every global input state. The conclusion holds jointly with every independent untouched matrix reference.

Proof. Every $T(b)$ belongs to the fixed algebra of $L$. Amplify that pointwise identity for the reference conclusion.

For the device class of Section sec:onelayer, an intervention localized in a region disjoint from the initial ancestor region cannot change the terminal unconditioned law. Hence

for the declared operational interventions. The inclusion can be strict.

Proof. Lemma lem:devicesupport makes the exterior device fix the ancestor algebra, and Theorem thm:ancestry places the terminal pullback there. Apply Theorem thm:nosignalling. The constant report in Example ex:constant supplies strictness.

The result neither requires nor implies product states, independent outcomes, or arbitrary independent preparation of regional snapshots. Proposition prop:sharedcenter exhibits a compatibility constraint even before correlations are considered. Stronger subsystem-independence statements, such as split-property results in AQFT, require additional assumptions Fewster.

Absence of signalling under a restricted set of exterior devices does not imply all-state snapshot determinacy. A device class containing only the identity tests nothing. A converse needs a sufficiently rich intervention or state-separation interface.

For states on any unital $C^{\ast }$-algebra use

For finite matrix or hybrid algebras this is trace distance. For classical finite algebras it is total variation.

Let $\tilde{T}:\mathfrak{B}\to \mathfrak{M}$ be UCP and let $T_0={\iota }_{I}K:\mathfrak{B}\to \mathfrak{M}$ be a UCP map factoring through $\mathfrak{A}(I)$. If

then states with equal restrictions on $\mathfrak{A}(I)$ have output distance at most

The same estimate holds at every independent reference size when the corresponding joint input restrictions agree.

Proof. The ideal output states agree. Each implemented output differs from its ideal output by at most $e$, because composition of a norm-one state with the difference map has norm at most $2e$. The triangle inequality gives $2e$. State distance is at most one. At matrix size $r$, the difference-map norm is bounded by its completely bounded norm.

For compatible UCP maps $T_j,{\tilde{T}}_j$,

Proof. Replace one factor at a time. Each UCP factor is a complete contraction, so every term in the telescoping sum is bounded by the corresponding local error.

On finite interfaces, this is the Heisenberg dual of the half-diamond accounting used in the supplied channel-certificate comparisons Watrous,CheungEn05A,CheungEn05B,CheungPsy04B. A small observational defect alone is not asserted to supply a nearby UCP regional factorization.

Let $\mathfrak{A}\subseteq \mathfrak{M}$ be a unital von Neumann subalgebra. If $h=h^{\ast }\notin \mathfrak{A}$, there exist normal states $\omega ,\nu$ on $\mathfrak{M}$ with equal restrictions to $\mathfrak{A}$ and $\omega (h)\ne \nu (h)$.

Proof. The self-adjoint part of $\mathfrak{A}$ is ultraweakly closed. Separation in the dual pairing $(\mathfrak{M},{\mathfrak{M}}_{\ast })$ gives a nonzero normal Hermitian functional $f$ vanishing on $\mathfrak{A}$ with $f(h)\ne 0$. Its Jordan components $f_{\pm }$ are normal. As in Lemma lem:separation,

Normalize them to obtain the stated normal states.

Suppose the ambient algebras and local algebras are von Neumann algebras, their inclusions are normal, and $T$ is normal UCP. The three conditions of Theorem thm:comparison remain equivalent when snapshot determinacy is tested on all normal global input states. The unique induced local map $K$ is normal UCP. The factorization holds on preduals and is stable under normal independent matrix amplification.

Proof. Use Lemma lem:normalseparation in place of Lemma lem:separation. The matrix-positivity proof of complete positivity is unchanged.

To prove normality, let $b_{\lambda }$ be a bounded increasing net of positive local output elements, with supremum $b$. Normality of $T$ gives

in the ambient input algebra. These elements belong to the input von Neumann subalgebra, where the same supremum is computed. Thus $K(b_{\lambda })\uparrow K(b)$, proving normality. Dualization gives the predual identity.

For a merely norm-closed subalgebra represented inside a von Neumann algebra, testing only normal states need not recover that norm-closed algebra. The normal theorem requires the regional algebra to be a von Neumann subalgebra.

For example, the algebra of compact operators plus scalars on an infinite-dimensional Hilbert space is ultraweakly dense in $\mathcal{B}(\mathcal{H})$. Two normal states agreeing there agree everywhere, so normal-state determinacy cannot detect membership in that proper norm-closed subalgebra.

No normal conditional expectation has been added to Theorem thm:normal. Its optional state-preserving existence continues to require a separate criterion, as discussed in Section sec:expectations.

This section gives an actual observable algebra, not a conclusion from commuting superoperators or convergent outcome numbers.

Let ${\mathfrak{M}}_n$ be unital $C^{\ast }$-algebras. Define

The latter is a closed two-sided ideal. Put

Write $[a_n]$ for a class. Its norm is

For nets ${\mathfrak{A}}_n(O)\subseteq {\mathfrak{M}}_n$, define

The following statements hold.

1. Each ${\mathfrak{A}}_{\infty }(O)$ is a unital $C^{\ast }$-subalgebra of ${\mathfrak{M}}_{\infty }$, and isotony passes to the limit.

2. If the two scale algebras commute for all sufficiently large $n$, their limit local algebras commute. More generally, the same conclusion follows if every pair of bounded local representative sequences has commutator norm tending to zero.

3. UCP maps $T_n:{\mathfrak{M}}_{\mathrm{out},n}\to {\mathfrak{M}}_{\mathrm{in},n}$ induce a UCP map

   $$T_{\infty }([a_n])=[T_n(a_n)].$$

4. If

   $$T_n({\mathfrak{A}}_{\mathrm{out},n}(Y))\subseteq {\mathfrak{A}}_{\mathrm{in},n}(I)$$

   for all sufficiently large $n$, the induced map has the same containment and the tensor-free comparison theorem applies.

5. Complete recorded instruments with fixed finite alphabets, independent matrix references, and matched equalities of device actions pass to this quotient.

Proof. The local bounded product is a $C^{\ast }$-algebra. Its intersection with the ambient null ideal is its own norm-null ideal. Formula (eq:quotientnorm) shows that its quotient embeds isometrically into the ambient quotient. Its image is therefore closed, proving (i).

For (ii),