Accessible Interference in Recorded Quantum Circuits: Exact Interfaces, Record Architecture, and Collapse-Model Hierarchies

We give a self-contained finite-dimensional theory of access to interference in nested-measurement circuits with undisturbed records. Internal recording is coherent copying, and established records are subsequently used only as copy sources or computational-basis controls. For a record set $J$, the discrepancy between a coherent state $\rho$ and its record-dephased model is the traceless Hermitian operator $X_J=\rho -{\Delta }_J\rho$. For every quantum channel $\Lambda$, agreement on all available effects is equivalent to $\Lambda (X_J)=0$, and the maximal probability bias is $\frac{1}{2}\Vert \Lambda (X_J){\Vert }_1$. Subsystem and algebraic interfaces follow as special cases.

A branch cross-operator calculus separates perfect-copy screening, residual distinguishability, operator-level cancellation, and question-specific agreement. Pairwise visibility is the fidelity of the inaccessible branch states. This gives exact product laws for independent noisy fragments and all-or-nothing behavior for correlated phase records. Operator-valued Möbius elements distinguish state, interface, and question orders. State and subsystem supports are down-closed; arbitrary channel supports need not be, and every prescribed support family is realizable by a binary measurement channel on a full-support tower. For the symmetric Frauchiger–Renner state, order two belongs to Wigner's question, not to the state.

We prove sequential agreement on the record-invariant algebra, diamond-norm perturbation bounds, a same-data counterexample to scalar visibility characterizations, and the codimension-two geometry of the complex-amplitude Frauchiger–Renner order-two stratum. The read-chain bias $2^{-c}$ is established for single product effects, while local parity measurements attain the unrestricted bias $1/2$. Standard hypothesis-testing theorems give a trace-distance Chernoff sandwich and a Pinsker lower bound for the Stein exponent. An embedded, hash-identifiable regression program is supplied without claiming an execution record.

Figure fig:scope-En01-A, Figure fig:split-En01-A, Figure fig:chain-En01-A, Figure fig:pair-En01-A, and Figure fig:gate-En01-A name the stop. Not a proof, and not an admission of a quantum coefficient.

Nested-observer experiments distinguish two statistical descriptions of the same recorded quantum process: a coherent description of the entire laboratory and a model in which selected records have been dephased. The Frauchiger–Renner experiment is a prominent example FrauchigerRenner2018. The present question is operational: which accessible measurements can distinguish these descriptions?

This article consolidates the subsystem and channel developments of two earlier drafts DraftSubsystem,DraftChannel. All definitions and proofs needed for the finite-dimensional results are included here. There is no dependence on an unproved companion theorem.

The organizing chain is

Its three stages describe interference content, accessible interference, and implementable probability bias, respectively.

The channel-interface theorem is the standard operational characterization of trace distance, applied to a structured family of state differences. We do not claim a new proof of the Born rule or a new state-discrimination principle. The circuit-specific results concern record anchoring, copy screening, cross-operator cancellation, Möbius localization, and the consequences of restricting the interface.

Several distinctions are essential.

1. Internal recording is a coherent isometry, not destructive readout. A nonselective dephasing channel is obtained by discarding a fresh coherent copy.

2. Agreement of one effect is weaker than agreement of every effect on a subsystem or after a channel.

3. The copy budget is necessary, not sufficient, for detection. Relative cross-operator phases are not determined by scalar visibilities.

4. Classicality order is explicitly indexed by the object being tested: state, interface, or question.

5. Exact vanishing is not generally stable under arbitrary perturbations. Quantitative smallness and thresholded support are stable in the senses proved below.

and mathematical boundary. The Born rule and the completely positive trace-preserving (CPTP) channel formalism are assumed. The dephased model is a specified statistical comparator, not an additional physical collapse mechanism. The probability calculations for the Frauchiger–Renner state do not reproduce, resolve, or replace the original theorem concerning agents' reasoning assumptions. No claim is made about destructive-primitive uniqueness, quantum field theory, gravitational coefficient admission, or a higher-order protocol calculus.

The new finite-dimensional statements are proved below. The asymptotic quantum Chernoff and Stein theorems are standard external results, invoked with their assumptions and references in Section sec:testing.

All Hilbert spaces are finite-dimensional. A state is a positive operator of trace one. An effect is an operator $0\le M\le 1$. We use

A finite qubit set $Q$ has Hilbert space ${\mathcal{H}}_Q={⨂}_{q\in Q}{\mathbb{C}}^2$. For $S\subseteq Q$, write $\overline{S}=Q\setminus S$.

For a measured factor $S$, an untouched factor $B$, and a computational basis $\{|z{⟩}_S\}$, coherent recording is the isometry

It retains $S$ and a quantum carrier $A$. It does not make $A$ a classical register in the sense of a joint block-diagonal output state.

By contrast, destructive readout is the channel

whose output does not contain $S$.

A represented nondestructive projective instrument has a third output:

It retains $S$, but its output record $R$ is classical. These three maps have different declared outputs and are not identified.

The circuits studied here use (eq:coherentcopy). The nonselective comparator arises by tracing its additional carrier, not by renaming (eq:coherentcopy) as (eq:destructive).

A compiled tower is a finite event sequence on $Q$, initialized in a pure lab state and fresh zero-state ancillas, with evolving pairwise disjoint copy classes $K_1,\dots ,K_N$, each having a designated root. The admissible events are:

1. (T1)

   Designation: a qubit outside all classes becomes the root of a new singleton class. This is bookkeeping, not a gate.

2. (T2)

   Perfect copy: a CNOT from a class member to a fresh ancilla in $|0⟩$; the target is added to the source's class.

3. (T3)

   Record-controlled processing: a unitary on qubits outside all classes, controlled in the computational basis by class members.

4. (T4)

   Residual processing: a unitary acting only on qubits outside all classes.

After designation or copy creation, a class member may appear only as a (T2) source or a (T3) control.

Write

The final pure state is $|\Psi ⟩$, with $\rho =|\Psi ⟩⟨\Psi |$.

The discipline means that an established record is not edited. It is a restriction on internal gates, not on an external detecting effect. An external reversal circuit is allowed only through an interface containing all factors on which that circuit acts.

A partition into observer levels may be supplied, but the results depend on the final classes, not on the level names. A record designated at one level and copied once at each of $d-1$ later levels has $c=d$ members. Throughout this paper a $c$-member read-chain contains one root and $c-1$ copying events.

Every compiled tower has the form

For supported branches $p_r>0$, define

Then

Phases are included in ${\hat{\xi }}_r$. All subsequent branch sums range over supported branches.

Proof. Induct on the event sequence. With no designated classes the assertion is immediate. For (T1), expand the residual vector in the computational basis of the new root, creating an additional branch bit. For (T2), a source in class $K_j$ is $|r_j⟩$ in branch $r$; the fresh target becomes another factor $|r_j⟩$. A (T3) gate acts on the residual vector by the unitary selected by $r$. A (T4) gate acts on each residual vector without touching the record factors. These exhaust the allowed events. Distinct branch labels differ on a class carrying orthogonal computational factors, proving orthonormality.

For a qubit set $A$, let

be computational-basis pinching on $A$, with identity factors elsewhere. It is CPTP, idempotent, and Hilbert–Schmidt self-adjoint. For $J\subseteq \mathcal{N}$, define the fixed channel

Thus ${\Delta }_J$ is defined on all operators, not just tower states.

Let $A\subseteq {\bigcup }_j{K}_j$, with $A\cap K_j\ne \varnothing$ exactly when $j\in J$. On a tower state,

In particular, pinching one member of each selected class is sufficient. The equality need not hold on an arbitrary operator outside the tower-supported subspace.

Proof. On a member $q\in K_j$, a branch cross term carries $|r_j⟩⟨{r^{\prime }}_j|$. Pinching that member preserves the term precisely when $r_j={r^{\prime }}_j$. The number and choice of members pinched within the class are therefore irrelevant on the branch expansion. The condition $A\subseteq {\bigcup }_j{K}_j$ excludes additional pinching of residual degrees of freedom.

For an effect $M$ and $J\subseteq \mathcal{N}$, define

The difference operator is

It is Hermitian and traceless, and the probability bias of $M$ is $\mathrm{tr}(M{X}_J)$.

Choose one anchor in each class $j\in J$. Let $C$ copy those anchors into a fresh register $R^{\prime }$. Then

Consequently, the dephased model is the coherent Born model of an extended tower when its last copies are inaccessible.

Proof. For one anchor,

Tracing the copy yields the factor ${\delta }_{b{b}^{\prime }}$, hence pinching. Repeat for each anchor and apply Lemma lem:anchor.

The proposition identifies two specified statistical descriptions within the Born formalism. It does not prove that exactly two probability maps are possible under all interpretations or assumptions. Likewise,

shows that external unitary processing can be absorbed into the effect; it does not constitute a uniqueness theorem for measurement laws.

For traceless Hermitian $Y$,

The positive-spectral projector $P_{+}(Y)$ attains the positive maximum.

Proof. Write $Y=Y_{+}-Y_{-}$, with positive parts of orthogonal supports. Since $\mathrm{tr}Y=0$, $\mathrm{tr}{Y}_{+}=\mathrm{tr}{Y}_{-}=\frac{1}{2}\Vert Y{\Vert }_1$. For an effect $M$,

Taking $M=P_{+}(Y)$ attains the upper endpoint.

This is the standard two-state discrimination optimization Helstrom1976,Watrous2018.

For a CPTP map $\mathcal{E}$ and Hermitian $Y$, $\Vert \mathcal{E}(Y){\Vert }_1\le \Vert Y{\Vert }_1$.

Proof. Positivity and trace preservation give

An interface channel is a CPTP map $\Lambda :\mathcal{B}({\mathcal{H}}_Q)\to \mathcal{B}(\mathcal{K})$. An input effect is available after $\Lambda$ if it equals ${\Lambda }^{\ast }(M)$ for an output effect $M$. Define

For a fixed tower, record set $J$, and channel $\Lambda$, the following are equivalent:

1. The coherent and $J$-dephased models agree on every effect available after $\Lambda$.

2. ${\delta }_J(\Lambda )=0$.

3. $\Lambda (X_J)=0$.

Moreover,

The supremum is attained by $P_{+}(\Lambda (X_J))$.

Proof. The operator $\Lambda (X_J)$ is Hermitian and traceless. For every output effect,

Apply Lemma lem:helstrom to $\Lambda (X_J)$. The three conditions are equivalent because the resulting maximum vanishes exactly when that operator vanishes.

For every further CPTP map $\Gamma$,

Exactly agreeing channels are closed under post-processing. For subsystem interfaces the agreement family

is a down-set.

Proof. Use Lemma lem:contract. For $S^{\prime }\subseteq S$, restriction to $S^{\prime }$ is a further partial trace after restriction to $S$.

Take ${\Lambda }_S={\mathrm{tr}}_{\overline{S}}$. Then

All effects on $S$ agree if and only if ${\mathrm{tr}}_{\overline{S}}{X}_J=0$. A Helstrom projector on $S$ attains the maximum.

Proof. The available effects after ${\mathrm{tr}}_{\overline{S}}$ are exactly $M_S\otimes 1_{\overline{S}}$. Apply Theorem thm:channel.

For a unital $\ast$-subalgebra $\mathcal{A}\subseteq \mathcal{B}({\mathcal{H}}_Q)$, the Hilbert–Schmidt orthogonal projection $E_{\mathcal{A}}$ onto $\mathcal{A}$ is CPTP, unital, and self-adjoint. Hermitian elements of $\mathcal{A}$ have spectral projectors in $\mathcal{A}$.

Proof. After a unitary change of coordinates, write

If $P_{\ell }$ are the block projectors, then

Block pinching, partial trace, and replacement by a maximally mixed factor are completely positive. The formula preserves trace and the identity and is the Hilbert–Schmidt orthogonal projection. Spectral projectors belong to $\mathcal{A}$ by finite-dimensional functional calculus.

For effects restricted to $\mathcal{A}$,

All such effects agree if and only if $E_{\mathcal{A}}(X_J)=0$. The corresponding Helstrom projector belongs to $\mathcal{A}$.

Proof. For $M\in \mathcal{A}$, self-adjointness gives $\mathrm{tr}(M{X}_J)=\mathrm{tr}(M{E}_{\mathcal{A}}(X_J))$. The spectral projector is admissible by Lemma lem:expectation. Apply Lemma lem:helstrom.

For $\mathcal{A}=\mathcal{B}({\mathcal{H}}_S)\otimes 1_{\overline{S}}$,

so the algebraic formula recovers the subsystem formula. Agreeing algebras and subsystems form ordered families; no general closure under joins or unions is asserted.

For a POVM $\mathsf{M}=\{M_o\}$, let

Then

For the binary POVM $\{M,1-M\}$, denoted by ${\Lambda }_M$,

Thus a fixed question is itself a particular interface channel.

Proof. The output difference is diagonal. For the binary case its two entries are $\mathrm{tr}(M{X}_J)$ and its negative.

Detector noise and classical post-processing can be included by composition of channels. They cannot increase ${\delta }_J$. This is a contraction statement about the total Hermitian difference; it does not imply that every general branch cross operator is multiplied by one universal scalar damping factor.

If $M$ commutes with every computational record projector defining ${\Delta }_J$, then $\mathrm{tr}(M{X}_J)=0$.

Proof. Such an $M$ satisfies ${\Delta }_J(M)=M$. Hilbert–Schmidt self-adjointness of pinching gives $\mathrm{tr}(M\rho )=\mathrm{tr}(M{\Delta }_J\rho )$.

For a fixed subsystem, individually agreeing effects form the closed convex slice

When the reduced difference is nonzero, this is a hyperplane slice of the effect body, not the whole body.

For distinct branch labels define the difference set

A pair is $J$-split if $D(r,r^{\prime })\cap J\ne \varnothing$. Pairs below are unordered unless an ordering is explicitly displayed.

For supported $r\ne r^{\prime }$, set

Then

For every pair,

Visibility is nondecreasing when $S$ is enlarged.

Proof. For a square matrix $A$, polar decomposition and trace duality give

Apply this identity to ${{\Gamma }^S}_{r{r}^{\prime }}$ and use the defining property of partial trace. The overlap of two unit vectors has modulus at most one. A unitary on a smaller interface extends by the identity to a larger one, proving monotonicity.

We use the square-root fidelity convention

For

one has

Proof. This is Uhlmann's variational identity Uhlmann1976. For completeness, write Schmidt decompositions

The overlap optimized in Lemma lem:variational is $\mathrm{tr}(UB)$, where

Its maximum is $\Vert B{\Vert }_1$. In the respective orthonormal Schmidt bases, the nonzero singular values of $B$ are those of the entrywise conjugate of the matrix of $\sqrt{{{\sigma }^{\overline{S}}}_r}\sqrt{{{\sigma }^{\overline{S}}}_{r^{\prime }}}$. Complex conjugation preserves singular values. Thus the two trace norms coincide.

For a perfect-record tower:

1. If some $j\in D(r,r^{\prime })$ has a member outside $S$, then ${{\Gamma }^S}_{r{r}^{\prime }}=0$.

2. If every member of every differing class lies in $S$, then the left and right supports of ${{\Gamma }^S}_{r{r}^{\prime }}$ are orthogonal, and

   $$\Vert {{\Gamma }^S}_{r{r}^{\prime }}+{{\Gamma }^{S†}}_{r{r}^{\prime }}{\Vert }_1=2v_{r{r}^{\prime }}(S).$$

Proof. An inaccessible differing member contributes $\mathrm{tr}|r_j⟩⟨{r^{\prime }}_j|=0$, proving the first assertion. Otherwise, the reduced cross operator retains the differing perfect record factors on $S$. They place its left and right supports in orthogonal computational subspaces. For an operator $P\Gamma Q$ with $PQ=0$, the Hermitian sum has block form

Its singular values are those of $B$, each occurring twice.

For every $J,S$,

If exactly two branches $r,r^{\prime }$ are supported, then

The equality in the first line uses perfect-record structure.

Proof. Apply the triangle inequality to (eq:cross-expansion) and use $\Vert \Gamma +{\Gamma }^{†}{\Vert }_1\le 2\Vert \Gamma {\Vert }_1$. For two branches, an unsplit pair contributes nothing to $X_J$. A split pair is either screened, making both sides zero, or has orthogonal supports as in Proposition prop:screening, giving equality.

If every supported $J$-split pair has an inaccessible perfect member of a differing class, then ${\delta }_J(S)=0$. Conversely, detection requires a supported $J$-split pair satisfying

In particular,

The minimum of an empty family is interpreted as $+\infty$.

Proof. Screening kills every summand of (eq:cross-expansion). If its sum is nonzero, at least one cross operator is nonzero. Proposition prop:screening then forces the stated containment.

The budget is a necessary threshold. Even after it is met, inaccessible residuals may lower visibility or different cross operators may cancel. Section sec:cancellation gives a same-data counterexample to sufficiency.

Start with a designated $|+⟩$ root and make $c-1$ perfect copies. The state is

For an interface containing $m<c$ class members, every effect has zero bias. At $m=c$,

The projector onto $|{\mathrm{GHZ}}_c⟩$ has coherent probability one and dephased probability $1/2$. Thus the perfect-copy transition is a hard step at the full budget. This is an example at every chain length, not a universal statement that every budget-complete interface has maximal discrepancy.

Nonorthogonal pointer states are not interchangeable with perfect computational records.

A fresh residual fragment $a$ may be prepared under control of a perfect record root $q$ by

The fragment is not added to a perfect copy class unless it is actually a computational-basis copy. A product-law assertion additionally assumes that the branch states retain the specified fragment tensor factors.

The map is an isometry because the source labels are orthogonal. However, a nonorthogonal fragment cannot generally be used as a perfect source that copies its underlying branch bit. For example, an isometry satisfying

would preserve an input overlap that is nonzero while producing zero output overlap. Thus a noisy-fragment “read-chain” must specify how each fragment is prepared; the independent construction below uses a retained perfect root.

Suppose the inaccessible branch states factorize as

Then

In particular, the overlap product alone is an upper bound, and is an equality when the residual fidelity is one.

Proof. Fidelity is multiplicative under tensor products: both the square root and the trace norm factorize. For pure factors, $F(|e⟩⟨e|,|e^{\prime }⟩⟨e^{\prime }|)=|⟨e^{\prime }|e⟩|$. Apply Theorem thm:fidelity.

Consider

Dephase the perfect root. If $S$ contains that root and $m-1$ fragments, then

For uniform ${\gamma }_a=\gamma$,

Complex overlap phases do not change this equality.

Proof. Tracing the inaccessible fragments yields

The accessible vectors $u,v$ are orthogonal because $S$ retains the perfect root. The Hermitian cross sum therefore has trace norm $2|z|$.

If the root is inaccessible, exact screening remains. Consequently the expression ${\gamma }^{c-m}$ is not an architecture-free formula for arbitrary choices of $m$ accessible qubits.

Let a retained perfect root control the two residual environment states

Prepare the environment with residual gates, then apply a record-controlled $Z$ to one fragment. If $k<c$ environment fragments are inaccessible, their two reduced states coincide, so $v=1$. If all $c$ fragments are inaccessible, the two environment states are orthogonal, so $v=0$. For $k=0$, the inaccessible environment is the one-dimensional trivial factor.

This differs sharply from independent redundant fragments. The relationship to environment-based classicalization Zurek2009 is architectural: fidelity measures information in the inaccessible remainder, not merely the number of fragments.

For $J\subseteq \mathcal{N}$, define

One has $Z_{\varnothing }=\rho$, and $Z_J$ is traceless for nonempty $J$.

For every $J$,

For nonempty $J$,

Equivalently,

Proof. For inversion, interchange the subset sums. The coefficient of ${\Delta }_L\rho$ is

which is one when $L=J$ and zero otherwise.

For localization, the diagonal terms cancel when $J\ne \varnothing$. A cross term with difference set $D$ receives the coefficient

The product formula follows by expanding the commuting factors.

For every $J$,

For a singleton,

Proof. Subtract the inversion formula from $Z_{\varnothing }=\rho$, then apply the triangle inequality.

For nonempty $J$,

Moreover,

Thus the state support, after adjoining the empty face, is the simplicial complex generated by supported branch difference sets.

Proof. The branch matrix units are Hilbert–Schmidt orthonormal:

Hence no distinct terms in (eq:localization) cancel globally. The norm formula and support characterization follow. Subset closure is immediate.

Set

and

with $\min \varnothing =\infty$. The state order is $\mathrm{ord}(\mathrm{id})$. An interface is fully classical for this hierarchy if its order is infinite.

For every channel $\Lambda$ and output effect $M$,

Post-processing can only increase order and decrease each weight. The state order is one whenever two supported branches coexist, and is infinite otherwise.

The question order is at least $m$ if and only if all dephasings of fewer than $m$ records leave its probability unchanged. The corresponding statement holds for interface order with agreement on every output effect.

Proof. If $Z_J=0$, then $\Lambda (Z_J)=0$. If $\Lambda (Z_J)=0$, then $\mathrm{tr}(M\Lambda (Z_J))=0$. Contractivity proves the post-processing statement.

Any two distinct labels differ on some $j$, and Proposition prop:statecomplex then gives $Z_{\{j\}}\ne 0$. With one supported branch all nonempty elements vanish.

Finally,

Vanishing below size $m$ implies adequacy below size $m$. Conversely, induction on $|J|$ recovers the vanishing of the coefficients from adequacy. The operator argument is identical after applying $\Lambda$.

For a measurement channel,

For a binary question and nonempty $J$,

Proof. Apply the measurement channel to $Z_J$. In the binary case its two diagonal entries are ${\eta }_M(J)$ and $-{\eta }_M(J)$.

The channel hierarchy and the subsystem hierarchy have different combinatorial restrictions.

For a perfect-record tower and fixed subsystem $S$, the family

is down-closed under nonempty subsets. In particular,

Proof. For each class meeting $S$, choose an accessible member and let $Q_j$ be pinching that member on ${\mathcal{H}}_S$. If $K_j\cap S=\varnothing$, put $Q_j=\mathrm{id}$. The $Q_j$ commute.

Inspect a branch cross term. If a class has an inaccessible differing member, that term is already zero after partial trace. Every surviving term therefore has the same survival rule under global class pinching and under its selected accessible anchor. A class wholly outside $S$ cannot distinguish surviving reduced terms. Thus, for all $K\subseteq \mathcal{N}$,

Consequently,

For $L\subseteq J$,

If the rightmost argument were zero, the result would be zero. Hence a nonzero $J$-element forces every nonempty subset element to be nonzero. A nonempty support therefore contains a singleton.

General channel supports need not be simplicial complexes. The order-two measurement channel in Section sec:fr already supplies a counterexample. In fact, there is no down-closure restriction for unrestricted binary measurement channels.

Fix $N\ge 1$. There is a full-support compiled tower such that, for every family

a binary measurement channel ${\Lambda }_M$ satisfies

Thus every order in $\{1,\dots ,N,\infty \}$ is realizable at the question and arbitrary-channel levels.

Proof. Prepare $|+{⟩}^{\otimes N}$ and designate all $N$ qubits. The classes are singleton, every branch is supported, and $p_r=2^{-N}$.

For each nonempty $D\subseteq \mathcal{N}$, define

where $1_D$ flips precisely the bits in $D$. These are nonzero Hermitian operators with mutually disjoint matrix-unit supports, hence are Hilbert–Schmidt orthogonal. Localization gives

The Boolean-lattice transformation on nonempty subsets is triangular with nonzero diagonal, so the $Z_J$, $J\ne \varnothing$, are linearly independent over the reals.

Their real Gram matrix $G_{JK}=\mathrm{tr}(Z_J{Z}_K)$ is therefore positive definite. Set $a_J=1$ for $J\in \mathcal{F}$ and $a_J=0$ otherwise. Choose a Hermitian

so that $\mathrm{tr}(H{Z}_J)=a_J$. Take $t>0$ with $t\Vert H{\Vert }_{\infty }\le 1$, and put

This is an effect. Since $Z_J$ is traceless for nonempty $J$,

Corollary cor:question-order gives the prescribed support. For the empty family, $M=1/2$ suffices.

Down-closure for arbitrary channel support is false, not an open question. Unrestricted support-family realizability is settled by Theorem thm:anysupport. Realizability with a prescribed subsystem, branch support, locality restriction, or implementability class is a different problem and is not settled by that theorem.

Use qubits $s_1,a_1,s_2,a_2,A$, initially zero. Apply:

then $X$ on $A$ controlled by $s_2=1$, and $\mathrm{diag}(1,e^{i\theta })$ on $A$ controlled by $s_1=s_2=1$. These are legal compiled events. The classes are

and the final state is

Let $J=\{1\}$, $S=K_1$, and $S^{\prime }=K_1\cup \{A\}$.

For the family (eq:phasefamily),

The copy classes, branch probabilities, and all pairwise visibilities on $S$ are independent of $\theta$. At $\theta =\pi$, every nonempty Möbius element vanishes on $S$, so that subsystem is fully classical for the hierarchy.

Proof. Use the normalized branch vectors

Pairs with different $b$-values are screened by inaccessible $K_2$. The two surviving split pairs are $(0b,1b)$, and

Each has trace norm one. Therefore

Its eigenvalues are $\pm |1+e^{-i\theta }|/4$.

On $S^{\prime }$, the two contributions occupy orthogonal $A$-blocks: