Hawk07-A Existence of Operational Wormhole Certificates A finite Enter model with a fixed receiver, nested access, and a positive-residual cutoff

We define and prove existence of operational wormhole certificates: complete finite encodings whose unknown logical input can be returned through one declared accessible interface, with a deterministic decoder and a worst-case reference-preserving error allowance. This terminology describes a recovery contract, not a spacetime wormhole.

The model has an unknown diary, an old-vault register maximally entangled with accessible early memory, and one input-independent scrambling unitary. A nested schedule hands over vault cells while retaining a positive-dimensional physical residual. We prove the exact complete-complement criterion, derive the finite Haar decoupling moment, and specify a deterministic decoder by a semidefinite program. One threshold supplies the same decoder core at every subsequent retained cut.

For every prescribed full-diamond tolerance $0<\epsilon <1$, an explicit finite parameter family has Haar measure at least $1/64$ on which three consecutive retained cuts admit the operational certificate. On the same devices and for one fixed pure diary input, late-radiation entropy rises and then falls within that window; the last cut retains one physical qubit. Append-only Enter records remain nondecreasing.

We correct two essential claims in the proposed analogy. Nested-access recovery error and complete residual leakage cannot peak at an intermediate Page cut: both are nonincreasing. Furthermore, measuring late radiation does not imply a record-only obstruction if an early quantum register survives; teleportation provides an exact counterexample. Terminal readout and forgetting equal partial trace on the returned interface, not physical annihilation or disappearance of the complete complement.

No metric, Einstein–Rosen bridge, Morris–Thorne throat, geometric ER $=$ EPR identification, or gravitational signal-crossing theorem is constructed.

of the title. The existence result concerns the operational objects defined below. It is not a proof that physical spacetime wormholes exist. Calling a recovery certificate a wormhole adds no geometry to it.

Cheung's Philosophy of Enter distinguishes four questions: what is the input, what is accessible, what is returned, and what additional identification is being claimed. Destructive readout, quantum return, thermodynamic implementation, and geometric interpretation are separate interfaces CheungPhilosophy.

The Enter calculus treats runtime qubit names as access rights. Enter consumes measured quantum names and returns classical outcomes; it is not the identity channel on the consumed quantum factor. Cheung's operational characterization fixes Born probabilities and projective residual rays within its specified strategy class, rather than deriving gravitational dynamics from measurement syntax CheungEnter,CheungI.

We work with ordinary finite-dimensional quantum channels and explicitly declared Born instruments. Enter supplies the access and return-type discipline. Haar scrambling, the entangled resource, the release schedule, and the receiver are additional model declarations.

Einstein–Rosen bridges, Morris–Thorne wormholes, and the ER $=$ EPR proposal concern geometric or gravitational structures EinsteinRosen,MorrisThorne,MaldacenaSusskind. Gravitational traversability constructions also specify interactions, matter stress, and backreaction GaoJafferisWall,MaldacenaStanfordYang. Those structures are not specified by an isometry $L\to R\otimes B$.

Here “mouth” is an overlay label for an output factor. “Throat” is an overlay label for the complete complementary channel and its input dependence. The complement is a channel, not a third tensor factor, a tunnel, or a proper-length quantity. “Operational traversability” means successful quantum return through the declared receiver. It is directional and does not claim a causal curve from one spatial mouth to another.

{}1.12

These are mathematical corrections, not changes in physical interpretation alone.

Fix

The variable input is the whole diary $L$. An arbitrary external reference is untouched. For Choi tests use $Q\simeq L$.

Before the unknown input is used, prepare

The preparation is independent of $L$ and its references. The vault is $LHF$, of dimension

Every purifier is retained: $E$ is accessible, and $Q$ belongs to the reference side of the test, not to either receiver mouth.

A fixed encoding unitary can be absorbed into a known, input-independent vault unitary $U$. For the random model, $U$ is one Haar sample on ${\mathbb{C}}^D$. It is sampled once, not independently at every cut.

Choose a tensor-cell presentation in advance. At cut $k$, regroup the vault as

The accessible receiver is

Previously released cells and $E$ are not subsequently acted on by the vault.

Release means handing over control of named tensor factors. It can also be implemented by SWAPs into reserved blank registers. That implementation adds fixed source placeholders, which must remain in the complete bookkeeping. The primary model needs no such additional placeholders: it changes control over existing laboratory cells. No qubit leaves the laboratory.

The complete prepared isometry is

regrouped as $R_k{B}_k$. Set

For qubits,

A positive-residual cutoff is a predeclared $K<n$. The physical factor $B_K$ remains present.

For states and channels use

The diamond norm includes external references WatrousBook. Write

Set

Entropies are in bits.

A certificate for the whole logical diary is a tuple

such that:

1. $V:L\to R\otimes B$ is a complete isometry;

2. the receiver accesses $R$, not physical $B$;

3. $\mathcal{D}:\mathcal{B}(R)\to \mathcal{B}(L_{\mathrm{out}})$ is a specified, deterministic CPTP quantum-return map;

4. $${\Vert \mathcal{DN}-{\mathrm{id}}_L\Vert }_{\diamond }\le \epsilon ;$$

5. $B$ remains the declared physical residual;

6. no two commuting, code-preserving star representations both intertwine $V$ with the same algebra $\mathcal{B}(L)$.

We call the certificate nonclassical when $\epsilon <2b_d$, below the record-only lower bound. The mouth and throat terminology is an overlay on this contract.

The definition introduces no metric or state trajectory through a tunnel. It does not define two-way communication.

This definition is a recovery certificate with an additional terminological overlay. It does not distinguish error-correcting encodings from geometric wormholes.

For example, with $U=I$, release all diary cells and retain $H$. Then

and $\mathcal{D}={\mathrm{Tr}}_E$ is exact. For $h\ge 2$, this has a positive residual and entangled output mouths, but no spacetime bridge has been constructed. The Haar theorem below adds a quantitative, nontrivially scrambled existence statement, not a gravitational identification.

For every complete isometry $V:L\to RB$,

is CPTP and left-inverts ${\mathrm{Ad}}_V$, for any fixed diary state ${\omega }_L$. The complete isometry preserves the trace norm of every operator, also with an external reference.

Proof. Both terms are CP. Their traces sum to $\mathrm{Tr}Y$, and substitution gives ${\mathcal{L}}_V(VX{V}^{†})=X$. An isometric embedding preserves all nonzero singular values, including after tensoring with a reference identity.

This is complete-output restoration, not a decoder restricted to $R$. It does not relocate the diary into $B$ CheungHawk04AB,CheungEn07B.

For $V:L\to RB$,

If $s=\mathrm{rank}{\tau }_B$, then

There are an isometry $T:L\otimes {\mathbb{C}}^s\to R$ and a state $\sigma$ such that

An exact decoder on the entire accessible space is

Proof. Suppose a decoder is exact and dilate it by $W:R\to LG$. The isometry $(W\otimes I_B)V$ dilates the identity. Its Choi operator has rank one, so every Kraus operator is proportional to identity. Therefore

with fixed $\chi$. Tracing $LG$ proves complement constancy.

Conversely, diagonalize

and write

The identities $\mathcal{C}(|i⟩⟨j|)={\delta }_{ij}{\tau }_B$ give

These $ds$ orthonormal vectors define $T$. Equations (eq:normalform) and (eq:exactdecoder) follow. The decoder is CP, its two trace contributions sum to $\mathrm{Tr}Y$, and it recovers every matrix unit.

This is the established quantum error-correction criterion KnillLaflamme. Cheung's supplied full-code and observable-target discussions retain the same distinction CheungOai280290,Hawk06A. The proof above does not import their model-specific channels.

The isometry

gives computational dephasing on both outputs. The $Z$ observable is readable, but the full qubit cannot be recovered from either factor. The complement is nonconstant. A successful $Z$ record does not establish the identity channel.

Every accessible channel has ${\epsilon }_{\mathrm{rec}}\le a_d$. A replacement accessible channel has ${\epsilon }_{\mathrm{rec}}=a_d$. Moreover,

Proof. A decoded replacement channel sends a Bell input to $I_Q/d\otimes {\sigma }_L$, with Bell overlap $d^{-2}$. Testing the Bell projector gives error at least $a_d$. Preparing $I_L/d$ attains it, since depolarization is the uniform average of $d^2$ Weyl conjugations, one being identity. This constant decoder also gives the universal upper bound. Equation (eq:lower) follows by contraction and the reverse triangle inequality.

The finite information–disturbance theorem, stated in full diamond normalization, gives

Substituting the half-diamond convention proves (eq:KSW).

A nearly constant complement gives an upper recovery bound. A nearly constant accessible channel gives a lower recovery bound. The two directions cannot be interchanged.

At cut zero in the declared model, for every $U$,

The initial complement is reversible when completely accessible.

Proof. A unitary on the vault does not change the early marginal:

The recovery benchmark follows from Lemma lem:KSW. The complement is

Undoing $U$ and tracing $HF$ left-inverts it.

For any replacement state, this left inverse maps its comparison to an identity-versus-replacement comparison, whose distance is at least $a_d$. Conversely, comparing with ${{\mathcal{C}}_0}^U(I_L/d)$ and using contraction gives distance at most $d_{\diamond }({\mathrm{id}}_L,{\mathcal{T}}_{I_L/d})=a_d$.

This supplies one rigorously unavailable phase. It does not show that every nonzero early cut is featureless.

In the release schedule of Section sec:model,

If

then $z_{k+1}(U)\le z_k(U)$.

Proof. Discarding the newly received cells maps ${{\mathcal{N}}_{k+1}}^U$ to ${{\mathcal{N}}_k}^U$. Any earlier decoder can be used later.

The later complement is a partial trace of the earlier complement. Partial trace takes a replacement channel to replacement, so contractivity proves the optimized leakage inequality. It also maps ${{\sigma }_{B_k}}^U$ to ${{\sigma }_{B_{k+1}}}^U$, proving the named-center inequality.

Thus an intermediate “neck” cannot be a new maximum of these quantities. Their initial values are already maximal in this model. This directly corrects Steps 2–3 of the proposed overlay.

For a qubit define

For $0<\theta <\pi /2$, the complement is nonconstant, but

The accessible channel tends to identity as $\theta \to 0$.

Indeed, the complement is a measurement in the logical basis followed by preparation of two pure states whose trace distance is $\sin \theta$. Their average gives the stated upper bound; the triangle inequality applied to the two outputs gives the matching lower bound. Appending arbitrarily large fixed residual spectators changes neither conclusion.

Residual size, complementary nonconstancy, accessible privacy, and decoder error are separate statements.

Use the Bell test

At a cut $D=rb$, set

For $D\ge 2$,

where

For $O=I_{P{P}^{\prime }}\otimes {\mathsf{F}}_{B{B}^{\prime }}$,

Proof. The averaged operator commutes with every $U\otimes U$. The symmetric and antisymmetric subspaces give its expression as a linear combination of identity and swap. Preservation of $\mathrm{Tr}O$ and $\mathrm{Tr}(O\mathsf{F})$ fixes the coefficients. For the specified $O$, those traces are $r^{2}b$ and $r{b}^2$.

For the complete entangled preparation,

Proof. After tracing $E$, the initial reference–vault state is

Its purity is $1/h$, its vault marginal has purity $1/(dh)$, and its reference marginal has purity $1/d$.

The swap identity and Lemma lem:twirl give

Since ${\mathrm{Tr}}_Q{{\omega }_{QB}}^U={{\sigma }_B}^U$,

The $\alpha$ terms cancel. Insert $\beta$ from (eq:twirl).

For channels with input dimension $d$,

Consequently,

where $z(U)=d_{\diamond }({\mathcal{C}}^U,{\mathcal{T}}_{{{\sigma }_B}^U})$.

Proof. The Bell input gives the lower inequality. Every pure referenced input with a sufficient reference dimension can be written as $(T\otimes I)|{\Phi }_d⟩$, with $\mathrm{Tr}(T^{†}T)=d$, hence ${\Vert T\Vert }_{\infty }^2\le d$. Multiplying the Choi difference by $T\otimes I$ on both sides gives the upper inequality; convexity handles mixed inputs. This is the usual normalized Choi-to-diamond conversion WatrousBook.

The Choi difference acts on dimension $db$, so

Take expectations and use (eq:moment). Finally,

which gives the last bound.

Discarding the factor $d$ would incorrectly promote a Bell-input estimate to a whole-code diamond estimate.

Fix $0<p<1$, $\zeta >0$, and a cutoff $K$. If a retained cut $k_{\ast }\le K$ satisfies

then, with Haar probability at least $1-p$, the same sampled unitary satisfies, for every $k_{\ast }\le k\le K$,

A sufficient threshold is

Proof. Markov's inequality gives

Proposition prop:nested propagates this single event to all later retained cuts. Lemma lem:KSW supplies recovery. Equation (eq:simplethreshold) follows from (eq:K).

The preparation and access resemble the finite information task studied by Hayden and Preskill HaydenPreskill. The displayed moment is proved here. Hawk06-A is a predecessor comparison, not a substitute for this calculation Hawk06A.

The theorem does not certify every $U$, or guarantee a threshold before an arbitrary chosen cutoff. It applies only when its dimensional inequality is satisfied.

For a map $\Phi$, use the unnormalized input-first Choi operator

At $k_{\ast }$, let $R=R_{k_{\ast }}$, $q_R=\dim R$. A decoder Choi matrix $C_{\mathcal{D}}$ on $R^{\prime }\otimes L_{\mathrm{out}}$ defines

Set

Choose a minimizing solution of

Among optimal decoder Choi matrices, choose the one minimizing the squared Frobenius norm. Call the resulting map ${\mathcal{D}}_{\ast }$.

The program defines a deterministic CPTP decoder on all of $R_{k_{\ast }}$. Its optimal value is ${\epsilon }_{\mathrm{rec}}({{\mathcal{N}}_{k_{\ast }}}^U)$. On the success event of Theorem thm:threshold,

For a later cut, let ${\Pi }_{k\to k_{\ast }}$ discard the additional received cells and retain the same early memory. Then the one fixed decoder core with policy

has

for every $k_{\ast }\le k\le K$. Thus the full-diamond error is at most $2e$ at every retained cut.

Proof. The decoder Choi constraints are exactly complete positivity and trace preservation. Composition with the fixed channel is linear in the decoder Choi matrix.

For a Hermiticity-preserving map $\Delta$, the diamond-norm dual is

This follows as the Hermitian specialization of the two-block dual: symmetrizing the two diagonal blocks gives $Z$, and diagonalizing the block matrix gives $Z\pm J(\Delta )\ge 0$. It proves the optimal-value assertion.

Decoder Choi matrices form a compact convex set. The continuous diamond error attains its minimum. The secondary squared-norm minimization is strictly convex and therefore selects a unique decoder matrix. The error estimate follows from Lemma lem:KSW.

Finally, ${\Pi }_{k\to k_{\ast }}{{\mathcal{N}}_k}^U={{\mathcal{N}}_{k_{\ast }}}^U$ by the nested schedule. This proves the fixed-policy identity.

The decoder depends on the declared device matrix $U$, not on the unknown diary or its test reference. The optimization is a mathematical prescription, not a reported execution or an efficiency theorem.

The decoder Choi rank obeys

It has a Stinespring isometry

With qubit spaces, a conservative realization at cut $k\ge k_{\ast }$ uses peak live quantum occupancy at most

This count includes early memory, all received cells, output, and allocated decoder environment. It excludes inaccessible $B_k$ and the untouched external reference.

Proof. Diagonalize the decoder Choi matrix to obtain at most $d{q}_R$ Kraus operators and stack them into $W_{\mathcal{D}}$. Allocate output and environment registers, then extend the isometry from its prepared subspace to a unitary. The displayed register count prices a nonoptimal implementation.

Arbitrary finite unitaries admit elementary decompositions when continuous one-qubit gates and CNOT are permitted Barenco. No efficient gate count or executed compilation is asserted.

There is no one-live-qubit or restricted streaming claim. Cheung's En02-B and En02-C distinguish batch availability from memory-constrained execution in their specified benchmarks CheungEn02BC.

A completed Petz map is a CPTP candidate, but its formula alone does not certify (eq:SDP)'s error. Similarly, a postselected map satisfying

is trace decreasing when $p_{\mathrm{succ}}<1$. A completed decoder with a failure output has unconditional error at most $1-p_{\mathrm{succ}}$ in half-diamond normalization, but not automatically zero. The success probability, reflections, failure flag, and resources in an amplitude-amplified implementation must be declared CheungEn02BC,YoshidaKitaev.

Suppose proxy and actual complete channels, after access-preserving identifications, differ by at most $\eta$ in half-diamond distance. Suppose the implemented receiver differs from the specified receiver by at most $\iota$. If the specified proxy decoder has half-diamond error $e$, then

A proxy complementary replacement allowance $\zeta$ gives actual leakage at most $\min \{1,\zeta +\eta \}$.

Proof. Partial trace contracts the complete comparison. Insert the same decoder and apply CPTP contraction and the triangle inequality. Do the same with the proxy replacement state.

Thus channel closeness can transfer a fixed decoder on a correctly identified common interface. It does not install the decoder, grant forbidden access, or prove its laboratory calibration. Comparisons mixing physical $B$ into the receiver are not access preserving CheungHawk04D.

Haar matrices and the ideal decoder may have arbitrary real coefficients. Exact finite-alphabet compilation is not presumed. Finite-precision implementations belong to the allowance $\eta +\iota$.

Fix one pure diary input $|\psi ⟩$, independent of $U$. After tracing $E$, the vault state is

Its late marginal is ${{\rho }_{P_k}}^U$. It is this late-radiation entropy, not necessarily the entropy of the entire receiver $P_{k}E$, that we plot.

At $D=rb$, define

Then

For computational-basis Enter on the released cells, with prefix probabilities ${p_y}^U$,

Writing $t=\min (r,bh)$, one has

Proof. Twirl ${\mathsf{F}}_P\otimes I_{B{B}^{\prime }}$. Its identity and swap coefficients are

The initial vault purity is $1/h$, giving $p_{\mathrm{q}}$.

For record probabilities, use ${\Pi }_y=|y⟩⟨y{|}_P\otimes I_B$, of rank $b$. The averaged two-copy vault state is

Since ${p_y}^U=\mathrm{Tr}({\Pi }_y{{\rho }_{\mathrm{vault}}}^U)$,

Sum over $r$ outcomes.

The late marginal has rank at most $\min (r,bh)$, and the record has at most $r$ outcomes. These prove the upper bounds. Entropy is at least minus the logarithm of purity; Jensen's inequality proves the lower bounds. Direct comparison gives $p_{\mathrm{q}}\le 1/r+1/(bh)$, proving the final estimate.

The coherent rank envelope is

Its balancing cut is $r=bh$, when that cut is retained. It is not generally $r=b$. Nor is it a balance between the augmented receiver dimension $rh$ and the residual dimension $b$.

These dimension comparisons are not throat area, proper length, or quantum extremal surfaces.

Only when $h=1$ does the declared fixed pure input become a Haar pure-state ensemble on the vault. Let $m_0=\min (r,b)$, $n_0=\max (r,b)$, and ${\mathsf{H}}_j={{\sum }_{u=1}}^j{u}^{-1}$. Then

The first is the established Page–Sen evaluation PageAverage,Sen. The second follows by differentiating the Dirichlet moment

and summing over the $r$ outcomes. These pure-ensemble comparisons do not replace Theorem thm:entropy's early-entangled calculation CheungHawk05BC,Hawk06A.

For a measured factor $A$ in a declared basis, define

With surviving quantum factors $E$, use

The name $A$ is consumed. The surviving $E$ is not.

For every fixed prescribed process whose complete record is append only,

The statement permits adaptive readout and does not require Haar scrambling.

For fixed-basis Enter on $P_k$,

where $\Delta$ denotes basis dephasing.

Proof. The first identity is the classical chain rule. Dephasing is an average of diagonal unitary conjugations, so entropy concavity gives the second inequality.

Measuring emitted cells affects conditional residual states. It does not change the unconditional later unmeasured marginals when subsequent vault operations do not act on the emitted cells. One must not identify conditional branch entropies with the coherent history in Theorem thm:entropy.

Take

Declare

The three cuts have

If

then a set of Haar unitaries of measure at least $1/64$ satisfies all of the following:

1. $z_k(U)\le \zeta$ for every $k_{\ast }\le k\le K$;

2. the fixed decoder policy (eq:fixedpolicy) has half-diamond error at most $\min \{a_{16},\sqrt{2\zeta }\}$ at all three cuts;

3. for the fixed pure diary input,

   $$S({P_{k_{\mathrm{peak}}}}^U)>{\log }_2(2h)\ge \max \{S({P_{k_{\ast }}}^U),S({P_K}^U)\};$$

4. every append-only destructive record is nondecreasing;

5. the cutoff retains one physical vault qubit.

The ensemble mean also rises and falls strictly at these three cuts.

Proof. At $k_{\ast }$, (eq:K) gives

Hence, under (eq:size),

At the intermediate cut,

Markov's inequality gives

Therefore an event of probability greater than $1/32$ has

At $k_{\ast }$, rank is at most $2h$. At $K$, rank is at most $b_{K}h=2h$. Their entropy upper bounds hold for every unitary.

Subtracting the leakage failure set from the turnover event leaves measure greater than $1/64$, hence at least $1/64$. Nested monotonicity and Theorem thm:decoder prove (i)–(ii). Proposition prop:record gives (iv), and $b_K=2$ gives (v).

Finally,

The mean entropy lower bound in Theorem thm:entropy is therefore strictly greater than ${\log }_2(2h)$ at the intermediate cut, whereas both neighboring mean entropies are at most that value.

The turnover is a same-model entropy result. It is not the decoder proof. In particular, leakage and optimal recovery error remain nonincreasing through this entropy peak.

For every $0<\epsilon <1$, choose

There is one finite model and a set of Haar unitaries of measure at least $1/64$ on which

with the same decoder core, strict coherent rise-and-fall, and a cutoff retaining one physical qubit.

Proof. The chosen $h$ satisfies (eq:size), and $2\sqrt{2\zeta }=\epsilon$. Apply Theorem thm:joint. The nonduplicate condition is proved in Section sec:nocopy. Since $\epsilon <1<2b_{16}$, the receiver is below the purely classical recovery floor.

Let

The size requirement is $h\ge 102400$. The vault has $21$ qubits, and the early receiver has $17$. The retained cuts are:

The full-diamond allowance is $0.2$, not $0.1$. The fixed decoder core has input dimension $2^{35}$; its Kraus rank is at most $2^{39}$. Equation (eq:workspace) gives a conservative batch peak of $80$ live decoder qubits at the last cut.

This is an exact parameter calculation. No Haar sample, decoder optimization, or hardware execution is reported.

The Haar existence statement is an ideal-control theorem. It does not assert that a generic Haar matrix has a finite exact description in a discrete gate alphabet.

A finite-description approximation can be obtained without changing the operational interpretation. Apply Theorem thm:joint with a stricter leakage target, for example $\zeta /2$. For a successful unitary, $z_{k_{\ast }}<\zeta$ has a margin, and the entropy inequalities have a strictly positive gap. Channels and finite-dimensional entropy are continuous in $U$; more explicitly,

Hence sufficiently accurate finite gate approximations preserve the leakage target and the strict entropy gap. Dense finite gate families provide such approximations Barenco, while Proposition prop:transfer prices the same decoder's return error.

This proves robustness and existence of finite-precision realizations, not an executed search or an efficient compilation.

For a purely classical returned channel on a $d$-dimensional diary, every deterministic decoder has

Equivalently, its full-diamond error is at least $2b_d$.

Proof. Decoding a classical channel is measure-and-prepare, hence entanglement breaking EntanglementBreaking. Its Bell-input output is separable. A normalized product vector has squared overlap at most $1/d$ with $|{\Phi }_d⟩$, and convexity extends the bound to separable states. Testing the Bell projector gives error at least $1-1/d$.

This applies if the returned interface is an album only, including the case in which both late cells and early memory are measured. The packet obstruction in Em03-F is a separately specified instance CheungEm03AF.

Take $d=2^{\ell }$ and prepare $\ell$ Bell pairs $HE$ before the unknown diary is supplied. A Bell transform on $LH$ gives the complete expansion

where $W_z$ are fixed Pauli corrections and $S$ is an untouched two-dimensional spectator.

Enter on $T$ returns the hybrid channel

The CPTP receiver

satisfies ${\mathcal{D}}_{\mathrm{tel}}\mathcal{M}={\mathrm{id}}_L$, including references.