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| \chapter{Continuous quantum fields and a compactness capacity law} | |
| \label{labcont:chapter} | |
| The spatial theorems based on Taylor expansion leave an apparent regularity barrier. A piecewise constant approximation to a continuous field has a small error at each site, but the sum of the operator norms of those errors can grow with the number of qubits. It is therefore insufficient to replace a curvature estimate by a continuity estimate in the previous proof. A different outer code is needed. | |
| This chapter supplies a specialized longitudinal code at the level of an existence theorem. The earlier noncommuting-field theorem is broader and uses no inner sector, while the present diagonal construction retains a sharper finite error exponent. Its inner constraint fixes only excitation number in each spatial cell. Its outer decoder corrects both the erased registers and a sublinear family of unknown phase errors. The bath may keep arbitrary quantum memory, and the field operators at different positions or times need not commute. What is required is longitudinal coupling on the physical registers, input-independent losses, and a common integrated modulus of spatial continuity. No derivative, regular lattice, dipole constraint, or finite-dimensional spatial coordinate system is required. | |
| The principal result is an operational capacity law on a compact metric domain. A sufficiently fine cover groups the registers into cells with almost identical environmental couplings. Compactness permits the number of cells to grow more slowly than the number of registers. Their fixed-charge subspaces then retain almost the entire physical Hilbert-space exponent. Spatial continuity makes the unprotected coupling weak at each site. A support expansion of the joint propagator and a completely positive domination argument convert that weakness into a universal final decoder. The resulting sharp leading rate is the ordinary independent-erasure value, $(1-2p)_+$. | |
| This specialized theorem is a statement about an ideal channel family and existence of codes. It neither provides an efficient capacity-achieving decoder nor certifies a new experimental memory. It also has a different noise boundary from the earlier extraction theorem: the present support argument assumes inert, longitudinal particle loss. Arbitrary charge-changing interventions on erased registers are not included. | |
| \section{Model, geometry, and the uniform continuity hypothesis} | |
| \label{labcont:model} | |
| Let $X$ be a metric space with distance $d$. The $N$ registers occupy known positions $x_1,\ldots,x_N\in X$. The registers are qubits, and their number operators are $n_j=\ket1\bra1_j$. Between prescribed removal times, the active registers interact with a shared bath according to | |
| \begin{equation} | |
| H_A(t)=I\otimes H_B(t)+\sum_{j\in A(t)}n_j\otimes B_N(x_j,t). | |
| \label{labcont:hamiltonian} | |
| \end{equation} | |
| The bath and its dimension may depend on $N$. Each $B_N(x,t)$ is Hermitian. All Hamiltonians are bounded and integrable over the storage interval so that their propagators exist. Neither a common eigenbasis of the bath operators nor a Markov approximation is assumed. The bath starts in a fixed state independent of the unknown logical input. A purification of this initial state may be absorbed into the bath whenever convenient. | |
| The sites removed during storage are retained as inaccessible registers in a dilation and traced out at the end. After removal, they have no further physical interaction. Their computational-basis populations consequently remain fixed throughout the enlarged unitary model. The final erased set is $E$, the survivor set is $R=E^c$, and every active set contains $R$. Loss locations are available to the decoder. Their selection is independent of the encoded state and the bath's quantum measurement outcomes. For the rate statement, each site belongs to $E$ independently with probability $p$; the removal times may otherwise be arbitrary. | |
| The following hypothesis is uniform over the channel family, rather than a separate continuity assertion for each finite array. | |
| \begin{definition}[Integrated continuity envelope] | |
| An integrated continuity envelope is a nondecreasing function $w(h)$ satisfying $w(h)\to0$ as $h\downarrow0$ and | |
| \begin{equation} | |
| \frac1\hbar\int_0^{T_N} | |
| \sup_{d(x,y)\leq h} | |
| \|B_N(x,t)-B_N(y,t)\|\,dt\leq w(h) | |
| \label{labcont:modulus} | |
| \end{equation} | |
| for every array size, permitted bath, field, and removal history under consideration. The storage time $T_N$ is allowed to depend on $N$ if this same envelope remains valid. | |
| \end{definition} | |
| The quantity $w(h)$ is dimensionless. The argument $h$ has the units of the metric, while the integrand has units of energy. The envelope controls integrated spatial variation, not the strength of the spatially constant field. An arbitrarily strong common component can be added to every $B_N(x,t)$ without changing \eqref{labcont:modulus}. Such a component will be absorbed exactly by the fixed-charge construction below. | |
| For a fixed field on a compact domain, norm continuity combined with an integrable dominating function often supplies an envelope by dominated convergence. This familiar sufficient condition must not be confused with the uniform family hypothesis. If the field develops finer spatial structure as $N$ grows, or if the storage interval grows without a compensating bound, a common envelope may fail to exist. A field that interpolates independent order-one couplings at increasingly close register positions is an example of this failure. | |
| \section{Spatial cells and the available Hilbert-space exponent} | |
| \label{labcont:sectors} | |
| Partition the sites into $C_N$ nonempty cells $\mathcal I_1,\ldots,\mathcal I_{C_N}$, each having diameter at most $h_N$. Let $\ell_c=|\mathcal I_c|$ and choose a representative position $y_c$ in each cell. Define its charge by | |
| \begin{equation} | |
| Q_c=\sum_{j\in\mathcal I_c}n_j. | |
| \end{equation} | |
| Use the charge sector $Q_c=q_c$, where $q_c=\lfloor\ell_c/2\rfloor$. The product projector and its rank are | |
| \begin{equation} | |
| P_N=\bigotimes_{c=1}^{C_N}P_{c,q_c},\qquad | |
| D_N=\operatorname{rank}P_N | |
| =\prod_{c=1}^{C_N}\binom{\ell_c}{\lfloor\ell_c/2\rfloor}. | |
| \label{labcont:projector} | |
| \end{equation} | |
| The projector is diagonal in the computational basis. This fact will later improve the random-coding estimate for the canonical phase-error channel. | |
| \begin{lemma}[Cell-charge redundancy] | |
| The dimension in \eqref{labcont:projector} satisfies | |
| \begin{equation} | |
| \log_2D_N\geq N-\Delta_N,\qquad | |
| \Delta_N=\sum_{c=1}^{C_N}\log_2(\ell_c+1) | |
| \leq C_N\log_2\left(1+\frac N{C_N}\right). | |
| \label{labcont:redundancy} | |
| \end{equation} | |
| Consequently $C_N=o(N)$ implies $\Delta_N=o(N)$, without any lower bound on the occupancy of each individual cell. | |
| \end{lemma} | |
| \begin{proof} | |
| The binomial coefficients of a cell sum to $2^{\ell_c}$, and the central coefficient is maximal among the $\ell_c+1$ terms. It is therefore at least $2^{\ell_c}/(\ell_c+1)$. Multiplication proves the first bound. Concavity of $\log_2(1+x)$ proves the second. Finally, if $u_N=C_N/N\to0$, then $u_N\log_2(1+1/u_N)\to0$. | |
| \end{proof} | |
| No arithmetic property of the positions is used. The constraint is the integer-valued occupation count, not an exact dipole or higher moment. Registers at irrationally related coordinates therefore present no special issue. Very small cells can carry little or no logical information, but the redundancy estimate accounts for them automatically. It is the total covering cost that matters. | |
| \section{Why the interaction picture remains local on the survivors} | |
| \label{labcont:interaction} | |
| The most delicate step is the removal of the large common bath coupling. In a general many-body noise model, conjugating a small local perturbation by the unperturbed propagator can spread its system support. A Dyson expansion in the resulting interaction picture would then give no low-weight error control. Here the longitudinal structure and the fixed cell charges prevent that spreading. | |
| Write the field in a cell as | |
| \begin{equation} | |
| B_N(x_j,t)=B_N(y_c,t)+\delta B_j(t), | |
| \qquad j\in\mathcal I_c. | |
| \end{equation} | |
| Let $L_c(t)\subseteq E\cap\mathcal I_c$ be the sites of cell $c$ already removed by time $t$. On the entire product charge sector, including the inert erased registers, | |
| \begin{equation} | |
| \sum_{j\in A(t)\cap\mathcal I_c}n_j | |
| =q_c I-\sum_{j\in L_c(t)}n_j. | |
| \label{labcont:chargeIdentity} | |
| \end{equation} | |
| Every term in the physical Hamiltonian commutes with the original full cell charges, so this sector is invariant at all times. Equation~\eqref{labcont:chargeIdentity} is therefore usable throughout the evolution, not merely on the initial vector. | |
| Absorb the common-cell part, the bath-only Hamiltonian, and all residual interactions on active sites that will ultimately be erased into an environmental Hamiltonian on $E+B$: | |
| \begin{align} | |
| H_{\rm env}(t)={}&I_E\otimes H_B(t) | |
| +\sum_c\left(q_cI_E-\sum_{j\in L_c(t)}n_j\right) | |
| \otimes B_N(y_c,t)\nonumber\\ | |
| &+\sum_{j\in E\cap A(t)}n_j\otimes\delta B_j(t). | |
| \label{labcont:environmentH} | |
| \end{align} | |
| The remaining interactions are $\sum_{j\in R}n_j\otimes\delta B_j(t)$. The resulting effective Hamiltonian agrees with the original one on the invariant sector. Outside that sector it is merely a convenient bounded Hermitian extension. | |
| The propagator of $H_{\rm env}$ is diagonal in the erased-register computational basis, although its conditional bath unitaries may be arbitrarily complicated. Moving to this interaction picture gives | |
| \begin{equation} | |
| K_I(t)=\sum_{j\in R}n_j\otimes C_j(t),\qquad | |
| C_j(t)=\sum_e\ket e\bra e_E\otimes C_j(e,t), | |
| \label{labcont:localInteraction} | |
| \end{equation} | |
| with $\|C_j(t)\|=\|\delta B_j(t)\|$. Thus each interaction still has support on a single surviving register. Its other support is entirely in the inaccessible environment. | |
| This argument explicitly uses the absence of charge-changing operations on $E$. Such operations can destroy the diagonal conditional form needed below. Arbitrary erased-site interventions were allowed in the exact extraction theorem of the earlier chapters, but they are not silently imported into the present theorem. The two results concern overlapping, rather than identical, noise families. | |
| By the cell diameter and \eqref{labcont:modulus}, every surviving site obeys | |
| \begin{equation} | |
| \frac1\hbar\int_0^{T_N}\|C_j(t)\|\,dt | |
| \leq\epsilon_N,\qquad \epsilon_N=w(h_N). | |
| \label{labcont:localBudget} | |
| \end{equation} | |
| The integrated norm of the full residual can be as large as $N\epsilon_N$. The proof does not require this quantity to vanish. Instead, it will correct a family of errors large enough to contain the dominant part of its support expansion. | |
| \section{A controlled low-weight approximation to the joint unitary} | |
| \label{labcont:dyson} | |
| For $q\geq0$ and an integer $s\geq0$, define | |
| \begin{equation} | |
| \tau_s(q)=\sum_{k=s+1}^{\infty}\frac{q^k}{k!}. | |
| \label{labcont:tail} | |
| \end{equation} | |
| The interaction-picture propagator has its usual time-ordered Dyson expansion. Let $V_s$ be its truncation after order $s$. Expanding each factor of \eqref{labcont:localInteraction}, every term of order $k$ has system support on at most $k$ survivor sites. Since $n_j=(I-Z_j)/2$, it follows that | |
| \begin{equation} | |
| V_s=\sum_{e}\ket e\bra e_E\otimes | |
| \sum_{\substack{S\subseteq R\\|S|\leq s}} | |
| Z_S\otimes A_S(e), | |
| \qquad Z_S=\prod_{j\in S}Z_j. | |
| \label{labcont:lowWeightExpansion} | |
| \end{equation} | |
| Repeated appearances of a site do not increase the support, and the empty set includes the identity contribution. No commutativity among the $A_S(e)$ is required. | |
| \begin{lemma}[Support-truncation bound] | |
| Let $U_I$ be the interaction-picture joint unitary and set $q_N=N\epsilon_N$. Then | |
| \begin{equation} | |
| \|U_I-V_s\|\leq\tau_s(q_N),\qquad | |
| \|V_s\|\leq1+\tau_s(q_N). | |
| \label{labcont:operatorTail} | |
| \end{equation} | |
| For $q>0$, | |
| \begin{equation} | |
| \tau_s(q)\leq e^q\frac{q^{s+1}}{(s+1)!} | |
| \leq e^q\left(\frac{eq}{s+1}\right)^{s+1}. | |
| \label{labcont:tailEstimate} | |
| \end{equation} | |
| \end{lemma} | |
| \begin{proof} | |
| The ordered integral of the product of $k$ residual norms is at most $q_N^k/k!$, because the integral of their sum is at most $N\epsilon_N$. Summing the omitted terms gives the first claim. The second follows from the unit norm of $U_I$. For the scalar tail, use $(s+1+j)!\geq(s+1)!j!$ and then $(s+1)!\geq[(s+1)/e]^{s+1}$. | |
| \end{proof} | |
| The approximation $V_s$ need not be unitary or trace preserving after environmental tracing. It will be used as a completely positive, generally trace-changing map. Keeping this distinction is necessary: a probability-of-failure argument that assumes the truncated map is a channel would not be justified. | |
| If $\epsilon_N\to0$, one convenient asymptotic choice is | |
| \begin{equation} | |
| s_N=\max\{\lceil\sqrt N\rceil, | |
| \lceil N\sqrt{\epsilon_N}\rceil\}, | |
| \label{labcont:choiceS} | |
| \end{equation} | |
| with a cap at $N$ for the finitely many initial cases where needed. Eventually $s_N/N\to0$ and $s_N<N/2$. Moreover $q_N/s_N\leq\sqrt{\epsilon_N}$, so \eqref{labcont:tailEstimate} implies $\tau_{s_N}(q_N)\to0$. Indeed, its logarithm divided by $s_N$ is at most $\sqrt{\epsilon_N}+1+\tfrac12\ln\epsilon_N+o(1)$ when $\epsilon_N>0$, which tends to minus infinity. If $\epsilon_N=0$, the tail is exactly zero. | |
| \section{A completely positive domination lemma} | |
| \label{labcont:domination} | |
| Fix the erased set $E$ and an integer $s$. Let | |
| \begin{equation} | |
| \mathcal Z_{R,s}=\{S\subseteq R:|S|\leq s\},\qquad | |
| m_{R,s}=|\mathcal Z_{R,s}|. | |
| \end{equation} | |
| Define the canonical channel | |
| \begin{equation} | |
| \Phi_{E,s}(\rho)=\frac1{m_{R,s}} | |
| \sum_{S\in\mathcal Z_{R,s}}Z_S\Tr_E(\rho)Z_S. | |
| \label{labcont:canonical} | |
| \end{equation} | |
| It is ordinary erasure followed by an equal mixture of all allowed low-weight phase operators. This channel is introduced to construct a decoder. It need not itself belong to the physical smooth-field family. | |
| For completely positive maps, write $\Psi\leq_{\rm CP}\Xi$ if $\Xi-\Psi$ is completely positive. The following bound does not introduce a factor proportional to the number of erased computational strings. | |
| \begin{lemma}[Domination without an erased-dimension penalty] | |
| \label{labcont:dominationLemma} | |
| Suppose $V_s$ has the form \eqref{labcont:lowWeightExpansion} and $\|V_s\|\leq v$. With a fixed initial bath state, trace out $E+B$ after applying $V_s$ and call the resulting completely positive map $\Psi_{E,s}$. Then | |
| \begin{equation} | |
| \Psi_{E,s}\leq_{\rm CP}m_{R,s}v^2\Phi_{E,s}. | |
| \label{labcont:CPorder} | |
| \end{equation} | |
| \end{lemma} | |
| \begin{proof} | |
| Purify the bath state to $\ket b$. For each erased computational string $e$, orthogonality of the survivor Paulis in normalized Hilbert--Schmidt inner product gives the operator Parseval identity | |
| \begin{equation} | |
| \sum_S A_S(e)^\dagger A_S(e) | |
| =2^{-|R|}\Tr_R\!\left[V_s(e)^\dagger V_s(e)\right] | |
| \leq v^2I_B. | |
| \label{labcont:Parseval} | |
| \end{equation} | |
| Here $V_s(e)$ is the conditional operator on $R+B$. Positivity of partial trace proves the inequality. The coefficient Gram matrix | |
| \begin{equation} | |
| G_e(S,T)=\bra b A_T(e)^\dagger A_S(e)\ket b | |
| \end{equation} | |
| is positive semidefinite and has trace at most $v^2$. Consequently $G_e\leq v^2I_{m_{R,s}}$. | |
| Tracing the erased register removes all cross terms between different strings $e$. Thus the $e$ contribution to $\Psi_{E,s}$ has the operator-sum coefficient matrix $G_e$ in the errors $Z_S\bra e_E$. The corresponding coefficient matrix for the $e$ contribution to $\Phi_{E,s}$ is $I_{m_{R,s}}/m_{R,s}$. Their difference is positive after multiplication by $m_{R,s}v^2$, which proves complete positivity of that difference for each $e$. Summing over $e$ proves \eqref{labcont:CPorder}. | |
| \end{proof} | |
| The absence of a second $2^{|E|}$ factor is important. A bound obtained by treating each erased string independently and then summing norms would generally lose the sharp erasure rate. The Gram matrices here are controlled separately by the norm of a conditional propagator, and their canonical erasure contributions already have the correct normalization. No union bound over the erased strings is taken. | |
| There is a direct operational use of CP order. For an encoding $W:\mathbb C^K\to\operatorname{ran}P_N$ and a chosen decoder $\mathcal D_E$, define the linear functional | |
| \begin{equation} | |
| \mathfrak f_E(\Lambda)= | |
| \Tr\!\left[(I-\ket{\Gamma_K}\bra{\Gamma_K}) | |
| (\id_F\otimes\mathcal D_E\Lambda\mathcal W) | |
| (\ket{\Gamma_K}\bra{\Gamma_K})\right], | |
| \label{labcont:failureFunctional} | |
| \end{equation} | |
| where $\ket{\Gamma_K}$ is maximally entangled and $\mathcal W(\rho)=W\rho W^\dagger$. This functional is nonnegative on completely positive maps, including maps that change trace. For a trace-preserving map it is the entanglement infidelity. Therefore CP domination transfers a decoder for the canonical channel to the unknown bath map. | |
| Finally, if a unitary $U$ and an approximant $V$ differ by at most $\tau$ in operator norm, the difference of their values under this functional is at most $\tau(2+\tau)$. To see this, pull the positive failure measurement through the decoder. The resulting joint operator $M$ satisfies $0\leq M\leq I$. For a normalized joint input vector, | |
| \[ | |
| |\langle U\psi|M|U\psi\rangle- | |
| \langle V\psi|M|V\psi\rangle| | |
| \leq\|U-V\|(\|U\|+\|V\|)\leq\tau(2+\tau). | |
| \] | |
| This justifies use of the truncated map without pretending that it preserves total probability. | |
| \section{Random coding for the canonical channel} | |
| \label{labcont:randomCode} | |
| Choose a Haar-random $K$-dimensional isometry into the rank-$D_N$ product charge sector. A Stinespring dilation of \eqref{labcont:canonical} has environmental dimension $2^t m_{R,s}$, where $t=|E|$: one factor records the erased qubits and the other records the phase operator. The standard two-copy calculation applies to the image of this isometry just as it does to a subspace of a physical tensor product. | |
| There is a useful improvement over using only this environmental dimension. Let $\Omega=P_N/D_N$. The mean receiver state of the canonical channel is | |
| \begin{equation} | |
| \Phi_{E,s}(\Omega)=\Omega_R=\Tr_E\Omega, | |
| \label{labcont:meanOutput} | |
| \end{equation} | |
| because $P_N$ is diagonal and every $Z_S$ commutes with it. Also $P_N\leq I$ implies $\|\Omega_R\|_\infty\leq2^t/D_N$, and hence $\Tr\Omega_R^2\leq2^t/D_N$. | |
| \begin{lemma}[Canonical-channel decoding bound] | |
| \label{labcont:canonicalCoding} | |
| Assume $D_N>1$. For a Haar-random $K$-dimensional code in $P_N$, the mean entanglement infidelity achievable by a decoder for $\Phi_{E,s}$ is at most | |
| \begin{equation} | |
| 2^t\sqrt{\frac{K m_{R,s}}{D_N}}. | |
| \label{labcont:canonicalError} | |
| \end{equation} | |
| \end{lemma} | |
| \begin{proof} | |
| Let $C$ denote the canonical dilation's environment and $O$ its receiver. The exact Haar variance of | |
| $\rho_{FC}-I_F/K\otimes\Omega_C$ is | |
| \[ | |
| \frac{D_NK-1}{K(D_N^2-1)} | |
| [D_N\Tr\Omega_O^2-\Tr\Omega_C^2] | |
| \leq\Tr\Omega_O^2. | |
| \] | |
| The coefficient of $\Tr\Omega_O^2$ is at most one because $K\leq D_N$. Cauchy--Schwarz on the $K\dim C$ dimensional space gives a mean trace-norm decoupling error at most | |
| \[ | |
| \sqrt{K\,2^t m_{R,s}\,\Tr\Omega_O^2} | |
| \leq2^t\sqrt{K m_{R,s}/D_N}, | |
| \] | |
| using \eqref{labcont:meanOutput}. Uhlmann's theorem and the fidelity--trace-distance inequality supply a receiver decoder whose entanglement infidelity is no larger than this trace-norm error. This decoder is chosen for the known canonical channel, not for an unknown field realization. | |
| \end{proof} | |
| Combining \eqref{labcont:CPorder} and \eqref{labcont:canonicalError} gives the factor $m_{R,s}^{3/2}$, rather than a penalty in the full erased Hilbert-space dimension. This factor can be expensive for a finite block. Its logarithm, however, is sublinear whenever $s=o(N)$, which is sufficient for the leading-rate theorem. | |
| \section{Finite bound and the continuous-field capacity theorem} | |
| \label{labcont:main} | |
| Define | |
| \begin{equation} | |
| m_{N,s}=\sum_{k=0}^{\min(s,N)}\binom Nk, | |
| \qquad q_N=N\epsilon_N, | |
| \qquad \tau_N=\tau_s(q_N). | |
| \end{equation} | |
| The integer $m_{N,s}$ upper-bounds the number of phase supports on every possible survivor set. For $s\leq N/2$, | |
| \begin{equation} | |
| \log_2m_{N,s}\leq N\hbin(s/N). | |
| \label{labcont:phaseEntropy} | |
| \end{equation} | |
| \begin{theorem}[Continuous longitudinal fields preserve the leading erasure rate] | |
| \label{thm:continuous-capacity} | |
| Consider \eqref{labcont:hamiltonian} with inert, input-independent independent erasures of probability $p$, the uniform envelope \eqref{labcont:modulus}, and a known cell partition of diameter $h_N$. For every $K\leq D_N$, integer $s\geq0$, and $\eta>0$, there is one encoding into the product cell-charge sector and a family of final loss-location-dependent decoders whose average entanglement infidelity is, uniformly over the allowed fields, bath memories, and removal times, at most | |
| \begin{align} | |
| e^{-2\eta^2N} | |
| &+(1+\tau_N)^2 | |
| 2^{[\log_2K+2N(p+\eta)+3\log_2m_{N,s}-\log_2D_N]/2} | |
| \nonumber\\ | |
| &+\tau_N(2+\tau_N). | |
| \label{labcont:finiteBound} | |
| \end{align} | |
| The right side may be capped at one. If $C_N=o(N)$ and $\epsilon_N=w(h_N)\to0$, the robust unassisted asymptotic entanglement-transmission rate equals $(1-2p)_+$. The encoding and decoder are independent of the actual field and bath history. No intermediate recovery or dynamical-decoupling pulse sequence is required. | |
| \end{theorem} | |
| \begin{proof} | |
| If $D_N=1$, then $K=1$ and the decoder's output is a one-dimensional logical system, so entanglement transmission is trivial and the claim holds. Assume henceforth $D_N>1$. | |
| For a fixed erased set, the effective environmental reduction and Dyson truncation give \eqref{labcont:lowWeightExpansion}, with norm error at most $\tau_N$. The environmental interaction-picture propagator has no effect after tracing $E+B$. The domination lemma therefore bounds physical decoder failure by | |
| \[ | |
| m_{R,s}(1+\tau_N)^2\, | |
| \mathfrak f_E(\Phi_{E,s})+\tau_N(2+\tau_N). | |
| \] | |
| This bound holds for every permitted field and history once the code and its canonical decoder are fixed. | |
| Call a loss pattern typical if $t\leq N(p+\eta)$. On such patterns, Lemma~\ref{labcont:canonicalCoding} and $m_{R,s}\leq m_{N,s}$ show that the Haar mean of the first term is at most | |
| \[ | |
| (1+\tau_N)^2\,2^{N(p+\eta)}m_{N,s}^{3/2} | |
| \sqrt{K/D_N}. | |
| \] | |
| Average this inequality over typical loss patterns and select one code with no greater weighted canonical failure. The corresponding decoder for each pattern is its canonical-channel decoder. This selection is performed once and does not depend on a bath realization. On atypical patterns, use the physical bound that entanglement infidelity is at most one, rather than applying CP domination to an unnecessarily large auxiliary estimate. Hoeffding's inequality gives the first term of \eqref{labcont:finiteBound}. These steps prove the finite claim and its uniformity. | |
| For the asymptotic statement, choose $s_N$ by \eqref{labcont:choiceS}. Then $s_N=o(N)$ and $\tau_N\to0$. Equation~\eqref{labcont:phaseEntropy} gives $\log_2m_{N,s_N}=o(N)$, while \eqref{labcont:redundancy} gives $\log_2D_N=N-o(N)$. For any fixed $R<1-2p$, choose a fixed $\eta>0$ with $R+2p+2\eta<1$ and take $K=\lfloor2^{NR}\rfloor$. Every term of \eqref{labcont:finiteBound} then vanishes. The field family includes zero coupling, so its worst-case rate cannot exceed the ordinary independent-erasure capacity. For $p\geq1/2$ that converse gives zero. This establishes the stated equality. | |
| \end{proof} | |
| The theorem requires only a vanishing per-site residual budget. The total budget $N\epsilon_N$ may diverge, and its corresponding uncorrected channel can be far from the identity. The outer phase-error code is what makes this different from a global Duhamel estimate. Conversely, the theorem does not claim that an arbitrary amount of local noise is harmless: if $\epsilon_N$ stays of order one, the phase-support entropy need not be sublinear and the argument supplies no full-rate conclusion. | |
| \section{Compact metric domains and geometric examples} | |
| \label{labcont:compactness} | |
| \begin{corollary}[Compactness capacity law] | |
| \label{labcont:compactCorollary} | |
| Suppose all registers lie in one fixed compact metric space $X$ and the field family has the uniform envelope \eqref{labcont:modulus}. Then, for arbitrary known register positions in $X$, there exist cell-charge encodings and final decoders attaining the robust rate $(1-2p)_+$ under the longitudinal, inert-loss model. | |
| \end{corollary} | |
| \begin{proof} | |
| For each positive integer $m$, choose a finite cover of $X$ by sets of diameter at most $h_m$, where $h_m\downarrow0$, and let $c_m$ be its size. Since every fixed $c_m$ is finite, a sequence $m_N\to\infty$ can be chosen sufficiently slowly that $c_{m_N}\leq\sqrt N$. Assign each register to the first covering set containing it, remove empty cells, and use those cells as the partition. Then $C_N\leq\sqrt N=o(N)$, while $w(h_{m_N})\to0$. Apply Theorem~\ref{thm:continuous-capacity}. | |
| \end{proof} | |
| Only total boundedness, rather than metric completeness, is used in this proof. Compactness is a convenient familiar sufficient condition. A more quantitative formulation uses the covering number: if a diameter-$h_N$ cover has $c_N=o(N)$ sets and $w(h_N)\to0$, the same conclusion follows even for a sequence of changing spatial domains. This is an entropy condition on the distinguishable environmental coupling regions. | |
| For a regular grid in a fixed $d$-dimensional box, take cells with diameter $h_N=N^{-\alpha}$ for any fixed $0<\alpha<1/d$. Their number is $O(N^{\alpha d})=o(N)$, so the result applies to every uniform continuity envelope, however slowly it decays. The positions need not be on a grid: a known cover of an irregular cloud provides the same proof. Empty portions of the physical domain only reduce the number of occupied cells. | |
| If the envelope has an integrated H\"older form $w(h)\leq Lh^\gamma$, with $\gamma>0$, then the residual budget is $O(N^{-\alpha\gamma})$. The explicit support choice \eqref{labcont:choiceS} already proves capacity preservation. A sharper choice, given next, provides better finite-size bounds. None of these sufficient estimates is asserted to be the optimal convergence rate to capacity. | |
| Compact geometry alone does not permit arbitrary independent microscopic noise. The uniform envelope excludes an order-one difference in couplings at positions whose separation tends to zero. If that assumption is removed, continuity of a different interpolating field at every finite $N$ gives no useful uniform guarantee. The distinction can be tested directly by whether the same $w(h)$ works for all array sizes. | |
| Nor can compactness simply be deleted while retaining only uniform continuity. On an infinite discrete space with all distinct points at distance one, every field has zero spatial modulus for $h<1$. Independent bath qubits can nevertheless couple to the separate registers and completely dephase each one in a fixed time, yielding an entanglement-breaking channel. The required small-diameter cover then has one cell per occupied site, so its size is $N$ and the dimension-redundancy condition fails. This example identifies precisely which geometric part of the proof prevents independent dephasing from being admitted as a harmless smooth field. | |
| \section{Sharper support budgets and explicit finite-size recipes} | |
| \label{labcont:rates} | |
| The square-root support choice in \eqref{labcont:choiceS} is convenient for an arbitrary continuity envelope, but it is not necessary. Let $c>0$ be fixed and use | |
| \begin{equation} | |
| s_N=\left\lceil e^2N\epsilon_N+c\ln N\right\rceil. | |
| \label{labcont:sharpSupport} | |
| \end{equation} | |
| For the initial cases in which this exceeds $N$, the finite bound can simply be replaced by one; asymptotically $\epsilon_N\to0$ ensures $s_N<N/2$. If $q_N>0$, then $eq_N/(s_N+1)\leq1/e$, and \eqref{labcont:tailEstimate} gives | |
| \begin{equation} | |
| \tau_{s_N}(q_N)\leq | |
| \exp[q_N-(s_N+1)]\leq N^{-c}. | |
| \label{labcont:polynomialTail} | |
| \end{equation} | |
| The same conclusion holds trivially for $q_N=0$. In particular, the correctable support fraction is of order $\epsilon_N+\ln N/N$, rather than its square root. | |
| Combining this choice with \eqref{labcont:phaseEntropy} yields | |
| \begin{equation} | |
| \frac{\log_2m_{N,s_N}}N | |
| =O\!\left(\epsilon_N\log\frac e{\epsilon_N} | |
| +\frac{\log^2N}{N}\right), | |
| \label{labcont:entropyCost} | |
| \end{equation} | |
| where $\epsilon_N\log(e/\epsilon_N)$ is interpreted as zero at $\epsilon_N=0$. Constants can depend on the fixed $c$. The logarithm in the first term is harmless for leading-rate purposes, but it should not be discarded when comparing finite resource choices. | |
| A complete polynomial-error coding recipe can now be stated. Set | |
| \begin{equation} | |
| \eta_N=\sqrt{\frac{c\ln N}{2N}} | |
| \end{equation} | |
| and choose the number of logical qubits no larger than the integer part of | |
| \begin{equation} | |
| N[1-2(p+\eta_N)]-\Delta_N | |
| -3\log_2m_{N,s_N}-2c\log_2N. | |
| \label{labcont:logicalBudget} | |
| \end{equation} | |
| For a fixed $p<1/2$, this number is positive for sufficiently large $N$ under the theorem's hypotheses. The first term of \eqref{labcont:finiteBound} is then at most $N^{-c}$. The canonical-decoder contribution is at most $(1+N^{-c})^2N^{-c}$, and the truncation contribution is at most $N^{-c}(2+N^{-c})$. A safe combined bound is $8N^{-c}$ for all sufficiently large $N$. | |
| Equation~\eqref{labcont:logicalBudget} keeps three different costs separate. The sector deficit comes from restricting the physical Hilbert space to fixed cell charges. The phase-support cost comes from correcting the weak spatial residual uniformly over the bath. The $\eta_N$ and logarithmic safety terms make the ordinary random erasure tail and decoupling error small. A sector-overhead calculation by itself would miss the last of these costs. | |
| As a conditional design-point calculation, take $N=10^6$, $p=0.1$, cells of $1000$ sites, a verified per-site residual budget $\epsilon_N\leq10^{-6}$, and $c=2$. Equation~\eqref{labcont:sharpSupport} gives $s_N=36$ and $\log_2m_{N,s_N}\simeq579.4413$. The exact product-sector dimension, rather than its coarse pigeonhole lower bound, gives | |
| \[ | |
| N-\log_2D_N | |
| =10^6-1000\log_2\binom{1000}{500}\simeq5309.0009. | |
| \] | |
| The logical budget may use this exact deficit wherever available. With $\eta_N\simeq0.00371692$, its integer value is $785{,}439$ logical qubits, and the sufficient error estimate is $8\times10^{-12}$. The unrestricted leading-rate benchmark at this $p$ is $0.8$ qubits per physical qubit; the displayed sufficient rate is $0.785439$. | |
| These numbers evaluate an existence bound under the stated residual budget. They are not a simulation of a million-qubit device, a constructed fast decoder, or an experimental performance estimate. In particular, the very large canonical error family and its erasure decoders are represented analytically. The example is useful because each cost can be recomputed from explicit formulas, and because a claimed finite-resource design can be rejected if its actual continuity envelope does not certify the assumed $\epsilon_N$. | |
| For an explicit spatial scaling, consider a regular array in a fixed $\nu$-dimensional box. Use cells of approximately $b$ qubits. Their diameter is of order $(b/N)^{1/\nu}$, up to a fixed geometric length scale. If $w(h)\leq Lh^\gamma$ for $0<\gamma\leq1$, then | |
| \begin{equation} | |
| \epsilon_N=O\!\left((b/N)^{\gamma/\nu}\right),\qquad | |
| \frac{\Delta_N}{N}=O\!\left(\frac{\log b}{b}\right). | |
| \end{equation} | |
| The phase-support entropy from \eqref{labcont:entropyCost} is of order $(b/N)^{\gamma/\nu}\log N$, plus a smaller logarithmic term. Balancing the two geometric contributions gives | |
| \begin{equation} | |
| b\asymp N^{\gamma/(\nu+\gamma)},\qquad | |
| \frac{\Delta_N+3\log_2m_{N,s_N}}N | |
| =O\!\left(N^{-\gamma/(\nu+\gamma)}\log N\right). | |
| \label{labcont:holderRate} | |
| \end{equation} | |
| The full sufficient rate deficit in the polynomial-error recipe also includes $O(\sqrt{\ln N/N})$ from the erasure-tail slack. For the displayed H\"older range and integer $\nu\geq1$, the geometric estimate is at least as large in order, but retaining that separate term makes the conclusion unambiguous and permits comparison with smoother variants. | |
| This support-correction method can also improve the earlier higher-moment constructions. More generally, suppose a diagonal projector fixes a finite list of commuting spatial moments in every cell, the exactly represented polynomial part of the field reduces to an environmental Hamiltonian as in \eqref{labcont:chargeIdentity}, and the per-site remainder budget is $\epsilon_N=O(h_N^r)$. The proof of the domination and canonical-coding lemmas is unchanged: all onsite residual terms remain longitudinal and preserve the moment sector, while the mean sector state is still diagonal. On a fixed-dimensional integer grid with a fixed Taylor order, counting the finitely many moment labels gives a cell redundancy $O(\log b)$, hence $O(\log b/b)$ per physical qubit. The same balance gives a sector-plus-phase overhead of order | |
| \begin{equation} | |
| N^{-r/(\nu+r)}\log N. | |
| \label{labcont:highOrderOverhead} | |
| \end{equation} | |
| This is a resource-overhead estimate, not a promise to beat the ordinary finite-block erasure dispersion. If $r>\nu$, the $O(\sqrt{\ln N/N})$ coding slack can dominate the displayed geometric overhead. Exact moment degeneracy also requires a suitable known grid or another verified joint-sector dimension bound; it is not supplied by smoothness alone. | |
| The practical distinction is useful even before an efficient outer decoder is available. A norm-only analysis requires the sum of all residual norms to vanish. The support analysis permits that sum to grow, chooses a controlled low-weight approximation, and pays for its error space with explicit logical redundancy. It provides a finite formula for deciding whether a proposed cell size improves the tradeoff, while leaving decoder complexity and hardware precision as separate, visible requirements. | |
| \section{Interpretation, scope, and what remains constructive} | |
| \label{labcont:scope} | |
| The result changes the regularity threshold of the passive-storage model. The earlier Taylor proof required a derivative modulus because it demanded a globally small channel perturbation. Here an uncorrected residual may have a large global norm budget. It is sufficient that each site has a vanishing budget, because the resulting support entropy is sublinear and an outer code can handle it at negligible leading-rate cost. This is the substantive distinction between the two proofs. | |
| Several ingredients have explicit finite descriptions. The cell cover and charge labels are known from geometry. A basis of each half-filled charge sector is specified by constant-weight binary strings; reversible combinatorial ranking embeds logical states in that sector. The canonical error family consists of a finite list of phase supports and known erasure flags. The random-code argument then proves existence of a single common encoding and a compatible family of decoders. It does not give a polynomial-time method to find or operate the outer code at capacity. Efficient decoding, finite gate error, and fault-tolerant implementation remain separate problems. | |
| The universality is over the stated longitudinal bath family, not over all open-system dynamics. Transverse couplings need not preserve the cell charges or the local interaction-picture structure. State-dependent loss selection can reveal or steer logical information through the loss record. Charge-changing operations on already erased registers invalidate the conditional diagonal Gram decomposition. Uncontrolled motion of the register positions can also invalidate the chosen cover or continuity budget. Each of these changes requires a separate theorem rather than an informal extrapolation. | |
| The noncommuting operator-envelope theorem in Chapters~\ref{openv:chapter} and~\ref{nonab:chapter} supplies a separate analysis for transverse fields with a common integrated spatial envelope and permits arbitrary erased-site/bath operations. The diagonal proof in this chapter alone does not establish those extensions. | |
| The result also says nothing favorable about a fixed ongoing particle-loss hazard over arbitrarily long storage. Its parameter $p$ is the total independent endpoint erasure probability. If that probability approaches one half or more during storage, the ordinary unassisted erasure converse already removes positive rate. The continuity envelope may likewise deteriorate with time. Any lifetime statement must track both effects, rather than treating spatial continuity as a source of unrestricted memory duration. | |
| Within those boundaries, the compactness law is broad: it allows arbitrary known placement in any fixed compact metric domain, a noncommuting quantum bath with memory, rough spatial fields with no derivative control, and repeated externally selected losses. The complete proof rests on the interaction-picture support property, the CP domination factor independent of erased dimension, and a direct growing-array decoupling argument. Its scientific priority and practical impact require comparison with the full literature; the theorem and its qualifications are the mathematical claims established here. | |