Memory-Induced Limits on Entanglement Distribution

Abstract

We investigate the operational limits of entanglement distribution through quantum channels reused without reinitialization. Building on an effective repeated-use channel description, we show that residual excitations left in the channel from previous uses act as a memory source that progressively suppresses coherent quantum transmission. To quantify this limitation, we derive an analytical lower bound on the quantum capacity and combine it with the Holevo-Werner semidefinite-programming upper bound. We also provide an analytical criterion to identify entanglement-breaking channels, with thresholds that depend jointly on chain length and reuse. For perfect-state-transfer chains, the resulting capacity window yields tighter estimates than previous bounds, with the relative improvement becoming substantially more pronounced closer to the entanglement-breaking threshold. Moreover, on the second use, our lower bound certifies a maximal relative timing-error tolerance across all chains with internal memory that lies just below 22.63% and is attained by the three-site chain. We further derive a fundamental relative timing-error threshold of 36.41%, above which we analytically guarantee zero unassisted quantum capacity for every chain with internal memory, independent of the number of uses. These results establish a clear benchmark model for repeated entanglement distribution, and identify timing precision, chain length, and memory accumulation as fundamental constraints on scalable quantum-network architectures.

The paper asks how residual excitations affect entanglement distribution when a spin-chain communication channel is reused without resetting its internal sites. It studies an existing effective-channel model for $U(1)$-symmetric quadratic-fermion dynamics, averaging over previous pure inputs. Each use gives a qubit channel comprising dephasing and generalized amplitude damping. The capacity analyzed is that of independent realizations of this fixed, averaged marginal channel—not the capacity of the correlated physical sequence. An analytical achievable-rate expression is obtained by maximizing coherent information over diagonal inputs. A geometric Rényi/PPT semidefinite-programming hierarchy supplies upper bounds, evaluated through level $l=10$ and compared with a previous damping/dephasing bottleneck bound. The exact entanglement-breaking condition is $(1-\gamma_n)(1-\lambda_n)\leq\gamma_n^2p_n(1-p_n)$, from positivity under partial transpose of the two-qubit Choi state. For engineered perfect-state-transfer chains with identical readout timing errors, $\gamma^{(N)}=1-\cos^{2(N-1)}(\pi\delta/2)$. Damping-induced antidegradability guarantees zero unassisted capacity above $\delta_*(N)=\frac{2}{\pi}\arccos(2^{-1/[2(N-1)]})$. For all chains with internal memory ($N\geq3$), the largest such threshold is $36.4057\%$. On the second use, the largest timing-error tolerance certified by the lower bound is $22.6243\%$, attained at $N=3$; its value is numerically optimized, while optimality across lengths is established analytically. Numerical landscapes for $N=3$–$10$ and uses $2$–$16$ track shrinking positive-capacity certification and entanglement-breaking boundaries. Intermediate regions remain unresolved. The conclusions exclude collective coding across physical uses and access to internal memory; they do not rule out recovery through memory control or active reset.