Bounds on concatenated entanglement-assisted quantum error-correcting codes

TL;DR

The paper investigates how concatenating entanglement-assisted quantum error-correcting codes (EAQECCs) affects bounds and performance, deriving a general expression for shared entanglement (ebits) that accounts for whether the outer length is divisible by the inner rate and how concatenation order influences the result. It shows that, in general, concatenation order matters for ebits and error thresholds, but identifies two broad families of code pairs for which the resultant ebits are invariant to order, and provides explicit constructions demonstrating order-dependent and order-independent effects. The authors prove that maximal-entanglement EAQECCs derived from classical quaternary Griesmer or Plotkin codes saturate the EA Singleton and EA Griesmer/linear EA Plotkin bounds, and extend this saturation to certain nonmaximal-entanglement EAQECCs via distance conditions, alongside an EA Griesmer-Rains bound on correctable errors. They also analyze bound saturation/violation under concatenation, showing that some EAQECCs can violate the EA Hamming bound in an order-dependent manner, while others preserve bound-saturating properties under self-concatenation or specific outer-inner pairings. Overall, the work provides a framework for designing long, entanglement-efficient EAQECCs with controlled bound behavior and highlights open questions about refinements of EA bounds and alphabet-size effects on bound violations.

Abstract

Code concatenation combines two or more component codes to design larger codes with greater noise resilience. Introducing entanglement assistance to concatenated codes provides a further advantage in terms of improved error rates and beating certain bounds on codes that would otherwise be unbeatable. First, we derive the general expression for the shared entanglement of a concatenated code and show that the number of ebits can depend on the order of concatenating the component entanglement-assisted quantum error-correcting codes (EAQECCs). We further construct families of pairs of EAQECCs such that the number of ebits of the resultant of concatenating the two codes in a given pair is order independent. Second, we derive conditions on code distance under which non-maximal-entanglement EAQECCs obtained from a classical quaternary Griesmer or Plotkin code saturate the entanglement-assisted (EA) Griesmer or linear EA Plotkin bound, respectively, extending the known result for maximal-entanglement EAQECCs. Furthermore, we present several families of such nonmaximal-entanglement EAQECCs. Third, we derive an EA version of the quantum Griesmer-Rains bound on the number of correctable errors for EAQECCs. Finally, we present families of pairs of EAQECCs such that the violation of the EA Hamming bound by the resultant of concatenating the two codes in a given pair is order dependent.

Figures (4)

• Figure 1: Schematic diagram of the operation of an EAQECC. Entanglement is pre-shared between Alice and Bob. Alice encodes the $k$-qubit state $\ket{\psi}$ using the $a \equiv n-k-c$ ancilla and her part of ebits, with the help of the encoder $U$ [Eq. (\ref{['eq: logicalstateeaqecc']})], and transmits the $n$ qubits via a noisy quantum channel. Bob measures the qubits that he received over the channel, together with his half of the ebits to determine the error syndromes, by which he recovers $\ket{\psi}$.
• Figure 2: Graph structure of a $[[12,1;9;5]]$ CEAQECC with an outer $[[3,1,3;2]]$ code and inner $[[4,1,3;1]]$ code. The outer code encodes one logical qubit (white ball) into three physical qubits (light red balls) using two ebits (green balls). The inner code has four physical qubits (blue balls) encoded into each three of the blocks of the outer code, with one ebit (green balls) in each block.
• Figure 3: Logical error probabilities for the concatenation of $[[5,1,3]]$ and $[[3,1,3;2]]$ repetition codes.
• Figure 4: Hamming efficiency plot for the two families of CEAQECCs $\mathcal{C}_{a} \rhd \mathcal{C}_{1}$ and $\mathcal{C}_{1} \rhd \mathcal{C}_{a}$, where $\mathcal{C}_{1} \equiv [[8,1,5;1]]$ and $\mathcal{C}_{a} \equiv [[n, 1, n; n-1]]$ (odd $n$). The former concatenated code violates the EA Hamming bound, while the latter concatenated code does not.

Theorems & Definitions (29)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• proof
• proof