A Novel Stabilizer-based Entanglement Distillation Protocol for Qudits

TL;DR

The paper addresses robust entanglement distillation for qudit systems by developing a general stabilizer-based framework in prime dimension and deriving a standard form for output states of recurrence protocols. It introduces a canonical two-copy encoding and a fidelity-increase-maximizing protocol (FIMAX) that, for Bell-diagonal inputs, yields maximal per-iteration fidelity gain and improved distillability compared with established recurrence schemes. The authors connect input states, stabilizers, and encodings through the coset structure of Weyl errors, enabling efficient fidelity calculations and protocol design. Numerical comparisons show FIMAX outperforms BBPSSW, DEJMPS, ADGJ, and P12 in many regimes, particularly at low initial fidelities, and the framework offers a path to systematic stabilizer-based protocol development. The work also discusses extensions to non-Bell-diagonal and non-prime-dimension settings and links to permutation-based schemes.

Abstract

Entanglement distillation, the process of converting weakly entangled states into maximally entangled ones using Local Operations and Classical Communication (LOCC), is pivotal for robust entanglement-assisted quantum information processing in error-prone environments. A construction based on stabilizer codes offers an effective method for designing such protocols. By analytically investigating the effective action of stabilizer protocols for systems of prime dimension $d$, we establish a standard form for the output states of recurrent stabilizer-based distillation. This links the properties of input states, stabilizers, and encodings to the properties of the protocol. Based on those insights, we present a novel two-copy distillation protocol, applicable to all bipartite states in prime dimension, that maximizes the fidelity increase per iteration for Bell-diagonal states. The power of this framework and the protocol is demonstrated through numerical investigations, which provide evidence for superior performance in terms of efficiency and distillability of low-fidelity states compared to other well-established recurrence protocols. By elucidating the interplay between states, errors, and protocols, our contribution advances the systematic development of highly effective distillation protocols, enhancing our understanding of distillability.

Figures (5)

• Figure 1: Hilbert space $\mathcal{H}^{\otimes N}$ of $N$-copies of a bipartite system $\mathcal{H} = \mathcal{H}_A \overset{AB}{\otimes} \mathcal{H}_B$.
• Figure 2: Protocol comparison of distillation efficiencies depending on the fidelity of isotropic input states.
• Figure 3: Efficiency comparison depending on the fidelity. States are grouped in bins of $1000$ states by their fidelity, rounded to two digits, and the mean efficiency for all protocols is determined.
• Figure 4: Protocol comparison of the iterative fidelity increase for a low fidelity isotropic input state.
• Figure 5: Distillation efficiency for off-line states in dependence on the fidelity (a). Fidelities and success probabilities for each FIMAX iteration for a low-fidelity off-line input state (b).

Theorems & Definitions (47)

• Definition 1: Weyl(-Heisenberg) operators, Weyl(-Heisenberg) errors
• Definition 2: Bell states
• Definition 3: Stabilizer group, generating operators, generating elements
• Lemma 1
• proof
• Definition 4: Error coset
• Definition 5: Codeword
• Definition 6: Encoding
• Definition 7: Symplectic product decomposition
• Lemma 2