Unitary Local Permutations on Bell Diagonal States of Qudits and Quantum Distillation Protocols

Stefano Facchini, Simon Perdrix

Héctor Bombín, Miguel Angel Martin-Delgado

We show that the analysis of entanglement distillation protocols for qudits of arbitrary dimension $D$ benefits from applying basic concepts from number theory, since the set ${\mathbf{Z}}_{D}^{n}$ associated with Bell diagonal states is a module rather than a vector space. We find that a partition of ${\mathbf{Z}}_{D}^{n}$ into divisor classes characterizes the invariant properties of mixed Bell diagonal states under local permutations. We construct a very general class of recursion protocols by means of unitary operations implementing these local permutations. We study these distillation protocols depending on whether we use twirling operations in the intermediate steps or not, and we study them both analytically and numerically with Monte Carlo methods. In the absence of twirling operations, we construct extensions of the quantum privacy algorithms valid for secure communications with qudits of any dimension $D$. When $D$ is a prime number, we show that distillation protocols are optimal both qualitatively and quantitatively.

Erik Hostens, Jeroen Dehaene, Bart De Moor

We show the equivalence of two approaches to the design of entanglement distillation protocols. The first approach is based on local unitary operations that yield permutations of tensor products of Bell states. The second approach is based on stabilizer codes.

J.P. McCarthy

Christopher Popp, Tobias C. Sutter, Beatrix C. Hiesmayr

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.

Moritz Weber