Quantifying the performance of bidirectional quantum teleportation

Abstract

Bidirectional teleportation is a fundamental protocol for exchanging quantum information between two parties by means of a shared resource state and local operations and classical communication (LOCC). In this paper, we develop two seemingly different ways of quantifying the simulation error of unideal bidirectional teleportation by means of the normalized diamond distance and the channel infidelity, and we prove that they are equivalent. By relaxing the set of operations allowed from LOCC to those that completely preserve the positivity of the partial transpose, we obtain semidefinite programming lower bounds on the simulation error of unideal bidirectional teleportation. We evaluate these bounds for three key examples: when there is no resource state at all and for isotropic and Werner states, in each case finding an analytical solution. The first aforementioned example establishes a benchmark for classical versus quantum bidirectional teleportation. We then evaluate the performance of some schemes for bidirectional teleportation due to [Kiktenko et al., Phys. Rev. A 93, 062305 (2016)] and find that they are suboptimal and do not go beyond the aforementioned classical limit for bidirectional teleportation. We offer a scheme alternative to theirs that is provably optimal. Finally, we generalize the whole development to the setting of bidirectional controlled teleportation, in which there is an additional assisting party who helps with the exchange of quantum information, and we establish semi-definite programming lower bounds on the simulation error for this task. More generally, we provide semi-definite programming lower bounds on the performance of bipartite and multipartite channel simulation using a shared resource state and LOCC.

Quantifying the performance of approximate teleportation and quantum error correction via symmetric two-PPT-extendibility

“…We can quantify the simulation error either in terms of the diamond distance [Kit97] or the channel infidelity [GLN05]. However, we prove here that the simulation error is the same, regardless of whether we use the channel infidelity or the diamond distance, when quantifying the deviation between the simulation and an ideal quantum channel (note that a similar result was found previously in [SW20] and we exploit similar techniques to arrive at our conclusion here). Next, in order to obtain a lower bound on the simulation error, and due to the fact that it is computationally challenging to optimize over oneway LOCC channels, we optimize the error over the larger set of two-extendible channels and show that the resulting quantity can be calculated by means of a semi-definite program.…”

“…where 𝑇 𝐵𝐵 is the partial transpose on systems 𝐵 and 𝐵 . We note that the C-PPT-P constraint has been used in prior work on bounding the simulation error in bidirectional teleportation [SW20]. See also [LM15,WXD18,WD16b,WD16a,BW18] for other contexts.…”

“…The standard metric for quantifying the distance between quantum channels is the normalized diamond distance [Kit97]. See the related paper [SW20] for discussions of the operational significance of the diamond distance (see also [KW20]). For channels N 𝐶→𝐷 and N 𝐶→𝐷 , the diamond distance is defined as…”

“…We thus consider our paper to offer two distinct, yet related contributions. The conceptual approach that we take here is linked to that of [SW20], which was concerned with a more involved protocol called bidirectional teleportation. Our approach has strong links as well with that taken in [BBFS21], the latter concerned with bounding the performance of approximate quantum error correction by means of semi-definite programming and the two-extendible channels originally introduced in [KDWW19, KDWW21] (more generally these works considered 𝑘-extendible channels for 𝑘 > 2 as well).…”

A smallest computable entanglement monotone

“…The significance of the set of completely PPT-preserving channels [35], [11], [40] is that it contains the operationally relevant set of local operations and classical communication (LOCC) channels [6], [41]. The constraints specifying the former set of channels are semi-definite, and it is thus much easier in a computational sense to optimize an objective function over this set of channels than to optimize over the set of LOCC channels (see, e.g., [42]). Generalizing completely PPT-preserving channels, physically well motivated, and of relevance for us here, is the set of selective PPT operations.…”