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The computational two-way quantum capacity

Johannes Jakob Meyer, Jacopo Rizzo, Asad Raza, Lorenzo Leone, Sofiene Jerbi, Jens Eisert

TL;DR

This work focuses on the computational two-way quantum capacity and showcases that it is closely related to the computational distillable entanglement of the Choi state of the channel, and shows a stark computational capacity separation.

Abstract

Quantum channel capacities are fundamental to quantum information theory. Their definition, however, does not limit the computational resources of sender and receiver. In this work, we initiate the study of computational quantum capacities. These quantify how much information can be reliably transmitted when imposing the natural requirement that en- and decoding have to be computationally efficient. We focus on the computational two-way quantum capacity and showcase that it is closely related to the computational distillable entanglement of the Choi state of the channel. This connection allows us to show a stark computational capacity separation. Under standard cryptographic assumptions, there exists a quantum channel of polynomial complexity whose computational two-way quantum capacity vanishes while its unbounded counterpart is nearly maximal. More so, we show that there exists a sharp transition in computational quantum capacity from nearly maximal to zero when the channel complexity leaves the polynomial realm. Our results demonstrate that the natural requirement of computational efficiency can radically alter the limits of quantum communication.

The computational two-way quantum capacity

TL;DR

This work focuses on the computational two-way quantum capacity and showcases that it is closely related to the computational distillable entanglement of the Choi state of the channel, and shows a stark computational capacity separation.

Abstract

Quantum channel capacities are fundamental to quantum information theory. Their definition, however, does not limit the computational resources of sender and receiver. In this work, we initiate the study of computational quantum capacities. These quantify how much information can be reliably transmitted when imposing the natural requirement that en- and decoding have to be computationally efficient. We focus on the computational two-way quantum capacity and showcase that it is closely related to the computational distillable entanglement of the Choi state of the channel. This connection allows us to show a stark computational capacity separation. Under standard cryptographic assumptions, there exists a quantum channel of polynomial complexity whose computational two-way quantum capacity vanishes while its unbounded counterpart is nearly maximal. More so, we show that there exists a sharp transition in computational quantum capacity from nearly maximal to zero when the channel complexity leaves the polynomial realm. Our results demonstrate that the natural requirement of computational efficiency can radically alter the limits of quantum communication.
Paper Structure (1 section, 7 theorems, 38 equations, 1 figure)

This paper contains 1 section, 7 theorems, 38 equations, 1 figure.

Table of Contents

  1. End Matter

Key Result

Lemma 3

Consider a quantum channel $\Phi_n\colon \mathcal{H}_n \to \mathcal{K}_n$ of polynomial complexity $C(\Phi_n) \leq O(\operatorname{poly}(n))$ such that $\log \dim\mathcal{H}_n \leq O(\operatorname{poly}(n))$. If the channel is teleportation-covariant, i.e. for any teleportation correction unitary $U the channel is efficiently Choi-stretchable.

Figures (1)

  • Figure 1: The computational two-way quantum capacity captures the limits of quantum information transmission with the help of bidirectional classical communication when the involved operations are of polynomial complexity. The difference to the traditional, unbounded setting can be quite dramatic, as we show there exists a channel with nearly maximal unbounded capacity but vanishing computational capacity.

Theorems & Definitions (19)

  • Definition 1: Computational two-way quantum capacity
  • Definition 2: Efficient stretchability
  • Lemma 3: Efficient Choi-strechability
  • Lemma 4: Lower bound to computational two-way quantum capacity
  • proof
  • Lemma 5: Upper bound for the computational two-way capacity
  • proof
  • Theorem 6: Characterization of the two-way capacity
  • proof
  • Lemma 7: Two-way quantum capacity of the dephasing quantum channel
  • ...and 9 more