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Superadditivity of Zero-Error Capacity in Noisy Classical and Perfect Quantum Channel Pairs

Ambuj, Anushko Chattopadhyay, Kunika Agarwal, Rakesh Das, Amit Mukherjee

Abstract

We demonstrate superadditivity of one-shot zero-error classical capacity in an asymmetric communication setting where a noisy classical channel is used in parallel with a perfect quantum channel. Each channel individually supports only a fixed number of perfectly distinguishable messages. Their joint use enables transmission of strictly more messages than permitted by the product of the individual capacities. We present explicit constructions achieving this enhancement and establish that replacing the perfect quantum channel with a perfect classical channel eliminates the effect. Finally, we identify a structural criterion on the noisy channel governing this effect and show that the quantum advantage is rooted in Kochen-Specker contextuality.

Superadditivity of Zero-Error Capacity in Noisy Classical and Perfect Quantum Channel Pairs

Abstract

We demonstrate superadditivity of one-shot zero-error classical capacity in an asymmetric communication setting where a noisy classical channel is used in parallel with a perfect quantum channel. Each channel individually supports only a fixed number of perfectly distinguishable messages. Their joint use enables transmission of strictly more messages than permitted by the product of the individual capacities. We present explicit constructions achieving this enhancement and establish that replacing the perfect quantum channel with a perfect classical channel eliminates the effect. Finally, we identify a structural criterion on the noisy channel governing this effect and show that the quantum advantage is rooted in Kochen-Specker contextuality.
Paper Structure (11 sections, 5 theorems, 1 figure)

This paper contains 11 sections, 5 theorems, 1 figure.

Table of Contents

  1. Appendix A: Encoding and decoding scheme for Remark \ref{['sic']}
  2. A family of vectors in $\mathbb{C}^3$.
  3. Orthogonality graph and contextuality properties.
  4. Construction of a noisy classical channel.
  5. Capacity parameters.
  6. Proof of Remark \ref{['sic']}.
  7. Encoding.
  8. Channel output.
  9. Decoding.
  10. Superadditivity.
  11. Appendix B: Proofs of Corollary \ref{['c1']} and \ref{['c2']}

Key Result

Proposition 1

Let $N_n$ be a noisy classical channel with confusability graph $G_n$ and one-shot zero-error capacity $\mathfrak{C}_0(N_n)=\alpha(G_n)$, and let $N_p$ be a $d$-level perfect classical channel with confusability graph $G_p$; then the one-shot zero-error capacity of their parallel composition $N_n \o

Figures (1)

  • Figure 1: Channel hypergraph associated with the noisy classical channel $N_n$ given in Theorem \ref{['scheme']}. The hypergraph has $18$ vertices, corresponding to the channel inputs, which are represented by vectors $v_i$ in $\mathbb{R}^4$ and listed in the table. There are 9 hyperedges corresponding to possible channel outputs. Each vertex is contained in exactly two hyperedges. Accordingly, for any given input, the noisy classical channel can produce either of the two associated outputs.

Theorems & Definitions (11)

  • Proposition 1: Shannon1956
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Corollary 1
  • Corollary 2
  • proof
  • ...and 1 more