# Bounding the quantum capacity with flagged extensions

Farzad Kianvash1, Marco Fanizza1,2, and Vittorio Giovannetti1

1NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy
2Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona) Spain

### Abstract

In this article we consider flagged extensions of convex combination of quantum channels, and find general sufficient conditions for the degradability of the flagged extension. An immediate application is a bound on the quantum $Q$ and private $P$ capacities of any channel being a mixture of a unitary map and another channel, with the probability associated to the unitary component being larger than $1/2$. We then specialize our sufficient conditions to flagged Pauli channels, obtaining a family of upper bounds on quantum and private capacities of Pauli channels. In particular, we establish new state-of-the-art upper bounds on the quantum and private capacities of the depolarizing channel, BB84 channel and generalized amplitude damping channel. Moreover, the flagged construction can be naturally applied to tensor powers of channels with less restricting degradability conditions, suggesting that better upper bounds could be found by considering a larger number of channel uses.

Quantum information can be protected from noise by encoding it in logical qubits. For several fundamental noise models in discrete variables, such as thermal attenuation and depolarizing noise, the optimal ratio of logical versus physical qubits (quantum capacity) is not known. We improve the upper bounds on the quantum capacity by constructing less noisy channels, where information on the noise acting is encoded in an auxiliary quantum register, and for which the quantum capacity can be computed.

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### Cited by

[1] Stefano Chessa and Vittorio Giovannetti, "Resonant Multilevel Amplitude Damping Channels", Quantum 7, 902 (2023).

[2] Abbas Poshtvan and Vahid Karimipour, "Capacities of the covariant Pauli channel", Physical Review A 106 6, 062408 (2022).

[3] Seid Koudia, Angela Sara Cacciapuoti, Kyrylo Simonov, and Marcello Caleffi, "How Deep the Theory of Quantum Communications Goes: Superadditivity, Superactivation and Causal Activation", arXiv:2108.07108, (2021).

[4] Marco Fanizza, Farzad Kianvash, and Vittorio Giovannetti, "Estimating Quantum and Private Capacities of Gaussian Channels via Degradable Extensions", Physical Review Letters 127 21, 210501 (2021).

[5] Felix Leditzky, Debbie Leung, Vikesh Siddhu, Graeme Smith, and John A. Smolin, "Generic nonadditivity of quantum capacity in simple channels", arXiv:2202.08377, (2022).

[6] Marco Fanizza, Raffaele Salvia, and Vittorio Giovannetti, "Testing identity of collections of quantum states: sample complexity analysis", arXiv:2103.14511, (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2023-02-03 20:34:44) and SAO/NASA ADS (last updated successfully 2023-02-03 20:34:45). The list may be incomplete as not all publishers provide suitable and complete citation data.