Detection and estimation of quantum channel capacities

This thesis investigates the subject of quantum channel capacities and is composed of two parts. 1. Detecting positive quantum capacities of quantum channels Determining whether a noisy quantum channel can be used to reliably transmit qubits at a non-zero rate is a challenging problem, since it requires computation of the channel's coherent information for an unbounded number of copies of the channel. In this thesis, we devise an elementary perturbative method to solve this problem in a variety of circumstances. We use this method to develop simple tests which can be used to detect positivity of quantum channel capacities simply by comparing the channels' input, output, and environment dimensions. In particular, we show that if a channel’s output space is larger than its environment, the coherent information of a single copy of the channel is generically positive. We also completely characterize a subset of zero quantum capacity channels that is defined by the property that the corresponding complementary channels also have zero quantum capacity, even if classical feedback assistance is allowed. We prove that such channels must necessarily be entanglement-breaking. Finally, we apply our method to detect positive quantum capacities of several physically relevant channels, such as the depolarizing and transpose-depolarizing channels (including the Werner-Holevo channel), dephasing channels, generalized Pauli channels, and multi-level amplitude damping channels. 2. Estimating capacities of quantum Markov semigroups In the second part, we analyze the capacities of quantum Markov semigroups acting on finite-dimensional quantum systems. We show that in the limit of infinite time, the capacities can be efficiently computed in terms of the structure of the peripheral space of the semigroup, are strongly additive, and satisfy the strong converse property. We also establish convergence bounds to show that the infinite-time capacities are reached after time scaling quadratically with the system dimension. From the perspective of data storage, our analysis provides tight bounds on the number of bits or qubits that can be reliably stored for long times in a quantum memory device that is experiencing Markovian noise. From a practical standpoint, we show that typically, a quantum memory with Markovian noise acting independently and identically on all qubits and a fixed time-independent global error correction mechanism becomes useless for storage after time scaling exponentially with the number of qubits. In contrast, if the error correction is local, the memory becomes useless much more quickly after time scaling logarithmically with the number of qubits. In the setting of point-to-point communication between two spatially separated parties, our analysis provides efficiently computable bounds on the optimal rate at which bits or qubits can be reliably transmitted via `long' Markovian communication channels, both in the finite block-length and asymptotic regimes.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2025

Author dc:creator

• Singh, Satvik

Advisor dc:contributor.advisor

• Datta, Nilanjana

• Quantum channels
• Quantum capacities
• Quantum Markov semigroups
• Quantum information theory

Licence

• https://www.repository.cam.ac.uk/bitstreams/d352b972-a66d-4391-ac18-dfd9155796c3/download
• http://purl.org/NET/rdflicense/allrightsreserved

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.122710

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/391660