Information-Theoretic Aspects of Correlated Quantum Channels and Quantum Correlations: A Comprehensive Study

Sk, Rajiuddin (2024) Information-Theoretic Aspects of Correlated Quantum Channels and Quantum Correlations: A Comprehensive Study. Masters thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

Quantum information theory plays a crucial role in the realms of quantum computing and quantum communication systems. In various scenarios, such as transmitting a quantum state from one party to another, establishing quantum entanglement between two parties, or securely exchanging keys through quantum communication, the utilization of a quantum channel becomes indispensable. Therefore, comprehending the properties of quantum channels is of utmost significance in the realm of communication. The primary objective of this thesis is two-fold. Firstly, it aims to investigate the dynamics of quantum correlations in the presence of decoherence caused by the environment, which is implemented through a quantum channel. The focus is on understanding the preservation and resilience of these quantum correlations against the detrimental effects of decoherence. Secondly, it endeavours to calculate the information capacity of a quantum channel, assessing its efficacy in transmitting and processing quantum information. The focus lies in understanding the channel’s capabilities and limitations in handling quantum data. In the beginning, our study focuses on the dynamical evolution of quantum coherence and genuine multipartite concurrence for the extended Werner-like states under the action of different correlated quantum channels, specifically amplitude damping (AD), phase damping (PD), and depolarizing (DP) channels. The findings demonstrate that the presence of a correlation between successive actions of the channel curtails the decay rate of coherence. It has been observed that the use of correlated channels can provide substantial protection against the fragility of genuine multipartite entanglement caused by decoherence. Interestingly, for even qubit Werner-like states, both coherence and entanglement exhibit a freezing phenomenon in perfectly correlated phase damping and depolarizing channels. This phenomenon ensures that coherence and entanglement remain unaffected by the decohering environment. In the case of multipartite GHZ-class states in a perfectly correlated channel, our studies have revealed that entanglement sudden death can be avoided in the amplitude damping channel, and interestingly, there is a sudden birth of entanglement for states with an odd number of qubit passing through the depolarizing channel. Moreover, we have derived an analytical relationship between coherence and entanglement for both completely uncorrelated and fully correlated quantum channels. This relationship provides valuable insights into the interplay between these two important aspects of quantum information processing. Building upon the motivation to preserve quantum correlations against decoherence discussed in the previous chapter, we then explore the protective role of the Stark shift effect. It is demonstrated that the Stark shift effect can effectively safeguard quantum correlations against decoherence induced by the environment. In this study, we analyze the dynamics of quantum correlations by deriving the exact expressions for various measures, namely, Bures distance entanglement, trace distance discord, and local quantum uncertainty. Here, the atoms undergo two-photon transitions via an intermediate virtual state. Each atom is independently coupled to a dissipative reservoir at zero temperature in the presence of the Stark shift effect. The dynamics of the atomic system have been investigated under two different initial conditions of the environment. In the first case, we consider the state of the environment to be in the vacuum state and in the other case, we assume the environment to be in a single photon state. The second initial condition holds particular significance as it highlights the role played by both Stark shift parameters, in contrast to only one Stark shift parameter in the first initial condition. Our findings indicate that quantum correlations can be maintained over a prolonged duration in the presence of the Stark shift effect, irrespective of whether the system is subjected to Markovian or non-Markovian reservoirs. The impact in the non-Markovian reservoir is more prominent than the Markovian reservoir, even for a minuscule value of the Stark shift parameter. Among the correlation measures examined, it is noteworthy that only the local quantum uncertainty exhibits a sudden change phenomenon. This phenomenon is characterized by an abrupt shift in the decay rate of the correlation measure, setting it apart from the other measures considered in the study. These measures allow us to gain insights into the behaviour and evolution of quantum correlations between two two-level atoms. In quantum information theory, the impact of noise on quantum communication is commonly evaluated by the channel capacities. These capacities represent the optimal rates at which quantum or classical information can be transmitted reliably, considering an infinite number of channel uses. The calculation of channel capacity is a challenging task in many instances, primarily due to the involvement of optimization of Holevo quantity and coherent information over multiple uses of the channels and their superadditivity property. In the next chapter, we tackle this challenge for a specific class of quantum channels known as multi-level amplitude damping (MAD) channels. We calculate the information capacity of a fully correlated threelevel amplitude damping channel. The degradability analysis of the maps associated with the three-level correlated amplitude damping channel has been conducted, and the range of the decay parameter, ensuring channel degradability, has been calculated. Then, we deduce the upper bound single-shot classical capacities and the exact expression of quantum capacities associated with a broad range of maps for the three-level system. Additionally, we compute the entanglement-assisted quantum and classical capacities. We have shown that the quantum capacity of MAD channels can be sufficiently increased in the presence of a correlation between successive applications of channels. The results demonstrate that correlated channels possess a non-zero capacity value for all decay parameter values, in contrast to the uncorrelated case.

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