Information transmission under Markovian noise (2409.17743v2)

Abstract: We consider an open quantum system undergoing Markovian dynamics, the latter being modelled by a discrete-time quantum Markov semigroup $(\Phi^n)_{n \in {\mathbb{N}}}$, resulting from the action of sequential uses of a quantum channel $\Phi$, with $n \in {\mathbb{N}}$ being the discrete time parameter. We find upper and lower bounds on the one-shot $\epsilon$-error information transmission capacities of $\Phi^n$ for a finite time $n\in \mathbb{N}$ and $\epsilon \in [0,1)$ in terms of the structure of the peripheral space of the channel $\Phi$. We consider transmission of $(i)$ classical information (both in the unassisted and entanglement-assisted settings); $(ii)$ quantum information and $(iii)$ private classical information.

• The paper establishes lower bounds on one-shot capacities for quantum, private, classical, and entanglement-assisted communications based on the peripheral structure of quantum channels.
• It derives upper bounds incorporating the spectral radius and convergence factors, quantifying finite error impacts over discrete time evolutions.
• These results inform improved error correction and protocol design strategies for robust quantum communication in noisy environments.

An Essay on "Information transmission under Markovian noise"

The paper "Information transmission under Markovian noise" by Satvik Singh and Nilanjana Datta investigates the communication capacities of open quantum systems subject to Markovian dynamics, specifically focusing on discrete-time quantum Markov semigroups (dQMS).

Summary of the Main Contributions

The central theme of the paper revolves around both upper and lower bounds on the one-shot, $$-error information transmission capacities for different types of information: classical, quantum, entanglement-assisted classical, and private classical information. The authors scrutinize these capacities under finite-time evolution$$n \in \mathbb{N}$and for errors$\in [0,1)$$.</p> <h3 class='paper-heading' id='main-results'>Main Results</h3> <ol> <li><strong>Lower Bounds</strong>: <ul> <li><strong>Quantum Capacity</strong>: The quantum capacity$$Q_{\epsilon}(\Phi^n)$for$\Phi^n$was found to be at least$\log(\max_k d_k)$, where$d_k$$corresponds to the block dimensions in the decomposition of the peripheral space$$\chi(\Phi)$$.</li> <li><strong>Private Classical Capacity</strong>: The private classical capacity$$C_{\epsilon}^{p}(\Phi^n)$$obeys the same lower bound as the quantum capacity, stemming from the relationship$$Q_{\epsilon}(\Phi^n) \leq C_{\epsilon}^{p}(\Phi^n)$$.</li> <li><strong>Classical Capacity</strong>: The classical capacity$$C_{\epsilon}(\Phi^n)$was lower-bounded by$\log(\sum_k d_k)$$, aggregating the block dimensions.</li> <li><strong>Entanglement-assisted Classical Capacity</strong>: For capacities assisted by entanglement, the lower bound is$$\log(\sum_k d_k^2)$$.</li> </ul></li> <li><strong>Upper Bounds</strong>: <ul> <li>The upper bounds on$$Q_{\epsilon}(\Phi^n)$,$C_{\epsilon}^p(\Phi^n)$,$C_{\epsilon}(\Phi^n)$, and$C_{\epsilon}^{ea}(\Phi^n)$were formulated to have an additional factor$\log(\frac{1}{1-\epsilon-\kappa\mu^n})$besides the intrinsic limits defined by$d_k$and$\sum_k d_k$$, dependent on the convergence properties of the semigroup, where$$\mu$is the spectral radius and$\kappa$$is a constant.</li> </ul></li> </ol> <h3 class='paper-heading' id='theoretical-implications'>Theoretical Implications</h3> <p>By establishing these bounds, the paper contributes significantly to the understanding of information-theoretic limits in quantum communication under realistic conditions. The bounds are centered on the peripheral eigenvalues of the semigroup, illuminating the degrees of freedom that persist in the long-term evolution of quantum systems. This insight could be foundational for future explorations in quantum information theory, particularly concerning systems plagued by environmental interactions.</p> <h3 class='paper-heading' id='practical-implications'>Practical Implications</h3> <p>The practical ramifications of these results span across the design and optimization of quantum communication protocols, particularly those reliant on sequential uses of quantum channels. Understanding these capacity bounds aids in crafting better error correction and mitigation strategies, potentially advancing quantum communication networks, cryptography, and computing technologies.</p> <h3 class='paper-heading' id='future-directions'>Future Directions</h3> <p>Future research may focus on the following avenues:</p> <ul> <li><strong>Tighter Bounds</strong>: Investigating if the current bounds can be made tighter by explicit constructions accommodating finite error$$\epsilon$.

Non-Markovian Dynamics: Exploring similar bounds for non-Markovian noise models, where memory effects introduce further complexity.

Alternate Channel Models: Extending the methodologies to other channel models, such as those with non-trivial temporal correlations or higher-dimensional systems.

Conclusion

This paper provides a detailed analysis of information transmission capacities for quantum Markov semigroups, both under finite and asymptotic regimes. The bounds derived are instrumental in understanding the limits of information transfer in quantum systems subject to Markovian noise, offering valuable insights that influence both theoretical investigations and practical implementations in quantum information processing.