[Paper Review] SPHERE PACKING AND ZERO-RATE BOUNDS TO THE RELIABILITY OF CLASSICAL-QUANTUM CHANNELS

Marco Dalai|arXiv (Cornell University)|Jan 25, 2012

Quantum Information and Cryptography19 references2 citations

TL;DR

This paper extends the sphere packing and zero-rate bounds from classical channels to general classical-quantum channels, establishing exact expressions for the reliability function at high rates for pure state channels and at zero rate for channels without zero-error capacity. The results close longstanding gaps by showing that the upper bound matches known lower bounds in both regimes, thereby determining the reliability function exactly in these cases.

ABSTRACT

In this paper, the sphere packing bound of Fano, Shannon, Gal- lager and Berlekamp and the zero-rate bound of Berlekamp are extended to general classical-quantum channels. The upper bound for the reliability func- tion obtained from the sphere packing coincides at high rates, for the case of pure state channels, with a lower bound derived by Burnashev and Holevo (21). Thus, for pure state channels, the reliability function at high rates is now ex- actly determined. For the general case, the obtained upper bound expression at high rates was conjectured to represent also a lower bound to the reliability function, but a complete proof has not been obtained yet. Finally, the obtained zero-rate upper bound to the reliability function of a general classical-quantum channel with no zero-error capacity coincides with a lower bound obtained by Holevo, thus determining the exact expression.

Motivation & Objective

• To extend the classical sphere packing and zero-rate bounds to the quantum domain for general classical-quantum channels.
• To determine the exact reliability function at high rates for pure state channels by matching upper and lower bounds.
• To investigate whether the derived high-rate upper bound also serves as a lower bound for general classical-quantum channels.
• To establish the exact reliability function at zero rate for channels with no zero-error capacity.
• To unify and generalize existing bounds from classical information theory to the quantum setting.

Proposed method

• Adapt the sphere packing bound of Fano, Shannon, Gallager, and Berlekamp to classical-quantum channels using quantum state distinguishability and trace distance metrics.
• Apply the zero-rate bound of Berlekamp to classical-quantum channels, leveraging the structure of quantum measurement and state overlap.
• Use quantum relative entropy and fidelity-based bounds to analyze the error probability in channel coding for quantum states.
• Compare the derived upper bounds with existing lower bounds from Burnashev and Holevo to assess tightness.
• Employ asymptotic analysis of error exponents in the high-rate and zero-rate regimes to evaluate the reliability function.
• Utilize the properties of pure state channels and general quantum channels to distinguish between cases where bounds match exactly.

Experimental results

Research questions

• RQ1 Does the sphere packing bound for classical channels extend precisely to classical-quantum channels, particularly for pure state channels?
• RQ2 Can the derived high-rate upper bound on the reliability function be proven to also serve as a lower bound for general classical-quantum channels?
• RQ3 How does the zero-rate bound for classical channels generalize to classical-quantum channels with no zero-error capacity?
• RQ4 Is the upper bound on the reliability function at zero rate tight, and does it match existing lower bounds in the quantum setting?
• RQ5 What conditions allow the reliability function to be exactly determined in the high-rate and zero-rate regimes for classical-quantum channels?

Key findings

• For pure state channels, the reliability function at high rates is exactly determined because the derived upper bound from sphere packing matches a known lower bound by Burnashev and Holevo.
• The upper bound on the reliability function at high rates for general classical-quantum channels is conjectured to also be a lower bound, though a complete proof remains unachieved.
• At zero rate, the upper bound for channels without zero-error capacity exactly matches a lower bound by Holevo, thus determining the reliability function exactly in this regime.
• The sphere packing bound extension to classical-quantum channels provides a tight characterization of error exponents in the high-rate regime for pure state channels.
• The zero-rate bound derived in this work coincides with Holevo’s lower bound, resolving the exact reliability function for such channels.
• The results establish a significant bridge between classical information theory and quantum channel coding by extending fundamental bounds to the quantum setting.