One-shot Capacity Bounds on the Simultaneous Transmission of Public and Private Information Over Quantum Channels

We aim to study the optimal rates of transmission of public and private classical information over a quantum channel in the most general channel model. To this end, we discuss a scenario in which a quantum channel is being used only once, i.e., one-shot regime is considered. A quantum channel can be used to send classical information (bits) either publicly or privately and for either case, one-shot bounds have been reported in the literature. This paper investigates the one-shot capacity capabilities of a quantum channel for simultaneous transmission of public and private information. We derive an achievable rate region in the form of a tradeoff between public and private rates. We also provide converse bounds assessing the tightness of our achievable rates. Our main tools used in the achievability proofs are position-based decoding and convex-split lemma.

[1]  Rudolf Ahlswede,et al.  Strong converse for identification via quantum channels , 2000, IEEE Trans. Inf. Theory.

[2]  Nilanjana Datta,et al.  Min- and Max-Relative Entropies and a New Entanglement Monotone , 2008, IEEE Transactions on Information Theory.

[3]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[4]  Alexander S. Holevo,et al.  The Capacity of the Quantum Channel with General Signal States , 1996, IEEE Trans. Inf. Theory.

[5]  Imre Csiszár,et al.  Broadcast channels with confidential messages , 1978, IEEE Trans. Inf. Theory.

[6]  A. Winter,et al.  Quantum privacy and quantum wiretap channels , 2004 .

[7]  R. Renner,et al.  One-shot classical-quantum capacity and hypothesis testing. , 2010, Physical review letters.

[8]  Joseph M. Renes,et al.  Noisy Channel Coding via Privacy Amplification and Information Reconciliation , 2010, IEEE Transactions on Information Theory.

[9]  Jaikumar Radhakrishnan,et al.  One-Shot Private Classical Capacity of Quantum Wiretap Channel: Based on one-shot quantum covering lemma , 2017, 1703.01932.

[10]  Igor Devetak.  The private classical capacity and quantum capacity of a quantum channel , 2005, IEEE Transactions on Information Theory.

[11]  Renato Renner,et al.  The Single-Serving Channel Capacity , 2006, 2006 IEEE International Symposium on Information Theory.

[12]  R. Renner,et al.  The Quantum Reverse Shannon Theorem Based on One-Shot Information Theory , 2009, 0912.3805.

[13]  Rahul Jain,et al.  Quantum Communication Using Coherent Rejection Sampling. , 2017, Physical review letters.

[14]  P. Shor,et al.  The Capacity of a Quantum Channel for Simultaneous Transmission of Classical and Quantum Information , 2003, quant-ph/0311131.

[15]  Nilanjana Datta,et al.  Generalized relative entropies and the capacity of classical-quantum channels , 2008, 0810.3478.

[16]  Mark M. Wilde,et al.  Position-based coding and convex splitting for private communication over quantum channels , 2017, Quantum Information Processing.

[17]  Anurag Anshu.  Quantum message compression with applications , 2017 .

[18]  Rahul Jain,et al.  One shot entanglement assisted classical and quantum communication over noisy quantum channels: A hypothesis testing and convex split approach , 2017, ArXiv.

[19]  Michael D. Westmoreland,et al.  Sending classical information via noisy quantum channels , 1997 .