New Lower Bounds to the Output Entropy of Multi-Mode Quantum Gaussian Channels

We prove that quantum thermal Gaussian input states minimize the output entropy of the multi-mode quantum Gaussian attenuators and amplifiers that are entanglement breaking and of the multi-mode quantum Gaussian phase contravariant channels among all the input states with a given entropy. This is the first time that this property is proven for a multi-mode channel without restrictions on the input states. A striking consequence of this result is a new lower bound on the output entropy of all the multi-mode quantum Gaussian attenuators and amplifiers in terms of the input entropy. We apply this bound to determine new upper bounds to the communication rates in two different scenarios. The first is classical communication to two receivers with the quantum degraded Gaussian broadcast channel. The second is the simultaneous classical communication, quantum communication and entanglement generation or the simultaneous public classical communication, private classical communication and quantum key distribution with the Gaussian quantum-limited attenuator.

[1]  V. Giovannetti,et al.  Multimode Gaussian optimizers for the Wehrl entropy and quantum Gaussian channels , 2017, 1705.00499.

[2]  Giacomo De Palma,et al.  Gaussian states minimize the output entropy of one-mode quantum Gaussian channels , 2016, Physical review letters.

[3]  A. Holevo,et al.  Quantum state majorization at the output of bosonic Gaussian channels , 2013, Nature Communications.

[4]  Igor Devetak,et al.  Quantum Broadcast Channels , 2006, IEEE Transactions on Information Theory.

[5]  R. Konig,et al.  The Entropy Power Inequality for Quantum Systems , 2012, IEEE Transactions on Information Theory.

[6]  Saikat Guha,et al.  Capacity of the bosonic wiretap channel and the Entropy Photon-Number Inequality , 2008, 2008 IEEE International Symposium on Information Theory.

[7]  Saikat Guha,et al.  Quantum trade-off coding for bosonic communication , 2011, ArXiv.

[8]  Dario Trevisan,et al.  Gaussian States Minimize the Output Entropy of the One-Mode Quantum Attenuator , 2016, IEEE Transactions on Information Theory.

[9]  Giacomo De Palma,et al.  Gaussian optimizers for entropic inequalities in quantum information , 2018, Journal of Mathematical Physics.

[10]  Giacomo De Palma,et al.  The conditional Entropy Power Inequality for quantum additive noise channels , 2018, Journal of Mathematical Physics.

[11]  V. Giovannetti,et al.  The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have GaussianMaximizers , 2016, Annales Henri Poincaré.

[12]  S. Olivares,et al.  Gaussian states in continuous variable quantum information , 2005, quant-ph/0503237.

[13]  S. Olivares,et al.  Gaussian States in Quantum Information , 2005 .

[14]  Seth Lloyd,et al.  Gaussian quantum information , 2011, 1110.3234.

[15]  Robert Koenig,et al.  Geometric inequalities from phase space translations , 2016, 1606.08603.

[16]  Mark M. Wilde,et al.  Capacities of Quantum Amplifier Channels , 2016, ArXiv.

[17]  Mark M. Wilde,et al.  Classical Codes for Quantum Broadcast Channels , 2011, IEEE Transactions on Information Theory.

[18]  Saikat Guha,et al.  The Entropy Photon-Number Inequality and its consequences , 2007, 2008 Information Theory and Applications Workshop.

[19]  V.W.S. Chan,et al.  Free-Space Optical Communications , 2006, Journal of Lightwave Technology.

[20]  G. De Palma,et al.  The Conditional Entropy Power Inequality for Bosonic Quantum Systems , 2017, Communications in Mathematical Physics.

[21]  Mark M. Wilde,et al.  Entanglement-Assisted Communication of Classical and Quantum Information , 2008, IEEE Transactions on Information Theory.

[22]  S. Lloyd,et al.  Multimode quantum entropy power inequality , 2014, 1408.6410.

[23]  Stephen M. Barnett,et al.  Methods in Theoretical Quantum Optics , 1997 .

[24]  A. Holevo Gaussian optimizers and the additivity problem in quantum information theory , 2015, 1501.00652.

[25]  G. Palma Gaussian optimizers and other topics in quantum information , 2017, 1710.09395.

[26]  Alexander S. Holevo,et al.  One-mode quantum Gaussian channels: Structure and quantum capacity , 2007, Probl. Inf. Transm..

[27]  A. Holevo,et al.  A Solution of Gaussian Optimizer Conjecture for Quantum Channels , 2015 .

[28]  Alexander Semenovich Holevo,et al.  Quantum Systems, Channels, Information: A Mathematical Introduction , 2019 .

[29]  Giacomo De Palma,et al.  Passive States Optimize the Output of Bosonic Gaussian Quantum Channels , 2015, IEEE Transactions on Information Theory.

[30]  A. Serafini Quantum Continuous Variables: A Primer of Theoretical Methods , 2017 .

[31]  On the notion of entanglement in Hilbert spaces , 2005 .

[32]  Mark M. Wilde,et al.  The quantum dynamic capacity formula of a quantum channel , 2010, Quantum Inf. Process..

[33]  Graeme Smith,et al.  Corrections to “The Entropy Power Inequality for Quantum Systems” [Mar 14 1536-1548] , 2016, IEEE Transactions on Information Theory.

[34]  Saikat Guha,et al.  Classical Information Capacity of the Bosonic Broadcast Channel , 2007, 2007 IEEE International Symposium on Information Theory.

[35]  Saikat Guha,et al.  Information trade-offs for optical quantum communication , 2012, Physical review letters.

[36]  J. Shapiro,et al.  Classical capacity of bosonic broadcast communication and a minimum output entropy conjecture , 2007, 0706.3416.

[37]  Saikat Guha,et al.  On the minimum output entropy of single-mode phase-insensitive Gaussian channels , 2016 .

[38]  Mark M. Wilde,et al.  Trading classical communication, quantum communication, and entanglement in quantum Shannon theory , 2009, IEEE Transactions on Information Theory.

[39]  Timothy C. Ralph,et al.  Quantum information with continuous variables , 2000, Conference Digest. 2000 International Quantum Electronics Conference (Cat. No.00TH8504).

[40]  J. Habif,et al.  Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver , 2011, Nature Photonics.

[41]  R. Schumann Quantum Information Theory , 2000, quant-ph/0010060.

[42]  V. Giovannetti,et al.  A generalization of the entropy power inequality to bosonic quantum systems , 2014, 1402.0404.