Abstract
We prove the conditional Entropy Power Inequality for Gaussian quantum systems. This fundamental inequality determines the minimum quantum conditional von Neumann entropy of the output of the beam-splitter or of the squeezing among all the input states where the two inputs are conditionally independent given the memory and have given quantum conditional entropies. We also prove that, for any couple of values of the quantum conditional entropies of the two inputs, the minimum of the quantum conditional entropy of the output given by the conditional Entropy Power Inequality is asymptotically achieved by a suitable sequence of quantum Gaussian input states. Our proof of the conditional Entropy Power Inequality is based on a new Stam inequality for the quantum conditional Fisher information and on the determination of the universal asymptotic behaviour of the quantum conditional entropy under the heat semigroup evolution. The beam-splitter and the squeezing are the central elements of quantum optics, and can model the attenuation, the amplification and the noise of electromagnetic signals. This conditional Entropy Power Inequality will have a strong impact in quantum information and quantum cryptography. Among its many possible applications there is the proof of a new uncertainty relation for the conditional Wehrl entropy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Coles P.J., Berta M., Tomamichel M., Wehner S.: Entropic uncertainty relations and their applications. Rev. Mod. Phys. 89, 015002 (2017)
Cover T.M., Thomas J.A.: Elements of Information Theory. Wiley, New York (2006)
De Palma, G.: Uncertainty relations with quantum memory for the Wehrl entropy. arXiv preprint arXiv:1709.04921 (2017)
De Palma, G.: The Wehrl entropy has Gaussian optimizers. Lett. Math. Phys. (2017). https://doi.org/10.1007/s11005-017-0994-3
De Palma G., Mari A., Giovannetti V.: A generalization of the entropy power inequality to bosonic quantum systems. Nat. Photon. 8(12), 958–964 (2014)
De Palma G., Mari A., Lloyd S., Giovannetti V.: Multimode quantum entropy power inequality. Phys. Rev. A 91(3), 032320 (2015)
De Palma G., Mari A., Lloyd S., Giovannetti V.: Passive states as optimal inputs for single-jump lossy quantum channels. Phys. Rev. A 93(6), 062328 (2016)
De Palma, G., Trevisan, D., Giovannetti, V.: One-mode quantum-limited Gaussian channels have Gaussian maximizers. arXiv preprint arXiv:1610.09967 (2016)
De Palma G., Trevisan D., Giovannetti V.: Passive states optimize the output of bosonic Gaussian quantum channels. IEEE Trans. Inf. Theory 62(5), 2895–2906 (2016)
De Palma G., Trevisan D., Giovannetti V.: Gaussian states minimize the output entropy of one-mode quantum Gaussian channels. Phys. Rev. Lett. 118, 160503 (2017)
De Palma G., Trevisan D., Giovannetti V.: Gaussian states minimize the output entropy of the one-mode quantum attenuator. IEEE Trans. Inf. Theory 63(1), 728–737 (2017)
De Palma, G., Trevisan, D., Giovannetti, V.: Multimode Gaussian optimizers for the Wehrl entropy and quantum Gaussian channels. arXiv preprint arXiv:1705.00499 (2017)
Dembo A., Cover T.M., Thomas J.: Information theoretic inequalities. IEEE Trans. Inf. Theory 37(6), 1501–1518 (1991)
Ferraro, A., Olivares, S., Paris Matteo G.A.: Gaussian states in continuous variable quantum information. arXiv preprint arXiv:quant-ph/0503237 (2005)
Guha, S., Erkmen, B., Shapiro, J.H.: The entropy photon-number inequality and its consequences. In: Information Theory and Applications Workshop, pp. 128–130. IEEE (2008)
Holevo A.S.: Quantum Systems, Channels, Information: A Mathematical Introduction. De Gruyter Studies in Mathematical Physics. De Gruyter, Berlin (2013)
Holevo A.S.: Gaussian optimizers and the additivity problem in quantum information theory. Uspekhi Mat. Nauk 70(2), 141–180 (2015)
König R.: The conditional entropy power inequality for Gaussian quantum states. J. Math. Phys. 56(2), 022201 (2015)
König R., Smith G.: The entropy power inequality for quantum systems. IEEE Trans. Inf. Theory 60(3), 1536–1548 (2014)
König R., Smith G.: Corrections to “The entropy power inequality for quantum systems”. IEEE Trans. Inf. Theory 62(7), 4358–4359 (2016)
Shannon C.E.: A mathematical theory of communication. ACM SIGMOBILE Mob. Comput. Commun. Rev. 5(1), 3–55 (2001)
Stam A.J.: Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inf. Control 2(2), 101–112 (1959)
Weedbrook C., Pirandola S., Garcia-Patron R., Cerf N.J., Ralph T.C., Shapiro J.H., Lloyd S.: Gaussian quantum information. Rev. Mod. Phys. 84(2), 621 (2012)
Wilde M.M.: Quantum Information Theory. Cambridge University Press, Cambridge (2017)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. M. Wolf
Rights and permissions
About this article
Cite this article
De Palma, G., Trevisan, D. The Conditional Entropy Power Inequality for Bosonic Quantum Systems. Commun. Math. Phys. 360, 639–662 (2018). https://doi.org/10.1007/s00220-017-3082-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-017-3082-8