A-unital Operations and Quantum Conditional Entropy
1Centre for Quantum Science and Technology, International Institute of Information Technology-Hyderabad, Gachibowli, Telangana-500032, India.
2Center for Computational Natural Sciences and Bioinformatics, International Institute of Information Technology-Hyderabad, Gachibowli, Telangana-500032, India.
3Department of Physics, Indian Institute of Technology Guwahati, Guwahati-781039, Assam, India.
4Department of Mathematics, Birla Institute of Technology and Science Pilani, Hyderabad Campus, Telangana-500078, India.
5Center for Security, Theory and Algorithmic Research, International Institute of Information Technology-Hyderabad, Gachibowli, Telangana-500032, India.
Published: | 2022-02-02, volume 6, page 641 |
Eprint: | arXiv:2110.12527v3 |
Doi: | https://doi.org/10.22331/q-2022-02-02-641 |
Citation: | Quantum 6, 641 (2022). |
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Abstract
Negative quantum conditional entropy states are key ingredients for information theoretic tasks such as superdense coding, state merging and one-way entanglement distillation. In this work, we ask: how does one detect if a channel is useful in preparing negative conditional entropy states? We answer this question by introducing the class of A-unital channels, which we show are the largest class of conditional entropy non-decreasing channels. We also prove that A-unital channels are precisely the completely free operations for the class of states with non-negative conditional entropy. Furthermore, we study the relationship between A-unital channels and other classes of channels pertinent to the resource theory of entanglement. We then prove similar results for ACVENN: a previously defined, relevant class of states and also relate the maximum and minimum conditional entropy of a state with its von Neumann entropy.
The definition of A-unital channels naturally lends itself to a procedure for determining membership of channels in this class. Thus, our work is valuable for the detection of resourceful channels in the context of conditional entropy.

Featured image: The image depicts an A-unital operation that is not separable: "Swap and prepare". The grey circles indicate a maximally mixed subsystem, and the squiggly lines indicate entanglement. The operation involves first swapping subsystems A and B (box 2), tracing out subsystem A (box 3), and finally, preparing a maximally mixed state on subsystem A (box 4).
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[1] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum entanglement. Rev. Mod. Phys., 81: 865–942, Jun 2009. 10.1103/RevModPhys.81.865. URL https://link.aps.org/doi/10.1103/RevModPhys.81.865.
https://doi.org/10.1103/RevModPhys.81.865
[2] Eric Chitambar and Gilad Gour. Quantum resource theories. Rev. Mod. Phys., 91: 025001, Apr 2019. 10.1103/RevModPhys.91.025001. URL https://link.aps.org/doi/10.1103/RevModPhys.91.025001.
https://doi.org/10.1103/RevModPhys.91.025001
[3] Eric Chitambar, Debbie Leung, Laura Mančinska, Maris Ozols, and Andreas Winter. Everything you always wanted to know about LOCC (but were afraid to ask). Communications in Mathematical Physics, 328 (1): 303–326, March 2014. 10.1007/s00220-014-1953-9. URL https://doi.org/10.1007/s00220-014-1953-9.
https://doi.org/10.1007/s00220-014-1953-9
[4] J. I. Cirac, W. Dür, B. Kraus, and M. Lewenstein. Entangling operations and their implementation using a small amount of entanglement. Phys. Rev. Lett., 86: 544–547, Jan 2001. 10.1103/PhysRevLett.86.544. URL https://link.aps.org/doi/10.1103/PhysRevLett.86.544.
https://doi.org/10.1103/PhysRevLett.86.544
[5] V. Vedral and M. B. Plenio. Entanglement measures and purification procedures. Phys. Rev. A, 57: 1619–1633, Mar 1998. 10.1103/PhysRevA.57.1619. URL https://link.aps.org/doi/10.1103/PhysRevA.57.1619.
https://doi.org/10.1103/PhysRevA.57.1619
[6] Michael Horodecki, Peter W. Shor, and Mary Beth Ruskai. Entanglement breaking channels. Reviews in Mathematical Physics, 15 (06): 629–641, August 2003. 10.1142/s0129055x03001709. URL https://doi.org/10.1142/s0129055x03001709.
https://doi.org/10.1142/s0129055x03001709
[7] C. Macchiavello and M. Rossi. Quantum channel detection. Phys. Rev. A, 88: 042335, Oct 2013. 10.1103/PhysRevA.88.042335. URL https://link.aps.org/doi/10.1103/PhysRevA.88.042335.
https://doi.org/10.1103/PhysRevA.88.042335
[8] Colin Do-Yan Lee and John Watrous. Detecting mixed-unitary quantum channels is NP-hard. Quantum, 4: 253, April 2020. 10.22331/q-2020-04-16-253. URL https://doi.org/10.22331/q-2020-04-16-253.
https://doi.org/10.22331/q-2020-04-16-253
[9] Ashley Montanaro and Ronald de Wolf. A survey of quantum property testing. Theory of Computing, 1 (1): 1–81, 2016. 10.4086/toc.gs.2016.007. URL https://doi.org/10.4086/toc.gs.2016.007.
https://doi.org/10.4086/toc.gs.2016.007
[10] N. Milazzo, D. Braun, and O. Giraud. Truncated moment sequences and a solution to the channel separability problem. Phys. Rev. A, 102: 052406, Nov 2020. 10.1103/PhysRevA.102.052406. URL https://link.aps.org/doi/10.1103/PhysRevA.102.052406.
https://doi.org/10.1103/PhysRevA.102.052406
[11] Sevag Gharibian. Strong np-hardness of the quantum separability problem. Quantum Info. Comput., 10 (3): 343–360, March 2010. ISSN 1533-7146. 10.26421/QIC10.3-4-11.
https://doi.org/10.26421/QIC10.3-4-11
[12] N. J. Cerf and C. Adami. Negative entropy and information in quantum mechanics. Phys. Rev. Lett., 79: 5194–5197, Dec 1997. 10.1103/PhysRevLett.79.5194. URL https://link.aps.org/doi/10.1103/PhysRevLett.79.5194.
https://doi.org/10.1103/PhysRevLett.79.5194
[13] Mark M. Wilde. Quantum Information Theory. Cambridge University Press, 2016. 10.1017/9781316809976. URL https://doi.org/10.1017/9781316809976.
https://doi.org/10.1017/9781316809976
[14] Harold Ollivier and Wojciech H. Zurek. Quantum discord: A measure of the quantumness of correlations. Phys. Rev. Lett., 88: 017901, Dec 2001. 10.1103/PhysRevLett.88.017901. URL https://link.aps.org/doi/10.1103/PhysRevLett.88.017901.
https://doi.org/10.1103/PhysRevLett.88.017901
[15] L Henderson and V Vedral. Classical, quantum and total correlations. Journal of Physics A: Mathematical and General, 34 (35): 6899–6905, Aug 2001. 10.1088/0305-4470/34/35/315. URL https://doi.org/10.1088.
https://doi.org/10.1088/0305-4470/34/35/315
[16] Chandrashekar Radhakrishnan, Mathieu Laurière, and Tim Byrnes. Multipartite generalization of quantum discord. Phys. Rev. Lett., 124: 110401, Mar 2020. 10.1103/PhysRevLett.124.110401. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.110401.
https://doi.org/10.1103/PhysRevLett.124.110401
[17] Michał Horodecki, Jonathan Oppenheim, and Andreas Winter. Partial quantum information. Nature, 436 (7051): 673–676, August 2005. 10.1038/nature03909. URL https://doi.org/10.1038/nature03909.
https://doi.org/10.1038/nature03909
[18] Michał Horodecki, Jonathan Oppenheim, and Andreas Winter. Quantum state merging and negative information. Communications in Mathematical Physics, 269 (1): 107–136, October 2006. 10.1007/s00220-006-0118-x. URL https://doi.org/10.1007/s00220-006-0118-x.
https://doi.org/10.1007/s00220-006-0118-x
[19] Charles H. Bennett and Stephen J. Wiesner. Communication via one- and two-particle operators on einstein-podolsky-rosen states. Phys. Rev. Lett., 69: 2881–2884, Nov 1992. 10.1103/PhysRevLett.69.2881. URL https://link.aps.org/doi/10.1103/PhysRevLett.69.2881.
https://doi.org/10.1103/PhysRevLett.69.2881
[20] D. Bruß, G. M. D'Ariano, M. Lewenstein, C. Macchiavello, A. Sen(De), and U. Sen. Distributed quantum dense coding. Phys. Rev. Lett., 93: 210501, Nov 2004. 10.1103/PhysRevLett.93.210501. URL https://link.aps.org/doi/10.1103/PhysRevLett.93.210501.
https://doi.org/10.1103/PhysRevLett.93.210501
[21] R. Prabhu, Arun Kumar Pati, Aditi Sen(De), and Ujjwal Sen. Exclusion principle for quantum dense coding. Phys. Rev. A, 87: 052319, May 2013. 10.1103/PhysRevA.87.052319. URL https://link.aps.org/doi/10.1103/PhysRevA.87.052319.
https://doi.org/10.1103/PhysRevA.87.052319
[22] Igor Devetak and Andreas Winter. Distillation of secret key and entanglement from quantum states. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461 (2053): 207–235, January 2005. 10.1098/rspa.2004.1372. URL https://doi.org/10.1098/rspa.2004.1372.
https://doi.org/10.1098/rspa.2004.1372
[23] Dong Yang, Karol Horodecki, and Andreas Winter. Distributed private randomness distillation. Phys. Rev. Lett., 123: 170501, Oct 2019. 10.1103/PhysRevLett.123.170501. URL https://link.aps.org/doi/10.1103/PhysRevLett.123.170501.
https://doi.org/10.1103/PhysRevLett.123.170501
[24] Mario Berta, Matthias Christandl, Roger Colbeck, Joseph M. Renes, and Renato Renner. The uncertainty principle in the presence of quantum memory. Nature Physics, 6 (9): 659–662, July 2010. 10.1038/nphys1734. URL https://doi.org/10.1038/nphys1734.
https://doi.org/10.1038/nphys1734
[25] Mahathi Vempati, Nirman Ganguly, Indranil Chakrabarty, and Arun K. Pati. Witnessing negative conditional entropy. Phys. Rev. A, 104: 012417, Jul 2021. 10.1103/PhysRevA.104.012417. URL https://link.aps.org/doi/10.1103/PhysRevA.104.012417.
https://doi.org/10.1103/PhysRevA.104.012417
[26] Nicolai Friis, Sridhar Bulusu, and Reinhold A Bertlmann. Geometry of two-qubit states with negative conditional entropy. Journal of Physics A: Mathematical and Theoretical, 50 (12): 125301, feb 2017. 10.1088/1751-8121/aa5dfd. URL https://doi.org/10.1088/1751-8121/aa5dfd.
https://doi.org/10.1088/1751-8121/aa5dfd
[27] Subhasree Patro, Indranil Chakrabarty, and Nirman Ganguly. Non-negativity of conditional von neumann entropy and global unitary operations. Phys. Rev. A, 96: 062102, Dec 2017. 10.1103/PhysRevA.96.062102. URL https://link.aps.org/doi/10.1103/PhysRevA.96.062102.
https://doi.org/10.1103/PhysRevA.96.062102
[28] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2009. 10.1017/cbo9780511976667. URL https://doi.org/10.1017/cbo9780511976667.
https://doi.org/10.1017/cbo9780511976667
[29] Marek Kuśand Karol Życzkowski. Geometry of entangled states. Phys. Rev. A, 63: 032307, Feb 2001. 10.1103/PhysRevA.63.032307. URL https://link.aps.org/doi/10.1103/PhysRevA.63.032307.
https://doi.org/10.1103/PhysRevA.63.032307
[30] Saronath Halder, Shiladitya Mal, and Aditi Sen(De). Characterizing the boundary of the set of absolutely separable states and their generation via noisy environments. Phys. Rev. A, 103: 052431, May 2021. 10.1103/PhysRevA.103.052431. URL https://link.aps.org/doi/10.1103/PhysRevA.103.052431.
https://doi.org/10.1103/PhysRevA.103.052431
[31] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels. Phys. Rev. Lett., 70: 1895–1899, Mar 1993. 10.1103/PhysRevLett.70.1895. URL https://link.aps.org/doi/10.1103/PhysRevLett.70.1895.
https://doi.org/10.1103/PhysRevLett.70.1895
[32] Christian B. Mendl and Michael M. Wolf. Unital quantum channels – convex structure and revivals of birkhoff's theorem. Communications in Mathematical Physics, 289 (3): 1057–1086, May 2009. 10.1007/s00220-009-0824-2. URL https://doi.org/10.1007/s00220-009-0824-2.
https://doi.org/10.1007/s00220-009-0824-2
[33] E. M. Rains. Bound on distillable entanglement. Phys. Rev. A, 60: 179–184, Jul 1999. 10.1103/PhysRevA.60.179. URL https://link.aps.org/doi/10.1103/PhysRevA.60.179.
https://doi.org/10.1103/PhysRevA.60.179
[34] E.M. Rains. A semidefinite program for distillable entanglement. IEEE Transactions on Information Theory, 47 (7): 2921–2933, 2001. 10.1109/18.959270.
https://doi.org/10.1109/18.959270
[35] Bartosz Regula, Kun Fang, Xin Wang, and Mile Gu. One-shot entanglement distillation beyond local operations and classical communication. New Journal of Physics, 21 (10): 103017, October 2019. 10.1088/1367-2630/ab4732. URL https://doi.org/10.1088/1367-2630/ab4732.
https://doi.org/10.1088/1367-2630/ab4732
[36] Min Jiang, Shunlong Luo, and Shuangshuang Fu. Channel-state duality. Phys. Rev. A, 87: 022310, Feb 2013. 10.1103/PhysRevA.87.022310. URL https://link.aps.org/doi/10.1103/PhysRevA.87.022310.
https://doi.org/10.1103/PhysRevA.87.022310
[37] Michael A. Nielsen. An introduction to majorization and its applications to quantum mechanics. 2002. URL https://michaelnielsen.org/blog/talks/2002/maj/book.ps.
https://michaelnielsen.org/blog/talks/2002/maj/book.ps
[38] Yuan Li and Paul Busch. Von neumann entropy and majorization. Journal of Mathematical Analysis and Applications, 408 (1): 384–393, December 2013. 10.1016/j.jmaa.2013.06.019. URL https://doi.org/10.1016/j.jmaa.2013.06.019.
https://doi.org/10.1016/j.jmaa.2013.06.019
[39] Sarah Brandsen, Isabelle J Geng, Mark M Wilde, and Gilad Gour. Quantum conditional entropy from information-theoretic principles. arXiv preprint arXiv:2110.15330, 2021. URL https://arxiv.org/abs/2110.15330.
arXiv:2110.15330
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[2] Tapaswini Patro, Kaushiki Mukherjee, Mohd Asad Siddiqui, Indranil Chakrabarty, and Nirman Ganguly, "Absolute fully entangled fraction from spectrum", The European Physical Journal D 76 7, 127 (2022).
[3] Sarah Brandsen, Isabelle J. Geng, Mark M. Wilde, and Gilad Gour, "Quantum conditional entropy from information-theoretic principles", arXiv:2110.15330, (2021).
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