A-unital Operations and Quantum Conditional Entropy

Mahathi Vempati; Saumya Shah; Nirman Ganguly; Indranil Chakrabarty

doi:10.22331/q-2022-02-02-641

A-unital Operations and Quantum Conditional Entropy

Mahathi Vempati^1,2, Saumya Shah³, Nirman Ganguly⁴, and Indranil Chakrabarty^1,5

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).

Popular summary

Some quantum states, when shared between distant parties, can enable the communication of more than n bits of information by transferring only n qubits between the parties. This process is known as superdense coding, and the quantum states that can be used for the process are called 'negative conditional entropy states'. In this work, we answer the question: what quantum operations can be used to prepare negative conditional entropy states?

► BibTeX data

@article{Vempati2022unitaloperations,
  doi = {10.22331/q-2022-02-02-641},
  url = {https://doi.org/10.22331/q-2022-02-02-641},
  title = {A-unital {O}perations and {Q}uantum {C}onditional {E}ntropy},
  author = {Vempati, Mahathi and Shah, Saumya and Ganguly, Nirman and Chakrabarty, Indranil},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {641},
  month = feb,
  year = {2022}
}

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[1] Komal Kumar and Nirman Ganguly, "Quantum conditional entropies and steerability of states with maximally mixed marginals", Physical Review A 107 3, 032206 (2023).

[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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