A-unital Operations and Quantum Conditional Entropy

Mahathi Vempati1,2, Saumya Shah3, Nirman Ganguly4, and Indranil Chakrabarty1,5

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.

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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.

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?

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Cited by

[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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