Quantum-optimal information encoding using noisy passive linear optics

Andrew Tanggara1,2, Ranjith Nair2, Syed Assad3,4, Varun Narasimhachar5,2, Spyros Tserkis3, Jayne Thompson5, Ping Koy Lam3,4, and Mile Gu2,1,6

1Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543.
2Nanyang Quantum Hub, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 639673.
3Centre for Quantum Computation and Communication Technology, Department of Quantum Science, Research School of Physics and Engineering, Australian National University, Canberra ACT, Australia 2601.
4A*STAR Quantum Innovation Centre (Q.InC), Institute of Materials Research and Engineering (IMRE), Agency for Science Technology and Research (A*STAR), 2 Fusionopolis Way, 08-03 Innovis, Singapore 138634.
5A*STAR Quantum Innovation Centre (Q.InC), Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), Singapore.
6CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, Singapore 117543.

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The amount of information that a noisy channel can transmit has been one of the primary subjects of interest in information theory. In this work we consider a practically-motivated family of optical quantum channels that can be implemented without an external energy source. We optimize the Holevo information over procedures that encode information in attenuations and phase-shifts applied by these channels on a resource state of finite energy. It is shown that for any given input state and environment temperature, the maximum Holevo information can be achieved by an encoding procedure that uniformly distributes the channel's phase-shift parameter. Moreover for large families of input states, any maximizing encoding scheme has a finite number of channel attenuation values, simplifying the codewords to a finite number of rings around the origin in the output phase space. The above results and numerical evidence suggests that this property holds for all resource states. Our results are directly applicable to the quantum reading of an optical memory in the presence of environmental thermal noise.

Optical systems have been used ubiquitously in information processing tasks such as information storage (e.g. optical storage drives) and transmission (e.g. fiber optics). With the growing interest in extending these tasks to the quantum regime, it is of great importance to investigate how one can encode the largest amount of classical information possible in optical quantum systems, especially in the presence of inevitable environmental noise. In this work, we characterize the best procedure that one can do to encode the largest amount of information into an optical quantum system that goes through a channel in a noisy thermal environment by modulating its parameters. We consider a family of linear optical channels with no internal source of energy, hence restricts energy source to the state of input quantum systems, and have found the best encoding procedure for many given resource quantum states with a given energy and a given environment temperature.

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