Classical superdense coding and communication advantage of a single quantum Ram Krishna Patra, 1 Sahil Gopalkrishna Naik, 1 Edwin Peter Lobo, 1 Samrat Sen, 1 Tamal Guha, 2 Some Sankar Bhattacharya, 3 Mir Alimuddin, 1 and Manik Banik 1 1 School of Physics, IISER Thiruvananthapuram, Vithura, Kerala 695551, India. 2 Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong. 3 International Centre for Theory of Quantum Technologies, University of Gdansk, Wita Stwosza 63, 80-308 Gdansk, Poland. We analyze the communication utility of a single quantum system when the sender and receiver share neither any entanglement nor any classical shared randomness. To this aim, we propose a class of two-party communication games that cannot be won with a noiseless 1-bit classical channel, whereas the goal can be perfectly achieved if the channel is assisted by classical shared randomness. This resembles an advantage similar to the quantum superdense coding phenomenon where pre- shared entanglement can enhance communication utility of a perfect quantum communication line. Quite surprisingly, we show that a qubit communication without any assistance of classical shared randomness can achieve the goal, and hence establishes a novel quantum advantage in the simplest communication scenario. In pursuit of a deeper origin of this advantage we show that an advantageous quantum strategy must invoke quantum interference both at the encoding step by the sender and at the decoding step by the receiver. A subclass of these games can be won deterministically if the sender communicates some non-classical toy systems described by symmetric polygonal state spaces. We then proceed to design a variant of the game that can be won neither with 1-bit communication nor with a polygon system, but 1-qubit communication yields a perfect strategy, establishing a strict quantum nature of the advantage. To this end, we show that the quantum advantages are robust against imperfect encodings-decodings, making the protocols implementable with presently available quantum technologies. I. INTRODUCTION Present day digital era depends crucially on reli- able transfer of information among distant locations through several advanced means, such as cellular, in- ternet, and satellite communications. Foundation of modern communication theory was established in the seminal work of Claude E. Shannon, who modeled the physical devices to store and transfer information as classical objects [1]. Advent of quantum information the- ory [2], also recognized as ’the second quantum revolu- tion’ [3], identifies novel uses of non-classical features of quantum systems to devise exotic information and com- munication protocols that are advantageous over their classical counterparts and in some cases impossible with classical resources [4–10]. Over the last few decades, hands-on uses of quantum resources have been reported in a number of crucial experiments (see [11–14] and ref- erences therein), and with every passing day quantum technology is stepping out of mere academic interests to more practical uses [15–17]. These consequently mo- tivate search for more-and-more novel communication protocols where quantum resources exhibit advantage over their classical counterparts. The simplest communication scenario involves two distant parties – a sender (Alice) and a receiver (Bob) – where Alice aims to transmit some message to Bob by sending some physical systems. The pioneering ‘quantum superdense coding’ protocol establishes the first nontrivial quantum advantage in such a communic- ation scenario by showing that quantum entanglement, preshared between sender and receiver, can double the classical communication capacity of a perfect qubit chan- nel [4]. This is quite striking, 1 as, in such a scenario, quantum entanglement by its own has no communic- ation utility, and according to the fundamental no-go theorem of Holevo [18], the communication capacity of a perfect qubit channel alone is the same as the capacity of a perfect one bit classical channel. Recently, the no-go implication of Holevo has been strengthened further by Frenkel & Weiner while evaluating the communication utility of a quantum system in absence of preshared entanglement [20]. While in Holevo’s theorem, com- munication utility is measured in terms of the mutual information between the random variable the sender intends to send and the random variable the receiver ob- tains after the channel action, Frenkel & Weiner quantify a channel’s utility through a generic reward function rather than only mutual information and still establish that the communication utility of an n-level quantum system is the same as that of an n-level classical system. Contribution of the present work starts with the fol- lowing pivotal observation: while quantum entangle- ment and classical shared randomness both have zero communication utility on their own, the no-go results 1 See the illuminating remark by the reviewer of ‘quantum superdense coding’ paper [19]. arXiv:2202.06796v1 [quant-ph] 14 Feb 2022