# Quantum Advantage for Shared Randomness Generation

Tamal Guha1, Mir Alimuddin2, Sumit Rout3, Amit Mukherjee4, Some Sankar Bhattacharya5, and Manik Banik2

1Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700108, India.
2School of Physics, IISER Thiruvanathapuram, Vithura, Kerala 695551, India.
3International Centre for Theory of Quantum Technologies (ICTQT), University of Gdańsk, 80-308 Gdańsk, Poland.
4S.N. Bose National Center for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700098, India.
5Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong.

### Abstract

Sharing correlated random variables is a resource for a number of information theoretic tasks such as privacy amplification, simultaneous message passing, secret sharing and many more. In this article, we show that to establish such a resource called shared randomness, quantum systems provide an advantage over their classical counterpart. Precisely, we show that appropriate albeit fixed measurements on a shared two-qubit state can generate correlations which cannot be obtained from any possible state on two classical bits. In a resource theoretic set-up, this feature of quantum systems can be interpreted as an advantage in winning a two players co-operative game, which we call the non-monopolize social subsidy' game. It turns out that the quantum states leading to the desired advantage must possess non-classicality in the form of quantum discord. On the other hand, while distributing such sources of shared randomness between two parties via noisy channels, quantum channels with zero capacity as well as with classical capacity strictly less than unity perform more efficiently than the perfect classical channel. Protocols presented here are noise-robust and hence should be realizable with state-of-the-art quantum devices.

A classical coin can simulate the input-output statistics obtained from a quoin – a two level quantum system. However, in their multiple temporal occurrences, such as generation of Bernoulli factory, a quoin exhibits advantage over its classical counterpart. Present work depicts such an advantage for their multiple spatial occurrence i.e. for a shared two quoin. The non-monopolizing social subsidy game, involving two distant non-communicating players, quantifies such an advantage. Quite counterintuitively, a noisy quantum transmission line turns out to be more effective than its perfect classical counterpart when used to distribute such a two level system. The present work is a potential candidate to understand the resource content of shared randomness.

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