Properties of dimension witnesses and their semi-definite programming relaxations

Abstract

In this paper we develop a method for investigating semi-device-independent randomness expansion protocols that was introduced in [Li et al., Phys. Rev. A $87$, 020302(R) (2013)]. This method allows to lower-bound, with semi-definite programming, the randomness obtained from random number generators based on dimension witnesses. We also investigate the robustness of some randomness expanders using this method. We show the role of an assumption about the trace of the measurement operators and a way to avoid it. The method is also generalized to systems of arbitrary dimension, and for a more general form of dimension witnesses, than it the previous paper. Finally, we introduce a procedure of dimension witness reduction, which can be used to obtain from an existing witness a new one with higher amount of certifiable randomness. The presented methods finds an application for experiments [Ahrens et al., Phys. Rev. Lett. $112$, 140401 (2014)].

arXiv comment14 pages, 5 figures

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• The Magical Number Seven: An Unexpected Dimensional Threshold in Quantum Communication Complexity
• Quantum randomness protected against detection loophole attacks
• Detection efficiency and noise in semi-device independent randomness extraction protocol
• Device and semi-device independent random numbers based on non-inequality paradox
• Bounds on semi-device-independent quantum random number expansion capabilities
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• Certification of three black boxes with unsharp measurements using $3\to 1$ sequential quantum random access codes
• Generalized measurements on qubits in quantum randomness certification and expansion
• Semi-device-independent randomness certification on discretized continuous-variable platforms