Strong connections between quantum encodings, non-locality and quantum cryptography

Encoding information in quantum systems can offer surprising advantages but at the same time there are limitations that arise from the fact that measuring an observable may disturb the state of the quantum system. In our work, we provide an in-depth analysis of a simple question: What hap-pens when we perform two measurements sequentially on the same quantum system? This question touches upon some fundamental properties of quantum mechanics, namely the uncertainty principle and the complementarity of quantum measurements. Our results have interesting consequences, for example they can provide a simple proof of the optimal quantum strategy in the famous Clauser-Horne-Shimony-Holt game. Moreover, we show that the way information is encoded in quantum systems can provide a different perspective in understanding other fundamental aspects of quan-tum information, like non-locality and quantum cryptography. We prove some strong equivalences between these notions and provide a number of applications in all areas. Quantum information studies how information is en-coded in quantum systems and how it can be observed through measurements. On one hand, the exponential number of amplitudes that describe the state of a quan-tum system can be used in order to encode a vast amount of classical information into the state of a quantum sys-tem. Hence, we can use quantum information to resolve many distributed tasks much more efficiently than with classical information [1–3]. On the other hand, quantum information does not always offer advantages, since ev-ery time an observer measures a quantum system its state may collapse and information may become irretrievable. For example, Holevo's theorem [4], asserts that one quan-tum bit can be used to transmit only one bit of classical information and no more. The intricate interplay between encoding information in quantum systems and measurement interference is at the heart of some fundamental results in quantum infor-mation, from Bell inequalities [5] to quantum key distri-bution [6]. Our goal is to deepen our understanding of the connections between quantum encodings, non-locality, and quantum cryptography and provide new insight on the power and limitations of quantum information, by looking at it through these various lenses. This paper links three seemingly unrelated concepts in quantum information (encodings, non-local games, and cryptographic primitives) via properties of sequen-tial non-commuting measurements. The technical part of this paper examines quantum encodings and bounds the success of sequentially measuring an encoding of two bits (or strings) to learn their XOR. We then show how these bounds can be used to study not only encodings, but non-local games and cryptographic tasks as well. The conceptual part of this paper discusses how the applica-tions we consider are all equivalent in some sense. When viewing each as extracting information from a quantum encoding, we are able to preserve the three notions: (1) hiding the XOR in the encoding, (2) providing perfect security in the cryptographic task, and (3) satisfying the non-signaling principle in the non-local game. In addition to providing philosophical insights towards each of these quantum tasks, we combine the technical and conceptual tools in this paper to give applications in all areas.