Random Numbers Certified by Bell's Theorem

Randomness is a fundamental feature in nature and a valuable resource for applications ranging from cryptography and gambling to numerical simulation of physical and biological systems. Random numbers, however, are difficult to characterize mathematically, and their generation must rely on an unpredictable physical process. Inaccuracies in the theoretical modelling of such processes or failures of the devices, possibly due to adversarial attacks, limit the reliability of random number generators in ways that are difficult to control and detect. Here, inspired by earlier work on nonlocality based and device independent quantum information processing, we show that the nonlocal correlations of entangled quantum particles can be used to certify the presence of genuine randomness. It is thereby possible to design of a new type of cryptographically secure random number generator which does not require any assumption on the internal working of the devices. This strong form of randomness generation is impossible classically and possible in quantum systems only if certified by a Bell inequality violation. We carry out a proof-of-concept demonstration of this proposal in a system of two entangled atoms separated by approximately 1 meter. The observed Bell inequality violation, featuring near-perfect detection efficiency, guarantees that 42 new random numbers are generated with 99% confidence. Our results lay the groundwork for future device-independent quantum information experiments and for addressing fundamental issues raised by the intrinsic randomness of quantum theory.Comment: 10 pages, 3 figures, 16 page appendix. Version as close as possible to the published version following the terms of the journa

Electric-field-induced coherent coupling of the exciton states in a single quantum dot

The signature of coherent coupling between two quantum states is an anticrossing in their energies as one is swept through the other. In single semiconductor quantum dots containing an electron-hole pair the eigenstates form a two-level system that can be used to demonstrate quantum effects in the solid state, but in all previous work these states were independent. Here we describe a technique to control the energetic splitting of these states using a vertical electric field, facilitating the observation of coherent coupling between them. Near the minimum splitting the eigenstates rotate in the plane of the sample, being orientated at 45{\deg} when the splitting is smallest. Using this system we show direct control over the exciton states in one quantum dot, leading to the generation of entangled photon pairs

Quantum memories: A review based on the European integrated project "Qubit Applications (QAP)”

We perform a review of various approaches to the implementation of quantum memories, with an emphasis on activities within the quantum memory sub-project of the EU integrated project "Qubit Applications”. We begin with a brief overview over different applications for quantum memories and different types of quantum memories. We discuss the most important criteria for assessing quantum memory performance and the most important physical requirements. Then we review the different approaches represented in "Qubit Applications” in some detail. They include solid-state atomic ensembles, NV centers, quantum dots, single atoms, atomic gases and optical phonons in diamond. We compare the different approaches using the discussed criteri

Electrical control of the exciton fine structure of a quantum dot molecule

We electrically control the coherent mixing of the optically bright spin states of excitons confined in InAs/GaAs quantum dot molecules. By tunnel coupling two quantum dots, using a vertical electric field, the exciton fine structure splitting and eigenstate orientation relative to the crystal lattice are tuned. We model the electric field dependent anisotropic electron-hole exchange interaction accurately and propose that the controllable mixing of the spin states will enable electrically controlled quantum operations on exciton spin qubits. © 2013 American Physical Society

All-electrical coherent control of the exciton states in a single quantum dot

We demonstrate high-fidelity reversible transfer of quantum information from the polarization of photons into the spin state of an electron-hole pair in a semiconductor quantum dot. Moreover, spins are electrically manipulated on a subnanosecond time scale, allowing us to coherently control their evolution. By varying the area of the electrical pulse, we demonstrate phase-shift and spin-flip gate operations with near-unity fidelities. Our system constitutes a controllable quantum interface between flying and stationary qubits, an enabling technology for quantum logic in the solid state. © 2010 The American Physical Society

LETTERS Random numbers certified by Bell's theorem

Randomness is a fundamental feature of nature and a valuable resource for applications ranging from cryptography and gambling to numerical simulation of physical and biological systems. Random numbers, however, are difficult to characterize mathematically 1 , and their generation must rely on an unpredictable physical process The characterization of true randomness is elusive. There exist statistical tests used to verify the absence of certain patterns in a stream of numbers These considerations are of direct relevance to applications of randomness, and in particular cryptographic applications. Imperfections in random number generators Here we establish a fundamental link between the violation of Bell inequalities and the unpredictable character of the outcomes of quantum measurements and show, as originally proposed in ref. 14, that the non-local correlations of quantum states can be used to generate certified private randomness. The violation of a Bell inequality 15 guarantees that the observed outputs are not predetermined and that they arise from entangled quantum systems that possess intrinsic randomness. For simplicity, we consider the Clauser-HornShimony-Holt (CHSH) form of Bell inequality 19 , but our approach is general and applies to any Bell inequality. We thus consider two separate systems that can each be measured in two different ways, with a measurement on each system resulting in one of two values where P(a 5 bjxy) is the probability that a 5 b when settings (x, y) are chosen, and P(a ? bjxy) is defined analogously. Systems that admit a local, hence deterministic 20 , description satisfy I # 2. Certain measurements performed on entangled states, however, can violate this inequality. In order to estimate the Bell violation, the experiment is repeated n times in succession. The measurement choices (x, y) for each trial are generated by an identical and independent probability distribution P(xy). We denote the final output string after the n runs r 5 (a 1 , b 1 ; … ; a n , b n ) and the input string s 5 (x 1 , y 1 ; … ; x n , y n ). An estimatorÎ of the CHSH expression, equation where N(a 5 b, xy) is the number of times that the measurements x, y were performed and that the outcomes a and b were found equal after n realizations, and where N(a ? b, xy) is defined analogously. The randomness of the output string r can be quantified by the min-entropy 21 H ' (RjS) 5 2log 2 [max r P(rjs)], where P(rjs) is the conditional probability of obtaining the outcomes r when the measurements s are made and the maximum is taken over all possible values of the output string r. We show (Supplementary Information A) that the min-entropy of the outputs r is bounded by *These authors contributed equally to this work