Quantum Probability Estimation for Randomness with Quantum Side Information

Abstract

We develop a quantum version of the probability estimation framework [arXiv:1709.06159] for randomness generation with quantum side information. We show that most of the properties of probability estimation hold for quantum probability estimation (QPE). This includes asymptotic optimality at constant error and randomness expansion with logarithmic input entropy. QPE is implemented by constructing model-dependent quantum estimation factors (QEFs), which yield statistical confidence upper bounds on data-conditional normalized Rényi powers. This leads to conditional min-entropy estimates for randomness generation. The bounds are valid for relevant models of sequences of experimental trials without requiring independent and identical or stationary behavior. QEFs may be adapted to changing conditions during the sequence and trials can be stopped any time, such as when the results so far are satisfactory. QEFs can be constructed from entropy estimators to improve the bounds for conditional min-entropy of classical-quantum states from the entropy accumulation framework [Dupuis, Fawzi and Renner, arXiv:1607.01796]. QEFs are applicable to a larger class of models, including models permitting measurement devices with super-quantum but non-signaling behaviors and semi-device dependent models. The improved bounds are relevant for finite data or error bounds of the form e −κs , where s is the number of random bits produced. We give a general construction of entropy estimators based on maximum probability estimators, which exist for many configurations. For the class of (k, 2, 2) Bell-test configurations we provide schemas for directly optimizing QEFs to overcome the limitations of entropy-estimator-based constructions. We obtain and apply QEFs for examples involving the (2, 2, 2) Bell-test configuration to demonstrate substantial improvements in finite-data efficiency.

Device-independent lower bounds on the conditional von Neumann entropy

“…B Proof of Lemma 2.7 concerning the Sylvester equation (11) We use a result of [39] to show the existence of a bounded operator Z satisfying (11). Given the positive operators ρ and σ on H, we construct the operators on the direct sum H ⊕ H: X = (1 − t)ρ 0 0 tσ and K = 1 2 ρ ρ ρ λσ .…”

“…In order to compute the rates of finite round DI-RE or DI-QKD protocols, one can bound the total conditional smooth min-entropy H ǫ min (AB|XYE) accumulated by the devices during the protocol [57]. There are two tools, the entropy accumulation theorem [9,10] and quantum probability estimation [11], that allow one to break down this total smooth min-entropy into smaller, easier to compute quantities. For instance, the entropy accumulation theorem roughly states that in an n-round protocol…”

“…Beginning with the works of [7,8], significant effort has been placed into developing new device-independent protocols and subsequently proving their security. For tasks like DI-RE and DI-QKD, security is now well understood, with tools like the entropy accumulation theorem (EAT) [9,10] and quantum probability estimation (QPE) [11] enabling relatively simple proofs of security against all-powerful quantum adversaries [12,13]. In both cases a security proof can be readily established once one has developed a quantitative relationship between the nonlocal correlations observed in the protocol and the quantity of randomness generated by the devices used.…”

Experimental Realization of Device-Independent Quantum Randomness Expansion

“…Device-independent quantum randomness expansion provides a randomness resource of the highest security level [1,2]. Here we report the first experimental realization of device-independent quantum randomness expansion secure against quantum side information established through quantum probability estimation [3]. We generate 5.47 × 10 8 quantum-proof random bits while consuming 4.39 × 10 8 bits of entropy, expanding our store of randomness by 1.08 × 10 8 bits at a latency of about 12.5 hours, with a total soundness error 4.6 × 10 −10 .…”

“…One is the development of cutting-edge single-photon detection with near unity efficiency [18], which makes entangled photons-based loophole-free Bell test experiments viable. The other is the development of theoretical protocols [3,4,[19][20][21][22][23], which allows for the efficient generation of randomness secure against quantum side information in device-independent experiments, such as the quantum probability estimation (QPE) method [3]. Below we briefly describe the spot-checking QPE method and our procedure to apply it to realize DIQRE.…”