Certified randomness from a two-level system in a relativistic quantum field

Abstract

Randomness is an indispensable resource in modern science and information technology. Fortunately, an experimentally simple procedure exists to generate randomness with well-characterized devices: measuring a quantum system in a basis complementary to its preparation. Towards realizing this goal one may consider using atoms or superconducting qubits, promising candidates for quantum information processing. However, their unavoidable interaction with the electromagnetic field affects their dynamics. At large time scales, this can result in decoherence. Smaller time scales in principle avoid this problem, but may not be well analysed under the usual rotating wave and single-mode approximation (RWA and SMA) which break the relativistic nature of quantum field theory. Here, we use a fully relativistic analysis to quantify the information that an adversary with access to the field could get on the result of an atomic measurement. Surprisingly, we find that the adversary's guessing probability is not minimized for atoms initially prepared in the ground state (an intuition derived from the RWA and SMA model).1 By relativistic, we mean here that the detector is locally coupled to a Lorentz covariant field. This excludes any possibility of superluminal signalling (present within the SMA and RWA) [10] and guarantees a proper description of high frequency modes relevant at short times.arXiv:1601.06166v1 [quant-ph]

Light, matter, and quantum randomness generation: A relativistic quantum information perspective

“…In between the preparation of the atom and the projective measurement, the atom interacts with the electromagnetic field which will generally correlate both, giving an adversary with access to the field means to make an educated guess on the result of the measurement. Contrary to intuition, the acquisition of correlations between the field and the state of the atom can happen even if the time between preparation and projection is small, and even if both atom and field start in the ground state [7,8]. These correlations serve as a bias which can be exploited by an adversary who has access to the field to infer the measurement outcome better than just by chance.…”

“…It is possible to temporally vary the coupling strength between a superconducting qubit and the electromagnetic field inside a microwave cavity. In that way one can design a range of switching functions [8,13]. For our purposes, we expand the electric field operator into plane-wave modes of momentum k and polarization s, with their respective creation and annihilation operatorsâ † k,s andâ k,s , satisfying the canonical equal time commutation relations.…”

“…The conditional min-entropy [8,14,15] will be used to quantify a lower bound on the extracted randomness by an adversary with access to the quantum field after the initial measurement and is defined as…”

“…However, this intuition, and the calculation that backs it up, are not revealing the full story: If the coupling strength between the atom and the field is strong enough (strong-coupling in QO [3][4][5] or ultra-strong coupling in superconducting circuits [6]), or if the time between preparation and measurement is short enough, the most common approximations in QO (that happen to violate the local covariance of the interaction and thus render QO non-relativistic [7]), such as the single mode approximation, break down. An idealized model on a fully relativistic footing has been analyzed already for the case of a massless scalar field in 1+1 dimensions [8]. There, a simplified atom is modeled as an Unruh-DeWitt detector [9,10].…”

“…Moreover it turned out that a superposition of ground and excited state is the optimal state in maximizing the randomness extracted. It is important to clarify how our work is distinguished from previous work on the same issue [8]. Namely, in this study we go beyond a simple scalar 1+1 dimensional field-atom toy model employed in past literature, and we consider a fully-featured hydrogen-like atom interacting with a quantum electromagnetic field via a dipole coupling in 3+1 dimensions, with the appropriate physical values for all the fundamental constants in the problem.…”