Quantum verification and estimation with few copies

Abstract

As quantum technologies advance, the ability to generate increasingly large quantum states has experienced rapid development. In this context, the verification of large entangled systems represents one of the main challenges in the employment of such systems for reliable quantum information processing. Though the most complete technique is undoubtedly full tomography, the inherent exponential increase of experimental and postprocessing resources with system size makes this approach unfeasible even at moderate scales. For this reason, there is currently an urgent need to develop novel methods that surpass these limitations. This review article presents novel techniques focusing on a fixed number of resources (sampling complexity), and thus prove suitable for systems of arbitrary dimension. Specifically, a probabilistic framework requiring at best only a single copy for entanglement detection is reviewed, together with the concept of shadow and selective quantum state tomography, which enables the estimation of arbitrary elements of an unknown state with a number of copies that is independent of the system's size. These hyper-efficient techniques define a dimensional demarcation for partial tomography and open a path for novel applications.

A perspective on few-copy entanglement detection in experiments

“…Importantly, unlike standard approaches, this few-copy method also bypasses the need for independent and identically distributed (i.i.d.) copies of the quantum state through to the use of random sampling of measurement operators [24,25]. As the generation of identical copies in each single experimental run is always challenging from a practical perspective, the fact that the i.i.d.…”

“…A general and formal method to extract these quantities is given in Refs. [23,25], which also contain many details and examples. In short, all is needed to apply this procedure is the definition of any so-called entanglement witness operator W tailored to the quantum state one wishes to probe.…”

“…Once W is defined, the so-called "witness translation method" outlined in Refs. [23,25] can be used to derive M and p s . Constructing such witnesses is nowadays a pretty standard technique, and in general many witnesses have been already derived for many different quantum states [22,[26][27][28].…”

Statistical Methods for Quantum State Verification and Fidelity Estimation

“…As seen from Eqs. (14,22), the performance of a verification strategy depends solely on ν(Ω). Therefore, to achieve an optimal strategy, we need to maximize ν(Ω) over all accessible measurements.…”

“…We are aware of the fact that statistical tools have found widespread applications in quantum information processing and our article can cover only a small aspect of that. There are already excellent review articles on quantum state discrimination [20,21] and, very recently, on entanglement tests using witnesses from a statistical perspective [22]. Furthermore, we encourage the reader to consult the original literature on related topics, such as the estimation of pure or mixed quantum states [23][24][25][26], the estimation of drift or change point detection [27][28][29], and sequential hypothesis testing [30][31][32].…”