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 and estimation of large entangled systems represent 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 post‐processing resources with system size makes this approach infeasible 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 selective quantum state tomography, which enables the estimation of arbitrary elements of an unknown state with a number of copies that is low and independent of the system's size. These hyper‐efficient techniques define a dimensional demarcation for partial tomography and open a path for novel applications.

Statistical Methods for Quantum State Verification and Fidelity Estimation

“…[62, 63]; see also the very recent review paper. [ 34 ] In their method, the verifier also perform a set of random pass or fail tests $$\lbrace \Omega _\ell ,\mathbb {1}-\Omega _\ell \rbrace$$, such that no separable state can pass the test with probability higher than $$q_s$$. Similar to Equation (22), if in actual experiments the frequency of pass instances f is larger than $$q_s$$, the verifier can conclude the state is entangled with the confidence $$1-\delta$$, where $$\begin{equation} \delta \le \mathrm{e}^{-D[f\Vert q_s]N} \end{equation}$$and $$D[\cdot \Vert \cdot ]$$ is the Kullback–Leibler divergence.…”

“…There are already excellent review articles on quantum state discrimination [ 32,33 ] and, very recently, on entanglement tests using witnesses from a statistical perspective. [ 34 ] Furthermore, we encourage the reader to consult the original literature on related topics, such as the estimation of pure or mixed quantum states, [ 35–38 ] the estimation of drift or change point detection, [ 39–41 ] sequential hypothesis testing, [ 42–44 ] the blind channel estimation, [ 45 ] and the estimation of quantum teleportation. [ 46,47 ] In addition, this review focuses on the problem of state verification in discrete‐variable quantum systems.…”

Few‐Copy Entanglement Detection in the Presence of Noise

“…copies of the quantum state through 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. assumption is not required constitutes a clear advantage.…”

“…Once W is defined, the so‐called “witness translation method” outlined in refs. [23, 25] can be used to derive $$\mathcal {M}$$ 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.…”

“…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.…”