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 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 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.

arXiv comment20 pages, 4 figures, review article, very close to the published version

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• Quantum networks self-test all entangled states
• Shadow tomography on general measurement frames
• Demonstration of Robust and Efficient Quantum Property Learning with Shallow Shadows
• Efficient verification of Affleck-Kennedy-Lieb-Tasaki states
• Efficient Verification of Ground States of Frustration-Free Hamiltonians
• Optimising quantum tomography via shadow inversion
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• A survey of universal quantum von Neumann architecture
• Holographic Classical Shadow Tomography
• Experimental Verification of Entangled States in the Adversarial Scenario
• Universal and Efficient Quantum State Verification via Schmidt Decomposition and Mutually Unbiased Bases