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arXiv:quant-ph/0302089 · Journal of Optics B Quantum and Semiclassical Optics

A tomographic approach to quantum nonlocality

Stefano Mancini, Vladimir I. Man’ko, Evgeny V Shchukin, Paolo Tombesi

Published Jun 1, 2003• 16 citations
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Abstract

We propose a tomographic approach to study quantum nonlocality in continuous variable quantum systems. On one hand we derive a Bell-like inequality for measured tomograms. On the other hand, we introduce pseudospin operators whose statistics can be inferred from the data characterizing the reconstructed state, thus giving the possibility to use standard Bell's inequalities. Illuminating examples are also discussed.

DOI
10.1088/1464-4266/5/3/366

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We propose a tomographic approach to study quantum nonlocality in continuous variable quantum systems. On one hand we derive a Bell-like inequality for measured tomograms. On the other hand, we introduce pseudospin operators whose statistics can be inferred from the data characterizing the reconstructed state, thus giving the possibility to use standard Bell's inequalities. Illuminating examples are also discussed.
There have been theoretical and experimental studies on quantum nonlocality for continuous variables, based on dichotomic observables. In particular, we are interested in two cases of dichotomic observables for the light field of continuous variables: One case is even and odd numbers of photons and the other case is no photon and the presence of photons. We analyze various observables to give the maximum violation of Bell's inequalities for continuous-variable states. We discuss an observable which gives the violation of Bell's inequality for any entangled pure continuous-variable state. However, it does not have to be a maximally entangled state to give the maximal violation of Bell's inequality. This is attributed to a generic problem of testing the quantum nonlocality of an infinite-dimensional state using a dichotomic observable.
In view of the tomographic-probability representation of quantum states, we reconsider the approach to quantumness tests of a single system developed by Alicki and Van Ryn (2008 J. Phys. A: Math. Theor. 41 062001). For qubits we introduce a general family of quantumness witnesses, which are operators depending on an extra parameter. Spin tomogram and dual-spin tomographic symbols are used to study qubit examples and the test inequalities which are shown to satisfy simple relations within the framework of the standard probability theory.
Tomographic measurements are the standard tool for characterizing quantum states, yet they are usually regarded only as means for state reconstruction or fidelity measurement. Here, we show that the same Pauli-basis measurements (X, Y, Z) can be directly employed for the certification of nonlocality at no additional experimental cost. Our framework allows any tomographic data - including archival datasets -- to be reinterpreted in terms of fundamental nonlocality tests. We introduce a generic, constructive method to generate tailored Bell inequalities and showcase their applicability to certify the non-locality of states in realistic experimental scenarios. Recognizing the stabilizer nature of the considered operators, we analyze our inequalities in the context of witnessing quantum magic - a crucial resource for quantum computing. Our approach requires Pauli measurements only and tests the quantum magic solely through the resources present in the state. Our results establish a universal standard that unifies state tomography with nonlocality certification and its application to quantum magic witnessing, thereby streamlining both fundamental studies and practical applications.