Abstract
We propose an ordered set of experimentally accessible conditions for detecting entanglement in mixed states. The k-th condition involves comparing moments of the partially transposed density operator up to order k. Remarkably, the union of all moment inequalities reproduces the Peres-Horodecki criterion for detecting entanglement. Our empirical studies highlight that the first four conditions already detect mixed state entanglement reliably in a variety of quantum architectures. Exploiting symmetries can help to further improve their detection capabilities. We also show how to estimate moment inequalities based on local random measurements of single state copies (classical shadows) and derive statistically sound confidence intervals as a function of the number of performed measurements. Our analysis includes the experimentally relevant situation of drifting sources, i.e. non-identical, but independent, state copies.
Highlights
In the past years, a considerable effort led to the building of larger and larger Noisy Intermediate-Scale Quantum (NISQ) devices[1,2,3].For the benchmarking of such devices comes the need for more scalable tools in order to characterize the underlying many-body quantum state
From the numerous theoretical sufficient conditions for entanglement that have been developed in the literature, many cannot be straightforwardly implemented experimentally, mainly because they require the knowledge of the full density matrix[8,9,10]. This is for instance the case of the celebrated PPT condition[5,11], which states that a separable state ρ always has a positive semi-definite (PSD) partial transpose (PT) ρΓ for any bipartite splitting of its subsystems
We propose different entanglement detection strategies depending on how many PT moments can be estimated
Summary
A considerable effort led to the building of larger and larger Noisy Intermediate-Scale Quantum (NISQ) devices[1,2,3]. From the numerous theoretical sufficient conditions for entanglement that have been developed in the literature, many cannot be straightforwardly implemented experimentally, mainly because they require the (exponentially expensive) knowledge of the full density matrix[8,9,10] This is for instance the case of the celebrated PPT condition[5,11], which states that a separable state ρ always has a positive semi-definite (PSD) partial transpose (PT) ρΓ for any bipartite splitting of its subsystems. The negativity, which resulted from this condition, is a highly used entanglement measure for mixed states[12,13,14] This powerful entanglement condition, which found many applications in theoretical works[15,16,17,18,19,20,21,22], is difficult to apply in experimental conditions as the PT spectrum is difficult to access. We show how source drifts in an experiment can be taken into account and how the quantities which are of interest here can be accurately estimated via local measurements on single copies of the state
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