Estimating localizable entanglement from witnesses

Abstract

Computing localizable entanglement for noisy many-particle quantum states is difficult due to the optimization over all possible sets of local projection measurements. Therefore, it is crucial to develop lower bounds, which can provide useful information about the behaviour of localizable entanglement, and which can be determined by measuring a limited number of operators, or by performing the least number of measurements on the state, preferably without performing a full state tomography. In this paper, we adopt two different yet related approaches to obtain a witness-based, and a measurementbased lower bounds for localizable entanglement. The former is determined by the minimal amount of entanglement that can be present in a subsystem of the multipartite quantum state, which is consistent with the expectation value of an entanglement witness. Determining this bound does not require any information about the state beyond the expectation value of the witness operator, which renders this approach highly practical in experiments. The latter bound of localizable entanglement is computed by restricting the local projection measurements over the qubits outside the subsystem of interest to a suitably chosen basis. We discuss the behaviour of both lower bounds against local physical noise on the qubits, and discuss their dependence on noise strength and system size. We also analytically determine the measurement-based lower bound in the case of graph states under local uncorrelated Pauli noise.

Scalable characterization of localizable entanglement in noisy topological quantum codes

“…Localizable entanglement (LE) [81][82][83]92] over a region Ω composed of the N −m qubits in a N -qubit state ρ is the maximum average entanglement that can be localized over Ω, by performing local projection measurements on the m qubits in Ω. Without any loss in generality, we assign the first m qubits in the set of N qubits labelled as 1, 2, · · · , m, m + 1, · · · , N to the region Ω and the rest in Ω, with Ω ∪ Ω representing the complete N -partite system with Ω ∩ Ω = ∅, and 1 ≤ m ≤ N − 2.…”

“…In order to extract useful information about the properties of LE in situations where computing the exact value of the quantity proves difficult, a possible approach is to determine computable lower bounds of LE. In [92], two avenues for constructing such lower bounds of the LE have been discussed -(1) a measurement-based lower bound (MLB) by performing a specific set of local Pauli measurements on the state itself, or on a state connected to the original state by local unitary transformation, and (2) an experimentally accessible entanglement witness-based lower bound (WLB) by using entanglement witness operators appropriate for the post-measurement states on the region Ω. We consider the state ρ to be a mixed one in general, originated from, for example, a stabilizer state ρ 0 due to application of noise ρ 0 → ρ = Λ(ρ 0 ), Λ(.)…”

“…On the other hand, a WLB of LE employs an entanglement witness operator [99][100][101][102][103][104][105][106], W, which indicates non-zero entanglement content in a quantum state ρ via a negative expectation value, i.e., Tr(Wρ) < 0. A lower bound of the entanglement content in a quantum state ρ can be determined using the set of expectation values of appropriately chosen entanglement witness operators as the solution of the optimization problem [92,[108][109][110][111][112] E min (ω) =inf E(ρ), subject to Tr (ρW) = ω, ρ ≥ 0,…”

“…Note that the analytical computation of E P ab (ρ ) depends also on the noise model Λ. As shown in [92], in the case of local uncorrelated Pauli noise, E P ab (ρ ) is analytically computable for arbitrary system size as long as the neighborhood of the region Ω ≡ ab in G and the noise present in this region is fully known.…”

Design and experimental performance of local entanglement witness operators

“…In this work we focus on local witness operators [70][71][72], i.e. witnesses that detect the entanglement among qubits inside subsets Ω of larger multi-qubit systems (see Fig.…”