Permutationally Invariant Quantum Tomography

Abstract

We present a scalable method for the tomography of large multiqubit quantum registers. It acquires information about the permutationally invariant part of the density operator, which is a good approximation to the true state in many relevant cases. Our method gives the best measurement strategy to minimize the experimental effort as well as the uncertainties of the reconstructed density matrix. We apply our method to the experimental tomography of a photonic four-qubit symmetric Dicke state.

Genuine multiparticle entanglement of permutationally invariant states

“…This offers the possibility that one can tailor or optimize the analysis tool for those more restricted sets. Such more efficient tomography protocols have been recently designed for generic states of low rank [6], particularly important low rank states like matrix product states [7] or multi-scale entanglement renormalization ansatz states [8], or for states which possess some further symmetry like permutation invariance [9].…”

“…Thus, in combination with the tomography protocol [9] (and its variants [21][22][23]) and its efficient state reconstruction algorithm [24], we develop an additional tool to analyze the data after such a quantum state tomography process. At this point, we would like to stress that the derived detection method does not rely on the fact that the underlying state indeed possesses this symmetry: If the permutationally invariant part of a quantum state is entangled, then the complete state must be entangled, too [9]. As a further result we prove that the criterion of PPT mixtures is necessary and sufficient to decide whether a given permutationally invariant three-qubit states is genuine multipartite entangled or not.…”

Deterministic generation of arbitrary symmetric states and entanglement classes

“…In order to detect the final state |Ψ 4 S after the protocol, one may apply recently developed techniques for symmetric state tomography [38]. The advantage of these techniques is that the number of observables one has to measure to perform the full tomography grows only poly-nomially in the number N of qubits.…”

Practical implementation of mutually unbiased bases using quantum circuits

“…This includes states with low rank [11][12][13], with special emphasis in some relevant cases as matrix product (MPS) [14,15] ,or multiscale entangled renormalization ansatz (MERA) states [16]. The specific but pertinent example of permutationally invariant qubits has been also examined [17][18][19][20], as they are of great import in diverse quantum information strategies [21][22][23][24][25][26][27].…”