Entanglement and the truncated moment problem

Bohnet-Waldraff, Fabian; Braun, Daniel; Giraud, Olivier

doi:10.1103/physreva.96.032312

Cited by 14 publications

(22 citation statements)

References 51 publications

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“…We now briefly summarize the TMS algorithm approach described in detail in [32], which will be the framework for the following sections. The basic idea is to map the quantum entanglement problem onto the mathematically well-studied truncated moment problem.…”

Section: Framework and Notationmentioning

confidence: 99%

“…The algorithm applies -at least in principleto arbitrary quantum states with arbitrary number of constituents and arbitrary symmetries between the subparts. The general case is dealt with in [32]; we only recall here the main key points for the case of symmetric states of qubits, defined as mixtures of symmetric pure states (the latter are invariant under any permutation of the qubits). To do so, we will use a convenient representation in terms of symmetric tensors which was introduced in [35], generalizing the Bloch sphere picture of spins−1/2.…”

Section: Framework and Notationmentioning

confidence: 99%

“…They can be solved by a semidefinite relaxation procedure. The algorithm proposed in [32] uses indeed semidefinite programming (SDP) and the concept of "extensions", already introduced in [36], but based on a matrix of moments and a theorem in the theory of moment sequences. In order to present more clearly the mathematical setting for the AK-TMS problem, we introduce a more compact notation for Eq.…”

Section: Framework and Notationmentioning

confidence: 99%

“…If, on the contrary, the SDP problem is "feasible", then the TMS admits a representing measure, and the corresponding quantum state is separable. The algorithm also extends to the case of non-symmetric states (see [32] for further detail).…”

Section: Framework and Notationmentioning

confidence: 99%

“…4). Perfectly suited for answering these questions is the formalism of "truncated moment sequences" (TMS) that we introduced in [32] to the analysis of entanglement. The TMS problem aims at finding a probability measure for which only some moments are known.…”

Section: Introductionmentioning

confidence: 99%

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Optimal measurement strategies for fast entanglement detection

Milazzo¹,

Braun²,

Giraud³

2019

Phys. Rev. A

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With the advance of quantum information technology, the question of how to most efficiently test quantum circuits is becoming of increasing relevance. Here we introduce the statistics of lengths of measurement sequences that allows one to certify entanglement across a given bi-partition of a multiqubit system over the possible sequence of measurements of random unknown states, and identify the best measurement strategies in the sense of the (on average) shortest measurement sequence of (multi-qubit) Pauli-measurements. The approach is based on the algorithm of truncated moment sequences that allows one to deal naturally with incomplete information, i.e. information that does not fully specify the quantum state. We find that the set of measurements corresponding to diagonal matrix elements of the moment matrix of the state are particularly efficient. For symmetric states their number grows only like the third power of the number N of qubits. Their efficiency grows rapidly with N , leaving already for N = 4 less than a fraction 10 −6 of randomly chosen entangled states undetected.

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Section: Framework and Notationmentioning

confidence: 99%

Section: Framework and Notationmentioning

confidence: 99%