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arXiv:1604.02202 · Physical Review A

Measurable genuine tripartite entanglement of (2⊗2⊗n) -dimensional quantum states via only two simultaneous copies

Chang‐shui Yu, Bao-qing Guo, Si‐ren Yang

Published Apr 4, 2016• 2 citations
arXivHTMLPDF

Abstract

Usually, the three-tangle of a tripartite pure state of qubits can be directly measured with the simultaneous preparation of a not-less-than fourfold copy of the state. We show that the exact genuine tripartite entanglement for ($2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}n$)-dimensional pure quantum states can be measured in a similar manner, provided that only two simultaneous copies of the state are available. Lower bounds are also proposed for more convenient experimental operations. As an example, a comprehensive demonstration of the scheme is provided for the three-tangle of a three-qubit state.

DOI
10.1103/physreva.93.042304

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Usually, the three-tangle of a tripartite pure state of qubits can be directly measured with the simultaneous preparation of a not-less-than fourfold copy of the state. We show that the exact genuine tripartite entanglement for (2\ensuremath⨂2\ensuremath⨂n2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}n2\ensuremath⨂2\ensuremath⨂n)-dimensional pure quantum states can be measured in a similar manner, provided that only two simultaneous copies of the state are available. Lower bounds are also proposed for more convenient experimental operations. As an example, a comprehensive demonstration of the scheme is provided for the three-tangle of a three-qubit state.
In this paper, we present a new approach to study genuine tripartite entanglement existing in (2×2×n)−(2\times 2\times n)-(2×2×n)−dimensional quantum pure states. By utilizing the approach, we introduce a particular quantity to measure genuine tripartite entanglement. The quantity is shown to be an entanglement monotone in 2-dimensional subsystems (semi-monotone) and reaches zero for separable states and (2×2×2)−(2\times 2\times 2)-(2×2×2)−dimensional WWW states, hence is a good criterion to characterize genuine tripartite entanglement. Furthermore, the formulation for pure states can be conveniently extended to the case of mixed states by utilizing the kronecker product approximation technique. As applications, we give the analytic approximation for weakly mixed states, and study the genuine tripartite entanglement of two given weakly mixed states.
A genuine tripartite entanglement monotone is presented for $(2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}n)$-dimensional tripartite pure states, obtained by introducing an entanglement measure for bipartite pure states. As an application, we consider the genuine tripartite entanglement of the ground state of the exactly solvable isotropic spin-1/21/21/2 chain with three-spin interaction. It is shown that the singular behavior of the genuine tripartite entanglement exactly signals a quantum phase transition.
We present a quantum circuit that transforms an unknown three-qubit state into its canonical form, up to relative phases, given many copies of the original state. The circuit is made of three single-qubit parametrized quantum gates, and the optimal values for the parameters are learned in a variational fashion. Once this transformation is achieved, direct measurement of outcome probabilities in the computational basis provides an estimate of the tangle, which quantifies genuine tripartite entanglement. We perform simulations on a set of random states under different noise conditions to asses the validity of the method.
We show that for tripartite quantum pure states of qubits, all the kinds of entanglement in terms of stochastic local operations and classical communications classification are experimentally measurable by simple projective measurements, provided that four copies of the composite quantum system are available. In particular, the entanglement of reduced density matrices, even though they are mixed states, can be exactly determined in experiment. Concurrence of assistance is also shown to be measurable by introducing an interesting equation with explicit physical meanings.