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Noiseless circuits for the concurrence and residual 3-tangle

Hilary A. Carteret

Published Sep 29, 2003• 0 citations
OpenAlex

Abstract

In this paper we describe a set of circuits that can measure the concurrence of a two qubit density matrix without requiring the addition of noise. We then show how this method can be extended to measure the 3-tangle for pure states of three qubits.

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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.
Quantum entanglement is the fundamental hallmark of quantum mechanics and a core resource for realizing long-distance quantum communication and scalable linear quantum computing. Accordingly, the precise detection and quantitative quantification of entanglement constitute a foundational and critical problem in quantum information theory. To date, researchers have proposed numerous sufficient conditions for entanglement detection as well as a variety of entanglement measures to characterize the entanglement strength of quantum states; nevertheless, efficient and direct measurement schemes for core entanglement parameters remain underdeveloped. Based on unitary transformations and auxiliary measurements, this paper proposes a set of quantum circuit schemes capable of directly measuring the bipartite concurrence and the tripartite 3-tangle entanglement measure. By introducing auxiliary qubits and constructing specific controlled unitary operations, the proposed scheme maps the analytical expressions of the two entanglement measures onto the measurement probabilities of output states from quantum circuits. It enables efficient and direct quantitative measurement of bipartite and tripartite entanglement without performing full quantum state tomography. This work provides a feasible technical route for the experimental characterization of entanglement properties and lays a groundwork for the practical deployment of multipartite entanglement resources in quantum information processing.
We present a new tripartite entanglement measure for three-qubit mixed states. The new measure tr(ρ)t_{\mathrm{r}}(ρ)tr​(ρ), which we refer to as the r-tangle, is given as a kind of the tangle, but has a feature which the tangle does not have; if we can derive an analytical form of $ t_{\mathrm{r}}(ρ)$ for a three-qubit mixed state ρρρ, we can also derive tr(ρ′)t_{\mathrm{r}}(ρ')tr​(ρ′) analytically for any states ρ′ρ'ρ′ which are SLOCC-equivalent to the state ρρρ. The concurrence of two-qubit states also satisfies the feature, but the tangle does not. These facts imply that the r-tangle trt_{\mathrm{r}}tr​ is the appropriate three-partite counterpart of the concurrence. We also derive an analytical form of the r-tangle trt_{\mathrm{r}}tr​ for mixtures of a generalized GHZ state and a generalized W state, and hence for all states which are SLOCC-equivalent to them.