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arXiv:1702.07295 · Physical Review A

Symmetric three-qubit-state invariants

Alexander Meill, DAVID A. MEYER

Published Dec 8, 2017• 8 citations
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Abstract

For pure symmetric three-qubit states there are only three algebraically independent entanglement measures; one choice is the pairwise concurrence $\mathcal{C}$, the three-tangle $\ensuremath{\tau}$, and the Kempe invariant $\ensuremath{\kappa}$. Using a canonical form for symmetric $n$-qubit states derived from their Majorana representation, we derive the explicit achievable region of triples $(\mathcal{C},\ensuremath{\tau},\ensuremath{\kappa})$.

DOI
10.1103/physreva.96.062310

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For pure symmetric three-qubit states there are only three algebraically independent entanglement measures; one choice is the pairwise concurrence C\mathcal{C}C, the three-tangle \ensuremathτ\ensuremath{\tau}\ensuremathτ, and the Kempe invariant \ensuremathκ\ensuremath{\kappa}\ensuremathκ. Using a canonical form for symmetric nnn-qubit states derived from their Majorana representation, we derive the explicit achievable region of triples (C,\ensuremathτ,\ensuremathκ)(\mathcal{C},\ensuremath{\tau},\ensuremath{\kappa})(C,\ensuremathτ,\ensuremathκ).
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.
Quantum states that are symmetric with respect to permutations of their subsystems appear in a wide range of physical settings, and they have a variety of promising applications in quantum information science. In this thesis, the entanglement of symmetric multipartite states is categorized, with a particular focus on the pure multi-qubit case and the geometric measure of entanglement. An essential tool for this analysis is the Majorana representation, a generalization of the single-qubit Bloch sphere representation, which allows for a unique representation of symmetric n-qubit states by n points on the surface of a sphere. Here this representation is employed to search for the maximally entangled symmetric states of up to 12 qubits in terms of the geometric measure, and an intuitive visual understanding of the upper bound on the maximal symmetric entanglement is given. Furthermore, it will be seen that the Majorana representation facilitates the characterization of entanglement equivalence classes such as stochastic local operations and classical communication (SLOCC) and the degeneracy configuration (DC). It is found that SLOCC operations between symmetric states can be described by the Möbius transformations of complex analysis, which allows for a clear visualization of the SLOCC freedoms and facilitates the understanding of SLOCC invariants and equivalence classes. In particular, explicit forms of representative states for all symmetric SLOCC classes of up to five qubits are derived. Well-known entanglement classification schemes such as the four qubit entanglement families or polynomial invariants are reviewed in the light of the results gathered here, which leads to sometimes surprising connections. Some interesting links and applications of the Majorana representation to related fields of mathematics and physics are also discussed.
A method is developed to derive algebraic equations for the geometric measure of entanglement of three-qubit pure states. The equations are derived explicitly and solved in the cases of most interest. These equations allow one to derive analytic expressions of the geometric entanglement measure in a wide range of three-qubit systems, including the general class of W states and states which are symmetric under the permutation of two qubits. The nearest separable states are not necessarily unique, and highly entangled states are surrounded by a one-parametric set of equally distant separable states. A possibility for physical applications of the various three-qubit states to quantum teleportation and superdense coding is suggested from the aspect of entanglement.
We construct an important missing piece in the entanglement theory of pure three-qubit states, which is a polynomial measure of W-entanglement. This works in parallel to the three-tangle, which is a polynomial measure of GHZ-entanglement, and to the bipartite concurrence, which is a polynomial measure of bipartite entanglement. We also show that these entanglement measures are ordered, the bipartite measure is larger than the W measure, which is larger than the GHZ measure. It is meaningful then to consider these three types of three-qubit entanglement, which are also ordered, bipartite is weaker than W, which is weaker than GHZ, in parallel to the order of the three kinds of equivalence classes of entangled three-qubit states.