We introduce a measure for the genuinely N-partite (all-party) entanglement of N-qubit states using the trace distance metric and find an algebraic formula for the Greenberger-Horne-Zeilinger (GHZ)-diagonal states. We then use this formula to show how the all-party entanglement of experimentally produced GHZ states of an arbitrary number of qubits may be bounded with only four measurements.
We introduce a measure for the genuinely N-partite (all-party) entanglement of N-qubit states using the trace distance metric and find an algebraic formula for the Greenberger-Horne-Zeilinger (GHZ)-diagonal states. We then use this formula to show how the all-party entanglement of experimentally produced GHZ states of an arbitrary number of qubits may be bounded with only four measurements.
The paradox of Greenberger-Horne-Zeilinger (GHZ) disproves directly the concept of EPR elements of reality, based on the EPR correlations, in an all-versus-nothing way. A three-qubit experimental demonstration of the GHZ paradox was achieved nearly 20 years ago, followed by demonstrations for more qubits. Still, the GHZ contradictions underlying the tests can be reduced to a three-qubit one. We show an irreducible four-qubit GHZ paradox, and report its experimental demonstration. The bound of a three-setting-per-party Bell-GHZ inequality is violated by $7\ensuremath{\sigma}$. The fidelity of the GHZ state was around 81, and an entanglement witness reveals a violation of the separability threshold by $19\ensuremath{\sigma}$.
A Multipartite entangled state has many different kinds of entanglements specified by the number of partitions. The most essential example of multipartite entanglement is the entanglement of multi-qubit Greenberger–Horne–Zeilinger (GHZ) state in white noise. We explicitly construct the entanglement witnesses for these states with stabilizer generators of the GHZ states. For an [Formula: see text] qubit GHZ state in white noise, we demonstrate the necessary and sufficient criterion of separability when it is divided into [Formula: see text] parties with [Formula: see text] for arbitrary [Formula: see text] and [Formula: see text]. The criterion covers more than a half of all kinds of partial entanglements for [Formula: see text]-qubit GHZ states in white noise. For the rest of multipartite entanglement problems, we present a method to obtain the sufficient conditions of separability. As an application, we consider [Formula: see text] qubit GHZ state as a codeword of the degenerate quantum code passing through depolarizing channel. We find that the output state is neither genuinely entangled nor fully separable when the quantum channel capacity reduces from positive to zero.
We find an algebraic formula for the N-partite concurrence of N qubits in an X matrix. X matrices are density matrices whose only nonzero elements are diagonal or antidiagonal when written in an orthonormal basis. We use our formula to study the dynamics of the N-partite entanglement of N remote qubits in generalized N-party Greenberger-Horne-Zeilinger (GHZ) states. We study the case in which each qubit interacts with a local amplitude damping channel. It is shown that only one type of GHZ state loses its entanglement in finite time; for the rest, N-partite entanglement dies out asymptotically. Algebraic formulas for the entanglement dynamics are given in both cases. We directly confirm that the half-life of the entanglement is proportional to the inverse of N. When entanglement vanishes in finite time, the time at which entanglement vanishes can decrease or increase with N depending on the initial state. In the macroscopic limit, this time is independent of the initial entanglement.
We introduce a real-valued functional I(n1,n2) of four three-qubit correlations that turns the Greenberger-Horne-Zeilinger (GHZ) algebraic paradox into a quantitative witness of genuine tripartite entanglement. For every three-qubit state ρ and every pair of directions, ∣I(n1,n2;ρ)∣≤2, with equality if and only if n1⊥n2 and ρ is locally unitarily equivalent to the GHZ state. On the five-parameter Acín family, I(x^,y^) depends only on λ0λ4 and is maximised by the GHZ state; for the W state I(x^,y^)=0 and max∣IW∣=35/27. Maximising over independent local orthonormal frames yields a local-unitary invariant measure EGHZ∈[0,1], computable from four three-body Pauli correlators, that equals one exactly on the GHZ orbit and takes the values 35/54 on the W state and 1/2 on product states. We show that EGHZ is not convex: an equal mixture of three biseparable states has EGHZ=14/27, larger than the average 1/2 of its components. We also relate EGHZ to GHZ fidelity witnesses, discuss its behaviour under local operations and classical communication, and outline a generalisation to three qudits via Heisenberg-Weyl operators.
Certifying entanglement of a multipartite state is generally considered a demanding task. Since an N qubit state is parametrized by 4^{N}-1 real numbers, one might naively expect that the measurement effort of generic entanglement detection also scales exponentially with N. Here, we introduce a general scheme to construct efficient witnesses requiring a constant number of measurements independent of the number of qubits for states like, e.g., Greenberger-Horne-Zeilinger states, cluster states, and Dicke states. For four qubits, we apply this novel method to experimental realizations of the aforementioned states and prove genuine four-partite entanglement with two measurement settings only.
The generalized n-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only pure states of n qubits that are not uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices of $n\ensuremath{-}1$ qubits. Thus, the generalized GHZ states are the only ones containing information at the n-party level.