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Physical Review A

Noise tolerance of Dicke states

Xiaoyu Chen, Li-zhen Jiang

Published Jan 9, 2020• 20 citations
Publisher

Abstract

Starting from explicitly constructing an entanglement witness for the mixture of a Dicke state with white noise (noisy Dicke state), we provide an entanglement criterion for any multiqubit state. The criterion is necessary and sufficient for noisy $W$ states of three, four, and five qubits. We demonstrate an entanglement criterion for any four-qubit state. The criterion is necessary and sufficient for the generalized noisy four-qubit Dicke states. We present the sufficient criterion of full separability for a noisy $2k$ qubit Dicke state with $k$ excitations. Then we generalize the entanglement witness and criterion to the mixture of white noise and generic $n$-qubit Dicke state with $k$ excitations based on strong numeric evidence. Apart from noisy Dicke states, we also consider the entanglement properties of simultaneously amplitude damped and dephased Dicke states, and depolarized multiqubit $W$ states.

DOI
10.1103/physreva.101.012308

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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.
Abstract A quantum entangled state is easily disturbed by noise and degenerates into a separable state. Compared to the entanglement with bipartite quantum systems, less progress has been made for the entanglement with multipartite quantum systems. For tripartite separability of a four-qubit system, we propose two entanglement witnesses, each of which corresponds to a necessary condition of tripartite separability. For the four-qubit GHZ state mixed with a W state and white noise, we prove that the necessary conditions of tripartite separability are also sufficient at W states side.
We construct an entanglement witness for many-qubit systems, based on symmetric two-body correlations with two measurement settings. This witness is able to detect the entanglement of some Dicke states for any number of particles and such detection exhibits some robustness against white noise and thermal noise under the Lipkin-Meshkov-Glick Hamiltonian. In addition, it detects the entanglement of spin-squeezed states, with a detection strength that approaches the maximal value for sufficiently large numbers of particles. As spin-squeezed states can be experimentally generated, the properties of the witness with respect to these states may be amenable to experimental investigation. We show that while the witness is unable to detect Greenberger-Horne-Zeilinger (GHZ) states, it is instead able to detect superpositions of Dicke states with GHZ states.
Entanglement detection is one of the important topics in quantum information theory. Wootters formula can be seen as a high level criterion for entanglement detection or entanglement quantification. However, it is restricted to two-qubit system, and its generalization is also limited. Based on a thorough study of a three-qubit Greenberger-Horne-Zeilinger (GHZ) state mixed with a W state and white noise, we find a connection between Wootters formula and the matched entanglement witness. Further study shows that for a four-qubit GHZ state mixed with a Dicke state and white noise, there is a new kind of Wootters formula.
We analyze robustness of correlations of the N -qubit GHZ and Dicke states against white noise admixture. For sufficiently large N , the Dicke states (for any number of excitations) lead to more robust violation of local realism than the GHZ states (e.g. for N > 8 for the W state). We also identify states that are the most resistant to white noise. Surprisingly, it turns out that these states are the GHZ states augmented with fully product states. Based on our numerical analysis conducted up to N = 8, and an analytical formula derived for any N parties, we conjecture that the three-qubit GHZ state augmented with a product of ( N − 3) pure qubits is the most robust against white noise admixture among any N -qubit state. As a by-product, we derive a single Bell inequality and show that it is violated by all pure entangled states of a given number of parties. This gives an alternative proof of Gisin’s theorem.
We consider properties of states of many qubits, which arise after sending certain entangled states via various noisy channels (white noise, colored noise, local depolarization, dephasing, and amplitude damping). Entanglement of these states and their ability to violate certain classes of Bell inequalities are studied. States which violate them allow a higher than classical efficiency in solving related distributed computational tasks with constrained communication. This is a direct property of such states---not requiring their further modification via stochastic local operations and classical communication such as entanglement purification or distillation procedures. We identify families of multiparticle states which are entangled but nevertheless allow the local realistic description of specific Bell experiments. For some of them, the ``gap'' between the critical values for entanglement and violation of Bell inequality remains finite even in the limit of infinitely many qubits.