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arXiv:0706.2765 · Journal of Physics A Mathematical and Theoretical

Detecting entanglement of random states with an entanglement witness

Marko Žnidarič, Tomaž Prosen, Giuliano Benenti, Giulio Casati

Published Oct 23, 2007• 24 citations
arXivHTMLPDF

Abstract

The entanglement content of high-dimensional random pure states is almost maximal, nevertheless, we show that, due to the complexity of such states, the detection of their entanglement using witness operators is rather difficult. We discuss the case of unknown random states, and the case of known random states for which we can optimize the entanglement witness. Moreover, we show that coarse graining, modeled by considering mixtures of m random states instead of pure ones, leads to a decay in the entanglement detection probability exponential with m. Our results also allow to explain the emergence of classicality in coarse grained quantum chaotic dynamics.

DOI
10.1088/1751-8113/40/45/017

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The entanglement content of high-dimensional random pure states is almost maximal, nevertheless, we show that, due to the complexity of such states, the detection of their entanglement using witness operators is rather difficult. We discuss the case of unknown random states, and the case of known random states for which we can optimize the entanglement witness. Moreover, we show that coarse graining, modeled by considering mixtures of m random states instead of pure ones, leads to a decay in the entanglement detection probability exponential with m. Our results also allow to explain the emergence of classicality in coarse grained quantum chaotic dynamics.
Entanglement is not only the most intriguing feature of quantum mechanics, but also a key resource in quantum information science. The entanglement content of random pure quantum states is almost maximal; such states find applications in various quantum information protocols. The preparation of a random state or, equivalently, the implementation of a random unitary operator, requires a number of elementary one- and two-qubit gates that is exponential in the number n_q of qubits, thus becoming rapidly unfeasible when increasing n_q. On the other hand, pseudo-random states approximating to the desired accuracy the entanglement properties of true random states may be generated efficiently, that is, polynomially in n_q. In particular, quantum chaotic maps are efficient generators of multipartite entanglement among the qubits, close to that expected for random states. This review discusses several aspects of the relationship between entanglement, randomness and chaos. In particular, I will focus on the following items: (i) the robustness of the entanglement generated by quantum chaotic maps when taking into account the unavoidable noise sources affecting a quantum computer; (ii) the detection of the entanglement of high-dimensional (mixtures of) random states, an issue also related to the question of the emergence of classicality in coarse grained quantum chaotic dynamics; (iii) the decoherence induced by the coupling of a system to a chaotic environment, that is, by the entanglement established between the system and the environment.
Quantum science and engineering increasingly target the exploitation and control of large-scale dynamical systems. As system size grows, novel phenomena and quantum advantages emerge, but the resulting many-body dynamics become correspondingly harder to monitor and characterize. In particular, the underlying Hamiltonian often lacks symmetries, and the induced many-body dynamics quickly become hard to track. Focusing on one of the most important quantum features---entanglement---we demonstrate that, for a broad class of common Hamiltonian models, one can design effective witnesses that detect both the presence and structure of entanglement over relatively long evolution times. Inspired by stabilizer-based simulation, we track the evolution of generalized stabilizers of the underlying state and use them to construct linear entanglement witnesses. Through numerical experiments on Heisenberg-type Hamiltonians with up to twelve qubits, we show that the required number of measurement settings can be kept at a modest scale as the system size increases, and entanglement detection remains effective over evolution times that significantly exceed those accessible to naive approaches. The corresponding calculations can be carried out largely symbolically. Furthermore, unlike accurate state simulation, the stabilizer-based design of entanglement witnesses can be performed in a much more coarse-grained, digitized manner. We hope these results will stimulate further exploration of the quantum control of large systems.
Abstract We develop a scheme for the detection of entanglement in any continuous variable system, by constructing an optimal entanglement witness from random homodyne measurements. To this end, we introduce a set of linear constraints that guarantee the necessary properties of a witness and allow for its optimisation via a semidefinite program. We test our method on the class of squeezed vacuum states and study the efficiency of entanglement detection in general unknown covariance matrices. The results show that we can detect entanglement, including bound entanglement, in arbitrary continuous variable states with fewer measurements than in full tomography. The statistical analysis of our method shows a good robustness to statistical errors in experiments.
Entanglement witness is an effective method to detect entanglement in unknown states without doing full tomography. One of the most widespread schemes of witnessing entanglement is measuring its fidelity with respect to a pure entangled state. Recently, a large class of states whose entanglement can not be detected with the fidelity witness has been discovered in Phys.Rev.Lett \textbf{124},200502(2020). They are called unfaithful states. In this paper we propose a new way to detect entanglement by calculating the lower bound of entanglement using measurement results. Numerical simulation shows our method can detect entanglement in unfaithful states with a small number of measurements. Moreover, we generalize our scheme to multipartite states and show that it can tolerate higher noise than previous entanglement witness operators with same number of measurement settings.