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arXiv:2308.07744

Information theoretical perspective on the method of Entanglement Witnesses

Paulo J. Cavalcanti, Giovanni Scala, Antonio Mandarino, Cosmo Lupo

Published Aug 15, 2023• 0 citations
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Abstract

We frame entanglement detection as a problem of random variable inference to introduce a quantitative method to measure and understand whether entanglement witnesses lead to an efficient procedure for that task. Hence we quantify how many bits of information a family of entanglement witnesses can infer about the entanglement of a given quantum state sample. The bits are computed in terms of the mutual information and we unveil there exists hidden information not \emph{efficiently} processed. We show that there is more information in the expected value of the entanglement witnesses, i.e. $\mathbb{E}[W]=\langle W \rangle_ρ$ than in the sign of $\mathbb{E}[W]$. This suggests that an entanglement witness can provide more information about the entanglement if for our decision boundary we compute a different functional of its expectation value, rather than $\mathrm{sign}\left(\mathbb{E}\right [ W ])$.

DOI
10.48550/arxiv.2308.07744

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We frame entanglement detection as a problem of random variable inference to introduce a quantitative method to measure and understand whether entanglement witnesses lead to an efficient procedure for that task. Hence we quantify how many bits of information a family of entanglement witnesses can infer about the entanglement of a given quantum state sample. The bits are computed in terms of the mutual information and we unveil there exists hidden information not \emph{efficiently} processed. We show that there is more information in the expected value of the entanglement witnesses, i.e. E[W]=⟨W⟩ρ\mathbb{E}[W]=\langle W \rangle_ρE[W]=⟨W⟩ρ​ than in the sign of E[W]\mathbb{E}[W]E[W]. This suggests that an entanglement witness can provide more information about the entanglement if for our decision boundary we compute a different functional of its expectation value, rather than sign(E[W])\mathrm{sign}\left(\mathbb{E}\right [ W ])sign(E[W]).
Entanglement witnesses provide tools to detect entanglement in experimental situations without the need of having full tomographic knowledge about the state. If one estimates in an experiment an expectation value smaller than zero, one can directly infer that the state has been entangled, or specifically multi-partite entangled, in the first place. In this paper, we emphasize that all these tests—based on the very same data—give rise to quantitative estimates in terms of entanglement measures: 'If a test is strongly violated, one can also infer that the state was quantitatively very much entangled'. We consider various measures of entanglement, including the negativity, the entanglement of formation, and the robustness of entanglement, in the bipartite and multipartite setting. As examples, we discuss several experiments in the context of quantum state preparation that have recently been performed.
Entanglement detection is one of the most fundamental and practical tasks for quantum information processing. The framework of entanglement witnesses provides an experimentally feasible method detecting entangled states. Although It is clear that no entanglement witness per se can detect all entangled states, little is known about how useful a single entanglement witness is. In this work, we show that an entanglement witness can construct another entanglement witness. This means that the same measurement outcomes can be repeatedly applied to constructing different entanglement witnesses.
We consider a possible detector-efficiency loophole in experiments that detect entanglement via the local measurement of witness operators. Here, only local properties of the detectors are known. We derive a general threshold for the detector efficiencies which guarantees that a negative expectation value of a witness is due to entanglement, rather than to erroneous detectors. This threshold depends on the local decomposition of the witness and its measured expectation value. For two-qubit witnesses, we find the local operator decomposition that is optimal with respect to closing the loophole.
Entanglement is an essential resource in quantum information science [1] and its presence in any quantum system can be experimentally detected through entanglement witness operators [2]. In particular, measuring a negative expectation value of a witness with high statistical confidence provides a necessary and sufficient condition to confirm the generation of a genuine multipartite [3] and/or d-level entangled state [4]. In recent years, the experimental generation of complex quantum states has intensified the need for witnesses that are capable of detecting such systems and are experimentally optimal at the same time. This means that the witness should require the least measurement effort (in terms of number and complexity of the measurement settings), include only projections on single qudits, while at the same time possessing a high noise tolerance (Fig. 1a). However, "experimentally-friendly" witnesses capable of accomplishing these tasks have not been derived yet.
Within the framework of Information Theory, the existence of correlations between two random variables means that we can obtain information about one of them, just by measuring or observing the other random variable. In certain cases, this kind of relationship allows obtaining information about a variable even when the other is separated by a very large distance, that is, the process of obtaining information is non-local, an example (if not the only) is the quantum entanglement. These features of correlations make it interesting and important to study, classify and quantify them. The correlations are classified into classical correlations and quantum correlations, in addition they are quantified through the mutual information. Here we will present a natural way to define classical mutual information and then we will generalize it to the quantum case. Furthermore, every term in the definitions of mutual information will be interpreted using the concepts of classical and quantum entropy.
Abstract Quantum entanglement is a key resource in quantum information. Teleportation, dense coding, and many other quantum protocols rely on the existence of entanglement, and would not be possible with just classical correlations. Furthermore, using entanglement between qubits that support computation, quantum computers can solve problems faster than classical computers. Bell's inequalities can sometimes help in discriminating between entangled and separable states. However, these inequalities are not always reliable — this is true in the sense that they may fail to detect genuinely entangled states. This question is addressed in the present chapter by focusing on an entanglement witness, a Hermitian operator winch helps us to decide whether a state is entangled or not. The basic idea is that the expectation value of the witness will be different for separable and entangled states. This chapter discusses entanglement witnesses, the Jamiolkowski isomorphism, and the Peres Horodecki criterion. More examples of entanglement witnesses are given.