A typical concept in quantum state analysis is based on the idea that states in the vicinity of some pure entangled state share the same properties, implying that states with a high fidelity must be entangled. States whose entanglement can be detected in this way are also called faithful. We prove a structural result on the corresponding fidelity-based entanglement witnesses, resulting in a simple condition for faithfulness of a two-party state. For the simplest case of two qubits faithfulness can directly be decided and for higher dimensions accurate analytical criteria are given. Finally, our results show that faithful entanglement is, in a certain sense, useful entanglement; moreover, they establish connections to computational complexity and simplify several results in entanglement theory.
A typical concept in quantum state analysis is based on the idea that states in the vicinity of some pure entangled state share the same properties, implying that states with a high fidelity must be entangled. States whose entanglement can be detected in this way are also called faithful. We prove a structural result on the corresponding fidelity-based entanglement witnesses, resulting in a simple condition for faithfulness of a two-party state. For the simplest case of two qubits faithfulness can directly be decided and for higher dimensions accurate analytical criteria are given. Finally, our results show that faithful entanglement is, in a certain sense, useful entanglement; moreover, they establish connections to computational complexity and simplify several results in entanglement theory.
A conceptually simple and experimentally prevalent class of entanglement witnesses, known as fidelity witnesses, detects entanglement via a state's fidelity with a pure reference state. While existence proofs guarantee that a suitable witness can be constructed for every entangled state, such assurances do not apply to fidelity witnesses. Recent results have found that entangled states that cannot be detected by a fidelity witness, known as unfaithful states, are exceedingly common among bipartite states. In this paper, we show that even among two-qubit states, which are the simplest of all entangled states, unfaithful states can be created through a suitable application of decoherence and filtering to a Bell state. We also show that the faithfulness is not monotonic to entanglement, as measured by the concurrence. Finally, we experimentally verify our predictions using polarization-entangled photons, and specifically we demonstrate a situation in which an unfaithful state is brought to faithfulness at the expense of further reducing the entanglement of the state.
One of the most widespread methods to determine if a quantum state is entangled, or to quantify its entanglement dimensionality, is by measuring its fidelity with respect to a pure state. In this Letter, we find a large class of states whose entanglement cannot be detected in this manner; we call them unfaithful. We find that unfaithful states are ubiquitous in information theory. For small dimensions, we check numerically that most bipartite states are both entangled and unfaithful. Similarly, numerical searches in low dimensions show that most pure entangled states remain entangled but become unfaithful when a certain amount of white noise is added. We also find that faithfulness can be self-activated, i.e., there exist instances of unfaithful states whose tensor powers are faithful. To explore how the fidelity approach limits the quantification of entanglement dimensionality, we generalize the notion of an unfaithful state to that of a D unfaithful state, one that cannot be certified as D-dimensionally entangled by measuring its fidelity with respect to a pure state. For describing such states, we additionally introduce a hierarchy of semidefinite programming relaxations that fully characterizes the set of states of Schmidt rank at most D.
Characterizing entanglement of systems composed of multiple particles is a very complex problem that is attracting increasing attention across different disciplines related to quantum physics. The task becomes even more complex when the particles have many accessible levels, i.e., they are of high dimension, which leads to a potentially high-dimensional multipartite entangled state. These are important resources for an ever-increasing number of tasks, especially when a network of parties needs to share highly entangled states, e.g., for communicating more efficiently and securely. For these applications, as well as for purely theoretical arguments, it is important to be able to certify both the high-dimensional and the genuine multipartite nature of entangled states, possibly based on simple measurements. Here we derive a novel method that achieves this and improves over typical entanglement witnesses like the fidelity with respect to states of a Greenberger-Horne-Zeilinger (GHZ) form, without needing more complex measurements. We test our condition on paradigmatic classes of high-dimensional multipartite entangled states like imperfect GHZ states with random noise, as well as on purely randomly chosen ones and find that, in comparison with other available criteria our method provides a significant advantage and is often also simpler to evaluate.
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.
Abstract Certifying entanglement for unknown quantum states experimentally is a fundamental problem in quantum computing and quantum physics. Because of being easy to implement, a most popular approach for this problem in modern quantum experiments is detecting target quantum states with fidelity-based entanglement witnesses. Specifically, if the fidelity between a target state and an entangled pure state exceeds a certain value, the target state can be guaranteed to be entangled. Recently, however, it has been realized that there exist so-called unfaithful quantum states, which can be entangled, but their entanglement cannot be certified by any fidelity-based entanglement witnesses. In this paper, by specific examples, we show that if one makes a slight modification to fidelity-based entanglement witnesses by combining multiple fidelities together, it is still possible to certify entanglement for unfaithful quantum states with this popular technique. Particularly, we will analyze the mathematical structure of the modified entanglement witnesses, and propose an algorithm that can search for the optimal designs for them.
Complex forms of quantum entanglement can arise in two qualitatively different ways: either between many qubits or between two particles with higher-than-qubit dimension. While both the many-qubit frontier and the high-dimension frontier are well established, state-of-the-art quantum technology is becoming increasingly able to create and manipulate entangled states that simultaneously feature many particles and high dimension. Here, we investigate generic states that can be considered both genuinely high-dimensional and genuine multiparticle entangled. We consider a natural quantity that characterizes this key property. To detect it, we develop three different classes of criteria. These enable us both to probe the ultimate noise tolerance of this form of entanglement and to make detection schemes using sparse or even minimal measurement resources. The approach provides a simple way of benchmarking entanglement dimensionality in the multiparticle regime and general, platform-independent, detection methods that readily apply to experimental use.