Entanglement is an important resource for quantum technologies that allow us to go beyond the classical world. Quantum systems can be entangled in various ways. Also, the degree of entanglement between the subsystems of a quantum system can be calculated in different ways. In our recent paper [Prog. Theor. Exp. Phys. (2022)], we directly measure the three-tangle of a pure three-qubit quantum state. In this paper, we extend this method to pure N-qubit quantum states (N is any odd number greater than 1) by considering polynomial invariant of degree 4 as a measure of entanglement. In this algorithm, we use quantum gates and output probabilities to calculate polynomial invariant of degree 4. The importance of this method lies in the possibility of experimental implementation of quantum calculations in the language of quantum gates.
What does measuring one qubit reveal about another?
K-networks as a directed diagnostic for quantum circuits
Many-qubit circuit states are hard to inspect directly, so they are often summarized by pairwise graph weights. Common pairwise weights report symmetric correlations, while many circuit questions are directed and basis-specific: if qubit
i is measured in a given basis, how strongly does the outcome reshape the conditional state of qubit
j? We define
Ki→j, a directed, basis-conditioned edge weight for this question. It is large when the two measurement outcomes occur with comparable probability and leave qubit
j in clearly different conditional states; it is zero when the source outcome is deterministic or the target states are indistinguishable. The scalar uses standard binary-ensemble distinguishability; the paper's contribution is to turn this conditional comparison into a directed network layer for circuit states. The resulting networks are computable from two-qubit reduced density matrices. They are diagnostic (not entanglement measures): for pure two-qubit states
K reduces to the tangle
C2 (squared concurrence)~\cite{WoottersConcurrence,CKWTangle}, while separable mixed states can reach
K=1. Examples on teleportation, Grover, QAOA, and random circuit families show the intended use:
K-networks map feed-forward, phase, and interaction-graph structure that symmetric or computational-basis summaries can leave weak or absent.
Quantum entanglement is the fundamental hallmark of quantum mechanics and a core resource for realizing long-distance quantum communication and scalable linear quantum computing. Accordingly, the precise detection and quantitative quantification of entanglement constitute a foundational and critical problem in quantum information theory. To date, researchers have proposed numerous sufficient conditions for entanglement detection as well as a variety of entanglement measures to characterize the entanglement strength of quantum states; nevertheless, efficient and direct measurement schemes for core entanglement parameters remain underdeveloped. Based on unitary transformations and auxiliary measurements, this paper proposes a set of quantum circuit schemes capable of directly measuring the bipartite concurrence and the tripartite 3-tangle entanglement measure. By introducing auxiliary qubits and constructing specific controlled unitary operations, the proposed scheme maps the analytical expressions of the two entanglement measures onto the measurement probabilities of output states from quantum circuits. It enables efficient and direct quantitative measurement of bipartite and tripartite entanglement without performing full quantum state tomography. This work provides a feasible technical route for the experimental characterization of entanglement properties and lays a groundwork for the practical deployment of multipartite entanglement resources in quantum information processing.