We present a quantum circuit that transforms an unknown three-qubit state into its canonical form, up to relative phases, given many copies of the original state. The circuit is made of three single-qubit parametrized quantum gates, and the optimal values for the parameters are learned in a variational fashion. Once this transformation is achieved, direct measurement of outcome probabilities in the computational basis provides an estimate of the tangle, which quantifies genuine tripartite entanglement. We perform simulations on a set of random states under different noise conditions to asses the validity of the method.
Abstract Understanding how to decompose quantum computations in the language of the shortest possible sequence of quantum gates is of interest to many researchers due to the importance of the experimental implementation of the desired quantum computations. We contribute to this research by providing a quantum circuit to directly measure the three-tangle of three-qubit quantum states. Direct measurement of outcome probabilities in the computational basis quantifies the three-tangle of the three-qubit quantum states.
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.
Usually, the three-tangle of a tripartite pure state of qubits can be directly measured with the simultaneous preparation of a not-less-than fourfold copy of the state. We show that the exact genuine tripartite entanglement for ($2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}n$)-dimensional pure quantum states can be measured in a similar manner, provided that only two simultaneous copies of the state are available. Lower bounds are also proposed for more convenient experimental operations. As an example, a comprehensive demonstration of the scheme is provided for the three-tangle of a three-qubit state.
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.
The geometric measure of entanglement of variational quantum states is studied on the basis of its relation with the mean value of spin. We examine n-qubit quantum states prepared by a variational circuit with a layer formed by the rotational gates and two-qubit controlled phase gates. The variational circuit is a generalization of that used for preparing quantum Generative Adversarial Network states. The entanglement of a qubit with other qubits in the variational quantum states is determined by the angles of rotational gates that act on the qubit and qubits entangled with it by controlled phase gates and also their parameters. In the case of one layer variational circuit, the states can be associated with graphs with vertices representing qubits and edges corresponding to two-qubit gates. The geometric measure of entanglement of a qubit with other qubits in the quantum graph state depends on the properties of the vertex that represents it in the graph, namely it depends on the vertex degree. The dependence of the geometric measure of entanglement of variational quantum states on their parameters is quantified on IBM's quantum computer.
The three-tangle,
τ3, is a measure of genuine tripartite entanglement that distinguishes GHZ-type from W-type entanglement in pure three-qubit states. Here, we identify the statistically optimal one-sided four-copy detector of the three-tangle and study its implementation cost. The optimal detector has click probability
τ32/48, while previous four-copy constructions achieve
τ32/256 and therefore require asymptotically
16/3 times as many experimental repetitions. Using the representation-theoretic structure of the problem, we construct an ancilla-free implementation of the optimal detector using 15 CNOT gates, together with an 11-CNOT detector with click probability
τ32/96. These results reveal a tradeoff between statistical efficiency and implementation cost, since reducing the number of repetitions requires increasingly expensive circuits. We then show that this tradeoff persists once the circuits are encoded. In a fully fault-tolerant setting, the increased click probability comes with a substantial non-Clifford synthesis cost, while an early fault-tolerant implementation based on
T gates and STAR rotations reduces this overhead at the expense of additional encoded resources. We quantify these costs using both the Steane and
[[8,3,2]] codes, providing explicit resource estimates for implementing the optimal detector across different computational regimes.