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arXiv:2502.03820 · Physica Scripta

Nonlocal characteristics of two-qubit gates from Argand diagram geometry

M. Karthick Selvan, S. Balakrishnan

Published Jan 6, 2026• 0 citations
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Abstract

Abstract We show that the geometrical features of the unimodular squared eigenvalues of the nonlocal part of a two-qubit gate in the complex plane can be used to study the nonlocal characteristics of the two-qubit gate. It has been shown that the entangling power of a two-qubit gate can be expressed in terms of the mean squared length of the chords connecting the squared eigenvalues of the nonlocal part with each other. In this paper, we provide similar expressions for the gate typicality and the linear entropy of two-qubit gates. The expression of gate typicality involves the squared lengths of three chords connecting a squared eigenvalue of the nonlocal part with the complex conjugate of the other three squared eigenvalues. The expression of linear entropy is obtained from the expressions of entangling power and gate typicality. We also express them in terms of the linear entropies of certain two-qubit states. In addition, we show that the normalized entangling power of a two-qubit gate can be provided exact description in the Argand diagram. Finally, we analyze the simplices enclosing the origin in the convex hull of the squared eigenvalues of the nonlocal part of the perfect entangling two-qubit gates to construct a pair of orthonormal product states that can be transformed into maximally entangled states, and divide the region of perfect entanglers in the Weyl chamber into different regions based on the uniqueness of the simplices enclosing the origin.

DOI
10.1088/1402-4896/ae2dd4

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•Apr 16, 2026•0 citations
Abstract We show that the geometrical features of the unimodular squared eigenvalues of the nonlocal part of a two-qubit gate in the complex plane can be used to study the nonlocal characteristics of the two-qubit gate. It has been shown that the entangling power of a two-qubit gate can be expressed in terms of the mean squared length of the chords connecting the squared eigenvalues of the nonlocal part with each other. In this paper, we provide similar expressions for the gate typicality and the linear entropy of two-qubit gates. The expression of gate typicality involves the squared lengths of three chords connecting a squared eigenvalue of the nonlocal part with the complex conjugate of the other three squared eigenvalues. The expression of linear entropy is obtained from the expressions of entangling power and gate typicality. We also express them in terms of the linear entropies of certain two-qubit states. In addition, we show that the normalized entangling power of a two-qubit gate can be provided exact description in the Argand diagram. Finally, we analyze the simplices enclosing the origin in the convex hull of the squared eigenvalues of the nonlocal part of the perfect entangling two-qubit gates to construct a pair of orthonormal product states that can be transformed into maximally entangled states, and divide the region of perfect entanglers in the Weyl chamber into different regions based on the uniqueness of the simplices enclosing the origin.
In this paper, we discuss the experimental determination of the nonlocal characteristics of two-qubit gates. Based on the recently derived expressions for the entangling power and gate typicality of two-qubit gates, we construct two-qubit quantum circuits to measure the entangling power and gate typicality of the two-qubit gates, which are generated by the elements of the su(4) Cartan subalgebra. These elements describe the native interactions in many quantum processors. Hence, these circuits can be used to determine the nonlocal characteristics of native gates of many quantum processors. In each circuit, the native gate is applied twice. In addition, each circuit consists of at least one CNOT gate. A set of six two-qubit circuits is constructed to measure the entangling power. The number of circuits is further reduced to three by increasing the nonlocal resources (the number of CNOT gates). To measure the gate typicality, a set of three two-qubit circuits is constructed. Measurement of gate typicality requires more nonlocal resources than the measurement of entangling power.
The capacity of a quantum gate to produce entangled states on a bipartite system is quantified in terms of the entangling power. This quantity is defined as the average of the linear entropy of entanglement of the states produced after applying a quantum gate over the whole set of separable states. Here we focus on symmetric two-qubit quantum gates, acting on the symmetric two-qubit space, and calculate the entangling power in terms of the appropriate local invariant. A geometric description of the local equivalence classes of gates is given in terms of the su(3)\text{su}(3)su(3) Lie algebra root vectors. These vectors define a primitive cell with hexagonal symmetry on a plane, and through the Weyl group the minimum area on the plane containing the whole set of locally equivalent quantum gates is identified. We give conditions to determine when a given quantum gate produces maximally entangled states from separable ones (perfect entanglers). We find that these gates correspond to one-fourth of the whole set of locally distinct quantum gates. The formalism developed here is applicable to general three-level systems. Via the Majorana representation, qutrit transformations can be regarded as having entangling power and hence classified as perfect and nonperfect entanglers and be grouped into local-equivalence classes of the associated symmetric two-qubit space. The results are illustrated by an anisotropic Heisenberg model, the Lipkin-Meshkov-Glick model, and two coupled quantized oscillators with cross-Kerr interaction, which we use to obtain three-level gates relevant in qutrit quantum computation.
We present a framework based on the determinantal geometry of two-qubit gates. Combining the Weyl chamber representation with operator Schmidt theory, we interpret gate synthesis as a distance problem to determinantal varieties. This gives an operational geometry to the Weyl chamber, quantifying nonlocal complexity. We show that the square root iSWAP gate is the closest perfect entangler to the variety of local operations, and that no perfect entangler can be approximated by a local gate with average gate fidelity above 79.8%. The three different determinantal costs form a synthesis-adapted coordinate system that encodes nonlocal complexity and generally reconstructs the Weyl chamber.
Nonlocal properties of an ensemble of diagonal random unitary matrices of order N2{N}^{2}N2 are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as lnNlnNlnN, in contrast to the lnN2ln{N}^{2}lnN2 behavior characteristic of random unitary gates. Entangling power of a diagonal gate UUU is related to the von Neumann entropy of an auxiliary quantum state $\ensuremath{\rho}=A{A}^{\ifmmode\dagger\else\textdagger\fi{}}/{N}^{2}$, where the square matrix AAA is obtained by reshaping the vector of diagonal elements of UUU of length N2{N}^{2}N2 into a square matrix of order NNN. This fact provides a motivation to study the ensemble of non-Hermitian unimodular matrices AAA, with all entries of the same modulus and random phases and the ensemble of quantum states $\ensuremath{\rho}$, such that all their diagonal entries are equal to 1/N1/N1/N. Such a state is contradiagonal with respect to the computational basis, in the sense that among all unitary equivalent states it maximizes the entropy copied to the environment due to the coarse-graining process. The first four moments of the squared singular values of the unimodular ensemble are derived, based on which we conjecture a connection to a recently studied combinatorial object called the ``Borel triangle.'' This allows us to find exactly the mean von Neumann entropy for random phase density matrices and the average entanglement for the corresponding ensemble of bipartite pure states.
Circuit cutting enables quantum computations to be decomposed into smaller subcircuits at the cost of additional sampling overhead. Gate-cutting provides one such approach by replacing nonlocal gates with quasiprobability decompositions of local operations. Starting from the known optimal two-qubit gate-cutting formula, we derive complete sharp lower and upper envelopes relating the quasiprobability extent γγγ to five established descriptors: entangling power, gate typicality, operator entanglement, Schmidt strength, and maximum product-input concurrence. We recast γγγ as a Rényi-1/21/21/2 functional of the operator-Schmidt spectrum and show how the physical constraints on two-qubit spectra sharpen generic entropy bounds and select a small set of recurring extremal Cartan families. None of the five descriptors generically determines γγγ alone, but each imposes exact constraints, revealing how substantially the optimal gate-cutting cost can vary between gates with similar nonlocal characteristics.
We present a framework based on the determinantal geometry of two-qubit gates. Combining the Weyl chamber representation with operator Schmidt theory, we interpret gate synthesis as a distance problem to determinantal varieties. This gives an operational geometry to the Weyl chamber, quantifying nonlocal complexity. We show that the √ iSWAP gate is the closest perfect entangler to the variety of local operations, and that no perfect entangler can be approximated by a local gate with average gate fidelity above 79.8%. The three different determinantal costs form a synthesis-adapted coordinate system that encodes nonlocal complexity and generally reconstructs the Weyl chamber.