Universal Entanglement Dynamics of Unitary Operators
Abstract
The entangling power of a unitary operator acting on a bipartite Hilbert space measures the entanglement it generates from product states, averaged over the inputs. A finite-dimensional unitary has a spectral decomposition U=∑a=1neiθaPa, where eiθa are the eigenvalues, Pa the corresponding eigen-projectors, and n is the number of distinct eigenvalues. After removing an overall phase, the entangling power is a function on the (n−1)-torus of relative eigenphases at fixed spectral projectors. We prove that this function is stationary at all 2n−1 points on the torus where every relative phase is $0$ or π, which we define as \textit{corners}. Up to an overall phase, U at each corner is a generalized reflection R=𝕀−2Q satisfying R2=𝕀, where Q is the sum of spectral projectors whose relative phase is π. At the corner the entangling power is expressed in terms of seven local-unitary invariants of Q. A unitary gate U can be realized as a corner of some projector family if and only if U2∝𝕀, a condition satisfied by many Clifford and non-Clifford gates. We illustrate the theorem with two-qubit gates, SU(N) channel decompositions, and two-site spin chains, obtaining examples of minima, maxima, and saddle points. In addition, a corner that is a saddle point on the full phase torus can appear as a local maximum or minimum along different time-evolution trajectories.
Related
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BibTeX
@misc{low2026universal,
title = {{Universal Entanglement Dynamics of Unitary Operators}},
author = {Ian Low and Navin McGinnis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09276},
eprint = {2609.09276},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09276},
}
RIS
TY - UNPB
TI - Universal Entanglement Dynamics of Unitary Operators
AU - Ian Low
AU - Navin McGinnis
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09276
UR - https://arxiv.org/abs/2609.09276
AB - The entangling power of a unitary operator acting on a bipartite Hilbert space measures the entanglement it generates from product states, averaged over the inputs. A finite-dimensional unitary has a spectral decomposition U=∑_a=1^ne^iθ_aP_a, where e^iθ_a are the eigenvalues, P_a the corresponding eigen-projectors, and n is the number of distinct eigenvalues. After removing an overall phase, the entangling power is a function on the (n−1)-torus of relative eigenphases at fixed spectral projectors. We prove that this function is stationary at all 2^n−1 points on the torus where every relative phase is $0$ or π, which we define as \textit{corners}. Up to an overall phase, U at each corner is a generalized reflection R=𝕀−2Q satisfying R^2=𝕀, where Q is the sum of spectral projectors whose relative phase is π. At the corner the entangling power is expressed in terms of seven local-unitary invariants of Q. A unitary gate U can be realized as a corner of some projector family if and only if U^2∝𝕀, a condition satisfied by many Clifford and non-Clifford gates. We illustrate the theorem with two-qubit gates, SU(N) channel decompositions, and two-site spin chains, obtaining examples of minima, maxima, and saddle points. In addition, a corner that is a saddle point on the full phase torus can appear as a local maximum or minimum along different time-evolution trajectories.
ER -