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What the theorem studies
A finite-dimensional unitary operator can be written in terms of its distinct eigenvalues and the projectors onto their eigenspaces. Remove an overall phase, which does not affect the gate’s action on states, and vary the remaining relative eigenphases while keeping the projectors fixed. Those phase variables form an (n−1)-dimensional torus for an operator with n distinct eigenvalues.
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The quantity being tracked is entangling power: the entanglement generated by applying the unitary to product-state inputs, averaged over those inputs. In the paper, it is treated as a function of the relative eigenphases for a fixed set of projectors and a chosen subsystem bipartition. The theorem applies across choices of projectors, subsystem dimensions, and bipartitions. The authors’ arXiv abstract and full paper give the formal statement.
Why there are 2^(n−1) corners
Each of the n−1 relative phases can take either of two values at a corner: 0 or π. There are therefore 2^(n−1) such configurations. Low and McGinnis prove that entangling power is stationary at every one of them. “Stationary” means the first-order change with respect to the phase variables vanishes there; it does not by itself say whether the point is a maximum, a minimum, or a saddle.
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What a corner looks like as an operator
At a corner, the unitary can be written, up to an overall phase, as a generalized reflection:
R = I − 2Q, with R² = I.
Here I is the identity operator, and Q is the sum of the spectral projectors associated with relative phase π. This form makes the corner condition more than a coordinate description: the resulting operator squares to the identity. The paper also states a converse characterization: a gate can be realized as a corner for some family of projectors if and only if its square is proportional to the identity.
Stationary does not mean optimal
The paper’s examples include minima, maxima, and saddle points. A saddle is stationary, but nearby changes can raise entangling power in some directions and lower it in others. The authors illustrate the theorem with two-qubit gates, SU(N) channel decompositions, and two-site spin chains. At a corner, the entangling power is expressed using seven local-unitary invariants of Q.
There is also an important distinction between the full phase space and a particular physical trajectory through it. A point that is a saddle on the entire torus can look like a local maximum or minimum when viewed only along a restricted time-evolution path. That trajectory-specific appearance does not change its classification on the full phase torus.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.What the result does—and does not—show
The theorem identifies a universal set of stationary configurations for the mathematical function defined by entangling power as relative eigenphases vary. Its scope is theoretical and depends on holding the spectral projectors fixed while varying those phases. It does not establish that a specific quantum processor reaches a corner, that a corner is best for a computing task, or that quantum hardware performance improves.
The work is an arXiv preprint by Ian Low and Navin McGinnis, submitted on 8 September 2026; the PDF is dated 10 September 2026. The cited record identifies it as a preprint, not a peer-reviewed journal publication. Read the arXiv record or the paper PDF.
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