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1Repair Windows errors before they cause bigger problems2Fix the driver behind crashes, sound loss and screen glitches3Clear out junk files and repair common Windows errorsA new mathematical result identifies a universal set of stationary points for the entangling power of a finite-dimensional unitary: when its relative eigenphases are each either 0 or π, the entangling power is stationary. Ian Low and Navin McGinnis call these configurations “corners.” The result characterizes possible stationary configurations; it is not a quantum-hardware experiment or evidence of improved computer performance.
What the theorem maps
Entangling power measures how much entanglement a unitary operator generates, averaged over product-state inputs. Low and McGinnis study how this quantity changes when the operator’s eigenphases vary while its spectral projectors stay fixed. Their theorem says that every corner in this phase space is a stationary point of entangling power.
For a unitary with n distinct eigenvalues, an overall phase can be removed without changing the entangling power. The remaining relative phases form an (n−1)-dimensional torus. Each relative phase can be 0 or π at a corner, giving 2n−1 such configurations. This is a mathematical count, not an experimental measurement. The arXiv abstract states the theorem; the full paper gives its derivation and examples.
Why corners have a reflection form
At a corner, the unitary can be written, up to an overall phase, as a generalized reflection:
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R = I − 2Q
Here, I is the identity operator and Q is the sum of the spectral projectors assigned relative phase π. Since Q is a projector, this form satisfies R² = I. The paper also gives a converse: a unitary can be realized as a corner for some projector family if and only if its square is proportional to the identity. At a corner, the entangling power is expressed using seven local-unitary invariants of Q.
Stationary does not mean maximum or minimum
A stationary point is a place where the first-order change in entangling power vanishes across the full phase space. It need not be a maximum or a minimum: the paper’s examples include minima, maxima, and saddle points.
This distinction matters when a physical evolution traces only one path through the phase torus. A saddle in the full space may look like a local maximum or minimum along a particular trajectory. That trajectory-specific appearance does not change the point’s classification in the full phase space.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Examples and scope
Low and McGinnis illustrate the result with two-qubit gates, SU(N) channel decompositions, and two-site spin chains. These are mathematical examples of the theorem, not competing implementations or hardware demonstrations. The claim is general across choices of spectral projectors, subsystem dimensions, and bipartitions: with projectors fixed, all 2n−1 corners are stationary.
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Clear out junk files and repair common Windows errorsFree Scan →Fix the driver behind crashes, sound loss and screen glitchesFind Drivers →Repair Windows errors before they cause bigger problemsFix Now →The work is an arXiv preprint by Ian Low and Navin McGinnis. The arXiv record lists its submission date as 8 September 2026, and the PDF is dated 10 September 2026; these sources identify it as a preprint, not as a peer-reviewed journal publication.
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