# Classification of nonlocal two-qubit gates using Schmidt number

## TL;DR

This paper investigates the classification of nonlocal two-qubit quantum gates based on their Schmidt number, revealing conditions for Schmidt number 2 and highlighting the unique entangling properties of CNOT gates.

## Contribution

It provides a geometric characterization of two-qubit gates with Schmidt number 2 and analyzes the entangling capabilities of controlled unitary gates, especially CNOT.

## Key findings

- Schmidt number of two-qubit gates is 2 or 4.
- CNOT is the only perfect entangler among controlled unitaries.
- Schmidt strength and entangling power are maximized for CNOT.

## Abstract

It is known from Schmidt decomposition that Schmidt number of nonlocal two-qubit quantum gates is 2 or 4. We identify conditions on geometrical points of a gate to have Schmidt number 2. A simple analysis reveals that Schmidt number 2 corresponds to controlled unitary gates with CNOT being the only perfect entangler. Further, it is shown that Schmidt strength and entangling power are maximum only for CNOT in the controlled unitary family.

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Source: https://tomesphere.com/paper/0907.5519