Constant rotation of two-qubit equally entangled pure states by local quantum operations

TL;DR

This paper investigates how local quantum operations can uniformly rotate the set of equally entangled two-qubit pure states, revealing limitations for non-maximally entangled states and extending results to three-qubit states.

Contribution

It characterizes the conditions under which local unitaries can rotate all equally entangled states by the same amount, including maximally entangled states and specific non-maximally entangled sets.

Findings

Maximally entangled states can be rotated uniformly by local unitaries.

No non-trivial local unitary can rotate all non-maximally entangled states equally.

Certain optimal sets of non-maximally entangled states can be rotated by the same amount.

Abstract

We look for local unitary operators $W_1 \otimes W_2$ which would rotate all equally entangled two-qubit pure states by the same but arbitrary amount. It is shown that all two-qubit maximally entangled states can be rotated through the same but arbitrary amount by local unitary operators. But there is no local unitary operator which can rotate all equally entangled non-maximally entangled states by the same amount, unless it is unity. We have found the optimal sets of equally entangled non-maximally entangled states which can be rotated by the same but arbitrary amount via local unitary operators $W_1 \otimes W_2$, where at most one these two operators can be identity. In particular, when $W_1 = W_2 = (i/\sqrt{2})({\sigma}_x + {\sigma}_y)$, we get the local quantum NOT operation. Interestingly, when we apply the one-sided local depolarizing map, we can rotate all equally entangled two-qubit pure states through the same amount. We extend our result for the case of three-qubit maximally entangled state.