Partition of 3-qubits using local gates

TL;DR

This paper classifies 3-qubit states based on local gate equivalence, analyzing their entanglement properties and how specific gates transform these classes, revealing a structured partition with five distinct entanglement levels.

Contribution

It introduces a detailed partition of 3-qubit states into five classes based on local gate equivalence and entanglement entropy, including real and Clifford-preparable states, with explicit orbit sizes and transformations.

Findings

The quotient space of Clifford states has 5 elements.

States in each orbit have specific entanglement entropy values.

CNOT gates map states within and between these orbits.

Abstract

It is well known that local gates have smaller error than non-local gates. For this reason it is natural to make two states equivalent if they differ by a local gate. Since two states that differ by a local gate have the same entanglement entropy, then the entanglement entropy defines a function in the quotient space. In this paper we study this equivalence relation on (i) the set $\mathbb{R}\hbox{Q}(3)$ of 3-qubit states with real amplitudes, (ii) the set $Q\mathcal{C}$ of 3-qubit states that can be prepared with gates on the Clifford group, and (iii) the set $Q\mathbb{R}\mathcal{C}$ of 3-qubit states in $Q\mathcal{C}$ with real amplitudes. We show that the set $Q\mathcal{C}$ has 8460 states and the quotient space has 5 elements. We have $\frac{Q\mathcal{C}}{\sim}=\{S_{0},S_{2/3,1},S_{2/3,2},S_{2/3,3},S_{1}\}$. As usual, we will call the elements in the quotient space, orbits. We have that the orbit $S_0$ contains all the states that differ by a local gate with the state $|000\rangle$. There are 1728 states in $S_0$ and as expected, they have zero entanglement entropy. All the states in the orbits $S_{2/3,1},S_{2/3,2},S_{2/3,3}$ have entanglement entropy $2/3$ and each one of these orbits has 1152 states. Finally, the orbit $S_{1}$ has 3456 elements and all its states have maximum entanglement entropy equal to one. We also study how the controlled not gates $CNOT(1,2)$ and $CNOT(2,3)$ act on these orbits. For example, we show that when we apply a $CNOT(1,2)$ to all the states in $S_0$, then 960 states go back to the same orbit $S_0$ and 768 states go to the orbit $S_{2/3,1}$. Similar results are obtained for $\mathbb{R}Q\mathcal{C}$. We also show that the entanglement entropy function reaches its maximum value 1 in more than one point when acting on $\frac{\hbox{$\mathbb{R}$Q(3)} }{\sim}$.