# LOCC operations

LOCC - local operations and classical communication - are certain type of transformations of a state in quantum information theory.

# Introduction

One of the tasks in quantum information theory is to distill or concentrate entanglement of a given state. Basically, there are two types of protocols, which aim to perform such tasks. The first one is based on non-local quantum measurements on many copies of the initial state. The others are concerned only with local operations with possible classical communication (LOCC) and performed on the only one copy of a state. Therefore they are of special interest from experimental point of view due to the fact, that local measurements can be performed in a more simple way, than the non-local ones.

From the point of view of quanutm communication LOCC protocols are important because there is no perfect communication channel in the real-world. Hence it is natural to ask how much entanglement can be obtained from the imperfectly entangled states, which arise for example during sharing of a perfect entangled state between two observers, using only LOCC.

Concerning fundamental questions of quantum information theory which such tasks as characterization and general understanding of entanglement belong to, LOCC operations are of importance because of their locality. As a concept of entanglement is strongly related to the nonlocal properties of a physical state, LOCC operations can not affect the intrinsic nature of entanglement. Using LOCC operations different equivalence classes of states can be defined. Representatives of each class can be used in experiments to perform same tasks, but with a different probability.

# LOCC operations (a very simple example)

Physical processes which are involved in LOCC operations become more plausible by considering the following simple example. Given two observers Alice and Bob, who share two Bell states:

$|\Phi^-\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B - |1\rangle_A \otimes |1\rangle_B)$
$|\Psi^-\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B)$

and provided some classical communication channel (a phone or internet). Alice and Bob can choose one of the two shared states, but the information, which state it exaclty is, is lacking. Using LOCC Alice and Bob can distinguish between these two states. To do so, Alice has just to measure her qubit and send the measurement's outcome to Bob. After receiving this, Bob has to perform a measurement on his qubit, after which Alice and Bob would certainly know which state they had. If, for example, Alice would measure 0 and Bob 1, they measured $|\Phi^-\rangle$.

# LOCC protocols (two qubit case)

The main task of LOCC protocols is to obtain a maximally entangled state with respect to some entanglement measure (entanglement of formation for example). LOCC protocol for two qubit case is a mapping of the form

$\rho^{\prime}=\frac{A\otimes B\rho A^{\dagger}\otimes B^{\dagger}}{Tr\left(A\otimes B\rho A^{\dagger}\otimes B^{\dagger}\right)}$, $A^{\dagger}A\leq\mathbb{I}$ , $B^{\dagger}B\leq\mathbb{I}$.
$Tr\left(A\otimes B\rho A^{\dagger}\otimes B^{\dagger}\right)$ is the probability that protocol succeeds.

Both local operations A and B can be written in form

$A = U_A\begin{pmatrix}\alpha_1 & 0 \\ 0 & \alpha_2\end{pmatrix}U^{\prime}_A$, where U's are unitary, $0<\alpha_{1,2}\leq 1$ and the whole A is invertible.
$F_A=\begin{pmatrix}\alpha_1 & 0 \\ 0 & \alpha_2\end{pmatrix}$ is also called filtering operation.

Considering of the most general protocol where the final state can consist of mixtures of states is redundant since mixing decreases the value of entanglement monotone.

Starting with some mixed state ρ with non-zero entanglement of formation and using LOCC protocol one can obtain some another state $\rho^{\prime}$ with maximum entanglement of formation. This state $\rho^{\prime}$ has a special form namely it is Bell diagonal:

$\rho^{\prime}_{r_1,r_2,r_3}=\frac{1}{4}\left(\mathbb{I}+\sum_{i=1}^3 r_i\sigma_i\otimes\sigma_i\right)$

All ri have the same sign and $r_1\leq r_2\leq r_3$. The Bell diagonal form of the state is unique up to local unitary trasformations and gives a characterization of locally equivalent entangled density matrices.

# Characterization of states via LOCC operations

## Two qubits

Interestingly LOCC operations for mixed states correspond to, what in physics called, Lorentz transformations. Using this correspondance one can show that there exist two different classes of all two qubit states, which can be transformed onto each other by LOCC operations. One of them is the class of states which can be brought into Bell diagonal form leaving the rank of the density matrix constant, another one consists of states which can be brought into Bell diagonal form with lower rank asymptotically.

## Three qubits

Each pure state of three entangled qubits can be converted either to the GHZ-State or to the W-state which leads to two inequivalent ways of entangling three qubits, which represented on the picture.

GHZ type of entanglement: three qubit entanglement is there, but no two qubit entanglement.
W-type of entanglement: no three qubit entanglement, all monotones give value 0. Tracing out a single party provides a Bell pair.

## Four qubits and discussion of general case

Looking at the orbits of LOCC operations one can define equivalence classes of entangled states, which any initial state can be transformed into. Each class of states will correspond to the way, how qubits in the state are entangled. There are nine essentially different classes for nine qubit pure states, but only one of them is generic, the other eight classes have W type entanglement.