# LOCC operations

LOCC - local operations and classical communication - are a certain type of transformation of states 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 other is concerned only with local operations, with possible classical communication (LOCC), performed on the only copy of the state. Therefore they are of special interest from an experimental point of view due to the fact that local measurements can be performed in a simpler way than the non-local ones.

From the point of view of quantum 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 the sharing of a perfectly entangled state between two observers using only LOCC.

Concerning fundamental questions of quantum information theory, of which tasks such as the characterization and general understanding of entanglement belong to, LOCC operations are of importance because of their locality. As the concept of entanglement is strongly related to the nonlocal properties of a physical state, LOCC operations cannot affect the intrinsic nature of entanglement. By using LOCC operations, different equivalence classes of states can be defined; representatives of each class can be used in experiments to perform the 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. Consider two observers, Alice and Bob, who share the 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 are provided some classical communication channel (a phone or internet). Alice and Bob can choose one of the two shared states, but the information about which state it is exactly, is lacking. By 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, then they measured ∣Ψ − .

## 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 the two qubit case is a mapping of the form

$$\rho^{\prime}=\frac{A\otimes B\rho A^{\dagger}\otimes B^{\dagger}}{Tr\left(A\otimes B\rho A^{\dagger}\otimes B^{\dagger}\right)}$$
,

where A † A ≤ I , B † B ≤ I, and Tr(A ⊗ BρA †  ⊗ B † ) is the probability that the protocol succeeds. Both local operations A and B can be written in the form

$$A = U_A\begin{pmatrix}\alpha_1 &amp; 0 \\ 0 &amp; \alpha_2\end{pmatrix}U^{\prime}_A$$
,

where the U's are unitary, $0&lt;\alpha_{1,2}\leq 1$ and the whole of A is invertible. $F_A=\begin{pmatrix}\alpha_1 &amp; 0 \\ 0 &amp; \alpha_2\end{pmatrix}$ is called the filtering operation.

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

Starting with some mixed state ρ with non-zero entanglement of formation and by using a LOCC protocol, one can obtain some other state ρ with maximum entanglement of formation. This state ρ 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 r1 ≤ r2 ≤ r3. 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 is called Lorentz transformations in physics. 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 whilst leaving the rank of the density matrix constant, and the other 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 into either the GHZ-State or the W-state, which leads to two inequivalent ways of entangling three qubits, as shown in the picture.

### 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 in which qubits in the state are entangled. There are nine essentially different classes for nine qubit pure states, but only one of them is generic, whilst the other eight classes have W type entanglement.