LOCC and Entanglement

The entire resource theory of entanglement rests on one operational axiom: LOCC cannot create entanglement. This lesson proves why, makes precise what LOCC can and cannot do, and states Nielsen's theorem — the exact condition for converting one pure entangled state into another.

What LOCC is

Local operations and classical communication is the class of protocols where spatially separated parties may:

apply any quantum operation (unitaries, measurements, ancillas, discarding) to their own subsystem, and
send each other classical messages, conditioning future local operations on what they hear.

What is forbidden is any global operation across the parties and any exchange of quantum systems. LOCC is the natural notion of "free" operations when the parties are distant and only a classical channel connects them. Entanglement is precisely the resource that LOCC cannot supply.

Why LOCC cannot create entanglement

The cleanest argument tracks the separable structure of states. Call a state separable if it is a convex mixture of products, $\rho = \sum_k p_k\,\rho_k^{A}\otimes\sigma_k^{B}\otimes\cdots$ . Start from a product (hence separable) state and check that every LOCC primitive maps separable states to separable states:

A local unitary $U_A$ on Alice acts as $\rho_k^A \mapsto U_A\rho_k^A U_A^\dagger$ inside each term — the product structure is untouched.
A local measurement by Alice with Kraus operators $\{M_a\}$ produces, on outcome $a$ , the state $\propto \sum_k p_k (M_a\rho_k^A M_a^\dagger)\otimes\sigma_k^B$ — still a sum of products.
Classical communication of the outcome $a$ lets Bob pick a local operation $\Lambda_a^B$ ; this acts term-by-term as $\sigma_k^B \mapsto \Lambda_a^B(\sigma_k^B)$ — again product structure inside each term.
Adding/discarding local ancillas tensors or traces a local factor — product structure preserved.

Composing these, any LOCC map sends $\rho = \sum_k p_k\bigotimes_j\rho_k^{(j)}$ to another state of the same separable form. So the set of separable states is closed under LOCC. Since a product state is separable and LOCC keeps it separable, no LOCC protocol can produce an entangled state from an unentangled one. Entanglement can only be consumed or redistributed, never manufactured.

LOCC vs. separable operations

A subtlety worth flagging: the set of LOCC maps is a strict subset of the separable operations — maps of the form $\rho \mapsto \sum_k (A_k\otimes B_k)\,\rho\,(A_k\otimes B_k)^\dagger$ . Every LOCC map is separable, but not every separable map is implementable by LOCC (the famous "nonlocality without entanglement" examples). Both classes cannot create entanglement; LOCC is simply the physically operational subclass, and it is notoriously hard to characterize because protocols may use unboundedly many rounds of back-and-forth communication.

Nielsen's theorem: when can pure states be converted?

For deterministic, single-copy conversion between pure bipartite states, Nielsen gave a complete answer in terms of majorization. Let $|\psi\rangle$ and $|\phi\rangle$ have Schmidt-coefficient vectors $\lambda_\psi$ and $\lambda_\phi$ (the eigenvalues of their reduced states, sorted in decreasing order). Then

|\psi\rangle \xrightarrow{\text{LOCC}} |\phi\rangle \quad\Longleftrightarrow\quad \lambda_\psi \prec \lambda_\phi,

i.e. $\lambda_\psi$ is majorized by $\lambda_\phi$ : $\sum_{i=1}^{k}\lambda_\psi^{\downarrow}(i) \leq \sum_{i=1}^{k}\lambda_\phi^{\downarrow}(i)$ for every $k$ (with equality at $k = d$ ). Two readings make this intuitive:

A more entangled state (flatter, more spread-out Schmidt vector) can be converted down to a less entangled one, but not vice versa — consistent with monotonicity.
Some pairs are incomparable: neither majorizes the other, so neither can be deterministically obtained from the other by LOCC. Entanglement is not even totally ordered for pure states.