LOCC Preview

LOCC stands for local operations and classical communication. It is the standard operational setting for asking which transformations are possible when separated parties can manipulate their own subsystems, exchange ordinary classical messages, but cannot send quantum systems to each other.

For this volume, LOCC has one main purpose: it explains why entanglement behaves like a resource. Separable states are free to prepare by local operations and shared randomness; entangled states are not. Once shared, entanglement can be transformed, degraded, concentrated, or consumed by protocols that use only local devices and classical messages.

The full theory of entanglement manipulation, teleportation, distillation, channel capacities, and quantum communication protocols belongs in quantum information. Entanglement Distillation gives a concrete branch-resolved example in which accepted Bell-pair fidelity rises while the success probability and consumed pairs preserve average resource accounting. This page supplies the structural preview needed to understand why bipartite entanglement measures are required to be monotone under local operations and classical communication.

Local Operations

Fix a bipartite decomposition

$$\mathcal H_{AB} = \mathcal H_A\otimes\mathcal H_B.$$

A local operation is a physically allowed quantum operation applied to one subsystem, possibly with a local apparatus or local ancilla, while the other named subsystem is not acted on directly.

For example, a nonselective local operation on $A$ can be represented in finite dimensions by operators $\{K_\alpha\}$ on $\mathcal H_A$ satisfying

$$\sum_\alpha K_\alpha^\dagger K_\alpha = I_A.$$

It acts on the joint state as

$$\rho_{AB} \longmapsto \rho'_{AB} = \sum_\alpha (K_\alpha\otimes I_B) \rho_{AB} (K_\alpha^\dagger\otimes I_B).$$

Local unitaries are the reversible special case with one Kraus operator $K=U_A$. Local measurements, local noise, adding and discarding local ancillas, and resetting a local system are also local operations.

The word local is relative to the specified subsystem split. An operation may be local for one decomposition and nonlocal for another. This is the same convention used in Local Unitary Equivalence and Entanglement Depends on a Decomposition.

Classical Communication

Classical communication means that the parties can send classical data, such as measurement outcomes, basis choices, or instructions. A message may influence which local operation is performed later.

A simple one-way adaptive protocol has this form:

1. Alice applies a measurement operation with outcome $a$.
2. Alice sends the classical label $a$ to Bob.
3. Bob chooses a local operation depending on $a$.

In a simplified single-Kraus-per-outcome notation, an unnormalized branch state has the form

$$\widetilde\rho_{ab} = (K_a\otimes L_{b\vert a}) \rho_{AB} (K_a^\dagger\otimes L_{b\vert a}^\dagger),$$

where $K_a$ acts on $A$ and $L_{b\vert a}$ acts on $B$. The branch probability is

$$p(a,b) = \operatorname{Tr}(\widetilde\rho_{ab}),$$

and, when $p(a,b)>0$, the normalized branch state is

$$\rho_{AB\vert ab} = \frac{\widetilde\rho_{ab}}{p(a,b)}.$$

Realistic instruments may have several Kraus operators for one recorded outcome. The essential point is unchanged: the operations act locally, while the choice of later operations may depend on classical records. The Conditional States page develops the state-update bookkeeping for outcome-conditioned states.

An LOCC protocol may have many rounds and may be adaptive in both directions. The precise mathematical closure of all possible LOCC protocols has subtleties, so this page uses LOCC as an operational class rather than as a formal taxonomy.

Why LOCC Cannot Create Entanglement

The key structural fact is:

$$\text{separable input} \quad \xrightarrow{\ \operatorname{LOCC}\ } \quad \text{separable output}.$$

First consider a product local channel $\mathcal E_A\otimes\mathcal F_B$. If

$$\rho_{AB} = \sum_r p_r\, \sigma_A^{(r)} \otimes \tau_B^{(r)}$$

is separable, then

$$(\mathcal E_A\otimes\mathcal F_B)(\rho_{AB}) = \sum_r p_r\, \mathcal E_A(\sigma_A^{(r)}) \otimes \mathcal F_B(\tau_B^{(r)}),$$

which is still a convex mixture of product states.

Measurements and classical messages only refine this argument. Suppose an LOCC branch labeled by $m$ has nonzero probability. Starting from a separable input, the unnormalized branch can always be written as a sum of product positive operators:

$$\widetilde\rho_{AB\vert m} = \sum_r q_{r,m}\, \widetilde\sigma_{A,m}^{(r)} \otimes \widetilde\tau_{B,m}^{(r)}, \qquad q_{r,m}\ge0.$$

After division by the branch probability, the conditioned branch remains separable:

$$\rho_{AB\vert m} = \frac{\widetilde\rho_{AB\vert m}} {\operatorname{Tr}\widetilde\rho_{AB\vert m}}.$$

Thus even postselection cannot turn a genuinely separable input into an entangled output by LOCC. Postselection can increase the entanglement of some branches when the input was already entangled, but the probabilities and the other branches must be included in any monotonicity statement.

This is the operational content behind the definition of Separable Mixed States: they are exactly the bipartite states that can be assembled from local quantum states with classical randomness.

Entanglement as a Resource

If LOCC operations are considered free, entangled states are nonfree resources. The resource is not energy or particle number; it is nonseparability across a specified split.

The resource-theoretic viewpoint explains several otherwise separate facts:

• Local unitaries preserve entanglement exactly because they are reversible LOCC operations.
• Local measurements can destroy entanglement because LOCC is allowed to discard information into classical records or environments.
• Classical communication can create classical correlation but cannot create entanglement from separable states.
• Shared entanglement can enable tasks that are impossible with LOCC alone, such as teleportation-like state transfer protocols.
• Any proposed entanglement measure must not increase under LOCC, at least on average over outcomes.

The slogan is:

$$\text{entanglement is what is unavailable for free under LOCC.}$$

This is more operational than simply saying “an entangled state is not separable.” Separability gives the structural definition; LOCC explains why the distinction has task-level consequences.

A Bell-Pair Measurement Example

Consider the Bell state

$$\lvert\Phi^+\rangle = \frac{1}{\sqrt2} \bigl( \lvert00\rangle+\lvert11\rangle \bigr).$$

If Alice measures her qubit in the computational basis and sends the outcome to Bob, the conditioned branch states are product:

$$\rho_{AB\vert 0} = \lvert00\rangle\langle00\rvert, \qquad \rho_{AB\vert 1} = \lvert11\rangle\langle11\rvert.$$

If the outcome is ignored but the measurement has occurred, the joint state becomes

$$\rho'_{AB} = \frac12 \lvert00\rangle\langle00\rvert + \frac12 \lvert11\rangle\langle11\rvert.$$

This state is separable but classically correlated. Alice’s classical message lets Bob sort the data into correlated branches, but it does not recreate the lost Bell entanglement. The example shows how LOCC can convert entanglement into ordinary shared classical information.

Entanglement Conversion Preview

For bipartite pure states, the Schmidt coefficients provide a sharp first glimpse of LOCC convertibility. Write

$$\lvert\psi\rangle = \sum_i \sqrt{\lambda_i^\psi}\, \lvert i_A i_B\rangle, \qquad \lvert\phi\rangle = \sum_i \sqrt{\lambda_i^\phi}\, \lvert i'_A i'_B\rangle,$$

where the Schmidt probability vectors are sorted in decreasing order:

$$\lambda_1\ge\lambda_2\ge\cdots, \qquad \sum_i\lambda_i=1.$$

Nielsen’s pure-state conversion theorem says that deterministic LOCC conversion

$$\lvert\psi\rangle \longrightarrow \lvert\phi\rangle$$

is possible exactly when

$$\boldsymbol\lambda^\psi \prec \boldsymbol\lambda^\phi,$$

meaning

$$\sum_{i=1}^k \lambda_i^\psi \le \sum_{i=1}^k \lambda_i^\phi \quad \text{for every }k, \qquad \sum_i\lambda_i^\psi = \sum_i\lambda_i^\phi =1.$$

The direction is important. More entangled pure states have more uniform Schmidt vectors. A uniform vector is majorized by a more peaked vector, so deterministic LOCC can degrade pure-state entanglement but cannot make it more uniform.

For two qubits,

$$\left( \frac12,\frac12 \right) \prec (1,0),$$

so a Bell pair can be converted to a product state by LOCC. The reverse conversion is impossible by LOCC because

$$(1,0) \not\prec \left( \frac12,\frac12 \right).$$

Similarly,

$$(0.6,0.4) \prec (0.8,0.2),$$

so a partially entangled two-qubit pure state with Schmidt vector $(0.6,0.4)$ can be deterministically converted into a less entangled one with Schmidt vector $(0.8,0.2)$.

Mixed-state conversion, probabilistic conversion, distillation, dilution, catalytic transformations, and asymptotic rates are substantially richer. They are quantum-information topics; the role of this page is only to show why Schmidt data have operational meaning.

LOCC Monotonicity

An entanglement monotone is a quantity that does not increase under LOCC. For a selective protocol with outcomes $m$, probabilities $p_m$, and output states $\rho_m$, the usual average monotonicity condition is

$$E(\rho) \ge \sum_m p_m E(\rho_m).$$

For pure bipartite states, the Entanglement Entropy is the central example:

$$E(\psi) = S(\rho_A) = S(\rho_B).$$

Concurrence for Two Qubits and Negativity and PPT Criterion are useful mixed-state diagnostics in their proper domains, but operational entanglement theory asks an additional question: how does the quantity behave under all allowed LOCC protocols?

Average monotonicity does not say that every selected branch must have no more entanglement than the input. A rare successful branch may be more entangled than the initial state, provided the success probability and the other branches keep the average from increasing. This is why postselection must be handled carefully.

What LOCC Is Not

LOCC is not the same as applying only local unitaries. Local unitaries are reversible LOCC operations; LOCC also includes irreversible operations such as measurements, noise, discarding systems, and conditioning on classical messages.

LOCC is not the same as no-signaling. There are mathematical operations that do not allow faster-than-light signaling but are not implementable by LOCC. Conversely, LOCC explicitly allows classical communication, so it is not a spacelike separated operation class.

LOCC is not the same as all separable operations. A separable operation has Kraus operators that can be written in product form, but not every separable operation is achievable by a finite-round LOCC protocol. The distinction matters in quantum information, though it is usually beyond what is needed for a first entanglement course.

LOCC is also not permission to send a quantum system. If Alice physically sends a qubit to Bob through a quantum channel, the operation is no longer LOCC. Shared entanglement plus classical communication can simulate some communication tasks when the right extra resources are present, but those protocols are not the same as creating entanglement from nothing.

Why the Full Theory Belongs Elsewhere

This page belongs in the composite-systems volume because it explains the operational constraint behind separability, entanglement measures, and Schmidt coefficients. The canonical home for a central two-way LOCC protocol is Entanglement Distillation.

Resource Theories owns the cross-theory conversion grammar; this page remains the structural preview of LOCC and entanglement monotonicity.

The full subject includes:

• teleportation, dense coding, and entanglement-assisted communication;
• entanglement cost and distillable entanglement;
• one-way versus two-way LOCC;
• separable operations and non-entangling operations;
• resource theories of states, channels, measurements, and reference frames;
• asymptotic conversion rates and finite-blocklength corrections.

Those topics require quantum channels, coding theorems, protocol models, and complexity distinctions that go beyond the structural role of entanglement in composite quantum mechanics.

Common Mistakes

• Thinking classical communication can create entanglement. It can coordinate local actions and create classical correlations, but it cannot turn a separable input into an entangled state.
• Confusing local-unitary equivalence with LOCC convertibility. Local unitaries are reversible; general LOCC transformations may be irreversible and probabilistic.
• Ignoring branch probabilities after postselection. A high-entanglement success branch does not by itself violate monotonicity.
• Treating every non-entangling mathematical map as LOCC. LOCC is an operationally restricted class.
• Forgetting that local means local relative to a specified tensor-product decomposition.
• Assuming mutual information is an entanglement monotone. It measures total correlation, including classical correlation.

Cross-Links

• Separable Mixed States
• Entangled States
• Local Unitary Equivalence
• Classical Correlation versus Entanglement
• Entanglement Depends on a Decomposition
• Bell States
• Conditional States
• Schmidt Decomposition
• Entanglement Entropy
• Concurrence for Two Qubits
• Negativity and PPT Criterion
• Entanglement Witnesses
• Mutual Information
• Entanglement Sharing
• Entanglement in Quantum Information
• State Update Rule

Exercises

1. Let

$$\rho_{AB} = \sum_r p_r\, \sigma_A^{(r)} \otimes \tau_B^{(r)}$$

be separable. Show that a local unitary $U_A\otimes U_B$ maps it to another separable state.

Solution

The transformed state is

$$(U_A\otimes U_B) \rho_{AB} (U_A^\dagger\otimes U_B^\dagger) = \sum_r p_r\, U_A\sigma_A^{(r)}U_A^\dagger \otimes U_B\tau_B^{(r)}U_B^\dagger.$$

Each term is still a product density operator, and the coefficients are still nonnegative and sum to one. Therefore the state remains separable.

2. Alice and Bob share $\lvert\Phi^+\rangle$. Alice measures in the computational basis and Bob learns the outcome. Compute the unconditioned post-measurement state and decide whether it is entangled.

Solution

The two outcomes occur with probability $1/2$. The conditioned states are $\lvert00\rangle\langle00\rvert$ and $\lvert11\rangle\langle11\rvert$. If the outcome is ignored, the state is

$$\rho'_{AB} = \frac12 \lvert00\rangle\langle00\rvert + \frac12 \lvert11\rangle\langle11\rvert.$$

This is a convex mixture of product states, so it is separable. It contains classical correlation but no entanglement.

3. Use Nielsen’s majorization condition to decide which deterministic LOCC conversions are possible:

$$(0.7,0.3) \to (0.9,0.1), \qquad (0.9,0.1) \to (0.7,0.3).$$

Solution

For two-component sorted probability vectors, $\lambda\prec\mu$ is equivalent to $\lambda_1\le\mu_1$. Since $0.7\le0.9$,

$$(0.7,0.3) \prec (0.9,0.1),$$

so the first conversion is possible by deterministic LOCC. The reverse would require $0.9\le0.7$, which is false, so the second conversion is impossible. The allowed direction reduces entanglement by making the Schmidt vector more peaked.

4. A protocol sometimes produces a maximally entangled pair from a partially entangled input, but succeeds only with probability $p<1$. Why does this not contradict LOCC monotonicity?

Solution

LOCC monotonicity is normally an average statement:

$$E(\rho) \ge \sum_m p_m E(\rho_m).$$

A successful branch may have more entanglement than the input, but the failed branches and the success probability must be included. The protocol is not creating free entanglement; it is probabilistically concentrating entanglement already present in the input.

5. Explain why sending a qubit from Alice to Bob is not an LOCC operation, even if Alice also sends classical messages.

Solution

LOCC allows local quantum operations and classical communication only. A transmitted qubit is quantum communication: it carries an unknown quantum state and can transmit entanglement or coherent superposition. That physical resource is outside LOCC. Classical messages can coordinate Bob’s local operations, but they cannot replace the act of sending a quantum system unless some additional resource, such as shared entanglement, is explicitly supplied in a protocol.

References

• C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating Partial Entanglement by Local Operations,” Physical Review A 53, 2046-2052, 1996.
• C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-State Entanglement and Quantum Error Correction,” Physical Review A 54, 3824-3851, 1996.
• M. A. Nielsen, “Conditions for a Class of Entanglement Transformations,” Physical Review Letters 83, 436-439, 1999.
• G. Vidal, “Entanglement Monotones,” Journal of Modern Optics 47, 355-376, 2000.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum Entanglement,” Reviews of Modern Physics 81, 865-942, 2009.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018.