| LOCC | |
|---|---|
| Name | Local Operations and Classical Communication |
| Field | Quantum information theory |
| Related | Quantum entanglement, Quantum communication |
LOCC
Local Operations and Classical Communication (LOCC) is an operational paradigm in Quantum information theory that constrains parties to perform quantum operations on their local subsystems and coordinate only by exchanging classical messages. LOCC is central to the study of quantum entanglement as it formalizes which state transformations are possible without global quantum control, thereby underpinning protocols in quantum teleportation, entanglement distillation, and distributed quantum computation.
LOCC denotes the set of protocols where two or more spatially separated agents (commonly called Alice and Bob) apply completely positive trace-preserving maps to their local Hilbert spaces and may adapt subsequent local actions using outcome-dependent classical communication. The model contrasts with global operations that act jointly on the composite Hilbert space and with purely classical communication channels such as those studied in classical information theory. Key primitive operations include local unitary operations, local measurements described by positive operator-valued measures (POVMs), and conditional branches based on exchanged classical bits. The notion of separable operations (SEP) and operations preserving positive partial transpose (PPT) are frequently used to compare operational power against LOCC.
Formally an LOCC protocol is a sequence of rounds. In a bipartite setting on Hilbert spaces H_A and H_B, each round k consists of a local instrument on one party, represented by Kraus operators {A_i^{(k)}} or {B_j^{(k)}}, whose outcome i is communicated classically and conditions the next instrument. The overall map is a quantum channel Λ that can be written as a sum of tensor-product Kraus terms conditional on communication histories. LOCC is not a closed convex set in the operator norm topology, which complicates its mathematical characterization; instead one often works with the larger convex set of separable maps defined by Kraus operators that factorize as sums of local operators. The distinction between one-way LOCC (communication from Alice to Bob only) and two-way LOCC (interactive classical communication) is important for convertibility and resource protocols. The concept of stochastic LOCC (SLOCC) allows probabilistic transformations with nonunit success probability and is mathematically associated with invertible local operators up to normalization.
LOCC defines the operationally relevant equivalence classes of entanglement: two states are considered equally useful for tasks if convertible by deterministic LOCC. Fundamental results such as Nielsen's theorem characterize deterministic pure-state transformations under LOCC via majorization relations on the vector of Schmidt coefficients, linking to majorization theory and matrix analysis. For mixed states, entanglement measures like entanglement of formation, distillable entanglement, and entanglement cost are defined with respect to LOCC asymptotic interconversions. LOCC plays a central role in the study of entanglement monotones—functions that do not increase under LOCC—and in demonstrating phenomena such as bound entanglement and irreversibility in entanglement manipulation.
LOCC is strictly contained within the set of separable (SEP) maps and within the set of maps preserving positive partial transpose (PPT) for bipartite systems, although these containments are strict: there exist SEP or PPT maps that are not implementable by LOCC. Comparisons between LOCC and global operations clarify resource gaps: global unitaries can create entanglement, while LOCC cannot increase entanglement on average. Studies by researchers at institutions such as MIT, Caltech, and University of Cambridge illustrate task separations where SEP outperforms LOCC in exact state discrimination or local cloning. The hierarchy LOCC ⊂ SEP ⊂ PPT is often used to bound achievable transformations and to design approximation schemes for LOCC protocols using semidefinite programming techniques in convex optimization.
LOCC underlies many foundational and practical protocols. In quantum teleportation classical communication plus pre-shared entanglement allows faithful transport of quantum states via LOCC. Entanglement distillation and dilution protocols exploit LOCC to concentrate or dilute entanglement for quantum key distribution protocols such as those studied in BB84-like and Ekert schemes. LOCC-constrained state discrimination and local tomography are crucial in distributed sensing and quantum networks (e.g., quantum repeaters) where quantum channels are costly. In quantum cryptography, LOCC constraints model adversarial limitations in device-independent or local operations-only settings, informing security proofs and resource accounting.
LOCC's limitations motivate resource-theoretic frameworks where entanglement is the resource and LOCC are the free operations. Resource theories formalize monotones, catalytic transformations, and asymptotic conversion rates under LOCC. Limitations include nonexistence of universal catalytic conversion under finite LOCC, irreversibility between entanglement cost and distillable entanglement, and the existence of transformations achievable by SEP but not LOCC. Convertibility criteria range from Nielsen's majorization for pure states to entanglement witnesses and semidefinite programs for mixed states; open problems include a complete operational characterization of LOCC protocols, the role of communication complexity in multi-party LOCC, and tight bounds on rates in entanglement manipulation. Work by authors associated with Peres, Bennett, Nielsen, Vidal, and research groups at IBM Research and QuTech continues to refine operational and mathematical boundaries of LOCC in practical quantum technologies.
Category:Quantum information theory Category:Quantum entanglement