| stochastic local operations and classical communication | |
|---|---|
| Name | Stochastic local operations and classical communication |
| Caption | Schematic of local operations and classical communication between parties sharing an entangled state |
| Type | Quantum operation class |
| Field | Quantum information science |
| Introduced | 1990s |
| Related | Local operations and classical communication, Entanglement theory, SLOCC classification |
stochastic local operations and classical communication
Stochastic local operations and classical communication (SLOCC) denotes the class of quantum operations in which spatially separated parties apply local quantum operations and conditional measurements, supplemented by unlimited classical communication of measurement outcomes, with the possibility of non-unit probability (i.e., success only on some branches). SLOCC is central to the study of entanglement convertibility and resource interconversion in Quantum information and Quantum computing, because it captures which pure states can probabilistically simulate one another under locality constraints.
In formal terms, an n-party SLOCC protocol consists of local completely positive trace-nonincreasing maps on each subsystem, implemented by Kraus operators {A_i^(k)} for party k, together with rounds of classical communication that condition subsequent local maps on earlier outcomes. A successful SLOCC transformation from state |ψ⟩ to |φ⟩ exists if there are invertible local operators G_k (not necessarily unitary) such that |φ⟩ ∝ (⊗_k G_k)|ψ⟩ up to normalization, corresponding to nonzero-probability branches of a local measurement sequence. The concept relies on the mathematics of Kraus operators, completely positive map, and stochastic matrix-like conditioning, and often uses notions from linear algebra (rank, determinant) and algebraic geometry (orbit classification).
SLOCC provides an equivalence relation coarser than deterministic LOCC equivalence: two pure multipartite states are SLOCC-equivalent if they can be converted into each other with nonzero probability by local operations and classical communication. This relation partitions the space of pure states into finite or infinite families of SLOCC classes, which serve as canonical labels for qualitatively different types of multipartite entanglement. Notable uses appear in analyses of entanglement measures like Schmidt rank, concurrence, tangle, and in proofs of nonlocality or convertibility constraints studied at institutions such as Perimeter Institute for Theoretical Physics and Institute for Quantum Computing.
For two-party pure states bipartite SLOCC classification reduces to the Schmidt decomposition: all nonseparable states with the same Schmidt rank form a SLOCC class. For three qubits, the seminal result by Dür, Vidal and Cirac identified two inequivalent genuine tripartite SLOCC classes represented by the GHZ state and the W state. For higher numbers of qubits or higher local dimension, SLOCC classification becomes richer and was studied in works by researchers at Caltech, Massachusetts Institute of Technology, and University of Cambridge using algebraic invariants, entanglement polytopes, and numerical algebraic geometry. SLOCC classes often correspond to orbits under the action of the group GL(d,C)^{⊗n} on the Hilbert space.
Operationally, SLOCC captures probabilistic state conversion protocols such as entanglement distillation, probabilistic entanglement swapping, and protocol branches that are postselected on particular measurement records. Example conversions include probabilistic conversion of a three-qubit GHZ-type state to a W-type state being impossible under SLOCC equivalence (they lie in distinct classes), while certain asymptotic or catalytic protocols studied in quantum resource theories can achieve transformations with vanishing failure probability in the limit. Practical implementations and demonstrations of SLOCC-inspired protocols have appeared in experiments using photonic quantum information platforms, trapped ions, and superconducting qubits.
Mathematically, SLOCC orbits are studied via representation theory and invariant theory for the group GL(d,C)^{⊗n}. Polynomial invariants such as the determinant, hyperdeterminant, and other entanglement invariants (for example, the three-tangle introduced by Coffman, Kundu, and Wootters) distinguish inequivalent SLOCC classes. SLOCC conversion criteria frequently reduce to rank conditions of reduced density matrices, vanishing of covariants, or existence of invertible local operators. The geometry of SLOCC orbits connects to Geometric invariant theory and stratification of projective Hilbert spaces into orbit closures, relevant to algorithms in computational algebraic geometry.
SLOCC is weaker than deterministic LOCC because SLOCC allows transformations that succeed only probabilistically, while LOCC requires deterministic or trace-preserving protocols. Conversely, SLOCC is stronger than separable operations in that SLOCC focuses on invertible local filters and group orbits rather than general separable maps. Comparative studies consider separable operations (SEP), positive partial transpose (PPT) operations, and nonlocal operations permitted by shared entanglement or quantum channels. The hierarchy LOCC ⊂ SEP ⊂ PPT is complemented by SLOCC as an equivalence notion rather than an operational closure.
SLOCC classification informs resource-based tasks: multipartite entanglement resource allocation, design of quantum error correction encodings, protocols for quantum communication like teleportation networks and secret sharing, and identification of states enabling universal measurement-based quantum computation (MBQC). In cryptography, SLOCC distinctions affect multiparty quantum key distribution protocols and thresholds for entanglement-based security. Theoretical results on SLOCC feed into algorithmic studies at research centers such as IBM Research, Google Quantum AI, and academic groups exploring entanglement transformations, resource interconversion, and the role of stochastic filtering in noisy intermediate-scale quantum (NISQ) devices.