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Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)
Eric Chitambar, Debbie Leung, Laura Mancinska, Maris Ozols, Andreas Winter
TL;DR
LOCC is physically important but mathematically difficult to characterize, especially relative to separable operations and when protocols use unbounded communication rounds. The paper formalizes LOCC through quantum instruments and related closure classes. It shows that LOCC is not closed, while fixed finite-round LOCC subsets are compact and an open neighborhood of the depolarizing map is LOCC; a two-qubit map is approximable but not exactly implementable.
Problem
LOCC has an intuitive operational description but lacks a precise mathematical characterization comparable to separable operations, particularly for unbounded-round and approximate protocols.
Method
The paper describes LOCC and related operational classes using quantum instruments, including finite-round, infinite-round, and closure constructions.
Results
LOCC is not closed, fixed finite-round finite-outcome LOCC subsets are compact, an open ball around the completely depolarizing map is LOCC, and a two-qubit map is approximable but not perfectly implementable.
Takeaways & Limitations
LOCC and its topological closure must be distinguished when analyzing whether quantum instruments are exactly implementable or only approximable.
Takeaways & Limitations
For quantum channels without classical registers, whether the same LOCC–SEP separation results hold remains unknown, and one coarse-grained channel is conjectured but not proved infeasible by LOCC.
Abstract
from arXiv · showhide
In this paper we study the subset of generalized quantum measurements on finite dimensional systems known as local operations and classical communication (LOCC). While LOCC emerges as the natural class of operations in many important quantum information tasks, its mathematical structure is complex and difficult to characterize. Here we provide a precise description of LOCC and related operational classes in terms of quantum instruments. Our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds. While the set of LOCC is not topologically closed, we show that finite round LOCC constitutes a compact subset of quantum operations. Additionally we show the existence of an open ball around the completely depolarizing map that consists entirely of LOCC implementable maps. Finally, we demonstrate a two-qubit map whose action can be approached arbitrarily close using LOCC, but nevertheless cannot be implemented perfectly.
1 Introduction
LOCC captures distributed quantum operations using local actions and classical communication, but its mathematical structure is substantially subtler than that of separable operations. The paper develops an instrument-based formalism to define finite-round, infinite-round, and closure classes precisely.
- Motivation: LOCC restricts distributed parties to local quantum operations while permitting classical communication, including shared randomness and prior measurement results.This makes LOCC a physically motivated subset of globally realizable quantum operations for studying correlations and resource transformations.
- Motivation: Teleportation illustrates LOCC resource transformation: one qubit can be transmitted using one ebit and two cbits.The protocol makes LOCC universal for implementing quantum operations when suitable entanglement resources are available.
- Why LOCC is subtle: LOCC can yield less accessible information than global measurements even for ensembles of bipartite product states, demonstrating nonlocality without entanglement.These examples distinguish nonlocality from entanglement and show that LOCC restrictions matter independently of shared entanglement.
- Why LOCC is subtle: Unlike SEP, whose maps are characterized by separable Choi matrices, LOCC lacks an equally simple mathematical characterization.The distinction between SEP and LOCC contributes to the difficulty of describing LOCC precisely.
- Paper approach: The paper uses quantum instruments to formalize LOCC, including unbounded-round protocols and LOCC-closure, and studies their topological relationships.It also investigates compactness, interior structure, and an example approximable by LOCC but not exactly implementable.
2 How to define LOCC?
The paper defines quantum instruments as outcome-indexed completely positive maps and builds LOCC classes by conditional local operations, coarse-graining, and convergence. This framework distinguishes finite-round, infinite-round, and closure-based notions of implementability.
- Quantum instruments: A quantum instrument is a finite- or countably indexed family of completely positive maps whose sum is trace-preserving.For input state ρ, each map gives an unnormalized postmeasurement state, with its trace equal to the outcome probability.
- Quantum instruments: Quantum-classical maps encode each instrument outcome in an orthonormal classical register, establishing a one-to-one correspondence with instruments.This correspondence permits distances between instruments to be defined through associated QC maps and the diamond norm.
- Quantum instruments: Coarse-graining combines outcome maps by a partition, corresponding physically to post-processing or discarding classical outcome information.A general instrument can be implemented by fine-grained measurement followed by coarse-graining.
- LOCC construction: LOCC-linked instruments apply conditional one-way local instruments after earlier outcomes, allowing the acting party to depend on those outcomes before coarse-graining.This conditional composition is the basic step used to build multiround protocols.
- LOCC classes: LOCC1 consists of one-way local instruments followed by coarse-graining; LOCCr extends this recursively to r rounds, while LOCCN contains all finite-round protocols.LOCC includes limits of finite-round sequences, and LOCC-closure is the topological closure of LOCCN.
- LOCC classes: Finite-round approximation can improve within one fixed LOCC protocol, whereas instruments in LOCCN \ LOCC require different protocols for different approximation accuracies.This distinction explains why the finite-round closure and LOCC itself are operationally different.
- Relations to SEP: Every separable instrument admits a stochastic LOCC implementation with nonzero success probability and a completely depolarizing failure outcome.The paper also provides a lower bound on that success probability.
3 What is the shape of LOCC?
The paper establishes several geometric properties of LOCC: convexity, a nonempty interior around the completely depolarizing instrument, and compactness for fixed finite-round, finite-outcome protocols. These results coexist with the non-closedness of broader LOCC classes.
- Topological distinctions: LOCC is not closed, so it is not compact, while finite-round fixed-outcome LOCC is compact.The paper uses this contrast to separate the topology of finite-round protocols from broader LOCC classes.
- Convexity and interior: LOCC is convex because globally accessible randomness can select between two LOCC protocols in the first round.The resulting random mixture implements the corresponding convex combination of instruments.
- Convexity and interior: LOCC with a finite outcome index set has a nonempty interior containing an open ball around the completely depolarizing instrument.The proof uses separability properties of operators near the identity and stochastic LOCC implementation.
- Convexity and interior: A separable instrument can be decomposed into a depolarizing component and another separable instrument, enabling an LOCC implementation near the depolarizing map.The construction combines stochastic LOCC with convexity and coarse-graining of the failure outcome.
- Compactness: For any r and m, the subset of m-outcome instruments in LOCCr is compact.The bounded parametrization reduces the feasible set to a closed and bounded collection of algebraic constraints.
- Compactness: Finite-round LOCC instruments admit implementations with bounded numbers of measurement outcomes per round, using at most mD^4(r−l+1) CP maps in round l.Here r is the number of rounds, m the final number of outcomes, and D the global dimension.
4 Is LOCC closed?
The paper constructs a two-qubit instrument that is the limit of LOCC instruments but is not itself implementable by LOCC. Its impossibility follows from an entanglement monotone that must decrease under a nontrivial local measurement, although the target transformation preserves it on average.
- 4 Is LOCC closed?: A bipartite instrument acting on two qubits lies in LOCC but not in LOCC.The authors construct LOCC instruments converging to it, then prove exact LOCC infeasibility.
- 4 Is LOCC closed?: The approximating instruments repeat local measurements until a nonzero joint outcome occurs or a maximum of ν iterations is reached.Alice and Bob share outcomes, stop on 01, 10, or 11, and repeat after 00 before coarse-graining outcomes.
- 4 Is LOCC closed?: The sequence converges to the target instrument through limits of the corresponding Choi matrices.Because the Hilbert-space dimension is four, entrywise Choi-matrix convergence is equivalent to convergence of the instruments.
- 4.2 Digression: Random Concurrence Distillation: Random concurrence distillation converts a W-class state into a bipartite pure state held by either of two pairs while maximizing expected concurrence.The task generalizes fixed-pair concurrence distillation and random EPR combing.
- 4.3 LOCC Impossibility: The target transformation has ˆC(W)=8/9 and final concurrence C(ω)=8/9 in both outcomes, so its average monotone value is preserved.Any first non-unitary local measurement would strictly decrease the monotone, proving the transformation and original instrument infeasible by LOCC.
5 What did we learn?
The paper clarifies LOCC's structure while identifying unresolved separations between LOCC, SEP, and POVM-based operational classes. It also proposes distance-based questions for future study.
- LOCC instruments on two qubits are not closed, resolving an open problem and underscoring LOCC's complexity in small systems.
- Known separations between LOCC and SEP rely on classical information obtained from quantum measurements.
- For quantum channels without classical registers, whether the same LOCC–SEP separation holds remains unknown.
- The authors conjecture that coarse-graining the three maps in Eq. (11) does not yield an LOCC-feasible channel, but their proof techniques cannot establish this.
- The paper raises whether LOCC and SEP coincide for POVMs, while related random-distillation separations depend crucially on quantum outputs.
- Future work includes measuring the distance from separable instruments to the closest LOCC instrument and relating that distance to required nonlocal resources.
A Proof of Theorem 2
The proof compresses general r-round LOCC protocols into bounded, measurement-ordered protocols by normalizing local maps and using convex-hull decompositions. The resulting protocol preserves the round count while limiting the number of CP maps at each level.
- Theorem 2: Theorem 2 bounds each round's instrument by at most mD^4(r−l+1) CP maps of the form M(·)M†.Here D is the product of the parties' local dimensions.
- Protocol representation: A general r-round LOCC protocol is represented as a tree with r levels, whose nodes encode measurement histories and one-way local instruments.
- Protocol normalization: Fine-graining and polar decomposition make intermediate maps dimension-preserving, moving isometries into the next level without increasing the number of rounds.
- Protocol normalization: Measurement-ordered expansion inserts trivial maps so one predetermined party acts non-trivially at every node of a level, after which compression reverses this expansion.The expansion uses no more than Nr levels.
- Operator representation: For the four-round Alice–Bob–Alice–Bob case, the instrument is rewritten using an operator representation with copied local Hilbert spaces and coarse-graining variables.
- Convex reduction: Carathéodory's theorem repeatedly replaces convex combinations with finite ones, bounding the required measurement branches while preserving trace preservation.The theorem represents points in a convex hull using at most n+1 elements.
B Proof of Lemma 2
Lemma 2 establishes that the function C does not increase on average under LOCC measurements and decreases strictly for specified non-trivial local measurements. The proof analyzes binary measurements across distinct W-class cases.
- Lemma 2: C is non-increasing on average under LOCC and strictly decreasing when party ⋆ or n1 performs a non-trivial measurement.
- Proof strategy: The proof reduces arbitrary local measurements to sequences of binary measurements and tracks the resulting W-class state parameters.
- Measurement constraints: Completeness imposes a1 + a2 = 1 and c1 + c2 ≤ 1, with equality exactly when both bλ coefficients vanish.
- Case analysis: When the measured party is ⋆, the average change in C reaches equality only for the trivial measurement.
- Case analysis: For unequal xn1 and xn2, sufficiently weak measurements preserve the party ordering, and ΔC is non-positive near (a1,c1) = (1/2,1/2).
- Case analysis: When xn1 = xn2, post-measurement ordering may change, but pλC(x⃗λ) increases with cλ and equality again requires a trivial measurement.