LOCC protocols with bounded width per round optimize convex functions

Abstract

We start with the task of discriminating finitely many multipartite quantum states using LOCC protocols, with the goal to optimize the probability of correctly identifying the state. We provide two different methods to show that finitely many measurement outcomes in every step are sufficient for approaching the optimal probability of discrimination. In the first method, each measurement of an optimal LOCC protocol, applied to a $d_{\mathrm{loc}}$-dim local system, is replaced by one with at most $2d_{\mathrm{loc}}^2$ outcomes, without changing the probability of success. In the second method, we decompose any LOCC protocol into a convex combination of a number of "slim protocols" in which each measurement applied to a $d_{\mathrm{loc}}$-dim local system has at most $d_{\mathrm{loc}}^2$ outcomes. To maximize any convex functions in LOCC (including the probability of state discrimination or fidelity of state transformation), an optimal protocol can be replaced by the best slim protocol in the convex decomposition without using shared randomness. For either method, the bound on the number of outcomes per measurement is independent of the global dimension, the number of parties, the depth of the protocol, how deep the measurement is located, and applies to LOCC protocols with infinite rounds, and the "measurement compression" can be done "top-down" -- independent of later operations in the LOCC protocol. The second method can be generalized to implement LOCC instruments with finitely many outcomes: if the instrument has $n$ coarse-grained final measurement outcomes, global input dimension $D_0$ and global output dimension $D_i$ for $i=1,...,n$ conditioned on the $i$-th outcome, then one can obtain the instrument as a convex combination of no more than $R={\sum }_{i=1}^n{D}_0^2{D}_i^2-D_0^2+1$ slim protocols; in other words, ${\log }_{2}R$ bits of shared randomess suffice.

arXiv commentsee related results by Scott Cohen

References in corpus (36)

• Mixed State Entanglement and Quantum Error Correction
• Quantum Nonlocality without Entanglement
• Unextendible Product Bases and Bound Entanglement
• Local Distinguishability of Multipartite Orthogonal Quantum States
• Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)
• Quantum Data Hiding
• Distinguishability of the Bell states
• Hiding bits in Bell states
• Distinguishability and indistinguishability by LOCC
• Local indistinguishability: more nonlocality with less entanglement
• Local indistinguishability and LOCC monotones
• Classical randomness in quantum measurements
• Hiding classical data in multi-partite quantum states
• Bounds on Multipartite Entangled Orthogonal State Discrimination Using Local Operations and Classical Communication
• Distinguishing Bipartitite Orthogonal States using LOCC: Best and Worst Cases
• Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding
• Distinguishability of maximally entangled states
• Four Locally Indistinguishable Ququad-Ququad Orthogonal Maximally Entangled States
• Bipartite subspaces having no LOCC-distinguishable bases
• Distinguishing Arbitrary Multipartite Basis Unambiguously Using Local Operations and Classical Communication
• Distinguishability of complete and unextendible product bases
• Distinguishability of Quantum States by Separable Operations
• Local distinguishability with preservation of entanglement
• LOCC distinguishability of unilaterally transformable quantum states
• Condition for unambiguous state discrimination with local operations and classical communication
• PPT-indistinguishable states via semidefinite programming
• Limitations on separable measurements by convex optimization
• Distinguishability of Quantum States by Positive Operator-Valued Measures with Positive Partial Transpose
• More nonlocality with less purity
• Any $2\otimes n$ subspace is locally distinguishable
• Local Transformations Requiring Infinite Rounds of Classical Communication
• Asymptotically perfect discrimination in the LOCC paradigm
• Distinguishing multi-partite states by local measurements
• On the Chernoff distance for asymptotic LOCC discrimination of bipartite quantum states
• General approach to quantum channel impossibility by local operations and classical communication
• Strong bounds on required resources for quantum channels by local operations and classical communication

Cited by in corpus (3)

• Limitations of Classically-Simulable Measurements for Quantum State Discrimination
• The Round Complexity of Local Operations and Classical Communication (LOCC) in Random-Party Entanglement Distillation
• Strong bounds on required resources for quantum channels by local operations and classical communication