Round Complexity in the Local Transformations of Quantum and Classical States

Abstract

A natural operational paradigm for distributed quantum and classical information processing involves local operations coordinated by multiple rounds of public communication. In this paper we consider the minimum number of communication rounds needed to perform the locality-constrained task of entanglement transformation and the analogous classical task of secrecy manipulation. Specifically we address whether bipartite mixed entanglement can always be converted into pure entanglement or whether unsecure classical correlations can always be transformed into secret shared randomness using local operations and a bounded number of communication exchanges. Our main contribution in this paper is an explicit construction of quantum and classical state transformations which, for any given $r$, can be achieved using $r$ rounds of classical communication exchanges but no fewer. Our results reveal that highly complex communication protocols are indeed necessary to fully harness the information-theoretic resources contained in general quantum and classical states. The major technical contribution of this manuscript lies in proving lower bounds for the required number of communication exchanges using the notion of common information and various lemmas built upon it. We propose a classical analog to the Schmidt rank of a bipartite quantum state which we call the secrecy rank, and we show that it is a monotone under stochastic local classical operations.

Submitted to QIP 2017. Proof strategies have been streamlined and differ from the submitted version

References in corpus (4)

• Entanglement irreversibility from quantum discord and quantum deficit
• Conditions for entanglement transformation between a class of multipartite pure states with generalized Schmidt decompositions
• Round elimination in exact communication complexity
• A Four-Round LOCC Protocol Outperforms All Two-Round Protocols in Reducing the Entanglement Cost for A Distributed Quantum Information Processing

Cited by in corpus (13)

• Quantum Resource Theories
• Entanglement of a bipartite channel
• Complexity of causal order structure in distributed quantum information processing and its trade-off with entanglement
• Bounds on Instantaneous Nonlocal Quantum Computation
• Interplay of entanglement structures and stabilizer entropy in spin models
• Measurement outcomes that do not occur and their role in entanglement transformations
• Hierarchy of quantum operations in manipulating coherence and entanglement
• A link between symmetries of critical states and the structure of SLOCC classes in multipartite systems
• More assistance of entanglement, less rounds of classical communication
• The Round Complexity of Local Operations and Classical Communication (LOCC) in Random-Party Entanglement Distillation
• Quantum network-entanglement measures
• Strong bounds on required resources for quantum channels by local operations and classical communication
• One-Shot Distributed Source Simulation: As Quantum as it Can Get