[Paper Review] Distributed Compression and Squashed Entanglement

Ivan Savov|ArXiv.org|Feb 5, 2008

Quantum Information and Cryptography74 references3 citations

TL;DR

This master's thesis introduces multiparty information and multiparty squashed entanglement as tools to characterize distributed quantum states, and proposes a multiparty distributed compression protocol based on the fully quantum Slepian-Wolf (FQSW) protocol. It establishes inner and outer bounds on the achievable rate region for m-party quantum communication, proving the first rigorous bounds for general multiparty distributed compression beyond separable states.

ABSTRACT

A single quantum state can be shared by many distant parties. In this thesis, we try to characterize the information contents of such distributed states by defining the multiparty information and the multiparty squashed entanglement, two steps toward a general theory of multiparty quantum information. As a further step in that direction, we partially solve the multiparty distributed compression problem where multiple parties use quantum communication to faithfully transfer their shares of a state to a common receiver. We build a protocol for multiparty distributed compression based on the fully quantum Slepian-Wolf protocol and prove both inner and outer bounds on the achievable rate region. We relate our findings to previous results in information theory and discuss some possible applications.

Motivation & Objective

• To develop a general theory of multiparty quantum information by introducing multiparty information and multiparty squashed entanglement.
• To solve the multiparty distributed compression problem where multiple senders transmit quantum states to a common receiver without prior classical communication.
• To extend the fully quantum Slepian-Wolf (FQSW) protocol to the m-party scenario and derive achievable rate regions.
• To relate the results to foundational problems in quantum gravity, particularly the black hole information paradox.
• To provide a rigorous mathematical framework for supermodular rate regions in quantum information theory.

Proposed method

• Constructs a multiparty FQSW protocol via sequential applications of the two-party FQSW protocol with careful tracking of quantum information and entropies.
• Uses the FQSW resource inequality to derive an inner bound on the achievable rate region for m parties.
• Applies convex geometry and facet inequalities in m-dimensional space to prove the inner bound, generalizing to supermodular rate regions.
• Defines multiparty squashed entanglement as a continuous, convex, and subadditive entanglement measure, generalizing the bipartite squashed entanglement.
• Derives an outer bound on the rate region using the new multiparty squashed entanglement, establishing a necessary condition for faithful compression.
• Applies the framework to the black hole information paradox, modeling information recovery from Hawking radiation using the FQSW-based protocol.

Experimental results

Research questions

• RQ1 How can multiparty information and multiparty squashed entanglement be rigorously defined to generalize bipartite quantum information measures?
• RQ2 What are the achievable rate regions for distributed quantum compression when m parties share a quantum state and must transmit it to a receiver without classical communication?
• RQ3 Can the fully quantum Slepian-Wolf protocol be extended to the m-party case, and what are the necessary and sufficient conditions for faithful compression?
• RQ4 How does the multiparty squashed entanglement relate to the outer bound on the rate region, and what does it reveal about the structure of quantum correlations?
• RQ5 Can this framework shed light on the black hole information paradox, particularly the delayed return of information from Hawking radiation?

Key findings

• The multiparty squashed entanglement is proven to be a continuous, convex, and subadditive measure of entanglement, satisfying desirable properties rare in the multipartite setting.
• An inner bound on the achievable rate region for m-party distributed compression is established using sequential FQSW protocols and convex geometry in m-dimensional space.
• An outer bound on the rate region is derived using the multiparty squashed entanglement, showing that the sum of rates must satisfy a condition involving the entanglement of the state.
• The protocol achieves faithful compression for separable states, with the outer bound matching the inner bound in this case.
• For states with significant correlations between subsystems, the information may not emerge quickly from the black hole, as shown by the condition log d_R > max{H(A), H(A) + ½I(B₂;L)}.
• The framework generalizes to other protocols, including state redistribution and quantum broadcast channels, via the universal FQSW building block.