Private algebras in quantum information and infinite-dimensional complementarity

Abstract

We introduce a generalized framework for private quantum codes using von Neumann algebras and the structure of commutants. This leads naturally to a more general notion of complementary channel, which we use to establish a generalized complementarity theorem between private and correctable subalgebras that applies to both the finite and infinite-dimensional settings. Linear bosonic channels are considered and specific examples of Gaussian quantum channels are given to illustrate the new framework together with the complementarity theorem.

20 pages

References in corpus (7)

• Quantum Error Correction of Observables
• Decoherence-full subsystems and the cryptographic power of a private shared reference frame
• Complementarity of Private and Correctable Subsystems in Quantum Cryptography and Error Correction
• Random subspaces for encryption based on a private shared Cartesian frame
• Quantum error correction on infinite-dimensional Hilbert spaces
• Secure Multi-party Quantum Computing
• Quantum Subsystems: Exploring the Complementarity of Quantum Privacy and Error Correction

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• Jones index, secret sharing and total quantum dimension
• Understanding holographic error correction via unique algebras and atomic examples
• State convertibility in the von Neumann algebra framework
• Fidelity of density operator in an operator-algebraic framework
• Nonperturbative gravity corrections to bulk reconstruction
• Quantum teleportation in the commuting operator framework
• Structure Theorem for Quantum Replacer Codes