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arXiv:1510.06672 · Journal of Mathematical Physics

Private algebras in quantum information and infinite-dimensional complementarity

Jason Crann, David W. Kribs, Rupert H. Levene, Ivan G. Todorov

Published Nov 16, 2015• 22 citations
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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.

DOI
10.1063/1.4935399

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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.
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We investigate the most general notion of a private quantum code, which involves the encoding of qubits into quantum subsystems and subspaces. We contribute to the structure theory for private quantum codes by deriving testable conditions for private quantum subsystems in terms of Kraus operators for channels, establishing an analogue of the Knill-Laflamme conditions in this setting. For a large class of naturally arising quantum channels, we show that private subsystems can exist even in the absence of private subspaces. In doing so, we also discover the first examples of private subsystems that are not complemented by operator quantum error correcting codes, implying that the complementarity of private codes and quantum error correcting codes fails for the general notion of private quantum subsystems.
We relate notions of complementarity in three layers of quantum mechanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices. Taking a more general categorical perspective of which the above are instances, we consider dagger monoidal kernel categories for (ii), so that (i) become (sub)endohomsets and (iii) become subobject lattices. By developing a ‘point-free’ definition of copyability we link (i) commutative von Neumann subalgebras, (ii) classical structures, and (iii) Boolean subalgebras.
We establish operator structure identities for quantum channels and their error-correcting and private codes, emphasizing the complementarity relationship between the two perspectives. Relevant structures include correctable and private operator algebras, and operator spaces such as multiplicative domains and nullspaces of quantum channels and their complementary maps. For the case of privatizing to quantum states, we also derive dimension inequalities on the associated operator algebras that further quantify the trade-off between correction and privacy.
Abstract. We present a generalization of quantum error correction to infinite-dimensional Hilbert spaces. The generalization yields new classes of quantum error correcting codes that have no finite-dimensional counterparts. The error correction theory we develop begins with a shift of focus from states to algebras of observables. Standard subspace codes and subsystem codes are seen as the special case of algebras of observables given by finite-dimensional von Neumann factors of type I. Our generalization allows for the correction of codes characterized by any von Neumann algebra and we give examples, in particular, of codes defined by infinite-dimensional algebras. 1.
We establish operator structure identities for quantum channels and their error-correcting and private codes, emphasizing the complementarity relationship between the two perspectives. Relevant structures include correctable and private operator algebras, and operator spaces such as multiplicative domains and nullspaces of quantum channels and their complementary maps. For the case of privatizing to quantum states, we also derive dimension inequalities on the associated operator algebras that further quantify the trade-off between correction and privacy.