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arXiv:2403.06733 · Lobachevskii Journal of Mathematics

On the Construction of a Quantum Channel Corresponding to Non-commutative Graph for a Qubit Interacting with Quantum Oscillator

Grigori Gennadievich Amosov, Мокеев Александр Сергеевич, Alexander Nikolaevich Pechen

Published Oct 1, 2021• 5 citations
arXivHTMLPDF

Abstract

Abstract We consider error correction, based on the theory of non-commutative graphs, for a model of a qubit interacting with quantum oscillator. The dynamics of the composite system is governed by the Schrödinger equation which generates positive operator-valued measure (POVM) for the system dynamics. We construct a quantum channel generating the non-commutative graph as a linear envelope of the POVM. The idea is based on applying a generalized version of a quantum channel using the apparatus of von Neumann algebras. The results are analyzes for a non-commutative graph generated by a qubit interacting with quantum oscillator. For this model the quantum anticlique which determines the error correcting subspace has an explicit expression.

DOI
10.1134/s1995080221100036

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Abstract We consider error correction, based on the theory of non-commutative graphs, for a model of a qubit interacting with quantum oscillator. The dynamics of the composite system is governed by the Schrödinger equation which generates positive operator-valued measure (POVM) for the system dynamics. We construct a quantum channel generating the non-commutative graph as a linear envelope of the POVM. The idea is based on applying a generalized version of a quantum channel using the apparatus of von Neumann algebras. The results are analyzes for a non-commutative graph generated by a qubit interacting with quantum oscillator. For this model the quantum anticlique which determines the error correcting subspace has an explicit expression.
Given a quantum channel it is possible to define the non-commutative operator graph whose properties determine a possibility of error-free transmission of information via this channel. The corresponding graph has a straight definition through Kraus operators determining quantum errors. We are discussing the opposite problem of a proper definition of errors that some graph corresponds to. Taking into account that any graph is generated by some POVM we give a solution to such a problem by means of the Naimark dilatation theorem. Using our approach we construct errors corresponding to the graphs generated by unitary dynamics of bipartite quantum systems. The cases of POVMs on the circle group Zn{\mathbb{Z}}_{n}Zn​ and the additive group R\mathbb{R}R are discussed. As an example we construct the graph corresponding to the errors generated by dynamics of two mode quantum oscillator.
Quantum error correction plays a key role for quantum information transmission and quantum computing. In this work, we develop and apply the theory of noncommutative operator graphs to study error correction in the case of a finite-dimensional quantum system coupled to an infinite-dimensional system. We consider as an explicit example a qubit coupled via the Jaynes-Cummings (JC) Hamiltonian with a bosonic coherent field. We extend the theory of noncommutative graphs to this situation and construct, using Gazeau-Klauder coherent states, the corresponding noncommutative graph. As the result, we find the quantum anticlique, which is the projector on the error-correcting subspace, and analyze it as a function of the frequencies of the qubit and the bosonic field. The general treatment is also applied to the analysis of the error-correcting subspace for certain experimental values of the parameters of the Jaynes-Cummings Hamiltonian. The proposed scheme can be applied to any system that possess the same decomposition of spectrum of the Hamiltonian into a direct sum as in JC model, where eigenenergies in the two direct summands form strictly increasing sequences.
We study the nature of the information preserved by a quantum channel via the observables which exist in its image (in the Heisenberg picture), and can therefore be simulated on the receiver's side. The sharp observables preserved by a channel form an operator algebra which can be characterized in terms of the channel's elements. The effect of the channel on these observables can be reversed by another physical transformation. These results generalize the theory of quantum error correction to codes characterized by arbitrary von Neumann algebras, which can represent hybrid quantum-classical information, continuous variable systems, or certain quantum field theories. The preserved unsharp observables (positive operator-valued measures) allow for a finer characterization of the information preserved by a channel. We show that the only type of information which can be duplicated arbitrarily many times consists of coarse-grainings of a single POVM. Based on these results, we propose a model of decoherence which can account for the emergence of a classical phase-space. This model supports the view that the quantum-classical correspondence is given by a quantum-to-classical channel, i.e. a POVM.
We tackle the dynamical description of the quantum measurement process by explicitly addressing the interaction between the system under investigation and the measurement apparatus, the latter ultimately considered as a macroscopic quantum object. We consider arbitrary positive-operator-valued measures (POVMs) such that the orthogonality constraint on the measurement operators is relaxed. We show that, as with the well-known von Neumann scheme for projective measurements, it is possible to build up a dynamical model holding a unitary propagator characterized by a single time-independent Hamiltonian. This is achieved by modifying the standard model so as to compensate for the possible lack of orthogonality among the measurement operators of arbitrary POVMs.
Alice and Bob receive a bipartite state (possibly entangled) from some finite collection or from some subspace. Alice sends a message to Bob through a noisy quantum channel such that Bob may determine the initial state, with zero chance of error. This framework encompasses, for example, teleportation, dense coding, entanglement assisted quantum channel capacity, and one-way communication complexity of function evaluation. With classical sources and channels, this problem can be analyzed using graph homomorphisms. We show this quantum version can be analyzed using homomorphisms on non-commutative graphs (an operator space generalization of graphs). Previously the Lovász ϑ number has been generalized to non-commutative graphs; we show this to be a homomorphism monotone, thus providing bounds on quantum source-channel coding. We generalize the Schrijver and Szegedy numbers, and show these to be monotones as well. As an application, we construct a quantum channel whose entanglement assisted zero-error one-shot capacity can only be unlocked using a non-maximally entangled state. These homomorphisms allow definition of a chromatic number for non-commutative graphs. Many open questions are presented regarding the possibility of a more fully developed theory.