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Journal of Physics Conference Series

Approximation of mutual information in a bipartite quantum state under single-party decoherence

S V Rozanov, Evgeniy O. Kiktenko

Published Nov 1, 2017• 0 citations
Publisher

Abstract

We consider the problem of approximate calculation of quantum mutual information in a bipartite quantum state where one of subsystems goes through a decoherence process (quantum channel). The approximation employs a reduced density matrix of the subsystem affected by decoherence and the initial value of mutual information before decoherence. We test the approximation on sets of randomly generated qubit-qudit mixed states for three types of qubit decoherence channels: dephasing, depolarizing, and damping. We consider how approximation accuracy is related to decoherence type, decoherence strength, and the dimension of the qudit subsystem.

DOI
10.1088/1742-6596/918/1/012006

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We consider the problem of approximate calculation of quantum mutual information in a bipartite quantum state where one of subsystems goes through a decoherence process (quantum channel). The approximation employs a reduced density matrix of the subsystem affected by decoherence and the initial value of mutual information before decoherence. We test the approximation on sets of randomly generated qubit-qudit mixed states for three types of qubit decoherence channels: dephasing, depolarizing, and damping. We consider how approximation accuracy is related to decoherence type, decoherence strength, and the dimension of the qudit subsystem.
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Decoherence is a fundamental obstacle to the implementation of large-scale and low-noise quantum information-processing devices. We suggest an approach for suppressing errors by employing preprocessing and postprocessing unitary operations, which precede and follow the action of a decoherence channel. In contrast to quantum error correction and measurement-based methods, the suggested approach relies on specifically designed unitary operators for a particular state without the need in ancillary qubits or postselection procedures. We consider the case of decoherence channels acting on a single qubit belonging to a many-qubit state. Preprocessing and postprocessing operators can be either individual, which is acting on the qubit effected by the decoherence channel only, or collective, which is acting on the whole multiqubit state. We give a classification of possible strategies for the protection scheme, analyze them, and derive expressions for the optimal unitary operators providing the maximal value of the fidelity regarding initial and final states. Specifically, we demonstrate the equivalence of the schemes where one of the unitary operations is individual while the other is collective. We then consider the realization of our approach for the basic decoherence models, which include single-qubit depolarizing, dephasing, and amplitude damping channels. We also demonstrate that the decoherence robustness of multiqubit states for these decoherence models is determined by the entropy of the reduced state of the qubit undergoing the decoherence channel.
Considering the quantum memory channels, we study the dynamical evolutions of quantum coherence and quantum mutual information as measures of quantum correlations under the actions of different decoherence channels on some bipartite initial states. Under any quantum operation or process occurring in a noisy environment, quantum correlations exhibit behavior that does not increase due to the system interacting with its environment. We state that for such a case the decrement of quantum correlations can be improved by the suitable choice of the initial states and by adjusting the parameters. Thus quantum correlations can be partially preserved against the action of the environment. It can be shown that optimal conditions to prohibit the partial loss in quantum coherence and quantum mutual information for performing any quantum information task may be generated by the memory.
Let X be an unbiased random bit, let Y be a qubit. whose mixed state depends on X, and let the qubit Z be the result of passing Y through a depolarizing channel, which replaces Y with a completely random qubit with probability p. We measure the quantum mutual information between X and Y by T(X; Y)=S(X)+S(Y)-S(X, Y), where S(...) denotes von Neumann's (1948) entropy. (Since X is a classical bit, the quantity T(X; Y) agrees with Holevo's (1973) bound /spl chi/(X; Y) to the classical mutual information between X and the outcome of any measurement of Y.) We show that T(X; Z) /spl les/ (1-p)/sup 2/T(X; Y). This generalizes an analogous bound for classical mutual information due to Evans and Schulman (1993), and provides a new proof of their result.
We determine the relative entropy of entanglement of a bipartite mixed state ρABρ_{AB}ρAB​ obtained by tracing out one subsystem of a tripartite Haar-random pure state ∣ψ⟩ABC|ψ\rangle_{ABC}∣ψ⟩ABC​, finding ER(ρAB)=log⁡dAdBmax⁡(dA,dB,dC)+O(1)E_R(ρ_{AB})=\log\frac{d_Ad_B}{\max(d_A,d_B,d_C)}+O(1)ER​(ρAB​)=logmax(dA​,dB​,dC​)dA​dB​​+O(1). Equivalently, the relative entropy of entanglement nearly saturates the smaller of the entanglement of formation EF(ρAB)E_F(ρ_{AB})EF​(ρAB​) and the mutual information I(A:B)I(A:B)I(A:B). The upper bound is achieved by an explicit separable state obtained through one-sided Schmidt dephasing, which is therefore approximately closest.
Quantum fisher information of a parameter characterizing the sensitivity of a state with respect to parameter changes. In this paper, we study the quantum fisher information of the W state for four, five, six and seven particles in decoherence channels, such as amplitude damping, phase damping and depolarizing channel. Using Krauss operators for decoherence channels components, we investigate the quantum fisher information quite analytically. We observe that for the W state, the amount of the quantum fisher information and consequently, the amount of entanglement are reduced by applying the quantum noises, which affect each particle in the same way. In some cases, a relative increase appears in the quantum fisher information for certain values of the decay strength p. We also show that the phase damping channel vanishes the value of the quantum fisher information for the W state.