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arXiv:1312.3655 · Physical Review A

Fidelity of Fock-state-encoded qubits subjected to continuous-variable Gaussian processes

Brian Julsgaard, Klaus Mølmer

Published Jan 29, 2014• 3 citations
arXivHTMLPDF

Abstract

When a harmonic oscillator is under the influence of a Gaussian process such as linear damping, parametric gain, and linear coupling to a thermal environment, its coherent states are transformed into states with Gaussian Wigner function. Qubit states can be encoded in the $|0\ensuremath{\rangle}$ and $|1\ensuremath{\rangle}$ Fock states of a quantum harmonic oscillator, and it is relevant to know the fidelity of the output qubit state after a Gaussian process on the oscillator. In this paper we present a general expression for the average qubit fidelity in terms of the first and second moments of the output from input coherent states subjected to Gaussian processes.

DOI
10.1103/physreva.89.012333

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When a harmonic oscillator is under the influence of a Gaussian process such as linear damping, parametric gain, and linear coupling to a thermal environment, its coherent states are transformed into states with Gaussian Wigner function. Qubit states can be encoded in the ∣0\ensuremath⟩|0\ensuremath{\rangle}∣0\ensuremath⟩ and ∣1\ensuremath⟩|1\ensuremath{\rangle}∣1\ensuremath⟩ Fock states of a quantum harmonic oscillator, and it is relevant to know the fidelity of the output qubit state after a Gaussian process on the oscillator. In this paper we present a general expression for the average qubit fidelity in terms of the first and second moments of the output from input coherent states subjected to Gaussian processes.
We present a general theoretical formalism to compute the fidelity of transformations of unknown quantum states, and we apply our theory to Gaussian transformations of continuous variable quantum systems. For the case of a Gaussian distribution of displaced coherent states, the theory is readily tractable by a covariance matrix formalism, and a wider class of states, exemplified by Fock states, can be treated efficiently by the Wigner function formalism. Given the distribution of input states, the optimum feedback gain is identified, and analytical results for the fidelities are presented for recently implemented teleportation and memory storage protocols for continuous variables.
The statistical properties of systems of coupled quantum-mechanical harmonic oscillators are analyzed. The Hamiltonian for the system is assumed to be an inhomogeneous quadratic form in the creation and annihilation operators, and is allowed to have an explicit time dependence. The relationship to classical theory is emphasized by expressing pure states in terms of the coherent-state vectors, and density operators by means of the PPP representation and an analogous representation involving the Wigner function. The state which evolves from an initially coherent state of the system is found, and equations governing the time evolution of the Wigner function and the weight function for the PPP representation are derived, in differential and integral form, for arbitrary initial states of the system. The results remain valid for couplings which do not preserve the vacuum state, and for cases in which the time dependence of the coupling parameters gives rise to large-scale amplification of the initial field intensities. The analysis is performed by first treating general linear inhomogeneous canonical transformations on the oscillator variables, and then specializing to the case in which these transformations represent the solutions for the Heisenberg operators in terms of their initial values. The results are illustrated within the context of a model of parametric amplification.
The linear and phase insensitive absorption of a single quanta via coherent interactions with a saturable system, even a single ground state qubit, is sufficient to deterministically generate quantum non-Gaussian states in an oscillator, even stimulated merely by increasing thermal oscillator energy. However, the resultant states only approach Fock states and therefore do not exhibit quantum coherence. Here we overcome this limitation using a minimal step: a nonlinear phase-insensitive absorption process added to the linear one. The coherent addition of such individually passive processes allows coherence to emerge and increase in phase space without an external drive and with minimal interaction requirements. The coherence of quantum non-Gaussian states emerges because the linear and nonlinear absorption processes are not mutually passive. In the simplest case rotationally symmetric Wigner functions of the oscillator Fock states convert their many negative regions to an extremely complex asymmetric structure in sharp contrast to the rotational symmetry of those obtained by the individual interactions. We extend this case to include an unsaturable absorber (oscillator) and analyse switching between linear and nonlinear absorptions, suitable for broad classes of experiments.
From the original definition of fidelity in a pure state,we first give the well defined expansion fidelity between two Gaussian mixed states.It is related to the variances of output and input states in quantum information processing.It could be convenient to quantify the quantum teleportation(quantum clone) experiment since the variances of input(output) state are measurable.Furthermore,we also show that the fidelity of a pure input state is smaller than that of a mixed input state in the same quantum information processing.